Analysis of optimal energy management strategies for the hybrid electric Formula 1 car

Contemporary Formula 1 racing cars feature a high-performance hybrid-electric powertrain. It consists of a turbocharged internal combustion engine and two electric motor-generators coupled to the engine crankshaft and the turbocharger shaft. Besides the fuel tank, the battery is a second on-board energy storage. The energy management strategy in terms of battery energy deployment must be carefully optimized in order to cover a lap on a given race circuit as fast as possible. In particular, the finite size of the battery must be taken into account, since the electric boosting and recuperation capabilities of the powertrain are restricted when the battery is depleted or fully charged, respectively. So far, this problem has scarcely been investigated in a time-optimal racing context. In this paper, we include path constraints on the battery energy trajectory in a previously developed off-line convex optimization framework that is based on a model of the powertrain and the car’s longitudinal dynamics. The resulting minimum lap time problem is formulated as a second-order cone program and is solved numerically with global optimality guarantees. With this tool, we conduct a case study for a sweep of initial conditions on the battery state-of-charge and different charge or discharge targets over the lap. Thereby, we study the optimal solution to the energy management problem when either the lower or the upper bound on the battery state-of-charge is attained. First, we show that the optimal operating strategy differs substantially for these two cases: Whilst it is optimal to hit the upper bound only in one particular time instant and then discharge the battery again, the battery state-of-charge can be kept at the lower bound for prolonged sections of the lap. We highlight that these differences are related to the particular topology of the Formula 1 power unit, and in particular to the interaction between the two motor-generator units. Second, based on Pontryagin’s minimum principle, we analyze the trajectory of the costate variable associated with the battery state-of-charge. Indeed, the costate variables are crucial for the parameterization of optimal control policies that can be implemented on the car for on-line control. Our findings show that the impact of battery deployment on the achievable lap time, and therefore the structure of the optimal control policy, varies over the course of the lap. In fact, the changes in the battery costate variable are linked to the limits on electric boosting and recuperation imposed by the Formula 1 regulations. The results underline the importance of considering all the relevant cross-couplings in this complex hybridized powertrain.