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