9/2004Strickler, et. al1 Reconstruction of Modular Coil Shape and Control of Vacuum Islands in NCSX D. Strickler, S. Hirshman, B. Nelson, D. Williamson, L. Berry, J. Lyon Oak Ridge National Laboratory 9/2004

Strickler, et. al2 Topics 1.Determination of modular coil shape from magnetic field measurements 2.Optimization of the vacuum magnetic field configuration with respect to coil orientation and currents.

9/2004Strickler, et. al3 1. Shape reconstruction of (as-built) modular coils from magnetic measurements Linear method –Find corrections to design coil coordinates –SVD Nonlinear method –Parameterization of coil centerline –Fourier or cubic spline representation –Use Levenberg-Marquardt to find coefficients

9/2004Strickler, et. al4 Measure B=(Bx,By,Bz) or │B│ on 3D grid enclosing design coil shape Example: NCSX modular coil M1 Uniform grid spacing of 5 cm in each coordinate direction ~3500 field measurement points within 20–25 cm of design coil centerline Side viewTop view

9/2004Strickler, et. al5 Test procedures on NCSX modular coil M1 with known shape distortion Sine, cosine coil distortions : ΔR = δr sin (mθ) Δφ = δφ sin (mθ) ΔZ = δz sin (mθ) Design coil (solid) Distorted coil (dashed) m=5 δr = δφ = δz = 5 mm Side view Top view Include random error in: Magnetic field meaurements Location of measurements

9/2004Strickler, et. al6 x 0 = [x 01,y 01,z 01, …,x 0n, y 0n, z 0n ] T ideal (design) coil coordinates B 0 = B(x 0 )magnetic field of ideal coil over 3D grid x 0 + Δxcoordinates of actual coil (manufactured) B 1 = B(x 0 + Δx)magnetic field of actual coil (measured) B 1 – B 0 = Δb ≈ B  ΔxB  = dB/dx is MxN Jacobian matrix B  = U∑V T SVD Δx = ∑ σn ≠ 0 (u n T Δb)v n / σ n least-squares solution for Δx Drop ‘small’ singular values σ n to stabilize solution, making it less sensitive to data vector Δb Linear Method Solve for coordinates of coil winding center

9/2004Strickler, et. al7 Linear shape reconstruction of distorted modular coil M1 sin(5θ) distortion with δr = δφ = δz = 2mm Random measurement error in field: δB/B max ≤ 1.e-5, grid : dx, dy, dz ≤ 0.1 mm Solution depends on number of singular values retained (N σ ) Error in coil coord.: χ coil = { [ ∑ (x n – x n (dist) ) 2 + (y n – y n (dist) ) 2 + (z n – z n (dist) ) 2 ] / N } 1/2 σ min /σ max = 0.001, N σ = 213 σ min /σ max = 0.015, N σ = 132 Field error χ = 2.281e-5 T Χ coil = 3.052e-3 Field error χ = 2.310e-5 T Χ coil = 7.766e-4 Distorted coil (dashed) Reconstruction (solid) Distorted coil (dashed) Reconstruction (solid)

9/2004Strickler, et. al8 d [mm] δB/B max χ field [T] χ coil [m] e e e e e e e e e e e e e e-3 Linear coil shape reconstruction from magnetic field │B│ with random error ≤ d in location of field measurements sine coil distortion with m=5, δr = δφ = δz = 0.002m σ min / σ max = 0.015, N σ = 132

9/2004Strickler, et. al9 δB/B max χ field [T] χ coil [m] 1.e e e-4 5.e e e-4 1.e e e-4 5.e e e-4 1.e e e-3 Linear coil shape reconstruction from magnetic field │B│ with random error ≤ δB in field measurement values sine coil distortion with m=5, δr = δφ = δz = 0.002m σ min / σ max = 0.015, N σ = 132 Error in measurement locations d ≤ 0.1 mm

