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Real-time simulation of large-scale HTS systems: multi-scale and homogeneous models using T-A formulation
Edgar Berrospe-Juarez, Victor M R Zermeno, Frederic Trillaud, Francesco Grilli
TL;DR
Large-scale HTS simulation is constrained by the limited realism of analytical models and the computational cost of FEM. The paper adapts multi-scale and homogenization strategies to the T-A formulation, achieving accurate real-time simulation for suitable slow cycles. The models reduce computational demands but remain limited by the T-A one-dimensional approximation and reduced-model accuracy.
Problem
Large-scale HTS systems are difficult to model because analytical models are limited and FEM can require excessive memory and computation time.
Method
The paper combines the T-A formulation with adapted multi-scale and homogenization strategies to estimate current density and hysteresis losses in large-scale HTS systems.
Results
0.28% and 2.23% losses relative error are obtained for the T-A multi-scale and homogeneous models, respectively, relative to the T-A full model.
Takeaways & Limitations
The approaches enable real-time simulation and analysis of large-scale 2G HTS systems under slow charge/discharge cycles.
Takeaways & Limitations
The approaches require a meaningful 1D approximation with negligible parallel-field influence and provide lower accuracy than full models.
Abstract
from arXiv · showhide
The emergence of second-generation high temperature superconducting tapes has favored the development of large-scale superconductor systems. The mathematical models capable of estimating electromagnetic quantities in superconductors have evolved from simple analytical models to complex numerical models. The available analytical models are limited to the analysis of single wires or infinite arrays that, in general, do not represent real devices in real applications. The numerical models based on finite element method using the H formulation of the Maxwells equations are useful for the analysis of medium-size systems, but their application in large-scale systems is problematic due to the excessive computational cost in terms of memory and computation time. Then it is necessary to devise new strategies to make the computation more efficient. The homogenization and the multi-scale methods have successfully simplified the description of the systems allowing the study of large-scale systems. Also, efficient calculations have been achieved using the T-A formulation. In the present work, we propose a series of adaptations to the multi-scale and homogenization methods so that they can be efficiently used in conjunction with the T-A formulation to compute the distribution of current density and hysteresis losses in the superconducting layer of superconducting tapes. The computation time and the amount of memory are substantially reduced up to a point that it is possible to achieve real-time simulations of HTS large-scale systems under slow ramping cycles of practical importance on personal computers.
1. Introduction
Large-scale HTS systems require models that reduce the computational burden of many-turn devices while retaining accurate current-density and hysteresis-loss estimates. This work adapts multi-scale and homogenization strategies to the T-A formulation for real-time simulation.
- Motivation: 2G HTS tapes enable large superconducting systems, but hysteresis losses make their design and operation thermally challenging.The paper focuses on hysteresis losses because extracting the resulting heat is difficult.
- Existing models: Analytical models are generally limited to single conductors or special conductor stacks and may disagree with experimental measurements.They also do not generally capture electromagnetic interaction between conductors.
- Existing models: FEM, especially the H formulation, supports larger systems but can become impractical because of excessive memory and computation-time requirements.The paper presents the T-A formulation as an efficient approach for simulating thin superconducting layers.
- Efficiency strategies: Homogenization represents tape stacks as homogeneous anisotropic bulks, while multi-scale modeling solves a reduced subset of analyzed tapes using the field from the full system.The multi-scale method applies the resulting magnetic field as a boundary condition for computing current density in the analyzed tapes.
- This work: The proposed T-A multi-scale and homogeneous strategies target 1D conductors such as 2G HTS tapes and estimate full-system current density from analyzed tapes or a bulk.The paper reports that these strategies can achieve real-time simulations of practical slow charge/discharge cycles.
- Paper structure: The paper presents the T-A formulation, validates it on a 2,000-turn racetrack coil, and then evaluates the proposed multi-scale and homogeneous models.The validation compares the T-A full model with an H-formulation model and examines element-order choices.
2. T-A formulation
The T-A formulation models thin HTS layers with a current vector potential in the layers and a magnetic vector potential across the bounded universe. It derives transport-current boundary conditions from the one-dimensional layer representation, whose applicability is restricted by field orientation and aspect ratio.
