RFX-mod 2009 Programme Workshop, 20th Jan 2009 #1/33 F. Villone, RWM modelling of RFX-mod with the CarMa code RWM modelling of RFX-mod with the CarMa code:

F. Villone, RWM modelling of RFX-mod with the CarMa code

RFX-mod 2009 Programme Workshop, 20th Jan 2009

#1/33

RWM modelling of RFX-mod with the CarMa code: results,

perspectives,possible experiments

Fabio VilloneAss. EURATOM/ENEA/CREATE, DAEIMI, Università di Cassino, Italy

With contributions of: Y.Q. Liu (UKAEA)

R. Albanese, G. Ambrosino, M. Furno Palumbo, G. Rubinacci, S. Ventre (CREATE)

T. Bolzonella, G. Marchiori, R. Paccagnella, A. Soppelsa (RFX-mod)



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Outline• Introduction• The CarMa code• Applications to RFX-mod• Electromagnetic modelling of RFX-mod• What next?• Conclusions



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What are RWM?• Linearized ideal MHD equations can

describe fusion plasmas in some situations– In some cases predict unstable modes on inertial

time scale (microseconds for typical parameters)– External kink is one of the most dangerous (e.g.

setting beta limits in tokamaks)– A sufficiently close perfectly conducting wall may

stabilize such mode thanks to image currents induced by plasma movements

– Due to finite wall resistivity, image currents decay (Resistive Wall Modes) the mode is again unstable but on eddy currents time scale (typically milliseconds or slower)

– Feedback active control becomes possible

Introduction



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How do we analyse RWM?

Introduction

• Solution of a coupled problem is needed in principle– Plasma evolution can be described by MHD

equations– Eddy currents equations need magneto-quasi-

static electromagnetic solvers– Usual stability codes (MARS-F, KINX, ETAW,

etc): MHD solver + a simplified treatment of wall (e.g. thin shell approximation, axisymmetric or cylindrical assumptions, single wall, etc.)

– Our approach: coupling of a MHD solver (MARS-F) to describe plasma with a 3D eddy currents formulation (CARIDDI) to describe the wall CarMa code



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Why modelling RWMs in RFX?

Introduction

• In ITER RWMs will set stringent limits to the plasma performance (beta limit) it is important to make reliable predictions via accurate modelling

• Open issues still remain in RWM modeling– Stabilization via rotation and damping– 3D effects of passive (vessel, shell) and active (feedback

control coils) conductors

• RFX-mod allows us to concentrate on second issue:– RWMs in RFPs share many features with tokamaks but are

not affected by plasma flow (current driven instabilities)– Most advanced feedback control system for MHD modes

• Some interesting theoretical points peculiar to RFPs



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3D eddy currents formulation /1

• Integral formulation assuming J as unknown– Well suited for fusion devices (only the conducting domain

Vc must be meshed)– This formulation is at the basis of the CARIDDI code, widely

used for electromagnetic computations on fusion devices– Volumetric conductors of arbitrary shape taken into

account (no thin shell approximation nor other simplifications)

– Electric vector potential J = T solenoidality of J – Non-standard two-component gauge (numerically

convenient)– Tree-cotree decomposition of the mesh minimum

number of discrete unknonws I– Edge elements Nk right continuity conditions on J– Automatic treatment of complex topologies and electrodes

The CarMa code



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3D eddy currents formulation /2

• Mathematical model– Eddy currents equation in the time domain– No magnetic materials (not a theoretical limitation:

easily taken into account)– Resistivity tensor to account for anisotropies

(e.g. for “equivalent” piecewise homogeneous materials)– Ohm’s law E = J imposed in weak form (Galerkin

approach)

The CarMa code

c cV V

ji dVdVjiL ''

)'()(

4),( 0

rr

rNrN

cV

ji dVjiR NN ),( cV

i dVU 00

i 4AN

VFUI

LIR dt

d

dt

d

jS

iji dSF nN ˆ,

Electrode with potential Vj

Magnetic vector potential due to current density representing

plasma

- Equations:



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MHD formulation• Single fluid linearized MHD equations

The CarMa code

jb

vv

Bvb

bJBjv

0

PPt

pt

pt

• p, v, : plasma pressure, velocity and density; : specific heats ratio

• Uppercase: reference values, lowercase: first-order perturbations

• A exp(j n ) dependence of the quantities is assumed in the toroidal direction (n : toroidal mode number)

• In the poloidal plane, Fourier decomposition is used along the poloidal angle, and a Galerkin-based finite element method is implemented on a staggered grid along the radial direction.

