Optimization of Timetables on the Bratislava–Žilina–Košice Route in the Period after the End of the COVID-19 Pandemic

Abstract

Quality and efficient long-distance transport as a key system of public passenger transport could be the basis for the national and international comprehensive integrated transport system. Especially due to the pandemic caused by the COVID-19 disease, the demand for long-distance transport has significantly decreased since March 2020 in the Slovak Republic on the busiest long-distance line Bratislava–Žilina–Košice. It is, therefore, necessary to propose various measures in order to increase this demand and bring passenger frequencies to at least the level of values before this period. This paper analyzes the impact of the measures on passenger frequencies in 2020, examining the amount of their fall in individual months of the first half of the year 2020 on the long-distance route Bratislava–Žilina–Košice. Furthermore, the paper deals with the long-term concept of national long-distance passenger rail transport on the mentioned line in the period after the end of the pandemic. It uses scientific methods enabling to rationalize and optimize current train timetables. The benefits of these proposals will be to offer passengers better transport services, with their frequencies expected to increase in the coming years.

1. Introduction

Long-distance passenger rail transport is significant for linking places within a single country or across several countries. In the period from about March 2020, the demand for long-distance transport has been significantly lower than the long-term trend so far, mainly due to the COVID-19 restrictions. Passengers have also started to prefer individual car transport to protect their health. After the end of the pandemic, it will therefore be very difficult to bring them back to public passenger transport, and thus to long-distance trains. For this reason, it is now necessary to take a number of measures for revitalization and systematic conceptual development of long-distance passenger rail transport, also in order to ensure better and more efficient transport services as such. National and international long-distance passenger rail transport is very important from a strategic point of view, and regional transport links depend very much on it. It is, therefore, necessary to constantly monitor passenger flows and also to analyze the trend in this segment. The contribution aims to point out the drop in passenger frequencies in 2020 on the most important and busiest national long-distance railway line in the Slovak Republic, namely the Bratislava–Žilina–Košice line. Subsequently, it proposes some measures how to improve the unfavorable situation and increase the passenger flow on the long-distance trains. It is also very important to improve the quality of the services offered. Therefore, it is necessary to rationalize and optimize the current timetables and propose certain modifications. To achieve the rationalization and optimization of timetables, we need to employ modeling methods used in transportation processes. Based on the outputs of these scientific and professional methods, the new concept of timetables for the Bratislava–Žilina–Košice railway line will be proposed [1].

2. Materials and Methods

For proper determination of the traffic service and timetable on the mentioned transport route, it is very important to know and explore current issues and research in the field of the train traffic diagrams, timetabling, traffic service determination, and transport processes in passenger railway transport. It is also important to research areas directly related to these issues, which can be the modal shift, sustainable transport system, mobility in regions and cities, traffic service and supporting public passenger transport in cities and regions, etc. There are lots of scientific publications which deal with this topic. For example, the role of passenger railway transport in population and employment changes has long been debated in the context of suburban sprawl, urban development, central cities’ decline, and inter-/intra-metropolitan accessibility [2]. There are many optimization methods that can be used in transport processes and railway transport optimization. For example, the employment of the Monte Carlo optimization method is described in the article [3]. This method was used to optimize the fleet capacity.

Infrastructure planning is also very important for traffic service determination. These problems relate to capacity issues. More details on the railway infrastructure capacity estimation can be found in [4,5]. There are also many articles and scientific studies which have been written about the traffic service of a certain area or region and other issues related to it. Publications [6,7] bring a model of serving the area by bus and railway transport within public transport. Based on mathematical models, there are attempts to maintain an even service of urban areas with optimal use of transitional links. Another publication [8] analyzes the possibility of innovating a sustainable transport system in a certain Italian region. It uses the 2009 economic crisis and the current COVID-19 pandemic as restrictive conditions. When creating a traffic service on a certain railway line, it is also necessary to know the concept of creating individual train connections. The use of empirical models for the creation and evaluation of train connections is elaborated in the publication [9]. In most cases, it is necessary to take into account the transport network and to evaluate the connections in the context of certain parameters of the transport network. This issue is elaborated on in detail in [10].

