Effects of Noise in Time Dependent RWM Feedback Simulations

O. Katsuro-Hopkins, J. Bialek, G. Navratil

(Department of Applied Physics and Applied Mathematics, ColumbiaUniversity, New York, NY USA)

Building on the successful experiments on HBT-EP and DIII-D, activefeedback control of the resistive wall mode (RWM) has emerged as anessential part of present and planned tokamak designs. In an effort to advanceour feedback model closer to actual experimental conditions, the VALENcode was modified by introducing noise into the closed loop control system.In practice, feedback system performance is limited by detection thresholdsand the stable operating range that is set by both systematic error and randomnoise in the measurement input to the feedback control loop. The results of aninitial survey analyzing the effects of systematic error and noise (white,Gaussian, 1/f, etc.) on the RWM feedback system performance is performedusing the newly developed time dependent capability in VALEN (see posterby J. Bialek).

Motivations

• Control of long-wavelength MHD instabilities using conducting walls

and external magnetic perturbations is a very promising route to

improved reliability and better performance of magnetic confinement

fusion devices.

• It is well known that control of these resistive wall slowed kink modes

above the no-wall beta limit is essential to achieve bootstrap current

sustained steady-state operation in a high gain tokamak fusion energy

systems.

• The ability to accurately model and predict the performance of active

MHD control systems is critical to present and future advanced

confinement scenarios and machine design studies. The VALEN

modeling code has been designed and bench marked to predict the

performance limits of MHD control systems.

• To enhance VALEN’s ability to model more realistic feedback

systems initial value, time dependent capability was added to the code.

• To advance our feedback model closer to actual experimental

conditions, the VALEN code was also modified by introducing noise

into the closed loop control system.

• Understanding of non-ideal effects in feedback loop including

systematic error and noise (white, Gaussian, 1/f, etc.) on the RWM

feedback system performance is important for improved performance

of tokamaks.

VALEN: A Reliable Computational Tool For RWM Passive and Active Control System Study

•VALEN is the only code available to model 3D external conducting structures withreal porthole penetrations and other complex geometric features. Modeling of these

features is crucial for accurately determining the passive stabilization properties of current and future devices

•VALEN models unstable plasma modes using an electrical circuit representationdeveloped by Boozer and plasma mode structure from DCON

•VALEN models arbitrary external conducting structures using a 3D finite element electromagnetic formulation developed by Bialek

•VALEN models arbitrary sensors, control coils and the control loop feedback logicthat connects them to determine the effectiveness of feedback systems•VALEN predicts the growth rate of the instability, feedback currents and controlsystem gain, and current distributions in the external passive structure to characterizeRWM feedback performance•VALEN models initial value, time dependent problem•VALEN models non-ideal effects in feedback loop including systematic error andnoise

The VALEN Equations

The VALEN matrix equations describing the conducting structure and modeand control coil geometry are for the unknowns {Iw}, {Id}, and {Ip} are:

Lww� � I w� �� Mwp� � I d

� �� Mwp� � I p� �� �w� � flux @ wall

flux @ plasma

stability equation

Mpw� � I w� �� Lp� � I d

� �� Lp� � I p� �� �� �

Lp� � I p� �� S� � �� �

��w� �� Rww� � Iw

� �� V� �

��� �� Rd� � I d

� �� 0� �

The equivalent circuit (induction) equationsdescribing the system mode growth are then:

Msw� � Msd� � Msp� �� �

I w� �

I d� �

I p� �

��

��

��

��

��

��

��

��

��

��

s� �

Where {V} depends on sensor signals {�s} via thefeedback loop equations:

• VALEN uses DCON ( A. Glasser ) results without a conducting wallto formulate the stability equation

• Energy change �W = �����i�i2 in plasma & surroundings has

negative eigenvalues �i if an instability exists, fi(�,�) diagonalizes �Wand defines the flux from the plasma instability

• Complex helical magnetic geometry is expressed in terms ofinductance and current Li = �i / Ii and the stability equation may beexpressed as Sij=(�ij+si�ij) where si = -�iLi and the �ij may be derived

from the fi(�,�)

�

� i � fi(� ,� )���

B � d��

a�

VALEN Models External MHD Modes As SurfaceCurrents

• The interaction of an external MHD plasma instability with surrounding conductorsand coils is completely described by giving �Bnormal at the surface of theunperturbed plasma.

