Calculating Screening Current Effects in ReBCO Devices and Coils · 2019. 5. 3. · Finite condition number Stability Model order reduction (e.g., [2, 3]) High aspect ratio Tapes

This work is supported by:

(*) The ‘Excellence Initiative’ of the German Government and by the Graduate School of Computational Engineering at TU Darmstadt;

(**) The Gentner program of the German Federal Ministry of Education and Research (grant no. 05E12CHA).

1 2 (*) (**)

L. Bortot1,2, M. Mentink1, S. Schöps2, J. Van Nugteren1, A. Verweij1

Calculating Screening Current Effects

in ReBCO Devices and Coils

Special Thanks:

B. Auchmann (PSI), F. Grilli (KIT), M. Maciejewski, M. Prioli (INFN), E. Ravaioli

5th Workshop on Accelerator Magnets in HTS11-13 April 2019 - Budapest, Hungary

Outline

A. Introduction

B. Motivation

C. Formulation

D. Verification

Theoretical references

E. Validation

Measurements from Feather2

F. Summary and Outlook

Feather2 magnet: net magnetic

flux density contribution due to

screening currents (first quadrant)

i(A)

t(s)

B(T)

Introduction: Screening (Eddy) Currents

ReBCO tape in an external magnetic density B:

External magnetic field change 𝜕tB:

Persistent screening currents Ԧjscreen, due to ρ → 0 Persistent magnetization Bscreen

Large filament size (~5 mm), significant persistent magnetization:

Relevant for field quality, especially at low field

Relevant for tape thermal behavior, dominant AC loss contribution

3

■ Copper

■ ReBCO

■ Substrate

~mm

~μm

ρ → 0

ԦJscreen

𝜕tB Bscreen

Motivation

The analysis of future HTS magnets for accelerators

needs to take into account the screening currents dynamics

At CERN, there is a need for a code:

• Fast and numerically stable

• Scalable to accelerator magnets

• Capable of accounting for iron

• Capable of field-circuit coupling

• Validated, reliable (maintainable)

Our contributions

• Investigation and extension of a suitable field formulation

• Implementation in a proprietary software (*), using the Finite Element Method

• Scaling of the implementation to an HTS magnet (Feather2)

4

(*) COMSOL Multiphysics® v. 5.3. www.comsol.com. Last access: 10/04/2018

Formulation: Overview

Assumptions:

• Meißner Effect neglected (Bc1 < 10 mT)

• No ferromagnetic substrate (μ = μ0)

Mixed potentials (*), starting from [1]

• A solved in Ω0 (air, iron), where σ → 0

• H solved in Ωc (conductors), where ρ → 0 :

Finite condition number Stability

Model order reduction (e.g., [2, 3])

High aspect ratio

Tapes as surfaces in ℝ3 (lines in ℝ2) Speed-up

5

A → Ω0, ℝ3

H → Ωc, ℝ3

𝐯𝐬

𝐢𝐬

+

−

[1] Bíró, O. "Edge element formulations of eddy current problems." CMAM 169.3-4 (1999): 391-405.

[2] Carpenter, C. J. "Comparison of alternative formulations of 3-dimensional magnetic-field and eddy-current problems at power frequencies." Proceedings of the

Institution of Electrical Engineers. Vol. 124. No. 11. IET Digital Library, 1977.

[3] Zhang, H., et al. "An efficient 3D finite element method model based on the T–A formulation for superconducting coated conductors.“ SuST, 30.2 (2016): 024005.

