Amie Albrecht

Rescheduling rail networks with maintenance disruptions using Problem Space Search

The creation of train timetables for long-haul single track networks is a challenging process. This task is more difficult if track maintenance disruptions are to be taken into account. This paper describes how the Problem Space Search (PSS) meta-heuristic can be used for large scale problems to create quality timetables in which both train movements and scheduled track maintenance are simultaneously considered. We show that the PSS meta-heuristic can rapidly generate a large number of alternative train timetables and then describe how the technique is generalized to construct an integrated timetable which includes track maintenance. We show how the technique can also be used as an operational tool where a revised schedule can be quickly generated to take into account the new state of a disrupted system. A case study for a single track rail network in Queensland Australia, which spans a distance of 480km, has 57 crossing loops and typically carries over 50 trains per day is discussed. Statement of scope and purpose: This paper details a fast and efficient heuristic for the simultaneous scheduling of trains and track maintenance in a large scale rail network. We also show how the heuristic can work in a dynamic environment in which disruptions occur.

Energy-efficient train control: The two-train separation problem on level track

Abstract When two trains travel in the same direction along the same track it is a common safety requirement that they must be separated by at least one signal. If the signals are located at fixed positions, they divide the track into separate sections and the safety requirement means that two trains cannot occupy the same section at the same time. Safe separation can be ensured by specifying supplementary section clearance times which define the latest allowed exit time for the leading train and the earliest possible entry time for the following train. The clearance times could initially be based on an existing timetable but we will show that adjusting these times can substantially decrease the total energy required by the trains. In this paper we find driving strategies that minimize total energy consumption and allow both trains to finish on time while adhering to the separation constraints imposed by the supplementary clearance times. We establish a new necessary condition to check whether a set of specified clearance times is optimal and discuss a heuristic procedure to find the optimal clearance times and the corresponding speed profiles. We illustrate our methods with a simplified but realistic case study.

Optimal train control: Analysis of a new local optimization principle

It is known that the optimal driving strategy for a train takes the form of a power-speedhold-coast-brake strategy unless the track contains steep grades. In such cases the predominant speedhold mode must be interrupted by phases of power on steep uphill sections and phases of coast on steep downhill sections. The Freightmiser device is used by Pacific National to provide on-board advice to train drivers about energy efficient driving strategies. Freightmiser uses a fast and efficient numerical algorithm to solve a key local energy minimization problem and hence find the optimal switching points. Although the numerical algorithm converges to a feasible solution there is no direct proof that the solution is unique. We explain the basic ideas behind the local energy minimization principle and use an extended perturbation analysis to derive various equivalent forms of the necessary conditions.

Optimal driving strategies for two successive trains on level track subject to a safe separation condition

When two trains travel along the same track in the same direction it is a common safety requirement that they should always be separated by at least one signal. If the signals are located at fixed positions dividing the track into different sections then the two trains cannot occupy the same section of track at the same time. It is desirable to find driving strategies that minimize the total energy consumption and allow the trains to reach their final destinations at the designated times. In this paper we impose a prescribed sequence of intermediate times that ensure adequate separation for two successive trains on level track. By varying these times we find necessary conditions defining the optimal sequence of prescribed intermediate times-a sequence that minimizes total energy consumption.

The cost-time curve for an optimal train journey on level track

In this paper, we show that the cost of an optimal train journey on level track over a fixed distance is a strictly decreasing and strictly convex function of journey time. The precise structure of the cost–time curves for individual trains is an important consideration in the design of energy-efficient timetables on complex rail networks. The development of optimal timetables for busy metropolitan lines can be considered as a two-stage process. The first stage seeks to find optimal transit times for each individual journey segment subject to the usual trip-time, dwell-time, headway and connection constraints in such a way that the total energy consumption over all proposed journeys is minimized. The second stage adjusts the arrival and departure times for each journey while preserving the individual segment times and the overall journey times, in order to best synchronize the collective movement of trains through the network and thereby maximize recovery of energy from regenerative braking. The precise nature of the cost–time curve is a critical component in the first stage of the optimization. doi:10.1017/S1446181116000092

Passing bays in an underground mine

Abstract Some mining companies are investigating the use of road trains to increase productivity in underground mines. Road trains require dedicated passing bays in declines. The spacing of these passing bays can have a significant impact on haulage productivity. This technical note describes the use of simulation to find the optimal spacing. If the distance between passing bays is sufficiently small then descending trains can be interleaved with ascending trains, which increases productivity. If the spacing is too small, however, productivity can decrease as descending trains wait in passing bays for ascending trains. For a real mine the spacing should be less than the theoretical critical distance to cope with variations in loading and unloading durations.

Learning from experience: the realities of developing mathematics courses for an online engineering programme

Rarely do university departments of mathematics redesign their basic mathematics courses. Through developing an online version of our associate degree in engineering in collaboration with Open Universities Australia, we redesigned the first in a sequence of five engineering mathematics courses. The online cohort proved different to our face-to-face experience. We embarked on a process of refining the unit using experiential learning and action research. The 13 week unit is delivered up to four times a year and this paper reviews the first 10 cycles of enhancements over 3 years and unpacks the layers of hypotheses underlying development decisions. Several category themes were identified with a focus on students, teachers and learning activities. Investment in online developments for mathematics can have multiple flow-on impacts for other teaching modes. Good curriculum design, regardless of environment, will always be a cornerstone of effective course development processes.

Using Maximum Power to Save Energy

Some rail operators discourage their drivers from using maximum power on the grounds that it has high fuel flow rates or high energy consumption rates. However, optimal control theory indicates that the most efficient driving strategies should use maximum power. Although the rate of energy use is higher while using maximum power, the duration of power phases is shorter and overall energy use is less. We use realistic examples to show how using more power when accelerating can reduce overall energy use.

Optimising passing bay locations and vehicle schedules in underground mines

In many underground mines, haulage vehicles carry ore from underground loading stations to the surface. Vehicles travel in narrow tunnels with occasional passing bays that allow descending empty vehicles to pull off the main path and wait for ascending laden vehicles to pass. The number of passing bays and their locations influence the delays to descending vehicles, and hence the haulage productivity of the mine. We formulate and solve a mixed integer programming (MIP) model to determine the optimal locations of passing bays to maximise haulage productivity for given numbers of vehicles and passing bays. The MIP also generates the corresponding vehicle schedule. Previous studies have only examined the placement of equally spaced bays. The results obtained from the MIP show that this is not always optimal. Furthermore, we observe that the best locations of passing bays are those that allow interleaving of vehicles without delays at bays.

Application of Origin-Destination Matrices to the Design of Train Services

Summary We consider two related problems in the design of train services on a linear rail network. In the first case, for a prescribed set of practical stopping plans, we determine the number of train services with each allowable stopping pattern that best meets the known demand. We establish fundamental results to define the concept of a maximal origin-destination demand matrix, and use this insight to formulate and solve an integer program that finds the best collection of train services. In the second case we discuss demand estimation from a collection of observed traffic counts. Our aim is to outline the fundamental procedures proposed in a celebrated paper by Van Zuylen & Willumsen (1980). These two problems arose during an Australian Mathematical Sciences Institute (AMSI) industry internship sponsored by Sydney-based company TTG Transportation Technology. These problems are well-suited as a basis of a senior level project-based mathematics course in which students build research skills and develop real-world technical experience through the study of industrial problems. The instructor may use the problems to motivate the study of deterministic mathematical programming and stochastic optimisation, and to introduce undergraduate mathematics students to important techniques in modern applied mathematics.