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 actual devices in real applications. The numerical models based on finite element method using the H formulation of the Maxwell's 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 recently 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.

The development of the high-temperature superconductors (HTS) has allowed the emergence of diverse superconductor devices. Some of these devices, like wind power generators and high-field magnets, are classified as large-scale HTS systems, because they are made of several hundreds or thousands of turns of conductors. The electromagnetic analysis of such systems cannot be addressed by means of the available analytical models. The finite-element method has been extensively used to solve the H formulation of the Maxwell's equations, thus far with great success. Nevertheless, its application to large scale HTS systems is still hindered by excessive computational load. The recently proposed T-A formulation has allowed building more efficient models for systems made of HTS tapes. Both formulations have been successfully applied in conjunction with the homogenization and multi-scaling methods, these advanced methods allow reducing the required computational resources. A new advanced method, called densification, is proposed here. The most important contribution of this article is the comprehensive comparison of the strategies emerged from the combined use of the two formulations and the three advanced methods.

In recent years, commercial HTS materials have gained an increasing interest for their use in applications involving large-scale superconductor systems. These systems are typically made from hundreds to thousands of turns of conductors. These applications can range from power engineering devices like power transformers, motors and generators, to commercial and scientific magnets. The available analytical models are restricted to the analysis of individual tapes or relatively simple assemblies, then it is not possible to apply these models to the study of large-scale systems and other simulation tools are required. Due to the large number of turns, the simulations of a whole system can become prohibitive in terms of computing time and load. Therefore, an efficient strategy which does not compromise the accuracy of calculations is needed. Recently, a method, based on a multi-scale approach, showed that the computational load can be lowered by simulating, in detail, only several significant tapes from the system. The main limitation of this approach is the inaccuracy of the estimation of the background magnetic field, this means the field affecting the significant tapes produced by the rest of the tapes and by external sources. To address this issue, we consider the following two complementary strategies. The first strategy consists in the iterative implementation of the multi-scale method. The multi-scale method solves itself a dynamic problem, the iterative implementation proposed here is the iterative application of the multi-scale method, and a dynamic solution is obtained at each iteration. The second strategy is a new interpolation method for current distributions. With respect to conventional interpolation methods, a more realistic current density distribution is then obtained, which allows for a better estimation of the background magnetic field, and consequently, a better estimation of the hysteresis losses. In contrast with previous works, here we do not focus only on the estimation of the hysteresis losses, but also the estimation of the current density distribution is addressed. This new method is flexible enough to simulate different sections of the system with a better level of detail while providing a faster computational speed than other approaches. In order to validate the proposed method, a case study is analyzed via a reference model, which employs the H-formulation of Maxwell's equations and includes all the system's tapes. The comparison, between the reference model and the iterative multi-scale model, shows that the computation time and memory demand are greatly reduced. In addition, a very good agreement with respect to the reference model, both at a local and global scale, is achieved.

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