This work is supported by:

Graduate School CE within the Centre for Computational Engineering at Technische Universität Darmstadt.

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

Lorenzo Bortot

Analysis and Simulation of Magneto-Thermal Dynamics

in High-Temperature Superconducting Magnets

19th Gentner Day, 28 April 2021, CERN

https://indico.cern.ch/event/1002599/

Supervisors:

Prof. Dr. rer. nat. Sebastian Schöps (TU Darmstadt)

Dr. Matthias Mentink (CERN)

Acknowledgments

I.C. Garcia, H. de Gersem

A. Verweij, J. van Nugteren, C. Petrone, G. Deferne, T.

Koettig, J.C. Perez, F.O. Pincot, G. de Rijk, S. Russenschuck,

G. Kirby, M. Pentella, M. Maciejewski

B. Auchmann

2

Contents from research activity over the past three years.

The Gentner program of the German

Federal Ministry of Education and Research

(grant no. 05E12CHA).

Outline

I. Dynamic Phenomena in High-Temperature Superconductors (HTS)

II. Coupled Field Formulation

III. Numerical Analysis

IV. Experimental Work

V. Summary and Outlook

3

HIGH TEMPERATURE SUPERCONDUCTORS

Copper oxides (CuO2) doped with rare earths (La, Bi-Sr-Ca, Y-Ga-Ba etc.)

Higher critical temperature and upper critical field with respect to the traditional

low-temperature superconductors (LTS), such as Nb-Ti or Nb3Sn

4

0

2

4

6

8

10

0 5 10 15 20 25 30 35 40

(kA

mm

-2)

Temperature (K)

J(T) @ 5T

Nb-TiNb3SnReBCO

0

2

4

6

8

10

0 5 10 15 20 25 30 35 40

(kA

mm

-2)

Magnetic flux density (T)

J(B) @ 1.9K

Nb-Ti

Nb3Sn

ReBCO

High-thermal load applications High-field applications

SCREENING CURRENTS

HTS tape in a time-dependent magnetic field 𝜕t𝐁:

𝜕t𝐁 Screening currents 𝐉screenρ → 0 Persistent magnetization 𝐁screen

Large filament size (5-12 mm) large 𝐁screen

Magnetic field quality and thermal behavior, as principal Joule loss contribution

Inhomogeneous current density distribution Detrimental impact on field quality

5

■ Copper

■ ReBCO

■ Substrate

~mm

~μm

ρ → 0

𝐉screen

𝜕t𝐁 𝐁screen

QUENCH

Local transition from superconducting to normal conducting state

Energy dissipated in the resistive zone

Potentially irreversible effects for high energy-density devices (accelerator magnets)!

HTS characteristics:

• low heat diffusivity

• low Ԧ𝑣quench, small resistive zone, difficult to detect

• high Thot−spot, potential damage in short time (tens of ms)

6

HTS tape

T

x

Thot−spot

Ԧ𝑣quench

resistive zone

DYNAMIC EFFECTS IN ACCELERATORS

Circuit (e.g. LHC mains)

7 km long, 154 Magnets

Topological: Switches, thyristors, diodes

Electrical: Propagative phenomena,

dynamic impedances (magnets) ,EM

crosstalk

• …

Magnets (e.g. MB magnet)

~102 coil turns, ~104 strands

Magnetic: magnetization, coupling currents, EM crosstalk…

Thermal: magnetization, coupling losses, quenches, coolant…

Mechanic: Lorentz forces, thermal strain…

7

Field-Circuit Coupling of multi-physics, multi-rate and multi-scale Systems!

FORMULATION

Magnet domain decomposition

ΩH = ΩH,s ∪ ΩH,c source region (coils):

• ΩH,s superconductors (𝜎 → +∞)

• ΩH,c normal conductors

ΩA = ΩA,c ∪ ΩA,i passive region (Iron, air):

• ΩA,c normal conductors

• ΩA,i insulators (𝜌 → +∞)

