Joint Fuel-optimal Control of the Velocity and Power-split of Hybrid Electric Vehicles

This paper explores the eco-driving problem of parallel hybrid electric vehicles, intended to drive a certain distance within a limited amount of time, where the longitudinal vehicle velocity and powertrain controls are optimized to minimize the fuel consumption. In particular, we incorporate Pontryagin's Minimum Principle (PMP) and singular control theory in an optimization framework to find the fuel-optimal velocity and power-split control policy for the prime mover and the electric machine with global optimality guarantees. In addition, we present reformulations and derivations, so that the same problem can be solved jointly using another framework based on convex optimization, with the same global optimality properties, employing methods originally derived for time-optimal control of race cars. Thereby, we formally show the equivalence between the eco-driving and the racing problem. We showcase both our frameworks with numerical solutions, drawing three comparisons: First, we solve the velocity and power-split problem, both sequentially and jointly, using the PMP framework. We show that the latter can improve the fuel consumption by 2.6 %. Second, we benchmark the PMP and the convex framework by solving the joint problem with both methods and observe a discrepancy of 0.14 % in terms of the resulting fuel energy consumption. Finally, in a numerical study addressing the performance of both methods individually, we observe that the efficiency of the PMP and the convex framework are strongly dependent on the stopping criteria and the discretization step size, respectively.