Domain decomposition

Model order reduction

H → Γc, ℝ2

H → Ωc, ℝ3

−

+

−

+

(*) Paper accepted to COMPUMAG 2019

Formulation: 2D Implementation

A = (0,0, Az) solved in Ω0

H = (Hx, Hy, 0) solved in Γc

A-H link via electric field E = (0,0, Ez) such that (*):

Ez = Es − EindEz = ρJzEind = 𝜕tAz

with Es = χzvs, 𝜕x,yEs = 0 :

• χz as winding density function [4]

• vs external voltage supply

• Input, if voltage driven model

• Lagrange multiplier, if current driven model

6

Ω0 : air

Γc : coil

[4] Schöps, S., et al. "Winding functions in transient magnetoquasistatic field-circuit coupled simulations." COMPEL: The

International Journal for Computation and Mathematics in Electrical and Electronic Engineering 32.6 (2013): 2063-2083.

𝛻 × μ−1𝛻 × A = 0

𝛻 × ρ𝛻 × H + μ𝜕tH = 0

i𝑠 = න

Ωc

χT 𝛻 × H dΩ

ρ =EcJc

J

Jc

n−1

x

y

z

(*) Paper accepted to COMPUMAG 2019

Outline

A. Introduction

B. Motivation

C. Formulation

D. Verification

Theoretical references

1. Critical State (Bean) Model

2. Skin Effect

3. Field Dependency

E. Validation

F. Summary and Outlook

Verification – 1. Critical State (Bean) Model

Scenario: magnetic field diffusion

n = ∞ (1e3), Jc = Jc0→ ρ = 0; ρc

Geometry: slab of infinite height, modelled

as stack of tapes

Source: boundary field Hs

Reference: Analytical solution, e.g., [5]:

Hsol x, t = x ∙ J x, tJ x, t = {0; ± Jc0(sign(H))}

8

Ωc

Ω0 : air

Ωc : conductor

Computational domain

Ω0

x

y

z

HsolHs

Ω0

Hs

Hs

t

Source term

[5] Russenschuck, S. Field computation for accelerator magnets: analytical and numerical methods for electromagnetic design

and optimization. John Wiley & Sons, 2011.

Verification – 1. Critical State (Bean) Model

Numerical Solution: magnetic field diffusion

9

Magnetic flux density and normalized current density distribution

-0.5

0

0.5

1

0 2 4 6 8 10

-0.5

0

0.5

1

0 2 4 6 8 10

-0.5

0

0.5

1

0 2 4 6 8 10

-0.5

0

0.5

1

0 2 4 6 8 10

-1.0

0.0

1.0

0 2 4 6 8 10

-1.0

0.0

1.0

0 2 4 6 8 10

-1.0

0.0

1.0

0 2 4 6 8 10

-1.0

0.0

1.0

0 2 4 6 8 10

B (

T)

J/J

c(-

)

x (mm) x (mm) x (mm) x (mm)

Profiles consistent with theory [5]

[5] Russenschuck, S. Field computation for accelerator magnets: analytical and numerical methods for electromagnetic design

and optimization. John Wiley & Sons, 2011.

Verification – 2. Skin Effect

Scenario: magnetic field diffusion

n = 1, Jc = Jc0→ ρ = const

Geometry: bulk material, modelled as

stack of tapes

Source: boundary field Hs

Reference: Analytical solution, e.g., [6]:

Hsol = Hs 1 − ferf ξ

ferf(ξ) Gaussian error function

ξ =𝑥

2 k(t−t∗)Similarity variable

k = ρμ0−1 Magnetic diffusivity

10

Hs

t

Ωc

Ω0 : air

Ωc : conductor

Computational domain

Ω0

x

y

z

HsolHs

t∗ = 0

Source term

[6] Knoepfel, H. E., Magnetic fields: a comprehensive theoretical treatise for practical use. John Wiley & Sons, 2008.

Verification – 2. Skin Effect

Numerical Solution: magnetic field diffusion

11

Magnetic flux density distribution

in the conductive slab

Magnetic flux density distribution

at x=1 mm, as function of time: numerical

solution (sol) and analytical solution (ref)

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50

B (

T)

t (s)

Field at x = 1 mm

sol

ref

Consistent with theory [6]

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-2 0 2 4 6 8 10

B (

T)

x (mm)

Field diffusion

t=0

t=1

t=5

t=10

t=50

[6] Knoepfel, H. E., Magnetic fields: a comprehensive theoretical treatise for practical use. John Wiley & Sons, 2008.

