Transient Nonlinear Electrothermal Adjoint Sensitivity Analysis for HVDC Cable Joints

TL;DR

The paper tackles sensitivity analysis for HVDC cable joints under transient electrothermal operation with nonlinear materials. An adjoint variable method (AVM) is developed for the coupled nonlinear PDEs, enabling efficient evaluation of $\frac{dG_k}{dp_j}$ without solving $N_P$ forward problems, particularly when $N_P \gg N_{QoI}$. The method is formulated for nonlinear conductivity curves of a field-grading material (FGM) with parameters $p_1,\dots,p_5$, implemented in the Pyrit FE framework, and validated against the direct sensitivity method and COMSOL. A 320 kV HVDC cable joint under switching impulse demonstrates accurate sensitivity predictions and highlights the benefits of a multi-rate time integration approach. This work enables efficient design optimization of HVDC cable joints by enabling rapid assessment of how parameter changes influence quantities such as Joule heating $\bm J \cdot \bm E$ and field distribution.

Abstract

Efficient computation of sensitivities is a promising approach for efficiently of designing and optimizing high voltage direct current cable joints. This paper presents the adjoint variable method for coupled nonlinear transient electrothermal problems as an efficient approach to compute sensitivities with respect to a large number of design parameters. The method is used to compute material sensitivities of a 320kV high voltage direct current cable joint specimen. The results are validated against sensitivities obtained via the direct sensitivity method.

Figures (9)

• Figure 1: Schematic of the investigated hvdc cable joint Hussain_2017aa in a cylindrical coordinate system $(\varrho,z)$. The numbers indicate the different materials as described in the text, the fgm layer is highlighted in green. The red line marks the interface between the fgm layer and the cable insulation. The cable joint is surrounded by a 30 cm thick layer of sand and buried 2 m beneath the ground.
• Figure 2: Nonlinear field- and temperature-dependent conductivity of the fgm. The fgm conductivity is modeled by the analytic function \ref{['Conductivity']}.
• Figure 3: Illustration of the weak multi-rate coupling scheme: With thermal dynamics spanning minutes to hours and electric phenomena occuring on the microsecond to millisecond scale, distinct time step sizes are adopted for the electric and thermal subproblems. The thermal time step, $\Delta t_\text{th}$, is set as a multiple of the smaller electric time step, $\Delta t_\text{el}$. As indicated by the dotted lines, the field distributions are exchanged between both subproblems after each thermal time step.
• Figure 4: The switching impulse over the simulated time span $[0, 30\,\text{ms}]$.
• Figure 5: The tangential electric field strength, $E_z$, along the interface of the xlpe and the fgm (red line in Fig. \ref{['Muffe Konfiguration']}) for different time instances.