RWM realistic modelling for feedback control design in fusion devices

14th IEA Workshop on RFP Research, Padova, 26-28 April 2010

RWM realistic modelling for feedback control design in fusion devices

Fabio Villone

Ass. EURATOM/ENEA/CREATE, DAEIMI, Università di Cassino, Italy

With contributions of:

Yueqiang Liu, CCFE

T. Bolzonella, G. Marchiori, R. Paccagnella, A. Soppelsa & RFX-mod team

R. Albanese, G. Ambrosino, M. Furno Palumbo, A. Pironti,

G. Rubinacci, S. Ventre & CREATE team

F. Villone, RWM realistic modelling for feedback control design in fusion devices #1/26

• Introduction


Outline

• The CarMa code

• Examples of applications

• Conclusions and perspectives


Introduction


What are RWM?

• Linearized ideal MHD equations can describe fusion plasmas in some situations

– In some cases predict unstable modes on Alfvénic 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 perturbations

– Due to finite wall resistivity, image currents decay

( Resistive Wall Modes )  the mode is still unstable but on eddy currents time scale (typically milliseconds or slower)

– Feedback active control becomes feasible


Introduction


How do we analyse RWM?

• 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,

MARS-K) to describe plasma with a 3D eddy currents formulation (CARIDDI) to describe the wall  CarMa code


Introduction


RWM realistic modelling

• In ITER RWMs will set stringent limits to the plasma performance ( beta limit )  modelling is needed to make predictions

• RWM realistic modelling : allowing a reliable extrapolation to future devices (ITER, JT60-SA)

– Including all significant features of the system

– Able to predict experimental evidence on existing devices

• Significant features : physics and engineering side

– Inclusion of plasma flow and damping

– Detailed description of conducting structures

(passive and active)

– Thorough representation of control chain

(Marchiori’s presentation on Monday)

• The CarMa code is able to pursue all this


Introduction


Plasma flow and damping

• Experiments can go above the beta threshold predicted by ideal MHD

[ Strait PRL 74(1995) ]

– This effect has been usually attributed to nonnegligible plasma flow

– It has been suggested that kinetic damping can act as “ energy sink ” and help to stabilize the mode

– Different physical models of damping have been proposed, each with distinct range of influence:

• Alfven continuum [Bondeson PRL 72(1994), Zheng PRL 95(2005)]

• sound wave [Betti PRL 74 (1995), Bondeson, PRL 72 (1994)],

• drift kinetic damping [Bondeson PoP 3 (1996), Hu PRL 93 (2004)]

– Work is still in progress in understanding the RWM damping physics in ITER [Liu NF 49 (2009)]


Introduction


Conducting structures

• 3D features of conducting structures may give conflicting contributions to passive stabilization

– Ports, holes, cuts are detrimental (faster growth rate)

– Blanket modules are beneficial (slower growth rate)

• On active stabilization different considerations apply

– Shielding effects that help passive stabilization make active stabilization more difficult!

– Active coils have a inherently 3D geometry and complex feeding schemes

– Nonlinearities (e.g. saturations) may have an important effect

• Detailed models computationally very demanding

– Fast/parallel techniques often needed



The CarMa code


The CarMa code


The CarMa approach

• The plasma/wall interaction is decoupled via a suitable surface S in between

– Inside S , MHD-kinetic equations

– Outside S , eddy currents equations

– On S suitable matching conditions plasma

S • Theoretically sound approach Resistive wall

– Independent theoretical validation on general geometry

[Pustovitov, PPCF and PoP]

– Analytical proof of the coupling scheme available in the cylindrical limit [Liu, PoP 15 (2008)]

– Many successful benchmarks in various limits and situations

(MARS-F, ETAW, KINX, STARWALL, VALEN, …)

– No fitting parameters, no tuning: true predictions


The CarMa code


3D wall description /1

• Integral formulation assuming J as unknown

– Well suited for fusion devices (only the conducting domain V c must be meshed)

– Volumetric conductors of arbitrary shape taken into account with a finite elements mesh (no thin shell approximation nor other simplifications)

– State-of-the-art fast methods and parallel computing techniques

– Anisotropic resistivity tensor (“equivalent” anisotropic materials to account for holes, slits,…)

– Inclusion of externally fed electrodes

– Automatic treatment of complex topologies


The CarMa code


3D wall description /2

• Some technicalities

– 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 N k

 right continuity conditions on J

– Both frequency- and time-domain simulations

U 



4



0



V c

 

N i



A

0 dV

R ( i , j )

 

V c

 

N i

    

N j dV

L ( i , j )





4



0

 

V c

V c

 

N i

( r )

  

N j

( r ' ) r

 r ' dV dV '

