Elsevier

Omega

Volume 96, October 2020, 102070
Omega

Mixed-fleet single-terminal bus scheduling problem: Modelling, solution scheme and potential applications

https://doi.org/10.1016/j.omega.2019.05.006Get rights and content

Highlights

  • We extend our previous mixed-fleet scheduling model with optimized memory requirements, addressing real-size instances.

  • We introduce an ad-hoc decomposition scheme to solve problems with an arbitrary number of trips, fleet size and composition.

  • We validate the proposed MILP and solution methods through two test cases arising in the city of Luxembourg.

  • We perform sensitivity analysis of the model's solutions to degree of decomposition, fleet size and fleet composition.

  • We numerically assess the proposed MILP and solution method's scalability capabilities.

Abstract

Reducing pollutant emissions and promoting sustainable mobility solutions, including Public Transport (PT), are increasingly becoming key objectives for policymakers worldwide. In this work we develop an optimal vehicle scheduling approach for next generation PT systems, considering the instance of mixed electric / hybrid fleet. Our objective is that of investigating to what extent electrification, coupled with optimal fleet management, can yield operational cost savings for PT operators. We propose a Mixed Integer Linear Program (MILP) to address the problem of optimal scheduling of a mixed fleet of electric and hybrid / non-electric buses, coupled with an ad-hoc decomposition scheme aimed at enhancing the scalability of the proposed MILP. Two case studies arising from the PT network of the city of Luxembourg are employed in order to validate the model; sensitivity analysis to fleet design parameters is performed, specifically in terms of fleet size and fleet composition. Conclusions point to the fact that careful modelling and handling of mixed-fleet conditions are necessary to achieve operational savings, and that marginal savings gradually reduce as more conventional buses are replaced by their electric counterparts. We believe the methodology proposed may be a key part of advanced decision support systems for policymakers and operators that are dealing with the on-going transition from conventional bus fleets towards greener transport solutions.

Introduction

The introduction of electrified mobility solutions, especially in the public transportation sector, is becoming a widespread policy choice all over the globe. The inherent advantages are considerable, especially in terms of reduced pollutant and noise emissions especially within densely populated areas. Moreover, many cities are adhering to EU regulations, following the 2011 white paper on transport [37], and establishing low emission zones, in which conventional internal combustion buses are forbidden to operate (See Fig. 1). Public Transport (PT) operators therefore face considerable challenges, having to adapt their current fleet to newer, greener solutions, shifting towards either hybrid electric or full electric buses. For the specific instance of the country of Luxembourg, following the ministerial guidelines for sustainable mobility [29], traditional combustion engines are expected to be phased out entirely from PT operations, in favour of electric ones, by 2025.

The recently concluded EU FP7 Zero Emission Urban Bus System (ZeEUS) project has highlighted, through field demonstrations carried out in nine cities across Europe, the potentials as well as the hardships related to urban public transport electrification. These tests, carried out in cooperation with key stakeholder groups of each city, have concluded that the technological side (batteries, charging stations, drivetrains) is sufficiently mature to consider widespread implementation; a key challenge remains that of ensuring a smooth relationship between the operators and the city councils, in order to promote quick and efficient installation of charging infrastructure.

Compared to conventional combustion, operating a fleet of electric or partially electric buses introduces additional challenges to the transit planning process. In relation to the classical four stages as discussed in [4] (line planning, timetabling, vehichle scheduling and crew rostering), electrification has impacts in terms of line planning (as lines might be redesigned at route level to be able to access charging infrastructure, or maximum total trip length might be limited due to battery range), timetabling (in order to include constraints related to charging times) and, especially, vehicle scheduling. This latter problem considerably increases in complexity, as not only must the different buses composing the fleet be dispatched in the least costly fashion - in order to ensure timetable adherence and minimise vehicle/passengers delays – but recharging of batteries must also be scheduled appropriately, in order to both ensure that the buses have sufficient charge to perform trips when necessary and to avoid conflicts at the charging infrastructure. This is especially true for the instance of full electric buses equipped with opportunity charging technology, which exploit on-route and at-terminal fast charging spots to achieve 100% EV operations. Mishandling of charging could cause considerable losses to operators, as electricity pricing policies from the grid operator could result in unforeseen expenses. The problem of scheduling electric vehicles with explicit consideration of charging constraints has therefore received increased attention in recent research, and several models and solution techniques have been proposed to address the problem, extending the standard vehicle scheduling problem formulations to consider homogenous fleets of electric vehicles [12,17,38].

