Lidia Piron Consorzio RFX, Euratom-ENEA Association, and University of Padova, Italy

1. Experimental characterization and numerical modeling of the active control of resistive MHD modes in RFX-mod Lidia Piron Consorzio RFX, Euratom-ENEA Association, and University of Padova, Italy

2. RFX-mod and its MHD active control system Feedback control of Tearing Modes (TMs) Avoid wall locking of TMs Error fields control Future plans: integration of a control system model and a plasma model Outline

3. RFX- mod and the MHD active control system

4. The largest reversed field pinch in the world 1.5MA plasma current up to now at low magnetic field B (a)< 0.1T (target  2MA) Multi-mode feedback control of MHD modes and Error Fields RFX-mod, Padova, Italy a=0.459m, R0=2m

5. 4(pol) x 48(tor) =192 saddle coils (Cycle latency <400 s) Sensors for br, bT and isassociated to each saddle coil Stabilizing shell : tshell~50ms RFX-mod active coils system The edge radial magnetic field br is controlled by saddle coils independently fed Assembly of saddle coils on vacuum vessel (mid’ 2004)

6. A large portion of BT and BP is generated by currents flowing in the plasma, through a dynamo mechanism The dynamo mechanism + ? =

7. BT and BP are generated by currents flowing in the plasma, through a dynamo mechanism The dynamo mechanism Edynamo + = • Edinamo is produced by resistive MHD modes, identified as Tearing Modes (TMs) or dynamo modes [H. Ji et S. C. Prager, Magnetohydrodynamics 38 (2002) 191 ]

8. TMs spectrum q(r) Safety Factor: m=1, n=-7 m=1, n=-8 m=1, n=-9 m=0, n=1,2,3,4, … r(m)

9. TMs effects m=1, n=-7 m=1, n=-8 m=1, n=-9 br (mT)  (r) r/a Core: confinement degradation Edge: Non axi-symmetric deformations of the Last Closed Flux Surface,  (r)

10. TMs effects •  (r) produces plasma-wall interaction, • which is affected by the phase-locking • and wall-locking phenomena • Controlling the edge radial field br: -  (r) is reduced and, as conseguence, the plasma wall interaction - transition to a QSH regime [P. Martin et al., PPCF 49 (2007) A177 ]

11. An algorithm for Feedback control: Virtual Shell Plasma bext Sensors To control br Power Amplifier Inputs: br Digital Controller Outputs: coil current reference [C. M. Bishop, Plasma Phys. Control. Fusion 31 (1989) 1179 ]

12. The problem of sidebands in VS • Discrete grid of coils (MxN, M=48, N=4) sideband harmonics • The VS scheme is applied to the raw measurements, which contain the high m-n sidebands • The sidebands are computed from the coils currents using a cylindrical vacuum model, and are real-time subtracted from the measurements

13. The problem of sidebands in VS • Discrete grid of coils (MxN, M=48, N=4) sideband harmonics • The VS scheme is applied to the measurements, which contain the high m-n sidebands • The sidebands are computed from the coils currents using a cylindrical vacuum model, and are real-time subtracted from the measurements 0 The Clean Mode Control algorithm [P.Zanca et al, Nucl. Fusion47 (2007) ]

14. The Clean Mode Control Scheme Power Supplies Digital Controller Irefm,n brm,n Icoilm,n Sidebands correction br, DFTm,n br br, DFTm,n bf, DFTm,n Plasma DFT Magnetic analysis

15. Effects of active control on TMs No control Virtual Shell Clean Mode Control • Reduction of m=1 deformation of the Last Closed Flux Surface,   (m) • The highest current for a RFP up to now • Significant increase of pulse length Clean Mode Control Ip (MA) Virtual Shell Ip (MA) No MHD active control time (s)

16. Effects of active control on TMs • Spontaneous transition to a quasi-single-helicity state (QSH) m=1, n=-7 < m=1, n=-8 to -16 > bT / Br (a) (%) time (s) • The magnetic chaos is reduced, i.e. the confinement is improved Te (eV) Radius (mm)

17. Avoid wall locking of TMs

18. Active control on TMs: effects on the mode-rotation • Effects not only in the amplitudes but also in the phase of the modes • TMs start to rotate in CMC Virtual Shell Phase (rad) Clean Mode Control Phase (rad) Freq (Hz) Distribution of the median of the rotation frequency for the m=1, n=-7 TM for two ensembles of VS and CMC discharges time (s) Phase dynamics of m=1, n=-7

19. Control of the direction of rotation: Complex Gain • The reference value for the applied field of the CMC algorithm for each mode is Complex proportional gain b-71,r (a) (mT) • The direction of the mode rotation is determined by the sign of the phase m,n on the dominant m=1,n=-7 mode Phase (rad) time (ms)

20. Experiments with Complex Gain: Multiple Modes • Complex gains of opposite sign have been set on TMs with n=-8 to n=-16 • The direction of the phase rotation follows the alternate pattern Freq (Hz) n

21. In the next campaign: Sweeping control • Mitigation of the wall-locking phenomena appling rotation perturbations with a sweeping frequency as suggested by results obtained in the DIII-D experiment [F. Volpe, 34º EPS conference, 2007 ] • Rotating fields unlock the mode and sustain rotation at up to 60 Hz Phase and amplitude of 2/1 NTM in DIII-D experiment Proposed by P. Piovesan et F. Volpe in RFX-mod

