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Optimization of current carrying multicables

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Abstract

Intense electric currents in cable bundles contribute to hotspot generation and overheating of essential car elements, especially in connecting structures. An important aspect in this context is the influence of the positioning of wires in cable harnesses. In order to find an appropriate multicable layout with minimized maximum temperatures, we formulate an optimization problem. Depending on the packing density of the cable bundle, it is solved via different optimization strategies: in case of loosely packed cable bundles solely by a gradient based strategy (shape optimization), densely packed ones by arrangement heuristics combined with a standard genetic algorithm, others by mixed strategies. In the simulation model, the temperature dependence of electric resistances and different parameter values for the multitude of subdomains are respected. Convective and radiative effects are summarized by a heat transfer coefficient in a nonlinear boundary condition. Finite elements in combination with an interior-point method and a genetic algorithm allow the solution of the optimization problem for a large number of cable bundle types. Furthermore, we present an adjoint method for the solution of the shape optimization problem. The jumps at the interfaces of different materials are essential for the Hadamard representation of the shape gradient. Numerical experiments are carried out to demonstrate the feasibility and scope of the present approach.

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Notes

  1. To avoid problems in the calculation with finite elements, we require in the numerical implementation that the distance between each single cable and multicable insulation respectively pairwise between two single cables has to be larger than \(\delta >0\). The parameter \(\delta \) is chosen such that the space in between is sufficiently large to create a feasible mesh.

  2. Note that assuming the cables to be surrounded by air is a standard in the norms for dimensioning of cables. Thus, we also suppose the multicable to be hanging in free air in our examples.

  3. Alternatively, the factor \(1/q\) and the exponent \(q\) could be dropped in the shape functional. Then, an approximation of the \(L^{\infty }\)-norm would be computed for large \(q\), with the drawback of a more complicated right hand side of the adjoint system.

  4. The measure of refinement is a convention used in COMSOL Multiphysics 3.5a.

  5. Even the exterior boundary condition is only linear in \(p\).

  6. IPOPT is a software package for large-scale nonlinear optimization that implements an interior-point line-search filter method (cf. [28–30]). In our options, we use a monotone strategy for the barrier parameter \(\mu \) of the logarithmic barrier function. If this barrier parameter is small enough and the KKT conditions are fulfilled satisfactorily, an optimum is reached. Otherwise, if no minimum is attained after a given number \(n_\text {max}\) of iterations, the procedure is interrupted.

  7. Each function evaluation includes the solution of state and adjoint system.

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Acknowledgments

The authors thank Prof. Thomas Apel, Benjamin Nolet and Max Winkler for their help in the implementation of the algorithms, for valuable hints concerning the problem definition resp. the optimization strategy and for proofreading of the article. This research has been supported in parts by the German National Science Foundation (DFG) through the Priority Program 1253 “Optimierung mit partiellen Differentialgleichungen” (Optimization with partial differential equations).

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Authors and Affiliations

  1. Mathematisches Institut, Universität Basel, Rheinsprung 21, 4051, Basel, Switzerland

    Helmut Harbrecht

  2. Institut für Mathematik und Bauinformatik, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, 85577, Neubiberg, Germany

    Florian Loos

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  1. Helmut Harbrecht
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  2. Florian Loos
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Correspondence to Florian Loos.

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Harbrecht, H., Loos, F. Optimization of current carrying multicables. Comput Optim Appl 63, 237–271 (2016). https://doi.org/10.1007/s10589-015-9764-2

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  • DOI: https://doi.org/10.1007/s10589-015-9764-2

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