Construction of Differentiated Periodic Freight Train Paths in Dense Mixed Traffic

Abstract

Freight rail transport plays key role in the transition to sustainable development. However, on European mainlines, freight trains have to cope with busy passenger operation—mostly in the form of (integrated) periodic timetables. Freight trains are characterized with very diverse parameters, so scheduling pre-arranged periodic freight train paths (PFTPs) on the basis of one sample freight train does not meet the needs of most freight operators. This article introduces new detailed framework process for hierarchized construction of differentiated (segmented) pre-arranged PFTPs. The process considers fluctuations in demand for capacity from freight rail operators, so the quality of a freight train path in terms of number of stops is related with its construction priority. This way, the process enhances competitiveness and decreases energy consumption of freight railway, as a factor for sustainable development. Correctness of the framework process is tested on the example of the Prague—Dresden mainline, in the context of prospective (denser) model passenger timetable. Results show that above 70% of real freight trains from the available historical data can fit into the proposed PFTPs. As a conclusion, the authors recommend reduction of service of the least frequent stops of regional trains to reduce number of scheduled stops for freight trains.

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sustainability
Article
Construction of Differentiated Periodic Freight Train Paths in
Dense Mixed Traffic
Michal Drábek * and Vít Janoš


Citation: Drábek, M.; Janoš, V.
Construction of Differentiated
Periodic Freight Train Paths in Dense
Mixed Traffic. Sustainability 2021,13,
8330. https://doi.org/10.3390/
su13158330
Academic Editors: Jozef Gašparík and
Davor Dujak
Received: 30 May 2021
Accepted: 21 July 2021
Published: 26 July 2021
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Copyright: © 2021 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
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conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
Department of Logistics and Management of Transport, Faculty of Transportation Sciences,
Czech Technical University in Prague, Horská3, 128 03 Prague, Czech Republic; janos@fd.cvut.cz
*Correspondence: xdrabek@fd.cvut.cz; Tel.: +420-721-971-072
Abstract:
Freight rail transport plays key role in the transition to sustainable development. However,
on European mainlines, freight trains have to cope with busy passenger operation—mostly in
the form of (integrated) periodic timetables. Freight trains are characterized with very diverse
parameters, so scheduling pre-arranged periodic freight train paths (PFTPs) on the basis of one
sample freight train does not meet the needs of most freight operators. This article introduces new
detailed framework process for hierarchized construction of differentiated (segmented) pre-arranged
PFTPs. The process considers fluctuations in demand for capacity from freight rail operators, so the
quality of a freight train path in terms of number of stops is related with its construction priority.
This way, the process enhances competitiveness and decreases energy consumption of freight railway,
as a factor for sustainable development. Correctness of the framework process is tested on the
example of the Prague—Dresden mainline, in the context of prospective (denser) model passenger
timetable. Results show that above 70% of real freight trains from the available historical data can fit
into the proposed PFTPs. As a conclusion, the authors recommend reduction of service of the least
frequent stops of regional trains to reduce number of scheduled stops for freight trains.
Keywords:
periodic freight train path (PFTP); differentiation; train path symmetry; active overtaking;
power-to-mass ratio (PMR); Integrated Periodic Timetable (IPT); sustainable transportation
1. Introduction
Rail transport plays key role in the transition to sustainable development, contributing
particularly to UN goal 11: Sustainable cities and communities—see the list [
1
], as stated in
the UN General Assembly Resolution 70/1 [
2
]. In the case of electric traction, which the
majority of the busiest European mainlines are equipped with, there are zero local emissions.
In 2018, the share of railway on direct (local) greenhouse gas emissions were only 0.4%
from total emissions of the transport sector [
3
]. This way, railways help meet ambitious
goals of emission reduction, as declared in the European Green Deal [
4
] and the related
regulation proposal [
5
]. Besides passenger transport, freight railway is important for
sustainable development as well. It offers high economies of scale, and with adequate
timetabling (as few a number of stops as possible), the energy consumption per net tonne-
kilometre can be comparably low. However, a typical European mainline is mixed-traffic,
and three segments of rail transport have to be scheduled there in a coordinated way:
long-distance, regional passenger trains and freight trains. Such coordination represents
a challenge, because different timetabling requirements of all segments have to be taken
into consideration. In Western and Central Europe, railway public transport is commonly
offered in the form of periodic timetables (sometimes referred to as clockface timetables).
In some countries [
6
–
10
], periodic service is enhanced to Integrated Periodic Timetable
(IPT), which offers travel chains all-day and during whole network, using periodic offer of
transfers between public transport lines. The best variant of such transfers is an IPT-node,
where (ideally) services of all adjoining public transport lines arrive few minutes before
symmetry time (usually the minute zero or few minutes before). Symmetry time is the
Sustainability 2021,13, 8330. https://doi.org/10.3390/su13158330 https://www.mdpi.com/journal/sustainability

Sustainability 2021,13, 8330 2 of 33
periodic time when services of a particular public transport line, while running in opposite
directions, meet each other (either stopping or not). Around the symmetry time, the node
station is full of trains. The passengers can transfer between any two public transport
lines [
11
]. Hence, competitiveness with cars—to a certain extent—is assured. IPT-nodes
and other periodic transfer connections create spatial availability of the offer. The period
creates temporal availability.
The problem that this article deals with is gradual, hierarchical construction of dif-
ferentiated periodic freight train paths (PFTPs) with unified symmetry, for the sake of
meeting the differentiated demand from the freight rail operators. Priority rules for the
construction of these PFTPs, in the context of busy passenger timetable in the form of IPT,
are formulated here as well. The more PFTPs per hour (with a certain level of quality) can
be constructed, the greater demand can be met. So, it is necessary to design PFTPs in such
a way that railway capacity is utilized as efficiently as possible.
Lindner and von Redern [
12
] proposed to leave periodic time windows free for freight
trains within IPT of passenger trains. In such windows, PFTPs can be scheduled in a
flexible way (in terms of maximum speed or alternative routing to more adjacent lines).
ˇ
Capek [
13
] and Klabes [
14
] used “bending” (i.e., intentional lengthening of runtime) of train
paths to harmonize speed profiles with other trains for the sake of more efficient utilization
of rail capacity. Janoš and Kˇríž [
15
] proposed to utilize otherwise useless capacity of a
railway line by active overtaking of a slow regional passenger train by a fast freight train,
given the speed ratio of both trains is high enough. Drábek [
16
–
18
] introduced, to the
authors’ knowledge, first concept of network-bound PFTPs, scheduled alternatively into
two directions from the node station. However, he considered only one or two different
segments of PFTPs.
Based on the research cited above, a gap can be observed—to the authors’ knowledge,
no research work has so far dealt neither with detailed segmentation of PFTPs for more than
two types of freight trains, nor with relative prioritization of construction of differentiated
PFTPs on the same railway line.
The proposed framework process considers two dimensions of segmentation. The first
one is division of the solved mainline into bottlenecks and sections with dense traffic on
the one hand, and remaining sections on the other hand, because of different conditions
and requirements for freight timetabling. The second dimension is segmentation of PFTPs,
which is reflected in their priority of scheduling. The higher priority, the lower number of
scheduled stops for overtaking by passenger trains. The authors are aware of no similar
approach. Contrary to mostly strictly mathematical approaches cited in the literature
review below, the presented approach strives for strengthening of the bridge between
advanced quantitative research and often empirical managerial practice in the field of
timetabling, with emphasis on offer of pre-arranged, attractive PFTPs.
The article is divided into six sections: Introduction, Literature Review, Materials and
Methods, Results, Discussion and Conclusions. Here we summarize the content of the
sections that follow. The Literature Review section proceeds in a broad-to-narrow logical
order towards the addressed research gaps (as far as possible due to large variedness
of the related research). For better clarity, the review is defined into four topics divided
according to their proximity to the topic of the article. In the Materials and Methods
section, the problem solved in the article is defined in a more detailed way. The scope of
both the proposed framework process and the following timetable experiment is clearly
stated. The new framework process is defined and supported with arguments. Then,
necessary materials are introduced and described. Parameters of sample freight trains for
the proposed differentiated (segmented) PFTPs are derived. The Results section describes
PFTPs used for the timetable experiment in terms of timetable and recovery margins by
section and direction. Besides, the benefit of the designed PFTPs is quantified in terms of
number of really running freight trains from available historical data that would fit into
the PFTPs. Sensitivity analysis with variable power-to-mass ratio (PMR) is added and its
results are briefly discussed. In the Discussion section, first of all, the proposed framework

Sustainability 2021,13, 8330 3 of 33
process and timetabling experiment are interpreted in the context of previous research.
Then, the proposed framework approach is discussed, mainly in terms of its potential
contributions to sustainable development. The last section, Conclusions, summarizes a
brief background of the introduced research, its contribution and major findings, followed
by recommendations for further research.
2. Literature Review
Railway transport, including freight, is a key factor for sustainable development.
However, to fully utilize the potential of freight railway to reduce the emissions per
net transported tonne, efficient timetabling of freight trains is necessary—in the context
of already prevailing IPT of passenger trains. The problem of construction (design) of
coordinated timetable of trains with various parameters is considerably complex. If the
timetable is periodic, the complexity is reduced to a certain period (30 min to 2 h as a rule).
Periodic Event Scheduling Problem, which can be applied also in the field of freight rail,
was formulated by Serafini and Ukovich [
19
]. The following literature review focuses on
problem of timetabling in a broad-to narrow logical order towards higher proximity to
the research gaps identified in this article. The review is divided into five subsections.
The first one focuses on timetabling of mixed rail traffic. The second one proceeds to
timetabling approaches specific for freight trains and to induced need to define sample
freight trains for the timetabling process. The third subsection deals with more integrated
timetabling approach (in the context of periodic timetable of passenger trains)—periodic
capacity for freight trains and PFTPs. The fourth subsection is the closest to the topic of
this article. It focuses on network-bound PFTPs and timetabling approaches for more
efficient capacity utilization. The fifth subsection summarizes the literature review and
subsequently identifies research gaps that delimit the problem solved in this article.
2.1. Operation, Capacity and Timetabling of Mixed Rail Traffic
Efficiency of railway operation and capacity planning, especially scheduling of IPT
and other forms of periodic timetable (predominantly for passenger trains) are favoured
research topics in the field of mathematics and operation research, since many sub-problems
that enable algorithm development or simulation of the problem, can be found there. Thus,
only the works that are most related with the topic of this article, are briefly cited here.
Serafini and Ukovich [
19
] proposed a mathematical model for scheduling activities of
periodic type—Periodic Event Scheduling Problem (PESP). Most of (but not all) quantitative
scientific works in the field of IPT were based on this model, for instance Liebchen [
20
] and
Opitz [21].
Caimi et al. [
22
] proposed mathematical formulation of partially periodic service—
some train paths can be scheduled in peak times only, and can, if need be, alternate with
another (such as freight) train paths—supposed that both train path types fit within fully
periodic (all-day) timetable pattern.
First edition of UIC Code 406 Capacity [
23
] considered train path heterogeneity (i.e.,
difference in scheduled section runtimes) one of four key parameters that influence railway
capacity (however, the capacity is determined in much more complex way, depending on
many factors—see Stoilova et al. [
24
]). This influence is negative—the higher heterogeneity,
the lower the line capacity, i.e., the fewer train paths can be scheduled per defined time
unit. It is obvious that slower trains have to be overtaken by faster ones, unless the capacity
requirements for the line are very low. The thing is, the longer is the section between two
stations where a slow train is overtaken, the more potential faster train paths are consumed
by a slow train path. Thus, the capacity estimation based on compression of subsequent
train paths closely together (with time supplements added to secure quality of operation)
gives various results, depending on length of the chosen line section—see the 2nd edition
of UIC Code 406 Capacity [
25
]. Positive influence of overtaking tracks on railway capacity
was demonstrated, for instance, by Gašparík et al. [
26
]. However, utilization of such tracks
in operation should be verified—ideally by microsimulation, as shown by ˇ
Camaj et al. [
27
],

