In the context of the High Field Magnet (HFM) program, 14 T dipole magnets are under development towards requirements for a proposed Future Circular Collider (FCC-hh). For evaluation of operating margins, a reliable critical current density scaling, J(c)(B,T), is required up to 18 T and for 1.9-10 K; whilst AC loss calculations for designing cooling and quench protection schemes require this to be extended down to an injection field of similar to 1 T or below at the operating temperature. Scaling laws for Nb3Sn wires have been extensively studied in the literature. However, for practical high I-c wires, it is rarely possible to access this full temperature and field range in measurements with a transport current, and a variety of procedural methodologies have been proposed to address divergences between transport and magnetization measurements. In this study, a recently-developed Coupled Axial and Transverse Currents (CATI) finite element method is applied for modeling the electrical and magnetic behavior of the RRP Nb3Sn wire used for HL-LHC MQXF magnets, incorporating its exact twisted composite geometry as parameterized by image analysis. Fitting parameters for the J(c)(B,T) of the superconductor are found to simultaneously match transport I-c (6-15 T, 1.9 K and 4.3 K) and magnetization (0-10 T, 1.9-16 K), and the assumptions required are discussed. The model is then used to explore the validity of common procedures applied to experimental data, including self-field correction and procedural fitting approaches.

No-insulation (NI) and metal-insulation (MI) high-temperature superconducting (HTS) magnets require three-dimensional (3D) models to describe the current distribution around critical current defects. In this work, we design and validate the EXTRA homogenisation method, standing for explicit turn resolution with anisotropic homogenisation method. It allows 3D magneto-thermal finite-element (FE) simulations of large-scale magnets to be performed with high accuracy at a reasonable computational cost. The method combines the anisotropic homogenisation of turn-to-turn contact layers (T2TCLs) and their neighbouring winding turns with the explicit resolution of specific T2TCLs. In particular, the inner- and outermost winding turns and adjacent contact layers are explicitly resolved to properly describe the current distribution near current leads. In addition, the method is able to simulate local J_c defects for a broad range of turn-to-turn contact resistances, provided the winding turns and T2TCLs next to the defect are explicitly resolved. For efficiency, the resolved T2TCLs are modelled using the surface contact approximation. The consistency of the proposed method is first verified on a 50-turn single pancake benchmark. It is shown to reproduce AC losses and temperature distributions obtained with a turn-resolved FE reference model, for both nominal operation and during thermal runaway. The computational efficiency of the EXTRA method is demonstrated with the simulation of a stack of three 150-turn pancake coils, for which computation time is reduced by a factor of up to 13 with respect to a turn-resolved FE reference model. Finally, the results of a large-scale 3D FE simulation, currently out of reach of turn-resolved models, are provided for an insert HTS magnet with 10,000 turns. The EXTRA method is open-source and input files to reproduce all results are made available.

High-temperature superconducting (HTS) coated conductors (CCs) can be wound into no-insulation (NI) coils, in which electrical current can partially bypass local normal zones via turn-to-turn contact layers (T2TCLs). Accurate magneto-thermal simulation of such coils, therefore, requires an efficient representation of the electrical and thermal behavior of the T2TCLs. This paper introduces a magneto-thermal surface contact approximation (SCA) for finite element analysis of NI HTS coils. The formulation is derived as a special case of the more general thin shell approximation (TSA) by introducing suitable approximations such as negligible tangential surface currents and eddy-current effects inside the T2TCL. The resulting SCA formulation replaces the thin volumetric contact layer with a dedicated surface weak formulation based on the electric contact resistance and thermal contact conductance. In contrast, the TSA formulation requires the definition of electric resistivities and thermal conductivities as well as the thickness of the T2TCL. The SCA is implemented in the Pancake3D module of the free and open-source Finite Element Quench Simulator. It is verified through transient magneto-thermal simulations of a model NI pancake coil. Numerical results are compared against the established TSA formulation. The results show that the SCA accurately reproduces the relevant electromagnetic and thermal behavior. For the TSA, there is a trade-off between choosing large (potentially unphysical) thicknesses with low resistivities leading to inaccurate results, or small thicknesses with large resistivities making the linear system harder to solve, increasing the computational effort. In contrast, the SCA, thanks to using contact resistances and conductances directly without the necessity to define a thickness, is easy to use and robust.