Moreover, Kleprlík [11] deals with scheduling in general. In his publications, he suggests using specific methods to solve transport optimization as an assignment problem. There are two subjects of the paper: Mathematical approaches mentioned above and the purpose of passenger railway transport. It is necessary to analyze the transport chain in passenger and freight transport [12].

It is also important to support public passenger transport. Public transport rationalization and timetabling are mentioned in other publications [13,14]. Timetabling is a crucial process of the traffic service. The principles of periodic timetabling are developed in studies [15] to ensure the connection quality is very important for timetabling process [16].

Issues of the Slovak Republic timetable are addressed in the publication [17]. It presents a new concept of the timetable for long-distance and regional passenger rail transport on the Slovak Railway. These suggestions are interesting, but they are not based on any scientific methods. There is also a need to present some more progressive timetable proposals in order to attract more passengers to trains during the COVID-19 pandemic. Last but not least, we drew some scientific and professional knowledge from the contributions focused on transport and economic processes [18,19,20,21,22,23].

There are many unexplored facts in the long-distance passenger railway transport and its timetabling. The scientific focus of the contribution will be, therefore, oriented to new progressive proposals in the field of transport science [24].

2.1. Passenger Frequencies during the Year 2020

Due to the COVID-19 problems, several measures have also been introduced by the national passenger rail operator, Železničná spoločnosť Slovensko, a. s. (ZSSK). During the spring of 2020, several train connections were reduced. It is possible to confirm that the Bratislava–Žilina–Košice railway line represents a significant connection of national importance. There are currently at least 11 pairs of train connections during one day on this transport route, which represents more than 3.5 million train kilometers per year. Based on analyses of internal materials of ZSSK, the number of passengers before the outbreak of the pandemic regularly exceeded two million per month. ZSSK recorded the peak transportation flow in November 2019, when 2,593,883 passengers used the connection. Looking at the picture, we can observe a regular decrease in passengers during the summer months and an increase in passengers in the autumn. The beginning of 2020 was marked by slow growth. Due to the outbreak of the pandemic in March and the introduction of related measures, the passenger numbers fell down below one million. The following month, the measures tightened, free transport for students and pupils was suspended, and production plants were closed throughout Slovakia, too. The fact is that the number of passengers across the country fell to 338,137. In May 2020, the measures gradually loosened, and the associated return of regular passengers to trains took place. The trend of loosening the measures and returning of passengers continued in June and July. ZSSK recorded the most passengers in August, when their number exceeded 1.5 million. Comparison of the number of passengers is shown in Figure 1. Percentage change compared to the previous year is shown in Figure 2 [25].

During the first months of 2020, the trend of sustainable growth in transport performance continued. With the start of the pandemic in March, we could observe a two-thirds drop in performance until the absolute bottom in April 2020 was reached. The trend of a renewed growth was the same as the trend of passenger return, but transport performance remained 32% to 31% lower during the summer months when compared to 2019, while passenger numbers were only 25% to 22% lower.

The reason for the lower performance might be the above-mentioned trend of longer trips during the summer, for example, by students from Bratislava traveling back to the east. As colleges remained closed during the first half of 2020, students traveling the entire length of the session remained at home and transport performance loss was higher [25,26].

In the first pandemic year, Slovakia adopted very strict COVID-19 measures in the spring months of 2020, which was also reflected in traffic performance. Subsequently, in the second wave, which began at the end of 2020 and lasted until May 2021, the measures were also relatively strict, but there was no significant reduction in long-distance rail transport in Slovakia. Traffic performance also did not fall as sharply as in spring 2020. However, this contribution mainly addresses the situation in long-distance rail passenger transport in the first COVID-19 wave in 2020. Data from 2021 have not yet been officially available on any relevant sources of the national carrier (ZSSK a. s.). Therefore, we will work with data from 2020 only.

2.2. Scientific Methods

As already mentioned in the introduction, problems with the drop in passenger frequencies will be addressed through the optimization of timetables. Modeling methods of timetable and train traffic diagrams, as well as other general scientific methods, will be used for this purpose. The main method that can be used in optimizing timetables is the method of inscribed n-angles. Within this chapter, its modification is proposed, which will be applicable to national long-distance passenger rail transport. In addition to this method, other complementary scientific methods are used [27].

