A large variety of supervision, data analysis and communication algorithms monitor trains, exploiting most of their available computational power. On-board eco-driving algorithms such as Driver Advisory Systems are no exception, as the computational power available can limit their complexity and features. This was the case of Roltijd, the in-house developed Driver Advisory System based on coasting advice of Nederlandse Spoorwegen (NS), the main Dutch passenger railway undertaking. This platform cal-culated the coasting curves at every second by integrating the equations of motion numerically, assuming that the track is fl at. However, the plans of NS regarding gener-ating more complex driving advice require replacing this coasting curve calculation by a more computationally-effi cient algorithm. In this article we propose a new coasting advice algorithm based on the analytical solutions of the train motion model’s diff er-ential equations that assumes the gradients and speed limits as piecewise constant functions of the train location. We analyze the qualitative properties of these solutions using the theory of dynamical systems, showing that bifurcations arise depending on the value of the gradient and the applied tractive eff ort. We validate the proposed algorithm by comparing its performance and accuracy against the previous method and a train trajectory optimizer based on a pseudospectral method, fi nding that our algorithm is accurate and can be 15 times faster than the previous method. This allows NS to implement the proposed method on the trains running in the Dutch railway network, contributing daily to the sustainable mobility of 1.3 million passengers.@en

Supervision, data analysis and communication algorithms monitor trains, exploiting most of their available computational power. On-board eco-driving algorithms such as Driver Advisory Systems (DAS) are no exception, as the computational power available limits their complexity and features. This was the case of Roltijd, the in-house developed DAS based on coasting advice of NS, the main Dutch passenger railway undertaking. This platform calculated the coasting curves at every second by integrating the equations of motion numerically, assuming that the track is flat. However, generating more complex driving advice required replacing this coasting curve calculation by a more computationally-efficient algorithm. In this article we propose a new coasting advice algorithm based on the analytical solutions of the train motion model, assuming that gradients and speed limits are piecewise constant functions of the train location. We analyse the qualitative properties of these solutions using bifurcation theory, showing that bifurcations arise depending on the value of the gradient and the applied tractive effort. We validate the proposed algorithm, finding that our algorithm is accurate and can be 15 times faster than the previous method. This allowed NS to implement our algorithm on their trains, contributing daily to the sustainable mobility of 1.3 million passengers.

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Railways in Europe need to reduce CO2 emissions and energy usage to contribute to sustainability. One of the measures that railway undertakings can apply with low investment cost and both high reductions in CO2 emission and energy consumption, is energy-efficient train trajectory optimization. Another efficient measure is incorporating energy-efficiency in the timetable design. The aim of this thesis is to incorporate energy-efficient train trajectory optimization in the timetable design in order to improve the potential for energy-efficiency of railways. The thesis develops and applies models for the energy-efficient train control problem for a single train over (multiple) stops with both mechanical and regenerative braking behavior. In addition, energy-efficient train trajectory optimization is incorporated in timetabling by formulating and developing algorithms for a multiple-objective optimization problem applied on a corridor with multiple interacting trains.@en

Energy-efficient train driving is an important topic to railway undertakings (RUs) for sustainability and cost reduction. The timetable affects the possibilities for energy-efficient train driving by the amount of running time supplements, which is the topic of energy-efficient train timetabling (EETT). The scientific literature on EETT focuses mainly on the balance between total running time and energy consumption. However, in practice RUs consider a trade-off between the total running time, the infrastructure occupation and the timetable robustness, while energy efficiency is not considered. In this paper we consider a multiple-objective timetabling problem at a microscopic infrastructure level that adds energy consumption to the other three objectives. We approach the multiple-objective problem by a brute force search algorithm, where we use two different methods to compute the optimal solution: A weighted sum method and a distance metric method. We apply the method to a Dutch case study on the corridor between the stations Arnhem Central and Nijmegen with alternating Intercity and Sprinter trains, without intermediate overtaking possibilities. The results indicate that there is a balancing relationship between the total running time and energy consumption, without influencing the infrastructure occupation and robustness. The results of the 10 Pareto-optimal solutions show a variation of 5% for the total running time, 18% for the energy consumption, 0.3% for the extended cycle time, and 0.8% for the buffer time. The shortest running time leads to 18% more energy consumption than the longest running time with 5% more running time supplement. In both cases the extended cycle time and buffer time are almost constant. On the other hand, reducing the infrastructure occupation leads to homogenization of the timetable. Therefore, including energy consumption in the multiple-objective can be used to balance the trade-off between total running time and capacity consumption.

