Perspectives of

tearing

modes control in RFX-mod

Paolo Zanca

Consorzio RFX, Associazione Euratom-ENEA sulla Fusione, Padova, Italy

RFX-mod contributions to TMs control (I)

• Demonstrated the possibility of the feedback control onto TMs

• Clean-Mode-Control (CMC) based on the de-aliasing of the measurements from the coils produced sidebands

RFX-mod contributions to TMs control (I)

• Demonstrated the possibility of the feedback control onto TMs

• Clean-Mode-Control (CMC) based on the de-aliasing of the measurements from the coils produced sidebands

• Not obvious results: phase-flip instability?

RFX-mod contributions to TMs control (I)

• Demonstrated the possibility of the feedback control onto TMs

• Clean-Mode-Control (CMC) based on the de-aliasing of the measurements from the coils produced sidebands

• Not obvious results: phase-flip instability?

• No-sign of phase-flip instability; equilibrium condition can be established where CMC induces quasi-uniform rotations of TMs

• Wall-unlocking of TMs with CMC

• In general, the feedback cannot suppress the non-linear tearing modes requested by the dynamo.

• The feedback keeps at low amplitude the TMs edge radial field

• Improvement of the magnetic structure: sawtooth of the m=1 n=-7 which produces transient QSH configurations

RFX-mod contributions to TMs control (II)

• Increase the QSH duration → recipes under investigation

• Which are the possibilities to reduce further the TMs edge radial field? → Model required

CMC optimizations

RFXlocking

• Semi-analitical approach in cylindrical geometry

• Newcomb’s equation for global TMs profiles

• Resonant surface amplitudes imposed from experiments estimates

• Viscous and electromagnetic torques for phase evolution

• Radial field diffusion across the shell(s)

• Feedback equations for the coils current

• It describes fairly well the RFX-mod phenomenology →L.Piron talk

General analysis of the TM control

plasma

Sensors CoilsVessel

Single-shell external coils

Normalized edge radial field

•The feedaback action keeps low the normalized edge radial field

)()(ˆ,

,,,nm

nmrsens

nmr

nmsens rbrbb

• At best b^sens can be made close but not smaller than the

ideal-shell limit

plasma

Sensors CoilsVessel

Feedback limit

plasma

Sensors CoilsVessel

Feedback limit

plasma

Sensors

br=0 everywhere: impossible

CoilsVessel

Feedback limit

Role of the Vessel

• The stabilizing effect of the vessel is crucial for having low b^

sens and moderate power request to the coils

• The shorter τw the faster must be the control system (fc=1/Δt) to avoid feedback (high-gain) induced instabilities

• Optimum range: τw >10ms better τw 100ms

plasma

Sensors Coils Vessel

Single-shell Internal coils

plasma

Sensors Coils Vessel

Single-shell Internal coils

Single-shell Internal coils

• Continuous-time feedback → solution ωω0 with br(rsens) 0 for large gains

•Discrete-time feedback : including the latency Δt the high-gain instability may occur

• The good control region is not accessible for realistic TM amplitudes.

• For stable gains b^sens is determined by the ideal-shell limit,

which is large due to the loose-fitting vessel required by the coils dimension

RFP design for good TM control

(a personal view)

Premise

• The passive stabilization provided by a thick shell does not solve the wall-locking problem

• In the thick-shell regime wall-locking threshold ~σ1/4

• Feedback is mandatory to keep TMs rotating

Design in outline • In-vessel coils not interesting

• Single structure (vessel=stabilizing shell) with the coils outside

• Close-fitting vessel to reduce the ideal-shell limit

• τw 10ms-100ms with Δt 10μs-100μs

RFX-mod perspectives (a personal view)

RFX-mod layout

• 3ms vacuum-vessel, 100ms copper shell, ~25ms mechanical structures supporting the coils

• The control limit is mainly provided by the 100ms copper shell

RFX-mod status

0

0,2

0,4

0,6

0,8

1

8 10 12 14 16

b^

a ideal shell

b^

a RFXlocking

b^

a experiment

-n

Gain optimization guided by RFXlocking simulations for the RFX-mod case

m=1 TMs

Optimizations

• Get closer to the ideal-shell limit (minor optimization)

• Reduce the ideal-shell limit by hardware modifications (major optimization)

Minor optimizations

• Increase the coils amplifiers bandwidth: maximum current and rensponse time

• Acquisition of the derivative signal dbr /dt in order to have a better implementation of the derivative control (to compensate the delay of the coils amplifiers)

• Compensation of the toroidal effects by static decoupler between coils and sensors only partially exploited

