Characteristics of SVD in ST Plasma Shape Reproduction Method Based on CCS

Kazuo NAKAMURA1, Shinji MATSUFUJI2, Masashi TOMODA2, Feng WANG3, Osamu MITARAI4, Kenichi KURIHARA5, Yoichi KAWAMATA5, Michiharu SUEOKA5, Makoto HASEGAWA1, Kazutoshi TOKUNAGA1, Kohnosuke SATO1, Hideki ZUSHI1,

Kazuaki HANADA1, Mizuki SAKAMOTO1, Hiroshi IDEI1, Shoji KAWASAKI1, Hisatoshi NAKASHIMA1 and Aki HIGASHIJIMA1

1Research Institute for Applied Mechanics, Kyushu University, Kasuga, Fukuoka, 816-8580 Japan, 2Interdisciplinary Graduate School of Engineering Sciences, Kyushu University,

Kasuga, Fukuoka, 816-8580 Japan 3Institute of Plasma Physics, Chinese Academy of Sciences, Hefei, 230031 China

4Tokai University, Toroku, Kumamoto, 862-8652 Japan 5Japan Atomic Energy Agency, Mukoyama Naka-shi, Ibaraki-ken, 311-0193 Japan

CUP2008, EAST, ASIPP, Mar. 2, 2009 (Mon.)

Abstract

LSM (Least Square Method) has been used to solve the inverse problem from flux loop measurement to the boundary values on the CCS (Cauchy Condition Surface). When the CCS method is applied to real experimental data, noise superposition is inevitable. By introduction of SVD (Singular Value Decomposition) and truncation of the least SV components, we can expect the shape reconstruction robustness against the noise. [1] K. Kurihara: Fusion Eng. Design, Vol. 51-52 (2000) 1049-1057.

[2] F. Wang, K. Nakamura, O. Mitarai, K. Kurihara, Y. Kawamata, M. Sueoka, K. N. Sato, H. Zushi, K. Hanada, M. Sakamoto, H. Idei, M. Hasegawa, et al.: Engineering Sciences Reports, Kyushu University, Vol. 29, No. 1 (2007) 7-12.

Flux (known)

Boundary values on CCS

(unknown)

Noise

Inverse problem

Normal calculation

Characteristics of SVD in ST Plasma Shape Reproduction Method Based on CCS

• Abstract

• Cauchy Condition Surface method– Observation equation– Least Square Method– Singular Value Decomposition– Magnetic surface and Largest Closed Flux Surface

• Plasma shape reproduction by SVD

• Robustness by least SV truncation

• Summary

bUVWx

bxUWV

bAAAx

bAxAA

bAx

t

t

tt

tt

)(

)(

)()(

)()(

1

1

-1000

-500

0

500

1000

10008006004002000

CCS（ Cauchy Condition Surface）Method

・ Unknown: Boundary values on CCS

・ Known: Magnetic sensor (flux loop, magnetic probe), PFC current

・ Solve the observation equation

・ Calculate vacuum magnetic surfaces

・ Identify LCFS as plasma boundary

observation equation

LSM

Error⇒(Condition number)2

SVD

Error ⇒ (Condition number)1

Condition number = max SV / min SV

[1] K. Kurihara: A new shape reproduction method based on the Cauchy-condition surface for real time tokamak reactor control, Fusion Eng. Design 51-52 (2000) 1049.

Vacuum

Plasma

We consider “vacuum region”.

We think “plasma” is outside of the “vacuum region”.

“Flux value” in the “vacuum region” is calculated from the boundary values on “CCS”.

