RWM CONTROL IN BP EXPERIMENTS:CONTROL COIL LOCATION, PASSIVE STABILIZER

GEOMETRY; FEEDBACK LOOP DYNAMICS

Gerald A. Navratil[with Jim Bialek, Allen Boozer & Oksana Katsuro-Hopkins]

Columbia University

Third Meeting of ITPA MHD, Disruption, and Control GroupIoffe Institute, St. Petersburg, Russia

15-17 July 2003

OUTLINE

• REVIEW OF RWM FEEDBACK

• CRITICAL ISSUES IN CONTROL DESIGN+ PASSIVE STABILIZER PERFORMANCE

+ CONTROL COIL-PLASMA-STABILIZER COUPLING

+ REACTIVE VS RESISTIVE CONTROL COIL

• DIFFERENCES WITH LIU & BONDESON+ TRANSITION TO IDEAL BRANCH IN DISPERSION RELATION

+ MAPPING OF bN TO DISPERSION RELATION LIMITS

+ EFFECTIVENESS OF ITER ERROR FIELD CORRECTION COILSFOR RWM CONTROL

• RESOLVING DIFFERENCES: BENCHMARKS

VALEN combines 3 capabilitiessee PoP 8 (5), 2170 (2001) � Bialek J., et al.

• Unstable Plasma Model ( PoP Boozer 98)• General 3D finite element

electromagnetic code• Arbitrary sensors, arbitrary control

coils, and most common feedbacklogic (smart shell and mode control)

X

Y

Z

VALEN Model

• All conducting structure, control coilsand sensors, are represented by afinite element integral formulation, wehave a matrix circuit equation: i.e.,

L[ ] I{ } + R[ ] I{ } = V(t){ }

• Unstable Plasma mode is modeled asa special circuit equation. We startwith a plasma equilibrium, use DCONwithout any conducting walls, toobtain δW, and the magneticperturbation represents the plasmainstability.

• The instability is represented via anormalized mode strength

s = −δW

LI2 / 2( ) , the equations are now

L' (s)[ ] I'{ } + R'[ ] I'{ } = V' (t){ }

DCON computation of mode structureNSTX - derived from EFITreconstruction of #106165βn = 6.154, Fp = 2.2, n = 1

6 . 05 . 04 . 03 . 02 . 01 . 00 . 00 . 0-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

Mode #1 @ 0.

DCON mode for beta n = 6.154mode at 0.0 deg ( even )

arc length

B-n

orm

al

( ar

b

un

its

)

6 . 05 .04 .03 .02 .01 .00 .00 .0-0.4

-0.2

0.0

0.2

0.4

Mode #1 @ 90.

DCON mode for beta n = 6.154mode at 90.0 deg ( odd )

arc length

B-n

orm

al

( ar

b

un

its

)

from dcon_surf_Fp2.2_bn6.154 24 by 72 Equivalent Surface Current which produces mode field

Input to VALEN for NSTX ’s’= 2.0590e-1

X Y

Z

from dcon_surf_Fp2.2_bn6.154 unwrapped inboard cut Equivalent Surface Current which produces mode vacuum field

Input to VALEN for NSTX ’s’= 2.0590e-1

X Y

Z

VALEN 03/14/02 NSTX half of VV removed

NSTX geometry ( including sensors and ideal control coils )

X

Y

Z

VALEN predicts growth rate forplasma instability as function of

the instability strength parameter 's'

• 's' is a normalized mode energy

s = −δW

LI2 / 2( )• computed dispersion relation of

growth rate vs. 's' is an eigenvaluecalculation

1 0 01 0 - 11 0 - 210 0

10 1

10 2

10 3

10 4

10 5

10 6

passive performance for NSTX structurebasic dispersion relation from VALENData from "NSTX.02.2002"

normalized mode energy s

gro

wth

rat

e (

1 /

sec

)

ideal branch

resistive branchdetermined by geometry &resistance of structure

difference producedby coupling toconducting structure

VALEN predicts growth rate forplasma instability as function of 's'

•s = −δW

LI2 / 2( ) a normalized energy

• great agreement for different startingplasma equilibria !