Conductor-on-round-tube (CORT) cables are a potential solution for carrying AC power in a small cross-section. Due to the geometry of the cable and the helical arrangement of the coated conductors (CC), the current follows a non-trivial pattern inside each CC. For instance, for the case of a single-layer cable, the current flow is mostly axial along the outer face of the CCs and mostly azimuthal along their inner face. Such a current distribution, known as the Garber current pattern, affects the transport AC losses. In numerical models, commonly adopted simplifications are either based on straight conductors or infinitely thin CCs. Such approaches neglect the Garber current pattern and thus misrepresent both the detailed current flow within the CC and the resulting 3D distribution of the fields. In this work, the detailed 3D current distribution in the CCs is investigated in a one-layer CORT cable, as a function of the cable geometrical parameters such as the conductor thickness, the pitch angle, and the gap between adjacent CCs. In particular, the impact of the Garber current pattern is studied on the two largest contributions to the AC losses, namely the surface losses (associated with the penetration of the component of the magnetic field parallel to the wide faces of the superconducting layer) and the edge losses (associated with the penetration of the perpendicular component of the magnetic field occurring in the vicinity of the gaps between the CCs). The detailed distribution of the currents in the CCs is examined and its relationship with the different AC loss mechanisms is established. This study is carried out by means of an effective 2D model that uses a system of coordinates conforming with the helical structure of the cable.

While the prediction of AC losses during transients is critical for designing large-scale low-temperature superconducting (LTS) magnets, brute-force finite-element (FE) simulation of their detailed geometry down to the length scale of the conductors is a computational challenge. Multi-scale methods, balancing between a coarse approximation of the fields at the scale of the magnet and a detailed description at the scale of the conductors, are promising approaches to reduce the computational load while keeping a sufficient accuracy. In this work, we introduce a neural network approach to accelerate multi-scale magneto-thermal simulations of LTS magnets by replacing costly single-turn FE models with neural network surrogates. The neural network architecture is presented and discussed, together with an automated procedure for generating simulation data for its training. The resulting accelerated multi-scale model is used to simulate current ramp-up procedures for the IBA S2C2 magnet. The surrogate-based multi-scale model is compared with a conventional multi-scale model based on a composite wire-in-channel FE model. The surrogate model is shown to reproduce single-turn filament hysteresis, inter-filament coupling, and eddy losses, while the computational time of the multi-scale method is reduced by a factor of 800.

In this paper, we present the distributed inter-strand coupling current (DISCC) model. It is a finite element (FE) model based on a homogenization approach enabling efficient and accurate simulation of the transient magnetic response of superconducting Rutherford cables without explicitly representing individual strands. The DISCC model reproduces the inter-strand coupling current dynamics via a novel mixed FE formulation, and can be combined with the reduced order hysteretic magnetization and flux models in order to reproduce the effects of internal strand dynamics: hysteresis, eddy, and inter-filament coupling currents, as well as ohmic effects. The DISCC model offers a massive reduction of the computational time compared to fully detailed FE models and still accounts for all types of loss and magnetization contributions. As a result, Rutherford cables homogenized with the DISCC model can be directly included in FE models of magnet cross-sections for efficient electro-magneto-thermal simulations of their transient response. We present two possible FE formulations for the implementation of the DISCC model, a first one based on the h-phi-formulation, and a second one based on the h-phi-a-formulation, which is well suited for an efficient treatment of the ferromagnetic regions in magnet cross-sections.

