H2-norm for mesh optimization with application to electro-thermal modeling of an electric wire in automotive context

• Published in: Communications in nonlinear science & numerical simulation Vol. 45; pp. 176 - 189
• Main Authors: Chevrié, Mathieu, Farges, Christophe, Sabatier, Jocelyn, Guillemard, Franck, Pradere, Laetitia
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.04.2017; Elsevier Science Ltd; Elsevier
• Subjects: [formula omitted]-norm; Algorithms; Approximation; Automatic Control Engineering; Automobiles; Computer Science; Conductors; Electric conductors; Electric wire; Electric wires; Finite element method; Mathematical models; Meshed model; Non integer model; Optimization; Production costs; Thermal analysis; Transfer functions; Wire; Electric wires; Meshed model; Non integer model; H2-norm; electricwires; meshedmodel; nonintegermodel
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2016.09.018

•An analytical dynamic model of the longitudinal temperature of an electric wire involving a special implicit transfer function with an exponential is presented.•A meshed model (spatial discretization) is proposed to obtain an accurate integer order approximation of this kind of transfer function.•The spatial discretization is optimized through a criterion based on the H2-norm of the error transfer function between the analytical model and its meshed approximation.•Accuracy of the meshed model is evaluated with measured data from dedicated benchmark and compared to other models obtained from classical frequency identifications. In automotive application field, reducing electric conductors dimensions is significant to decrease the embedded mass and the manufacturing costs. It is thus essential to develop tools to optimize the wire diameter according to thermal constraints and protection algorithms to maintain a high level of safety. In order to develop such tools and algorithms, accurate electro-thermal models of electric wires are required. However, thermal equation solutions lead to implicit fractional transfer functions involving an exponential that cannot be embedded in a car calculator. This paper thus proposes an integer order transfer function approximation methodology based on a spatial discretization for this class of fractional transfer functions. Moreover, the H2-norm is used to minimize approximation error. Accuracy of the proposed approach is confirmed with measured data on a 1.5  mm2 wire implemented in a dedicated test bench.