Magnetic Control of Perturbed Plasma Equilibria

Magnetic Control of Perturbed Plasma Equilibria Nikolaus Rath February 17th, 2012 N. Rath (Columbia University) Magnetic Control of Perturbed Plasma Equilibria February 17th, 2012 1 / 19

The HBT-EP Tokamak S. Angelini, J. Bialek, P. Byrne, B. DeBono, P. Hughes, J. Levesque, B. Li, M. Mauel, G. Navratil, Q. Peng, D. Rhodes, D. Shiraki, C. Stoafer N. Rath (Columbia University) Magnetic Control of Perturbed Plasma Equilibria February 17th, 2012 2 / 19

Recent upgrade allows high-resolution magnetic measurement and control Inside the HBT-EP vacuum chamber N. Rath (Columbia University) Magnetic Control of Perturbed Plasma Equilibria February 17th, 2012 3 / 19

Plasma dynamics are complicated f ( α t + v α f α e E + 1 c v B) f α p = 0 B = 4π j c + 1 E c t, E = 1 c E = 4πρ, B = 0 ρ = e (f i f e ) d 3 p j = e (f i f e ) v d 3 p p/m α v α = 1 +p 2 /m 2 αc 2 B t N. Rath (Columbia University) Magnetic Control of Perturbed Plasma Equilibria February 17th, 2012 4 / 19

MHD makes things easier With a few not-quite-appropriate approximations: ρ d u dt = ( B) B p d B dt = ( u B) d dt dρ = (ρ u) dt ) (1) = 0 (2) ( p ρ γ But even better, in a Tokamak: u B µ0 ρ meters (3) microseconds d u dt 0 (4) N. Rath (Columbia University) Magnetic Control of Perturbed Plasma Equilibria February 17th, 2012 5 / 19

Tokamak plasmas transition through MHD equilibria Plasma reaches force-balance within microseconds Longer lived plasmas must thus be in equilibrium Evolution is caused by all our approximations The plasma always has minimum MHD energy, but the properties of the minimum-energy-state are changing slowly N. Rath (Columbia University) Magnetic Control of Perturbed Plasma Equilibria February 17th, 2012 6 / 19

Tokamak equilibria are tractable This talk is about: ( B) B = p (5) N. Rath (Columbia University) Magnetic Control of Perturbed Plasma Equilibria February 17th, 2012 7 / 19

In Tokamaks, the plasma state is axisymmetric with a 3d perturbation Solving ( B) B = p in axisymmetry is easy so we assume someone did it. Solving ( B) B = p in 3d is hard, so we don t want to do it. So will solve ( B) B = p with just small 3d perturbation - Tokamaks are great! A small 3d displacement ζ must minimize (Boozer, 1999 & 2003) dw( ζ) = ζ F( ζ)dv (6) plasma F is known from the axisymmetric solution, minimization is just number crunching. Just need boundary conditions on plasma surface N. Rath (Columbia University) Magnetic Control of Perturbed Plasma Equilibria February 17th, 2012 8 / 19

The permeability matrix defines the plasma response to external fields The plasma shape follows magnetic field lines The non-axisymmetric field on the plasma surface is the boundary conditions for the plasma displacement. The field has components from both the plasma and the environment, write it as b( ζ) = b plasma +b external (7) For easier math, write surface functions as vectors of expansion coefficients, e.g. b(θ,φ) = i Φ i f i (θ,φ) = Φ f(φ,θ) (8) Define permeability matrix P by: P. Φ external = Φ (9) N. Rath (Columbia University) Magnetic Control of Perturbed Plasma Equilibria February 17th, 2012 9 / 19

Plasma response example External field N. Rath (Columbia University) Magnetic Control of Perturbed Plasma Equilibria February 17th, 2012 10 / 19

Plasma response example Total field N. Rath (Columbia University) Magnetic Control of Perturbed Plasma Equilibria February 17th, 2012 10 / 19

Half-Time Summary We want to control the plasma In Tokamaks, the plasma is mostly in MHD equilibrium In Tokamaks, the equilibrium is also approximately axisymmetric The axisymmetric part is constant, the 3d perturbation changes in time The axisymmetric part is easily measured and calculated, we assume it is known The 3d perturbation is uniquely defined by magnetic field due to external currents on the plasma surface To control Tokamak plasmas, we therefore have to measure, track and control external currents N. Rath (Columbia University) Magnetic Control of Perturbed Plasma Equilibria February 17th, 2012 11 / 19

The system model consists of plasma, wall, control coils, and sensors Control Coils: I c Wall: I w (finite element model) Plasma: I p Sensors: Φ s External Currents I x = ( I c, I x ) N. Rath (Columbia University) Magnetic Control of Perturbed Plasma Equilibria February 17th, 2012 12 / 19

