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Numerical models for AC loss calculation in large-scale applications of HTS coated conductors

Loïc Quéval, Víctor M. R. Zermeño, Francesco Grilli

arXiv:1509.05560v1cond-mat.supr-con

TL;DR

Models developed for individual HTS tapes or small assemblies may not scale to devices containing hundreds or thousands of tapes. This paper compares homogenized and multi-scale models, reporting below 1% loss error at 50–70× speedup for homogenization and up to 1600× speedup with good accuracy for multi-scale modeling at 28 A.

  • Problem

    Existing HTS tape models work for individual tapes or small assemblies, but their applicability to devices with hundreds or thousands of tapes remains questionable.

    Source: 1) Modeling strategy · p. 5, 3) Discussion · p. 8

  • Method

    The paper compares a homogenized coil model using anisotropic bulks and equal-turn-current constraints with a multi-scale approach that models selected tapes and interpolates to others.

    Source: 1) Modeling strategy · p. 5, 3) Discussion · p. 7

  • Results

    Below 1% loss error at 50–70× speedup was reported for homogenization, while multi-scale modeling reached up to 1600× speedup with good accuracy at 28 A.

    Source: 1) Modeling strategy · p. 5, 3) Discussion · p. 7

  • Takeaways & Limitations

    For large HTS coils, homogenized modeling offers the best AC-loss estimates, while multi-scale modeling has low memory requirements and supports parallel tape submodels.

    Source: 3) Discussion · pp. 7–8

    The homogenized model applies only to tape-like conductors with thin superconducting layers and cannot account for parallel-field magnetization currents or magnetic materials.

    Source: 3) Discussion · p. 6

Abstract

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Numerical models are powerful tools to predict the electromagnetic behavior of superconductors. In recent years, a variety of models have been successfully developed to simulate high-temperature-superconducting (HTS) coated conductor tapes. While the models work well for the simulation of individual tapes or relatively small assemblies, their direct applicability to devices involving hundreds or thousands of tapes, as for example coils used in electrical machines, is questionable. Indeed the simulation time and memory requirement can quickly become prohibitive. In this article, we develop and compare two different models for simulating realistic HTS devices composed of a large number of tapes: 1) the homogenized model simulates the coil using an equivalent anisotropic homogeneous bulk with specifically developed current constraints to account for the fact that each turn carries the same current; 2) the multi-scale model parallelizes and reduces the computational problem by simulating only several individual tapes at significant positions of the coil's cross-section using appropriate boundary conditions to account for the field generated by the neighboring turns. Both methods are used to simulate a coil made of 2000 tapes, and compared against the widely used H-formulation finite element model that includes all the tapes. Both approaches allow speeding-up simulations of large number of HTS tapes by 1-3 orders of magnitudes, while keeping a good accuracy of the results. Such models could be used to design and optimize large-scale HTS devices.

1) H-formulation

The section describes a 2D finite-element H-formulation model for superconductors, using a power-law resistivity and an anisotropic critical-current description. Integral constraints impose conductor currents, while outer boundary conditions apply external magnetic fields.

1) H-formulation

  • The model uses a 2D finite-element H-formulation implemented in COMSOL Multiphysics 4.3a PDE mode, with superconducting resistivity modeled by a power law.The model uses the power-law parameters J_c, E_c, and n.Source: 1) H-formulation · p. 3
  • An anisotropic Kim-like model describes critical-current density dependence on magnetic field, and its angle dependence can also be included.The model is described as a reasonable account of HTS coated-conductor anisotropy without artificial pinning.Source: 1) H-formulation · p. 3
  • Integral constraints impose an explicit transport current in each conductor, while outer boundary conditions apply an external magnetic field.Source: 1) H-formulation · p. 3
  • A 4 mm YBCO coated-conductor tape is parameterized with a self-field critical current of 99.2 A [Zermeno2013b].The simulation parameters are summarized in Table I.Source: 1) H-formulation · p. 3, Table I · p. 3

2) Test case

The test case is a symmetric quarter-coil model of a 2,000-tape coil carrying 50 Hz AC transport current with no applied external field. Two current amplitudes probe partial and full penetration, and hysteretic losses are averaged over the second half of one simulated period.

2) Test case

  • The modeled coil contains 10 pancake coils of 200 turns each; symmetry reduces the 2D planar model to one quarter, or 100×5 tapes.The tape array is indexed from the bottom-left corner by (i,j); its layout and numbering appear in Fig. 2, and layout parameters are summarized in Table II.Source: 2) Test case · p. 4, Table II · p. 4
  • At 11 A, central tapes in the outermost column (i=1,j=5) have no virgin regions, while at 28 A they are fully penetrated by transport current.At 11 A, these tapes contain only transport or magnetization currents; at 28 A, magnetization currents have been displaced, and the coil critical current is defined as 28 A.Source: 2) Test case · p. 4
  • Average hysteretic losses per cycle, in W/m, are computed from the second half of one simulated period.T denotes the cycle period and Ω the superconductor domain.Source: 2) Test case · p. 4

3) Reference model

The reference model uses the H-formulation to simulate all tapes in the coil, with the superconducting layers modeled at their real thickness. The passage describes its geometry and mesh.