Sustainability 2021,13, 8330 4 of 33
so that investment in an additional track can be justified. Ljunggren et al. [
28
] introduced an
algorithm that maximized the timetable (operational) robustness for bottlenecks of railway
network by maximizing the temporal distance to neighbouring train paths in the timetable.
2.2. Freight Timetabling and Sample Freight Trains
Contrary to extensive research in passenger, especially periodic, timetabling, freight
timetabling was solved mostly merely as an additional problem. However, some works
dedicated to freight railway timetabling problems emerged as well. Both types of works
are cited here.
Haldeman [
29
] described rules for recovery margins (time reserves) in Switzerland.
A relative margin was 11% of the runtime of freight trains. In addition, one-minute margin
was added for each 30 min of runtime. Further, Special Operational Supplements were
added, for instance, in busy node areas.
Vromans [
30
] compared approaches for determination of recovery margins in chosen
countries—for passenger, as well as for freight trains. In Netherlands, the supplements
were proportionally allocated with respect to the minimum runtime. He stated that in
practice the freight trains were often faster and lighter than planned (and scheduled).
Vromans [30] explained that the choice whether to stop at a station or not provided much
flexibility to his model, and that this option could be used also for determining the optimal
location of stations for possible overtaking of freight trains.
Müller [
31
] verified IPT for wagonload transport in Germany. Marshalling yards were
supposed to correspond to IPT-nodes. Various automation levels of wagons (automatic
coupling, autonomous drive etc.). were considered as well. The result was a calculation
that has shown no significant time- or cost-saving potential for such concept with use of
conventional freight wagons.
Some authors focused on improvement of process of inserting additional freight train
paths into previously determined passenger timetable. Cacchiani et al. [
32
] proposed to
introduce as many new freight trains as possible by assigning them timetables that were as
close as possible to the ideal ones (from the operators’ viewpoint).
Bablinski [
33
] carried out a set of simulation games based on a real-world scenario
where preferential treatment of freight operators was included. However, the author
admitted that benefits for freight operators were achieved at the cost of loss of rail system
efficiency due to increased level of runtime heterogeneity.
Kuckelberg et al. [
34
] introduced clustering algorithm for gaining of sample trains,
that was implemented in analytical LUKS
®
system for railway operation research. They clus-
tered all trains, even with different routes, at the same time, and therefore they had to
compare several key train parameters, which they enumerated in descending order of
importance considering train similarity—firstly, the kind of traction (electric or diesel),
secondly, the general kind of train (passenger, commuter or freight). Thus, more groups of
fundamentally different trains were processed together.
The third parameter was the route of the train, including the stopping regime. The fourth
was mass and tractive forces of the train. The fifth one was the train length. The last one was
the maximum speed.
The clustering algorithm included calculation of similarity (“distance”) between two
trains, which was the product of parameter-specific distances which comprised parameter-
specific weights.
From each cluster, a sample train was formulated as follows: the most frequent route
with the most common stopping regime along it, and the most frequent braking regime
and percentage. Further, besides other things, maximum recovery margin, speed of trainset
and minimum number of wagons were chosen. In addition, 0.8 quantiles of maximum
speed, mass and length were chosen. The selected stopping regime (i.e., used overtaking
tracks) and the train length should match together.
Opitz [
21
] modelled PFTPs with chosen sample freight trains, using “sample train
atoms”, later referred to as “snippets” by Pöhle [
35
]. With the help of such building blocks,

Sustainability 2021,13, 8330 5 of 33
Opitz was able to distinguish different capacity consumption of otherwise identical freight
trains that did or did not stop in a particular station.
Woodburn [
36
] investigated variability in number of wagons in a freight train of
particular category or origin-destination pair. His survey was based on a sample of almost
3000 individual freight trains, with analysis at four levels of disaggregation, from the
commodity groupings used in official statistics down to individual services. He identified
considerable variability even at fairly high levels of disaggregation.
Another research question is whether there can be scheduled any “typical” freight
train path, which can be used as a pre-arranged PFTP.
Drábek [
16
] worked out origin-destination matrix of freight train paths, according to
official timetable at that time. He gathered scheduled section runtimes and compared them
with desired runtimes for the proposed PFTPs. Determination of desired runtimes was
iterative, influenced by scheduled freight train runtimes. If any section runtime was too
short for about half of freight trains, the desired runtime was lengthened (i.e., relaxed) to
the detriment of runtime of neighbouring, non-critical section. The aim of this iteration
was prevention of scheduling of too many stops for overtaking for the PFTPs. However,
some longer sections uphill became critical, and resulted in 70% of scheduled freight trains
that were able to fit in PFTPs (scheduled runtimes for sections between neighbouring
stations were assessed, if there was scheduled at most 1 stop). So, remaining 30% of freight
trains were supposed to be scheduled at night or within the remaining capacity.
2.3. Periodic Capacity for Freight Trains, Periodic Freight Train Paths
Lindner and von Redern [
12
] summed up problems with construction of freight train
paths in IPT. They found that it was necessary to let periodic time windows (“canals”)
free for freight trains, giving them sufficient capacity. The result of their proposal was
not a periodic freight timetable, but PFTPs—or merely periodic time windows—e.g.,
1 or 2 freight
train paths for the speed of 90 km/h instead of two or three freight train paths
for the speed of 100 km/h. These time windows should be preferably connected together
in nodes (if needed, among more than two lines to enable use by freight trains with various
origin/destination stations). A freight train can also use some PFTP only partially.
Opitz [
21
], within his model of IPT, modelled PFTPs as well. R˚užiˇcka designed
partially periodic freight train paths for public additional locomotive service for freight op-
erators in his bachelor thesis [
37
]. Because of different operation of additional locomotives
in both directions, unified symmetry axis was not maintained.
Chýle [
38
] designed two types of PFTPs on the mainline between Prague and Sum-
merau (for lighter and faster vs. for heavier and slower trains), both in 4-h periods.
However, due to partially periodic passenger services (some services fell out in off-peak
times) and long single-track sections, he had to do many adjustments, so the resulting
catalogue freight train paths have turned into practically aperiodic—especially in single-
track sections.
R˚užiˇcka designed PFTPs for busy mainline between Pˇrerov and Petrovice u Karviné
in his diploma thesis [
39
]. Along the whole line, two groups of PFTPs, both with 2-h
period, were designed. Due to imperfect symmetry and many additional peak services in
long-distance passenger railway, only one group out of two was constructed with unified
(zero) symmetry.
2.4. Network-Bound Periodic Freight Train Paths and Timetabling Approaches for More Efficient
Capacity Utilization
Drábek [
16
] introduced, to the authors’ knowledge, the first concept of network-
bound PFTPs, scheduled alternatively into two directions from the node station. Given
node station with level crossing (without flyovers), timetabling conflict was avoided by
different runtime margins, depending on a direction that the freight train should proceed
to, so blocking of the whole switch region at one time did not exclude blocking of the
whole switch region by the train path between another directions, which was scheduled at
another time. The approach proposed by Drábek, which preserves train path symmetry

Sustainability 2021,13, 8330 6 of 33
at the end of sections adjoining to the node station, is an analogy to passenger IPT-node,
but tailor-made for freight railway. The unified period and (at least outer) unified symmetry
remained common for both concepts. But transfers in passenger transport were replaced
by possibility of direct passing through a node (without stopping) by a freight train.
Both concepts enable free combination of adjoining directions, but there is another key
difference. Passenger IPT offers periodic services, which can be used by passengers (in the
part of the service route as a rule, or not at all). A network-bound system of PFTPs offers
capacity to train operators. A freight train can use only a needed part of a particular PFTP.
Based on this pilot study, Drábek developed in his doctoral thesis [
18
] (and published
previously in 2011 [
17
]) a framework process, which was intended as a building set of mea-
sures (i.e., only the appropriate ones should be applied in any particular case). This frame-
work process (Figure 1without the red frame) came out from the above-mentioned freight
analogy to IPT.
As a conclusion, two measures for efficient capacity utilization were proposed: ho-
mogenization of freight trains with the most frequent speed segment of passenger trains
(already in the timetabling/allocation phase, unless faster running of freight trains was
technically impossible) and avoidance of stopping freight trains in bottlenecks and uphill
sections if possible.
Michl et al. [
40
], following up on Drábek‘s framework process, proposed systematic
homogenization of heterogeneous train paths—already in the phases of capacity allocation
and timetabling. Moreover, they proposed speed bundling of homogeneous train paths
with the use of bending of freight and (if necessary) long-distance train paths towards
longer section runtime. Thus, more efficient speed bundling with the most frequent train
segment (suburban trains as a rule) could be enabled.
ˇ
Capek [
13
] and Klabes [
14
] used “bending” of train paths to harmonize speed profiles
(i.e., section runtimes) as a part of their approaches, so that efficiency of railway capacity
utilization can be improved.
Michl et al. [
40
] segmented train paths based on two criteria—periodicity and degree
of advance. Periodic train paths can be either previously allocated ones (based on the
operators’ requests) or catalogue (pre-arranged) ones—mostly for typical freight trains,
whose departure days and times cannot be determined in the timetabling phase, but such
trains are reasonably expected to run, based on ad hoc capacity requests.
Individual train paths—both previously allocated and ad hoc ones—must not disrupt
all-day (or rush hour) periodic timetable pattern.
Besides bundling, an appropriate sequence of heterogeneous train paths can save
capacity as well. Thus, the fastest train should run as first, then a middle-fast train and the
slowest train in the end. IPT-based symmetry causes that in each direction this sequence is
exactly opposite [40].
Janoš and Kˇríž [
15
] proposed to utilize lost capacity by active overtaking of a slow
regional passenger train by a fast freight train, given the speed ratio of both trains was high
enough. They analysed possibilities of such overtaking on Prague—Dˇeˇcín mainline. If such
overtaking becomes a routinely repeated process, then adequate real-time dispatching will
be necessary. One of most serious obstacles to it on most of European rail networks is
deficiency in standardized data exchange that would enable accurate prediction of train
running, as highlighted by Kamenický[41].

Sustainability 2021,13, 8330 7 of 33
Figure 1.
Framework process for construction of network-bound PFTPs by Drábek [
17
,
18
] with
research gaps (displayed in red) added by the authors.
By construction of PFTPs one cannot avoid some inherent asymmetry (items 1 to 3
come from Drábek [42]):
1. Local overtaking-based asymmetry in case of steep gradient;

Sustainability 2021,13, 8330 8 of 33
2.
Local asymmetry based on adding and releasing of assistance locomotive in the uphill
direction only (for steep gradient);
3.
Asymmetric scheduled stop (only in one direction) of a freight train for crossing on a
single-track section;
4.
In case of sequence of heterogeneous trains, mentioned above, freight train can depart
after fast train almost immediately. Contrary, in the opposite direction, some appro-
priate buffer time between arrivals of freight train and fast train is necessary.
The asymmetry should be maintained as local only—between two nearest node
stations where PFTPs branch to more directions. In these stations, PFTPs should be
symmetric again, so the asymmetry propagation further through the network is avoided.
Only few minutes deviation in symmetry in nodes is permissible, provided that:
1.
It is necessary for conflict-free PFTPs scheduling through level crossing (see introduc-
tion of a freight IPT-node by Drábek [18]);
2. PFTPs can symmetrically proceed further to/from next nodes where PFTPs branch.
Drábek [
18
] examined modification of passenger timetable on a section between
two node stations, for the sake of capacity for catalogue PFTPs on a double-track line
with mixed traffic. As a timetable pattern, prospective timetable from Czech research
project on configuration of IPT-nodes [
43
] was used, with seven long-distance and two
regional passenger train paths per hour (30 min period) and direction. Intercity train paths
were kept untouched. Regional train paths were provisionally removed. Fast (regional
express) train paths (30 min period) were shifted closer to (i.e., bundled with) intercity
train paths. Then, PFTPs (with at most one stop per PFTP) and a regional train path
were iteratively constructed. The main result, besides three PFTPs with one overtaking
only, was a significant deterioration of regional railway offer—one train per hour only
and service cancellation for three stops. Such a solution would be very likely politically
unacceptable if applied in practice.
Šimral [
44
], as co-founder of freight railway undertaking METRANS Rail Ltd. and
a locomotive driver himself, pointed out that some railway undertakings did not equip
their trains with appropriate locomotives, so the PMR (power to hauled gross mass ratio
[kW/t]) of such trains was very low. Or the drivers were not motivated to keep allowed
(and scheduled) speed. Thus, real runtimes were considerable higher than scheduled,
and such trains could have caused delay of another trains. He proposed that keeping of
scheduled speed should be legally enforced and that timetabling process in the Czech
Republic should consider PMR as a priority factor by train path allocation.
Chýle [
38
] proposed sample freight trains with comparable high PMR (3.76 and 5.74)—
combination of powerful modern locomotive and restricted gross mass. Such sample trains
enabled construction of PFTPs with relatively low capacity consumption.
2.5. Summary of Literature Review
The literature review shows that the majority of related research in the field of mixed
and freight railway timetabling and operation can be characterized by quantitative ap-
proach (mathematical modelling, network timetable optimization, various optimization of
partial problems, etc.).
There are few exceptions with qualitative (in the sense “not quantitative”) approach:
Lindner and von Redern [
12
], doctoral thesis and related research by Drábek [
16
–
18
,
42
],
and empirical statement by Šimral [
44
]. The related research is, in accordance with its
relative proximity to the research presented in this article, sorted in Table 1—from left to
right towards closer research.

