YQ Liu, Peking University, Feb 16-20, 2009 Effects of 3D Conductors on RWM Stability and Control...

YQ Liu, Peking University, Feb 16-20, 2009

Effects of 3D Conductors on RWM Stability and Control

Yueqiang Liu

UKAEA Culham Science Centre

Abingdon, Oxon OX14 3DB, UK


Outline1. Why important?

2. CarMa code

3. CarMa modelling and experiments RFX ITER

4. Outlook

5. Overall Summary (selected key notes)


Why important ? RWM is an external mode

External mode produces magnetic field perturbations in vacuum region, leading to interaction with external (3D) conducting structures

n=0 vertical instability is another example

3D structure may couple n>0 RWM with n=0 vertical control system Geometrical coupling

Realistic prediction for RWM stability and control in ITER


... in more details Many present fusion devices have essential 3D feature

for the conductors (partial walls, poloidal/toroidal cuts, coils, etc.)

Realistic prediction of RWM stability and control performance in ITER requires 3D modelling of conducting structures

Importance of 3D geometry already demonstrated by comparing passive growth rates of RWM between RFX experiments and CarMa simulations [Villone08]

Accurate simulation of RFA under ac conditions also requires 3D modelling of conducting structures

3D simulations and benchmarks are gaining momentum during recent years [VALEN, STARWALL, CarMa]



2. CarMa code


4. Outlook



CarMa code Couples MARS-F (MHD) and 3D eddy current

code CARIDDI (EM)

Formulation Forward coupling (from MHD to EM) Backward coupling (from EM to MHD)

Benchmark CarMa

Other RWM codes with 3D conductors VALEN (Columbia, US) STARWALL (IPP, Germany) Typhoon+KINX (Russia)


CarMa formulation: overview

MARS-F basically solves single fluid linear MHD

vv

jb

Bvb

bJBjv

ppp

p

0

)(

CARIDDI solves 3D eddy current problem (quasi-magnetostatic Maxwell) Using integral

formulation and FEM (curl-conforming edge elements)

State-of-the-art fast computing techniques

Discretized equationFV

UILRI

dt

d

dt

d

eddy current plasma electrode

Other features (rotation, resistivity, feedback, kinetic extentions, etc.) not shown here

Need to couple I to

U: U=U(I)


S

Resistie wall

S

Resistie wall

The plasma (instantaneous) response to a given magnetic flux density perturbation on S is computed as a plasma response matrix.

plasma

S

Resistie wall

Using such plasma response matrix, the effect of 3D structures on plasma is evaluated by computing the magnetic flux density on S due to 3D currents. The currents induced in the 3D structures by plasma are computed via an equivalent surface current distribution on S providing the same magnetic field as plasma outside S.

Forward coupling procedure

S

S

S

Albanese IEEE Trans. Mag. 44 1654(2008)Portone PPCF 50 085004(2008)Liu PoP 15 072516 (2008)Pustovitov PPCF 50 105001(2008)


QSLL*

VFUI

LIR dt

d

dt

d

eqIMU

Mutual inductance matrix between 3D structures and equivalent surface currents

Induced voltage on 3D structures

Equivalent surface currents providing the same magnetic

field as plasma

IQBKI 1Eneq

Matrix expressing the effect of 3D current density on plasma

VFIRI

L* dt

d

VBIAI

dt

d

Modified inductance matrix

Dynamical matrix

N h matrix h N matrix

h << N

h =DoF of magnetic field on S

N =DoF of current in 3D structure

Forward coupling procedure


Forward coupling has difficulty to include plasma inertia, flow, and kinetic effects

Can be overcomed using backward coupling scheme [Liu PoP 15 072516(2008)]

Start again with replacing plasma current by equivalent surface current

Backward coupling procedure

eddy current equation:

total field at coupling surface S:

sensor flux:

feedback current:


With algebraic combinations of previous equations, it is possible to obtain the following linear relations (w.r.t. the eigenvalue)

Linear boundary condition for MHD code, with computational boundary at coupling surface

Sensor flux for RFA or feedback

Similar BC can be derived even for coupling to a nonlinear MHD code

Backward coupling procedure


Key features of CarMa Accurate for RWM calculations. Coupling scheme

analytically proven [Liu PoP 15 072516(2008)]. Well benchmarked against MARS-F for 2D walls, and against other similar codes for 3D walls

The coupling matrices assemble responses from all poloidal Fourier Harmonics. Hence the final system contains all unstable/stable RWM (multimode approach)