2.2.1. Method of Inscribed n-Angles

The proposal of routes to serve an area presents a complex and multi-criteria analysis. A line can be characterized as several consecutive recurring services serving a predetermined specific district. This method was used because it is a specific method used to coordinate and optimize selected transport processes. It is a relatively well-known modern progressive method. Several experts and scientists (including mentioned [4,6,7,8]) use this method in their contributions and books. This method was also called ‘’the Žilina Circle’’ (Žilinská Kružnica in Slovak language). Its other advantages are that it is relatively simple, suitably practically applicable, and based on exact mathematical principles. This method can be used to optimize the timetables of individual lines in passenger rail, bus, and public transport. When operating a common section with several routes, there is a problem with coordinating the timetables of these routes. It is essential to keep the time lost for passengers, who can use any route, to a minimum. The solution to this problem is based on a geometric representation. The basic principle is that there will be three lines in operation on the common section. Departure intervals from the starting node will be for No. 1 line—12 min, for No. 2 line—8 min, and for No. 3 line—6 min. The smallest common multiple of these intervals is 24 min, which represents the period after which the mutual distribution of moments of departures will be repeated. This mutual distribution can be illustrated in the following way: On a circle divided into 24 equal parts, the known moments of departures of the No. 1 line are depicted by vertices of a regular inscribed two-angle object (i.e., a line segment), the moments of departures of the No. 2 line are depicted by vertices of an inscribed equilateral triangle, and the moments of departures of the No. 3 line are depicted by vertices of an inscribed square. The basic principle of this method is shown in the Figure 3 [27].

When formulating the criterion of optimality of the mutual position of the above-mentioned figures, which in the general case represent regular n-angles inscribed in a common circle, we proceed from the requirement that the minimum mutual distance of adjacent vertices of n-angles on the circle should be as high as possible. Generally, this task has several solutions and, depending on the circumstances, it is possible to choose the one that better suits some additional requirements. Figure 4 shows two solutions. In both cases, we assume that the square against the triangle cannot be rotated so that the distance between the nearest vertices (measured on the circle) is greater than 1. By turning the line in solution (b), when compared to solution (a), it is possible to reduce the longest time between connections from 6 to 5 min, but at the cost of worsening a certain ‘’uniformity’’ of other intervals [27]. It is important to note that this methodology is only a general formulation of an issue that has been proposed in the past by professionals and experts in the field of providing transport services to the study area. Otherwise, the method can also be called rhythmic coordination in transport.

The mentioned scientific method can also be applied to the coordination of time positions of individual lines and connections in long-distance passenger railway transport. However, due to different conditions in comparison with urban and suburban transport, it is necessary to adjust the parameters of individual points on the circle, possibly vertices of polygons. The circle represents the whole day in the range of 24 h, while each point on it represents the whole hour. At the same time, the individual points are connected in such a way that they represent the optimally recommended departure times of individual connections from the starting station, while the connections of the same line are marked with the same color. Figure 5 shows a proposed modification of the method of inscribed n-angles, which can be practically applied to the long-distance line Bratislava–Žilina–Košice [28].

The individual connections are represented with the following colors:

• Brown color—Inter City train line on the (Vienna)—Bratislava–Žilina–Košice route,
• Blue color—Express train line on the Bratislava–Žilina–Košice route,
• Green color—Fast train line on the Bratislava–Žilina–Košice route,
• Yellow color—Fast train line on the Bratislava–Žilina–Prešov route.

This basic concept with four proposed long-distance lines represents an optimal distribution of time positions of individual connections, which respect the basic principles of the method of inscribed n-angles. In practice, however, it will be necessary to move the ‘’yellow line’’ connections one hour earlier because a collision with the ‘’blue line’’ connections would occur in the Žilina–Košice section, as the connections of these lines are not equally fast. If both dashed lines lead to a certain point on the circle, then there is no connection within the given line at this time, and it is possible to point out the so-called ‘’hole in the cycle’’ at night saddle time.

