Optimal trajectory planning for trains using mixed integer linear programming (original) (raw)
Related papers
2012
The optimal trajectory planning problem for trains under operational constraints is considered, which is essential for the success of the real-time operation and the rescheduling process for railway networks. The operational constraints caused by the timetable, real-time operation, or rescheduling often include target points and target window constraints. The approach proposed in this paper can take such constraints into account. In addition, the varying maximum traction force is approximated using a piecewise affine function and included in the trajectory planning problem. The optimal control problem is recast as a mixed integer linear programming problem, which can be solved efficiently by existing solvers. A case study is used to demonstrate the performance of the proposed approach.
Control Engineering Practice, 2014
The optimal trajectory planning problem for multiple trains under fixed block signaling systems and moving block signaling systems is considered. Two approaches are proposed to solve this optimal control problem for multiple trains: the greedy approach and the simultaneous approach. In each solution approach, the trajectory planning problem is transformed into a mixed integer linear programming (MILP) problem. In particular, the objective function considered is the energy consumption of trains and the nonlinear train model is approximated by a piece-wise affine model. The varying line resistance, variable speed restrictions, and maximum traction force, etc. are also included in the problem definition. In addition, the constraints caused by the leading train in a fixed or moving block signaling system are first discretized and then transformed into linear constraints using piecewise affine approximations resulting in an MILP problem. Simulation results comparing the greedy MILP approach with the simultaneous MILP approach show that the simultaneous MILP approach yields a better control performance but requires a higher computation time. Moreover, the performance of the proposed greedy and the proposed simultaneous MILP approach is also compared with that of the greedy and the simultaneous pseudospectral method, where the pseudospectral method is a state-of-the-art method for solving optimal control problems. The results show that the energy consumption and the end time violations of the greedy MILP approach are slightly larger than those of the greedy pseudospectral method, but the computation time is one to two orders of magnitude smaller. The same trend holds for the simultaneous MILP approach and the simultaneous pseudospectral method.
A survey on optimal trajectory planning for train operations
Proceedings of 2011 IEEE International Conference on Service Operations, Logistics and Informatics, 2011
Because of the rising energy prices and environmental concerns, the calculation of energy-optimal reference trajectories for trains is significant for energy saving. On the other hand, with the development automatic train operation (ATO), the optimal trajectory planning is significant to the performance of train operation. In this paper, we present an integrated survey of this field. First, a nonlinear continuoustime train model and a continuous-space model of train operations are described, after which the optimal trajectory planning problem is formulated based on these two models. The various approaches in the literature to calculate the reference trajectory are reviewed and categorized into two groups: analytical solutions and numerical optimization. Finally, a short discussion of some open topics in the field of optimal trajectory planning for train operations are given.
Optimal trajectory planning for trains under a moving block signaling system
2013 European Control Conference (ECC), 2013
The optimal trajectory planning problem for trains under a moving block signaling system is considered. This optimal trajectory planning problem is significant for punctuality, energy consumption, passenger comfort, etc. In a moving block signaling system, the minimum distance between two successive trains is the instantaneous braking distance required by the following train plus a safety margin. The constraints caused by the moving block signaling system are described as nonlinear inequalities, which can be transformed into linear inequalities using piecewise affine approximations. The optimal trajectory planning problem is subsequently recast as a mixed integer linear programming problem, which can be solved efficiently by existing solvers. A case study is used to demonstrate the performance of the proposed approach.
Real-time train routing and scheduling through mixed integer linear programming: Heuristic approach
In railway traffic management, when an unexpected event perturbs the system, finding an effective train routing and scheduling in real-time is a key issues. Making the right routing and scheduling decisions may have a great impact on the efficiency of the system in terms of delay propagation. However, the time available for making these decisions is quite short: in few minutes a viable set of routes and schedules must be delivered to the dispatching system. In this paper, we assess the performance of a mixed integer linear programming (MILP) formulation exploited as a heuristic approach: we seek for the best feasible solution given a limited and predefined computation time. We run an experimental analysis on instances representing traffic in the Lille Flandres station, France. The results show that the approach tested is very promising, often finding the optimal solution to the instances tackled. Moreover, we show how the performance can be improved by tuning the parameters of the MILP solver.