- Core assumption: The T-A formulation assumes thin superconducting layers can be represented as one-dimensional elements, enabling real-time modeling of large-scale systems.Transport current is imposed by modifying the potentials at the layer extremities.
- A formulation: The A formulation uses magnetic permeability μ and current density J, with magnetic flux density defined by B = ∇×A.This equation governs the magnetic field calculation across the modeled universe.
- T formulation: The T formulation uses resistivity ρ and defines current density through J = ∇×T.In two dimensions, the thin-layer approximation reduces T to a single scalar component.
- Boundary conditions: Integrating current density over the superconducting cross-section supplies boundary conditions for T that enforce the tape transport current.The resulting surface current density is K = J·δ and is impressed into the A formulation through a boundary condition.
- Domain and variables: The magnetic vector potential A is solved over the entire bounded universe, while the current vector potential T is computed only in the HTS layers.The surrounding medium is treated as non-conductive, with current confined to the superconducting layers.
- Validity and limitation: The 1D approximation is suitable for large-aspect-ratio layers when the parallel magnetic-field influence is negligible, but it cannot model parallel-field penetration.Cases such as long solenoids, where that influence cannot be neglected, are outside the formulation’s scope.
3. Case study and full models
The case study models a quarter of a 2,000-turn racetrack coil using H and T-A full formulations, then evaluates T-A accuracy as tape-width mesh resolution changes. Sixty width elements provide the selected balance between accuracy and computation time.
- Case study: The modeled quarter contains 5 pancakes with 100 tapes each, representing a 10-pancake coil of 200 turns per pancake through symmetry.The analyzed section contains 500 tapes in total.
- Full models: The H full model is treated as the reference for validating the T-A full model and proposed modeling strategies.The choice follows the stated widespread use and experimental validation of H-formulation models.
- Full models: The H full model uses 100 structured elements across each tape width, whereas the T-A full model uses 60.Both models use structured unit-cell meshes with one element through the tape thickness.
- Validation metrics: The models compare average hysteresis-loss relative error and the coefficient of determination R^2 for current-density distributions.R^2 evaluates the spatial and temporal agreement of the J distributions, while losses are averaged over the second half of the cycle.
- Mesh convergence: 60 width elements were selected because the losses error is below 1% and R^2 exceeds 0.99, while 150 elements take longer than the H full model.Increasing the width-element count improves accuracy asymptotically but increases computation time.
- Full-model results: At peak transport current, T-A reproduces local magnetic-field and current-density quantities accurately, with nearly identical losses across the coil.The largest losses occur in pancake 5 and are almost three orders of magnitude above those in pancake 1.
4. T-A multi-scale and homogeneous approaches
The paper develops T-A multi-scale and homogeneous strategies that reduce model resources while preserving accurate current-density, magnetic-field, and hysteresis-loss estimates for large-scale HTS systems.
- The refined T-A strategies use fewer resources than full models while retaining similar accuracy for hysteresis losses, current density, and magnetic-field distributions.
- T-A multi-scale approach: The T-A multi-scale model computes T and A in one model, solving current density in analyzed tapes and interpolating it in non-analyzed tapes.Surface current K from both tape groups is impressed into the A formulation.
- T-A homogeneous approach: The homogeneous model replaces each tape stack with an anisotropic bulk, defines T only inside the bulk, and neglects the magnetic-field component parallel to tape surfaces.Boundary conditions are applied at the bulk edges, and scaled current density is used as an A-formulation source term.
- T-A multi-scale approach: The multi-scale case analyzes 30 tapes, concentrating analyzed tapes in pancake regions where full-model losses vary more strongly.PCHIP interpolation estimates losses in non-analyzed tapes.
- T-A multi-scale approach: Second-order A elements are used near analyzed tapes to avoid oscillations in current density, while first-order elements are used elsewhere and for T.
- T-A homogeneous approach: The homogeneous case uses five bulks and interpolates losses from six center lines per bulk across the pancake tapes.The bulk mesh uses six unequal elements across its width and 60 elements along the tape width.
5. Real-time simulations
The T-A multi-scale and homogeneous models are evaluated for a slow three-hour charge/discharge cycle. Both reproduce the full model sufficiently for real-time simulation, with the homogeneous model trading accuracy for speed.
