基于多节拍组合的城际铁路列车运行图优化

为了使得城际铁路列车车站到发时刻既能具有严格等时间间隔的周期性规律，又能灵活适应城际铁路客流需求的时间分布，首先提出了列车多节拍组合运行组织模式，并将其与单节拍、非节拍运行组织模式进行特征对比分析. 进而基于列车多节拍组合运行模式，以最小化所有节拍单元列车总旅行时间为优化目标，以同节拍单元列车等时间间隔运行、各类作业安全时间间隔要求等为约束，构建了城际铁路列车多节拍协同运行图优化模型. 在通过确定模型中复杂约束、引入拉格朗日乘子将其松弛的基础上，结合次梯度法设计基于拉格朗日松弛的优化算法求解模型. 以京津城际为背景的算例优化获得由4个节拍列车组合的列车运行图，其中各节拍列车分别以17、34、34、35 min等时间间隔周期性运行，该方案目标函数与其最优下界的相对差距仅为2.55%. 

Optimization of Train Timetable for Intercity Railway Based on Coordinated Operation of Multi-periodic Trains

To ensure regularity of train arrival and departure times at stations operating with strict train periodicity and timing schedules, a novel methodology for coordinated operation of multi-periodic trains is proposed. The proposed method aims to ensure a better fit for the time distribution of passenger demands in the intercity railway system and is further compared with the existing aperiodic and single periodic train operation patterns to analyze their characteristic differences. On this basis, an optimisation model aimed at minimising the total travel times of all periodical operating trains is built to solve the train scheduling problem; this involves the coordinated operation of multi-periodic trains with identical time interval constraints for each period type of trains and all safety intervals. An algorithm based on Lagrangian relaxation framework is designed and combined with the gradient method by relaxing the complex constraints via Lagrangian multiples. The proposed model is implemented numerically for Beijing-Tianjin intercity railway using its operating timetable for four periodical operating trains with intervals of 17 min, 34 min, 34 min, and 35 min. The analysis shows that their objective values differ from the optimised lower boundary by 2.55%.