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Finite Element Modeling of Power Cables using Coordinate Transformations

Albert Piwonski, Julien Dular, Rodrigo Silva Rezende, Rolf Schuhmann

TL;DR

This paper tackles the computational cost of 3-D eddy-current simulations for helicoidally twisted power cables by introducing a coordinate transformation to helicoidal coordinates and embedding it into the magnetic vector potential based A-v finite element formulation to reduce the problem to 2-D. The authors derive the helicoidal A-v weak form, exposing anisotropic, w-invariant material tensors that encode the geometry, and provide implementation details using GetDP and Gmsh. Validation against a 3-D CST reference model and comparison with a prior H-phi approach show excellent agreement in field distributions and losses, demonstrating substantial computational savings without sacrificing accuracy. The work enables efficient analysis of complex multi-conductor cables and points toward extensions to non-ideal symmetries and non-circular cross-sections under mechanical loading.

Abstract

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.

Finite Element Modeling of Power Cables using Coordinate Transformations

TL;DR

This paper tackles the computational cost of 3-D eddy-current simulations for helicoidally twisted power cables by introducing a coordinate transformation to helicoidal coordinates and embedding it into the magnetic vector potential based A-v finite element formulation to reduce the problem to 2-D. The authors derive the helicoidal A-v weak form, exposing anisotropic, w-invariant material tensors that encode the geometry, and provide implementation details using GetDP and Gmsh. Validation against a 3-D CST reference model and comparison with a prior H-phi approach show excellent agreement in field distributions and losses, demonstrating substantial computational savings without sacrificing accuracy. The work enables efficient analysis of complex multi-conductor cables and points toward extensions to non-ideal symmetries and non-circular cross-sections under mechanical loading.

Abstract

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 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 formulation.
Paper Structure (8 sections, 7 equations, 5 figures)

This paper contains 8 sections, 7 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Coordinate transformations
  3. TEXT formulation in different coordinates
  4. Weak formulation in Cartesian coordinates
  5. Weak formulation in helicoidal coordinates
  6. Space discretization and implementation details
  7. Results
  8. Conclusion & outlook

Figures (5)

  • Figure 1: Generic model of a cable's inner conductor: Modeled using a1.
  • Figure 2: Geometric objects represented in Cartesian (left, arbitrary units) & helicoidal coordinates (right, same arbitrary units): Helical objects appear straight ($w$-invariant), which is in general not true for straight lines (e.g., see the dashed line between $[-3,\,-3,\,0]^\top$ and $[3,\,3,\,2]^\top$). Further, note that the $(u,\,v,\,w)$ coordinate system must be understood here as non-directional since it is non-orthogonal ($w$-axis is not orthogonal to the $uv$-plane).
  • Figure 3: Coordinate transformation $\boldsymbol{\phi}$, its inverse $\boldsymbol{\phi}^{-1}$ and identities on $\Omega_{xyz}$ resp. $\Omega_{uvw}$ summarized as commutative diagram, i.e., $\boldsymbol\phi^{-1} \circ \boldsymbol\phi \equiv \mathrm{id}_{\Omega_{xyz}}$ and $\boldsymbol\phi \circ \boldsymbol\phi^{-1} \equiv \mathrm{id}_{\Omega_{uvw}}$.
  • Figure 4: Cross-section view of the 3-D reference model at center $z=0.1\,\mathrm{m}$: Highlighted points along $x$-axis.
  • Figure 5: Absolute of $\mathbf{J}_{xyz}$ and $\mathbf{H}_{xyz}$ along $x$-axis.