•Toroidal plasma flow and various kinetic effects (to simulate the damping) are presently not included in CarMa• The MHD solver used is MARS-F



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Coupling strategy• Plasma mass is neglected

– Good approximation if the time scale is much longer than inertial times (i.e. >> microseconds)

– The plasma response is instantaneous and can be characterized by a response matrix to unit total normal field perturbations on coupling surface

• No plasma rotation is considered– Worst case analysis– Not important for current-driven modes (like in RFP’s)

• A coupling surface S is chosen – Any surface in between plasma and conducting structures– The plasma-wall interaction is decoupled via S

The CarMa code



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(De-)coupling surface S

The CarMa code

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

plasma

S

Resistive wall

S

Resistive wall

S

Resistive 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.



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QSLL*

Overall model

The CarMa code

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



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Several possible uses…

The CarMa code

• Growth rate calculation – Unstable eigenvalue of the dynamical matrix– Standard routines (e.g. Matlab) or ad hoc computations– Beta limit with 3D structures (when the system gets

fictitiously stable)

• 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.)



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Theoretical validations• Theoretically sound approach

– Independent theoretical validation on general geometry (V. Pustovitov, PPCF and PoP)

– Analytical proof of the coupling scheme available in the cylindrical limit (Y.Q. Liu et al, PoP)– Many successful benchmarks in various limits and situations (MARS-F, ETAW, KINX, STARWALL, …)– No fitting parameters, no tuning, no normalizations– “Genuine” MHD computations with 3D geometries are possible

The CarMa code



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Benchmarks

The CarMa code

• MARS-F as reference – Axisymmetric geometry (although with a 3D mesh)– stable and unstable modes with toroidal mode number n = 1– degenerate pairs of eigenvalues (modes shifted of 90° toroidally)



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ITER results /1

The CarMa code

• High level of details:– Double shell– Nested port extensions– Outer Triangular

Support with copper cladding

– Non- axisymmetric control coils



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The CarMa code

ITER results /2

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 measurement

s per 3 sectors)

48 magnetic Fourier

components

RWM feedback controller



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Other significant results

The CarMa code

• Best achievable performances – Rigorous computation of the best performance achievable by any RWM controller for given voltage and current

limitations in the actuators– Applied to ITER

• “Fast” computations – Fast SVD-based sparsification techniques for ameliorating the computational scaling with the number of unknowns

simulation with an unprecedented level of geometrical details– Applied to ITER

These volumetric cases (360° in the toroidal direction) cannot be analysed

with standard computational tools due to memory overflow!



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Multimodal analysis /1

Application to RFX-mod

• 3D structure cause multimodal coupling – With an axisymmetric structure linear MHD predicts that

every different n evolves separately – A 3D structure can couple different n ’s even in linear

MHD

• RFP’s particularly suitable– Rich toroidal spectrum

• RFX-mod has a dedicated control system – 192 independently fed saddle coils– Selective control of non-axisymmetric modes– Excellent magnetic measurement coverage



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Multimodal analysis /2• RFX-mod shell with gaps

– A small but noticeable multimodal coupling effect– Mode degeneracy removed




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Multimodal analysis /2• RFPs have multiple unstable RWMs

– Even with the same n value– This corresponds to positive n and negative n in the

cylindrical limit




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Experimental validation

3D effects are important on growth rate!

Purely axisymmetric estimates of growth rates are largely underestimated




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White: vesselRed: copper shellGreen: mech.

structureBlue: saddle coilsBlack: meas. coils

Details of conducting structures

Electromagnetic modelling

Detailed electromagnetic modelling of the main conducting structures of RFX-mod



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Coils/sensors interactions /1• Comparison of mutual inductances of coil/sensor

pairs– Very good amplitude prediction– Error in phase at “high” frequencies (>10° when >50 Hz)

(inaccurate modelling of mechanical structure?)