3 3 31 1

1( ) ( , ) ) ( , ) ) ( )

2

M M

F i i B i i C PFi i

x W x z z W x z t z W x I

2 2 21 1

( ) ( , ) ( ) ( , ) ( ) ( )M M

B F B i i B B i i C B PFi i

Bt x W x z z W x z t z W x I

1 1 11 1

( ) ( , ) ( ) ( , ) ( ) ( )M M

f F f i i B f i i C f PFi i

x W x z z W x z t z W x I

4 4 41 1

( ) ( , ) ( ) ( , ) ( ) ( )M M

F i i B i i C PFi i

x W x z z W x z t z W x I

Flux values at flux loop:

Magnetic field at magnetic probe:

Boundary values on CCS:

Outside of plasma:

Magnetic surface → LCFS → Plasma boundary

ECfC IxW )(8

ECfC IxW )(5

ECfC IxW )(6

ECfC IxW )(7

Boundary values on CCS (unknown)

Eddy current (unknown)

Least Square Method and Singular Value Decomposition

yAAAx

yAxAA

yxA

LSM

~~

~~

1

yUΔVx

yxVΔU

yxA

SVD

~

~

1

⊿: Diagonal component is SV

Observation euation

Normal equation

Multiply the transposed matrix

Multiply the inverse matrix

bUVWx

bxUWV

bAIAAx

bAxIAA

xxbAxbAx

bAAAx

bAxAA

bAxbAx

bAx

t

t

tt

tt

tt

tt

tt

t

)(

)(

)()(

)()(

)()min(

)()(

)()(

)()min(

1

1

1

Observation equation

Minimize the error

LSM

Error⇒(Condition number)2

Minimize the error and the norm

LSM

Error⇒(Condition number)2

SVD

Error ⇒ (Condition number)1

11111

11

1

11

321

3

2

1

321

333222111

00001

0

0

0

0

1

,

00

00

00

,

vVWVUWVuAu

uUUWvUWVAv

vvvVWuuuU

vuvuvuUWVA

ttttt

t

tttt

Singular Value Decomposition

Eigen (Characteristic) Value Decomposition

11111

11

1

11

321

3

2

1

321

333222111

00001

0

0

0

0

1

,

00

00

00

,

0det

0

uUUUUuAu

uUUuUUAu

uuuUuuuU

uuuuuuUUA

IA

uIA

uAu

ttttt

t

tttt

Definition of eigen value and eigen vector

Eigen (Characteristic) equation must hold good.In order that this equation has non-trivial solution,

Larger eigen-value component has larger information.

Larger singular-value component has larger information.

Least Square Method and Singular Value Decomposition

yAAAx

yAxAA

yxA

LSM

~~

~~

1

yUΔVx

yxVΔU

yxA

SVD

~

~

1

⊿: Diagonal component is SV

Matrix is multiplied only one time.Matrices are multiplied two times.

bUVWx

bxUWV

bAIAAx

bAxIAA

xxbAxbAx

bAAAx

bAxAA

bAxbAx

bAx

t

t

tt

tt

tt

tt

tt

t

)(

)(

)()(

)()(

)()min(

)()(

)()(

)()min(

1

1

1

Observation equation

Minimize the error

LSM

Error propagation ⇒ (Condition number)2

Minimize the error and the norm

LSM

Error propagation⇒(Condition number)2

SVD

Error propagation ⇒ (Condition number)1

11111

11

1

11

321

3

2

1

321

333222111

00001

0

0

0

0

1

,

00

00

00

,

vVWVUWVuAu

uUUWvUWVAv

vvvVWuuuU

vuvuvuUWVA

ttttt

t

tttt

Condition number = max SV / min SV

3 3 31 1

1( ) ( , ) ) ( , ) ) ( )

2

M M

F i i B i i C PFi i

x W x z z W x z t z W x I

2 2 21 1

( ) ( , ) ( ) ( , ) ( ) ( )M M

B F B i i B B i i C B PFi i

Bt x W x z z W x z t z W x I

1 1 11 1

( ) ( , ) ( ) ( , ) ( ) ( )M M

f F f i i B f i i C f PFi i

x W x z z W x z t z W x I

4 4 41 1

( ) ( , ) ( ) ( , ) ( ) ( )M M

F i i B i i C PFi i

x W x z z W x z t z W x I

At flux loop:

At magnetic probe:

On CCS:

Outside of plasma:

Magnetic surface → LCFS → Plasma boundary

ECfC IxW )(8

ECfC IxW )(5

ECfC IxW )(6

ECfC IxW )(7

Cauchy condition (unknown)

Eddy current (unknown)

(Largest Closed Flux Surface)

bUVWx

bxUWV

bAAAx

bAxAA

bAx

t

t

tt

tt

)(

)(

)()(

)()(

1

1

-1000

-500

0

500

1000

10008006004002000

CCS（ Cauchy Condition Surface）Method

・ Unknown: CC on CCS

・ Known: Magnetic sensor (flux loop, magnetic probe), PFC current

・ Solve the observation equation

・ Calculate vacuum magnetic surfaces

・ Identify LCFS as plasma boundary

observation equation

LSM

Error⇒(Condition number)2

SVD

Error ⇒ (Condition number)1

Condition number = max SV / min SV

[1] K. Kurihara: A new shape reproduction method based on the Cauchy-condition surface for real time tokamak reactor control, Fusion Eng. Design 51-52 (2000) 1049.

(Largest Closed Flux Surface)

-0.4

-0.2

0.0

0.2

0.4

Flux

loop

val

ue (

mW

b)

403020100

Outer---Upper---Inner Flux Loop---Lower---Outer

Flux value measured Singular Vector Singular Vector1 Singular Vector2

-0.4

-0.2

0.0

0.2

0.4

Flux

loop

val

ue (

mW

b)

403020100

Outer---Upper---Inner Flux Loop---Lower---Outer

Flux value measured Singular Vector3 Singular Vector4 Singular Vector5

SV of coefficient matrix in observation equation

Flux loop value and lower modes

Flux loop value and higher modes

8

6

4

2

0

Sin

gula

r Val

ue

6543210-1Singular Value No

Even Even Even

Odd Odd Odd

Larger singular-value component has larger information.

-0.4

-0.2

0.0

0.2

0.4

Flu

x lo

op v

alue

(m

Wb)

403020100

Outer---Upper---Inner Flux Loop---Lower---Outer

Flux value measured Singular Vector Singular Vector2 Singular Vector4

Flux loop value and even modes

SV of coefficient matrix in observation equation

-0.4

-0.2

0.0

0.2

0.4

Flu

x lo

op v

alue

(m

Wb)

403020100

Outer---Upper---Inner Flux Loop---Lower---Outer

Flux value measured Singular Vector1 Singular Vector3 Singular Vector5

Flux loop value and odd modes

8

6

4

2

0

Sin

gula

r Val

ue

6543210-1Singular Value No

Even Even Even

Odd Odd Odd

Larger singular-value component has larger information.

Error and contribution fraction (singular value)^2

Flux loop value and even-mode accumulation

-0.4

-0.2

0.0

0.2

0.4

Flu

x lo

op v

alue

(m

Wb)

403020100

Outer---Upper---Inner Flux Loop---Lower---Outer

Flux value measured Singular Vector0 Singular Vector2 Singular Vector4

-0.4

-0.2

0.0

0.2

0.4

Flu

x lo

op v

alue

(m

Wb)

403020100

Outer---Upper---Inner Flux Loop---Lower---Outer

Flux value measured Singular Vector0 Singular Vector02 Singular Vector024

bUVWx

bUxVW

bxUWV

bAx

t

tt

t

)(

)()(

)(

1

-1000

-500

0

500

1000

10008006004002000

W: Diagonal component is SV.

