Vertical Displacement Events: Dynamics and consequences - … · R. Albanese: "Vertical...

R. Albanese: "Vertical Displacement Events: Dynamics and consequences”

"Global and Local Control of Tokamak Plasmas"10th Course: ERICE-SICILY: 8 - 14 Nov. 2005

F. Gnesotto, R. Albanese,Scuola Nazionale Dottorandi di Elettrotecnica“Ferdinando Gasparini”, Torino, 14/6/2006

Vertical Displacement Events: Dynamics and consequences

R . AlbaneseENEA / CREATE / Univ. Mediterranea di Reggio Calabria

1. Introduction2. Elongated plasmas3. MHD instabilities 4. Stability analysis with the linearized MHD model5. Rigid displacement model6. Perturbed equilibrium approach7. Excitation of the unstable mode8. Diagnostics for the control of the vertical instability9. Feedback stabilization10. Experimental observations11. References




Introduction• VDEs:

– Vertical Displacement Events may occur for elongated, vertically unstable plasmas

– Elongated plasmas are stabilized by active control systems

– However, in some cases, the installed power is not sufficient to keep the plasma stable

– An instability (triggered by some perturbation, e.g. an ELM) grows up exponentially

– The plasma motion is typically axisymmetric and vertical

– Eventually the plasma hits the wall, with consequent thermal loads on the wall, rise of halo currents, and finally plasma current quench

– These disruptions are quite dangerous because the particular nature of the mechanical loads.

– Even more dangerous are the Asymmetric VDEs because of the concentration of the loads.

This lecture is focused on:

• plasma modelling for the analysis and the control of the vertical instabilities : – full MHD equations– rigid displacement– perturbed equilibrium

• phenomenological description of VDEs and their effects




• To guarantee radial equilibrium an external vertical field has to be applied according to virial theorem: a sector is subjected to a net radial force due to the total pressure p+B2/2µ0 from the rest of the plasma

Elongated plasmas (1)

.22

3aR8

lnR4I

B ipol

0

0

P0V

++−=l

βπ

µ

ΩΩ

ϕ dJIp

p ∫= 200

Vpol IpRdVp4

p

µβ ∫= ( ) 200

V0

2i IpRdV2/B4

p

µµ∫=l

• A large R0/a aspect ratio plasma would naturally tend to have a circular cross section: b/a ≈ 1 with a uniform external vertical field (slightly larger for common values of R0/a, e.g. b/a = 1.05 )

rφ

z r

R0 a

r

z

b

Ip

BV

dFextϕdRIBdF 0pVext ≈

dVpV

extext ∫ ×= BJF

orientation according to the standard right-hand-rule

dF




• Vertically elongated plasmas (b/a>1), have to be used to maximize the performance-to-cost ratio (large volume and pressure)

• To elongate the plasma an additional force distribution is needed (with zero total force):

• This is obtained with a quadrupole field:


r

z

IpdFext

∆FL

∆FU

∆FU+∆FL=0

r

z

IpdFext

vertical field (affecting the radial position)

quadrupole field (affecting the elongation)

r

z

IpdFext

∆FL

∆FU

+ = rIp

z

vertically unstable configuration




• Field index:

– a slight plasma displacement directed upwards δz>0 would give rise to a net force directed upwards δFz>0 → instability

– active stabilization needed


rIp

z

r

z

Ip

Br externally applied radial field

zB

BR

n r

V

0

∂∂

−= n > 0 ⇒ vertical instability

pr0z IBR2F δπδ −≈ zInB2F pVz δπδ ≈




• ideal MHD assumptions– single plasma fluid– σ→∞ and Ei→0 in the plasma

⇒ E+v×B=0

quasi-stationary Maxwell’s equations

)(

0

iEBvEJHB

BJH

tBE

+×+==

=⋅∇=×∇

∂∂

−=×∇

σµ

with constitutive laws:

0=⋅∇+ vρDtD ρ

( ) 0pρDtD

pBJDtDv

ρ

γ =

∇−×=

−

thermo-fluid-dynamics

MHD Model

MHD instabilities (1)




• MHD equilibrium assumptions:– stationary conditions: ∂/∂t=0– static conditions: v=0

quasi-stationary Maxwell’s equations

)(

0

iEBvEJHB

BJH

tBE

+×+==

=⋅∇=×∇

∂∂

−=×∇

σµ

with constitutive laws:

0=⋅∇+ vρDtD ρ

( ) 0pρDtD

pBJDtDv

ρ

γ =

∇−×=

−

thermo-fluid-dynamics

MHD Model

pBJ ∇=×equilibrium equation

p: kinetic pressure





• Linearized stability equations in the plasma:– linearization around equilibrium conditions (v=0)– basic variable: Eulerian displacement δξ

• FORCE


MHD Model: pt

∇−×=

∇⋅+

∂∂ BJvvvρ,0JB µ=×∇

Linearized MHD Model

ptt

δδδδδδρδρ ∇−×+×=

∇⋅+∇⋅+

∂∂

+

∇⋅+

∂∂ BJBJvvvvvvvv

,0 JB δµδ =×∇

p)()( 10

10 δδµδµρδ ∇−××∇+××∇= −− BBBBξ&&

.equilat

dt

0v

ξvvξ

=

=⇒= ∫ &δδδδ




• MAGNETIC FIELD


MHD Model: ,0BvE =×+ BE &−=×∇

Linearized MHD Model.equilat

dt

0v

ξvvξ

=

=⇒= ∫ &δδδδ

0BvBvE =×+×+ δδδ BE &δδ −=×∇

)()( BBBEB ××∇=⇒××∇=×−∇= δξδξδδδ && (after time integration withzero initial conditions)

MHD Model vvv

⋅∇−=∇⋅+⇒

=

⋅∇−=

− ppp0

Dt)p(D

DtD

γρ

ρρ

γ&

Linearized MHD Model: ξξ &&& δγδδ ⋅∇−=∇⋅+ ppp (discarding terms multiplied byunperturbed velocity)

ξξ δγδδ ⋅∇−∇⋅−= ppp

• PRESSURE

(after time integration withzero initial conditions)




• LINEARIZED FORCE BALANCE IN TERMS OF PLASMA DISPLACEMENT


)( ξξ δρδ F=&&

with

(F: force operator depending on equilibrium distributions of B and p)

)pp()()()]([)( 10

10 ξξBξBBBξξ δγδδµδµδ ⋅∇+∇⋅∇+××∇××∇+×××∇×∇= −−F

to solve the above PDE equation:– coupling to Maxwell and circuit equations outside the plasma– interface conditions at plasma boundary

• e.g., continuity of magnetic field and pressure at plasma boundary• plasma boundary: outermost closed magnetic surface that does not intersect solid walls

simplest case: plasma in contact with a perfect conducting wall:– B⋅n=0, E×n=0 ⇒ v⋅n=0 hence ξ⋅n=0– PDE system:

∂=⋅

=

p

p

on0

in)(

Ωδ

Ωδρδ

nξ

ξξ F&&




• Eigenvalue analysis in the frequency domain using weighted residuals:

Stability analysis with the linearized MHD model (1)

∂=⋅

=

p

p

on0

in)(

Ωδ

Ωδρδ

nξ

ξξ F&&

wnξξwξw ∀∂=⋅⋅=⋅ ∫∫ ,on0with,d)(d ppp

ΩδΩδΩδρΩΩ

F&&

PDE problem:

weighted residuals:

Galerkin‘smethod: iip

N

1kkk ,on0with,)(c)( uwnξrurξ =∂=⋅= ∑

=

Ωδδ

ΩΩρΩΩ

d)(F,dMwithpp

kiikkiik ∫∫ ⋅=⋅= uuuu FcFcM =&&ODE problem:

=

== − 0FM

I0A,

cc

xwithAxx 1&&eigenvalue

problem:check real part of the eigenvalues of A matrix




• Extensively used for the analysis of internal (fixed boundary) instabilities

• External (moving boundary) instabilities (including vertical instability) strongly interacting with the conductors outside the plasma

• Detailed description of external circuits and passive conductors therefore needed especially for n=0 (axisymmetric) external modes ⇒ complex coupled systems of PDE equations and circuit equations proposed,applied and still being developed

• Simplified treatment of plasma more commonly and efficiently used for the analysis of vertical instabilities:– rigid displacement model– perturbed equilibrium approach

Stability analysis with the linearized MHD model (2)




• Plasma treated as a single filamentary current (or a rigid set of filaments carrying constant currents) with a single d.o.f.: the vertical position zp

• System equations for a filamentary plasma:

Rigid displacement model (1)

(circuit equations)(vertical force balance)