9/2004Strickler, et. al10 x = [x(t), y(t), z(t)]coordinates of coil centerline x(t) = a x0 + ∑ a x,k cos(2πkt) + b x,k sin (2πkt), …Fourier representation c = [a x0,a x1,b x1, …,a zn, b zn ] T vector of coefficients B = B(c)magnetic field over 3D grid Minimize χ 2 = ║B(c) – B 1 ║ 2 B 1 = field measurements on 3D grid Nonlinear Method Solve for coefficients in parameterization of coil winding center Levenberg-Marquardt method to solve for coefficients Initial guess for c based on fit to design coil data Option in code for cubic spline representation: x(t) = ∑ c x,k B k (t), …

9/2004Strickler, et. al11 Sensitivity of solution to initial approximation Define design coil by Fourier series (e.g., fit to NCSX modular coil 1 coordinates, total N c =159 coefficients) Create distorted coil by changing single coefficient ( e.g., a y,1 = a y,1 (design) + Δa y ) Reconstruct distorted coil shape from magnetic measurements with initial approximation to solution given by: a x,m = a x,m (design), b x,m = b x,m (design), etc.., except for a y,1 = a y,1 (design) + (1 – α) Δa y Let χ x = [ ∑ (a x,m – a x,m (design) ) 2 + (b x,m – b x,m (design) ) 2 ] 1/2, etc..

9/2004Strickler, et. al12 Sensitivity to initial approximation (continued) α Field error(T) Δa y,1 (m)Χ x (m)Χ y (m)Χ z (m) e e e e e e e e e e e e e e e e-2 Design coil + Distorted X Design coil + Distorted X m m amam amam α = 0.01 α = 1.0

9/2004Strickler, et. al13 Design coil (solid) Distorted (dashed) Distorted coil (dashed) Reconstruction (solid) Distorted coil (dashed) Reconstruction (solid) α = 1.0 α = 0.01

9/2004Strickler, et. al14 Avg. error in reconstructed coil geometry Distorted coil described by N segments, unit tangent vectors e d Evaluate solution (approximating field of distorted coil) at N points x For each point x d on distorted coil, find nearest solution point x*, and perpendicular distance from solution point to distorted coil: x ║ = [( x* – x d )∙e d ]e d x┴ = x* – x d - x ║ d = 1/N ∑ i ║x i ┴║

9/2004Strickler, et. al15 Nonlinear shape reconstruction of distorted modular coil M1 sin(5θ) distortion with δr = δφ = δz = 2mm Random measurement error in field: δB/B max ≤ 1.e-5, grid : dx, dy, dz ≤ 0.1 mm Fourier representation of coil centerline with total of N c variable coefficients Distorted coil (dashed) Reconstruction (solid) Field error χ = 2.323e-5 T Distorted coil (dashed) Reconstruction (solid) Field error χ = 2.076e-4 T N c = 81N c = 159

9/2004Strickler, et. al16 Nonlinear shape reconstruction of distorted modular coil M1 sin(5θ) distortion with δr = δφ = δz = 2mm Random measurement error in field: δB/B max ≤ 1.e-5 Random error ≤ d in measurement locations : dx, dy, dz Distorted coil (dashed) Reconstruction (solid) Field error χ = 2.323e-5 T N c = 159, d = 1.0 mmN c = 159, d = 0.1 mm Distorted coil (dashed) Reconstruction (solid) Field error χ = 2.368e-4 T

9/2004Strickler, et. al17 d [mm] δB/B max χ field [T]║c – c coil ║ e e e e e e e e e e e e e e-2 Nonlinear coil shape reconstruction from magnetic field │B│ with random error ≤ d in location of field measurements sine coil distortion with m=5, δr = δφ = δz = 0.002m c coil = vector of coefficients in fit to distorted coil coordinates c = solution vector in nonlinear fit to field of distorted coil

9/2004Strickler, et. al18 Next Apply to multifilament coil with leads Test methods on racetrack coil

9/2004Strickler, et. al19 2. Optimization of the Vacuum Field Configuration Vacuum field constraint used in QPS design optimization Recent STELLOPT modifications for vacuum field optimization Examples

