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 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.

Today, Formula 1 race cars are equipped with complex hybrid electric powertrains that display significant cross-couplings between the internal combustion engine and the electrical energy recovery system. Given that a large number of these phenomena are strongly engine-speed dependent, not only the energy management but also the gearshift strategy significantly influence the achievable lap time for a given fuel and battery budget. Therefore, in this paper we propose a detailed low-level mathematical model of the Formula 1 powertrain suited for numerical optimization, and solve the time-optimal control problem in a computationally efficient way. First, we describe the powertrain dynamics by means of first principle modeling approaches and neural network techniques, with a strong focus on the low-level actuation of the internal combustion engine and its coupling with the energy recovery system. Next, we relax the integer decision variable related to the gearbox by applying outer convexification and solve the resulting optimization problem. Our results show that the energy consumption budgets not only influence the fuel mass flow and electric boosting operation, but also the gearshift strategy and the low-level engine operation, e.g., the intake manifold pressure evolution, the air-to-fuel ratio or the turbine waste-gate position.

Since 2014, the Fédération Internationale de l'Automobile has prescribed a parallel hybrid powertrain for the Formula 1 race cars. The complex low-level interactions between the thermal and the electrical part represent a non-trivial and challenging system to be controlled online. We present a novel controller architecture composed of a supervisory controller for the energy management, a feedforward cylinder deactivation controller, and a track region-dependent low-level nonlinear model predictive controller to optimize the engine actuators. Except for the nonlinear model predictive controller, the proposed controller subsystems are computationally inexpensive and are real time capable. The framework is tested and validated in a simulation environment for several realistic scenarios disturbed by driver actions or grip conditions on the track. In particular, we analyze how the control architecture deals with an unexpected gearshift trajectory during an acceleration phase. Further, we demonstrate how an increased maximum velocity trajectory impacts the online low-level controller. Our results show a suboptimality over an entire lap with respect to the benchmark solution of 49 ms and 64 ms, respectively, which we deem acceptable. Compared to the same control architecture with full knowledge of the disturbances, the suboptimality amounted to only 2 ms and 17 ms. For all case studies we show that the cylinder deactivation capability decreases the suboptimality by 7 to 8 ms.