Milliken-type conductors are the predominant design for the inner conductor of high-voltage AC power cables beyond a certain cross-sectional area. Their complex geometric structure gives rise to multiscale problems that pose significant challenges in the numerical modeling of their electromagnetic behavior. The latter is essential to investigate the influence of the design parameters on the conductor performance. We present a computationally efficient finite element model that exploits helicoidal symmetries and homogenization techniques to handle the multiscale nature of the problem. The homogenized finite element model is constructed in such a way that relevant design parameters such as distinct pitch lengths at the different levels can still be taken into account. We verify the proposed method by comparing it with brute-force 3-D computations using established commercial simulation software.

The simulation of transient effects in large-scale superconducting systems with the finite element method is computationally expensive. A Reduced Order Hysteretic Magnetization (ROHM) model has been recently proposed for the computation of the magnetization and loss of composite superconductors. It accounts for the interplay between hysteresis, eddy, and coupling effects, without a need to model the detailed current density distribution, leading to a substantial reduction of simulation time. The ROHM model naturally fits in finite element formulations written in terms of the magnetic field such as the h-phi- or phi-formulation, but these formulations are not always the optimal choice. For example, in the presence of ferromagnetic materials, one may prefer formulations written in terms of the magnetic flux density. In this context, we introduce in this paper a mixed a-h-formulation that implements the ROHM model. The main advantage of this formulation is the direct use of the constitutive relation defined by the ROHM model, without the need for its inversion. We discuss the computational efficiency of the new formulation compared to a conventional phi-formulation, based on a model of a composite superconducting strand.

Magnesium diboride (MgB $_{2}$ ) presents a promising solution for efficient, reliable, and cost-effective long-distance direct current (DC) power transmission. Its low cost, high critical temperature (39 K), and compatibility with liquid hydrogen cooling are particularly advantageous. However, High Voltage Direct Current (HVDC) superconducting cables face electrodynamic losses due to current ripples from AC/DC conversion, necessitating meticulous cryogenic system design to mitigate these losses. This study employs Finite Element Method (FEM) analysis to investigate the electrodynamic losses in MgB $_{2}$ wires under varying current conditions, utilizing a 2D Helicoidal Transformation Method-based model and the open-source solver GetDP. The analysis includes the nonlinear ferromagnetic properties of the wire's Nickel and Monel matrix, revealing that loss dynamics are significantly influenced by the electromagnetic behavior of the stabilizers. By assessing loss contributions across different materials at operational frequencies typical for DC cables, the study provides insights for optimizing MgB $_{2}$ superconducting wires in DC power transmission applications.

In this paper, we present the Reduced Order Hysteretic Flux (ROHF) model to describe the relationship between time-varying transport current and internal magnetic flux for composite superconductors. The ROHF model is parametrized using reference simulations of the conductor response, after which it enables the computation of macroscopic quantities such as voltage and power loss without requiring detailed electromagnetic field solutions. It is therefore suitable for the homogenization of transport current effects in large-scale superconducting magnets, avoiding a fine discretization of the composite strand small scale structures, allowing to drastically reduce the computational effort. The approximation can be implemented either (i) as a rate-independent model, neglecting eddy current effects in the normal conducting matrix, or (ii) as a rate-dependent model, including those eddy current effects. In this paper, the complete modeling workflow, including parameter identification, coupling with the other fields (temperature and magnetic field), and post-processing of the results, is described and verified using a twisted multifilamentary strand as an example.

The coupled axial and transverse currents (CATI) method was recently introduced to model the AC loss and magnetization in twisted composite superconducting strands with low computational cost and high accuracy. This method involves two-dimensional finite element (FE) models coupled with circuit equations representing the periodicity of the strand. In this paper, we propose to adapt the CATI method to Rutherford cables, which are periodic structures made of transposed superconducting strands. We focus on reproducing the interstrand coupling currents flowing across contact resistances between the strands and we analyze the associated AC loss. We show that results of a reference three-dimensional FE model are accurately reproduced with a strongly reduced computational cost.