22. Error fields

23. Error Fields • Control system acts not only on TMs but also on Error Fields m=1, n=-7 m=1, n=-6 phase (rad) phase (rad) Fr(a) FT Fr(a) FT  phase (rad) phase (rad)  time (s) time (s)

24. Effects of the shell axisimmetries on TMS locking • Clear effect of the gap of the shell on TMs locking • Non uniformities of the passive structures must be taken into account to improve the control of TMs p.d.f. Toroidal position Poloidal gap

25. Modeling

26. Electromagnetic Model • Dynamic ElectroMagnetic (EM) model [G. Marchiori, Fusion Eng. Des. 82 (2007) 1015] - State space rapresentation - Accurate description of the passive structures and of the mutual inductance between sensor and active coils Inputs: 48 x 4 voltages applied to the saddle coils Outputs: 48 x 4 magnetic fluxes measured by the sensor coils

27. Comparison EM model - experiment • Evolution of m=0, n=4 mode and the predictions by the EM model • The EM model reproduces rather well the experimental behavior, with an accuracy of 5% in vacuum Simulation Measure

28. Torque balance Model • The evolution of the amplitudes and phases of several TMs can be explained by a torque balance model in cylindrical geometry [P. Zanca et al., Nucl. Fusion 47 (2007) 1425 ] • The model is based on - Newcomb equation solution for TMs radial profiles - the thin shell dispersion relation to describe the effects of homogeneous passive structures - simplified one pole transfer function of the power supply and saddle coils dynamics

29. Torque balance Model • The TM phase evolution is ruled by a balance between EM torque: due to feedback coils and image currents Viscous torque: due to fluid motion • A threshold of the proportional gain is found after which the • modes begin to rotate

30. Torque balance Model • Qualitative comparison between the simulation and the experimental trend Simulation Measure m=1, n=-7 m=1, n=-8 (rad/s) (rad/s) kp Kp/2400

31. Next step • Limits in models: - EM model ignores plasma dynamics - Torque balance model uses an “idealized” thin shell • Necessary integration of the EM model with the torque balance model: Interaction between TMs dynamics and the control system non-uniformity has to be taken into account in order to reduce the phase and wall locking phenomena

32. Next step • With this integration we aim to: • - Optimize the control system • - Test new algorithms before the real-time implementation • - Improve the system performances both in terms of plasma wall • interaction and reduction of the core stochasticity as a secondary effect

33. Slides di riserva

35. Sideband harmonics- Nyquist theorem A net of MxN saddle coils covering a torus can produce radial fields with helicities up to m=M/2 and |n| up to N/2 together with an infinite number of sideband harmonics Nyquist’s sampling theorem states that the DFT harmonics (i.e. the Fourier coefficients of a discrete periodic sequence) correspond to Fourier harmonics only if the spectrum is contained within the Nyquist frequency. If this condition is violated the aliasing phenomenon occurs: i.e. Fourier harmonics with high toroidal number appear in the DFT spectrum at a lower toroidal number Es: if the system of 48x4 RFX-mod coils is generating an m=1, n=-7 radial field, all the sideband harmonics, e.g. the m=1,n=-55, m=1,n=41, etc., will all be aliased into the m=1,n=-7 DFT coefficient

36. PID feedback law The field harmonic produced by the saddle coils is given by a PID feedback law Cleaned harmonic The derivative action is performed on a one pole low filtered version of the cleaned harmonic, with a cut-off frequency of 300Hz

37. Experiments with Complex Gain: Multiple Modes The deformation of the LCFS tends to show a secondary local maximum of similar amplitude. The maximum of the deformation jumps between these two locations. Consequently, complex gains were set only on low amplitude high n (n<-12) TMs, while different proportional and proportional-derivative gains were set for dominant modes, in order to vary the modes’ phase locking.

38. Feedback equations Feedback acquired signal The CMC feedback law (PID): gives the cut-off of the low-pass filter applied to the derivative gain (5ms) Delays introduced by the digital acquisition, feedback operations and coils power supply modelled as one-pole filter law, with a cut-off frequency of 80Hz, corresponding to a delay 2ms

39. Feedback equations Feedback acquired signal: Power supply equation modelled as

40. Sideband effect The (MN) DFT harmonics must not be confused with the Fourier modes Fourier modes are defined by the analytical series DFT harmonics are affected by aliasing: The shape factor is due to the finite extent of the sensors

41. Vacuum formulas for sidebands The radial field produced by the coil in the absence of plasma is: It is obtained using the standard cylindrical vacuum solution for the m.f. in terms of the modified Bessel function and adopting the thin shell dispersion relation

42. Vacuum formulas for sidebands In the Newcomb’s equation, the plasma terms scale as Are small for sidebands, due to to their large poloidal and toroidal mode numbers (M=48, N=4) Vacuum solution Newcomb’s solution

43. Shell region A diffusion equation valid in the limit dw<<rw is adopted according to C.G.Gimblett, Nucl. Fusion 26 (1986) 617 More general than the thin-shell relation since the variation of the radial field across the shell is taken into account.