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In the last decade, pseudospectral methods have become popular for solving optimal control problems. Pseudospectral methods do not need prior knowledge about the optimal control structure and are thus very flexible for problems with complex path constraints, which are common in optimal train control, or train trajectory optimization. Practical optimal train control problems are nonsmooth with discontinuities in the dynamic equations and path constraints corresponding to gradients and speed limits varying along the track. Moreover, optimal train control problems typically include singular solutions with a vanishing Hessian of the associated Hamiltonian. These characteristics make these problems hard to solve and also lead to convergence issues in pseudospectral methods. We propose a computational framework that connects pseudospectral methods with Pontryagin's Maximum Principle allowing flexible computations, verification and validation of the numerical approximations, and improvements of the continuous solution accuracy. We apply the framework to two basic problems in optimal train control: minimum-time train control and energy-efficient train control, and consider cases with short-distance regional trains and long-distance intercity trains for various scenarios including varying gradients, speed limits, and scheduled running time supplements. The framework confirms the flexibility of the pseudospectral method with regards to state, control and mixed algebraic inequality path constraints, and is able to identify conditions that lead to inconsistencies between the necessary optimality conditions and the numerical approximations of the states, costates, and controls. A new approach is proposed to correct the discrete approximations by incorporating implicit equations from the optimality conditions. In particular, the issue of oscillations in the singular solution for energy-efficient driving as computed by the pseudospectral method has been solved.

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The driving strategy of train drivers has a large impact on the energy consumption. In recent studies the focus was on calculating the optimal eco-driving strategy, and measuring the exact amount of energy used during train runs. However, energy consumption is not the only key performance indicator that affects the operational performance of train or freight operating companies. In this study we define a set of key performance indicators, relevant to train operation, that are specific, measurable, assignable, realistic and time-related, and that are influenced by the driving strategy of the driver. These key performance indicators are safety, timeliness, energy consumption, workload of the driver, the environment, cost of maintenance and brand image. We chose four driving strategies that are most used in daily practice or most studied in the literature to assess these key performance indicators. Per key performance indicator we defined evaluation criteria to measure the impact of a driving strategy. We then defined key characteristics (e.g. track length, gradients), and conditions (e.g. speed restrictions, and load factor) based on which we defined test scenarios for three different train types. We then used the Radau Pseudospectral Method for solving the various optimal train control problems to compute the effect of the driving strategies on most of the key performance indicators. Our findings show amongst others, that a maximal coasting strategy causes the least environmental pollution, and in most scenarios its energy consumption coincided with the optimal energy-efficient train control strategy or it had an energy efficiency close to the optimal one. Furthermore, we found that on other key performance indicators there are differences between the driving strategies (e.g. in cost of maintenance), which should be considered when choosing a preferred driving strategy. Our results enable train and freight operating companies to make an informed decision when choosing a preferred driving strategy for their drivers, or when choosing a Driver Advisory System that supports this preferred driving strategy.

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Energy-efficient train control (EETC) has been studied a lot over the last decades, because it contributes to cost savings and reduction of CO₂ emissions. The aim of EETC is to minimize total traction energy consumption of a train run given the running time in the timetable. Most research is focused to apply mechanical braking on this problem. However, current trains are able to use regenerative braking, which leads to another optimal driving strategy compared to mechanical braking. Research on EETC with a realistic nonlinear bounded model for regenerative braking or a combination between regenerative and mechanical braking is limited. The aim of this paper is to compare the difference between the EETC with regenerative and/or mechanical braking. First, we derive the optimal control structure for the problems with different braking combinations. Second, we apply the pseudospectral method on different scenarios where we investigate the effect of varying speed limits and gradients on the different driving strategies. Results indicate that compared to pure mechanical braking, combined regenerative and mechanical braking leads to a driving strategy with higher energy savings, a lower optimal cruising speed, a shorter coasting phase and a higher speed at the beginning of the braking phase. In addition, a nonlinear bounded regenerative braking curve leads to a different driving strategy compared to a constant braking rate that is commonly used in literature. We show that regenerative braking at a constant braking rate overestimates the total energy savings.