Maxwell s equations allow to represent all system components as vectors Maxwell: the normal magnetic field on a closed surface uniquely defines the external field in the enclosed volume Plasma sees all external currents as Φ x = M px. I x External circuits see plasma as Φ p = (P 1). Φ x. We can express all interactions as matrices. The complete system obeys R [ ] x. I x + L x +M xp.l 1 p.(p 1).M px. d I x dt = V x (10) M and L matrices encapsulate all geometric information. P depends on the axisymmetric equilibrium and contains information about the plasma. N. Rath (Columbia University) Magnetic Control of Perturbed Plasma Equilibria February 17th, 2012 13 / 19

Controlling the plasma To control the solution of we need to control a system of the form ( B) B = p (11) Required steps: d I x dt = A. I x +B. V x (12) N. Rath (Columbia University) Magnetic Control of Perturbed Plasma Equilibria February 17th, 2012 14 / 19

Controlling the plasma To control the solution of we need to control a system of the form Required steps: ( B) B = p (11) d I x dt = A. I x +B. V x (12) 1 Subtract axisymmetric fields from measurements 2 Use measurements to estimate system state 3 Compare current plasma shape with target shape 4 Find control coil voltages that bring shape closer to target N. Rath (Columbia University) Magnetic Control of Perturbed Plasma Equilibria February 17th, 2012 14 / 19

Controlling the plasma To control the solution of we need to control a system of the form Required steps: ( B) B = p (11) d I x dt = A. I x +B. V x (12) 1 Subtract axisymmetric fields from measurements 2 Use measurements to estimate system state 3 Compare current plasma shape with target shape 4 Find control coil voltages that bring shape closer to target (not discussed in this talk) N. Rath (Columbia University) Magnetic Control of Perturbed Plasma Equilibria February 17th, 2012 14 / 19

Equilibrium fields are subtracted using continuous polynomial prediction Measurement errors in the (large) axisymmetric contribution easily exceed non-axisymmetric fluctuations. Solution: Continuously fit last n measurements of every sensor to quadratic polynomial, subtract predicted amplitude. N. Rath (Columbia University) Magnetic Control of Perturbed Plasma Equilibria February 17th, 2012 15 / 19

SVD of measurement matrix determines relative visibility of states Sensors measure magnetic flux: Φ s = M sx. I x +M sp. I p [ = M sx +M sp.l 1 p.(p 1).M px ]. I x =: C. (13) I x Singular value decomposition of C gives 1 v 1 U.S.V = u 1 u 2. s s 2. v 2 (14).... v i is the basis of directly measurable states s i is the relative visibility of a state We could drop invisible states and invert, but there is a better solution N. Rath (Columbia University) Magnetic Control of Perturbed Plasma Equilibria February 17th, 2012 16 / 19

The system model is used to estimate the state Control system continuously calculates: Error in calculated I x obeys d I x dt = A. I x +B. V x +K. ( Φ s C. ) I x (15) d e dt = (AT C T.K T ). e (16) Could pick K to obtain specific eigenvalues but which exactly? Better choice: let R be measurement uncertainties and S mode visibilities, and choose K to minimize S. e(t) 2 + R.K T. e 2 dt (17) 0 More visible states will converge faster, more reliable sensors will be used preferentially N. Rath (Columbia University) Magnetic Control of Perturbed Plasma Equilibria February 17th, 2012 17 / 19

Experimental results: wall model and shot reproducibility needs to be improved HBT-EP plasmas are short-lived, so equilibrium needs to be computed offline Variations in major radius change equilibrium considerably between shots, preventing pre-computation Plan: address by extending control to plasma major radius Vacuum testing show reasonably agreement, better wall model may give further improvements. Work in progress to add vacuum vessel. N. Rath (Columbia University) Magnetic Control of Perturbed Plasma Equilibria February 17th, 2012 18 / 19

Summary 1 The evolution of Tokamak plasmas can be approximated as a transition through a sequence of MHD equilibria. 2 These equilibria are approximately axisymmetric and consist of a static, axisymmetric part and a changing, 3-dimensional perturbation. 3 Transitions caused by changing external fields can be modeled as a linear, time invariant system. This system can be controlled with standard techniques from control theory, using magnetic sensors and control coils. 4 Experimental tests show reasonable agreement for vacuum model. Plasma tests were impaired by low shot reproducibility, which will be addressed by control of plasma major radius. N. Rath (Columbia University) Magnetic Control of Perturbed Plasma Equilibria February 17th, 2012 19 / 19