Field equations

8

𝛻 × ρ𝛻 × 𝐇 + μ𝜕t𝐇 + 𝛻 × 𝛘 us = 0 𝐇 formulation in ΩH

𝛻 × ν 𝛻 × 𝐀⋆ + σ𝜕t𝐀⋆ = 0 𝐀⋆ formulation in ΩA

ρmCp𝜕tT − 𝛻 ⋅ k𝛻T − 𝐉 ⋅ ρ𝐉 = 0 Heat balance equation T in Ω

ΩH𝛘 ⋅ 𝛻 × 𝐇dΩ = is Current constraint

𝛘 = −𝛻ξ, ξ ∶ 𝛻 ⋅ σ𝛻ξ = 0 Voltage distribution function

+

ΩA,c

𝐧

Ω ∈ ℝ3

isus−

ΩH,c

𝐧

ΓHA

ΓE

ΓJ

ΩH,s

ΓA

ΩA,i

SEMI-DISCRETE PROBLEM

Semi-discrete representation for numerical implementation (FE method)

Finite material properties, 𝜌 in (super)conductors, 𝜎 in insulators Solver stability

Electric ports ΓJ and ΓE as connections with the external circuit

𝐮𝐬 = 𝑓(𝐢𝐬) magnet as one-port component

9

Kν+Mσd

dt−𝐐 𝟎 𝟎

𝐐𝐓 K𝝆+M𝝁d

dt−𝚾 𝟎

𝟎 𝚾𝐓 𝟎 𝟎

𝟎 𝟎 𝟎 Kκ+Mρd

dt

a

h

us

t

𝟎

𝟎

is

q ⋅

=

𝐀⋆ form

𝐇 form

Field coupling

Heat Balance

Circuit coupling

⋅

𝐮𝐬 ≈ 𝐑𝐢𝐬 t + 𝐋d

dt𝐢𝐬 t

COSIMULATION FRAMEWORK

10

Middle Layer

• Communication bus,

common interface

• Standard set of rules for data

exchange

Bottom Layer

• Interpreter (framework vs.

simulation tools)

• Tool-oriented (single adapter

for multi-models)

Top Layer

• Hierarchical co-simulation

algorithm

• Waveform relaxation, Gauss-

Seidel scheme

Lean, Modular, Expandable

Waveform Relaxation, Gauss Siedel

Communication Bus

Tool Adapter𝑗

API𝑗

Tool𝑗CoupledProblem

Tool Adapter𝑘

API𝑘

Tool𝑘

Hierarchical Co-Simulation

User Input Output

Model1

Model𝑛

Model𝑚

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

(*)

ANALYSIS: FIELD QUALITY (1/3)

11

2D FEM model

Iron

Wing deckCentral deck

Feather-M2 HTS insert dipole magnet

Main features:

• Aligned coil concept

• Coil made of two central and wing decks

• Roebel cable, fully transposed tapes

2D model:

• Layer jumps 4-quadrant

• 48 turns, 720 tapes

Cable cross section

Photos courtesy of J. Van Nugteren

ANALYSIS: FIELD QUALITY (2/3)

Computational time

0.5 h (*) for about 120k DoF

Current density plots: same source current, different time steps

12

Normalized current density in the coil (first quadrant)

a) b) c)

(*) CPU: Intel Core i7-3770 @ 3.40GHz. RAM: 32 Gb. OS: Win 10

ANALYSIS: FIELD QUALITY (3/3)

Magnetic field quality

Multipole expansion series as function of current, for different temperatures

13

Very good agreement with measurements

HTS solenoid protected by quench heaters

Electro-magneto-thermal coupling between the solenoid and the quench heaters!

Solenoid supply Heaters supply

ANALYSIS: QUENCH (1/3)

14

(*) Material properties available at https://gitlab.cern.ch/steam/steam-material-library

(**) Circuit parameters in appendix

Iron core

Holder

Coil

Insulation

Heater strips

Rendering of the HTS solenoid (*) Electrical layout (**)

0

20

40

60

80

100

0

500

1000

1500

0 0.05 0.1 0.15I_

str

ip(A

)

I_co

il(A

)

time (s)

Coil cosim Coil mono Heater

ANALYSIS: QUENCH (2/3)

Circuital currents Peak temperature

15

Current decay in the coil (left) and current

discharge in the heater strips (right)

Induced quench

100

200

300

400

0 5 10 15 20

T_hots

pot(K

)

Quench detection time (ms)

Max. Temperature

Max. detection time

Adiabatic hotspot temperature in the

coil, as a function of the quench

detection time

≈ 15 msTime windows

ANALYSIS: QUENCH (3/3)

Temperature in the coil cross section

16

100 ms 150 ms50 ms20 ms 30 ms10 ms

Temperature distribution in the superconducting coil, as a function of time

Tem

pera

ture

(K)

Quench Heat diffusionHeater strips

EXPERIMENT (1/3)

Working principle

HTS tape geometry:

• Thin superconducting layer,≈ 1 μm

• High aspect ratio, ≈ 1 × 103

Thin shell approximation

HALO

Harmonics-Absorbing Layered Object

Key features:

1. Magnetic field lines shaped by

persistent screening currents

2. Cancellation only of undesired field

components (static and dynamic)

3. Passive and self-regulating

17

HTS ReBCO

σ → +∞

𝜕t𝐁𝜕t𝐁∥

𝜕t𝐁⊥

𝐁screen

𝜕t𝐁⊥ canceled

𝜕t𝐁∥ unchanged

[T]

Quasi-perfect electric wallSelective field-cancellation!