L dI dt



R I

 dU dt



V


The CarMa code


(





(



 in



) ξ

 in



) v



 v





(



ξ p



Linearized MHD



  j



) R

B

2  



J



Q

MARS-K

[Liu, PoP 15 (2008)]

 

||

|

 k

||



|

[ 2



Z v th , i

[ v



 v b





( v

( ξ





 

) R

2  

]



) V

0

 b ] b

• Shear toroidal rotation

(



(



 in



) Q in



) p

  

( v



 p

B )



( Q

 p



  

) R

2

    

(

 j )



0







 v



 b

 j

  v

• Parallel sound wave damping p

 p I

 p

|| kinetic

( ξ



) bb

 p kinetic



( ξ



)( I

 bb

• Self-consistency (non-perturbative)

)

• Kinetic inclusion

• Kinetic integration in full toroidal geometry

• Kinetic effects due to particle bounce resonance and precession drift resonance, both transit and trapped particles, both ions and electrons (where appropriate)

• Bulk thermal particle resonances (Maxwellian distribution)


The CarMa code


Two different approaches

• “ Backward ” coupling

[Liu PoP 15 (2008)]

– Eddy currents equations are “condensed” inside MHD

– Advantages:

• The 3D structure dynamics is exactly taken into account

• In principle “readily” applicable to nonlinear MHD

• “ Forward ” coupling

– MHD equations are “condensed” inside eddy currents

– Advantages:

• Easy multimodal modelling (multiple n ’s)

• Possibility of accounting for control non-idealities (saturations…)

• Both can be useful ( complementarity )


The CarMa code


Forward coupling

• The plasma response to a given magnetic flux density perturbation on S is computed as a plasma response matrix, solving MHD equations inside S.

• 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.

S

S

S plasma

F. Villone, RWM realistic modelling for feedback control design in fusion devices

Resistive wall

Resistive wall

Resistive wall

#14/26

The CarMa code


Overall model

R I



L d I dt

  d U dt



F V

Modified inductance matrix L

* d I dt



R I



F V d I



A I



B V dt

Induced voltage on 3D structures

U



M

Dynamical matrix

Matrix expressing the effect of

3D current density on plasma

I eq

N  h matrix h  N matrix

Mutual inductance matrix between

3D structures and equivalent surface currents

I eq



K

Equivalent surface currents providing the same magnetic field as plasma



1

B

En

Q I L

* 

L



S

h << N

Q

h DoF of magnetic field on S

N DoF of current in 3D structure


The CarMa code


Several possible uses…

• 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): Matlab, Simulink, …

• Time domain simulations

– Controller validation

– Inclusion of non-ideal power supplies (voltage/current limitations, time delays, etc.)



Examples of application




Plasma flow and damping /1

• Benchmark case: circular tokamak

• 3D wall cases: holes in the conducting structures

• Drift kinetic damping




Plasma flow and damping /2

• Realistic analysis :

– 3D wall, feedback, plasma flow, sound wave damping

Nyquist diagram of the plasma response transfer function showing the synergistic effect of rotational stabilization with active feedback also in presence of

3D wall

[Liu & Villone, PPCF

2009]




ITER results /1

State-of-the-art fast and parallel techniques allow CarMa to study RWM including a realistic description of ITER passive structure, including thick blanket modules

(mesh spans 360°)




ITER results /2

Holes are pessimistic - port extensions allow the current to "bypass" the hole along a conducting path

In terms of passive stability analysis, the detrimental effect of ports is compensated by the favourable stabilizing effect of blanket modules .

For active stabilization, the presence of blanket modules is not beneficial (shielding effect of the magnetic field produced by active coils).

The overall effect is not obvious  work in progress




ITER results /3

Realistic description of

• active coils (in-vessel and ex-vessel)

• measurement system (position, orietation)

• feeding system (saturations) in view of a ITER RWM feedback controller design and simulation




RFX-mod results /1

• Realistic modelling of feedback control loop thanks to favourable model properties

(state-space representation)

CarMa model




RFX-mod results /2

• A fundamental goal is being attained: successful prediction of experimental behaviour on an existing device

(RFX-mod: leading edge device for MHD control)

Other results in Marchiori’s presentation on Monday



Conclusions



Conclusions

• The CarMa model is able of realistic RWM modelling providing confidence in making extrapolations

Thank you for your attention

blanket modules, port extensions, active coils, …)

(thanks to state-of-the-art fast/parallel techniques)

– Reproduction of experimental results

• Many developments are expected in the near future

– Application to ITER of various damping models

– RWM feedback controller for latest ITER configurations

– Further applications and experiments on RFX-mod

– Applications to JET and JT60-SA are ongoing


RWM realistic modelling for feedback control design in fusion devices

RWM realistic modelling for feedback control design in fusion devices

Outline

What are RWM?

How do we analyse RWM?