calculating screening current effects in rebco devices and coils · 2019. 5. 3. · finite...
TRANSCRIPT
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