However, an important consideration has so far been largely neglected in literature: the shift towards full-electric vehicles will be gradual in nature, and methodologies and solutions dealing with mixed-fleet conditions will therefore be necessary, both to evaluate the impact of partial electrification and to ensure cost-efficient operations. The coexistence of two entirely different propulsion technologies, each bearing its own characteristics and requirements, can indeed considerably influence the bus scheduling process. First and foremost, conventional internal combustion buses (and, similarly, hybrid powertrain buses such as plug-in hybrids) can typically perform a full day's operation requiring no refueling, meaning that the vehicle scheduling process can be performed assuming that all buses perform their trips at any time, with conflicts arising solely due to the way trips are timetabled. In contrast, when considering electric buses, a single overnight charging will not realistically be sufficient to perform a full day's schedule (not even assuming high capacity battery packs yielding 200+kWh), meaning that within-day charging becomes an essential portion of the scheduling process. This yields additional complexity, as charging stations are a capacitated resource. The technological discrepancy is also reflected in terms of maximum range: whereas for internal combustion engines no such constraint arises from a single day's operations, unless explicitly included for the sake of, e.g., reducing wear&tear or enable efficient driver rostering [11], electric buses are limited in range due to the battery pack's capacity, and require extensive consideration of this aspect upon scheduling, warranting risk-averse policies in terms of evaluating how expensive a given scheduled trip might be in terms of energy (for example, road gradient along the given route is a strong determinant of whether or not a bus with a given residual battery capacity could feasibly be scheduled to perform a trip or not).

When handling a mixed fleet, optimal scheduling policies must therefore seek to take as much advantage as possible from both coexisting technologies (see Fig. 2): this could be achieved by, on the one hand, exploiting the lower cost per km of operations of e-buses, while on the other hand still leveraging internal combustion to perform trips that are conflicting with constraints arising for e-bus operations (conflicts due to recharging, residual range limitations). This must at the same time be balanced with optimal handling of recharging operations, both in terms of scheduling and energy costs (avoiding queueing at charging points, spreading charging operations both in time and space to avoid peak pricing). Careful modelling is therefore strongly required in order to empower practitioners facing the transition towards greener bus fleets, enabling them to both effectively minimize their operational costs and to assess the impact of design variables, such as fleet size and fleet composition.

To fill the current gap in literature, in this paper we formulate an extension to the Single Depot Vehicle Scheduling Problem (SDVSP), which considers how to best schedule available vehicles to trips originating from and returning to a single common terminal. Our proposed extension explicitly considers charging and discharging dynamics of a mixed fleet of fully electric (within-day opportunistic recharging) and hybrid-electric (overnight recharging) buses. The resulting mixed-integer linear program, dubbed Single Depot Electric Vehicle Scheduling Problem (SDEVSP), is highly parametrised, allowing to consider exogenous parameters such as consumption rates resulting from electric vehicle consumption models, as well as dynamic cost components related both to bus operations and to electricity pricing.

The problem we are extending, SDVSP, has been investigated through several modelling techniques, and has been recognized, in its simplest form, as solvable in polynomial time with complexity ranging between O((n+1)3) and O(n3) depending on the chosen modelling approach [7]. If one considers multiple vehicle types, however, the SDVSP has been shown to be NP-hard [6], implying that its computational complexity increases exponentially as more variables are added to the problem, suffering from poor scalability. By extending it with an additional layer of scheduled access to restricted resources, i.e. charging stations, we are further increasing the overall computational burden. In order to effectively and efficiently tackle real-life sized problems, we therefore develop an ad-hoc decomposition scheme, which exploits the time-modular structure of the problem at hand to considerably reduce computational efforts while, as will be shown later in our paper, maintaining a satisfactory level of optimality, i.e. finding solutions with a small optimality gap, in a reasonable computational time.