Sustainability 2021,13, 8330 9 of 33
Table 1.
Related research, sorted left-to-right by its increasing proximity to the problem researched in this article. Within
each category, literature is ordered chronologically and alphabetically.
Timetabling/Capacity/Operation
of Mixed Rail Traffic, ITF
Sample Freight Train/Freight
Timetabling
PFTPs, Periodic Capacity for
Freight Trains
Timetabling Approaches for
More Efficient Utilization of
Railway Capacity,
Network-bound PFTPs
Serafini and Ukovich, 1989 [19] Müller [31], 1999 Lindner and von Redern, 1989 [12]ˇ
Capek, 2004 [13]
Forschungsgesellschaft für
Straßen- und Verkehrswesen, e. V.,
2001 [11]Haldeman, 2005 [29] Opitz, 2009 [21] Drábek, 2009 [16]
UIC, 2004 [23] Vromans, 2005 [30] R˚užiˇcka, 2018 [37] Klabes, 2010 [14]
Liebchen, 2006 [20] Drábek, 2009 [16] Chýle, 2021 [38] Drábek, 2011 [17]
Caimi et al., 2011 [22] Opitz, 2009 [21] R˚užiˇcka, 2021 [39] Drábek, 2013 [42]
UIC, 2013 [25] Cacchiani et al., 2010 [32]Šimral, 2013 [44]
Kamenický, 2016 [41] Kuckelberg et al., 2015 [34] Drábek, 2014 [18]
Ljunggren et al., 2020 [28] Woodburn, 2015 [36] Janoš and Kˇríž, 2016 [15]
Stoilova et al., 2020 [24] Bablinski, 2016 [33] Michl et al., 2016 [40]
ˇ
Camaj et al., 2021 [27]Pöhle, 2016 [35] Chýle, 2021 [38]
Gašparík et al., 2021 [26]
Although Drábek introduced integrated concept of freight timetabling, as, to a certain
degree, an analogue to operational concept (timetable layout) of passenger railway trans-
port ordered by public sector, he considered only two, rather similar, sample freight trains
in his study of network-bound PFTPs for Prague node and adjoining lines [18].
Another problem that, to the authors’ knowledge, no research has focused on so far,
is internal priority of timetable construction (scheduling) of PFTPs, according to their
significance on particular mainline, as well as parameters of sample freight trains for
such PFTPs.
Based on the research cited above, two research gaps can be observed—detailed
differentiation (segmentation) of PFTPs for more than two types of freight trains and
internal (relative) prioritization of construction of differentiated PFTPs on the same railway
line. The research gaps are, for the sake of better comprehensibility, displayed in
Figure 1
in red colour, on the background of framework process for systematic construction of
network-bound PFTPs proposed by Drábek in his doctoral thesis [18].
3. Materials and Methods
3.1. Definition of the Problem
Railway capacity is limited, and on busy mainlines scarce, resource. It is difficult to
quantify, since it depends from its utilization—number and heterogeneity (difference in
average speeds) of the trains [
23
]. In the case of heterogeneous, mixed traffic (intercity,
regional or suburban, and freight trains together), lower service period of the fastest or the
slowest train segment (e.g., 30 min) even deteriorates the efficiency of capacity utilization—
the frequent alternation of faster and slower trains decreases potential of speed bundling,
i.e., scheduling homogeneous trains (with same or similar average speed) immediately
after each other. The ideal speed bundling occurs in underground or dedicated suburban
rail traffic—the result is the highest possible capacity utilization, and highest capacity in
terms of number of trains per hour and direction. In some cases, line capacity can be
additionally restricted by signaling system with long departure headway, but for majority
of Czech double-track mainlines, including the mainline researched below, it is not the case.
In this article, “freight-unfriendly” passenger timetable has been intentionally chosen—
heterogeneous trains with comparably low service periods (30 min or 1 h). Further, to
fill the above defined research gaps, one or two optimal sample freight trains are not
enough, since the data on really running freight trains show (as displayed in Section 3.5)
high diversity of freight train parameters—length, speed and power-to-mass ratio (PMR).
The last parameter is defined by total power of the locomotive(s) divided by gross hauled
mass of the train (i.e., mass of the wagons plus mass of the freight). PMR determines
relative ability of acceleration of particular freight train, compared with other freight trains.

Sustainability 2021,13, 8330 10 of 33
Thus, segmentation of periodic freight train paths (PFTPs) for more types of trains (in
terms of maximum allowed speed, length and PMR) is necessary, to meet at least prevailing
demand for capacity from freight rail operators. Of course, at night there is almost no
passenger traffic, and almost all the capacity is available for freight trains. However,
rail freight is often a part of multimodal supply chain, with increasing pressure on speed
and reliability of delivery time. So, many freight trains cannot wait for the end of daily
passenger traffic.
Hence, the problem solved in this article can be defined as follows: Within given
mainline and passenger timetable, first of all, divide the mainline into bottlenecks and
other sections with busy traffic on the one hand and other sections on the other hand. Then,
define and construct differentiated PFTPs (each one should maintain unified symmetry
in both directions), based on available demand data, and formulate framework process
for priority of their scheduling (construction). Try to avoid scheduling of any unnecessary
stopping. Try not to essentially reduce maximum length of a freight train, derived from
the shortest station track that is scheduled for overtaking. Finally, assess the solution by
comparison of number of really running freight trains that are suitable for the PFTPs with
number of trains that are not suitable for them.
3.2. A New Framework Process for Scheduling of Differentiated (Segmented) PFTPs
The introduced framework process for construction of symmetric PFTPs can advanta-
geously complement the framework process proposed by Drábek [
17
,
18
] (Figure 1), fills the
above defined research gaps and observes following generic principles:
(1)
Hierarchy of the offer of PFTPs and related hierarchy of the construction process;
(2)
Segmentation of the solved mainline (or network) into:
•Bottlenecks;
•Other busy sections (with dense, or strongly heterogeneous, traffic);
•Rest of the line/network.
(3)
Speed bundling whenever it saves capacity and does not lead to unnecessary extra
overtaking or waiting;
(4)
Active overtaking of or by a passenger train for the express PFTPs;
(5)
Scheduling of overtaking for subsequent PFTPs in various stations, so the capacity is
utilized efficiently (the acceleration of a freight train is the most capacity-consuming
phase of its running) [40].
As the 2nd edition of UIC Codex 406 [
25
] implies, increase of capacity utilization leads
to decrease of train path quality, and vice versa. The authors assume high daily fluctuations
in utilization of PFTPs by freight trains (see Section 3.5). Thus, they propose to offer enough
PFTPs per hour as a reserve in hourly capacity and segmentation of global (international)
PFTPs in terms of quality—express, fast and “common” with more scheduled stops for
overtaking. In many cases, only the best global PFTP would be used. Then, the second
one can be (if need be) used for a domestic freight train, given sufficient PMR and speed.
The third one would be used only if the first two ones cannot satisfy the freight operators’
demand for capacity in particular day, hour and direction.
The priority criteria for generic framework process for hierarchical construction of
PFTPs are arranged in descending order as displayed in Figure 2.
The framework process begins with delimitation of the solved area. This can be a
mainline or some chosen railway (sub)network. Here, the right segmentation of the area
into bottlenecks, other sections with dense (or strongly heterogeneous in terms of runtimes)
traffic and the remaining sections is crucial.

Sustainability 2021,13, 8330 11 of 33
Figure 2. Proposed generic framework process for hierarchical construction of PFTPs. Source: authors.
The second step is crucial for the efficiency of the following steps. On the one hand,
looser parameters of sample freight trains can increase the number of real freight trains that
would fit into the designed PFTPs. On the other hand, such looser parameters are likely
to deteriorate quality of particular PFTP in the form of increase in number of scheduled
stops for overtaking. The most important parameter to be wisely chosen is minimum
required PMR of the freight train. Further, maintaining of unified symmetry should be
considered (which can result in different train parameters for each direction), otherwise the
proceeding of the PFTPs to neighbouring lines would become extremely complicated
due to scheduling of PFTPs within passenger IPT, which maintains the unified symmetry.
Statistical evaluation of historical data on freight trains or other relevant data is helpful
for this step. However, some timetabling “common sense” is relevant as well—at least
verification of really achievable section runtimes in relation to the timetable of passenger
trains and a resulting breaking point, where any lower allowed PMR means necessity of
additional stop for overtaking.
The third step is formally equal to the same step in the original Drábek’s framework
process [
18
], but in the proposed new process it is more complex due to possible different
section runtimes of the differentiated PFTPs. As a rule, the below mentioned prioritization
of scheduling of the PFTPs should be maintained. At the same time, PFTPs with longer
runtimes through bottleneck sections (where any stop should be avoided if possible) should
be scheduled so that they do not waste railway capacity. This step is an indirect assessment
of available capacity, i.e., maximum number of PFTPs per hour and direction.

Sustainability 2021,13, 8330 12 of 33
Further, the actual (hierarchized) construction of PFTPs starts. First of all, the global
PFTPs are scheduled (during the whole mainline or with the longest distance within the
mainline or network)—with one exception. As the first one from them, the express PFTP
is scheduled. Its time position is determined by possibilities of active overtaking (of or
by the particular passenger train—see the explanation below) and by non-stop passing
through bottlenecks.
The PFTP with second priority is the global fast one. Active overtaking is not supposed
there. So, the quality is ensured by trade-off between as low number of scheduled stops as
possible and as high length limit as possible.
The third priority is reserved for the local PFTP with low PMR, so that number of
scheduled stops can be kept at a reasonable level. Each stop of a train with low PMR leads
to high secondary consumption of railway capacity, due to higher section runtime after the
stop caused by low acceleration.
The fourth priority is reserved for the global PFTP with high PMR, because more
frequent stopping of such trains leads to comparably low secondary consumption of
railway capacity.
The last two priorities are reserved for scheduling of local PFTPs in the remaining
sections, beginning with those with low PMR. Scheduling of global PFTPs with low PMR
is, from the authors’ point of view, not desired, because of disproportionate capacity
consumption. So, any “global” freight train, that can run during daytime, must have
parameters corresponding to at least one local PFTP in each section.
Then, the extension of PFTPs up to nodes in terms of solving of timetabling conflicts
there follows. This step lies already outside the proposed framework process.
In some special cases, lost capacity in the sense of the 1st edition of UIC 406 Codex [
23
]
can be utilized by active overtaking, i.e., without stopping of the overtaken train. This train
is scheduled to an opposite track, given that it is not occupied by traffic in the opposite
direction. Because of comparably high train frequency on a typical mainline, only a section
between neighbouring stations can be used for this purpose. While constructing PFTPs
this way, one has to pay attention whether another potential good PFTP is not avoided
(see Figure 3). Given PFTPs in both directions are constructed with a unified symmetry,
the mentioned active overtaking has to be scheduled symmetrically in both directions as
well. Thus, occupation of the opposite track is fundamentally impossible in the minutes
00 and 30, or 15 and 45 in addition (for 30 min period), and few minutes before and after
these times. The exact width of “forbidden timespan” depends on particular section and
occupation time of the opposite track by the overtaken train.
Figure 3.
Illustration of unusable capacity for the opposite direction (for not stopping trains).
Horizontal lines stand for neighbouring stations. A PFTP is displayed in blue colour. Source: authors.

Sustainability 2021,13, 8330 13 of 33
Freight trains with extremely low PMR, speed, or extremely long ones are supposed to
be scheduled either in freight off-peak hours, provided that one such train can run without
disturbing neither passenger nor “standard” freight trains, or at night, after scheduling of
“standard” freight trains.
The construction of all relevant PFTPs ends either when the expected peak demand
can be reasonably satisfied or when usable capacity is exhausted. As a rule of thumb,
if there is necessary to schedule overtaking in more than 33% of the stations on the railway
line, the quality of a PFTP is very low, so it should not be scheduled unless justified by
sufficient expected demand. The standard recovery margin for PFTPs should be 10% of
the calculated theoretical runtime (as usually used in timetabling studies by the authors,
see also Haldeman [29]).
The output of implementation of this framework process is a coordinated periodic
timetable pattern of passenger and freight train paths with unified symmetry that can be
extended into all-day exact timetable (with some minor adjustments—passenger off-/peak
times etc.). PFTPs should be offered to the railway undertakings in following descending
priority order:
(1)
Pre-arranged paths (PaPs) for European Rail Freight Corridors (as defined by Regula-
tion No. 913/2010 [45]) if relevant;
(2)
Annual timetable (e.g., combined transport);
(3)
Ad hoc regime (few days before actual running of the train);
(4)
Dispatching (delay or another irregularity).
The research question can be formulated as follows: Is it possible, with the help of
the framework process introduced above, to schedule such system of differentiated PFTPs
within dense and heterogeneous passenger symmetrical periodic timetable (or IPT), that at
least 50% of really running freight trains (in each direction) can fit into?
3.3. Materials Required for the Timetabling Experiment
For construction of PFTPs with the help of the framework process introduced above,
following inputs are necessary:
•
Track profile—number of tracks, stations with lengths of relevant tracks, speed and
gradient profiles, block signals, etc.;
•
Model timetable of passenger transport—its periodic pattern including peak only
services;
•Data indicating demand for PFTPs (at least past demand not older than 7 years).
For each pair of PFTPs (or group of more pairs of PFTPs), following sample train
parameters are necessary as an input:
•
Maximum allowed speed of the trainset (i.e., minimum of maximum allowed speeds
of vehicles);
•Minimum PMR—relevant for the acceleration phase and section runtimes;
•Specific speed profile of the train (if relevant).
3.4. Chosen Prospective Passenger Timetable and Railway Mainline
For the timetabling experiment—railway capacity model with PFTPs, where the
method introduced above was applied, such passenger timetable was chosen that was
considered unsuitable for freight train paths for the following reasons:
•Low period (30 min) of fastest (almost non-stop) trains;
•Suburban and regional trains with frequent stopping;
•
Low arrival and departure headways between following fast and regional trains (and
vice versa) that mostly did not enable to schedule a freight train path between them.
The authors used the timetable concept designed in diploma thesis of Rudolf
Vávra [
46
,
47
]—variant “Interposition”. This timetable in the thesis, as well as the fol-
lowing timetabling experiment, were elaborated in FBS-iPLAN software [
48
] that the
authors’ faculty was licensed to use for academic purposes. The timetabling experiment

