Requirements for Active Resistive Wall Mode (RWM) Feedback Control

Yongkyoon In1

In collaboration with M.S. Chu2, G.L. Jackson2, J.S. Kim1, R.J. La Haye2, Y.Q. Liu3, L. Marrelli4, M. Okabayashi5, H. Reimerdes6, and E.J. Strait2

1FAR-TECH, Inc., San Diego, CA, USA 2General Atomics, San Diego, CA, USA 3UKAEA, Culham Laboratory, UK 4Consorzio RFX, Padua, Italy 5Princeton Plasma Physics Laboratory, Princeton, NJ, USA 6Columbia University, New York, NY, USA

14th Workshop on MHD Stability Control, November 9-11, 2009

Princeton Plasma Physics Laboratory, Princeton, New Jersey Active feedback control safeguards high performance plasmas against the uncertainty of RWM stability boundary • The RWM stabilization is essential to sustain high performance plasmas in ITER or reactor- grade devices • The uncertainty of stability boundary of pressure-driven RWM challenges the evaluation of RWM stabilization process • Even if RWM is passively stabilized, a coupling of marginally stable RWM with other MHD activities readily leads to the unstable RWM, requiring active feedback control1 1M. Okabayashi et al, Nucl. Fusion (2009)

1 Reproducible current-driven RWM helps us clarify both physics requirements and control specifications for feedback control

• Highly reproducible current-driven RWM enables us to explore the RWM feedback control (no or little external torque source) – The RWM feedback control can be assessed using a reproducible RWM target at q95 ~ 4. • A systematic thorough investigation of various control parameters allows us to benchmark both RWM theory and control aspects • BUT, the optimization of RWM feedback control is ultimately determined by the minimal level of plasma fluctuation beyond magnetics

2 Outline • Introduction • Schematic of active RWM feedback control • Optimized feedback control (limited to PD controller for now) – Gain scans – Bandwidth requirements – Phase-shifts – Efficacy of active feedback control • Physics requirements – Internal vs external feedback coils – Direct RWM feedback vs error field correction (EFC) – Simplified Feedback control model – Mode structure – Possibility of 2nd least-stable RWM • Discussion • Conclusions and future works

3 The helical structure of the feedback is configured to match the structure of the n=1 RWM

4 Active feedback control system in DIII-D fully stabilizes current-driven RWM in parallel with EFC

• Tools • Ohmic discharge with high – Internal coils (“I-coils”): current ramp-up rate Direct Feedback + Dynamic error field • Feedback loop (τp << τw) correction (EFC) n=1 n=1 – External coils (“C-coils”): VFB (τw,τL/R, δIFB ) Feed-forward EFC Power Supply Plant (DIII-D/RWM) n=1 δB + ? Controller 1 ⎡ Gsd ⎤ Ks()=+⎢ G τ d ⎥ 11++ssp τ p ⎣⎢ τ d ⎦⎥

Gp,d : Gain τp,d : time constant where p - proportional, and d - derivative ? : Unknown n=1 error field 5 Current-driven RWM was obtained, diagnosed, and controlled with no external torque source present Plasma shape, p, q, and • Upper Single Null Ωφ @ 405 ms on 133021 (close to Double Null shape) • Monotonic pressure and q- profiles • Plasma rotation profile is nearly flat according to CER measurement using carbon-VI line [nearly “zero”:

(Ω0/ωA)< 0.1 %)] 6 Complete feedback stabilization of current-driven

RWM at q95~4 has been achieved in DIII-D

133021 133018 6 • RWM is observed to be q No feedback vs feedback 95 i) non-rotating or moving

slowly in co-Ip direction

5 at its birth, but δB n=1 (G) 4 p 3 ii) flips to the counter-I 30 p 20 direction as the mode 10 grows up

0 n=1 180 φ (deg) • Balanced beam-pulse 0 500 IU (A) -180 30 every 100 msec for 12 0 400 420 440 460 msec duration to -500 60 msTime 480 500 520measure q-profile and (ms) plasma rotations

7 Optimized feedback control

–Gain scans (Gp only vs Gp/Gd) –Bandwidth requirements –Phase-shifts –Efficacy of active feedback control

8 An optimized gain has been found in the vicinity of

“unit” flux gain (applied δΨr∼ measured δΨθ)

Maximum Mode Amplitudes (Gauss) vs G p 25

20

15

10 Optimized 5 Unit flux gain

Optimized 0 0 50 100 150 Proportional Gain G p • As the gain approaches the optimal level, the mode growth rate decreases as expected • The higher coil currents at low gains are primarily attributable to an inadequate EFC to unknown error field

9 The optimized RWM feedback control minimizes the plasma fluctuations beyond the plasma boundary

No feedback (133021) Feedback (133018)

n=1 Amp. [G]

[keV] Nov.08,2008 10:06:26 e 0.05 1.8 1.8

1.7 R Core maj 1.7

1.6 1.6

1.5 0 R [m] R [m] 1.5 q~2 1.4 1.4

1.3 δTe 1.3

1.2 1.2 −0.05 Edge 0.44 0.45 0.46 0.47 0.48 0.49 0.5 0.44 0.45 0.46 0.47 0.48 0.49 0.5 Time [s] Time [s] • RWM-induced edge fluctuations are also suppressed.

