The “Waterfilling Algorithm”—An Efficient Approach for Vehicle Velocity Planning With Varying Velocity Limits

A variety of longitudinal control systems employing predictive information provide a long-term velocity profile that minimizes the cost term consisting of the superposition of energy, ride comfort and travel time. The profile can be determined by discrete dynamic programming (DDP) with a three-dimensional state space, resulting in unacceptable high computation times for real-time applications. Previous studies have used DDP with two dimensions to reduce computation time. This approach, however, leads to suboptimal solutions when the maximal velocity varies. We present an iterative algorithm called waterfilling that finds the same solution as three-dimensional DDP for arbitrary velocity limitations, while its computational complexity is orders of magnitude lower that that of DDP with two dimensions. The waterfilling algorithm minimizes the energy at the wheels and reaches the upcoming traffic light at a given point in time. For vehicles powered by an internal combustion engine, the waterfilling algorithm is extended by an efficient traffic light approach unit that avoids low vehicle velocities to increase engine efficiency. The long-term planning unit is succeeded by a short-term planning unit that handles dynamic traffic and incorporates the fuel-saving pulse and glide driving strategy. The longitudinal control system developed in this paper is validated within a dynamic simulation environment with variable traffic density and compared with a longitudinal control system employing model-predictive control. For mild-hybrid electric vehicles, the developed system achieves fuel savings of $20\,\%$ at low traffic densities. Activation of pulse and glide adds between $2.4\,\%$ to $3.7\,\%$ to the savings, depending on traffic density.