Sustainability 2021,13, 8330 14 of 33
was then elaborated on the basis of FPL graphic timetable files from Vávra’s thesis [
46
],
with kind permission of the author and in accordance with the license of the thesis.
Table 2shows frequencies of fast and regional segments of passenger transport (per
section) in Vávra’s thesis [
46
,
47
], compared to frequencies in real actual timetable during
daytime (5:00 to 21:00) [
49
]. The authors added for the purpose of the experiment a pair
of hourly regional express trains between Prague and Kralupy, scheduled in interposition
with fast trains.
Table 2. Numbers of passenger trains per hour and direction for actual and prospective timetables.
Section
Actual Timetable (2020/2021) [49] Prospective Timetable [46,47]
Fast Regional Fast 2Regional
Praha—Roztoky u Prahy 12 4 4 4
Roztoky u Prahy—Kralupy n. V. 2 2 4 2
Kralupy n. V.—Roudnice n. L. 2 0.5 3 1
Roudnice n. L.—Ústínad Labem 2 1 3 1
Ústínad Labem—Dˇeˇcín2.5 1 2.5 1
Dˇeˇcín—Schöna 0.5 1 1 1
Schöna—Pirna 0.5 1 1 2
Bad Schandau—Pirna 0.5 2 1 2
Pirna—Dresden 0.5 0 1 0
1
From freight node station Praha-Libeˇn, passenger services occur gradually in three different stations or junctions.
For the sake of simplicity, this fact is neglected (relevant sections are very short).
2
including hourly regional
express services, scheduled by the authors.
Periodic arrival and departure times (minutes) are displayed in Figure 4, in the form
of a netgraph. It is the scheme of public transport lines, where the line style corresponds
to service period. Here, a single full line stands for hourly service, and double full line
stands for the 30 min service period. 2-h period is displayed by dashed line. The time
descriptors are on the side of travel direction from the edge—on the right. The descriptor
closer to a node shows the arrival time, the one further from the node shows the departure
time. Minutes at odd hours are written in italics. Intermediate stoppings are displayed on
particular lines either as a single point, or as a number of stoppings in a circle—for more
detailed explanations see Michl et al. [50].
Figure 4. Netgraph of periodic passenger services. Source: authors. Data: [46].

Sustainability 2021,13, 8330 15 of 33
Based on the suitable passenger timetable, the railway line 090/091 Praha-Libeˇn—
Praha-Holešovice—Ústínad Labem—Dˇeˇcín—Bad Schandau—Dresden Hauptbahnhof
(Hbf, main station) was chosen. This mainline is important for passenger transport—
long-distance connection of Prague with eastern and middle Germany, and with northern
and western Bohemia. In addition, Bohemian-Saxon Switzerland is an important tourist
attraction for both neighbouring countries. The mainline runs through Prague and Dresden
agglomerations, and closely together located cities of Ústínad Labem and Dˇeˇcín. Thus,
suburban and regional transport are significant as well—majority of the mainline runs
through catchment area of some city stated above [51].
There is considerable freight traffic on this mainline as well. International traffic is
strong, and consists of various goods, for instance, but not only, of intermodal containers
from German maritime ports and other terminals to terminals in the Czech Republic
or Slovakia. Another significant freight is coal. It is mostly transported from North
Bohemian Basin (westwards from Ústínad Labem). The destinations are coal power
plants—e.g., Mˇelník Power Plant, linked to DolníBeˇrkovice and Hnˇevice stations on this
mainline, or power plants eastwards from Prague. In northwestern Bohemia, there is
located chemical industry as well, and the chemicals are transported by railway. Last group
of goods are wagonload trains with mixed goods. Marshalling yards are situated in stations
Praha-Libeˇn, Kralupy nad Vltavou, Hnˇevice, Lovosice, Ústínad Labem západ (slightly
aside the mainline, but within Ústínad Labem node) and Dˇeˇcín. The stations listed above
mostly enable overtaking of 740 m long trains, as required by Article 39, Paragraph 2a of
the Regulation (EU) No 1315/2013 [
52
]. However, maximum train length is restricted by
the Czech infrastructure manager Správa železnic to 658 m [53–55].
It is apparent that during early morning and late evening, the passenger demand is
considerably lower, and so not all periodic passenger services are scheduled. The results of
analysis of current passenger timetable [
49
] displayed in Tables 3and 4show that there are
two main breaking points in frequency of passenger services: 5:00 and 21:00. Consequently,
daytime with considerable passenger traffic that creates a significant constraint for freight
timetabling, can be considered between 5:00 and 20:59.
Table 3.
Numbers of scheduled passenger train paths per segment, direction and timespan. Section
Praha-Podbaba—Roztoky u Prahy (direction to Dresden). Source: authors Data: [49,56].
Segment of Trains Numbers per Timespan Hourly Numbers
0:00 to 4:59 5:00 to 20:59 21:00 to 23:59 5:00 to 20:59 21:00 to 4:59
Intercity 0 16 1 1.00 0.13
Fast 0 14 2 0.88 0.25
Regional/Suburban 3 62 6 3.88 1.13
Total 3 92 9 5.75 1.50
Table 4.
Numbers of scheduled passenger train paths per segment, direction and timespan. Section
Roztoky u Prahy—Praha-Podbaba (direction to Prague). Source: authors Data: [49,56].
Segment of Trains Numbers per Timespan Hourly Numbers
0:00 to 4:59 5:00 to 20:59 21:00 to 23:59 5:00 to 20:59 21:00 to 4:59
Intercity 0 16 1 1.00 0.13
Fast 1 13 1 0.81 0.25
Regional/Suburban 2 62 6 3.88 1.00
Total 3 91 8 5.69 1.38
3.5. Freight Railway Transport Data
Parameters of freight trains on the mainline Praha—Ústín. L. were derived from
analytical part of research project Optimization of Development of the Railway System of
the Czech Republic in Terms of Transport Needs [
57
]. Within this project, data on trains
that really ran during March 2015 and March 2016, gathered and provided to researchers

Sustainability 2021,13, 8330 16 of 33
by Správa železnic, were analysed with the help of hierarchical cluster analysis by Kˇríž.
For each researched mainline, a representative section between neighbouring stations
was chosen. For the mainline Praha—Ústín. L., a section with the highest frequency of
trains—between Praha-Bubeneˇc and Roztoky u Prahy—was chosen.
Cluster analysis was elaborated on the basis on hauled gross mass and total power of
locomotives (both data sets were transformed to standard score). Ward’s method clustering
with Euclidean distance was chosen. Data of representative trains for each cluster, shown in
Table 5, were determined as 0.8 quantile of values of particular parameter from particular
cluster. The clusters are displayed in Figure 5.
Table 5. Parameters of clusters of freight trains between Praha-Bubeneˇc and Roztoky u Prahy [57]3.
No. No. of Trains Hauled Gross Mass [t] Power [kW] PMR [–]
No. of Trains by Max. Speed [km/h]
90 100 Other
1 865 1685 6000 3.56 31 830 4
2 1109 1865 3060 1.64 674 384 51
3 708 705 3060 4.34 324 267 117
3Based on data on trains that really ran during March 2015 and March 2016, by Správa železnic.
Figure 5.
Clusters of freight trains between Praha-Bubeneˇc and Roztoky u Prahy. Based on data on
trains that really ran during March 2015 and March 2016, by Správa železnic [57].
Table 6shows numbers of scheduled freight train paths (FTPs) in the Czech 2020/2021
railway timetable [
58
]. The authors counted all regularly scheduled FTPs, except local
(manipulation) freight trains and locomotive trains. For each section displayed in the
table, maximum number of FTPs per one section between neighbouring stations within the
displayed section (chosen for each direction separately) was considered.
Table 6.
Numbers of scheduled regular freight train paths (FTPs) per section and daytime hour.
Source: authors. Data: [58].
Section Daily No. of FTPs
There
Daily No of FTPs.
Back
Average Daily No.
of FTPs
No. of FTPs per
DayTime Hour 4
Praha—Ústínad
Labem 54 54 54 1.69
Ústínad
Labem—Dˇeˇcín50 39 45 1.41
Dˇeˇcín—Dresden 107 116 112 3.5
4Divided by 32.

Sustainability 2021,13, 8330 17 of 33
For more accurate estimation of average freight demand for capacity during daytime,
when passenger and freight trains have to share the capacity, following assumptions were
made:
(1)
Daytime with considerable passenger traffic lasts 16 h (between 5:00 to 21:00—see
numbers of scheduled passenger trains in Tables 3and 4);
(2)
Outside daytime, twice as many FTPs can be scheduled, compared to daytime (see
average numbers of passenger train paths per hour and direction during the chosen
daytime, shown in Tables 3and 4).
Thus, average numbers of FTPs per section were divided by (16 + 8
×
2), i.e., by 32.
This way, average hourly numbers of FTPs during daytime, per section and direction,
were estimated.
For further verification of the proposed framework process and timetabling experi-
ment, slightly different data from the same data set [
57
] were chosen. The section Lovosice
mesto—MaléŽernoseky was chosen, which also lays within the section Praha—Ústí, where
the construction of PFTPs is the most complex and challenging. The reason for the choice
was that in this section there ran most freight trains (all trains from the south to Ústíand
from Lovosice terminal of combined transport northwards). The total number of all really
running freight trains through this section during March 2015 and March 2016, except for
53 trains removed from the data set, was 4173 [
57
]. There were three reasons for removal
(extreme PMR in all cases): trains composed from locomotive(s) only with PMR above 20,
other trains with extreme PMR (above 30) and incorrect data (PMR equal to zero).
Figure 6shows an example of considerable fluctuations in number of really running
freight trains per hour and direction. The day with the largest number of really running
freight trains (119)—29 March 2015—was chosen.
Figure 6.
Example of hourly fluctuations in numbers of really running trains between Lovosice mˇesto
and MaléŽernoseky, on 29 March 2015, 5:00 to 20:59. Source: authors Data: [57].
Figures 7and 8show considerable fluctuations in number of really running freight
trains per day (years 2015 and 2016 are displayed separately).

Sustainability 2021,13, 8330 18 of 33
Figure 7.
Daily fluctuations in numbers of really running trains between Lovosice mˇesto and Malé
Žernoseky, March 2015. Numbers on the horizontal axis stand for the weeks in the month. Source:
authors Data: [57].
Figure 8.
Daily fluctuations in numbers of really running trains between Lovosice mˇesto and Malé
Žernoseky, March 2016. Numbers on the horizontal axis stand for the weeks in the month. Source:
authors Data: [57].
3.6. Chosen Sample Freight Trains and PFTPs to Be Scheduled in the Experiment
Parameters of sample freight trains, displayed in Table 7, were, however, determined
by the authors as looser (especially in terms of PMR) to enable more freight trains to use
designed PFTPs.
For sample trains of the PFTPs, defined in Table 7, locomotive classes, which are
common on the researched mainline, were used: Class DB (German Railways) 185 for
international express freight trains, Class ˇ
CDC (Czech Railways Cargo) 372 for freight
trains from/to “right-shore” mainline 072/073 to Dresden and DC only locomotives for
local PFTPs between Prague and Dˇeˇcín—a rather obsolete, but still common Class ˇ
CD
(Czech Railways) 121 and more powerful Class ˇ
CD 163. The power values displayed in
Table 7are cited from FBS-iPLAN locomotive database [48].
For the timetabling experiment, only electric locomotives were considered. However,
freight trains with diesel locomotive(s) can use such PFTPs as well, given sufficient PMR.
For the global PFTPs between Prague and Dresden, comparably higher PMRs (between
2.2 and 2.5) were chosen, so that better quality of a PFTP—or, at least, higher acceleration
after stop for an overtaking—can be ensured.

