Fig 1.
Examples of large-scale disruptions.
Railway delays for strongly disrupted situations in four European countries (shown only are the delays larger than two minutes in colours; see SI section A for data description and sources). Panel (a): near-simultaneous occurrence of several problems in the Italian railways in March 2015—a major one around Rome, affecting mostly intercity trains, and one between Milan and Venice. Panel (b): effect of cyclone ‘Niklas’ (31 March 2015) on the German railways. In particular, a specific train near Pegnitz (center-south) was severely damaged by a fallen tree and the rooftop of the Munich station was destroyed, along with multiple smaller incidents across the country. The high risk of more accidents and delays caused the Deutsche Bahn to cancel most of its train activity throughout the day, leaving passengers stranded in major cities like Hannover, Frankfurt, Kassel and Berlin. Panel (c): aftermath of storm ‘Friederike’ in January 2018 in the Netherlands, coinciding with an accident in the north of the country. Fallen trees and damaged overhead lines made the fire department force the Dutch railways to close at multiple stations—resulting in no train activity between the end of the morning and 14:00. A combination of the many disruptions with the lack of resources overview limited the possibility of mitigating delay at crucial corridors. The smaller scale and high density of the railway system in the Netherlands can be recognised also in Switzerland [panel (d)], where in January 2018 (coinciding with storm Burglind/Eleanor in the north-west of Europe) a strong disruption in near Zürich (north) rapidly propagated towards the rest of the country.
Fig 2.
Illustration of delay cascading mechanism in the Dutch railways.
Panels (a)-(c): Routes in the Dutch railways of (a) the train service 3028 from Nijmegen to Alkmaar via Amsterdam, (b) a rolling stock unit used in part of this service and (c) a crew member (partly) executing this service. The schedules for 3 December 2017 are used, a day selected randomly from the dataset. Dark brown lines mark the route they share between Nijmegen and Alkmaar, after which they go their separate ways—marked in light brown lines and yellow arrows. While the service continues along its service route in panel (a) towards Den Helder, the rolling stock unit is coupled in Alkmaar onto another service to the south-east (leaving the service with only part of its original rolling stock)—via Schiphol Airport back to Nijmegen shown in panel (b). Panel (c) shows that the crew member transfers towards another service to the south-west—via Amsterdam to Leiden and The Hague, proceeding via Utrecht to Leiden and eventually ending in Utrecht. If service 3028 would have an initial delay dinit > 0, so does the rolling stock and crew executing the service; meaning that if scheduled buffer times for the resource transfers in Alkmaar would be exceeded by the delay (and no replacement resource would be available), then the subsequent services of these resources would become delayed as well. In other words, service 3028’s delay will potentially be transmitted to other services, and subsequently carried to other geographical regions. Panel (d) shows the dinit-γ plot: γ remains zero for dinit < 23 minutes as the entire delay is absorbed by scheduled buffer times for resource transfers. However, with dinit = 23 minutes, a first resource delay overcomes the transfer buffer and adds delay to another service: namely the rolling stock unit in Alkmaar, going towards Nijmegen. As dinit grows, more and more transfer buffers are overcome and delay is added to many service lines throughout the country. Panel (e) contains the same information as in panel (d), for for a log-linear plot, revealing the near-exponential increase of γ as a function of dinit. Abbreviations used in the panel depict Alkmaar (Amr), Nijmegen (Nm), Schiphol Airport (Shl), Hoofddorp shunting yard (Hdfo) and Den Helder (Hdr). The dotted black lines in panels (d-e) correspond to γ = dinit; γ becomes larger than dinit for dinit ≥ 38 minutes.
Table 1.
An example calculation of delay propagation in case of resource transfers.
All values are stated in seconds.
Fig 3.
Model simulations of a real situation.
Comparison of simulation and observed data on 11 December 2017. All models were initialised with the system snapshot at 19:00h. Panels (a) and (b) show the spatial distribution of delay at 20:00h in the real data (a) and simulation outcome of the trilayer model (b). Panel (c) shows the total delay evolution in time for the observed data and simulation outcomes of the three models. The trilayer model predicts the total delay well up to 120 minutes, after which it decays while in reality the total delay increased again. The differences between the monolayer model and the bi-/trilayer models stem from delay cascading, built in the latter ones. Only delays larger than 3 minutes are shown for visualisation purposes.
Fig 4.
Total (summed) delay jumps, sorted by various mechanisms that cause them, and subdivided in ‘labels’, for Panel (a) one case study (11 December 2017), and panel (b) their proportions averaged over four days each for four day classes (‘Black’, ‘Red’, ‘Neutral’ and ‘Green’) for the Dutch railways. The origins of the delay jumps were identified by comparing observed data to model output. Magnitudes [panel (a)] and relative magnitudes [panel (b)] were calculated for time windows of 5 minutes, with a 30-minutes smoothening window used for display purposes. Four types of delay jumps that act as delay sources are distinguished: (I) delay cascading due to crew transfers (purple), (II) delay cascading due to rolling stock transfers (red), (V) other larger incidents (blue) and the positive part of (VI) net noise. Three types of delay jumps that act as delay recovery are distinguished: (III) mitigated cascading due to rescheduling (yellow), (IV) mitigated cascading due to cancellations (green), and the negative part of (VI) net noise. The positive part and the negative part are plotted separately in a cumulative sense—up- and downward, respectively. In panel (b), the total identified delay cascading (observed plus mitigated) is highlighted in hours.
Fig 5.
Predictability performance of the model.
Panel (a), above diagonal: Mitigation measure (blue), where both the model initialisation time t0 and crew activity time t are read from the horizontal axis. Panel (a), below diagonal: Model performance
(red), plotted similarly to
, but times are read from the vertical axis. See also text for details. The contours
and
are marked respectively in blue and red lines. The data in this panel is smoothened using a Gaussian-averaging for visualisation purposes. Panel (b): Horizons for P (blue) and C (red) for the same three values of c and p as in panel (a), measured as the horizontal and vertical distances to the diagonal of the P and C contours, respectively. Panel (c)-(f): Model performance
(horizontal) versus crew schedule invariance
(vertical) on multiple instances of the day (t = 11:00h, 12:00h, …, 22:00h), with various lead times (t0 up to 1.5 hours before t), for the Green, Neutral, Red and Black days also analysed in Fig 4—16 days in total. Colours depict lead time: red indicates small lead times (i.e., predictions are closely up front), blue indicates large lead times. Averages on 15 min intervals are shown in large circles, with extra emphasis on the C values of these averages for 15 min and 30 min lead times (marked at bottom of each panel). Calculations of
and
are performed at t0 = 6-minutes time resolution [15 minutes for (c)-(f)], with a 30-minutes window around t.