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

9 October 2008, European Doctorate in Fusion Science and Engineering

Lidia Piron

Consorzio RFX, Euratom-ENEA Association, andUniversity of Padova, Italy

Experimental characterization and numerical modeling of the active control

of resistive MHD modes in RFX-mod


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



a=0.459m, R0=2m

RFX-mod, Padova, Italy

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


The edge radial magnetic field br is controlled by saddle coils independently fed

Assembly of saddle coils on vacuum vessel (mid’ 2004)

RFX-mod active coils system

4(pol) x 48(tor) =192

saddle coils (Cycle latency <400 s)

Sensors for br, bT and is associated to each saddle coil

Stabilizing shell : tshell~50ms


The dynamo mechanism

A large portion of BT and BP is generated by currents flowing in the plasma, through a dynamo mechanism

€

E

€

η J+ =?


The dynamo mechanism

BT and BP are generated by currents flowing in the plasma, through a dynamo mechanism

€

E

€

η J+ =

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 ]

Edynamo

€

˜ v × ˜ B


q(r)

r(m)

TMs spectrum

m=1, n=-7

m=0, n=1,2,3,4, …

m=1, n=-8m=1, n=-9

€

q(r) =rBT

RBp

Safety Factor:


TMs effects

Core: confinement degradation Edge: Non axi-symmetric deformations of the Last Closed Flux Surface, (r)

m=1, n=-7m=1, n=-8m=1, n=-9

r/a

b r (m

T)

(r)


TMs effects

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 ]

(r) produces plasma-wall interaction,

which is affected by the phase-locking

and wall-locking phenomena


Inputs: br

Outputs: coil current

reference

To control br

Plasma

Digital Controller

Power Amplifier

An algorithm for Feedback control: Virtual Shell

[C. M. Bishop, Plasma Phys. Control. Fusion 31 (1989) 1179 ]

Sensors

bext


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

€

br,cm +l M ,n +k N (rs)

l ,k{ }∈Ζ2 −0

∑


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

The Clean Mode Control algorithm [P.Zanca et al, Nucl. Fusion 47 (2007) ]

0

€

brm,n (rs) = br,DFT

m,n − br,cm +l M ,n +k N (rs)

l,k{ }∈Ζ2 −0

∑


Sidebands correction

brbr, DFT m,n bf, DFT m,n

Irefm,n

Icoilm,n

The Clean Mode Control Scheme

Plasma

Digital Controller PowerSupplies

DFT Magnetic analysis

br, DFT m,n

brm,n


Effects of active control on TMs

Reduction of m=1 deformation of the Last Closed Flux Surface,

Clean Mode Control

Virtual Shell

No MHD active control

time (s)

Ip (M

A)

Ip (MA)

(m

)

The highest current for a RFP up to now

Significant increase of pulse length

Virtual ShellClean Mode Control

No control


Effects of active control on TMs

Spontaneous transition to a quasi-single-helicity state (QSH)

The magnetic chaos is reduced, i.e. the confinement is improved

time (s)

b T / B

r (a)

(%)

Radius (mm)

Te (e

V)

m=1, n=-7 < m=1, n=-8 to -16 >



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

Distribution of the median of the rotation frequency for the m=1, n=-7 TM for two ensembles of VS and CMC discharges

Phase dynamics of m=1, n=-7

Phas

e (ra

d)

time (s)

Freq (Hz)

Phas

e (ra

d)

Virtual Shell

Clean Mode Control


Control of the direction of rotation: Complex Gain

The reference value for the applied field of the CMC algorithm for each mode is

The direction of the mode rotation is determined by the sign of the phase m,n on the dominant m=1,n=-7 mode

€

bm,ncoil (t) = −KP ,m,n exp(iϕ m,n )br,m,n (t)

Complex proportional gain

Phas

e (ra

d)

time (ms)

b-7 1,

r (a) (

mT)


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


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

Proposed by P. Piovesan et F. Volpe in RFX-mod

Phase and amplitude of 2/1 NTM in DIII-D experiment



Error Fields

Control system acts not only on TMs but also on Error Fields

time (s)

pha

se (r

ad)

m=1, n=-7

m=1, n=-6

time (s)

pha

se (r

ad)

phas

e (ra

d)

phas

e (ra

d)

r(a)T

r(a)T


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

Poloidal gap

Toroidal position

p.d.

f.