Verification – 3. Field Dependency

Scenario: AC loss due to screening currents

1 < n < ∞, Jc = Jc0→ ρ as power law

Geometry: single tape, orthogonal to the field

Source: boundary field Hs

Reference: Analytical solution, e.g., [7]:

QAC AC loss (J/cycle), Hp penetration field

if H < Hp: QAC ∝ H4, QAC ∝ f0

if H > Hp: QAC ∝ H1

12

Ωc

Ω0 : air

Ωc : conductor

Ω0

x

y

z

Hs

QAC

Hs

t

Source term

[7] Brandt, E.H., "Superconductors of finite thickness in a perpendicular magnetic field: Strips and slabs." Physical review B 54.6 (1996): 4246.

Computational domain

Verification – 3. Field Dependency

Numerical Solution: AC loss due to screening currents

13

Ac loss per cycle, as function

of the applied field

Ac loss per cycle, as function of

frequency, for H < Hp

1.E-10

1.E-08

1.E-06

1.E-04

1.E-02

1.E+00

1.E+02

1.E+04

1.E-04 1.E-02 1.E+00 1.E+02

Q (

J/m

m3/c

ycl

e)

B (T)

AC loss / Cycle

n = 5

n = 20

n = 50

1.E-04

1.E-03

1.E-02

1.E-04 1.E-02 1.E+00 1.E+02

Q (

J/m

m3/c

ycl

e)

f (Hz)

AC loss / Cycle

n = 5

n = 20

n = 50

n = 100

Consistent with theory [7]

∝ H4 ∝ H1

∝ f0

[7] Brandt, E H., "Superconductors of finite thickness in a perpendicular magnetic field: Strips and slabs." Physical review B 54.6 (1996): 4246.

Outline

A. Introduction

B. Motivation

C. Formulation

D. Verification

E. Validation

Measurements on Feather2: Field quality assessment

4. Magnetic Field Quality

5. Screening Current Effects

F. Summary and Outlook

Ω0 : air

Ωc : coil

Ωm : iron

Measurements on Feather2

Scenario: magnetic field quality in the

Feather M2 insert dipole magnet [8]

n = 20, Jc T, B, θ

→ ρ as anisotropic power law,

derived from data @ 77 K

Resulting uncertainty in material

properties (annex slide 28)

Geometry: tapes modelled as lines

Source: measured current

Reference: measurements done by

C. Petrone (CERN)

15

[8] Van Nugteren, J., et al. "Powering of an HTS dipole insert-magnet operated standalone in helium gas between 5 and 85

K." SuST 31.6 (2018): 065002.

Feather2 magnet. Courtesy of J. Van Nugteren

x

y

z

Ωc

Ω0

Ωm

Ωc

Is

t

Source term

Computational domain

Field Quality Assessment

Pre-cycle → first magnetization

Steps of 250 A, plateaus of 120 s:

decay of inductive effects

Evaluation points {p𝑖,up, p𝑖,dn}

1. FEM simulation

2. Magnetic field quality calculation

3. Persistent screening currents contribution:

Calculation of change in magnetic field

quality (assuming screening currents as

dominant mechanism)

16

0

0.5

1

1.5

2

-1000 -500 0 500 1000 1500 2000 2500

I (k

A)

t (s)

p𝑖,up p𝑖,dn

Total magnetic flux density (T),

1 quadrant

B(T)

Results – FEM Simulation

17

Normalized current density in the coil, shown for the first quadrant

a) b) c)

Current density distribution (*) in the coil:

• same external current

• different time stepsa) b) c)

i∗

t (s)

(*) Current sharing resolved using a binary search algorithm (bisection method). Internal call for each time step