I: PF circuit currents Ψ: fluxes linked with the circuits U: applied voltages (zero for passive circuits)R: resistance matrixmp , zp , rp : plasma mass, vertical and radial positionIp: plasma currentBr : radial field acting on the filament

prppp IBr2zm π−==+

&&

& URIΨ




• Linearized equations:Rigid displacement model (2)

(circuit equations)(vertical force balance)

prppp IBr2zm δπδδδδ

−==+

&&

& UIRΨ

• in terms of δI and δzp …

pVprpp

Tpprpp

ppp

IBn2)zB(Ir2

)z(I)B(Ir2

zIz,

ππ

π

=∂∂−

∂∂=∂∂−

∂∂=∂∂=∂∂

MI

MΨLIΨ• where …

pprpprpppp

pp

z)zB(Ir2)B(Ir2zm

z)z()(

δπδπδ

δδδδ

∂∂−∂∂−=

=+∂∂+∂∂

II

UIRΨIIΨ&&

&&

r

zz+dz

dΨ = - Br 2πr dz

Br

using divergence-free condition …

L: circuit inductance matrixM: plasma-circuit mutual inductances




• Linearized equations:


pprpprpppp

pp

z)zB(Ir2)B(Ir2zm

z)z()(

δπδπδ

δδδδ

∂∂−∂∂−=

=+∂∂+∂∂

II

UIRΨIIΨ&&

&&

rewritten as:

pT

pp

p

z'Fzm

z

δδδ

δδδδ

+=

=++

Ig

UIRfIL

&&

&&

L, R: circuit inductance & resistance matrixf=g: stabilizing efficiencies Ip(∂M/∂zp)mp: plasma massF’: destabilizing force 2πnBVIp

δI: variation of circuit currents (note δIp=0)δzp: variation of plasma vertical position

LINEARIZED RIGID DISPLACEMENT MODEL




• Massy vs massless plasma model

• MASSY MODEL


pT

pp

p

z'Fzm

z

&&&&&

&&

δδδ

δδδδ

+=

=++

Ig

UIRgIL

UU

IRIggLIRIL

δδ

δδδδ

+−=

=+−+−−&&

&&&&&&

)'F/m(

)'F/()'F/m()'F/m(

p

Tpp multiplying

vertical force equation by g

and eliminating δzp• MASSLESS MODEL

UIRIggL δδδ =+− &)'F/( T

setting mp=0

Singular perturbation:highest derivatives

involved• IMPLICATIONS

0R)'F/ggL()'F/Rm()'F/Lm()(P T2p

3p =+−+−−= λλλλ

sample case: eigenvalues

with a single circuit(MASSY MODEL)

there exist a real unstable eigenvalue if F’>0 and R>0 :

(mp and L intrinsically positive) +∞→−∞→−≈

>=

λλλ for)'F/Lm()(P

0R)0(P3

p





0R)'F/gL()'F/Rm()'F/Lm()(P 22p

3p =+−+−−= λλλλ (MASSY MODEL)

Sample case: eigenvalues with a single circuit

no-wall limit: F’ ≤0, i.e., n≤0

no-wall growth rate (g=0): γ = λu≈(F’/mp)½

ideal (R=0) wall limit: L≤g2/F’

stability margin: ms=g2/LF’

F’ = 2πnBVIp,, mp=<ρ>Vp

n≈1, BV=1T, Ip=3 MA, <ρ>=10-6 Kg/m3 , Vp=100 m3

γ = 0.5⋅10-6 s-1 ,τg = 1/ γ =2 µsNO ACTIVE STABILIZATION

FEASIBLE ON ALFVEN TIME SCALE

0R)'F/gL()(P 2 =+−= λλ (MASSLESS MODEL)

growth time and growth rate on the resistive time scale:

τg =(ms-1)L/R, γ =1/ τg

massless model applicableonly for F’ ≤0 or ms≥1

with F’>0 and ms<1Alfven instability artificially hidden




Rigid displacement model (6)STABILITY MARGIN IN THE MULTI-CIRCUIT CASE

two alternative definitions of ms:

'F/* TggLL −= modified inductance matrix

1) ms= largest real part of the eigenvalues of M=− L* L-1

(all currents weighted independently from resistance)

2) ms= − (xuT L* xu)/(xu

T L xu)(resistances and unstable mode taken into account)

0* =+ IRIL δδ & open loop system:

−=

== − RLA

IxAxx 1*)(

,δ

&

uuu xAx λ= unstable (growth) mode: 0u >= λγ





MAIN LIMITATIONS OF RIGID DISPLACEMENT MODEL– plasma as a single filament → multifilament rigid plasma– axisymmetric conductors → coupling to 3D eddy currents– rigid displacement non-consistent with local MHD equilibrium

– uncorrect estimation of growth rate (especially for triangular plasmas)– cumbersome prediction of the excitation of the unstable mode– uncorrect modelling for VS magnetic diagnostics

→ perturbed equilibrium approach




Perturbed equilibrium approach (1)Plasma

Assumptions:• inertial effects neglected (plasma in equilibrium at each time instant) • the plasma is axisymmetric, and its equilibrium evolution is determined only by the

magnetic field averaged along the toroidal angle; • plasma current density profile parameterized with 3 degrees of freedom: total plasma

current Ip, poloidal beta βpol and internal inductance li.

Circuits and Conducting StructuresAssumptions:

• The mathematical model for the conducting structures is the standard eddy current model, i.e., the quasi-stationary Maxwell equations (∂D/∂t→0)

• The time evolution of the coil currents is described by the standard circuit equations (with zero applied voltages for passive circuits)

• The use of integral formulations allows for a unified treatment of circuits and eddy currents (even in the 3D case)




Perturbed equilibrium approach (2)

• usually derived from evolutionary MHD equilibrium:• dΨ/dt + R I = U (circuit equations)• [Ψ, Y] T =η (I,W) (Grad-Shafranov constraint in symbolic form)

I: including PF circuit currents and the plasma current IpΨ : fluxes linked with the above circuitsU: applied voltages (zero for passive circuits)R: resistance matrix

W: poloidal beta and internal inductance [βpol, li ]T

Y: including quantities of interest (gaps, magnetic signals, etc.)

• Grad-Shafranov constraint imposed numerically (for given geometry, circuit connections and materials, the only inputs to an MHD code are I and w)




Perturbed equilibrium approach (3)• nonlinear form:

dΨ/dt + R I = U (circuit equations)[Ψ, Y] T =η (I,W) (Grad-Shafranov constraint in symbolic form)

• linearized form (assuming I=I0+i, U=U0+u, W=W0+w, Y=Y0+y)L* di/dt + R i = u – LE* dw/dt (L* = ∂Ψ/∂I, LE* = ∂Ψ/∂W)y = C i + F w ( C = ∂Y/∂I, F = ∂Y/∂W )

• state-space form (assuming x=i, A=−(L*)-1R, B= (L*)-1 , E= −(L*)-1LE*)dx/dt = A x + B u + E dw/dty = C i + F w

• L* is an inductance matrix modified by the presence of the plasma• linearization carried out using any MHD equilibrium code using incremental ratios

(e.g., ∆Ψ/∆I) or Jacobiam matrix (readily available with Newton’s method)• state-space matrices strongly depending on the plasma configuration: time-varying in

various phases of scenario and even discontinuous in time in limited-diverted transition




Perturbed equilibrium approach (4)• growth rate and stability margin

– same definitions (and problems) as rigid massless model:• growth rate:

• stability margin: ms= − (xuT L* xu)/(xu

T L xu)– more realistic predictions than rigid massless model:

• rigid model predictions inaccurate especially for triangular plasmas (more elongated inwards than outwards → non-rigid motion)

0,uu >= γγ xAx

ITER start of burn configurationhigh δ high b/a

(Ip=15 MA, li=0.85, βpol =0.65)




Perturbed equilibrium approach (5)• Example of application to the JET tokamak




Excitation of the unstable mode (1)

How is a vertical instability triggered?By any disturbance exciting the unstable mode, e.g.• an externally applied field

– a radial field “kick” from an external circuit• a change in the current density profile

– an ELM (edge localized mode) – a soft disruption

ALL MODES IN PRINCIPLE EXCITED - ELMs do not excite vertical instability of up-down symmetric configuration- possible existence of neutral points also for unsymmetric configurations

ALL MODES STABLE EXCEPT ONE (THE GROWTH MODE)




Excitation of the unstable mode (2)

Modelling:• external circuits included in the model • ELMs modeled as equivalent jumps of β pol and li

e.g., ∆w = [∆β pol = –10% β pol, ∆li = +5% βpol ]

Quantitative estimation (useful for controller design):• find initial conditions:

dx/dt = A x + B u + E dw/dt ⇒ x(0+) = E ∆w• remove all stable components from the initial conditions:

x(0+) = αuxu + Σk αskxsk• evaluate a physical parameter corresponding to the unstable

mode at 0+, e.g. the vertical displacement δz0u≠ δz0(0+):δz0u = αucz

Txu (czT: row of output matrix C)




Diagnostics for vertical stabilization• Measurable current moments

(linear combinations of fluxesand fields:

• Vertical speed:

• Problems:• time derivatives noisy• non-measurable surrounded currents

∑∑ k kk kk IandIz

( ) ∑∑ −−= ccppk kkpp IzIzIzdtdzI &&&

Any signal might be used as controlled variable, provided that it is sensitive to the growth mode (which depends on the configuration) and is scarcely sensitive to ELMs, radial plasma motion and external currents




Feedback stabilization (1)

PRINCIPLES• open loop:

• closed loop, e.g. with derivative action :

• Various philosophies: PD, PID, bang-bang, adaptive, etc.

pT

p

z'F0

z

δδ

δδδδ

+=

=++

Ig

UIRgIL &&UIRIggL δδδ =+− &)'F/( T

p

pT

p

z

z'F0

z

&

&&

δδ

δδ

δδδδ

kU

Ig

UIRgIL

=

+=

=++ [ ] 0'F/)( T =+−+ IRIggkL δδ &

(character of closed loop L* may change)




Feedback stabilization (2)INSTALLED POWER REQUIRED

Superpose open loop (free) evolution and (forced) response to active voltage

• starting from:– connections of active and passive circuits– equilibrium configuration– applied disturbance (to get initial conditions)– maximum tolerable displacement

• apply a voltage step of given amplitude to the active circuit and find:– if it is able to stop the unstable motion (invert speed) before exceeding

tolerable displacement– values of time, currents and displacement at speed inversion

• determine lower bounds for installed voltage and current

• take safety margins

t

δzp

free

forced

superpos.




Feedback stabilization (3)AN INDICATION OF ACTIVE CIRCUIT PERFORMANCE

Bode Diagrams for the dBr/dt (radial field derivative) output channel in JET and ITER. The input channel is the stabilizing voltage (VFRFA for JET, VS1 for ITER).

Both transfer functions can be approximated as 2nd order systems.




Experimental observations (1)GROWTH RATE

• notice stable modes at the beginning of the VDE• equilibrium value of ZPDIP = 0• applied control voltage VFRFA = 0• techniques available also for nonzero equilibrium conditions and applied voltages

Estimation of the growth rate in JET pulse #54839: exponential fitting of ZPDIP (the controlled variable)




Experimental observations (2)ELMs

•

JET pulse #60682

Ip ≈ 2 MA

β ≈ 0.75

li ≈ 0.85

WDIA ≈ 4.4MJ




Experimental observations (3)ELMs with VDEs avoided

• ∆Ip ≈ 0 (???)(then small variation)

• ∆zp jump (>0 or <0)(then recovery)

•∆rp jump (<0)(due to loss of pressure energy)

• ∆βpol jump (<0)(loss of pressure energy)

•∆WDIA ≤ 10% WDIA

• ∆li jump (>0)(due to radial motion / loss of outercurrent layer)

Do we believe measurementson short time scale (Rogowski coils alsosurround passive currents) ???

complex phenomenon, BUT equivalent model for VS

∆βp ≈ –10% βpol ∆li ≈ +5% βpol ∆Ip ≈ 0

JET pulse #60682




Experimental observations (4)A GIANT ELM FOLLOWED BY A VDE

ZPDIP

ZP

IFRFA

VFRFA

Current limit reached:

Voltage switched off

Plasma in open loop (VDE)

Active control system pushing instead of counteracting the vertical instability in the initial phase

Spike of ZPDIP with opposite sign

In present JET operations VFRFA is switched off for hundreds of microseconds by the time an ELM is detected (via H-ALPHA radiation signal)

The JET stabilization system including power supply, coil connections, control algorithm and detection system is currently being enhanced

JET pulse #52314

+20

+16

-20

-16

FRFA




Experimental observations (5)FINAL PHASE OF VDE: HALO CURRENTS & CURRENT QUENCH (a)

JET pulse #54283 (γ=410 s-1)

ZPDIP≈d(zpIp)/dt

(Ip<0)

The plasma hits the wall after a ≈12 cm displacement:•diverted-limited transition•plasma response discontinuity

