A Stochastic Theory of Longitudinal Dynamics and Energy Consumption of Road Vehicles

Detailed longitudinal dynamics simulations may be used to predict the energy performance of road vehicles. However, including uncertainty in the operating conditions often implies high computational costs. Model-based formulations, in conjunction with statistical methods, may obviate this limitation by directly accounting for stochasticity, thus eliminating the need for simulating large populations of driving and operating cycles. To this end, leveraging directly the methods of stochastic calculus, this work presents a novel theory of longitudinal vehicle dynamics and energy consumption, where the vehicle's speed varies stochastically depending on the characteristics of the operating environment. In particular, the proposed formulation, consisting of stochastic differential equations (SDEs) governing the longitudinal motion of road vehicles, inherently accounts for the statistical variation connected with uncertainties in the driver's behavior and road properties, including, e.g., topography and legal speed. A Fokker-Planck partial differential equation (PDE) that describes the time evolution of the joint probability density function (PDF) of the vehicle's speed, position, and road parameters is also derived from the SDEs established in the paper. The SDE and Fokker-Planck-based approaches enable statistical estimation of important quantities like speed fluctuations, instantaneous power requests, and energy consumption. The developed models may be used to assess the energy performance of road vehicles for different combinations of road transport missions. This is applicable at the early stages of the development, virtual testing, and certification processes, without the need to perform computationally expensive simulations, as corroborated by the virtual experiments conducted in the paper.