Time-optimal Control of Electric Race Cars under Thermal Constraints

This paper presents a quasi-convex optimization framework to compute the minimum-lap-time control strategies of electric race cars, accurately accounting for the thermal limitations of the electric motor (EM). To this end, we leverage a previously developed thermally-unconstrained framework and extend it as follows: First, we identify a thermal network model of an interior permanent magnet EM comprising its shaft, rotor, magnets, stator, windings and end-windings, including their individual loss-models. Second, we devise a convex battery model capturing the impact of the state of energy on the battery losses. Third, in order to cope with the nonlinearities stemming from the transcription of the problem from time-domain to a position-dependent representation, we leverage an iterative algorithm based on second-order conic programming to efficiently compute the solution. Finally, we showcase our framework on the Le Mans racetrack. A comparison with high-fidelity simulations in Motor-CAD reveals that our proposed model can accurately capture the temperature dynamics of the EM, revealing the end-windings and the magnets to be the limiting components in a cold-start and a long-run operation scenario, respectively. Furthermore, our numerical results underline the considerable impact of the EM thermal dynamics on lap time, while suggesting that using a continuously variable transmission could significantly improve lap time with respect to a fixed-gear transmission.