Timetabling optimization of a mixed double- and single-tracked railway network
Applied Mathematical Modelling, 2011
The paper deals with the timetabling problem of a mixed multiple-and single-tracked railway network. Out of all the solutions minimizing the maximum relative travel time, the one minimizing the sum of the relative travel times is selected. User preferences are taken into account in the optimization problems, that is, the desired departure times of travellers are used instead of artificially planned departure times. To find the global optimum of the optimization problem, an algorithm based on the bisection rule is used to provide sharp upper bounds of the objective function together with one trick that allows us to drastically reduce the number of binary variables to be evaluated by considering only those which really matter. These two strategies together permit the memory requirements and the computation time to be reduced, the latter exponentially with the number of trains (several orders of magnitude for existing networks), when compared with other methods. Several examples of applications are presented to illustrate the possibilities and excellences of the proposed method. The model is applied to the case of the existing Madrid-Sevilla high-speed line (double track), together with several extensions to Toledo, Valencia, Albacete, and Málaga, which are contemplated in the future plans of the high-speed train Spanish network. The results show that the computation time is reduced drastically, and that in some corridors single-tracked lines would suffice instead of double-tracked lines.
Constrained Energy Optimal Control of Trains: Conceptual Implementation
Minimizing the energy consumption of trains brings large benefits to the railway operator. In this paper we propose an implementation concept for constrained energy optimal control of trains. In particular we exploit the structure of the pre-computed optimal control law which takes the form of a series of look-up tables for distinct destination station reaching times and train masses. Based on the currently estimated train mass and required time to reach the destination station, the correct look-up table is selected in the railway operator management centre thus ensuring evalution of the optimal train traction force based on the current train velocity and traversed path. The resulting optimal value of the train traction force is communicated to the train and either directly autonomously implemented or suggested for implementation to the driver via a driver-assistance system.
Discrete optimization in public rail transport
Mathematical Programming, 1997
Many problems arising in traffic planning can be modelled and solved using discrete optimization. We will focus on recent developments which were applied to large scale real world instances.
Multiple-phase train trajectory optimization with signalling and operational constraints
Transportation Research Part C: Emerging Technologies, 2016
The train trajectory optimization problem aims at finding the optimal speed profiles and control regimes for a safe, punctual, comfortable, and energy-efficient train operation. This paper studies the train trajectory optimization problem with consideration of general operational constraints as well as signalling constraints. Operational constraints refer to time and speed restrictions from the actual timetable, while signalling constraints refer to the influences of signal aspects and automatic train protection on train operation. A railway timetable provides each train with a train path envelope, which consists of a set of positions on the route with a specified target time and speed point or window. The train trajectory optimization problem is formulated as a multiple-phase optimal control model and solved by a pseudospectral method. This model is able to capture varying gradients and speed limits, as well as time and speed constraints from the train path envelope. Train trajectory calculation methods under delay and no-delay situations are discussed. When the train follows the planned timetable, the train trajectory calculation aims at minimizing energy consumption, whereas in the case of delays the train trajectory is recalculated to track the possibly adjusted timetable with the aim of minimizing delays as well as energy consumption. Moreover, the train operation could be affected by yellow or red signals, which is taken into account in the train speed regulation. For this purpose, two optimization policies are developed with either limited or full information of the train ahead. A local signal response policy ensures that the train makes correct and quick responses to different signalling aspects, while a global green wave policy aims at avoiding yellow signals and thus proceed with all green signals. The method is applied in a case study of two successive trains running on a corridor with various delays showing the benefit of accurate predictive information of the leading train on energy consumption and train delay of the following train.