• Compensation of the shell non-homogeneities requires dynamic decoupler (work in progress)

Major optimization

• Approach the shell to the plasma edge possibly simplifying the boundary (removing the present vacuum vessel which is 3cm thick)

• Moving the τw =100ms shell from b=0.5125m to b=0.475m (a=0.459) a factor 3 reduction of the edge radial field is predicted by RFXlocking

Conclusions• CMC keeps TMs into rotation

• Edge radial field: ideal-shell limit found both with the in-vessel and out-vessel coils → br(a)=0 cannot be realized

• The vessel=shell must be placed close the plasma → coils outside the vessel. Is a close-fitting vessel implementable in a reactor?

• The feedback helps the vessel to behave close to an ideal shell → τw cannot be too short

spare

Edge radial field control by feedback

0

5

10

15

0 0,02 0,04 0,06 0,08 0,1

w=100ms

rwi

=0.475m rf=r

wi

rwi

=0.5125m rf=r

wi

RFX-mod experiment

max

[1(

)] (

mm

)

time(s)

RFXlocking .vs. experiment

0

20

40

60

0

0,05

0,1

0,02 0,04 0,06 0,08 0,1

m=1, n=-7

brs

b^

a

mT

time(s)

Normalized edge radial field: weak brs dependence

br(rm,n) vs br(a) experimental

0

0,01

0,02

0,03

0,04

0,05

0,06

0

0,002

0,004

0,006

0,008

0,01

0,012

0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,1

br1_mn(7) br1_ra(7)br

1_m

n(7)

br1_ra(7)

time

Locking threshold

The present analysis valid for w<<rw cannot be extrapolated to very long w

2

6

10

1 10 100

Wall-locking threshold m=1, n=-7 (mT)

w (ms)

300

0,22

0,23

0,24

0,25

0,26

0,27

0,28

0,29

0 0.5 1 1.5 2 2.5 3

m=1 n=-8

b^

a

c(ms)

Edge radial field .vs. current time constant

a = 0.459m

rw i = 0.475m

c = 0.5815m

Single mode simulations: external coils

0,1

1

10-4 10-3 0,01 0,1 1

rf=a

1.1252.253672

b^

f

d/

w

ideal shell0.03

-Kpxa/(0.96 )=

0,04

0,06

0,08

0,1

0,3

0,001 0,01 0,1

Kp=10.4 w=

w/0.1x3mm

Kp=10.4 paper

b^ a

w(s)

brs

1,7 =12mT Kd=

cK

p r

f=r

wi

Single-mode analysis: feedback performances dependence on w

1000

104

105

106

107

0,001 0,01 0,1

Kp=10.4 w=

w/0.1x3mm

Kp=10.4 paper

abs(

IcV

c) (

W)

w(s)

brs

1,7 =12mT Kd=

cK

p r

f=r

wi

Single-mode analysis: feedback performances dependence on w

1000

104

105

106

107

0,05 0,052 0,054 0,056 0,058 0,06

rf=r

w

w=0.1s;

d=0.1ms

rf=a

w=0.1s;

d=0.1ms

rf=a

w=0.01s;

d=0.01ms

max

i,j(P

i,j)(

W)

time(s)

Multi-mode analysis: power dependence on w

Edge radial field: w dependence

Data averaged on 0.1s simulation

0,04

0,06

0,080,1

0,3

8 10 12 14 16 18

rf=r

wi

ideal shell

w=100ms

w=3ms

w=10ms

w=200ms

b^ a

-n

m=1

Normalized edge radial field: rwi dependence

0,04

0,06

0,080,1

0,3

0,5

8 10 12 14 16 18

rf=r

wi

w=100ms

ideal shell rwi

=0.475m

ideal shell rwi

=0.5125m

rwi

=0.475m

rwi

=0.5125m

b^ a

-n

m=1

Normalized edge radial field: no rf dependence

0,04

0,06

0,080,1

0,3

8 10 12 14 16 18

w=100ms

ideal shellrf=r

wi=0.475m

rf=a=0.459m

b^ a

-n

m=1

0

0.5

1

1.5

2

0,05 0,055 0,06 0,065

maxi,j(br

i,j) clean

maxi,j(br

i,j) raw

mT

time(s)

Out-vessel coils: signals

4x48 both for coils (c = 0.5815m) and sensors (rwi = 0.475m )