To simplify the understanding of the methodology used, the diagram shows Bratislava Main Station. The circle further shows 24 vertices representing the number of hours in one day. The solid lines associated with the peaks represent the proposal of the rail connection that could be available to the Bratislava–Žilina–Košice line, and that could be used by potential passengers. The types of trains are shown above and are distinguished in the graph by the appropriate color shown. However, if there is a dashed line connection in any of the vertices, regardless of color, it means that there is no possible connection at that time, which demonstrates a hole in the night-time seat times. These trains are classified in the proposal as international trains, which connect with the section Košice–Žilina–Bratislava, also Vienna (Austria).

The train categories Expres ‘’Ex’’ and EuroNight ‘’EN’’ are based on a similar principle as the trains of category R. However, the trains of category ‘’EN’’ are operated during night hours only.

2.2.2. Additional Methods

The following methods were used in the proposal:

• Synthesis method—this method, based on experience or logic, proceeds from the simplest principles to more complex ones by merging and connecting individual parts into a whole. In the case of proposals and outputs of this paper, it is a combination of partial proposals of individual connections and lines on the route representing a complete 24-h systematic timetable;
• Deduction method—it is a scientific method in which specific, special and less general elements and conclusions are derived from general outputs and conclusions. In this case, it is the application of general standards of a transport service to specific draft timetables;
• Brainstorming method—it is known as a creative method used to solve various problems using the generation of progressive ideas and thoughts. The result should be an original and unique solution to a specific problem, which also represents the proposals and outputs listed in the chapter;
• Delphi method—it is an expert, in other words, prognostic method looking for a group solution to a certain problem, based on the opinions, estimates, and solutions of a group of selected experts whose knowledge was also used in the given proposals [29].

2.3. Draft Standards of a Transport Service in National Long-Distance Transport

At present, it can be stated there is no systematic concept based on which it would be possible to ensure the quality conceptual development of national and international long-distance passenger rail transport. The current existing standards are not sufficient. For this reason, it is necessary to propose new standards for long-distance transport services that will be universally applicable to all long-distance routes, while fully respecting transport links in regional and suburban transport. Several methods presented in the previous chapter are also used in the process of creating the standards. Within the research in the field of optimization and rationalization of national long-distance passenger rail transport, these are the following five standards [30].

2.3.1. Fast, High-Quality, All-Day Transport

It aims to set up a high-speed train system on individual long-distance routes so that they are more time-efficient than individual car transport and competitive with air transport—over significantly longer distances, and it is highly recommended to operate them with modern electric long-distance train units possibly with high cruising speed stopping on the mentioned connections only in important centers (settlements) and railway hubs. The operation of these connections should be ensured daily without any significant restrictions in the regular clock mode, usually between 5:00 and 23:00/24:00, on most lines, even at night via at least one connection. The basic principal of integrated clock schedule is shown in Figure 6 [30].

2.3.2. Symmetrical Time Positions of Individual Connections

In order to establish a regular systematic operation of individual connections, in addition to setting up a regular clock operation, it is necessary to ensure their symmetry. This is important in order to be able to provide the same transport service in both directions of a particular route (what works in one direction must also work in the opposite direction), and at the same time, to ensure a more efficient turnover of individual sets. To ensure an efficient symmetrical operation, it is necessary to correctly determine a so-called axis of symmetry and subsequently to set the time positions of individual connections and their so-called symmetrical parts. In practice, it is recommended to introduce the symmetry axis at times 2:00 and 14:00 or 2:30 and 14:30. Therefore, for example, if a certain train connection leaves station A at 5:05 and arrives at station B at 13:50, then in the opposite direction, it leaves station B at 15:10 and subsequently arrives at station A at 23:55. Two examples of symmetry axes are shown in Figure 7, with the symmetry axis of times 2:00 and 14:00 in green and the symmetry axis of times 2:30 and 14:30 in blue color [30].

