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NORMAL MODE APPROACH TO MODELING OF FEEDBACK STABILIZATION OF THE RESISTIVE WALL MODE

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). Acknowledgement to A. Bondeson, Y.Q.Liu. Nucl. Fusion Vol. 43 , 441 (2003). MOTIVATION.

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NORMAL MODE APPROACH TO MODELING OF FEEDBACK STABILIZATION OF THE RESISTIVE WALL MODE

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  1. 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) Acknowledgement to A. Bondeson, Y.Q.Liu Nucl. Fusion Vol. 43, 441 (2003) Feedback Workshop, Austin

  2. 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 Feedback Workshop, Austin

  3. 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 DW Kinetic Energy Coil Excitation Energy Wall Dissipation Plasma WP, K EC Vacuum WV Feedback Workshop, Austin

  4. 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 Feedback Workshop, Austin

  5. 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 DW Plasma WP EC=0 Vacuum WV Feedback Workshop, Austin

  6. 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 Feedback Workshop, Austin

  7. CURRENTS ON VACUUM VESSEL REPRESENTED AS A SET OF DISSIPATION EIGENFUCTIONS • Flux leaking through the resistive wall excites dissipation eigenfunctions Odd Induced by toroidal efffect even Poloidal position along the resistive wall Feedback Workshop, Austin

  8. GRAD-SHAFRANOV SOLVER (TOQ) AND DCON ANALYSIS DETERMINES RWM STABILITY BOUNDARIES Equilibrium Flux Function Pressure W from Dcon Safety factor Plasma Vacuum Total W Feedback Workshop, Austin

  9. EDDY CUURENTS OF OPEN LOOP STABILITY EIGENFUNCTIONS 2nd Stable Mode • Computed also by MARS Unstable RWM Poloidal angle Toroidal angle 3rd Stable Mode 1st StableMode Feedback Workshop, Austin

  10. CHARACTERISTICS EQUATION OF CLOSED LOOP SYSTEM DETERMINES RWM FEEDBACK • Closed loop feedback stability described by a compact set of equations for open loop amplitudes iplus coil currents IC • Diagonalization of the open loop response allow reduction of the dynamical variables to (I, Ic) Characteristics Equation Response to Feedback Coils Open Loop Eigenfunction Excitation Matrix Identity Matrix Sensor Matrix Gain Matrix Feedback Workshop, Austin

  11. SINGLE INPUT AND SINGLEOUPUT CAN BE ANALYZED USING NYQUIST DIAGRAM 0 = No Wall 1 = Ideal wall • 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 Poloidal Sensor C = 10% Radial Sensor Less Effective Im[P(j)] Im[P(j)] 22% C-Coils 38% 67% 82%   -1 Re[P(j)] Re[P(j)] -1 Feedback Workshop, Austin

  12. Feedback Workshop, Austin

  13. I-Coils couple more effectively to the unstable RWM since closer to plasma EIand EC are elements of excitation matrix FEEDBACK MODELING SHOWS INTERNAL I-COILS ARE MORE EFFECTIVE THAN EXTERNAL C-COILS Ratio of Effectiveness C-coil / I-coil I-Coils 5.0 EI / EC 2.5 C-Coils 0.0 0.0 0.5 1.0 I-Coils C Feedback Workshop, Austin

  14. COUPLING OF FEEDBACK COIL TO STABLE MODES IMPEDES STABILIZATION Nyquist Diagram Ri f Ri for all stable modes f=1 C=42% C=83% f=3/4 f=1 f=1/2 f=3/4 f=1/4 f=1/2 f=1/4 f=1/8 f=1/16 (-1,0) (-1,0) Feedback Workshop, Austin

  15. FOR REAL SYSTEM THE TIME CONSTANT OF THE EXTERNAL CIRCUIT IS IMPORTANT • Solution of characteristic equation C=83% c=.03 w f=.15 f=1 30 RWM Stable Modes 0 w Circuit -30 Voltage Amplification Feedback Workshop, Austin

  16. SCOPING STUDY FOR C-COIL EXTENSIONS All Three Coils • Radial Sensor, Ideal Feedback Upper extension 1 30 C-Coil Upper+ Lower f C-Coil w 0 0 Lower extension 0 1 C Feedback Workshop, Austin

  17. 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 Feedback Workshop, Austin

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