1 001 0- 11 0 - 2100

101

102

103

104

105

106

g (5.182)

gamma - 5.182

gamma-6.154

gamma-7.112

g (6.154)

g (7.112)

passive performance for NSTX structureData from "NSTX.02.2002"

s

gro

wth

ra

te

( 1

/se

c

)

beta n = 5.182

beta n = 6.154

beta n = 7.112

resistive branchdetermined by geometry &resistance of structure

ideal branch

VALEN s = 0.31623 gamma = 573.3 betan(6.154)driver 03/01/02 10:05:40 EST artemis executable: xvps6

eddy currents ( passive only ) in NSTX structure

X

Y

Z

VALEN predicts growth rate forplasma instability as function of 's'

• examine limit of perfect conductors• connect s to βn

1 0 01 0 - 11 0 - 2100

101

102

103

104

105

106

passive performance for NSTX structurebasic dispersion relation from VALENData from "NSTX.02.2002"

normalized mode energy s

gro

wth

rat

e (

1 /

sec

)

ideal branch

r e s i s t i v ebranch

i d e a ll i m i t

beta n = 5.182

beta n = 6.154

beta n = 7.112

ColumbiaUniversity

ITERPASSIVE STABILIZER AND ACTIVE CONTROL COILS

Inner Vacuum Vessel Wall (6 cm-thick stainless steel)

Outer VV Wall

The ITER vessel is modeled as two walls.

. Three sets of six saddle coils, outside the vessel, but the upper and lower coils couple weakly to the RWM. .

ITER Base Case Feedback Control System Geometry

1 21 086420-6

-4

-2

0

2

4

6

r

z

• The ITER vacuum vessel is modeled as a double wall configurationusing design data provided by Gribov, with feedback control provided by3 n=1 pairs of external control coils on the mid-plane.

ITER Vacuum Vessel Mode Analysis

n ~ 0 and m ~ 0

toroidal

polo

idal

outboardmidplane

inb oardmidplane

inb oardmidplane

• Modeled with 3906 elements.

• Longest Time Constant: 0.39897 s

ITER Vacuum Vessel Mode Analysis

n ~ 1 and m ~ 1(2 fold degenerate)

toroidal

pol

oida

l

or toroidal

pol

oida

l

• Modeled with 3906 elements.

• 1/1 Time Constant: 0.18762 s.

Whole Family of ITER Vessel Modesthe n=0, m=0 'OH response mode' 0.39897 s, 0.39804 s

[red = 3906 elements ; black = 2970 elements]

toroidal

polo

idal

outboardmidplane

inb oardmidplane

inb oardmidplane

the n=1, m=0 horizontal field mode ( 2 fold degenerate ) 0.31114 s, 0.31729 s

toroidal

pol

oida

l

toroidalpo

loid

al

the n=0, m=0 toroidal flux response 0.24513 s, 0.25216 s

toroidal

po

loid

al

the n=0, m=1 response to vertical plasma displacement 0.23113 s, 0.22753 s

toroidal

pol

oida

l

the n=2, m=0 mode ( which is 2 fold degenerate ) 0.19377 s , 0.20166 s

toroidal

po

loid

al

toroidal

polo

idal

the n=1, m=1 mode ( which is 2 fold degenerate ) 0.18762 s, 0.18937 s

toroidal

pol

oida

l

toroidal

pol

oida

l

VALEN Model of ITER Vessel and Control Coils:Base Case Feedback Control System

• Vacuum Vessel Modeled with and without wall penetrations.

DCON Calculation of ITER RWM:B-normal vs Poloidal Angle

1 81 61 41 21 086420-1

0

1

Bn#1@0

Data from "iter_at_129.sgfile"

arc length

Bn

#1

@0

Use B-normal to Compute EquivalentPlasma Surface Current

RWM Induces Stabilizing Image CurrentsLargely on the Inner Vessel Wall

• Vacuum Vessel Modeled with and without wall penetrations.

ITER Vacuum Vessel Penetrations HaveLittle Effect on Passive RWM Stabilization

1 0 01 0 - 11 0- 210 - 1

10 0

10 1

10 2

10 3

10 4

10 5

10 6

nbl_3.0

g-passive no ports 3.0

Data from "ITER.10.2002"Scen4_bn3.0 ( 129 by 129 )

s - normalized mode energy

gro

wth

ra

te

( 1

/s

)

no blanketsno ports

no blankets45 ports

• Relatively small reduction in the ideal wall beta limit.

Include ITER Blanket Modules in PassiveRWM Stabilization Model

• Modeled as set of isolated plates above theinner vessel wall.

• Each blanket module adjusted to have 9 msradial field penetration time constant.