Capable of treating volumetric conductors (no thin shell approximations)

State-of-the-art fast techniques for solving EM problems allow very detailed modelling of conductor geometry [Rubinacci JCP 228 1562(2009)]

CarMa with backward coupling allows inclusion of inertia, rotation, kinetic effects, and feedback


Growth rate calculation Unstable eigenvalue of the dynamical matrix Standard routines (e.g. Matlab) or ad hoc computations

(e.g. inverse iteration: see the following…) Beta limit with 3D structures

Controller design state-space model (although with large dimensions and

with many unstable modes)

Time domain simulations Controller validation Inclusion of non-ideal power supplies (voltage/current

limitations, time delays, etc.)

What CarMa can do ?


Benchmark coupling scheme and CarMa

Choose a plasma with circular cross section, and aspect ratio =5

Assuming an axi-symmetric complete thin wall (2D wall), run MARS-F to compute growth/ damping rates of unstable/stable RWM

Run CarMa with 3D discretization of 2D wall

Compare results

q-profile

pressure profile


Both growth and damping rates agreeMARS-F [s-1] Coupling surface 1 [s-1] Coupling surface 2 [s-1]

Unstable eigenvalue 292.7 291.2 j 3.6e-4 293.3 j 7.5e-4

Stable eigenvalue #1 -165.7 -159.2 -159.3






CarMa results independent of choice of coupling surface


Benchmark mode structure Eddy current density distribution along the wall,

computed by MARS-F alone (line) and by CarMa (circle)

For the unstable mode


Positive comparison with other 3D RWM codes (STARWALL, VALEN)

Courtesy of J. Bialek and

E. Strumberger

Benchmark with other codes



2. CarMa code


4. Outlook



RWM study for RFX: equlib. & geometry

RFX upgrade: rw=1.1a, R/a=2m/0.459m

Typical unstable RWM: m=1, |n|=2,...,6

•Coils•Mechanical structure•Vessel•Shell


RWM stability with MARS-F RWM growth rates are well measured in RFX experiments MARS-F with a 2D wall reproduces exp. growth rates for a large

range of plasma parameters and various n’s Including other structures tends to underestimate growth rates

MARS-F computes two unstable RWM for some n’s (=2,3)


RWM stability with CarMa (3D structure) For RFP plasmas, all three codes: ETAW(cylindrical

Newcomb solver), MARS-F, CarMa(2D) agree well, as shown below for one equilibrium

Gaps in conducting wall destabilize RWM. However, mechanical structures and outer shells give additional stabilization

3D wall structures (gaps) split two otherwise identical eigenvalues, as well as in tokamak cases

γ [s-1]

Cylinder(ETAW) MARS-F CarMa(2D) CarMa (3D)

n=2 <02.45

0.431.81

0.371.94

0.45, 0.462.40, 2.48

n=3 1.821.90

2.082.16

1.912.49

2.58, 2.622.96, 3.04

n=4 4.09 4.04 4.27 5.46, 5.53

n=5 6.81 6.89 7.45 9.62, 9.73

n=6 11.8 11.7 12.9 17.0, 17.2


3D effects are important on growth rate!

Purely axisymmetric estimates of growth rates are largely underestimated on RFX-mod

CarMa modelling vs. RFX experiments

Villone PRL 100 255005(2008)


Eddy current flow modified by wall gaps

2D wall

3D wall with gaps

CarMa computed wall eddy current pattern

For an unstable mode with n=3,m=1


ITER modelling: detailed 3D wall geometry

Examples: mesh A: OTS+bypass mesh B:

OTS+holes+extensions +blanket

mesh A

outer triangular support (OTS)

mesh B


Simple hole approximation for ITER walls leads to too pessimistic prediction for RWM

stability Eddy current patterns significantly affected

by tubular extensions... that allow better imagine current flow, hence

more stablising effect


Major 3D wall effects are holes and tubular extenstions

3D holes roughly double growth rates Tubular extensions reduce growth rates to a

level as with 2D complete walls

γ [s-1 ] N 2.676 2.830 2.985 3.301 3.461

MARS-F γ 8.495 15.82 29.63 165.8 577.2

Mesh#1:2D γ 8.462 15.56 28.35 127.2 334.9

Mesh#2:2D+OTS γ 7.896 14.34 25.54 99.26 206.8

Mesh#3:2D+OTS+bypass γ 7.903 14.37 25.63 100.0 208.9

Mesh#4:3D+OTS+holes γ 15.83 32.60 71.94 N.A. N.A.