The efficacy of MgB2-based cables for DC power transmission applications is well-established due to several advantageous properties such as the natural abundance and cost-effectiveness of precursor powders and the innate compatibility with liquid hydrogen, which has a boiling point of 20 K. Notably, MgB2 has a critical temperature T-c of 39 K. However, transitioning this technology to alternating current (AC) applications, like electric motors or AC power transmission, poses challenges primarily due to the material's susceptibility to AC losses when subjected to alternating fields. Therefore, it is crucial to conduct a thorough investigation into the dissipative phenomena affecting MgB2 wires under AC conditions to develop design strategies for optimizing these wires for such applications. A detailed study involving both numerical simulations and experimental analyses has been conducted to understand the complex electrodynamics of MgB2 wires under sinusoidal time-varying external field conditions. Remarkably, a comprehensive numerical model has been developed, incorporating all dissipative contributions in a multifilamentary wire in a single numerical tool. This model also includes the ferromagnetic loss contributions due to the presence of ferromagnetic materials like Nickel and Monel in the stabilizing matrix. The monolithic nature of this model allows for a general treatment of AC losses in multifilamentary wires, accounting for the influence of ferromagnetic materials within the stabilizing matrix.

In this paper, we propose the Reduced Order Hysteretic Magnetization (ROHM) model to describe the magnetization and instantaneous power loss of composite superconductors subject to time-varying magnetic fields. Once the parameters of the ROHM model are fixed based on reference simulations, it allows to directly compute the macroscopic response of composite superconductors without the need to solve the detailed current density distribution. It can then be used as part of a homogenization method in large-scale superconducting models to significantly reduce the computational effort compared to detailed simulations. In this contribution, we focus on the case of a strand with twisted superconducting filaments subject to a time-varying transverse magnetic field. We propose two variations of the ROHM model: (i) a rate-independent model that reproduces hysteresis in the filaments, and (ii) a rate-dependent model that generalizes the first level by also reproducing dynamic effects due to coupling and eddy currents. We then describe the implementation and inclusion of the ROHM model in a finite element framework, discuss how to deduce the model parameters, and finally demonstrate the capabilities of the approach in terms of accuracy and efficiency over a wide range of excitation frequencies and amplitudes.

High-temperature superconductor (HTS) coated conductors (CC) are often wound into pancake coils with electrical insulation in-between the turns. The copper terminals are used for current injection and conduction cooling. An inherent variation of the critical current along the CC length results from its manufacturing process. This variation causes non-uniform heat generation, particularly when the coil is operated at a high fraction of the nominal critical current or when large critical current defects are present. The temperature distribution resulting from the balance between cooling and heating, in combination with the magnetic field and critical current distributions, determines whether a thermal runaway occurs. Accurately predicting the level of critical current defects that can be tolerated during conduction-cooled operation is difficult and requires a three-dimensional (3D) coupled electromagnetic and thermal simulation. This paper presents the results of simulations that are performed with the open-source Finite Element Quench Simulator (FiQuS) tool developed at CERN as part of the STEAM framework. The 3D coupled magnetodynamic-thermal simulations are based on the H- phi formulation and use thin shell approximations, a CC homogenization and conduction-cooling. The critical current can be varied along the CC length. The effect of a single defect specified as a reduction of the critical current along the CC length is investigated in terms of the coil's ability to reach and maintain the operating conditions. The critical current and length of the defect that results in a thermal runaway are analyzed in terms of defect location in the coil. These defect locations are compared in terms of the voltage signal available for quench detection and the implications are discussed.

Segmented conductors are a common choice for high voltage cables with large copper cross sections. Their complex geometries lead to multi-scale problems when modeling their electromagnetic behaviour through eddy current boundary value problems. We propose an finite element model that combines advantages of certain coordinate transformations and homoge-nization techniques to make computational costs manageable.

Power cables have complex geometries in order to reduce their ac resistance. Although there are many different cable designs, most have in common that their inner conductors' cross-section is divided into several electrically insulated conductors, which are twisted over the cable's length (helicoidal symmetry). In previous works, we presented how to exploit this symmetry by means of dimensional reduction within the $\mathbf{H}-\varphi$ formulation of the eddy current problem. Here, the dimensional reduction is based on a coordinate transformation from the Cartesian coordinate system to a helicoidal coordinate system. This contribution focuses on how this approach can be incorporated into the magnetic vector potential based $\mathbf{A}-v$ formulation.

