This paper presents models and control algorithms for the energy management of the Formula 1 race car in qualifying scenarios, whereby the car's performance is solely determined by the lap time achieved over a single lap. Specifically, we first extend an existing convex model of the Formula 1 car to include the waste-gate action, and analytically derive its lap-time-optimal control policy. Second, we combine nonlinear programming techniques with convex optimization to numerically compute the optimal solution for an out-lap-flying-lap sequence. Third, we devise equivalent lap time minimization strategies, whereby three PI controllers adapt the optimal control policy implemented on the car in real-time, in order to track precomputed control strategies in a minimum-lap-time fashion. Finally, we benchmark the control algorithms presented against the optimal solution and validate their performance with high-fidelity simulations. The results obtained show the optimality and the robustness of the presented approach.

The optimization of the energy management of modern hybrid-electric or fully electric race cars for minimum lap time requires a description of the vehicle dynamics performance envelope, that is, of the tires grip limit in corners, braking zones and during acceleration. In this paper, we present a computationally efficient performance envelope model in the form of convex constraints on the achievable longitudinal and lateral acceleration, on the assumption that the path on the track is given. The proposed acceleration limits are modeled velocity-dependent to take into account the effect of aerodynamic downforce present in many circuit race cars. The formulation as linear equality, inequality and second-order cone constraints allows to embed the model in a convex energy management optimization framework. To showcase the approach, we identify the model with data obtained from a state-of-the-art hybrid-electric Formula 1 car and present results for the Silverstone and Spa-Francorchamps circuits. The optimal energy management strategies can be evaluated with a computational time of less than 1 s. The optimal velocity profile subject to the performance envelope constraints is close to the measured one. The good agreement between the optimal solution and the measurement data shows that the proposed model captures the vehicle dynamics accurately enough for the purposes of energy management optimization.

The hybrid-electric powertrain currently used in Formula 1 race cars draws its energy from the car's fuel tank and battery. The usable battery size is limited, and refueling during a race is forbidden by the regulations of the Formula 1 race series. From a strategic point of view, lap-by-lap targets for the fuel and battery consumption must be chosen and imposed on the energy management controller of the car. This task is non-trivial due to the influence of the on-board fuel mass on the achievable lap time, as well as the cross-couplings between the electric and the combustion part of the powertrain. A systematic approach is thus required to compute the energy allocation strategy that minimizes the total race time. In this paper, we devise an optimization framework in the form of a non-linear program, yielding the optimal battery and fuel consumption targets for each lap of the race. The approach is based on maps that capture the achievable lap time as a function of car mass and allocated battery and fuel energy. These maps are generated beforehand with a model-based single-lap optimization framework and fitted using artificial neural network techniques. To showcase the approach, we present three case studies: First, we compare the optimal strategy to a heuristic method. The improvement of 2s over the entire race is substantial, given that the difference only lies in the energy allocation, but not in the overall consumption. It underlines the importance of optimizing the energy allocation. Second, we leverage the framework to compute the optimal fuel load at the beginning of the race. Finally, we apply the developed non-linear program in a shrinking-horizon fashion. Our simulation results show that the resulting model predictive controller correctly reacts to disturbances that frequently occur during a race.