EXPERIMENT (2/3)

Proof of concept design Experimental setup

18

[T]

2

3

1

4 4

5

1. Iron yoke

2. Coil

3. Iron bar (field error)

4. HTS screen

5. Rotating probe

3

77 K

296 K

Experimental setup at cold

HTS screen Screen holder

EXPERIMENT (3/3)

Experimental and numerical result

Magnetic field quality (total harmonic

distortion) in the magnet aperture, without

and with HALO

19

0

10

20

30

40

50

60

70

1 2 3 4Fi

eld

err

or

[un

its]

Scenarios

Magnetic field quality

MEAS 300K SIM 300K MEAS 77K SIM 77K

2~4 factor in field quality improvement

(*)

(*) L. Bortot et al., https://arxiv.org/abs/2103.14354 preprint.

Field quality for the four scenarios

1)

2)

3)

4)

No HALO HALO

CONCLUSIONS AND OUTLOOK

Conclusions

1. Transient effects in accelerators: multi physics, scale and rate phenomena

2. Specialized models + Co-simulation (divide et impera)

3. A-H coupled field formulation Field quality and quench protection simulations

4. HALO as a potential game-changer for magnet design

Outlook

1. Field quality optimization

2. Quench protection studies

3. HALO demonstrator in a full-scale magnet

20

Thank you for your attention!Contact: lorenzo.bortot@cern.ch

Ohmic loss distribution

in a HTS solenoid

W

m2

normalized induced

current in a HTS bulk

Persistent magnetization

in a block-coil dipole

21

BACKUP

OUTPUT

Publications

1. L. Bortot, M. Mentink, C. Petrone, J. Van Nugteren, G. Deferne, T. Koettig, G. Kirby, M. Pentella,J. Perez, F. Pincot, et al. “Proof of Concept of High-

Temperature Superconducting Screens for Magnetic Field-Error Cancellation in Accelerator Magnets.” arXiv preprint arXiv:2103.14354. Apr.2021.url:https://arxiv.org/pdf/2103.14354.pdf.

2. [A1]L. Bortot, M. Mentink, C. Petrone, J. V. Nugteren, G. Kirby, M. Pentella, A. Verweij, and S. Schöps. “Numerical Analysis of the Screening Current-Induced

Magnetic Field in the HTS Insert Dipole Magnet Feather-M2.1-2”. In:Superconductor Science and Technology33.12 (Oct. 2020). issn: 0953-2048.doi:10.1088/1361-6668/abbb17. arXiv:2005.09467.

3. [A2]L. Bortot, B. Auchmann, I. Cortes Garcia, H. De Gersem, M. Maciejewski, M. Mentink, S. Schöps, J. Van Nugteren, and A. Verweij. “A Coupled A-H

Formulation for Magneto-Thermal Transients in High-Temperature Superconducting Magnets”. In: IEEE Transactions on Applied Superconductivity30.5 (Aug.2020). issn: 1051-8223.doi:10.1109/TASC.2020.2969476. arXiv:1909.03312.

4. [A11]L. Bortot, B. Auchmann, M. Maciejewski, M. Prioli, S. Schöps, I. Cortes Garcia, and A. Verweij. “A 2-DFinite-element Model for Electrothermal Transients inAccelerator Magnets”. In: IEEE Transactions on Magnetics54.3 (Mar. 2018), pp. 1–4.issn: 0018-9464.doi:10.1109/TMAG.2017.2748390.arXiv:1710.01187.

5. [A12]L. Bortot, M. Maciejewski, M. Prioli, A. M. Fernandez Navarro, J. B. Ghini, B. Auchmann, and A. Verweij. “A Consistent Simulation of Electro-thermalTransients in Accelerator Circuits”. In: IEEE Transactions on Applied Superconductivity27.4 (June 2017).issn: 1051-8223.doi:10.1109/TASC.2016.2639585.