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• Comparison of mutual inductances of coil/sensor pairs– Correct qualitative and quantitative representation of

interactions due to toroidal and poloidal gaps in the shell and in the mechanical structure

Each pixel represents an interaction

Colours represent the log10 value of the normalized mutual interaction


Coils/sensors interactions /2



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Modal mutual inductances• Comparison of modal inductances

– Pure (m,n) harmonic current distribution in coils– Corresponding (m,n) harmonic in measured flux– The ratio is the modal inductance


Results similar to the previous case



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Time domain simulations• Experimental currents fed to the coils and

experimental fluxes compared to model predictions




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“Flight simulator” /1• Putting the state-space CarMa model inside the

overall Simulink® model of RFX-mod• Detailed time-domain simulation of given shots• Prediction of behaviour (e.g. controller gains for

stability margin)• A-priori model-based controller design (maybe not

necessary for RFX-mod itself…but a fundamental demonstration for ITER!)

What next?



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“Flight simulator” /2

What next?

CarMa model



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Plasma flow• We are currently developing a strategy for

including the effects of plasma flow (e.g. kinetic damping) and plasma mass

• Theoretically challenging: the plasma response changes qualitatively (from static to dynamic)

• May be of interest for RFX-mod (coupling 3D structures to other physical models, e.g. NTM) ?

• Again, RFX-mod could be an ideal test-bed for models and techniques

What next?



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Experimental fall-outs• Modelling contribution to the experimental

programme of RFX-mod– Help in the analysis/interpretation of experiments

• More complete coverage and understanding of “classical” (F, scans) or “innovative” (rotation?) RWM experiments

– Help in planning of future experiments• Prediction of “optimal” gains for given purposes (e.g. stability

margin)

– Flag the experiments of potential interest for model validation

• “RFA-like” experiments (steps or sinusoids as excitations)

– Link to ITER requests/needs from the point of view of MHD mode control

• Test of a-priori model/based MHD mode feedback controllers

What next?



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Conclusions

Conclusions /1• The CarMa code can self-consistently analyse

RWMs with 3D conducting structures– Theoretically sound (analytical proofs available)– Successful benchmarks on axisymmetric cases and on

3D geometries– Allows “pure” MHD comparisons– Highly modular code– Experimental validation on RFX-mod– Multimodal modelling possible– Quantification of 3D effects on RFX-mod (gaps in shell)

and ITER (port extensions, non axisymmetric control coils,…)

– Huge models handling allows an unprecedented level of details in geometrical description



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Conclusions

Conclusions /2• Several different computations are possible

– Growth rate computation– Time domain simulation – Feedback controller design

• RFX-mod contribution is very significant for MHD mode control– Model validation– Test of methods/techniques of interest for ITER



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Thank you for your attention!

F. Villone et al., Proc. of 34th EPS Conference, Warsaw (2007), P5.125

R. Albanese, Y. Q. Liu, A. Portone, G. Rubinacci, and F. Villone, IEEE Trans. Mag. 44, 1654 (2008).

A. Portone, F. Villone, Y. Q. Liu, R. Albanese, and G. Rubinacci, Plasma Phys. Controlled Fusion 50, 085004 (2008)

Y. Q. Liu, R. Albanese, A. Portone, G. Rubinacci, and F. Villone, Phys. Plasmas, 15, 072516 (2008)

F. Villone, Y. Q. Liu, R. Paccagnella, T. Bolzonella, and G. Rubinacci, Phys. Rev. Lett. 100, 255005 (2008)

F. Villone et al., Proc. of 35th EPS Conference, Hersonissos (2008), P2.067

G. Marchiori et al., Proc. of 35th EPS Conference, Hersonissos (2008), P5.047

A. Soppelsa et al., Proc. of SOFT Conference (2008)

Some references…

Please contact: [email protected]

Conclusions

Documents

RFX-mod 2009 Programme Workshop, 20th Jan 2009 #1/33 F. Villone, RWM modelling of RFX-mod with the CarMa code RWM modelling of RFX-mod with the CarMa code: