November 3-5, 2003 Feedback Workshop, Austin

NORMAL MODE APPROACH TO MODELING OF FEEDBACK

STABILIZATION OF THE RESISTIVE WALL MODE

By

M.S. Chu(GA), M.S. Chance(PPPL),

A. Glasser(LANL), and M. Okabayashi(PPPL)

Nucl. Fusion Vol. 43, 441 (2003) Acknowledgement toA. Bondeson, Y.Q.Liu

November 3-5, 2003 Feedback Workshop, Austin

MOTIVATION

• To develop a model for understanding results from experiments (DIII-D) on feedback stabilization and to evaluate performance of future devices (ITER)

• To develop a model beyond the usual model which includes only the geometrical effects from the slab or cylindrical geometry, i.e. Grad-Shafranov equilibrium

• To compare and benchmark with results from other codes

November 3-5, 2003 Feedback Workshop, Austin

NORMAL MODE APPROACH (NMA) BASED ON ENERGY CONSERVATION OF GENERAL

PLASMA EQUILIBRIUM• Perturbation energy of RWM for ideal

plasma

– General plasma equilibrium: axi-symmetric or helical

– General plasma perturbation: axisymmetric or helical

– Frequency dependent non-self-adjoint

€

δWp +δK +δWV + DW +δEC = 0

Plasma

δWP, δK

VacuumδWV

DW

δEC

Kinetic Energy Wall Dissipation Coil

Excitation Energy

November 3-5, 2003 Feedback Workshop, Austin

NMA BASED ON THE NORMAL MODES OF THE OPEN LOOP OPERATION

• NMA applicable if open loop system can be represented as a set of normal modes

– No plasma rotation– No plasma dissipation– A more conservative model than MARS-F

• The details of the system is completely described

– Does not rely on Pade approximation

November 3-5, 2003 Feedback Workshop, Austin

THREE STEPS FOR FULL SOLUTION

• Open loop stability: Generalization of the ideal MHD stability problem (no feedback)

• Evaluate the excitation and sensor matrices of the feedback geometry

• Evaluate feasibility of feedback based on Nyquist diagram or characteristics equations

€

δEC = 0

Plasma

δWP

VacuumδWV

DW

δEC=0

November 3-5, 2003 Feedback Workshop, Austin

NMA IMPLEMENTED BY COUPLING DCON + VACUUM + TANK

• DCON expresses plasma free energy in terms of perturbed magnetic field at plasma boundary

• Extended VACUUM expresses vacuum energy in terms of perturbed magnetic field at plasma boundary and the vacuum tank

• Tank evaluates the energy dissipation in terms of the perturbed

€

δWp =δWp (δBp ,δBp )

€

δWv =δWv1(δBp ,δBp )+δWv2 (δBp ,δBt )+δWv 3(δBt ,δBt )

€

Dw = Dw(δBt ,δBt )

November 3-5, 2003 Feedback Workshop, Austin

CURRENTS ON VACUUM VESSEL REPRESENTED AS A SET OF

DISSIPATION EIGENFUCTIONS

• Flux leaking through the resistive wall excites dissipation eigenfunctions

Odd

even

Poloidal position along the resistive wall

Induced by toroidalefffect

November 3-5, 2003 Feedback Workshop, Austin

GRAD-SHAFRANOV SOLVER (TOQ) AND DCON ANALYSIS DETERMINES RWM

STABILITY BOUNDARIES

Equilibrium Flux Function

Safety factor

Pressure

δW from Dcon

Plasma Vacuum Total δW

November 3-5, 2003 Feedback Workshop, Austin

EDDY CUURENTS OF OPEN LOOP STABILITY EIGENFUNCTIONS

• Computed also by MARS

Toroidal angle

Unstable RWM

1st StableMode

2nd Stable Mode

3rd Stable Mode

Poloidal angle

November 3-5, 2003 Feedback Workshop, Austin

CHARACTERISTICS EQUATION OF CLOSED LOOP SYSTEM DETERMINES RWM FEEDBACK

• Closed loop feedback stability described by a compact set of equations for open loop amplitudes i plus coil currents IC

• Diagonalization of the open loop response allow reduction of the dynamical variables to (I, Ic)

€

B = α ii

∑ {Bpi ,Bw

i }+ IcBc

∂α i

∂t− γ iα i = Ic

c∑ Ei

c

∂Ic

∂t+

1τ c

Ic = Gcl

l∑ Fl

c ({α i},{Ic })

€

D(s) = s I↔

− Γ↔

− E→

G→

F→

s I↔

− L↔ = 0

Response to Feedback

Coils

Open Loop Eigenfunction

Excitation Matrix

Sensor MatrixGain Matrix

Characteristics Equation

Identity Matrix

November 3-5, 2003 Feedback Workshop, Austin

SINGLE INPUT AND SINGLE OUPUT CAN BE ANALYZED USING NYQUIST DIAGRAM

• Stablized if transfer function P() encircles (-1,0)

• Radial sensors are less effective and stabilize lower range of N

• Poloidal sensors stabilize the whole computed range of N

€

P(ω) =Feedback Signal

Sensor Signal= Σ

i

FiEi

jω − γ i

= Σi

Ri

jω − γ i

-1 -1

C = 10%

22%

38%

67%

82%

Poloidal SensorRadial Sensor Less Effective €

Cβ =βN − βN

NW

βNIW − βN

NW

Re[P(j)] Re[P(j)]

Im[P

(j

)]

Im[P

(j

)]

0 = No Wall 1 = Ideal wall

C-Coils

November 3-5, 2003 Feedback Workshop, Austin

November 3-5, 2003 Feedback Workshop, Austin

FEEDBACK MODELING SHOWS INTERNAL I-COILS ARE MORE

EFFECTIVE THAN EXTERNAL C-COILS

C-Coils

• I-Coils couple more effectively to the unstable RWM since closer to plasma• EI and EC are elements of excitation matrix

I-Coils

I-Coils

Ratio of Effectiveness

C-coil / I-coil5.0

0.0

2.5

C

0.0 1.0

0.5

EI / EC

November 3-5, 2003 Feedback Workshop, Austin

COUPLING OF FEEDBACK COIL TO STABLE MODES IMPEDES STABILIZATION

f=1 f=1f=3/4

f=1/2

f=1/4

f=1/8

f=3/4

f=1/2

f=1/4

f=1/16

Ri f Ri for all stable modes

C=42% C=83%

(-1,0) (-1,0)

Nyquist Diagram

November 3-5, 2003 Feedback Workshop, Austin

FOR REAL SYSTEM THE TIME CONSTANT OF THE EXTERNAL CIRCUIT IS IMPORTANT

• Solution of characteristic equation

0

30

-30Voltage Amplification

w

RWM

Circuit

Stable Modes

f=1 f=.15C=83%c=.03 w

November 3-5, 2003 Feedback Workshop, Austin

SCOPING STUDY FOR C-COIL EXTENSIONS

• Radial Sensor, Ideal Feedback

Upperextension

Lower extension

C-Coil

C

0 1

w

0

30

f

0

1

All Three Coils

C-Coil

Upper+ Lower

November 3-5, 2003 Feedback Workshop, Austin

SUMMARY / CONCLUSION• Feedback with ideal plasma response formulated for general

plasma equilibrium through energy conservation. • Phase space of feedback system reduced to the normal

modes of open loop eigenfunctions and currents in feedback coils (NMA)

• For tokamak geometry NMA has been implemented by coupling DCON with extended VACUUM to study RWM feedback stabilization – Poloidal sensors are more effective than radial sensors – I-Coils are more effective than C-coil

• MARS-F benchmarked against NMA for ideal plasma