9/2004Strickler, et. al20 A vacuum field constraint in the STELLOPT / COILOPT code led to a robust Quasi-Poloidal compact stellarator configuration [1] with improved vacuum properties Minimize the normal component of vacuum magnetic field at the full-pressure plasma boundary - Target  B = w B |Bn|/|B|, where: n is normal to the full beta VMEC plasma boundary B is the magnetic field due to the coils Last-closed vacuum magnetic flux surface encloses a plasma volume exceeding that of the targeted high  equilibrium Aspect ratio is maintained as  is increased [1] D.J. Strickler, S.P. Hirshman, D.A Spong, et al., Fusion Science and Technology 45 (Jan. 2004)

9/2004Strickler, et. al21 QPS vacuum magnetic surface quality improvement Full-beta (VMEC) plasma boundary Coil winding surface Full  plasma boundary and vacuum surfaces of QPS configuration without the vacuum constraint Optimized QPS configuration with vacuum constraint has larger volume of good flux surfaces Does not target vacuum island size

9/2004Strickler, et. al22 Module (VACOPT) added to stellarator optimization code (STELLOPT) to target resonances in vacuum magnetic field Minimize size of vacuum islands resulting from winding geometry errors or displacements due to magnetic loads / joule heating by varying position of modular coils in array. Additional variables include rigid-body rotations, shifts about coil centroid, and vacuum field coil currents. Targets include residues of prescribed resonances, bounds on variables, and constraints on position of the island O-points. Input list now contains poloidal mode number of targeted islands, initial values of vacuum field coil currents, shifts and rotations, and initial positions of control points for each modular coil. Output includes positions of control points following optimal shifts / rotations of the modular coils.

9/2004Strickler, et. al23 Magnetic field line is described by dx/ds = B(x) In cylindrical coordinates (and assuming B φ ≠ 0), these are reduced to two field line equations: dR/dφ = RB R /B φ, dZ/dφ = RB Z /B φ Integrating the field line equations, from a given starting point (e.g. in a symmetry plane), over a toroidal field period, produces a return map X  = M(X) (X = [R,Z] t ) An order m fixed point of M is a periodic orbit: X = M m (X) The dynamics of orbits in the neighborhood of a fixed point are described by the tangent map: δX  = T(δX) (whereT 11 = ∂M R /∂R,T 12 = ∂M R /∂Z, T 21 = ∂M Z /∂R,T 22 = ∂M Z /∂Z) Field Line Equations and Tangent Map

9/2004Strickler, et. al24 Targeting Magnetic Islands in Vacuum Field Optimization Locate O-points of islands (order m fixed points of return map) X i = (R i,0) Following [1], compute residues of targeted islands Res i = [2 – trace(T(X i ))]/4 Optimization Vary coil currents to minimize targeted residues: min. χ 2 = Σ w i Res i 2, Subject to possible constraints, e.g.: Σ I MOD + I TF = constant R min ≤ R i ≤ R max, (in prescribed toroidal plane v = v i ) Bounds on modular coil currents [1] Cary and Hanson, Phys. Fluids 29 (8), 1986

9/2004Strickler, et. al25 Reference coil Currents. Res = Min. Residue, Subject to: R O ≤ 1.18 m. Res = Min. Residue. Res = ρ (m) ι Reference Res = Example 1: Minimization of m=7 Residue In NCSX 1.7T, HB Scenario, t=0.1 s (vacuum) by varying coil currents.

9/2004Strickler, et. al26 Example 2: Vacuum field optimization by shifts and rotations about the centroids of modular coils Coil currents from 1.7T, high beta scenario at t=0.1s (vacuum) Minimize m=7 residue Constrain radius of O-point by R ≤ 1.18m (Z=0) Res = Res = 0.004

9/2004Strickler, et. al27 Δx (m)Δy (m)Δz (m)θφα M M M Example 2 (cont’d): shifts and rotations about modular coil centroids for constrained minimization of the m=7 residue α m easured with respect to unit rotation vector centered at coil centroid with sperical coordinate angles θ, φ. Maximum changes in the control points are 1.17cm (M1), 1.58cm (M2), and 0.67cm (M3).

9/2004Strickler, et. al28 Next Test vacuum optimization effect on high beta physics targets Correct for non-stellarator symmetric field errors