adaptive extremum seeking control of eccd for ntm stabilization

1APS DPP 2006October 31 2006

Adaptive Extremum Seeking Control of ECCD

for NTM StabilizationL. Luo1, J. Woodby1, E. Schuster1

F. D. Halpern2, G. Bateman2, A. H. Kritz2

1Department of Mechanical Engineering2Department of Physics

Lehigh University, Bethlehem, PA 18015

48th Annual Meeting of the Division of Plasma PhysicsAmerican Physical Society

30 October – 3 November 2006Philadelphia, Pennsylvania


Abstract

Neoclassical Tearing Modes (NTMs) drive magnetic islands to grow to their saturated widths, at which they can persist stably in the plasma. The presence of magnetic islands leads to a local flattening of the current density and pressure profiles, which degrade plasma confinement. Since the bootstrap current density is proportional to the pressure gradient, this current is nearly absent within each island. One common method of stabilizing NTMs and therefore shrinking the island widths involves replacing the lost current via Electron Cyclotron Current Drive (ECCD). In order for ECCD to be successful at shrinking the island widths, the current must be driven at the flux surfaces that contain the islands. Moreover, in order to shrink each island with minimal ECCD power, the current must be deposited as close to the center of the island as possible. The difficulty lies in determining the locations of both the island flux surface and the ECCD deposition in real time. The Extremum Seeking feedback method is considered in this work for non-model based optimization of ECCD suppression of NTMs in tokamaks. ECCD steering change will be considered as mechanisms to maximize in real-time the alignment between the island flux surface and the current deposition location, and thus to minimize the ECCD power required for NTM stabilization. Theoretical analysis is done byWoodby [5].


Objectives

• Use BALDUR and ISLAND code to simulate NTM• Find a approximation model of the current drive. The

shrinking effect is determined by the position, width and strength of the current drive.

• Modify BALDUR and ISLAND to incorporate the current drive model

• Introduce a feedback control on the current drive using extremum seeking scheme

• Numerical simulations


References

1. ISLAND module from NTCC module library: http://w3.pppl.gov/ntcc2. Background, finding saturated magnetic island widths, ISLAND:

– G. Bateman and R. Morris, Phys. Fluids 29 (3) (1986)– F. Halpern, Physics of Plasmas 13 (2006) 062510

3. Similar work expressing current drive in Hamada coordinates:– Giruzzi et al., Nuclear Fusion 39 (1999) 107-125– C. Hegna and J. Callen, Physics of Plasmas 4 (1997) 2940

4. Computing elliptic integrals: www.netlib.org5. Dependence of NTM Stabilization on Location of Current Drive

Relative to Island– J. Woodby, APS 2006 Philadelphia, Poster Session JP1.00143


• Using Hamada-like coordinate system

(V is any quantity which is constant over a flux surface, such as volume)• Get set of coupled ODEs which describe change in background current and pressure profiles due to presence

of island• Implemented in ISLAND module, implemented in BALDUR, which computes saturated magnetic island widths

• NTM=neoclassical tearing mode, magnetic “islands” result from

tearing and reconnection of ideally nested magnetic flux surfaces • Starting from force-balance equations

Background


Current drive modelStart by assuming that the current drive has the following form


Current applied in u-coordinates gets spread over magnetic flux surfaces

Current drive model

Please see Ref. #5 for detailed deviation

J0

u

α


• Averaged driving current distribution:

• Taking the derivative

where

Current drive model

K is the complete elliptic integral of the first kind; E is the complete elliptic integral of the second kind.


The superposed current density derivative

Current drive model

• A current drive is determined by three parameters– location (a, in u coordinate)– width (b, in u coordinate) – strength (J0)

• JEC is positive.• A FORTRAN module is developed for ISLAND to handle current

drive


Current density profile without current driveDIII-D 2/1 island test (no current drive)• without any island (left)• with island (right)

x is the plasma minor radius (x=0 at the center of the plasma and x=1 at the edge of the plasma); j is the current density. Variables are non-dimensional.


Current density profile with current drive

The effect of current drive on the current density profile• b=0.8, J0=1, a=0 (left)• b=0.8, J0=1, a=2 (right)


Dependence of island width on the location of the current drive

Island half width as a function of location (a)• Narrow drive (b=0.8, left)• Wide drive (b=1.5, right)


Dependence of island width onthe current drive strength

Island half width as a function of current drive strength (J0)• Narrow drive (a=0, b=0.8, left)• Wide drive (a=0, b=1.5, right)


EXTREMUM SEEKING – HOW DOES IT WORK?

