An energy-efficient train trajectory describing the motion of a single train can be used as an input to a driver guidance system or to an automatic train control system. The solution for the best trajectory is subject to certain operational, geographic, and physical constraints. There are two types of strategies commonly applied to obtain the energy-efficient trajectory. One is to allow the train to coast, thus using its available time margin to save energy. The other one is to control the speed dynamically while maintaining the required journey time. This paper proposes a distance-based train trajectory searching model, upon which three optimization algorithms are applied to search for the optimum train speed trajectory. Instead of searching for a detailed complicated control input for the train traction system, this model tries to obtain the speed level at each preset position along the journey. Three commonly adopted algorithms are extensively studied in a comparative style. It is found that the ant colony optimization (ACO) algorithm obtains better balance between stability and the quality of the results, in comparison with the genetic algorithm (GA). For offline applications, the additional computational effort required by dynamic programming (DP) is outweighed by the quality of the solution. It is recommended that multiple algorithms should be used to identify the optimum single-train trajectory and to improve the robustness of searched results.

Regenerative braking improves the energy efficiency of railway transportation by converting kinetic energy into electric energy. This paper proposes a method to apply the Bellman-Ford (BF) algorithm to search for the train braking speed trajectory to increase the total regenerative braking energy (RBE) in a blended braking mode with both electric and mechanical braking forces available. The BF algorithm is applied in a discretized train-state model. A typical suburban train has been modeled and studied under real engineering scenarios involving changing gradients, journey time, and speed limits. It is found that the searched braking speed trajectory is able to achieve a significant increase in the RBE, in comparison with the constant-braking-rate (CBR) method with only a minor difference in the total braking time. An RBE increment rate of 17.23% has been achieved. Verification of the proposed method using BF has been performed in a simplified scenario with zero gradient and without considering the constraints of braking time and speed limits. Linear programming (LP) is applied to search for a train trajectory with the maximum RBE and achieves solutions that can be used to verify the proposed method using BF. It is found that it is possible to achieve a near-optimal solution using BF and the solution can be further improved with a more complex search space. The proposed method takes advantage of robustness and simplicity of modeling in a complex engineering scenario, in which a number of nonlinear constraints are involved.

In this paper a mixed integer quadratic programming (MIQP) is proposed to solve the dynamic economic dispatch (DED) with valve-point effect (VPE) where the non-linear and non-smooth cost caused by VPE is piecewise linearized. However if the DED with VPE is directly solved by the MIQP in a single step, the optimization suffers convergence stagnancy and will run out of memory. In this paper the multi-step method, the warm start technique and the range restriction scheme are combined with the MIQP. The optimization process can then break the convergence stagnancy and the computation efficiency can be greatly improved. When the system loss is considered, the loss formula is piecewise linearized. A post-processing procedure is proposed to eliminate the approximation error caused by linearization of the loss formula. The effectiveness of the proposed method is demonstrated by seven cases and the results are compared with those obtained by the previous published methods.

For improving the energy efficiency of railway systems, on-board energy storage devices (OESDs) have been applied to assist the traction and recover the regenerative energy. This paper aims to address the optimal sizing problem of OESDs to minimize the catenary energy consumption for practical train operations. By employing a mixed integer linear programming (MILP) model based on energy flow and the law of conservation of energy, three types of widely used OESD: supercapacitors, Li-ion batteries and flywheels have been studied in a real-world case of Beijing Changping line. Results show that without the constraints of capital cost and volume, the supercapacitor, Li-ion battery and flywheel with optimized capacity can save the catenary energy consumption by 23.6%, 22.9% and 23.7% compared to the cases without OESDs respectively. The minimum catenary energy consumption for each type of OESD has also been found with the constraints of capital cost and volume. The study shows that with a volume constraint less than 0.6 m3 and a higher allowed capital cost more than 20 k$, flywheel tends to achieve the least catenary energy consumption. When the volume is relaxed to go beyond 0.6 m3, supercapacitors always achieve the minimum catenary energy consumption disregarding the constraint of capital cost.

The optimization of train trajectory and timetable offer two effective methods of energy-efficient operations. Most traditional research optimizes them separately and thus the global optimality cannot be achieved. Previous work has mainly concentrated on minimizing the mechanical energy consumption of trains, not the electrical energy. This paper aims to develop an integrated model to optimize the integrated timetable, which includes both the timetable and the train trajectory, achieving the minimum electrical energy consumption. The AC power supply system model is first proposed to calculate the parameters of the AC power supply network. Then, a model is developed to calculate various energy consumptions using power flow calculation. Furthermore, an integrated timetable optimization model for minimum total energy consumption of AC railway system is proposed and the optimal integrated timetable is obtained by the proposed hybrid GA-PSO algorithm. Finally, 5 real-world case studies based on the Lanzhou Xinjiang high-speed railway are presented. The results show that the proposed integrated model can achieve a reduction in total energy consumption for the entire line up to 14.3% compared with the previous optimization model. The hybrid GA-PSO algorithm achieves the best results compared with the results achieved by the GA and PSO algorithms applied along.

Rail transportation is used extensively in urban areas to reduce CO₂ emission and increase road capacity. As a result, the energy efficiency of rail transportation is becoming a popular research topic. Energy-efficient train operation involves four types of control: maximal traction, cruising, coasting, and maximal braking. With the rapid development of energy storage devices (ESDs), this paper aims to develop an integrated optimization model to obtain the speed trajectory with the constraint of on-board ESD properties such as capacity, initial state of energy (SOE), and the degradation of the on-board ESD. Mixed integer linear programming is implemented on the discretized distance-based model to find the solution. In addition to the optimal speed trajectory, the on-board ESD discharge/charge operation strategy will be obtained. Results show that with high on-board ESD capacity, a train tends to apply more braking to recover more energy by using ESD; high initial SOE leads to more traction operation but undermines the energy-recovering process; degradation of on-board ESD influences the train operation and net energy consumption by affecting discharge/charge frequency. Additionally, in the case with on-board ESD, more than 11.6% of net energy consumption can be reduced compared with the one without on-board ESD. This paper indicates that the proposed method is an effective and robust way to study how on-board ESD influences optimal train operation as well as to achieve optimal train speed trajectory with on-board ESD.