VALEN Model of ITER Double Wall Vessel and Blanket Modules

XY

Z

ITER Blanket Modules Substantially Increasethe Passive RWM Stabilization Limits

1 0 01 0 - 11 0 - 210 - 1

10 0

10 1

10 2

10 3

10 4

10 5

10 6

passive character different modelsData from "ITER.10.2002"based on Scen4_bn4.0 ( 129 x 129 )from Y. Gribov

s - normalized mode energy

gro

wth

ra

te

( 1

/se

c

)

no blankets & no ports

no blankets45 ports

with blankets45 ports

ports are:18 top18 midplane9 divertor

• Offers prospect of much higher beta limit with anoptimized feedback control coil system.

Use DCON Computation of dW to Calibrate s to bN

0.60.40.20.0-0.2-0.22.0

2.5

3.0

3.5

4.0

4.5

beta-nbeta-n

s

beta

-n

• This calibration can then be used to replot passive response.

ITER Passive Dispersion Relation

4.03.02.010- 2

10- 1

100

101

102

103

104

105

106

beta-n

grow

th r

ate

[ 1

/ s

]

Without Blanket Modules

With BlanketModules

• No wall bN limit is 2.5; Ideal Wall Limit With Blanket is 3.7

Amplifier:

Gp and Gd

Control Coils PlasmaResponse Bp Sensors

V B-radial ψψψψ

RWM

Basic Feedback Control Loop with Voltage Amplifiersand Sensors Uncoupled to Control Coils

no-coupling

Feedback Volts/Weber

ITER Basic Coils: Scan of Proportional & Derivative Gain

4.03.53.02.52.02.010- 2

10- 1

100

101

102

103

104

105

106

g passive10^8 g10^8 10^8 g10^8 10^9 g10^9 10^10 gnbl_3.0

beta-n

grow

th r

ate

[ 1

/ s

]

stable < 2.643

stable < 2.743

stable < 2.759

• Feedback Saturates at bN ~ 2.76 for Gp=108 V/W & Gd=109 V/V• Gp=108 V/W is Liu’s Ki=0.32 and Gd=109 V/V is Liu’s Kp=15.6

RWM Dispersion Relation withMode Control Feedback*

* A. Boozer, Phys. Plasmas 5, 3350 (1998)

Apply Voltage to Control Coil, Vf(t) = - Lf

Mfp [gw Gp + Gd

ddt ] Fsensor

Gp = Proportional Gain Gd = Derivative Gaingw = Rwall/Lwall gf = Rcontrol coil/Lcontrol coil t= feedback delay

a3g3 + a2g2 + a1g + a0 = 0

a0/gw = – gf + gw Gp

a1 = gf D(s) + gw [Gd + cf Gp – s]/sa2 = D(s) + cf Gd/s a3 = t D(s)

For Stability all four Coefficients must be Positive!D(s) = c[(1+s)/s] -1 where c = [MpwMwp]/[LmodeLwall]

At Ideal Wall b Limit: D(scrit) = 0Feedback Coupling Constant, cf = 1 – [MpwMfw]/[LwallMfp]

For Feedback to Stabilize up toIdeal Wall b Limit cf must be ≥ 0

Want small Mfw and large Mfp to insure cf > 0

If Control Coils Outside Stabilizer then:Mwf > Mfp and cf < 0

Why is Basic ITER Control Coil Set aPoor Feedback System?

D(s) = c[(1+s)/s] -1 where c = [MpwMwp]/[LmodeLwall]

At Ideal Wall b Limit: D(scrit) = 0

ITER Basic System has scrit = 0.35 [or bN ~ 3.7]Therefore c = 0.26 for ITER

Feedback Coupling: cf = 1 – [MpwMfw]/[LwallMfp]Boozer shows that feedback fails when

D(s) + cf = 0

Using VALEN model results shows ITERFeedback Saturates ats = 0.063 therefore:

ITER Basic System: cf = - 3.39

Physically: cf = 1–[Vplasma from Iw]/[Vplasma from If]

Says Plasma Mode is more than 4 times bettercoupled to wall eddy currents thanexternal Basic ITER Control Coils.