Mesh#5:mesh#4 refined γ 15.65 32.06 70.20 N.A. N.A.

Mesh#6:mesh#5 smaller holes γ 13.71 27.13 55.78 1230 N.A.

Mesh#7:3D+OTS+holes+ext. γ 9.366 17.63 33.22 184.4 878.7


RWM feedback study for new ITER in-vessel coils reveals requirement on the coil current

Consider an ITER plasma with Use 3x9 ELM control coils for RWM

feedback Multivariable controller based on LQG

technique satisfying certain specification requirements

N = 3.17

Actual limiting factor is current Assuming 20kA current

limit (ELM control off), RWM can be stabilised for field perturbation within 300Gauss

Assuming 250A current limit (ELM control on), field perturbation within 5Gauss

20kA current limit

1 sec. settling time


Detail: LQG controlMultivariable controller designed by

using the LQG technique, based on the following requirements

Obtain a closed loop null controllable region as close as possible to the ideal result (BAP)

Allow to recover from a disturbance (initial condition on the unstable plane) as soon as possible (within current/voltage limits)

Avoid to generate a n1 magnetic field

Stabilize all the modes with growth rates lower than reference equilibrium (i.e. lower N)

Use a balanced truncation technique to obtain a sufficiently low order controller (five)


Plasma/circuit model

V(t) y(t)TIN

TOUT

y1(t)

-

V1(t)

K(s)

27 input voltages (3 coils per 9 sectors)

3 voltage Fourier components

144 magnetic outputs

(48 measurements per 3 sectors)

48 magnetic Fourier

components

RWM feedback controller

Detail: control diagram


The BAP is in the range of perturbations of tens of mTThe BAP is in the range of

perturbations of fractions of mT

ELM control off: current limits 20 kAELM control on:

current limits 250 A

Bk(t) are N=18 measurements of the vertical magnetic field

in the outboard region at equally spaced toroidal angles

k

The interior of the polygon corresponds to stabilizable

perturbations

11

2cos

N

k kk

y t B tN

21

2sin

N

k kk

y t B tN

-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

y1 [T]

y 2 [T

]

-5 -4 -3 -2 -1 0 1 2 3 4 5

x 10-4

-4

-3

-2

-1

0

1

2

3

4

x 10-4

y1 [T]

y 2 [T

]

Another approach of control study: best achievable performasnce (BAP)


Control coils voltage-current distribution

40 80 120 160 200 240 280 320 360-3

-2

-1

0

1

2

3

Toroidal angle [deg]

Upp

er C

oil V

olta

ge [

V]

at t

=0

40 80 120 160 200 240 280 320 360-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Toroidal angle [deg]M

id C

oil V

olta

ge [

V]

at t

=0

40 80 120 160 200 240 280 320 360-3

-2

-1

0

1

2

3


Low

er C

oil V

olta

ge [

V]

at t

=0

40 80 120 160 200 240 280 320 360-250

-200

-150

-100

-50

0

50

100

150

200

250


Upp

er C

oil C

urre

nt [

A]

at t

=0.

0075

s

40 80 120 160 200 240 280 320 360-200

-150

-100

-50

0

50

100

150

200


Mid

Coi

l Cur

rent

[A

]at

t=0.

0075

s

40 80 120 160 200 240 280 320 360-250

-200

-150

-100

-50

0

50

100

150

200

250


Low

er C

oil C

urre

nt [

A]

at t

=0.

0075

s


Outlook For RWM, geometrical coupling of different n’s

(including n=0!), via 3D conductors, is probably more important than physics coupling due to nonlinear MHD

Nonlinear MHD coupling should be more important for other, more localised modes such as TMs and ELMs

Extensive work going on to include 3D geometrical effects in RFA simulations RWM feedback stabilisation (in particular for ITER) Nonlinear (quasi-linear) MHD modelling for RWM (e.g.

nonlinear interplay between mode damping and momentum damping)


Overall summary (key notes)1. RWM research important for ITER

2. Sensor optimisation crucial for RWM feedback

3. Understanding RWM physics calls for hybrid MHD-kinetic description

4. RFA tests RWM damping physics

5. State-of-the-art in RWM modelling: Damping physics + 3D structures + feedback


[email protected]

YQ Liu, Peking University, Feb 16-20, 2009 Effects of 3D Conductors on RWM Stability and Control...

Documents

Transcript of YQ Liu, Peking University, Feb 16-20, 2009 Effects of 3D Conductors on RWM Stability and Control...