SVD

2.0

1.5

1.0

0.5Sum

of Fl

ux L

opp̂

2

543210Singular Value No

1.00

0.95

0.90

0.85

0.80

Contribution from

SV

Sum of Flux Loop Diff^2 Contribution from SV

0.20

0.15

0.10

0.05

0.00

-0.05

CC

S fl

ux

6543210-1

CCSM

Flux value on CCS Singular Vector0 Singular Vector2 Singular Vector4

Flux value and eigen vectors of even modes on CCS

out ← up ← in in ← down ← out

-1000

-500

0

500

1000

10008006004002000

-1000

-500

0

500

1000

10008006004002000

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

Z(m

)

0.60.40.20.0

R(m)

CCS ECODE

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

Z(m

)

0.60.40.20.0

R(m)

CCS ECODE

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

Z(m

)

0.60.40.20.0

R(m)

CCS ECODE

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

Z(m

)

0.60.40.20.0

R(m)

CCS ECODE

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

Z(m

)

0.60.40.20.0

R(m)

CCS ECODE

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

Z(m

)

0.60.40.20.0

R(m)

CCS ECODE

1 only 1 - 2 1 - 3

1 - 4 1 - 5 1 - 6

bUVWx

bxUWV

bAIAAx

bAxIAA

xxbAxbAx

bAAAx

bAxAA

bAxbAx

bAx

t

t

tt

tt

tt

tt

tt

t

)(

)(

)()(

)()(

)()min(

)()(

)()(

)()min(

1

1

1

Observation equation

Minimize the error

LSM

Error⇒(Condition number)2

Minimize the error and the norm

LSM

Error⇒(Condition number)2

SVD

Error ⇒ (Condition number)1

11111

11

1

11

321

3

2

1

321

333222111

00001

0

0

0

0

1

,

00

00

00

,

vVWVUWVuAu

uUUWvUWVAv

vvvVWuuuU

vuvuvuUWVA

ttttt

t

tttt

Least Square Method and Singular Value Decomposition

yAAAx

yAxAA

yxA

LSM

~~

~~

1

yUΔVx

yxVΔU

yxA

SVD

~

~

1

⊿: Diagonal component is SV

Shape difference increases in proportional to noise of flux value

Noise of flux value is simulated by random noise from ±1% to ±5% and shape difference is defined as the maximum difference of plasma shape.

RLRL ZZRRdiff ,,,max

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

Flux Loop Diff

Max/FL

20151050Measure Noise [%]

SV0 to SV1

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

Flux Loop Diff

Max/FL

20151050Measure Noise [%]

SV0 to SV1 SV0 to SV3 SV0 to SV5

oddevenoddevenoddeven

SVSVSVSVSVSV 543210

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

Flux Loop Diff

Max/FL

20151050Measure Noise [%]

SV0 to SV5

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

Flux Loop Diff

Max/FL

20151050Measure Noise [%]

SV0 to SV3

oddevenoddevenoddeven

SVSVSVSVSVSV 543210

25

20

15

10

5

0Sum

of Fl

ux L

opp

Diff

2̂

20151050Measure Noise [%]

SV0 to SV1 SV0 to SV3 SV0 to SV5

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

Flux Loop Diff

Max/FL

20151050Measure Noise [%]

SV0 to SV1 SV0 to SV3 SV0 to SV5

oddevenoddevenoddeven

SVSVSVSVSVSV 543210

Summary: Characteristics of SVD in ST Plasma Shape Reproduction Method Based on CCS

• Ideal equilibrium magnetic surface by equilibrium code– CCS (Cauchy-Condition Surface) method is a numerical approach

to reproduce plasma shape, which has good precision in conventional tokamak and also in spherical tokamak.

– When the CCS method is applied to real experimental data, noise superposition is inevitable.

– SVD may be effective, since error is proportional to condition number, though the error is proportional to the square in LSM

– By introduction of SVD (Singular Value Decomposition) and truncation of the least SV components, we can expect the shape reconstruction robustness against the noise.

– It is expected that noise effect decreases though detailed magnetic surface information is lost, if smaller SV is neglected.

– The truncation simulation of the less SV component shows shape reproduction error less than no truncation, when the measurement error exceeds a certain value.

– In this configuration, the ST plasma is symmetrical with respect to the equatorial plane and the number of free parameters on CCS is effectively half of 6.

– When unsymmetrical-shape plasma is considered or the degree of freedom is increased, the certain value for inversion may decrease.