Single-shell: discrete feedback

ttt jj 1 Δt = latency of the system

;1,,

1,,

1,

jnm

rnm

djnm

rnm

pjjnm

c tbdt

dKtbKttV

External coils: discrete feedback τw=100ms

0,03

0,04

0,05

0,06

0,1

0,2

2 6 10 14

w=100mscontinuous

t=10-6

t=10-5

t=10-4

t=10-3

< b

^ f >

-Kp rwi

/(0.96)

a)

ideal-shell limit

External coils: discrete feedback τw=10ms

0,03

0,04

0,05

0,06

0,1

0,2

2 6 10 14

w=10ms

continuous

t=10-6

t=10-5

t=10-4

< b

^ f >

-Kp rwi

/(0.96)

a)

ideal-shell limit

External coils: discrete feedback τw=1ms

0,03

0,04

0,05

0,06

0,1

0,2

2 6 10 14

w=1ms

continuous

t=10-6

t=10-5

< b

^ f >

-Kp rwi

/(0.96)

ideal-shell limit

a)

a=rf

plasma shellcoils grid

c rwi0

rm,n

w

0

m,n(r)

The in-vessel coils

Single mode simulations: frequency

τw= 1ms100ms

0,001

0,1

10

1000

105

107

0,0001 0,01 1 100 104

d1,7/dt

0( ra

d/s

)

Kd / K~

d

Single mode simulations: Ic, Vc

10

100

1000

104

105

|Ic

1,7| (A)

|Vc

1,7| (V)

|Ic

1,7| formula (74)

|Vc

1,7| formula (75)

Kd / K~

d

110-2 102 104

Single mode simulations: edge br

)()(ˆ,

,,,nm

nmrf

nmr

nmf rbrbb

10-8

10-6

10-4

0,01

1

0,0001 0,01 1 100 104

Kd / K~

d

b^

f

ideal shell

Multi-mode simulations: frequencies

1000

104

105

7 9 11 13 15 17 19

1 3 5 7 9 11

d1,n/dt

m=1

d0,n/dt

m=0

rad

/s

n (m=1)

n (m=0)

Averages over the second half of the simulation

Multi-mode simulations: plasma surface distortion

0

2,5x10-4

5x10-4

0 0,002 0,004 0,006 0,008

max[1

max[0

(cm

)

time(s)

Multi-mode simulations: no phase-locking

Ideal shell feedback

1,11,1 nn

Multi-mode simulations: no phase-locking

dt

d

dt

d

dt

d nnnn 1

,11,1

1,11,1

nn

nn

dt

d

dt

d ,10

,11,1

0

1,1

Incompatible with

Internal coils: discrete feedback stable solutions

10-5

0,0001

0,001

0,01

10-7 10-6 10-5 0,0001

K~

d / K~

d+

t(s)

Internal coils: discrete feedback stable solutions

0,01

0,1

1

10-5 0,0001 0,001 0,01

b^ f

K~

d / K~

d+

ideal shell limit

b)

The MHD model: Ψwi, Ψwe

wewi

nmnm

rrrrt

,;2

,2,

0

Boundary conditions from Newcomb’s solution

The MHD model: Ψs

)(,,0

, )()( tnminms

nms ett

trmtrndt

dnmnm

nm

,, ,,

,

From experiment

No-slip condition

The MHD model: Ωθ, ΩΦ

nmnm

nmEM rrrR

TS

rr

rrt ,

0

30

2

,

4

1

04

1,

00

32

,3

3

nm

nm

nmEM

Drr

Rr

T

n

mS

rr

rrt

The MHD model: δTEM

nmnmnmnmnm cskji

nmnmnm

nmk

nmj

nmi

nms

nmc

nm

nmcsnm

EM

kji

R

rnm

EnRT

,2,21,12,2,1,1 ,,,

,2,2,1,1

2,21,1,

,,

20

2,22

,

0

02

,

,,Im

Im8

The MHD model: Ψc

nmcc

cr

cr

nm InmR

cnmi

rr ,

20

222

0, ),(

Further variable: Icm,n

The MHD model: Ic

termsaddIIdt

dI nmf

nmc

nmcnm

c .,Re

,,

,

Further variable: IREFm,n

RL equation for the plasma-coils coupled system

rjijic

REFjicji dt

dIRIRV ,,,,

The MHD model: IREF

nmr

nmd

nmref w

dt

dKI ,,,

Acquired by the feedback

Why a pure derivative control?

When | cm,n|>>1, from the RL equation one gets

nmc

nmc

nmc

nmc

nmf IiIiI ,,,,,

Re 1

nmrnm

c

nmdnm

c wK

I ,,

,,

nmr

nmd

nmf wKiI ,,,

Re