•plasma speed jump•growth rate jump

•rise of halo currents:•vertical speed slowing down•plasma temperature decreasing•impurities increasing•higher plasma resistivity•time constant of poloidal eddy currents

•plasma disruption:•temperature quench•plasma current quench




Experimental observations (6)FINAL PHASE OF VDE: HALO CURRENTS & CURRENT QUENCH (b)

JET pulse #54283 (γ=410 s-1)

ZPDIP≈d(zpIp)/dt

Ip

The plasma hits the wall after a ≈12 cm displacement (t ≈64.015 s):

•diverted-limited transition•plasma response discontinuity

•rise of halo currents (at t ≈64.020 s):•vertical speed slowing down•plasma temperature decreasing•impurities increasing•higher plasma resistivity•time constant of poloidal eddy currents

•plasma disruption (64.025<t<64.045s):•temperature quench•plasma current quench halo currents flow

partly in the plasma, partly in the wall

halo measured by TF unbalance inside the vessel or currents flowing through FW tiles

halo current measurements




Experimental observations (7)Consequences of VDEs and AVDEs

• VDEs are more dangerous than other disruptions:– Large vertical forces (due to both eddy and halo currents)– Force concentration in upper (or lower) part of the structures– Halo currents: large thermal loads on the wall

• Asymmetric VDEs are even more dangerous:– Not yet clearly understood– Net horizontal (sidaways, not radial) forces due to combined

tilting and horizontal displacement (coming from the interaction with the toroidal field) superposed to vertical motion

– Force concentration in some toroidal sectors (toroidal peaking factors to be considered)

– Halo current concentration: large thermal loads on the wall (empirical toroidal peaking factors)

upwardVDE

or

downwardVDE

AVDE




References (1)MHD Stability

• J. P. Freidberg, Ideal Magneto-Hydro-Dynamics”, Plenum Press, New York, 1987, pp. 231-245.• G. Bateman, MHD Instabilities, MIT Press (1978)

Vertical instability• D.C. Robinson and A.J. Wootton, “An experimental study of tokamak plasmas with vertically elongated

cross-sections,” Nuclear Fusion, vol. 18, no. 11, pp. 1555-1567, Nov. 1978.• M. Mori, N. Suzuki, T. Shoji, I. Yanagisawa, T. Tani, Y. Matsuzaki, “Stability limit of feedback control

of vertical plasma position in the JFT-2M tokamak,” Nuclear Fusion, vol. 27, no. 5, pp. 725-734, May 1987.

• M. Okabayashi and G. Sheffield, “Vertical stability of elongated tokamaks,” Nuclear Fusion, vol. 14, no. 2, pp. 263-265, Apr. 1974.

• K. Lackner and A.B. Macmahon, “Numerical study of displacement instability in elongated tokamak,” Nuclear Fusion, vol. 14, no. 4, pp. 575-577, Sept. 1974.

Rigid displacement model• D.A. Humphreys and I.H. Hutchinson, “Filament-circuit model analysis of Alcator C-Mod vertical

stability,” Int. Rep. PFC/JA-89_28, Massachusetts Institute of Technology, 1989.• E.A. Lazarus, J.B. Lister, and G.H. Neilson, “Control of the vertical instability in tokamaks,” Nuclear

Fusion, vol. 30, no. 1, pp. 111-141, Jan. 1990.• S.C. Jardin and D.A. Larrabee, “Feedback stabilization of rigid axisymmetric modes in tokamaks,”

Nuclear Fusion, vol. 22, no. 8, pp. 1095-1098, Aug. 1982.




References (2)Perturbed equilibrium model

• R. Albanese, E. Coccorese, G. Rubinacci, “Plasma modeling for the control of vertical instabilities,” Nuclear Fusion, vol. 29, no. 6, pp. 1013-1023, June 1989.

• D.J. Ward and S.C. Jardin, “Effects of plasma deformability on the feedback stabilization of axisymmetric modes in tokamak plasmas,” Nuclear Fusion, vol. 32, no. 6, pp. 973-993, June 1992.

• D.A. Humphreys and I.H. Hutchinson, “Axisymmetric magnetic control design in tokamaks using perturbed plasma equilibrium response modeling,” Fusion Technology, vol. 23, no. 2, pp. 167-184, Mar. 1993.

• R. Albanese and F. Villone, “The linearized CREATE-L plasma response model for the control of current, position and shape in tokamaks,” Nuclear Fusion, vol. 38, no. 5, pp. 723-738, May 1998.