10 The use of derivative gain broadened the effective gain range for RWM feedback stabilization

Maximum Mode Amplitudes vs Gp

Unit flux gain

• Similarly finite amplitude of the coil currents at various

Gp values are associated with the EFC portion necessary for effective RWM stabilization.

11 The addition of Gd minimizes the phase lag in time between RWM and the applied field

Without Gd (133017) With Gd (133014)

[keV]

e 0.05

1.8 1.8 1.7 Core 1.7 Rmaj 1.6 1.6 0

R [m] 1.5

R [m] 1.5

1.4 1.4 Gp=40 1.3 δTe 1.3 Edge 1.2 1.2 −0. 05 0.44 0.45 0.46 0.47 0.48 0.49 0 .5 0.44 0.45 0.46 0.47 0.48 0.49 0.5 Time [s] Time [s]

• A value of Gd =10Gp is chosen to use voltage controller ‘effectively’ as current controller,

based on τd and τL/R of the feedback system.

12 The voltage controller used for RWM feedback could work as current controller using derivative gain =>Still the bandwidth is limited -1 -1 by minimum (τp , τd ) δI Plant,P(s) δB

Controller,K(s) δV K K =>The high frequency role-off L/R PCS due to L/R time could be 1 where Ks()= and compensated by the LR/ 1+s τ L/R choice of Gd, which helps 1 ⎡ Gsd ⎤ the system function as Ks()=+⎢ G τ d ⎥ PCS11++ss p current controller. τ pd⎣⎢ τ ⎦⎥

13 Impact of DEFC on RWM feedback can be fully explored by the bandwidth scan

• Transition from DEFC to RWM-dominant regime

- RWM: τp < τw

- Dynamic EFC: τp > τw – Feed-forward EFC: pre- programmed

• The low frequency EFC is necessary but not sufficient for RWM stabilization

14 The RWM feedback action should be taken faster than the mode growth time, as predicted1

τg : 3 - 4 ms

• When τp < τg : effective , while τp > τg :ineffective

• The used fixed gain provides δΨr ~ 5 x δΨθ on the midplane magnetic sensor 1 T. Strait et al, Nucl. Fusion (2003) 15 A phase-shifted n=1 field in the direction of co-Ip rotation is more effective than in the opposite direction

Maximum Mode Amplitudes [G] vs δφFB Phase-shift dependency in MARS-F

Stabilized

• A range of preferred toroidal phase shifts ahead of the RWM exists for effective feedback, consistent with theory

16 The efficacy of the feedback stabilization needs to be cautiously assessed based on magnetics alone

Gd= 10Gp

Gp= 80 (133011) Gp= 160 (133012) Gp= 320 (133013) e e e 0.05

1.8 .8 .8

1.7 Rmaj .7 .7 Core 1.6 .6 .6

0

R [m] 1.5 .5 .5 q~2 1.4 .4 .4

1.3 δT .3 .3 Edge e

1.2 .2 .2 −0.05 0.44 0.45 0.46 0.47 0.48 0.49 0.50.44 0.45 0.46 0.47 0.48 0.49 0.50.44 0.45 0.46 0.47 0.48 0.49 0.5 Time [s] Time [s] Time [s] • Similar plasma responses based on magnetics may not reflect the detailed evolution of the RWM-associated internal structures.

• The choice of Gd may not be optimal yet (likely due to mode coupling to coil currents,≠ ) (τ L //RLR)eff τ

17 Physics Requirements

– Internal vs external feedback coils – Direct RWM feedback vs error field correction (EFC) – Simplified feedback control model – Mode structure – Possibility of 2nd least-stable RWM

18 The RWM feedback coils external to the vessel was found ineffective in stabilizing RWM

• The internal coils (“I-coils”) merit more effective coupling of the feedback field due to the proximity to the plasmas than the external coils (“C-coils”) • This suggests that the error-field (EF) correction coils in ITER might not be a good choice for the purpose of RWM feedback control, though the investigation was not exhaustive. RWM growth time, τ : 3 ms g 19 Even after RWM stabilization, finite coil currents remain working as error field correction • While the gain K increases, the δB gets smaller, but the time-averaged δIrarely changes • Both DEFC and RWM feedback are working inside closed-loop but their roles are distinctive - DEFC: minimize the lack of axisymmetry of external DEFC field (low freq.) - RWM feedback: nullify the perturbed δB coming from unstable RWM (high freq.) 20 RWM feedback cannot be replaced by error-field correction (EFC)

RWM growth is not influenced by EFC (i.e. no slope change)

RWM growth time, τg : 3 - 4 ms 21 Simplified feedback model is based on idealized current controller • Measured perturbed field

δB = δBRWM+δBEF+AδBFB

where δBRWM: RWM,

δBEF : Error field, A : Plasma response (amplification),

and δBFB = -K δB = -αIcoil

K: Normalized gain (=Gp/Gp0) Icoil : feedback current (=IEFC+δIRWM), contributing to EFC and direct RWM FB

22 The key components in feedback control can be explicitly represented in modeling • Assumptions

– Open loop K=0 shows finite δB = δBRWM+ δBEF

where δBRWM > 0 only for unstable RWM (δBRWM = 0 for stable RWM)

– δBEF > 0 can remain finite even when δBRWM = 0 – Feedback is configured to oppose measured δB. 1 δδδBBB=+{} ∴ 1 + KA RWM EF α K IBB=+{}δδ coil(1+ KA ) RWM EF

23 Even after RWM stabilization, a finite IEFC is required to minimize the non-axisymmetric EF • Closed loop: 1 WhenKB>> 1,δδ{} B , while δ B 0 KA EF RWM δ B I ~ EF coil α A This suggests that when RWM is fully suppressed at high gain K, the DEFC could remain finite, regardless of the magnitude of K. • Indeed, the feedback coil currents remained finite, even after the RWM was fully suppressed. This is attributable to EFC, not interacting with RWM directly.

24 Modeling shows the remained EF would not be sensitive to gain increase, consistent with experiments

DEFC

• Even after RWM is fully suppressed, a finite amount of coil currents is required to provide EFC, whose role can be substituted by other slowly varying actuators

25 The internal mode structure observed in HFS is not necessarily the same as in LFS

m ⎧ 0:LFS ξξnn()rrmwhere==∑ cos( () θθ ), ⎨ m ⎩ : HFS Displacement calculation from MARS-F

LFS π

HFS

• The global structure of the current-driven RWM is unique in ideal MHD calculations, clearly different from tearing mode • Rotational threshold for passive stabilization of current- driven RWM is expected to be similar to RFP

26 The low frequency oscillatory behavior in high gain is indicative of the presence of 2nd least-stable mode Based on the poles calculated in MARS-F for pressure-driven RWM

Im(γτw)

Re(γτw)

• The coupling of the 1st and 2nd least-stable modes has been predicted in theory for a long time. • Need to distinguish the system pole and plasma pole (in progress)

27 MHD spectroscopy before and after current-driven RWM shows noticeable plasma responses at f= 400 Hz.

Midplane probes

I-coils

• Need to understand the plasma responses with respect to 1st or 2nd least-stable mode, as well as the RWM feedback algorithm

28 MHD spectroscopy prior to current-driven RWM shows noticeable plasma responses at f= 400 Hz • Strong plasma response (30-40 % amplified) in the off-midplanes, but not at midplane: Why?

• Another peak near 700 Hz (not a second harmonic of driving frequency nor spurious): What is it?

• Responses from 1st or 2nd least-stable modes under marginally stabilized conditions?

29 Requirements of RWM feedback control

Items Recommendations

Optimized • Gain scans •Advantage of Gd feedback • Bandwidth use -1 control requirements •Bandwidth > τw

• Phase-shifts •Co-Ip phase-shifts • Efficacy of active •Minimal plasma feedback control perturbations Physics •Internal vs external •Internal coils requirements feedback coils desirable • Direct RWM feedback •Dual systems for low vs error field correction frequency EFC and (EFC) high frequency FBK • Mode structure •Need of further • Possibility of 2nd investigation

least-stable RWM 30 The requirements of RWM feedback control are established, taking into account theoretical and experimental aspects

• Direct RWM feedback cannot be replaced by error field correction (EFC) • The bandwidth of the RWM feedback control should be wide enough to deliver the feedback action faster than the RWM growth time

• The benefits of Gd and phase-shifted feedback need to be taken into account • There was an indication of the presence of a second-least stable RWM which had been theoretically predicted but never identified in experiments. (cont’d)

31 The understandings of RWM feedback control are being enhanced, having clarified several key issues • The proximity of the feedback coils to the plasmas determines the degree of the effectiveness of RWM feedback • Simplified feedback modeling shows that a finite amplitude of coil current is still required for EFC, even after the RWM is stabilized. • Both modeling and experimental results are consistent, in that such finite amplitudes of DEFC would not be sensitive to gain values • The indication of the 2nd least stable RWM needs to be investigated to see if the optimized control parameters based on a single mode assumption are still valid

32