Sustainability 2021,13, 8330 19 of 33
Table 7. Parameters of sample freight trains for PFTPs. Sources: authors [48].
Catalogue PFTP 41xxx 45xxx 44xxx 42xxx 46xxx 48xxx 65xxx 67xxx
Route Praha—Dresden Dˇeˇcín P. Ž.—Dresden
Praha—Ústín. L.
Ústín. L.—Dˇeˇcín
Locomotive class DB 185 DB 185 ˇ
CDC 372 DB 185 ˇ
CDC 372 ˇ
CDC 372 ˇ
CDC 121 ˇ
CDC 163
Power [kW] 5600 5600 3080 5600 3080 3080 2032 3060
Hauled gross mass [t] 2250 2430 1360 1800 1988 2544 2070 2067
PMR [–] 2.49 2.30 2.26 3.11 1.55 1.21 0.98 1.48
Length [m] 514 554 262 415 358 466 380 382
Maximum speed [km/h] 100 100 100 100 90 90 90 90
Braked weight percentage 100 100 100 100 90 80 80 90
Brake regime 5P P P P P G G G
Timetabling priority 1. 2. 4. 7. 6. 5. 3. 3.
5P—passenger train braking, G—freight train braking.
The local PFTPs between Prague and Ústíwere, on the other hand, designed for trains
with lower PMR (between circa 1 and 1.5)—coal or mixed (freight) trains.
Between Dˇeˇcín-ProstˇredníŽleb, where the “right-shore” mainline 072/073 from Kolín
with busy freight traffic connects to the researched mainline, and Dresden, PFTPs for
various freight trains have to be considered. So, the chosen PMRs vary from circa 1.2 to 3.1.
Quantiles (semi-deciles) of PMR from the data set [
57
] are displayed in Table 8. If PMR
was the only limiting parameter of a freight train, circa 80% of real freight trains from
the data set [
57
] would fit into the local PFTPs and 55% into the global PFTPs. However,
there remains maximum speed of a train—lower values than 90 km/h would prevent a
train from any PFTP. Another limiting parameter—train length—will be defined after the
timetabling experiment. The length limit will depend on the shortest station track used for
overtaking for particular PFTP and direction (see Section 4.2).
Table 8.
Semi-deciles of PMR of really running freight trains (March 2015 and March 2016). Source:
authors. Data: [57].
x x Quantile of PMR
0 0.38
0.05 0.82
0.1 0.83
0.15 0.87
0.2 1.08
0.25 1.29
0.3 1.54
0.35 1.87
0.4 2.20
0.45 2.30
0.5 2.39
0.55 2.46
0.6 2.72
0.65 3.16
0.7 3.46
0.75 3.79
0.8 4.14
0.85 4.65
0.9 5.84
0.95 7.84
1 27.84
Compared with estimated numbers of freight train paths per daytime hour, section and
direction (see Tables 3and 4), the authors have decided to schedule approximately double
hourly numbers of PFTPs to meet the demand fluctuations (see Figures 6–8).
Maximum speeds of the sample trains for PFTPs were distributed in accordance with
their relative frequencies in the data set in particular directions [
57
]. For the direction from
Prague to Dresden, relative frequencies of speeds 90/95 and 100 km/h (or higher) are
35.36% and 58.36% For the direction from Dresden to Prague, relative frequencies of speeds
90/95 and 100 km/h (or higher) are 52.40% and 40.03%.

Sustainability 2021,13, 8330 20 of 33
4. Results
4.1. Construction of PFTPs
The timetabling experiment was elaborated in FBS-iPLAN software [
48
], on the basis
of FPL graphic timetable files from Vávra’s thesis [
46
], with kind permission of the author
and in accordance with license of the thesis. Further, academic licenses of Microsoft
Excel
®
[
59
] and MATLAB [
60
] software tools were used for processing of the freight train
data [57,58].
The timetabling experiment has resulted in scheduling of eight pairs of hourly PFTPs
with unified symmetry (for parameters, see Table 7). Three pairs of global PFTPs proceeded
during the whole solved mainline, three pairs between Dˇeˇcín-ProstˇredníŽleb and Dresden
(to connect Dresden node with the “right-shore” mainline 072/073),one between Prague
and Ústí, and one between Ústíand Dˇeˇcín (both for trains with low PMR). The netgraph
with periodic passenger and freight train paths is displayed in Figure 9. Recovery margins
(surcharges) per section and direction are displayed in Table 9. Active overtaking (PFTP
41xxx) took place between Nelahozeves and Vraˇnany, and between Ústíand Povrly (in both
directions). Resulting graphical timetable (train diagram) is attached as Supplementary
Material Figure S1.
Table 9.
Recovery margins (displayed as a percentage of minimum runtime, as an input for runtime
calculation by FBS software [
48
]) for particular PFTPs and sections (direction there/back). Source: authors.
Section Border 41xxx 45xxx 44xxx 42xxx 46xxx 48xxx 65xxx 67xxx
Praha-Libeˇn 20/20 18/18 25/25 10/10
Výh. Praha-Bubeneˇc 20/20 18/18 25/25 10/10
Roztoky u Prahy 20/20 18/18 25/25 10/10
Libˇcice nad Vltavou 20/20 18/18 25/25 10/10
Kralupy nad Vltavou 37/32 10/10 25/25 10/10
Nelahozeves 37/50 12/16 50/6 10/6
Vraˇnany 21/11 12/16 50/6 10/6
DolníBeˇrkovice 21/11 12/16 50/6 10/6
Hnˇevice 21/11 18/18 50/6 6/6
Roudnice nad Labem 21/11 18/18 50/20 6/6
Hrobce 21/11 18/18 50/20 4/6
Bohušovice nad Ohˇrí21/24 18/18 50/20 4/6
Lovosice 21/24 25/30 50/20 4/6
Prackovice nad Labem 21/24 25/30 15/50 74/6
Ústínad Labem hl. n. 2/4 10/10 50/50 710/10
Povrly 10/10 14/50 50/50 710/10
Dˇeˇcín hl. n. jih 10/10 50 6/50 650/50 7
Dˇeˇcín-ProstˇredníŽleb 10/10 50 6/50 650/50 710/10 6/6 30/40
Schöna 10/10 10/10 50/50 10/40 10/10 30/10
Bad Schandau 10/10 10/10 50/50 10/10 10/10 10/10
Kurort Rathen 10/10 10/10 50/35 10/10 10/10 10/10
Pirna 10/10 10/10 10/10 10/10 10/10 10/10
Dresden Hauptbahnhof
6
Additional 5% mass margin (surcharge), as an input for calculation by FBS software.
7
Additional 20% mass
margin (surcharge), as an input for calculation by FBS software.

Sustainability 2021,13, 8330 21 of 33
Figure 9. Netgraph of periodic passenger services (data: [46]) and resulting PFTPs. Source: authors.
It is evident that more passenger (and freight) train paths per hour cannot enable lower
average number of scheduled stops for overtaking per one freight train path. However,
all PFTPs are supposed to be utilized only in absolute peaks of freight operators’ demand,
which is few days in a month, and only for some hours during such a day
(see Figures 6–8).

Sustainability 2021,13, 8330 22 of 33
Average numbers of stops per freight train path in 2020/2021 timetable [
58
] and in the
presented experiment are compared in Table 10.
Table 10. Comparison of absolute and relative numbers of scheduled stops [53–55,58]. Source: authors.
Section
Timetable 2020/2021 [58] (Daily) Timetabling Experiment (Hourly)
There Back
No of Stops
per Train
Path
There Back
No of Stops
per Train
Path
Praha—Ústínad Labem 33 31 0.59 6 8 1.75
Ústínad Labem—Dˇeˇcín45 45 1.01 1 1 0.25
Dˇeˇcín—Dresden 5 13 0.08 1 1 0.17
The above stated numbers of scheduled stops would have been higher, if extensive
“bending” (artificial runtime lengthening) of PFTPs had not been implemented. Such “bend-
ing” is evident from Table 9—see the recovery margins over 10%. The first section, where all
PFTPs (except 65xxx with comparably low PMR) had to be bent, was from Prague node to
Kralupy. The reason was homogenization of train paths with frequently stopping suburban
passenger trains (otherwise there would have been lack of railway capacity for PFTPs).
Between Kralupy and Ústí, the bundled PFTPs 41xxx and 45xxx were slightly bent
for enabling necessary synchronization runtime of 41xxx—between two cases of active
overtaking. During the first one (between Nelahozeves and Vraˇnany), 41xxx was even more
bent to have enough long sectional runtime to be overtaken by nonstop fast train to Cheb.
On the other hand, standard recovery margin of 41xxx was reduced between Ústíand
Povrly to enable overtaking of a frequently stopping regional passenger train. The above
described solution is functionally symmetrical for both directions (and difference between
recovery margins in direction there and back is comparably low).
For the pair of PFTPs 65xxx, no bending in any direction was implemented. On the
contrary—the standard recovery margin had to be reduced from Hnˇevice to Ústíand from
Ústíto Nelahozeves, otherwise there would have been necessary more scheduled stops
for overtaking. The pair of short PFTPs 67xxx had standard recovery margin 10% in both
directions.
The timetabling experiment has demonstrated that high level of external unified sym-
metry of PFTP could be maintained. The only exception was the pair of PFTPs 65xxx with
low PMR, scheduled between Prague and Ústí. From Prague, most probably thanks to ride
down the stream of Vltava and Labe rivers, only two stops for overtaking were scheduled.
In the opposite direction, there were scheduled four stops for overtaking between Ústíand
Prague. This discrepancy has influenced asymmetric arrival and departure time from/to
direction Ústínad Labem západ node station (in the junction Ústínad Labem jih next to
Ústímain station).
For all other PFTPs, external unified symmetry was maintained. For almost all of
them, internal unified symmetry was maintained as well. The only exception was the pair
of PFTPs 42xxx between Dˇeˇcín-ProstˇredníŽleb and Dresden. As a pair of PFTPs (for given
distance) with the lowest PMR, it was chosen to be stopped in Dˇeˇcín-ProstˇredníŽleb in the
direction Dˇeˇcín východ (and in Bad Schandau Ost in the opposite direction) for the purpose
of synchronization with free symmetric time slots on the single-track section between the
stations Dˇeˇcín-ProstˇredníŽleb and Dˇeˇcín východ (which lays on the “right-shore” mainline
072/073).
For practical reasons (actual passenger timetable—see Figure 4), the timetable symme-
try axis of PFTPs was not the minute zero, but the minutes circa 56 to 58.
The proposed catalogue PFTPs can be used by freight operators in different ways, de-
pending on actual capacity utilization on particular day and at particular hour. The default
case is that each PFTP is used by a train with corresponding parameters.
When at least one PFTP at particular hour and direction is not used, some changes
in ad-hoc (daily) timetabling or train dispatching are possible. For example, a suitable

Sustainability 2021,13, 8330 23 of 33
train with shorter route can be scheduled in a part of the corresponding PFTP. Or, if the
parameters of the particular train are suitable for two different PFTPs, an additional stop
can be scheduled, and the train can use both PFTPs partially.
If more PFTPs remain free, the best one (in terms of number of scheduled stops) can
be allocated for the train—given the train fulfils sufficient parameters to fit into it.
4.2. Limitations of Train Length
Table 11 shows maximum length of a freight train, including locomotive(s). It was
calculated from the usable length of the shortest station track scheduled for overtaking for
particular PFTP. From this length (between departure signals in opposite directions as a
rule [
58
]), 20 m was subtracted to ensure visibility of the signal from the driver’s cab and
to ensure smooth arrival and stopping of a freight train on the track.
Table 11.
Maximum length of a train for particular PFTP and direction [m] and scheduled numbers
of intermediate stops. Source: authors. Data: [53–55,58].
PFTP There (to Dresden) Back (to Prague) Stops There Stops Back
41xxx 658 8658 80 0
45xxx 635 630 2 2
44xxx 635 627 3 3
42xxx 580 666 1 1
46xxx 695 8,9 695 8,9 0 0
48xxx 695 8,9 695 8,9 0 0
65xxx 615 593 2 4
67xxx 695 8695 80 0
8
Limitation of train length, defined by Správa železnic [
53
–
55
].
9
Train length is likely to be further limited by
adjoining “right-shore” mainline 072/073.
For some PFTPs that are not scheduled to stop (41xxx, 46xxx and 48xxx), the only limit
for length of a train was defined by the infrastructure manager—Správa železnic [53–55].
4.3. Numbers of Really Running Freight Train That Would Fit into Designed PFTPs
The benefit of the above proposed offer of differentiated (segmented) PFTPs is mea-
surable by share of real freight trains (for given timespan) whose parameters would be
suitable for at least one PFTP in particular direction. It gives us information, which share
of trains can run during daytime (given that demand in particular hour and direction is
not exceeded) and which share of trains has to be scheduled at night.
Estimation of share of real freight trains that would fit into proposed PFTPs was
elaborated, based on the data from March 2015 and March 2016, provided by Správa
železnic for the analytical part of research project Optimization of Development of the
Railway System of the Czech Republic in Terms of Transport Needs [
57
], since there are
legal obstacles for obtaining more recent data so far (commercial sensitivity). The proposed
framework process is generic, and thus independent of particular annual timetable. So,
a prospective “freight-unfriendly” passenger timetable can be for the sake of the verification
of the framework process combined with few years old historical data on freight trains.
The section Lovosice mesto—MaléŽernoseky was chosen, which lays within section
Praha—Ústí,
where the construction of PFTPs was the most complex and challenging.
In this section, there should run the most freight trains (all trains from the south to Ústí
and from Lovosice terminal of combined transport northwards).
Due to differentiated parameters for each PFTP and even direction, determination of
all suitable trains is comparably complex. Freight trains that either are not suitable at all
because of low maximum speed, low PMR or both, and freight trains that fit only into one
PFTP, are the easiest to determine and calculate.
For all other freight trains, their maximum speed and PMR correspond to two or
more PFTPs in particular direction, So, if they fit into at least one of the corresponding
length limits, they are suitable for the proposed PFTPs. Table 12 shows numbers of suitable

