F. Villone, RWM modelling of RFX-mod with the CarMa code
RFX-mod 2009 Programme Workshop, 20th Jan 2009
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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)
F. Villone, RWM modelling of RFX-mod with the CarMa code
RFX-mod 2009 Programme Workshop, 20th Jan 2009
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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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RFX-mod 2009 Programme Workshop, 20th Jan 2009
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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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RFX-mod 2009 Programme Workshop, 20th Jan 2009
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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:
F. Villone, RWM modelling of RFX-mod with the CarMa code
RFX-mod 2009 Programme Workshop, 20th Jan 2009
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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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RFX-mod 2009 Programme Workshop, 20th Jan 2009
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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.
F. Villone, RWM modelling of RFX-mod with the CarMa code
RFX-mod 2009 Programme Workshop, 20th Jan 2009
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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
F. Villone, RWM modelling of RFX-mod with the CarMa code
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Multimodal analysis /2• RFX-mod shell with gaps
– A small but noticeable multimodal coupling effect– Mode degeneracy removed
Application to RFX-mod
F. Villone, RWM modelling of RFX-mod with the CarMa code
RFX-mod 2009 Programme Workshop, 20th Jan 2009
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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
Application to RFX-mod
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RFX-mod 2009 Programme Workshop, 20th Jan 2009
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Experimental validation
3D effects are important on growth rate!
Purely axisymmetric estimates of growth rates are largely underestimated
Application to RFX-mod
F. Villone, RWM modelling of RFX-mod with the CarMa code
RFX-mod 2009 Programme Workshop, 20th Jan 2009
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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?)
Electromagnetic modelling
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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
Electromagnetic modelling
Coils/sensors interactions /2
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RFX-mod 2009 Programme Workshop, 20th Jan 2009
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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
Electromagnetic modelling
Results similar to the previous case
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RFX-mod 2009 Programme Workshop, 20th Jan 2009
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Time domain simulations• Experimental currents fed to the coils and
experimental fluxes compared to model predictions
Electromagnetic modelling
F. Villone, RWM modelling of RFX-mod with the CarMa code
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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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RFX-mod 2009 Programme Workshop, 20th Jan 2009
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“Flight simulator” /2
What next?
CarMa model
F. Villone, RWM modelling of RFX-mod with the CarMa code
RFX-mod 2009 Programme Workshop, 20th Jan 2009
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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?
F. Villone, RWM modelling of RFX-mod with the CarMa code
RFX-mod 2009 Programme Workshop, 20th Jan 2009
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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