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


Comparison EM model - experiment

SimulationMeasure


Torque balance ModelThe 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


The TM phase evolution is ruled by a balance between

Torque balance Model

€

δTEMm,n + δTV

m,n = 0

A threshold of the proportional gain is found after which the

modes begin to rotate

Viscous torque:due to fluid motion

EM torque:due to feedback coilsand image currents


Qualitative comparison between the simulation and the experimental trend

Measure

(

rad/

s)

kp

m=1, n=-7

Simulation

Kp/2400

m=1, n=-8

(

rad/

s)

Torque balance Model


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

Next step


Slides di riserva


R0=2, a=0.459

r(m)

plasmavessel

rvi=0.475 rve=0.505

τv=3ms

copper shell rwi=0.5125, τw=0.1s, δ

w=3mm

coils grid

c=0.58


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

Sideband harmonics- Nyquist theorem


PID feedback law

The field harmonic produced by the saddle coils is given by a PID feedback law

€

bm,ncoil (t) = −KPbm,n

r (t) − K I dt0

t

∫ bm,nr (t) − KD

d

dtℑ (bm,n

r (t), fcut )

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


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.

Experiments with Complex Gain: Multiple Modes

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.


Feedback equations

Feedback acquired signal wiffnm

r rratrb ≤≤);,(,

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

The CMC feedback law (PID):

€

IDFTm,n (t) =

KP + (iω / (1+ iωτ d ))KD + (1 / iω)K I

1+ iωΔtbr

m,n (rf , t)

€

τ d gives the cut-off of the low-pass filter applied to the derivative gain (5ms)


Feedback equations

Feedback acquired signal:

wiffnm

rnm

rnm

rd rratrbwwdt

d≤≤=+ );,(,,,τ

Power supply equation modelled as

msIIIdt

dc

nmREF

nmc

nmcc 5.0;,,, ==+ ττ


The (MN) DFT harmonics must not be confused with the Fourier modes

Sideband effect

( ) ( )θθ nmi

nm

nmr

r erbrb +

Ζ∈∑=,

, )(,,

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

€

br,DFTm,n = br

p,q (rs) f (p,q)l,k{ }εZ

∑

€

f ( p,q) =sin(q(Δφ / 2))

q(Δφ / 2)

sin(p(Δφ / 2))

p(Δφ / 2)€

p = m + lM

q = n + kN


( ) ( ) ( ) Ζ∈⋅++ℑ= ∫ −++ ++kldIeNknMlmrtrb

t

t

nmDFT

tANknMlmcr

NknMlm

,,)(,,,

0

, ,,, ξξξ

( ) ( ) ;,,, ,20

2

000

nmcm

cm Anmf

R

rn

R

rnI

R

rnKnmr ⎟⎟

⎠

⎞⎜⎜⎝

⎛′⎟⎟

⎠

⎞⎜⎜⎝

⎛′=ℑ μ

;11

00

2

0

2

0

2

,

⎟⎟⎠

⎞⎜⎜⎝

⎛′⎟⎟

⎠

⎞⎜⎜⎝

⎛′

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

Rbn

IRbn

KRbn

Rbn

mA

mmb

nm

τ

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 shelldispersion relation

€

τ b = μ0bδbσ b


Vacuum solution Newcomb’s solution

( )222/1 εnm +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 formulas for sidebands


Shell region

wiww

nmr

nmr r

r

b

t

b δστσ 0

,,

0 ; =∂∂

=∂∂

More general than the thin-shell relation since the variation of the radial field across the shell is taken into account.

wwir

wir

nmr

wir

nmr

w r

b

t

bδ

δσ+

∂∂

=∂∂ ,,

0

A diffusion equation valid in the limit δw<<rw is adopted according to C.G.Gimblett, Nucl. Fusion 26 (1986) 617

Documents

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