Sustainability 2021,13, 8330 24 of 33
trains per (sub)category and direction. Table 13 shows numbers of unsuitable trains in the
same way.
Table 12.
Numbers of freight trains which are suitable for the proposed PFTPs per (sub)category and
direction. Source: authors. Data: [57].
Necessary Conditions Number of Suitable Trains per Direction
Max. Speed V[km/h] PMR [–] Length l[m] There (to Dresden) Back (to Prague)
V≥90 0.98 ≤PMR < 2.26 l≤615 354
l≤593 306
90 ≤V< 100 PMR ≥2.26 l≤635 454
l≤627 238
V≥100 2.26 ≤PMR < 2.3 l≤635 68
l≤627 4
V≥100 2.3 ≤PMR < 2.49 l≤635 268
l≤630 48
V≥100 PMR ≥2.49 l≤658 698 560
Total Numbers of Suitable Trains per Direction 1842 1156
Shares of Suitable Trains per Direction [%] 88.26 55.42
Total Number of Suitable Trains 2998
Share of Suitable Trains [%] 71.8
Table 13.
Numbers of freight trains which are unsuitable for the proposed PFTPs per (sub)category
and direction. Source: authors. Data: [57].
Group of Trains by Their Parameters Number of Unsuitable Trains
per Direction
Max. Speed V[km/h] PMR [–] Length l[m] There (to Dresden) Back (to Prague)
V< 90 131 158
PMR < 0.98 43 712
Subtraction of Numbers of Trains with Both Limitations −8−33
V≥90 0.98 ≤PMR < 2.26 l> 615 62
l> 593 74
90 ≤V< 100 PMR ≥2.26 l> 635 2
l> 627 0
V≥100 2.26 ≤PMR < 2.3 l> 635 2
l> 627 0
V≥100 2.3 ≤PMR < 2.49 l> 635 1
l> 630 5
V≥100 PMR ≥2.49 l> 658 12 14
Total Numbers of Unsuitable Trains per Direction 245 930
Shares of Unsuitable Trains per Direction [%] 11.74 44.58
Total Number of Unsuitable Trains 1175
Share of Unsuitable Trains [%] 28.2
The data displayed in Tables 12 and 13 show that the most limiting parameter of
freight trains that prevents them from fitting into any PFTP is low PMR. It predominantly
affects trains in the direction to Prague. Further limiting parameter is maximum speed
lower than 90 km/h. For some trains with sufficient speed and PMR higher or equal to 0.98,
but lower than 2.26, their high length prevents them from fitting into any PFTP. In total,
much higher share of unsuitable trains in the direction to Prague is caused mostly by trains
with PMR lower than 0.98.

Sustainability 2021,13, 8330 25 of 33
The research question from the end of Section 3.2 can be responded positively—even
the worse result for the direction to Prague is higher than 50% (55.4% of really running
freight trains in this direction are suitable for the PFTPs designed according to the proposed
framework process).
4.4. Sensitivity Analysis—Variable Lowest Allowed PMR for the PFTP 65xxx
The results presented above should be verified with the help of sensitivity analysis.
The purpose of conducting a sensitivity analysis is changing an input value to see the extent
of changes in the output (results). Thus, it is necessary to choose variable(s), whose input
values will be changed.
Only 289 trains from the data set [
57
], which is 6.93% of all researched trains, would not
fit into proposed PFTPs because of allowed speed, which is lower than 90 km/h. Al-
lowed maximum length of a train is determined by the shortest station track, where stop-
ping (for overtaking by a faster train) is scheduled, so neither speed nor length are suitable
variables for sensitivity analysis. That leaves only PMR.
755 trains would not fit into proposed PFTPs because of PMR lower than 0.98. Af-
ter subtraction of 41 trains with speed lower than 90 km/h, there remain 714 trains that,
however, may as well not fit into the PFTPs because of length. The PFTP 65xxx with the
lowest minimum PMR is restricted by maximum train length 615 m, if scheduled to Dres-
den, and 593 m, if scheduled to Prague. From 714 trains, there remain 35 trains to Dresden
(no train exceeds the allowed length) and 676 trains to Prague (only three additional trains
exceed the allowed length). As a result, there are 711 trains which are restricted only by
little PMR.
The authors propose following sensitivity analysis. The only variable which is suitable
for changing its values (as an input) is PMR, which leads to changing one-factor-at-a-time.
Since PMR values occur in the data set in a discrete way, the steps were chosen to be
discrete as well. Let the minimum allowed PMR for the PFTP gradually change from 0.5 to
1.5, with the step 0.1. The reference values (one for each direction) are numbers of trains
that would fit (only) into the PFTP 65xxx, given the minimum allowed PMR is equal to
0.98 (as defined in the timetabling experiment above). For the direction from Prague to
Dresden, reference value is 354 trains. For the direction from Dresden to Prague, reference
value is 306 trains (see first three rows of Table 12). Absolute differences are calculated as
subtraction between number of trains that would fit (only) into the PFTP 65xxx in particular
direction and reference value for this direction. Maximum allowed PMR is always lower
than 2.26, so no trains considered in the sensitivity analysis can fit into any another PFTP.
Then, for each step, new share of suitable trains for particular direction is calculated
with following formulas (one for each direction) as percentual share of all suitable trains in
particular direction with the lowest PMR ito all researched trains in particular direction
(sums of suitable trains for the lowest PMR 0.98 and sums of all researched trains per each
direction can be derived from the sums per direction in Tables 12 and 13):
Si,PD =100 Di,PD +1842
2087 (1)
for the direction from Prague to Dresden, where
•
D
i,PD
is absolute difference in number of suitable trains for the direction from Prague
to Dresden and lowest PMR i;
•Si,PD is share of suitable trains for the same direction and lowest PMR iin percent.
and:
Si,DP =100 Di,DP +1156
2086 (2)
for the direction from Dresden to Prague, where
•
D
i,DP
is absolute difference in number of suitable trains for the direction from Dresden
to Prague and lowest PMR i;
•Si,DP is share of suitable trains for the same direction and lowest PMR iin percent.

Sustainability 2021,13, 8330 26 of 33
Results of sensitivity analysis for each direction are displayed in Tables 14 and 15.
Default PMRs with zero difference and corresponding reference shares of suitable trains
are displayed in the sixth rows of both tables.
Table 14.
Sensitivity analysis of the lowest allowed PMR for the PFTP 65xxx from Prague to Dresden [
57
].
Lowest PMR [kW/t] Absolute Difference in
Number of Suitable Trains Share of Suitable Trains [%]
0.5 32 89.79
0.6 28 89.60
0.7 28 89.60
0.8 25 89.46
0.9 8 88.64
0.98 0 88.26
1 0 88.26
1.1 −28 86.92
1.2 −48 85.96
1.3 −78 84.52
1.4 −85 84.19
1.5 −111 82.94
Table 15.
Sensitivity analysis of the lowest allowed PMR for the PFTP 65xxx from Dresden to Prague [
57
].
Lowest PMR [kW/t] Absolute Difference in
Number of Suitable Trains Share of Suitable Trains [%]
0.5 676 87.82
0.6 675 87.78
0.7 652 86.67
0.8 606 84.47
0.9 72 58.87
0.98 0 55.42
1−6 55.13
1.1 −45 53.26
1.2 −80 51.58
1.3 −119 49.71
1.4 −141 48.66
1.5 −193 46.16
Sensitivity analysis for this direction shows little differences in number of suitable
trains, whose absolute values slightly increase when the lowest allowed PMR increases.
Sensitivity analysis for the direction from Dresden to Prague shows a strong leap
in number of suitable trains when the lowest allowed PMR reaches the value 0.8—see
the fourth row in Table 15. This finding corresponds with large difference in numbers
of suitable trains in comparison to another direction. Considering known origins and
destinations of freight rail transport on and around the researched mainline, the authors
believe that most of the trains with PMR around 0.8 are loaded coal trains, running from
coal mines in Northwestern Bohemia southwards. Not only their gross hauled mass
mostly exceeds 2400 t, but the tractive power of their locomotives is comparably low—
approximately 2000 kW. Such trains are hauled mostly by obsolete electric locomotives,
so their replacement by newer and more powerful ones is only a matter of time.
5. Discussion
5.1. Interpretation in the Context of Previous Studies
The presented framework process makes Drábek’s approach from his doctoral the-
sis [
18
] more accurate in terms of capacity offer for various freight trains and related
hierarchy of the train path construction process. The PFTPs are constructed in suitable time
windows (as recommended by Lindner and von Redern [
12
]), with maintained unified

Sustainability 2021,13, 8330 27 of 33
symmetry axis for both directions. Outside bottlenecks (Prague—Kralupy in the exper-
iment) and other busy sections (Kralupy—Ústíin the experiment), flexibility in freight
train stopping (as referred to by Opitz [
21
] and Vromans [
30
]) is possible, provided that
not all PFTPs are allocated for particular time and direction, and the particular freight
train fulfils the limits of all utilized PFTPs. The presented approach proposes to locally
homogenize PFTPs with passenger train paths in bottlenecks and other busy sections as
much as possible. For the express PFTPs (41xxx in the experiment) the approach proposes
to implement active overtaking in suitable sections and times, as proposed by Janoš and
Kˇríž [
15
]. The presented approach is holistic, generic and this way brings new quality into
scheduling and allocation of periodic (pre-arranged) freight capacity.
5.2. Implications for Sustainable Transportation
The main benefit of the proposed framework process is possibility to create stable,
predictable and differentiated (segmented) offer of capacity (in the form of PFTPs) to freight
operators. Such offer helps to create transparent, discrimination-free market environment
that is one of important incentives for modal shift in freight transport from road to rail.
If the demand from the operators for particular hour and direction is lower, they can
use PFTPs with lower numbers of scheduled stops—even the non-stop one—provided that
the particular train fulfils the parameters of such PFTP.
The proposed concept motivates the operators to improve parameters of their freight
trains—speed and PMR, and not exceed certain length. However, the more attractive PFTP,
the higher length limit thanks to lower number of overtaking stations. On the other hand,
substantial recovery margins in many sections (see Table 9) enable the operators to save
traction energy by coasting (running without tractive power). As proven for instance by
Lieskovskýet al. [
61
], approximately 35% of traction energy can be saved by coasting of a
seldom-stopping train.
Pre-arranged, differentiated PFTPs can also help to better identify bottlenecks from
freight rail point of view—heterogeneous passenger train paths that force freight trains
to stop for overtaking and short overtaking tracks. It is not likely to make fast train paths
slower (although in short suburban sections it may be necessary—see Michl et al. [
40
]).
But regional train paths can be made a little faster by alternate service of stops with
little passenger demand (i.e., only one out of two subsequent regional trains would stop
there). IPT of passenger transport has helped to formulate targeted requirements for
infrastructure improvements. PFTPs, if they emerge into stable, multi-year offer of capacity,
can be effective in the same way. For instance, some stations in some direction would be
periodically used for overtaking of freight trains. So, if there is no built-up land around,
overtaking tracks can be lengthened to enable smooth operation of 740 m long trains,
as required by Regulation (EU) No 1315/2013 [
52
], during the nearest modernization
of particular mainline. Obviously, it is not economically sustainable to lengthen many
stations in such a way, so the knowledge about right targeted improvement is necessary.
On the other hand, seldomly used infrastructure can be reduced and the unused one can
be cancelled.
5.3. Discussion of the Proposed Approach and Resulting Timetable
From Prague node to Kralupy, there is a classical (sub)urban bottleneck where time
windows for PFTPs are strictly determined. However, by construction of each PFTP,
exact time window can be chosen in accordance with priority criteria formulated in the
presented framework process (see Section 3.2).
From Kralupy to Ústí, there is an “overtaking zone”, which is caused by high train
path heterogeneity (from regional trains with low average speed, through PFTPs, to non-
stop long-distance trains in 30 min period, running up to 160 km/h in the middle of this
section). The “tight” periodic pattern of passenger timetable, where regional trains are
overtaken as well, and no time windows that remain for the freight trains to run through
without a stop, further deteriorate the quality of PFTPs. Such passenger timetable pattern