Feedback stabilization• G. Ambrosino and R. Albanese, “Magnetic control of plasma current, position, and shape in tokamaks”,

IEEE Control Syst. Mag., vol. 25, no. 5, pp. 76-92, Oct. 2005• S.C. Jardin and D.A. Larrabee, “Feedback stabilization of rigid axisymmetric modes in tokamaks,” Nuclear

Fusion, vol. 22, no. 8, pp. 1095-1098, Aug. 1982.• S. Moriyama, K. Nakamura, Y. Nakamura, Y., and S. Itoh, “Analysis of optimal feedaback control of

vertical plasma position in a tokamak system,” Jpn. J. of Appl. Phys., vol. 24, no. 7, pp. 849-855, 1985.• M.M.M. Al-Husari, B. Hendel, I.M. Jaimoukha, E.M. Kasenally, D.J.N. Limebeer, A. Portone, “Vertical

Stabilization of Tokamak Plasmas,” Proc. 30th IEEE CDC, Brighton, UK, 1991, pp. 1165-1170• M. Lennholm, D. Campbell, F. Milani, S. Puppin, F. Sartori, B. Tubbing, “Plasma vertical stabilisation at

JET using adaptive gain adjustment,” Proc. 17th IEEE/NPSS Symposium on Fusion Engineering, San Diego, CA, 1997, vol. 1, pp. 539-542.

• D.J. Ward and F. Hofmann, “Active feedback stabilization of axisymmetric modes in highly elongated tokamak plasmas,” Nuclear Fusion, vol. 34, no. 3, pp. 401-415, Mar. 1994.

• D.A. Humphreys and I.H. Hutchinson, “Axisymmetric magnetic control design in tokamaks using perturbed plasma equilibrium response modeling,” Fusion Technology, vol. 23, no. 2, pp. 167-184, Mar. 1993.




References (3)Eddy current treatment

• R. Albanese, R. Fresa, G. Rubinacci, and F. Villone, “Time evolution of tokamak plasmas in the presence of 3D conducting structures,” IEEE Transactions on Magnetics, Vol. 36, no. 4, pp. 1804-1807, July 2000.

• I. Senda, T. Shoji, T. Tsunematsu, T. Nishino, and H. Fujieda, “Approximation of eddy currents in three dimensional structures by toroidally symmetric models, and plasma control issues,” Nuclear Fusion, vol. 37, No. 8, pp. 1129-45, Aug. 1997.

ELMs• Zohm H. 1996 Edge localized modes (ELMs) Plasma Phys.Control. Fusion 38 105–28• Connor J.W. 1998 A review of models for ELMs Plasma Phys.Control. Fusion 40 191–213

VDEs, AVDEs and halo currents• E.J. Strait, L.L.Lao, J.L. Luxon, E.E.Reis, “Observation of poloidal current flow to the vacuum vessel

during vertical instabilities in the DIII-D Tokamak”, Nuclear Fusion, Vol. 31, No. 3 (1991) pp. 527-534• L.Lao, T.H. Jensen, “Magnetohydrodynamic equilibria of attached plasmas after loss of vertical stability in

elongated tokamaks, Nuclear Fusion, Vol. 31, No. 10 (1991), pp.1909-1922• R. Albanese, M. Mattei, F. Villone, "Prediction of the growth rates of VDEs in JET", Nucl. Fus. , Vol. 44,

999-1007 (2004)• V. Riccardo, S. Walker, P. Noll, “Modelling Magnetic Forces during Asymmetric Vertical Displacement

Events in JET”, Fus. Eng. Des., 47, p.389-402 (2000).• Y. Neyatani, R. Yoshino, T. Ando, Effect of halo current and its toroidal asymmetry during disruptions in

JT60-U, Fusion Technol. 28 (1995) 1634.• R.S. Granetz, I.H. Hutchinson, J. Sorci, B. Labombard, D. Gwinn, Disruptions and halo currents in Alcator

C-MOD, Nucl. Fusion 36 (1996) 545.• T.E. Evans, A.G. Kellman, D.A. Humphreys, M.J. Shaffer, P.L. Taylor, D.G. Whyte, et al., Measurements

of non-axisymmetric halo currents with and without ‘killer’ pellets during disruptive instabilities in DIII-D tokamak, J. Nucl. Mater. 241–243 (1997) 606.

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