6. [A9]L. Bortot, B. Auchmann, I. Cortes Garcia, A. M. Fernando Navarro, M. Maciejewski, M. Mentink, M. Prioli, E. Ravaioli, S. Schöps, and A. Verweij. “STEAM:

A Hierarchical Co-Simulation Framework for Superconducting Accelerator Magnet Circuits”. In: IEEE Transactions on Applied Superconductivity28.3 (Apr.2018). issn: 1051-8223.doi:10.1109/TASC.2017.2787665.

Awards

22

IEEE ASC Viktor Keilin Memorial Prize - Large Scale Applications (2020)

Best student paper for innovations in magnet science and technology

IEEE CSC Graduate Study Fellowship (2019)

Recognition of academic excellence and achievement in the area of applied superconductivity

(A) SIMULATION SETUP

23

Rcrow + Rm discharge the

current in the solenoid, limiting

the peak temperature in the

superconducting coil

Rm determined by the quench

dynamics within the solenoid

A. B.

C.

0t (ms)

5 10 30

Quench

detected

A.

Crowbar

resistance

inserted

B.

Quench

heaters

powered

C.

Solenoid

quenchedSolenoid at

nominal conditions

…

Quench

(A) TIME STEPPING

24

Maximum time step for each solver, for each time window

Crowbar

Quench

Quench heaters

Small steps required for:

1. Quench heater powering

2. Quench transition

HTS SOLENOID: CIRCUIT PARAMETERS

25

CONVERGENCE RATE

26

Definitions, at iteration 𝑖:

• 𝑥𝑖 signal (current in the magnet)

• 𝜀𝑎𝑏𝑠 𝜀𝑟𝑒𝑙 absolute & relative error

• 𝜀𝑖 convergence error

• Fconv convergence flag

Enforcement of at least three

iterations per time window

𝜀𝑖 = 1

Quench

𝜀𝑎𝑏𝑠 = 0.001𝜀𝑟𝑒𝑙 = 0.25

Quench as abrupt change in resistivity

High influence on the solenoid current

More iterations needed!

𝜀𝑖 = max𝑥𝑖 − 𝑥𝑖−1

𝜀𝑎𝑏𝑠 + 𝑥𝑖 𝜀𝑟𝑒𝑙, 𝑖 ≥ 2

Fconv = ቊ0, if 𝑖 < 2𝜀𝑖 < 1, if 𝑖 ≥ 2

DYNAMIC EFFECTS IN ACCELERATORS (2/2)

Simulation challenges

• Complex components

• Cutting-edge technologies

• Physical size of devices

• Extreme ambient conditions

• Extremely fast phenomena

(e.g. beam dynamics)

• …

• Mutual electro-thermo-dynamic

interaction between circuit, power

converters, magnets and protection

systems

27

Thermal and fluid dynamics of coolant

Electrodynamics of circuits,

switches, crowbars,

energy extractors

Time (s)

Scale (m)

10−6 10−3 100 103

10−6

10−3

100

103

Thermal dynamics

of diodes

Electro-thermal dynamics

mechanical dynamics

of magnets

Thermal

dynamics of

quench

phenomena

Power

converters

control

multi-physics multi-scale and multi-rate

FIELD-CIRCUIT COUPLING

Interface derivation

Schur complement applied in the semi-discrete problem

Assumption Kν+𝝀Mσ positive-definite (true for gauged 𝐀⋆) Invertible

Interface derived as optimal Schwarz transmission condition for linear systems

𝐙 jω = 𝐗T 𝐊ρ + jω𝐌μ + 𝐐T 𝐊ν + jω𝐌σ −1 𝐐 −1𝐗 −1

Approximation of derivatives in time domain (e.g. Taylor expansion series)

𝐙 jω ≈ 𝐙 0 + jω ቤ𝜕𝐙 jω

𝜕jωjω=0

𝐮𝐬 ≈ 𝐑𝐢𝐬 t + 𝐋d

dt𝐢𝐬 t

28

resistive term 𝐇-flux 𝐀⋆-flux eddy currents term

Multirate systems

• Fast System small steps, slow System large steps

• Different time steps for efficiency

• Different solvers since different nature of the problem

TIME-DOMAIN COUPLING

29

tendt t

tendttend

t

tend

S1

S2

One-way coupling / Parameter extraction Strong coupling

Weak coupling Waveform relaxation / Dynamic iteration

S1

S2

S1

S2

S1

S2