VALEN Model Geometry: Resistive Wall & Control CoilSimple 1-turn Control Coil: Examine Control Fields with Wall Behind & in Front of Coil

z=0

z=-0.1

z=0.5z=-0.5

plate 130.e-08 ohm m2 x 2 x 0.0254 thick

coilR=0.5 m

sensor #50.01 x 0.01

sensor #80.01 x 0.01

Frequency Dependence of Control FieldMagnitude of Control Field vs Frequency

10 410 310 210 110 010 - 5

10 - 4

10 - 3

10 - 2

BB #5 gauss/ampBB #8 gauss/amp

Data from "FRdemo2a"t = 0.0254 rho = 130.e-08 ohm m

h z

gaus

s (

at s

enso

r )

/ (a

mp

in c

oil

)

sensor #8( 0.5 m left of coil )plate between coil & sensor

sensor #5( 0.5 m right of coil )

Phase of Control Field vs Frequency

10 410 310 210 110 0-270

-240

-210

-180

-150

-120

-90

-60

-30

0

BB ph - driverBB ph max plateBB #5 phaseBB #8 phase

Data from "FRdemo2a"t = 0.0254 m rho = 130.e-08 ohm m

hz driving frequency

phas

e re

lati

ve

to

driv

ing

volt

age

coil,

pla

te,

r. &

l.

sens

or(d

egre

es)

current in coil

sensor #5right of coil

sensor #8left of coil

current in plate

At High Frequency: Destabilizing Wall Image CurrentsPhase of Control Field vs Frequency

10 410 310 210 110 0-270

-240

-210

-180

-150

-120

-90

-60

-30

0

BB ph - driverBB ph max plateBB #5 phaseBB #8 phase

Data from "FRdemo2a"t = 0.0254 m rho = 130.e-08 ohm m

hz driving frequency

phas

e re

lati

ve

to

driv

ing

volt

age

coil,

pla

te,

r. &

l.

sens

or(d

egre

es)

current in coil

sensor #5right of coil

sensor #8left of coil

current in plate

Magnitude of Currents vs Frequency

10 410 310 210 110 0.1

1

10

100

BB I - driverBB max plate I

Data from "FRdemo2a"t= 0.0254 rho = 130.e-08 ohm m

hz driving frequency

curr

ent

in

coil

(am

p)

net

curr

ent

in p

late

( a

mp)

currentin coil

net currentin plate

Optimizing Resistive Wall Mode Control: FIRE Approach

Allows Ideal Beta Limit to be Achieved thru Cf > 0 Improved Plasma/Coil Coupling

1st Vacuum Shell

2nd Vacuum Shell

Copper Stabilizing Shell(backing for PFCs)

horizontal port (1.3 m x 0.65 m)

port shield plug (generic)

resistive wall modestabilization coil

(embedded in shield plug)

ITER Internal Coils in Port Plugs Easily Reach Ideal Limit

4.03.02.010- 2

10- 1

100

101

102

103

104

105

106

g passiveg wbl 10^8 cs5_3.0nbl_3.0

beta-n

grow

th r

ate

[ 1

/ s

]

stable < 3.647

• Ideal Beta Limit Reached with pure Proportional Gain Gp=108 V/W• Control Coils use only three n=1 pairs in 6 port plugs!

Time Dependent Feedback Model of ITER Internal Coils

0.040.030.020.010.000.0e+0

2.0e-8

4.0e-8

6.0e-8

8.0e-8

1.0e-7

sensor 6 #2s 6 #4 G=10^7s 6 #3 G=10^8

Data from "ITER.cs5"

time ( s )

sens

or 6

flux

( v*

s)

passiveonly cs#5

Gp=10^7

cs#5Gp=10^8

turn FBon at10 milli s

0.040.030.020.010.00-1000

-500

0

500

1000

1500

2000

4287 #64288 #64289 #6

Data from "ITER.cs5" s = 0.35 Gp = 10^8

time ( s )

curr

ent

in c

ontr

ol c

oils

[ a

mp

]

• At Ideal Beta Limit simple damped suppression in 20 to 30 ms• Peak Current in Control Coils reaches peak of only 1.5 kA• Peak Voltage on single turn control coils is only 5 volts• Reactive power requirements only 7.5 kW in each coil pair

Time Dependent Feedback Model of Basic ITER Coils

0.30.20.10.00.0e+0

2.0e-8

4.0e-8

6.0e-8

8.0e-8

flux s#5

Data from "Untitled Data #1"

time [ s ]

flux

in B

p se

nsor

#5

[ v

*s ]

0.30.20.10.0-1000

-750

-500

-250

0

250

500

750

1000

e 4287e 4288e 4289

Data from "Untitled Data #1"

time [ s ]

curr

ent

in c

ontr

ol c

oils

[

amp

]

• Beta chosen to be near predicted limit of bN ~ 2.7 which isonly about 20% between no-wall limit and ideal limit.