J/Jc (-) J/Jc (-) J/Jc (-)

i (A)

Validation – 4. Magnetic Field Quality

b𝑖 field multipoles, as function of the normalized staircase time

18

0

1

2

3

0.0 0.5 1.0

0

200

400

600

0.0 0.5 1.0

0

50

100

0.0 0.5 1.0

0

1

2

3

0.0 0.5 1.0

0

200

400

600

0.0 0.5 1.0

0

50

100

0.0 0.5 1.0

0

1

2

3

0.0 0.5 1.0

0

200

400

600

0.0 0.5 1.0

0

50

100

0.0 0.5 1.0

B1

(T

)b

3 (

un

its)

b5

(u

nit

s)

4.5 K 25.0 K 68.0 K

t (s) t (s) t (s)

Very good agreement with measurements

Measurement

Simulation

∆b𝑖 field multipole variations, as function of current (in kA)

-10

10

30

0 2 4 6

-10

10

30

50

0 2 4 6

-2

3

8

0 2 4 6

-10

10

30

0 2 4

-10

10

30

50

0 2 4

-2

3

8

0 2 4

-10

10

30

0 1 2

-10

10

30

50

0 1 2

-2

3

8

0 1 2

Validation – 5. Screening Currents

19

kA kA kA

Good agreement with measurements, considering uncertainty in Jc T, B, θ

B1

(T

)b

3 (

un

its)

b5

(u

nit

s)

4.5 K 25.0 K 68.0 K

Measurement

Simulation

Summary and Outlook

A-H weak formulation:

• Developed and implemented in FEM

• Scalable and fast (few hours for ~104 tapes in 2D models)

• Includes iron nonlinearity

• Excellent agreement with theory

• Consistency with Feather2 measurements

20

Thank you for your attention!

Outlook

• Get better data for Jc T, B, θ

• Include heat balance equation

• Develop field-circuit coupling interface

• Run field quality analysis and optimization

• Run quench protection studies (e.g., Feather2 in FRESCA2)

• Include the models in the STEAM framework (*)

(*) https://cern.ch/steam

22

Annex

Validation – D. Current dependency

Scenario: AC loss due to screening currents

1 < n < ∞, Jc = Jc0→ ρ as power law

Geometry: single tape

Source: External current I𝑠

Reference: previous work, e.g. [ref1]:

if Is < Ic: QAC ∝ Is3

if Is > Ic: QAC ∝ Is𝑛+1

field fully penetrated, J const

23

Is

Ωc

Ω0 : air

Ωc : conductor

Ω0

x

y

z

QAC Is

t

Source term

[ref1] Grilli, F, et al. "Computation of Losses in HTS Under the Action of Varying Magnetic Fields and Currents." IEEE

Transactions on Applied Superconductivity 24.1 (2014): 78-110.

Computational domain

Validation – D. Current dependency

Solution of the transport + screening currents problem

24

Ac loss per cycle, as function of frequency,

for I < Ic. Trends are highlighted with

dashed lines

1.E-12

1.E-09

1.E-06

1.E-03

1.E+00

1.E+03

1.E+06

1.E+00 1.E+02 1.E+04

Q (

J/m

m3/c

ycl

e)

I (A)

AC loss / Cycle

n = 5

n = 20

n = 50

Ac loss per cycle, as function of the applied

current. Trends are highlighted with dashed

lines

∝ I3

1.E-12

1.E-09

1.E-06

1.E-03

1.E+00

1.E+03

1.E+06

1.E-03 1.E-01 1.E+01 1.E+03

Q (

J/m

m3/c

ycl

e)

f (Hz)

AC loss / Cycle

I/Ic=10%

I/Ic=50%

I/Ic=75%

Ic

Ic

∝ f0

Consistent with previous research

∝ In+1

Project Overview

• Protection of HTS-based high-field accelerator magnets

• CERN

• Technische Universität Darmstadt

• Graduate School of Excellence Computational Engineering (GCE)