- Slow charge/discharge cycles in large-scale HTS systems can be simulated in real time using the T-A multi-scale and homogeneous models.Real-time simulation means computation finishes faster than the physical phenomenon occurs.
- The multi-scale and homogeneous models successfully reproduce the full model’s current-density distribution, while their magnetic-field plots are visually identical.Figure 10 compares magnetic flux density, current density, and cycle-integrated hysteresis losses.
- 0.28 % and 2.23 % are the losses relative errors for the multi-scale and homogeneous models, respectively, versus the T-A full model.Both R values exceed 0.95, while the multi-scale model is more accurate for losses and current distributions.
- 1h 13 min and 37 min are the computation times for the multi-scale and homogeneous models, respectively, both below the magnet’s 3 h physical cycle time.The homogeneous model is faster but less accurate.
- Reducing analyzed tapes or bulk elements can lower degrees of freedom and computation time, but excessive reduction compromises accuracy.The multi-scale model can reduce analyzed tapes especially in central pancakes, where losses are lower.
6. Conclusions
The work adapts multi-scale and homogeneous modeling to the T-A formulation, yielding fast, accurate large-scale HTS simulations. The approaches support real-time analysis but remain bounded by T-A and reduced-model assumptions.
- The study develops multi-scale and homogeneous techniques for use with the T-A formulation to provide quick and accurate results in large-scale HTS systems.
- The T-A multi-scale method computes T and A simultaneously in one model instead of iterating several dynamic simulations of two coupled submodels.This single-model structure also makes the model easier to build than the H multi-scale alternative.
- The approaches significantly reduce computation time while maintaining high accuracy in current-density distributions and the resulting losses.The paper identifies real-time simulation, faster parametric simulations, and digital twins as supported possibilities.
- The methods are limited to systems where the T-A 1D approximation is meaningful and the parallel magnetic-field component is negligible.Full models remain preferable when the best possible accuracy is required and computational resources permit.
Appendix A
Appendix A identifies the element-order combination that removes spurious current-density oscillations while preserving agreement with the H formulation and reducing computation time.
- Element-order validation: The tested T-A full models use first-order or second-order elements for T and A across one 11 A, 50 Hz transport-current cycle.The three combinations are first/first, second/second, and first/second order for T/A, respectively.
- Element-order validation: Spurious oscillations occur when T and A use the same element order, but disappear with first-order T and second-order A.The oscillations appear at subcritical current-density values; their period is twice as long for first-order elements than for second-order elements when both variables share an order.
- Accuracy and fit: T-A loss estimates agree with the H full model with less than 1% error, while the first-order T and second-order A model has the lowest relative error.The spurious oscillations have negligible impact on hysteresis losses because they occur at subcritical current-density values.
- Accuracy and fit: The first-order T and second-order A model achieves the largest R² value and is the only tested combination reported without oscillations.The lower R² value for second-order elements in both variables reflects larger oscillation amplitudes.
- Computational cost: All three T-A full models require significantly less computation time than the H full model.Using second-order elements for both variables takes almost three times as long as using first-order T and second-order A because the degrees of freedom increase.
Appendix B
Appendix B compares H and T-A full models under increasingly slow sinusoidal transport currents. The T-A model maintains nearly constant simulation time, whereas the H model becomes dramatically slower at low frequency.
- Benchmark setup: The benchmark uses a 20-tape stack with individual tapes modeled in both H and T-A formulations.The mesh is structured with 60 elements along the tape width.
- Benchmark setup: The simulations apply sinusoidal transport currents with amplitude 0.5*Ic = 150 A and frequencies from 5e-5 Hz to 50 Hz.Hysteresis losses per cycle and computation time are evaluated as functions of frequency.
- Hysteresis losses: The losses per cycle show a small dependence on frequency in the benchmark results.The losses-versus-frequency comparison is presented in Figure B1.
- Computation time: The T-A full model requires around 300 s per sinusoidal cycle independently of frequency across the tested range.The computation-time trend is shown in Figure B2.
- Computation time: At 5e-5 Hz, the H full model requires up to 36.6 h to simulate one cycle.Only the lower computation time for each case is presented in Figure B2.