3) Reference model

  • The reference model is an H-formulation simulation of every tape, a formulation previously verified against experiments for single conductors and coils [Grilli2007a, Grilli2007b, Nguyen2009, Nguyen2010, Zhang2012, Wang2015, Celebi2015].The paper states that this supports using the model to estimate real losses.Source: 3) Reference model · p. 4
  • The model includes surrounding air and superconducting layers at their real thickness, while omitting other tape layers.Source: 3) Reference model · p. 4
  • The superconducting layer uses a mapped mesh with 100 width elements distributed symmetrically in an arithmetic sequence and one element through its thickness.A mapped mesh is also used between the tapes; Fig. 3 shows one array cell.Source: 3) Reference model · p. 4

4) Results

At an 11 A transport current, magnetic flux is shaped by magnetization currents that hinder field penetration, while hysteretic losses vary strongly across rows but remain similar within columns.

  • Within a row, tape losses span almost 3 orders of magnitude, whereas losses within a column are very similar.The loss comparison is for the simulated coil's tapes.Source: 4) Results · p. 5
  • At the 11 A peak-current instant (t = 0.015 s), magnetization currents prevent magnetic-field penetration, particularly at low transport currents.The current-density and magnetic-flux-density distributions are shown for this case; tape thickness was expanded vertically for visualization.Source: 4) Results · p. 5

1) Modeling strategy

The homogenized model extends earlier HTS tape models to large arrays using anisotropic homogeneous bulks and current constraints; its losses closely match the reference model with substantial speedup.

1) Modeling strategy

  • The model represents the coil as equivalent anisotropic homogeneous bulks with current constraints ensuring each turn carries the same current.Taking advantage of the vanishing edge effect toward the stack center permits a very coarse mesh and reduces computing time.Source: 1) Modeling strategy · p. 5
  • Building on prior models for stacks of up to 64 tapes and 50-turn racetrack coils, this work extends the homogenized approach to large tape arrays.Unlike other homogenized models [Zhang2014], it makes no a priori assumptions about current distribution and permits overcritical local currents.Source: 1) Modeling strategy · p. 5
  • For both transport-current cases, homogenized-model loss error versus the reference model is less than 1%, with a speed-up factor of 50–70.The homogenized model reproduces the overall current and magnetic-field distributions, although its field distribution is smoother because homogenization washes out the tape layout.Source: 1) Modeling strategy · p. 5

3) Discussion

The discussion presents homogenized and multi-scale models as complementary ways to reduce the cost of simulating large HTS devices while estimating AC losses in good agreement with the full-device reference. Their tradeoffs concern accuracy, computational speed, and the types of conductors they can represent.

Discussion

  • The homogenized model closely matches local losses across the coil, despite values spanning almost 3 orders of magnitude, and is stable and straightforward to implement in commercial finite-element software.Its coarser mesh at stack centers reduces degrees of freedom and computing time; it slightly overestimates losses in the first stack.Source: 3) Discussion · pp. 5–6
  • The homogenized model is limited to thin-layer tape-like conductors without magnetic materials, whereas the multi-scale model can in principle represent multifilamentary conductors and tapes or wires with magnetic materials.The multi-scale model has low tape-submodel setup effort, though transferring the coupling-boundary field requires data processing outside the COMSOL interface.Source: 3) Discussion · pp. 6–8
  • The multi-scale model estimates a background field with a coil model, then computes losses for selected individual tapes and interpolates the remaining values; its two submodels are coupled through boundary fields.This article extends the approach with field-dependent critical current density and a refined background-field estimate that partially accounts for neighboring superconducting tapes.Source: 3) Discussion · p. 6
  • At 11 A, the J0 infinite-array approach yields a 6% loss error, while using reference-model J0 predicts losses accurately.Uniform J0 poorly reproduces the reference field and underestimates hysteretic losses; the infinite-array approximation improves agreement, especially near the coil center, but adds computation and implementation cost.Source: 3) Discussion · pp. 6–7
  • At 28 A, uniform J0 predicts losses accurately with speedup up to 1600 because the reference current distribution is nearly uniform.Five logarithmically distributed tape samples per stack differ from computing all tapes by about 1%, and the multi-scale approach supports parallel computation with low memory requirements.Source: 3) Discussion · p. 7
  • 50–70 times faster than the reference, the homogenized model gives the best AC-loss estimates, while both approaches reduce computation and retain good agreement with full-device results.The multi-scale model is fastest when parallelized; the homogenized model offers a favorable balance of computing and setup time.Source: 3) Discussion · p. 8
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