Canted Cos-Theta (CCT) magnets can use conducting formers in which eddy currents are induced during current change. This is relevant during magnet ramping or discharging after a quench, particularly when using energy extraction. CCT magnets provide substantial flexibility in terms of generating the required field shape, including the combined function field in a curved bore. However, this flexibility results in magnet geometry that often requires three-dimensional (3D) quench simulations. We propose a cooperative simulation (co-simulation) developed at CERN as part of the STEAM framework with finite element (FiQuS) and finite difference (LEDET) tools. It allows for a 3D quench simulation of a CCT magnet with great geometrical detail while maintaining reasonable computational cost. FiQuS simulates the transient electromagnetics of the entire magnet and the transient thermal behavior of the formers. The thin 3D insulation layers between the coil windings and formers are collapsed into two-dimensional surfaces, thus considerably reducing the meshing complexity and solution time. LEDET is used for 3D simulation of a quench transient, including heat diffusion in the windings, ohmic loss, and inter-filament coupling loss. Using the finite difference tool for resolving temperatures in coil windings with many turns greatly improves the simulation efficiency. In the co-simulation communication, data exchange, and convergence is controlled and achieved programmatically. A complete numerical approach is proposed, and the CCT magnet simulation results are presented and discussed.

High-temperature superconductor (HTS) coated conductors (CC) are often wound into pancake coils with electrical insulation in-between the turns. The copper terminals are used for current injection and conduction cooling. An inherent variation of the critical current along the CC length results from its manufacturing process. This variation causes non-uniform heat generation, particularly when the coil is operated at a high fraction of the nominal critical current or when large critical current defects are present. The temperature distribution resulting from the balance between cooling and heating, in combination with the magnetic field and critical current distributions, determines whether a thermal runaway occurs. Accurately predicting the level of critical current defects that can be tolerated during conduction-cooled operation is difficult and requires a 3D coupled electromagnetic and thermal simulation. This paper presents the results of simulations that are performed with the open-source Finite Element Quench Simulator (FiQuS) tool developed at CERN as part of the STEAM framework. The 3D coupled magnetodynamic-thermal simulations are based on the H-phi formulation and use thin shell approximations, a CC homogenization and conduction-cooling. The critical current (Ic) is varied along the CC length. The effect of a single defect specified as a reduction of Ic along the CC length is investigated in terms of the coil's ability to reach and maintain the operating conditions. The Ic and length of the defect that results in a thermal runaway are analyzed in terms of defect location. In addition, a classical 1D scenario with a quench heater is studied. Both the local defect and the heater cases are compared in terms of the voltage signal available for quench detection. These cases result in very different requirements for quench detection, and their implications are discussed.

A new module, Pancake3D, of the open-source Finite Element Quench Simulator (FiQuS) has been developed in order to perform transient simulations of no-insulation (NI) high-temperature superconducting (HTS) pancake coils in 3-dimensions. FiQuS/Pancake3D can perform magnetodynamic or coupled magneto-thermal simulations. Thanks to the use of thin shell approximations, an $\vec{H}-\phi$ formulation, and anisotropic homogenization techniques, each turn can be resolved on the mesh level in an efficient and robust manner. FiQuS/Pancake3D relies on pre-formulated finite-element (FE) formulations and numerical approaches that are programmatically adjusted based on a text input file. In this paper, the functionalities and capabilities of FiQuS/Pancake3D are presented. The challenges of FE simulation of NI coils and how FiQuS/Pancake3D addresses them are explained. To highlight the functionalities, the results of a magneto-thermal analysis of a double pancake coil with 40 turns per pancake with local degradation of the critical current density are conducted and discussed. Furthermore, a parameter sweep of the size of this local degradation is presented. All the FiQuS input files for reproducing the simulations are provided.