• VALEN uses this information in a circuit formulation of unstable plasma modes developed by Boozer to generate a finite element surface current representation of the unstable mode.

543210-0.1

0.0

0.1

Arc Length

�Bnormal calculated by DCONfor unstable plasma mode

VALEN finite element circuit repesentation of the

unstable plasma modestructure

This methodology allows VALEN to use output plasma mode informationfrom other instability physics codes (DCON, GATO, PEST or others)

VALEN's 3D Finite Element Capability Is Important In Accurately ModelingPassive Wall Stabilization Limits and Active Feedback Performance

• Correct representation of the geometricdetails of vacuum chambers with portholes and passive stabilizing plates is required to determine RWM control limits

• VALEN calculates these effects and allowsthe design of optimized control systemswith complicated real-world machinegeometry

Eddy current pattern induced in the wall of the DIII-D tokamak due to an unstable n=1

RWM [top and side view]

Eddy current pattern induced in thecontrol coils in the DIII-D tokamak

DIII-D New Internal Control Coils are an Effective Tool for Pursuing Active and Passive Stabilizations of the RWM

• Inside vacuum vessel: faster time response for feedback control

• Closer to plasma: more efficient coupling

Internal Coils Provide RWM Stabilization by Feedback Control

• I-coil experiments have confirmed theoretical predictions that RWA is stabilizedby direct feedback control

• Feedback performance is improvedwith internal poloidal field sensors

– Faster time response

– Decoupled from radial field ofcontrol coils

• Improved feedback performance is predicted for internal coils

– Faster time response

– Improved coupling to plasma

• Feedback stabilization up to ideal-wall limit requires that coil-wall coupling isnot too large:

[Boozer, Phys.Plasmas 1998; Okabayashi, Nucl.Fusion 1998 (RWM), Lazarus, Nucl.Fusion 1990 (Zp)]

– In cylindrical model: for external coils

for internal coils C �

rc

rm

��

��

��

��

��

��

2mC �1

C �

Mpw MwcLw M pc

�1

RWM Noise Data on DIII-D

Q i kTi ™ d TIFF (U d) d d d t thi i t

Noise on the poloidal field sensors in the midplane. The signals are

corrected for DC offsets. The power spectral density is shown as root-mean square amplitude per 10Hz frequency bin.

Simulated RWM Noise on DIII-D w/o ELMs

Low level noise was modeled as Gaussian random number withstandard deviate 1.5 about 0 mean and frequency 10kHz

Simulated RWM Noise on DIII-D with ELMs

0.01

0.10

1.00

0.01 0.10 1.00 10.00

f(kHz)

To the low level noise ELMs (Edge Localized Modes) were addedas small group of Gaussian random numbers from 6 to 16 Gaussapproximately every 0.01 sec with different signs +/- chosen with

50% probability

Resonant Field AmplificationA.Boozer predicted Resistive Wall Modes would amplify magnetic field(e.g. errors see Physics of Plasmas 10, pg 1458 (2003)) The VALEN formulation handles this via an extra field source.

Lww� � Iw� �� Mwp� �I

d� Mwp� �I

p� MwD� � ID

� �� �

w� �

Mpw� � Iw� �� LI d

� LI p� MpD� � I D

� �� �

LI p� 1� s� ��

field amplification in the sensors is given by

Msw� �� Msp� �1� s� �

sLMpw� �

��

��

��

��

Iw� �� Msp� �� Msp� �

1� s� �

sLL��

��

��

��

I d�

MsD� �� Msp� �1� s� �

sLMpD� �

��

��

��

��ID� ��

s� �

We anticipate that noise signal applied to marginally unstable plasma will be amplified through the phenomena of Resonant Field Amplification