2.3.3. Rationalization of Transfer Links to Lower Category Trains

Long-distance train connections should link regions as well as integrated transport systems. To achieve this, it is necessary to ensure the best and the most efficient transfer links to interregional, regional, and suburban connections. The ideal scenario is for higher category trains to cross at important centers and major hubs so that it is possible to switch to lower category trains in a short period of time. The recommended concept is shown in Figure 8. There are virtual railway line and virtual stations (A, B, C) [30].

2.3.4. Rationalization of a Band Service in Long-Distance Transport

A band type of service is usually planned in regional and suburban passenger timetables. However, the experience of certain developed agglomerations shows the advantageous application of this concept also in national long-distance transport. Therefore, it is recommended to introduce the so-called band express trains, which in the first band near the mayor area, will be run as express trains with a minimum number of stops, and in the next band, they will be run as classic express trains with a standard number of stops typical for the expressway, while at the railway hub at the border of these two bands a transient link will be provided from the first band to another express link. The overall benefit of these trains is positive, as assessed by the aggregate time spent by passengers in the transport process. This time is lower after the introduction of band express trains. The basic principle of the band service is shown in Figure 9, where the ‘’blue’’ line operates in the band A–B as an express train that does not stop anywhere, and it operates in the band B–C as a stop train, while in hub B, there is a transfer from the stop ‘’green’’ line [32,33].

These standards are better than others for long-distance transport services because current standards are insufficient, as they address only the basic principles of transport services in the area. These proposed standards point to more detailed specifications of the time positions of particular long-distance rail passenger trains. They use the basic mathematical principles of temporal symmetry. After correcting these standards of implementation, we will achieve more advantageous transport links between individual trains, thus achieving more advantageous transport services for passengers. The draft timetable in the third chapter, which is much better than the current situation, serves as proof.

3. Results

Following the draft standards for long-distance transport, it is possible to proceed to the creation of the concept of national long-distance transport on the specific line. Within this chapter, a proposal for a long-term strategic concept of domestic long-distance rail passenger transport on the Bratislava–Žilina–Košice route is presented. The proposal is implemented in accordance with the standards listed in the second chapter, and it will be possible to implement it in practice after the modernization of the Bratislava–Žilina–Košice line section, which means achieving the required systematic train running times. The main idea of the proposal is to introduce a four-hour clock for so-called band express trains on the Bratislava–Košice route, which in the Bratislava–Žilina section, will stop only in Trnava and Trenčín, and from Žilina to Košice subsequently in all express stations. The mentioned connections will be supplemented in a four-hour cycle by classic IC trains on the Vienna–Košice route [34].

As the timetable is constructed on the line Bratislava–Žilina–Košice on the territory of the Slovak Republic, this timetable includes, apart from international trains, also mainly trains of national significance with Slovak national designation.

The category of ‘R’ trains includes the designation of fast trains on this line, which means that the train does not stop at every station and stop, and it is one of the faster railway connections among the offer of trains of ZSSK, in the public interest services.

The category of InterCity ‘’IC’’ trains is assigned to the carrier ZSSK a. s. commercial trains, which means that they are not performed in the public interest, but an offer of trains that the repairer operates at its own commercial operational risk. This group of selected trains of category ‘’IC’’ can also be defined as a group of international trains since apart from the Košice–Žilina–Bratislava route, it also serves the station Vienna (Austria) and passengers have the opportunity to use this direct connection with eastern Slovakia.

The basic reasons for this method of dividing time result from the basic principle of the proposed modified method of inscribed n-angles, which can be adapted in long-distance transport. The most significant benefit will be to provide better and more efficient and attractive transport services on this transport route. The introduction of the band expresses is now a modern traffic concept. A great benefit is fast passenger transport from/to the capital to/from Žilina, and from there, it is possible to travel without a transfer to other smaller towns directly without a transfer. There are also connections to/from other smaller towns in Žilina station on the Bratislava–Žilina transport route. These proposals are new, as there is currently no such traffic type on the railway line. This schedule is judged by the aggregate time a passenger spends in the transport process. Thus, the essential benefit of the zonal timetable is mainly that the passenger spends less time in the vehicle during the journey, and therefore, has the opportunity to travel faster to the destination station.

