1. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #1/33 F. Villone, RWM modelling of RFX-mod with the CarMa code RWM modelling of RFX-mod with the CarMa code: results, perspectives,possible experiments Fabio Villone Ass. EURATOM/ENEA/CREATE, DAEIMI, Università di Cassino, Italy With contributions of: Y.Q. Liu (UKAEA) R. Albanese, G. Ambrosino, M. Furno Palumbo, G. Rubinacci, S. Ventre (CREATE) T. Bolzonella, G. Marchiori, R. Paccagnella, A. Soppelsa (RFX-mod)

2. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #2/33 F. Villone, RWM modelling of RFX-mod with the CarMa code Outline Introduction The CarMa code Applications to RFX-mod Electromagnetic modelling of RFX-mod What next? Conclusions

3. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #3/33 F. Villone, RWM modelling of RFX-mod with the CarMa code What are RWM? Linearized ideal MHD equations can describe fusion plasmas in some situations –In some cases predict unstable modes on inertial time scale (microseconds for typical parameters) –External kink is one of the most dangerous (e.g. setting beta limits in tokamaks) –A sufficiently close perfectly conducting wall may stabilize such mode thanks to image currents induced by plasma movements –Due to finite wall resistivity, image currents decay (Resistive Wall Modes)  the mode is again unstable but on eddy currents time scale (typically milliseconds or slower) –Feedback active control becomes possible Introduction

4. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #4/33 F. Villone, RWM modelling of RFX-mod with the CarMa code How do we analyse RWM? Introduction Solution of a coupled problem is needed in principle –Plasma evolution can be described by MHD equations –Eddy currents equations need magneto-quasi-static electromagnetic solvers –Usual stability codes (MARS-F, KINX, ETAW, etc): MHD solver + a simplified treatment of wall (e.g. thin shell approximation, axisymmetric or cylindrical assumptions, single wall, etc.) –Our approach: coupling of a MHD solver (MARS-F) to describe plasma with a 3D eddy currents formulation (CARIDDI) to describe the wall  CarMa code

5. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #5/33 F. Villone, RWM modelling of RFX-mod with the CarMa code Why modelling RWMs in RFX? Introduction In ITER RWMs will set stringent limits to the plasma performance (beta limit)  it is important to make reliable predictions via accurate modelling Open issues still remain in RWM modeling –Stabilization via rotation and damping –3D effects of passive (vessel, shell) and active (feedback control coils) conductors RFX-mod allows us to concentrate on second issue: –RWMs in RFPs share many features with tokamaks but are not affected by plasma flow (current driven instabilities) –Most advanced feedback control system for MHD modes Some interesting theoretical points peculiar to RFPs

6. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #6/33 F. Villone, RWM modelling of RFX-mod with the CarMa code 3D eddy currents formulation /1 Integral formulation assuming J as unknown –Well suited for fusion devices (only the conducting domain V c must be meshed) –This formulation is at the basis of the CARIDDI code, widely used for electromagnetic computations on fusion devices –Volumetric conductors of arbitrary shape taken into account (no thin shell approximation nor other simplifications) –Electric vector potential J =   T  solenoidality of J –Non-standard two-component gauge (numerically convenient) –Tree-cotree decomposition of the mesh  minimum number of discrete unknonws I –Edge elements N k  right continuity conditions on J –Automatic treatment of complex topologies and electrodes The CarMa code

7. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #7/33 F. Villone, RWM modelling of RFX-mod with the CarMa code 3D eddy currents formulation /2 Mathematical model –Eddy currents equation in the time domain –No magnetic materials (not a theoretical limitation: easily taken into account) –Resistivity tensor  to account for anisotropies (e.g. for “equivalent” piecewise homogeneous materials) –Ohm’s law E =  J imposed in weak form (Galerkin approach) The CarMa code Electrode with potential V j Magnetic vector potential due to current density representing plasma - Equations:

8. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #8/33 F. Villone, RWM modelling of RFX-mod with the CarMa code MHD formulation Single fluid linearized MHD equations The CarMa code p, v,  : plasma pressure, velocity and density;  : specific heats ratio Uppercase: reference values, lowercase: first- order perturbations A exp(j n  ) dependence of the quantities is assumed in the toroidal direction  (n : toroidal mode number) In the poloidal plane, Fourier decomposition is used along the poloidal angle, and a Galerkin- based finite element method is implemented on a staggered grid along the radial direction. Toroidal plasma flow and various kinetic effects (to simulate the damping) are presently not included in CarMa The MHD solver used is MARS-F

9. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #9/33 F. Villone, RWM modelling of RFX-mod with the CarMa code Coupling strategy Plasma mass is neglected –Good approximation if the time scale is much longer than inertial times (i.e. >> microseconds) –The plasma response is instantaneous and can be characterized by a response matrix to unit total normal field perturbations on coupling surface No plasma rotation is considered –Worst case analysis –Not important for current-driven modes (like in RFP’s) A coupling surface S is chosen –Any surface in between plasma and conducting structures –The plasma-wall interaction is decoupled via S The CarMa code

10. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #10/33 F. Villone, RWM modelling of RFX-mod with the CarMa code (De-)coupling surface S The CarMa code The plasma (instantaneous) response to a given magnetic flux density perturbation on S is computed as a plasma response matrix. plasma S Resistive wall S S Using such plasma response matrix, the effect of 3D structures on plasma is evaluated by computing the magnetic flux density on S due to 3D currents. The currents induced in the 3D structures by plasma are computed via an equivalent surface current distribution on S providing the same magnetic field as plasma outside S.

11. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #11/33 F. Villone, RWM modelling of RFX-mod with the CarMa code Overall model The CarMa code Mutual inductance matrix between 3D structures and equivalent surface currents Induced voltage on 3D structures Equivalent surface currents providing the same magnetic field as plasma Matrix expressing the effect of 3D current density on plasma Modified inductance matrix Dynamical matrix N  h matrix h  N matrix h << N h DoF of magnetic field on S N DoF of current in 3D structure

12. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #12/33 F. Villone, RWM modelling of RFX-mod with the CarMa code Several possible uses… The CarMa code Growth rate calculation –Unstable eigenvalue of the dynamical matrix –Standard routines (e.g. Matlab) or ad hoc computations –Beta limit with 3D structures (when the system gets fictitiously stable) Controller design –state-space model (although with large dimensions and with many unstable modes) Time domain simulations –Controller validation –Inclusion of non-ideal power supplies (voltage/current limitations, time delays, etc.)

13. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #13/33 F. Villone, RWM modelling of RFX-mod with the CarMa code Theoretical validations Theoretically sound approach –Independent theoretical validation on general geometry (V. Pustovitov, PPCF and PoP) –Analytical proof of the coupling scheme available in the cylindrical limit (Y.Q. Liu et al, PoP) –Many successful benchmarks in various limits and situations (MARS-F, ETAW, KINX, STARWALL, …) –No fitting parameters, no tuning, no normalizations –“Genuine” MHD computations with 3D geometries are possible The CarMa code

14. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #14/33 F. Villone, RWM modelling of RFX-mod with the CarMa code Benchmarks The CarMa code MARS-F as reference –Axisymmetric geometry (although with a 3D mesh) –stable and unstable modes with toroidal mode number n = 1 –degenerate pairs of eigenvalues (modes shifted of 90° toroidally)

15. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #15/33 F. Villone, RWM modelling of RFX-mod with the CarMa code ITER results /1 The CarMa code High level of details: –Double shell –Nested port extensions –Outer Triangular Support with copper cladding –Non- axisymmetric control coils

16. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #16/33 F. Villone, RWM modelling of RFX-mod with the CarMa code The CarMa code ITER results /2 Plasma/circuit model V(t) y(t) T IN T OUT y 1 (t) - V 1 (t) K(s) 27 input voltages (3 coils per 9 sectors) 3 voltage Fourier components 144 magnetic outputs (48 measurements per 3 sectors) 48 magnetic Fourier components RWM feedback controller

17. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #17/33 F. Villone, RWM modelling of RFX-mod with the CarMa code Other significant results The CarMa code Best achievable performances –Rigorous computation of the best performance achievable by any RWM controller for given voltage and current limitations in the actuators –Applied to ITER “Fast” computations –Fast SVD-based sparsification techniques for ameliorating the computational scaling with the number of unknowns  simulation with an unprecedented level of geometrical details –Applied to ITER These volumetric cases (360° in the toroidal direction) cannot be analysed with standard computational tools due to memory overflow!

18. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #18/33 F. Villone, RWM modelling of RFX-mod with the CarMa code Multimodal analysis /1 Application to RFX-mod 3D structure cause multimodal coupling –With an axisymmetric structure linear MHD predicts that every different n evolves separately –A 3D structure can couple different n ’s even in linear MHD RFP’s particularly suitable –Rich toroidal spectrum RFX-mod has a dedicated control system –192 independently fed saddle coils –Selective control of non-axisymmetric modes –Excellent magnetic measurement coverage

19. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #19/33 F. Villone, RWM modelling of RFX-mod with the CarMa code Multimodal analysis /2 RFX-mod shell with gaps –A small but noticeable multimodal coupling effect –Mode degeneracy removed Application to RFX-mod

20. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #20/33 F. Villone, RWM modelling of RFX-mod with the CarMa code Multimodal analysis /2 RFPs have multiple unstable RWMs –Even with the same n value –This corresponds to positive n and negative n in the cylindrical limit Application to RFX-mod

21. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #21/33 F. Villone, RWM modelling of RFX-mod with the CarMa code Experimental validation 3D effects are important on growth rate! Purely axisymmetric estimates of growth rates are largely underestimated Application to RFX-mod

22. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #22/33 F. Villone, RWM modelling of RFX-mod with the CarMa code White: vessel Red: copper shell Green: mech. structure Blue: saddle coils Black: meas. coils Details of conducting structures Electromagnetic modelling Detailed electromagnetic modelling of the main conducting structures of RFX-mod

23. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #23/33 F. Villone, RWM modelling of RFX-mod with the CarMa code Coils/sensors interactions /1 Comparison of mutual inductances of coil/sensor pairs –Very good amplitude prediction –Error in phase at “high” frequencies (>10° when >50 Hz) (inaccurate modelling of mechanical structure?) Electromagnetic modelling

24. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #24/33 F. Villone, RWM modelling of RFX-mod with the CarMa code Comparison of mutual inductances of coil/sensor pairs –Correct qualitative and quantitative representation of interactions due to toroidal and poloidal gaps in the shell and in the mechanical structure Each pixel represents an interaction Colours represent the log10 value of the normalized mutual interaction Electromagnetic modelling Coils/sensors interactions /2

25. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #25/33 F. Villone, RWM modelling of RFX-mod with the CarMa code Modal mutual inductances Comparison of modal inductances –Pure (m,n) harmonic current distribution in coils –Corresponding (m,n) harmonic in measured flux –The ratio is the modal inductance Electromagnetic modelling Results similar to the previous case

26. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #26/33 F. Villone, RWM modelling of RFX-mod with the CarMa code Time domain simulations Experimental currents fed to the coils and experimental fluxes compared to model predictions Electromagnetic modelling

27. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #27/33 F. Villone, RWM modelling of RFX-mod with the CarMa code “Flight simulator” /1 Putting the state-space CarMa model inside the overall Simulink® model of RFX-mod Detailed time-domain simulation of given shots Prediction of behaviour (e.g. controller gains for stability margin) A-priori model-based controller design (maybe not necessary for RFX-mod itself…but a fundamental demonstration for ITER!) What next?

28. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #28/33 F. Villone, RWM modelling of RFX-mod with the CarMa code “Flight simulator” /2 What next? CarMa model

29. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #29/33 F. Villone, RWM modelling of RFX-mod with the CarMa code Plasma flow We are currently developing a strategy for including the effects of plasma flow (e.g. kinetic damping) and plasma mass Theoretically challenging: the plasma response changes qualitatively (from static to dynamic) May be of interest for RFX-mod (coupling 3D structures to other physical models, e.g. NTM) ? Again, RFX-mod could be an ideal test-bed for models and techniques What next?

30. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #30/33 F. Villone, RWM modelling of RFX-mod with the CarMa code Experimental fall-outs Modelling contribution to the experimental programme of RFX-mod –Help in the analysis/interpretation of experiments More complete coverage and understanding of “classical” (F,  scans) or “innovative” (rotation?) RWM experiments –Help in planning of future experiments Prediction of “optimal” gains for given purposes (e.g. stability margin) –Flag the experiments of potential interest for model validation “RFA-like” experiments (steps or sinusoids as excitations) –Link to ITER requests/needs from the point of view of MHD mode control Test of a-priori model/based MHD mode feedback controllers What next?

31. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #31/33 F. Villone, RWM modelling of RFX-mod with the CarMa code Conclusions Conclusions /1 The CarMa code can self-consistently analyse RWMs with 3D conducting structures –Theoretically sound (analytical proofs available) –Successful benchmarks on axisymmetric cases and on 3D geometries –Allows “pure” MHD comparisons –Highly modular code –Experimental validation on RFX-mod –Multimodal modelling possible –Quantification of 3D effects on RFX-mod (gaps in shell) and ITER (port extensions, non axisymmetric control coils,…) –Huge models handling allows an unprecedented level of details in geometrical description

32. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #32/33 F. Villone, RWM modelling of RFX-mod with the CarMa code Conclusions Conclusions /2 Several different computations are possible –Growth rate computation –Time domain simulation –Feedback controller design RFX-mod contribution is very significant for MHD mode control –Model validation –Test of methods/techniques of interest for ITER

33. RFX-mod 2009 Programme Workshop, 20th Jan 2009 #33/33 F. Villone, RWM modelling of RFX-mod with the CarMa code Thank you for your attention! F. Villone et al., Proc. of 34th EPS Conference, Warsaw (2007), P5.125 R. Albanese, Y. Q. Liu, A. Portone, G. Rubinacci, and F. Villone, IEEE Trans. Mag. 44, 1654 (2008). A. Portone, F. Villone, Y. Q. Liu, R. Albanese, and G. Rubinacci, Plasma Phys. Controlled Fusion 50, 085004 (2008) Y. Q. Liu, R. Albanese, A. Portone, G. Rubinacci, and F. Villone, Phys. Plasmas, 15, 072516 (2008) F. Villone, Y. Q. Liu, R. Paccagnella, T. Bolzonella, and G. Rubinacci, Phys. Rev. Lett. 100, 255005 (2008) F. Villone et al., Proc. of 35th EPS Conference, Hersonissos (2008), P2.067 G. Marchiori et al., Proc. of 35th EPS Conference, Hersonissos (2008), P5.047 A. Soppelsa et al., Proc. of SOFT Conference (2008) Some references… Please contact: villone@unicas.it Conclusions