• Voltage limited to 40 V/turn times 28 turns = 1120 Volts• Peak Current in Control Coils reaches peak of 28 kA-Turns• Reactive power requirements exceed 1 MW per coil pair

ITER Basic Coils: Use Liu & Bondeson Gain Parameters

4.03.02.010- 2

10- 1

100

101

102

103

104

105

106

g passivenbl_3.0g L&B

beta-n

grow

th r

ate

[ 1

/s ]

stable < 2.674

• Liu uses Vf = - Lf bs/bos[Ki/s+Kp]; bo=28x10-7T/A; Lf=0.04H; bs=Fs/As

• VALEN uses Vf = - Lf/ boAs [Ki+Kpd/dt ] Fs = -1.4x108[Ki+Kpd/dt ]Fs

• Gp= 6x108 V/W is Liu’s Ki=4.318 and Gd= 2x108 V/V is Liu’s Kp=1.5• These values reduce best bN ~ 2.67

Time Dependent Feedback With Liu, et al. Parameters

0.40.30.20.10.0-6.0e-8

-4.0e-8

-2.0e-8

0.0e+0

2.0e-8

4.0e-8

6.0e-8

8.0e-8

1.0e-7

s 5

Data from "ITER.tran.07.14.2003"

time [ s ]

flux

in B

p se

nsor

#5

[

v*s

]

0.40.30.20.10.0-400

-200

0

200

400

600

e 4287e 4288e 4289

Data from "ITER.tran.07.14.2003"

time [ s ]

curr

ent

in c

ontr

ol c

oils

[

amp

]

• Beta chosen to be near predicted limit of bN ~ 2.6 which is only about 10%between no-wall limit and ideal limit. Vmax ~ 200 V

• Time history similar to Liu, et al.

Physics of Ideal Kink Transition Seems Absent in Liu, et al.

4.03.02.010- 2

10- 1

100

101

102

103

104

105

106

beta-n

grow

th r

ate

[ 1

/ s

]

Without Blanket Modules

With BlanketModules

• Note absence of Ideal Kink Branch Transition when Cb ~ 1• In Liu, et al. twall ~ 0.188 s so g at Cb~1 is between 53 to 120 s-1

• In VALEN modeling g at Cb~1 is between 1000 to 104 s-1

• Growth rate disparity consistent with g twall in Liu, et al. toosmall for claimed values of Cb

Use DCON Computation of dW to Calibrate s to bN

0.60.40.20.0-0.2-0.22.0

2.5

3.0

3.5

4.0

4.5

beta-nbeta-n

s

beta

-n

• Liu, et al. analysis seems constrained to small s values.

Limitation of Performance Due to cf Effects Seems Absent

4.03.53.02.52.02.010- 2

10- 1

100

101

102

103

104

105

106

g passive10^8 g10^8 10^8 g10^8 10^9 g10^9 10^10 gnbl_3.0

beta-n

grow

th r

ate

[ 1

/ s

]

worse than passive

stable < 2.643

stable < 2.743

stable < 2.759

• cf =1–[MpwMfw]/[LwallMfp] = 1–[Vplasma from Iw]/[Vplasma from If]• For ITER Basic External Coils is clear that cf < 0 [est. was –3.4]• Effect of Coil Toroidal Discreteness + Wall Modes May Play Role• Growth rate disparity consistent with g twall in Liu, et al. too

small for claimed values of Cb

Summary & Conclusions• Basic ITER External Control Coils with Double-wall Vacuum

Vessel in ITER Reduces Effectiveness of Feedback System:Stable only up to ~ 20% above No-wall Beta Limit. Voltagerequirements at coil operating limits (1.1 kV) degrade feedbackperformance and MW reactive power required.

• Inclusion of Blanket Modules Significantly Increases Ideal WallBeta Limit from about bN of ~2.7 to ~3.7

• Use of Single Turn Modular Coils in 6 of the 18 ITER MidplanePorts allows the feedback system to reach the Ideal Wall BetaLimit for the double wall ITER vacuum vessel plus blanketmodules. Time dependent modeling shows only 5 Volts at 1.5kA of current or 7.5 kW of reactive power needed.

• Unresolved discrepancies exist between VALEN analysis andLiu & Bondeson MARS analysis of Basic ITER ExternalControl Coils:

+ Program to benchmark two codes against common equilibria andgeometry needs to be established.

+ A common set of transfer functions needs to be established.