In addition to this line, it is proposed to introduce one pair of new express trains on the Munich–Košice route and back, hauled by the RailJet vehicles. An express connection from the direction of Bratislava and Košice will be provided to individual express trains and IC trains in Žilina. These express connections will be supplemented with three pairs of express trains operating in a four-hour cycle on the Bratislava–Žilina–Prešov route and back. The proposed concept is supplemented with a night pair of express trains on the Košice–Vienna route, an EN category train on the Bratislava–Humenné route and back, and two-hour cycle express connections on the Bratislava–Trnava–Prievidza/Nitra route and back. The proposal also includes certain connections, which are marked in red. These are ‘’interfering’’ trains that run irregularly as needed in chosen days, for example on Fridays from Bratislava and on Sundays to Bratislava. The draft timetable on the listed route in both directions is shown in Figure 10 and Figure 11.

4. Discussion and Conclusions

This paper deals with the proposal of a long-term strategic concept in the field of national long-distance passenger rail transport on the main railway line Bratislava–Žilina–Košice. In the first chapter, a study of this issue is processed, where the current state of the solution to the issue is analyzed. The second chapter contains an analysis and development of transport performance on the mentioned transport route in the year 2020, while pointing out their fluctuations during the COVID-19 pandemic. The third chapter describes the current scientific methods that are used in the process of optimization and rationalization of timetables, while proposing a modification of the method of inscribed n-angles so that it can be used in the case of long-distance passenger transport. The fourth chapter proposes a specific long-term concept of transport services on the Bratislava–Žilina–Košice route and back. It is a conceptual strategic proposal in which the mentioned scientific methods are practically applicable. The proposals may be practically applicable to both national and private carriers. The main benefit of the mentioned proposals is a better and a more efficient offer of connections on the mentioned transport route. The proposals can be compared with the current state. For effective comparison, it is appropriate to compare four selected indicators (the average travel time, the average travel speed, the number of connections, the average interval between connections). The values of these indicators are compared and shown in

Table 1. Indicator comparison of the proposed connections.

Indicators Comparison	Current Status	Proposal	Differences
Average travel time (hours)	5.47	4.96	−0.51
Average travel speed (km/h)	80.99	89.31	+8.32
Number of connections [-]	22	34	+12
Average interval between connections (hours)	1.69	1.14	−0.55

[35,36].

These figures show that the average travel time was reduced by more than 30 min when compared to the current situation. The average travel speed increased by more than 8 km/h, the total number of connections in both directions increased by 12, and the average interval between connections was also reduced by more than 30 min. These indicators become better than the previous optimization. It means that travel utility of mass on railway system will increase after optimization timetable has been implemented. These proposals should help to revitalize and subsequently further develop long-distance passenger rail transport in order to increase the number of passengers transported during and after the COVID-19 pandemic. Therefore, unless a national or even supranational integrated transport system is created in Slovakia, the mentioned concept of long-distance passenger transport must constitute a kind of bridge that will connect the regions and integrated transport systems in individual countries [37].

The traffic performance and travel demand in long-distance transport also along the Bratislava–Žilina–Košice transport route is still very low because of mistrust and fear of potential infection in public passenger transport. There might be a long time for mass psychology to eliminate the contagious fear of COVID-19. However, through certain measures to increase the attractiveness and safety of public passenger transport, this process should be accelerated. This is mainly because undertakings providing rail passenger services should start to provide a high-quality and time-efficient service to the customer at an acceptable price. Considering its ecological nature and big volume of passenger transport, railway passenger transport represents an irreplaceable sustainable system for the future. Within the framework of the common transport policy is the quality of rail transport services and customer satisfaction with the services is very important. Currently, customers are more demanding and increasingly demand higher quality and in the case of unfulfilled requirements, they expect compensation [38].

However, the problem nowadays is to identify the quality of the transport not only during the transport but also before the transport itself and after the transport has been completed. In the provision of rail passenger services, it is important to consider the fact that passenger requirements and demands may change over time. The use and explanation of the methodology of inscribed n-angles are beneficial for the creation of a more attractive timetable on the line (Vienna)—Bratislava–Žilina–Košice.