Analytical Solution to the Minimum Energy Consumption Based Velocity Profile Optimization Problem With Variable Road Grade

Abstract

The importance of eco-driving in reducing cumulative fuel consumption of on road vehicles is a well known issue. However, so far a generic algorithm that can globally solve the non-linear optimization problem and still implementable in to the current state of computing on board units is not present. In this study, we examine one aspect of the problem by incorporating the effects of road grade to the optimization problem and generate an optimal velocity trajectory for a given road grade profile. We developed simple yet accurate vehicle and fuel consumption models and employed the models as the objective function and state trajectory constraint of the optimization problem. The necessary conditions derived by employing the calculus of variation theory require to separately solve the differential equations with a set of interior point constraints for each road grade interval. As the control became linear to the Hamiltonian function we defined a set of singular arcs and derived the state and optimal control input trajectories along the arcs. We have tested the analytical solution in two example problems and compared the results with a dynamic programming (DP) solution and constant speed cruise operation. The results have shown that the analytical and DP solutions generate very close velocity trajectories which are around 8 – 10% more efficient than the constant cruise speed control case for the given examples. Moreover the calculation time of the analytical solution is significantly shorter than the DP solution rendering it possible to real-time on board implementations.