• Dr. Matthias Mentink (TE-MPE-PE)

• Prof. Dr. Sebastian Schöps (TU Darmstadt)

• 05.2018 – 05.2021

• Gentner Programme at CERN (grant no. 05E12CHA)

• ‘Excellence Initiative’ of the German Government

• GCE at TU Darmstadt

Magnetic field dynamics in superconductors

Type I: Meißner Effect

• Thermodynamical state, reversible

• London equation

∆B − λ−2B = 0, if B < Bc1

Type II: Abrikosov fluxons

• Flux pinning and motion, irreversible

• Power-law (phenomenological)

ρ B, T =Ec

Jc(B, T)

J

Jc(B, T)

n−1

• Faraday Law (eddy currents)

•

∆H − μρ−1𝜕tH = 0

26

Bc1

Bc2

Tc

B

T

Type I

Type II

B-T Diagram

Formulations

Conductivity σ − based:

• 𝐀: magnetic vector potential

ill-conditioned mass-conductivity matrix (∞ condition number)

Resistivity ρ − based:

• 𝐇: magnetic field strength

ρ ≠ 0 everywhere, unphysical eddy currents , computationally inefficient

• 𝐓-𝛀: current vector potential-scalar magnetic potential

cohomology basis functions for net currents in multiply connected domains

Mixed fields (from literature)

• 𝐀-𝐇: magnetic vector potential + magnetic field strength

Developed for 2D rotating machinery. Current driven, no external coupling

• 𝐓-𝐀: current vector potential + magnetic vector potential

Current driven, no external coupling, gauge and interface conditions not given,

Potentially inconsistent voltage balance

27

Modelling of Jc (T, B, θ)

• Jc data from Fujikura [9]

• Fit from [10]

• Jc available only for T = 77 K → Lift factor, calibrated with the measured Ic• Lift factor assumed consistent with θ = 30°

28

[9] Fujikura, Y-based high temperature superconductor. Company leaflet at ICEC (2014).

[10] Fleiter, J., and A. Ballarino. "Parameterization of the critical surface of REBCO conductors from Fujikura." CERN internal

note, EDMS 1426239 (2014).`

Measured Ic(T, B)Jc(T, B, θ) fit in the model

θ

n B

Pic. From [6]

Anisotropy included in the model, but uncertainty on material properties

Formulation - Mixed potentials

𝐀 − 𝐇 formulation, weak form:

1. Ω0 → Ampere-Maxwell Law

2. Ωc → Faraday Law

3. Ωc → Constraint on transport current

1.

2.

3.

29

𝐌𝜈 reluctance

𝐐 current

𝐌𝜌 resistance

𝐌𝜇 flux

𝐗 voltage

Advantages

Mρ ρ → finite conditon number

A → magnetostatic problem

Drawback

Weak form to be implemented

A → Ω0, ℝ3

H → Ωc, ℝ3

𝐯𝐬

𝐢𝐬

+

−

High-Temperature Superconductors

Cuprate compounds (CuO2) doped with rare earth elements (La, Bi-Sr-Ca, Y-Ga-Ba …)

• Higher Tc and Bc0 respect to the traditional LTS competitors

• Higher performance comes with higher prices! $HTS ≈ 1𝑒2 $LTS.. But in the early 2000s it was ≈ 1𝑒3

30

0

2

4

6

8

10

0 5 10 15 20 25 30 35 40

(kA

mm

-2)

(K)

J_eng(T) @ 5T

Nb-Ti

Nb3Sn

ReBCO

0

2

4

6

8

10

0 5 10 15 20 25 30 35 40

(kA

mm

-2)

(T)

J_eng(B) @ 1.9K

Nb-Ti

Nb3Sn

ReBCO

Tapes and Cables

Tapes

• ReBCO - Rare Earth Barium Copper Oxide tape

• Batches of ~102 m and beyond

• Cost driven by production process

Features

• Multi-layer, multi material

• Aspect ratio ~102 (tape), ~103 (HTS layer)

• HTS as anisotropic, nonlinear mono-filament Jc(B, T)

• AC losses: eddy currents

Cables

• Roebel geometry (1912)

• “Coil-able”, bended on the long edge

• Fully transposed: even current distribution

• Aligned-coil concept against AC losses

31

■ Copper

■ ReBCO

■Substrate

Source: CDS. Coiled Roebel cable (Henry

Barnard, CERN).