Sustainability 2021,13, 8330 28 of 33
was intentionally chosen to demonstrate limits of freight timetabling. The resulting PFTPs
show clearly that this aim was achieved. Without scheduling of active overtaking in two
sections, no non-stop PFTP could have been designed within this passenger timetable.
The section between Ústíand Dˇeˇcín nodes is very short. It contains only one inter-
mediate station between these nodes—Povrly. There is only one pair of scheduled stops
for overtaking—the PFTP 45xxx. The overtaking here represents synchronization of two
advantageous time windows—from Prague and short overtaking in Nelahozeves by non-
stop passenger train, which actively overtakes the PFTP 41xxx, to narrow time window
between ending regional and beginning suburban trains in Bad Schandau—and vice versa
in the opposite direction.
The section from Dˇeˇcín to Pirna is tricky—number of passenger trains is comparably
low, but the regional ones stop very frequently—especially between Dˇeˇcín and Bad Schan-
dau. Resulting low average speed leads to necessity of intensive “bending” of PFTPs (44xxx
and 45xxx) to avoid further scheduled stops—see Table 9. To a lesser degree, this constraint
proceeds to Pirna, where suburban trains leave the researched mainline to a separate
parallel double-track line.
From Pirna to Dresden Hbf, PFTPs are limited only by hourly EuroCity trains (which
run faster, but the section is comparably short) and mutual coordination.
Further from Dresden Hbf freight trains enter a separate double-track line outside the
researched mainline, up to marshalling yard Dresden-Friedrichstadt.
6. Conclusions
Railway plays a key role in the transition towards sustainable transportation system.
However, coexistence of busy passenger and freight traffic on the same infrastructure
represents a challenge to the timetabling process. In Central Europe, periodic timetable
or IPT prevail in passenger transport, so planning of freight capacity in a periodic pattern
with unified symmetry can simplify the timetabling process and make it more transparent.
The data used for verification of the timetabling experiment [
57
] has clearly shown
that parameters of freight trains, even on the same mainline, were very diverse. Thus,
one sample freight train for construction of PFTPs was definitely not enough to meet at
least majority of demand from the freight rail operators. Further, the same data have
shown considerable fluctuations of numbers of really running freight trains per direction,
during both week and day.
To the authors’ knowledge, no proposal for such detailed differentiation (segmenta-
tion) of PFTPs has been published so far. Due to prevalence of quantitative timetabling
research with few criteria to be optimised and considerable simplifications (especially in
the field of freight train paths), there still remains a gap between sophisticated theory and
practice in the freight train path allocation. The authors developed the proposed framework
process, which included design of differentiated PFTPs with various construction priority
with the aim of helping to fill this gap. The proposed process can, for instance, advanta-
geously complement the framework process proposed in Drábek’s doctoral thesis [18].
The resulting symmetrical timetable of hourly PFTPs evinces strong variedness, de-
pending on section of the solved mainline. From Prague to Kralupy, a bottleneck full of
mixed passenger traffic forces PFTPs to be strictly parallel and bundled by two subsequent
ones. So, the PFTP 65xxx with the lowest PMR determines the “gradient angle” of all PFTPs
in both directions. Other PFTPs in the bottleneck have to be slowed accordingly, otherwise
the train path heterogeneity would increase, and, as a result, capacity would decrease.
The timetabling experiment (case study) has demonstrated correctness of the presented
framework process, as well as its limits given by density and heterogeneity of passenger
railway (the latter caused mainly by speed of long-distance trains and frequency of stops of
regional trains). The research question, whether at least 50% of really run freight trains from
the available data set (in each direction) can fit into PFTPs designed in accordance with
the proposed framework process, was responded positively. In the direction to Dresden,
88.26% of the researched trains were suitable. In the direction to Prague, only 55.42% of

Sustainability 2021,13, 8330 29 of 33
researched freight trains were suitable, due to significant share of loaded coal trains from
Ústínad Labem southwards, characterized with low power-to-mass ratio (PMR) due to
obsolete electric locomotives. However, their replacement by more powerful ones is only a
matter of time.
The resulting line periodic timetable has clearly shown that freight railway suffers not
directly from frequency of passenger trains, but from their high heterogeneity in terms of
average speed. Regional train paths are obviously considerably more heterogeneous to
PFTPs than the long-distance ones. Keeping present frequency of long-distance offer is not
advisable, as long-distance train ridership has been growing [
62
]. Thus, the only sustainable
solution is restriction of regional train stopping in the stops with low passenger demand.
Since this measure is extremely politically sensitive, following measure is recommended—
regional trains should serve such stops only in 2-h period all-day (as in some sections
and times nowadays). The saved time windows should be scheduled so that at least
some PFTPs can be scheduled with one less stopping. The peak times, in reasonable
timespans, should make an exception, which would lower PFTP quality to the above
proposed timetable.
For the sake of not exceeding reasonable level of complexity, the authors did not
include in the proposed framework process any modifications of passenger timetable.
This limitation makes creation of new time windows for passage of a freight train between
two passenger trains impossible.
All PFTPs were scheduled to stop in Praha-Libeˇn marshalling yard. There is a busy
triple-track mainline, which proceeds eastwards. There is also a single-track freight by-
pass line which leads further southwards or westwards. However, this line crosses the
busy triple-track mainline without a flyover. Thus, before there will be built a double-track
flyover, the capacity is very low. This is the main limitation of the proposed concept
of PFTPs.
During the timetabling experiment, another significant limitation has emerged—loaded
coal trains running from Ústísouthwards (direction to Prague)—see Section 4.4. Since they
were hauled by obsolete locomotives, their PMR was mostly 0.8 or slightly higher. Thus,
hundreds of such trains did not fit even into the PFTP 65xxx with the lowest allowed
PMR—which cannot be set lower because of passenger timetable constraints (and likely
additional overtaking(s) as a result). However, train data were from the years 2015 and 2016.
On the other hand, the passenger operational concept with 30 min period of the fastest train
segment has been not implemented in 2020/2021 timetable [
49
]. So, practical impact of this
limitation is likely to be lower, thanks to gradual replacements of the obsolete locomotives
by new, more powerful, ones. This problem occurs practically only in one direction, since the
same trains run back unloaded, and consequently with enough high PMR.
Another significant limitation of the timetable experiment is that the freight train data
are related to the section Prague—Ústí. However, PFTPs for other sections have mostly
looser length limits and numbers of scheduled stops are considerably lower. Their PMR
limits are higher, but for German electrification system 15 kW 16.7 Hz, which begins
at the Czech-German border, only the more powerful locomotives (at least circa 3 MW)
are available.
The presented timetabling experiment, as well as similar experiments in Drábek’s
doctoral thesis [
18
] have shown that some particular sections (or, more broadly, some sec-
tions between two stops for overtaking) are limiting parameters of a freight train—PMR,
given some minimum recovery margin (surcharge, buffer time) to technical (theoretical)
runtime, and maximum length of the train, depending on the shortest utilized overtaking
track. A similar symmetric periodic timetable, coordinated for both passenger and freight
trains, with more balanced distribution of such limitations would enable an increase in
freight railway transport capacity, without costly construction. However, it is necessary
to formulate a suitable mathematical approach, whose application would not deteriorate
other criteria, relevant for the timetable.

Sustainability 2021,13, 8330 30 of 33
Another persisting problem, with overlap to jurisprudence, that does not have impact
only on freight, but also on passenger railway, is excessive frequency of stopping of
suburban and regional trains, which is in many cases not justified by sufficient passenger
demand. So, the settlements that generate the least demand, can be in most cases served by
bus instead. The problem consists in penalization of small group of inhabitants (in terms of
longer travel time and/or extra transfer) for the sake of increase of mainline capacity (by
decrease of train path heterogeneity in the sense of 1st edition of UIC 406 Codex [23]).
Gradual realization of parallel high-speed railway lines will on the one hand relieve
the conventional mainlines from majority of long-distance passenger trains. On the other
hand, there will occur new dimension of complexity of the network timetabling process.
Due to anticipated increase in total passenger traffic, new bottlenecks are likely to occur,
especially in agglomeration node areas, such as Prague or Ústínad Labem. For the freight
railway, higher capacity outside bottlenecks will be counterweighted by necessity of more
accurate scheduling (and real-time dispatching) through major bottlenecks. These problems
will likely further emphasize the network-oriented and multicriterial aspects of the railway
operation research.
The presented approach which enables the composition of passenger train paths in
IPT and differentiated, hierarchically constructed symmetric PTFPs could serve as a basic
input methodology for capacity analyses, which are necessary for increasing the share of
rail freight transport on the transport market and for the whole concept of sustainable
freight transport.
Supplementary Materials:
The following are available online at https://www.mdpi.com/article/10
.3390/su13158330/s1, Figure S1: Timetable Praha-Libeˇn—Dresden Hauptbahnhof in A3 size.
Author Contributions:
Conceptualization, M.D. and V.J.; methodology, M.D.; software, V.J.; vali-
dation, V.J. and M.D.; formal analysis, V.J.; investigation, M.D.; resources, V.J.; data curation, M.D.;
writing—original draft preparation, M.D.; writing—review and editing, V.J.; visualization, M.D.;
supervision, V.J.; project administration, M.D.; funding acquisition, V.J. All authors have read and
agreed to the published version of the manuscript.
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement:
Data sharing not applicable. All data analysed in this article, that are
not available online, are business sensitive (if not published as aggregate data).
Acknowledgments:
Publishing of this paper in Open Access was supported by Czech Technical
University in Prague. The authors hereby acknowledge the funding sponsor.
Conflicts of Interest: The authors declare no conflict of interest.
References
1.
United Nations. Sustainable Development Goals. Available online: https://www.un.org/sustainabledevelopment/sustainable-
development-goals/ (accessed on 7 April 2021).
2.
United Nations General Assembly. Resolution Adopted by the General Assembly on 25 September 2015. 70/1. Transforming Our
World: The 2030 Agenda for Sustainable Development, 2nd ed.; United Nations General Assembly: New York, NY, USA, 2015; p. 14.
Available online: https://undocs.org/A/RES/70/1 (accessed on 7 April 2021).
3.
European Commission. GHG emissions from transport EU-27–by mode (shares %). In EU Transport in Figures–Statistical Pocketbook
2020; Publications Office of the European Union: Luxembourg, 2020; p. 137, ISBN 978-92-76-17565-0, ISSN 2363-2739.
4.
European Commission. Communication from the Commission to the European Parliament, the European Council, the Council, the
European Economic and Social Committee and the Committee of the Regions; The European Green Deal. COM/2019/640 Final;
European Commission: Brussels, Belgium, 2019. Available online: https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=
COM%3A2019%3A640%3AFIN (accessed on 7 April 2021).