We assess both the model's capability to correctly capture the desired dynamics as well as the decomposition scheme's performance based on a real-life case study arising in the city of Luxembourg, located in the eponymous country. Since 2015, several opportunity charging stations have been equipped in terminals scattered throughout the city, where the local PT providers (Sales Lentz S.A., Autobus de la Ville du Luxembourg) are currently operating a fleet of plug-in hybrid electric vehicles and a recent trial involving full electric (autonomous) minibuses is also underway. To assess whether the city's mid-term plan considering a full electric switch by 2025 is indeed economically sustainable from the point of view of the operator, this work (including the models and algorithms developed therein) fits in a large national project [10] coordinated by Mobilab https://mobilab.lu/, whose objective is that of promoting development of next-generation PT systems, considering both electrification and ITS-enhanced solutions. This project involves key public and private stakeholders, including both aforementioned PT providers as well as Volvo Bus Corporation.

This paper is organised as follows. Section 2 provides a literature review focussing on the vehicle scheduling problem and its formulations in PT management, as well as considerations related to Mixed Integer Linear Programming (MILP) decomposition schemes. In Section 3 we first present our methodology, detailing the model formulation with its underlying assumptions, discussing its overall computational complexity and proposing a few key assumptions to reduce it as far as possible. We then discuss the proposed decomposition scheme and its inner workings and limitations. Afterwards, we introduce the experimental setup and case studies based on the city public transport network of Luxembourg in Section 4. We detail the different results obtained through model validation for the specific scenarios in Section 5, including an extensive sensitivity analysis to the main decisional variables (fleet size, fleet composition and decomposition policy); the proposed methodology's scalability properties are also assessed numerically. Concluding remarks on the potential applications of the proposed methodology are drawn in Section 6.

Section snippets

Literature review

Vehicle scheduling problems in the context of public transportation have been studied extensively as part of the “full operational planning process”, as defined in [4]. From a modelling perspective, such problems are usually formulated as Mixed-Integer Linear Programs (MILP), due to their inherently discrete nature (see e.g. the recent works of [13], [33], [39]). For the specific instance of bus scheduling, this problem has been approached by operational researchers under the name of

Methodology

In this Section we will first introduce our mixed-integer linear programming formulation, extending the work presented in [31]. Compared to our previous contribution, in this model we identify dominated constraints and merge otherwise redundant equations, which allows us to successfully reduce the total number of variables by about ½, while gaining better modelling capabilities by explicitly allowing the two mixed-fleet components (electric buses vs hybrid buses) to bear arbitrary, independent

Experimental setup

To validate our model, we focus on a real-life case study arising from Luxembourg City. We consider two large bus terminals within the city: “Gare Centrale”, which is the main train station and transport hub and “Bouillon P + R”, which is a secondary transport hub located southwest of the city centre, next to the main motorway connection to the city. We consider six lines departing from Gare Centrale, namely lines 9, 11, 14, 13, 27 and 28, and four lines from Bouillon, namely lines 1, 12, 17

Computational results

In this section, we discuss the results obtained for the two case studies, evaluating whether our proposed model and decomposition scheme are suitable to represent and optimize the dispatching of mixed fleet buses. We perform four sets of tests, designed to evaluate the following aspects of the proposed model and decomposition scheme. The first set of tests, whose results are detailed in Section 5.1, is designed to evaluate the loss of optimality introduced by our proposed decomposition scheme.

Conclusions

In this paper we presented a new mathematical formulation and solution methods aimed at correctly representing and solving the dynamics of mixed-fleet vehicle scheduling, consisting of electric buses featuring within-day charging and plug-in hybrid / conventional combustion buses. The latter only need overnight recharging (refuelling). The developed approach is aimed at supporting operators switching from conventional bus fleets towards greener transport solutions, specifically in terms of

Acknowledgements

The authors would like to acknowledge the financial support of the FNR-CORE project eCoBus C16/IS/11349329 and the precious and fruitful contributions of all the project partners.

References (39)

Cited by (0)

This paper was processed by Associate Editor Yagiura.

View full text