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Energy-efficient train driving can lead to a significant reduction in both CO₂ emissions and energy consumption. A lot of research has been done on the topic of energy-efficient train control that minimizes total traction energy consumption. Furthermore, the running time supplements in the timetable determine the possibilities for energy-efficient train driving. However, research on the distribution of these supplements in order to maximize the potential for energy-efficient train driving is limited. This paper considers the multiple-section train trajectory optimization problem, which aims at finding the optimal distribution of running time supplements over multiple stops for a single train given a total scheduled running time in order to maximize the potential for energy-efficient train driving. In addition, we compare two different methods to compute the energy-efficient train control: an indirect solution method and a direct solution method. We applied both methods to a Dutch case study for an Intercity train between the stations Utrecht Central and Arnhem Central. The running time supplement distribution that minimizes the total energy use for a train has the same cruising speed on each section. Furthermore, the shorter the distance between stops, the larger the relative running time supplement. The indirect solution method generates results very quickly compared to the direct solution method and can be used for real-time control. The direct solution method is able to provide direct feedback on the necessary optimality conditions and does not require a specialized code to generate the optimal speed trajectory profile, therefore, it is useful for exploring new variants of the optimal control problem.

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Energy-efficient train driving can lead to a significant reduction in both CO2 emissions and energy consumption. A lot of research has been done on the topic of energy-efficient train control that minimizes total traction energy consumption. Furthermore, the running time supplements in the timetable determine the possibilities for energy-efficient train driving. However, research on the distribution of these supplements in order to maximize the potential for energy-efficient train driving is limited. This paper considers the multiple-section train trajectory optimization problem, which aims at finding the optimal distribution of running time supplements over multiple stops for a single train in order to maximize the potential for energy-efficient train driving. In addition, we compare two different methods to compute the energy-efficient train control: an indirect solution method and a direct solution method. We applied both methods to a Dutch case study for an Intercity train between the stations Utrecht Central and Arnhem Central. The results indicate that the optimal running time supplement distribution means that the cruising speed in each section is the same. Furthermore, the shorter the distance between stops, the larger the relative running time supplement. The indirect solution method generates results very quickly compared to the direct solution method and can be used for real-time control. The direct solution method is able to provide direct feedback on the necessary optimality conditions and does not require a specialized code to generate the optimal speed journey profile, therefore, it is useful for exploring new variants of the optimal control problem. @en

The energy consumption of trains is highly efficient due to the low friction between steel wheels and rails, although the efficiency is also influenced largely by the driving strategy applied and the scheduled running times in the timetable. Optimal energy-efficient driving strategies can reduce operating costs significantly and contribute to a further increase of the sustainability of railway transportation. The railway sector hence shows an increasing interest in efficient algorithms for energy-efficient train control, which could be used in real-time Driver Advisory Systems (DAS) or Automatic Train Operation (ATO) systems. This paper gives an extensive literature review on energy-efficient train control (EETC) and the related topic of energy-efficient train timetabling (EETT), from the first simple models from the 1960s of a train running on a level track to the advanced models and algorithms of the last decade dealing with varying gradients and speed limits, and including regenerative braking. Pontryagin’s Maximum Principle (PMP) has been applied intensively to derive optimal driving regimes that make up the optimal energy-efficient driving strategy of a train under different conditions. Still, the optimal sequence and switching points of the optimal driving regimes are not trivial in general, which led to a wide range of optimization models and algorithms to compute the optimal train trajectories and more recently to use them to optimize timetables with a trade-off between energy efficiency and travel times.@en

Train operating companies try to reduce energy consumption. One of the measures taken by them is energy-efficient train control or EETC, in which the traction energy consumption of a train is minimized. Most research on this topic focuses on EETC without regenerative braking. However, modern rolling stock may have the ability to apply regenerative braking instead of mechanical braking in order to re-use braking energy, which changes the optimal solution to the EETC problem. Therefore, this paper presents a model that determines EETC with regenerative braking including the efficiency at the catenary system for main line passenger trains. We apply a Gauss Pseudospectral Method to solve the optimal train control problem, and tested the model in a Dutch case study with different scenarios (flat track and fixed speed limit). Regenerative braking leads to a lower optimal cruising speed, later coasting, and earlier braking at a higher speed to generate energy. For lower running time supplements or smaller speed limits the speed profiles become more similar, although EETC with regenerative braking will always regenerate energy in the final braking regime. Thus, EETC regenerative braking is more energy-efficient than EETC with mechanical braking, even considering the transmission losses over the catenary system.@en

An important topic to reduce the energy consumption in railways is the use of energy-efficient train control (EETC). Modern trains allow regen-erative braking where the released kinetic energy can be reused. This regener-ative braking has an effect on the optimal driving regimes compared to trains which can only use mechanical braking. This paper compares the impact of regenerative braking on the optimal train control strategy on level track. The energy-efficient train control problem is modelled as an optimal control problem over distance and solved using a Pseudospectral Method. Three different braking strategies are compared: mechanical braking only, a combination of mechanical and regenerative braking, and regenerative braking only. The result of a case study show that energy savings of at least 28% are possible by including regenerative braking.@en