Sustainability 2021,13, 8330 31 of 33
5.
European Commission. Proposal for a Regulation of the European Parliament and of the Council Establishing the Framework for Achieving
Climate Neutrality and Amending Regulation (EU) 2018/1999 (European Climate Law); COM/2020/80 Final; European Commission:
Brussels, Belgium, 2020. Available online: https://eur-lex.europa.eu/legal-content/EN/TXT/?qid=1588581905912&uri=CELEX:
52020PC0080 (accessed on 7 April 2021).
6.
SMA and Partners Ltd. Netgraph Switzerland 2021. Available online: https://www.sma-partner.com/images/Downloads/
NGCH-2021.pdf (accessed on 24 June 2021).
7. Deutsche Bahn. Available online: https://www.bahn.com/en (accessed on 24 June 2021).
8. ÖBB-Personenverkehr AG. Available online: https://www.oebb.at/en/ (accessed on 24 June 2021).
9.
Železnice Slovenskej Republiky. CestovnýPoriadok 2020–2021. Available online: https://www.zsr.sk/pre-cestujucich/cestovny-
poriadok-2020-2021/ (accessed on 24 June 2021).
10.
Správa Železnic. Trat’ovéJízdníˇ
Rády PlatnéOd 6. Dubna 2021. Available online: https://www.spravazeleznic.cz/cestujici/
jizdni-rad (accessed on 24 June 2021).
11.
Forschungsgesellschaft für Straßen- und Verkehrswesen, E.V. Merkblatt Zum Integralen Taktfahrplan; FGSV-Verlag GmbH: Cologne,
Germany, 2001; pp. 18–20.
12. Lindner, H.-R.; von Redern, H.W. Güterzüge im taktfahrplan–Möglichkeiten und grenzen. Die Bundesbahn 1989,10, 867–874.
13.
ˇ
Capek, K. FlexFahrplan–A New Approach to Railway Operational Planning. Diploma Thesis, Czech Technical University in
Prague, Prague, Czech Republic, 2004.
14.
Klabes, S.G. Algorithmic Railway Capacity Allocation in a Competitive European Railway Market. Ph.D. Thesis, RWTH Aachen,
Aachen, Germany, 14 January 2010.
15.
Janoš, V.; Kˇríž, M. Infrastructure parameters affecting capacity of railways in TEN-T. In Acta Polytechnica CTU Proceedings, Prague,
Czech Republic, 4 May 2016; Jírová, J., ˇ
Rezníˇcková, J., Eds.; ˇ
CeskáTechnika-Nakladatelstvíˇ
CVUT: Prague, Czech Republic, 2016;
Volume 5, pp. 22–25, ISSN 2336-5382. ISBN 978-80-01-06022-3. [CrossRef]
16.
Drábek, M. Systematickékatalogovétrasy pro nákladnívlaky. In Proceedings of the EURO-Žel 2009, Žilina, Slovakia, 3–4 June
2009; Nagy, P., Ed.; University of Žilina: Žilina, Slovakia, 2009; Volume 2, pp. 141–148, ISBN 978-80-554-0024-2.
17.
Drábek, M. Network-bound periodic freight train paths-a qualitative generic approach for railway capacity management. In
Proceedings of the 4th International Seminar on Railway Operations Modelling and Analysis RailRome 2011, Rome, Italy, 16–18
February 2011; Universita La Sapienza: Rome, Italy, 2011; pp. 1–17.
18.
Drábek, M. Periodic Freight Train Paths in Network. Ph.D. Thesis, Czech Technical University in Prague, Prague, Czech Republic,
6 June 2014. Available online: https://takt.fd.cvut.cz/cargo/Drabek_thesis.pdf (accessed on 22 March 2021).
19.
Serafini, P.; Ukovich, W. A mathematical model for periodic scheduling problems. SIAM J. Disc. Math.
1989
,2, 550–581. [CrossRef]
20.
Liebchen, C. Periodic Timetable Optimization in Public Transport. Ph.D. Thesis, Technische Universität Berlin, Berlin, Germany,
22 March 2006.
21.
Opitz, J. Automatische Erzeugung und Optimierung von Taktfahrplänen in Schienenverkehrsnetzen. Ph.D. Thesis, Technische
Universität Dresden, Dresden, Germany, 2009.
22.
Caimi, G.; Laumanns, M.; Schüpbach, K.; Wörner, S.; Fuchsberger, M. The periodic service intention as a conceptual frame for
generating timetables with partial periodicity. Transp. Plan. Technol. 2011,4, 323–339. [CrossRef]
23. International Union of Railways (UIC). UIC Code 406. Capacity, 1st ed.; UIC: Paris, France, 2004; pp. 3–16. ISBN 2-7461-0802-X.
24.
Stoilova, S.; Munier, N.; Kendra, M.; Skrúcaný, T. Multi-criteria evaluation of railway network performance in countries of the
ten-t orient–east med corridor. Sustainability 2020,12, 1482. [CrossRef]
25.
International Union of Railways (UIC). UIC Code 406. Capacity, 2nd ed.; UIC: Paris, France, 2013; pp. 30–34. ISBN 978-2-7461-
2159-1.
26.
Gašparík, J.; ˇ
Cechoviˇc, L.; Blaho, P.; Peˇcený, L. Capacity of corridor lines after modernization. Transp. Res. Procedia
2021
,53,
159–166. [CrossRef]
27.
ˇ
Camaj, J.; Nedeliaková, E.; Šperka, A.; Ližbetinová, L. The planning of investment activities in field of railway transport with
support of simulation tools. Transp. Res. Proc. 2021,53, 39–49. [CrossRef]
28.
Ljunggren, F.; Persson, K.; Peterson, A.; Schmidt, C. Railway timetabling: A maximum bottleneck path algorithm for finding an
additional train path. Public Transp. 2020. [CrossRef]
29. Haldeman, L. Automatische Analyse von IST-Fahrplänen. Master’s Thesis, ETH Zürich, Zürich, Switzerland, 2005.
30.
Vromans, M. Reliability of Railway Systems. Ph.D. Thesis, Erasmus University Rotterdam, Rotterdam, The Netherlands,
6 July 2005.
31.
Müller, A. Möglichkeiten Eines Cargo-Takt-Systems zur Verbesserung der Transportqualität im Schienengüterverkehr. Ph.D.
Thesis, RWTH Aachen, Aachen, Germany, 9 July 1999.
32.
Cacchiani, V.; Caprara, A.; Toth, P. Scheduling extra freight trains on railway networks. Transp. Res. Part B Methodol.
2010
,44,
215–231. [CrossRef]
33.
Bablinski, K. A game-based analysis of freight paths allocation with a case study on Great Britain Brighton Main Line. Transp. Res.
Procedia 2016,13, 196–208. [CrossRef]
34.
Kuckelberg, A.; Gepper, A.; Janecek, D. Pattern train extraction for analytical railway operations research. In Proceedings of
the EURO–ŽEL 2015, Žilina, Slovakia, 2–3 June 2015; Krištofová, L., Fabian, P., Eds.; University of Žilina and Tribun EU: Žilina,
Slovakia, 2015; Volume 2, pp. 58–65, ISBN 978-80-263-0936-9.

Sustainability 2021,13, 8330 32 of 33
35.
Pöhle, D. Strategische Planung und Optimierung der Kapazität in Eisenbahnnetzen unter Nutzung von Automatischer Takt-
fahrplanung. Ph.D. Thesis, Technische Universität Dresden, Dresden, Germany, 22 March 2016.
36.
Woodburn, A. An empirical study of the variability in the composition of british freight trains. J. Rail Transp. Plan. Manag.
2015
,5,
294–308. [CrossRef]
37.
R˚užiˇcka, J. Návrh PostrkovéSlužby v Oblasti Vysoˇciny. Bachelor Thesis, Czech Technical University in Prague, Prague, Czech
Republic, 12 September 2018. Available online: https://dspace.cvut.cz/handle/10467/80188 (accessed on 23 March 2021).
38.
Chýle, M. KatalogovéTrasy Pro NákladníVlaky na IV. Tranzitním Železniˇcním Koridoru. Diploma Thesis, Czech Technical
University in Prague, Prague, Czech Republic, 17 May 2021. Available online: https://dspace.cvut.cz/handle/10467/95005
(accessed on 29 May 2021).
39.
R˚užiˇcka, J. KatalogovéTrasy ve Smíšeném Provozu. Diploma Thesis, Czech Technical University in Prague, Prague, Czech
Republic, 16 January 2021. Available online: https://dspace.cvut.cz/handle/10467/92608 (accessed on 23 March 2021).
40.
Michl, Z.; Janoš, V.; Drábek, M.; Sojka, M.; Pospíšil, J.; Kˇríž, M.; Vašíˇcek, R. Optimalizace VyužitíTratís Vyˇcerpanou Kapaci-
tou; Methodology certified by Ministry of Transport; Czech Technical University in Prague: Prague, Czech Republic, 2016;
pp. 26–31, Unpublished.
41.
Kamenický, D. Traffic management system in terms of data exchange. In Proceedings of the Acta Polytechnica CTU, Prague,
Czech Republic, 4 May 2016; Jírová, J., ˇ
Rezníˇcková, J., Eds.; ˇ
Ceskátechnika-nakladatelstvíˇ
CVUT: Prague, Czech Republic, 2016;
Volume 5, pp. 26–28. [CrossRef]
42.
Drábek, M. On Symmetry of Periodic Freight Train Paths in Network. In Proceedings of the Horizons of Railway Transport 2013,
Streˇcno, Slovakia, 26–27 September 2013; University of Žilina: Žilina, Slovakia, 2013; pp. 120–128, ISBN 978-80-554-0764-7.
43.
Kopecký, F.; Baudyš, K.; Janoš, V.; Pospíšil, J. Redakˇcnˇe UpravenáZávˇereˇcnáZpráva Projektu CG723-138-190–Konfigurace Taktových
Uzl˚u v ŽelezniˇcníSíti ˇ
CR.; Research Project for Czech Ministry of Transport; KPM CONSULT, a.s. and Czech Technical University
in Prague: Prague, Czech Republic, 2009; Unpublished.
44.
Šimral, P. JAK Získat Potˇrebnou Kapacitu Dráhy Pro NákladníVlaky Hned a Bez Významných a Nákladných Opatˇrení?METRANS Rail
s.r.o.: Prague, Czech Republic, 2013; Unpublished.
45.
European Commission. Consolidated Text: Regulation (EU) No. 913/2010 of the European Parliament and of the Council of 22 September
2010 Concerning a European Rail Network for Competitive Freight; European Union: Brussels, Belgium, 2010. Available online:
https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=CELEX%3A02010R0913-20140101 (accessed on 22 March 2021).
46.
Vávra, R. DálkováŽelezniˇcníOsobníDoprava v Relacích Praha–Drážd’any/Cheb. Diploma Thesis, Czech Technical University
in Prague, Prague, Czech Republic, 18 June 2018. Available online: https://dspace.cvut.cz/handle/10467/77201 (accessed on 22
March 2021).
47.
Vávra, R.; Janoš, V. Long-distance Railway Passenger Transport in Prague-Dresden/Cheb Relations. In Modern Traffic Engineering
in the System Approach to the Development of Traffic Networks; Macioszek, E., Sierpi´nski, G., Eds.; Springer Nature Switzerland AG:
Basel, Switzerland, 2020; Volume 1083, pp. 257–274, ISSN 2194-5365/978-3-030-34069-8.
48.
Institut für Regional- und Fernverkehrsplanung, E.K. The Timetable Construction System FBS. Available online: http://www.en.
irfp.de/the-timetable-construction-system-fbs.html (accessed on 23 March 2021).
49.
Správa Železnic. Jízdníˇ
Rády (platnédo 5. 4. 2021). 090 Praha–Ústínad Labem–Dˇeˇcín. Prague, Czech Republic, 2021.
Available online: https://www.spravazeleznic.cz/documents/50004227/115131601/k090.pdf/2c5d2750-f47b-45ef-bce8-f5c8
89ffca5f (accessed on 23 March 2021).
50.
Michl, Z.; Drábek, M.; Vávra, R. Netgraph-an efficient tool for periodic timetabling and capacity planning. Acta Polytech. CTU
Proc. 2017,11, 29–30. [CrossRef]
51.
Ministerstvo Dopravy. Plán DopravníObsluhy ÚzemíVlaky CelostátníDopravy. Zásady Objednávky DálkovéDopravy Pro Období
2017–2021; Ministerstvo Dopravy: Prague, Czech Republic, 2016; Available online: https://www.mdcr.cz/getattachment/
Dokumenty/Verejna-doprava/Financni-ucast-statu/Plan-dopravni-obsluhy-uzemi-vlaky-celostatni-dopra/Plan-dopravni-
obsluhy-uzemi-2017-2021.pdf.aspx (accessed on 22 March 2021).
52.
European Commission. Regulation (EU) No 1315/2013 of the European Parliament and of the Council of 11 December 2013 on Union
Guidelines for the Development of the Trans-European Transport Network and repealing Decision No 661/2010/EU.; Text with EEA
Relevance; European Union: Brussels, Belgium, 2013; Available online: https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=
CELEX:32013R1315 (accessed on 24 March 2021).
53.
Správa Železnic. Tabulky Trat’ových Pomˇer˚u, TTP 526B. Tabulka 9. ZákladníProvozní Údaje Pro Jízdu Vlaku. Úˇcinnéod: 15. 03. 2021;
Správa Železnic: Prague, Czech Republic, 2021; Unpublished.
54.
Správa Železnic. Tabulky Trat’ových Pomˇer˚u, TTP 527A. Tabulka 9. ZákladníProvozní Údaje Pro Jízdu Vlaku. Úˇcinnéod: 15. 03. 2021;
Správa Železnic: Prague, Czech Republic, 2021; Unpublished.
55.
Správa Železnic. Tabulky Trat’ových Pomˇer˚u. TTP 544A. Tabulka 9. ZákladníProvozní Údaje Pro Jízdu Vlaku. Úˇcinnéod: 15. 03. 2021;
Správa Železnic: Prague, Czech Republic, 2021; Unpublished.
56.
Správa Železnic. Jízdníˇ
Rády (Platnéod 6. 4. 2021). 091 Praha-Hostivaˇr–Kralupy nad Vltavou. Prague, Czech Republic, 2021.
Available online: https://www.spravazeleznic.cz/documents/50004227/115131601/k091_od_2021-04-06_oprava.pdf/757a1
ad9-9ff7-421c-8b91-70e2e6fc4139 (accessed on 24 June 2021).

Sustainability 2021,13, 8330 33 of 33
57.
Michl, Z.; Janoš, V.; Drábek, M.; Kˇríž, M.; Pospíšil, J.; Sojka, M.; Vašíˇcek, R. Optimalizace Rozvoje Železniˇcního Systému ˇ
Cr z Hlediska
Pˇrepravních Potˇreb. Analytickáˇ
Cást. Zpráva o Dosažených Výsledcích Projektu TB0300MD013 ˇ
Rešeného s FinanˇcníPodporou TA ˇ
CR.;
Taktici.cz, s.r.o. and Czech Technical University in Prague: Prague, Czech Republic, 2016; pp. 118–121, Unpublished.
58.
Správa Železnic. Sbírka Služebních Pom ˚ucek Pro Jízdníˇ
Rád 2020/2021; Správa Železnic: Prague, Czech Republic, 2020; Unpublished.
59. Microsoft Excel. Available online: https://www.microsoft.com/microsoft-365/excel (accessed on 24 June 2021).
60. MathWorks®. MATLAB. Available online: https://www.mathworks.com/products/matlab.html (accessed on 24 June 2021).
61.
Lieskovský, A.; Myslivec, I.; Pavel, M. Advanced Energy Efficient Automatic Train Operation in Commercial Operation; AŽD Praha
s.r.o.: Prague, Czech Republic, 2013; p. 9. Available online: https://www.azd.cz/admin-data/storage/get/1005-ict-milano-2013
-def.pdf (accessed on 24 June 2021).
62.
Ministerstvo Dopravy. Plán dopravníobsluhy územívlaky celostátnídopravy. In Zásady Objednávky DálkovéDo-
pravy Pro Období2012-2016; Ministerstvo Dopravy: Prague, Czech Republic, 2012; p. 33. Available online: https:
//www.mdcr.cz/getattachment/Dokumenty/Verejna-doprava/Financni-ucast-statu/Plan-dopravni-obsluhy-uzemi-vlaky-
celostatni-dopra/plan-dopravni-obsluhy.pdf.aspx (accessed on 24 March 2021).