Survey Feedback as Function of �normal

100

10-1

10-2

10 -1

10 0

10 1

102

10 3

104

10 5

106

g passive

g ideal

g 10^8

Data from "DIII-D.10.2003.newR&L"

s normalized mode strength

passive

ideal

I-coils

Gp=10^8 [ v/weber]

R=60.e-6 H, R = 30.e-3 ohm

10 010 -110

-1

10 0

101

102

10 3

10 4

105

10 6

g passive

g ideal

g 10^8

g 3.1623e8

g 3.9811e8

g 5.0119e8

g 6.3096e8

g 10^9

gain scan

L = 10.e-6 H, R = 200.e-3 ohm

Data from "DIII-D.10.2003.newR&L"

s normalize mode energy

passive

ideal

wall

Conversion from s to �

DIII-D I-Coil Feedback model for the Control Coils L=60 �H and R=30 mOhm

with Proportional Gain Gp=7.2Volts/Gauss

-400

-300

-200

-100

0

100

200

300

0.00 0.05 0.10 0.15 0.20 0.25

time [sec]

-1200

-1000

-800

-600

-400

-200

0

200

400

600

800

0.00 0.05 0.10 0.15 0.20 0.25

time [sec]

Control coil current

-20

-15

-10

-5

0

5

10

15

0.00 0.05 0.10 0.15 0.20 0.25

time [sec]

Sensor Flux

Voltage applied to control coil

Maximum control coil current and voltageas function of �normal

Proposed improved DIII-D I-Coil Feedback model for the Control Coils L=10 �H and R=200 mOhm with

ELMs and with Proportional Gain Gp=36 Volts/Gauss

-1300

-800

-300

200

700

1200

0.00 0.05 0.10 0.15 0.20 0.25time [sec]

-1200-1000

-800-600-400-200

0200400600800

1000

0.00 0.05 0.10 0.15 0.20 0.25

time [sec]

Control coil current

Sensor Flux

-8

-6

-4

-2

0

2

4

6

8

0.00 0.05 0.10 0.15 0.20 0.25

time [sec]

-8

-6

-4

-2

0

2

4

6

8

0.00 0.05 0.10 0.15 0.20 0.25

time [sec]

Voltage applied to control coil

Maximum control coil current and voltageas function of �normal

Proposed improved DIII-D I-Coil Feedback model for the Control Coils L=10 �H and R=200 mOhm w/o

ELMs and with Proportional Gain Gp=36 Volts/Gauss

-700

-500

-300

-100

100

300

500

700

0.00 0.05 0.10 0.15 0.20 0.25

time [sec]

-700

-500

-300

-100

100

300

500

700

0.00 0.05 0.10 0.15 0.20 0.25

time [sec]

Control coil current

Sensor Flux

Voltage applied to control coil

-7

-5

-3

-1

1

3

5

7

0.00 0.05 0.10 0.15 0.20 0.25

time [sec]

-7

-5

-3

-1

1

3

5

7

0.00 0.05 0.10 0.15 0.20 0.25

time [sec]

Maximum control coil current and voltageas function of �normal

Effects of Noise on Feedback Dynamics for L=60 �Hand R=30 mOhm DIII-D I-Coil Feedback model

with Proportional Gain Gp=7.2Volts/Gauss

Sensor Flux

-10

-5

0

5

10

15

20

25

0.000 0.002 0.004 0.006 0.008 0.010

time [sec]

0

100

200

300

400

500

600

700

800

0.000 0.002 0.004 0.006 0.008 0.010

time [sec]

-60-40-20

020406080

100120140

0.000 0.002 0.004 0.006 0.008 0.010

time [sec]

Control coil current Voltage applied to control coil

Conclusions and Future Work

• Plasma resonant RWM response to noise increases closed loop current. Ideal limit can be approached but cannot be reached.

• Noise on the sensor flux does not effect stabilization dynamic of feedback.

• Voltage for the proposed improved DIII-D I-Coil feedback model is dominated by resistance of the connection cables,lowering resistance will significantly reduce the powerrequirements.

• Future Work:– investigate effect of noise on feedback dynamic with time delay

– include noise in a full PID feedback control model

– add current and voltage limits on feedback closed loop

– add high and low frequency bandwidth limits to feedback closedloop.