μm

mm

Quench Protection

Facts [1]

• Top ≥ 10 K in He gas, Tcrit = 93 K

• Cp Tcrit ~ 102 Cp Top

• Thermal load dominated by Jeddy

• “Smooth” quench transition (Power law E(J) , 𝑛HTS ≈ 20, cf. 𝑛LTS ≈ 40)

w.r.t. to LTS:

• Slower quench propagation, harder detection, potentially irreversible

• Available quench protection systems potentially inadequate, due to massive MQE

32

Quench propagation 𝑣q 𝑣q, HTS ≤ 1𝑒−2 𝑣q, LTS

Quench resistance 𝑅q 𝑅q, HTS ≪ 𝑅q, LTS

Hotspot temperature Ths Ths, HTS ≫ Ths, LTS

Minimum quench energy MQE MQEHTS ≥ 1𝑒3 MQELTS

[1] Van Nugteren, Jeroen. High temperature superconductor accelerator magnets. Diss. Twente U., Enschede, Enschede,

2016.

Accurate prediction of persistent currents crucial

for both, field quality and quench issues

Formulation - Crosscheck

Reference model [*] based on the H formulation,

provided by http://www.htsmodelling.com

2D model of a Single HTS tape in self-field

Source: Is = I0sin(2πft), I0 = 2Icrit t ∈ [0; 1]

33

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

0.0 0.5 1.0 1.5 2.0 2.5

(J)

Normalized current (-)

Ohmic loss per cycle

1 Hz

10 Hz

100 Hz

1000 Hz

Markers: H (ref)

Lines: A − H

J

Jcrit

[*] V. M. Rodriguez-Zermeno et al., "Towards Faster FEM Simulation of Thin Film Superconductors: A Multiscale

Approach," in IEEE Transactions on Applied Superconductivity, vol. 21, no. 3, pp. 3273-3276, June 2011.

Formulation - Benchmark

Forecasts on expected computational time:

34

1 tape

5 tapes

10 tapes

Same physics…

Increased computational cost

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1 10 100 1000 10000

(ho

urs

)

tapes (-)

Computational time

H

A-H

A-H opti

Optimization of:

Mesh, solver, numerical implementation

8 h

Formulation - PEC Limit Behavior

35

[1] Knoepfel, Heinz E. Magnetic fields: a comprehensive theoretical treatise for practical use. John Wiley & Sons, 2008.

1.E-16

1.E-12

1.E-08

1.E-04

1.E+00

1.E-04 1.E-02 1.E+00 1.E+02 1.E+04

J

(Hz)

AC loss / Cycle

1.E-5

1.E-10

1.E-15

1.E-20

1.E-25-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

m/m

Max

(-)

B/Bmax (-)

m(B) - normalized

1.00E-05

1.00E-15

1.00E-20

∝ f1

∝ f−1

Field multipoles without eddy currents

Staircase Scenario at 4.5 K

• “Eddy” considers the HTS tape dynamics

• “No Eddy” assumes a homogeneous current density in the tapes

36

0

1

2

3

0.00 0.50 1.00

B1

meas

eddy

no eddy0

200

400

600

0.00 0.50 1.00

b3

meas

eddy

no eddy

0

20

40

60

80

0.00 0.50 1.00

b5

meas

eddy

no eddy0

5

10

15

0.00 0.50 1.00

b7

meas

eddy

no eddy