0" ,*2

''* 2 J

JJJ

function to be minimizedJ*J minimum of the static map"J second derivative (if positive J() has a minimum)

estimate of

adaptation gain amplitude of the probing signal frequency of probing signalh cutoff frequency of high-pass filter

* unknown parameter that minimizes J / modulation/demodulation

J

* *J

hs

s

s

k sin ksin

Plant

Low-PassFilter

High-PassFilter

J


EXTREMUM SEEKING

0" ,*2

''* 2 J

JJJ

Any C2 function J() can be approximated locally in this way. The assumption is made without loss of generality. If J”<0 , we just replace ( > 0) in the figure with -. The purpose of the algorithm is to make -* as small as possible, so that the output J() is driven to its minimum J*. The perturbation signal αcos(k) helps to get a measure of gradient information of the static map J().

J

* *J

hs

s

s

k sin ksin

Plant

Low-PassFilter

High-PassFilter

J


EXTREMUM SEEKING

ˆ~ *Let

kkkkk ~sin*sinˆ* Thus

2~sin

2

''* kk

JJkJkJ Which gives

Estimation Error

J

* *J

hs

s

s

k sin ksin

Plant

Low-PassFilter

High-PassFilter

J


EXTREMUM SEEKING

2~sin

2

''* kk

JJkJkJ

kJ

kkJkJJ

JkJkJ 2cos4

''sin

~''

~

2

''

4

''*

22

2

kJ

kkJkJ

k 2cos4

''sin

~''

~

2

'' 22

kk 2cos1sin2 2

J

* *J

hs

s

s

k sin ksin

Plant

Low-PassFilter

High-PassFilter

J


EXTREMUM SEEKING

kJ

kkJkJ

k 2cos4

''sin

~''

~

2

'' 22

kkJ

kkJkkJ

k sin2cos4

''sin

~''sin

~

2

'' 222

kkJ

kkJ

kkJ

kJ

k sin~

2

''3sinsin

8

''2cos

~

2

''~

2

'' 22

kkkk sin3sinsin2cos2 kk 2cos1sin2 2

J

* *J

hs

s

s

k sin ksin

Plant

Low-PassFilter

High-PassFilter

J


EXTREMUM SEEKING

kkJ

kkJ

kkJ

kJ

k sin~

2

''3sinsin

8

''2cos

~

2

''~

2

'' 22

kJ

kkkJ

qk ~

2

''ˆ1ˆ~

2

''

1ˆ

J

* *J

hs

s

s

k sin ksin

Plant

Low-PassFilter

High-PassFilter

J


EXTREMUM SEEKING

kJ

kkkJ

zk ~

2

''ˆ1ˆ~

2

''

1ˆ

kaJ

kk ~

2

''~1

~ *ˆ0

~

kkkk ~1

~ˆ1ˆ

0"aJ

Stable System

ˆ~ *

J

* *J

hs

s

s

k sin ksin

Plant

Low-PassFilter

High-PassFilter

J


EXTREMUM SEEKING

)1(cos1ˆ1

ˆ1ˆ

sin

11

kkk

kkk

kkk

kJkJkhk

In our case

• θ is the position (a)

• J is the half island width

Iteration relations

J

* *J

hs

s

s

k sin ksin

Plant

Low-PassFilter

High-PassFilter

J


Extremum seeking results

Position (a) progression (b=1.5, J0=1.0)• The position of current drive eventually converges at the center of the island• Oscillation is caused by the probing signal


Extremum seeking results

Cost function J (half island width) progression (b=1.5, J0=1.0)


Conclusion

• The island width is dependent on the location, width and strength of the proposed current drive

• The modified ISLAND module gives estimation of island width and current density profile for different width and strength

• Extremum seeking appears an effective method to steering the current drive and to maximize the island shrinking


Future Research

• A more accurate current drive model• Implementation of off-center current drive model in

ISLAND/BALDUR• Extremum seeking feedback stabilization in time-

dependent simulations• Code optimization for better performance

adaptive extremum seeking control of eccd for ntm stabilization

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