Multi-Objective Trajectory Optimization for Freight Trains Based on Quadratic Programming

Abstract

The overspeed protection, smooth driving, punctuality, and energy efficiency of freight trains largely depend on their trajectory optimization. This paper proposes a multi-objective optimization model, which maximizes the weighted sum of energy efficiency, punctuality, and driving smoothness. Model constraints systematically cover many practical conditions, including varying line resistance, overspeed protection, discrete neutral zones, and nonlinear traction and electric braking characteristics. Electric braking and pneumatic braking are distinguished in the freight train model, and the utilization of feedback braking energy is also considered. By nonlinear approximation, the proposed multi-objective optimization model is solved by the quadratic programming (QP) algorithm, and optimized trajectories are obtained. Numerical simulation demonstrates the correctness and effectiveness of the proposed method. Comparisons with two actual trials show that the energy efficiency and driving smoothness of the proposed method are better than that of the drivers’ operation with the same journey time. In addition, the algorithm has a short computation time, which has the potential to be integrated for on-board devices such as the driver advisory system (DAS).

Multiobjective Optimization and Weight Selection Method for Heavy Haul Trains Trajectory

“…After these definitions and derivations, the multi-objective optimization model of the heavy haul train can be transformed into the standard QP form. The interior point method can be used to solve the quadratic programming problem [28]. With the protection of the speed limit, the train can reach the destination consistently, so this QP problem should be solved with a numerical solution.…”

“…The numerical optimization method refers to discretizing the dynamic system into a problem with finite variables, and then directly solving the problem through the mathematical programming method [24]. Many scholars hold the view that the speed trajectory planning of the heavy haul train is associated with quadratic programming problems [25]- [28], which equips with mature solver with high computational efficiency [29]. This method obtains an approximate optimal solution at a shorter time scale.…”

Optimal running time supplement for the energy‐efficient train control considering the section running time constraint

“…Zhou et al [11] proposed a dynamic programming (DP) solution algorithm to find the speed profile solutions with dualized train headway and power supply constraints. Bai et al [12] proposed a multiobjective optimization model to optimize the trajectory of the heavy-haul trains, the energy efficiency, punctuality, and driving smoothness were considered and the problem was solved by a quadratic programming (QP) algorithm. Xun et al [13] focused on utilization of regenerative energy for a couple of trains, and an approximate dynamic programming (ADP) approach was proposed to adjust the speed curve of the accelerating train.…”

“…Bai et al. [12] proposed a multi‐objective optimization model to optimize the trajectory of the heavy‐haul trains, the energy efficiency, punctuality, and driving smoothness were considered and the problem was solved by a quadratic programming (QP) algorithm. Xun et al.…”

Energy‐efficient train control considering the traction system efficiency

“…The direct solution methods mainly include dynamic programming (DP) [13], quadratic programming [14], pseudo‐spectral method [15], and mixed‐integer linear programming [16]. According to the direct solution method applied, the EETC problem will be converted into different mathematical models, and the feasible suboptimal solution will be obtained by using standard NLP solvers [17].…”