Time-optimal gearshift and energy management strategies for a hybrid electric race car

Modern Formula 1 race cars are hybrid electric vehicles equipped with an internal combustion engine and an electric energy recovery system. In order to achieve the fastest possible lap time, the components’ operation must be carefully optimized, and the energy management must account for the impact of the gearshift strategy on the overall performance. This paper presents an algorithm to calculate the time-optimal energy management and gearshift strategies for the Formula 1 race car. First, we leverage a convex modeling approach to formulate a mathematical description of the powertrain including the gearbox, preserving convexity for a given engine speed trajectory. Second, we devise a computationally efficient algorithm to compute the energy management and gearshift strategies for minimum lap time, under consideration of given fuel and battery consumption targets. In particular, we combine convex optimization, dynamic programming and Pontryagin's minimum principle in an iterative scheme to solve the arising mixed-integer optimization problem. We showcase our algorithm with a case study for the Bahrain racetrack, underlining the interactions between energy management and gear selection. Finally, we use our approach as a benchmark to evaluate the sub-optimality of a heuristic gearshift rule. Our results show that using an optimized engine speed threshold for upshifts can yield close-to-optimal results. However, already deviations smaller than 4% from the best possible threshold can increase lap time by more than 100ms, highlighting the importance of jointly optimizing energy management and gearshift strategies.