A fuzzy Petri net model to estimate train delays

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Abstract

Even with the most accurate timetable, trains often operate with delays. The running and waiting times for trains can increase unexpectedly, creating primary delays that cause knock-on delays and delays for other trains. The accurate estimation of train delays is important for creating timetables, dispatching trains, and planning infrastructures. In this work, we proposed a fuzzy Petri net (FPN) model for estimating train delays. The FPN model with characteristics of hierarchy, colour, time, and fuzzy reasoning was used to simulate traffic processes and train movements in a railway system. The trains were coloured tokens, the track sections were termed places, and discrete events of train movement were termed transitions. The train primary delays were simulated by a fuzzy Petri net module in the model. The fuzzy logic system was incorporated in the FPN module in two ways. First, when there were no historical data on train delays, expert knowledge was used to define fuzzy sets and rules, transforming the expertise into a model to calculate train delays. Second, a model based on the Adaptive Network Fuzzy Inference System (ANFIS) was used for systems where the historical data on train delays were available (from detection systems or from the train dispatcher’s logs). The delay data were used to train the neuro-fuzzy ANFIS model. After the results of the fuzzy logic system were verified, the ANFIS model was replicated by a fuzzy Petri net. The simulation was validated by animating the train movement and plotting the time-distance graph of the trains. Results of the simulation were exported to a database for additional data mining and comparative analysis. The FPN model was tested on a part of the Belgrade railway node.

Highlights

► The fuzzy Petri net (PN) model was used to estimate train delays on the Belgrade railway node. ► The high-level PN model was used to simulate traffic processes. ► The trains were coloured tokens, the sections were places, and the train movements were transitions. ► The initial train delays were calculated using a fuzzy Petri net (FPN) module. ► The FPN module was defined by expert knowledge (fuzzy logic) or by detected train delays (ANFIS).

Introduction

Train delays are one of the most commonly used parameters for determining timetables and solving infrastructure problems. The delay data are also important for train dispatching and railway traffic operations. Day-to-day or even hour-to-hour train delays can occur within the same day, and the unpredictability of the delays makes efficient planning of railway operations very difficult even for a short period of time.

Primary delays are train delays caused by external stochastic disturbances. When primary delays develop inside the observed network, they are called original delays. If the buffer times between trains are less than the length of the primary disturbance, delay is propagated to other trains. The primary delay of one train can cause delays on other trains and create knock-on or secondary delays. It is very difficult to calculate and predict secondary delays because they depend on the length of primary delays, the timetable of the trains, and the infrastructure (such as single or double track, station layouts, and interlocking) [10]. Primary delays are often caused by technical failures, lower-than-scheduled running speeds, prolonged alighting and boarding times of passengers, and bad weather conditions [8]. Distribution of primary delays can be obtained by a statistical analysis of existing empirical data. Yuan and Hansen [24] proposed an analytical stochastic model of propagation delays in the stations to calculate secondary delays.

Three common approaches to determine train delays are typically used in the literature [14]: analytical methods, microsimulation methods and statistical analyses based on empirical data.

The analytical model uses the queuing theory; it does not require a large amount of input data and usually applies simplifying assumptions on the system. Queuing models estimate the total and average waiting time of trains at platform tracks or junctions. These models are applied during strategic planning to evaluate the impact of increasing train frequencies and the impact of modifying infrastructure and train characteristics on the waiting time. Simulation models are detailed representation of a railway system, where different trains interact with each other and with the infrastructure. They require data regarding the infrastructure, the timetable and the performance of the trains [9]. If one of these pieces of data is unknown, assumptions would be required; therefore, the simulation results would depend on the quality of the input data. Microscopic simulation tools can be used to model the propagation of train delays in large railway networks, but their use require extensive work to model the infrastructure topology, signalling and timetables using simulation software such as RailSys and OpenTrack [3], [9].

Statistical analysis is primarily used for modelling the occurrence of primary delays. The observed delays can be used to establish empirical relationships between how well the capacity is utilised and the secondary delays, given the prevailing level of primary delays. This can be applied in railway systems that are well regulated and operate in stable conditions. In systems where there are many possible sources of disruptions and a relatively high probability of external influences that could induce primary delays, it is difficult to find a relationship that can be used to calculate train delays.

In the railway system of Serbian Railways, statistical analysis of arrival delay data suggests that many factors influence train delays; additionally, large disturbances are present in the system. Train delays are very difficult to predict because of the many factors involved.

Fuzzy logic is a mathematical tool used to model traffic processes that are distinguished by subjectivity, uncertainty, ambiguity and imprecision [22], [23]. Many authors use predictive modelling systems with fuzzy logic. Fay [7] used fuzzy logic to model a dispatcher support system for railway operation and control. This model was defined as a fuzzy Petri net model that combined expert knowledge of fuzzy systems and the graphical power of Petri nets, making the model easy to design, test, improve and maintain. Cheng and Yang [5] proposed a fuzzy Petri net model that used the professional knowledge of dispatchers to create database rules for testing a system in case of disorder.

In this paper we present two approaches to modelling the primary delays of arriving trains in Serbian Railways network. The first model was created for systems without any historical data on train delays that require additional information to determine the occurring frequency and length of train delays. For most technical systems, the solution to a problem is determined using the experience of experts who have dealt with these problems for an extended period of time. The fuzzy logic model used to calculate train delays in this work considered the expertise, experience and knowledge of railway personnel who directly participated in regulating the traffic in the system. Data from personnel interviews and timetable information were used to define the parameters of a fuzzy system in a fuzzy Petri net for forecasting trains delays. In this manner, the parameters of the fuzzy logic system were different between specific cases. Knock-on delays were calculated by the model using simulation results. A second model based on the Adaptive Network Fuzzy Inference System (ANFIS) [12] was used for systems where the historical data on train delays were available (from detection systems or from the train dispatcher’s logs). The delay data were used to train the neuro-fuzzy ANFIS model. After the results were verified, the ANFIS model was replicated by a fuzzy Petri net (FPN).

Simulation of train movement in railway systems is very complex because of the many parameters and relations required to describe such a complex system. Complex system can be divided into specific interrelated modules (subsystems). These modules can be easily modelled. In modelling complex systems, we can use abstraction, polymorphism, hierarchy, and monitoring [2]. The principle of monitoring provides the information about the model behaviour and observation data of the changes in parameter values of model’s objects, and evaluation of the current state and markings in the model. The simulation tool must be able to construct a model that incorporates all interlocking principles, operating rules and data. Petri nets are used for graphical and mathematical modelling of various systems. High level Petri nets (HLPNs), which consider the time, colour, stochastic and hierarchical characteristics, are used to model complex system, and they can present the model graphically. Many simulation models found in the literature presented analyses of various railway systems with focus on train delays. Basten et al. [4] created a simulation model for the analysis of interlocking specification using collared Petri nets in the software Expect [11]. Aalst and Odijk [1] proposed the interval-timed coloured Petri nets to model and analyse railway stations; the train delays were specified by an upper and lower bound, i.e., an interval. Daamen et al. [6] developed coloured Petri nets to identify route conflicts and estimate knock-on delays.

In our previous work [15], we have shown that a fuzzy logic model that uses expert knowledge can be used to calculate primary train delay. In further research, as presented in this paper, we use the data detected from the real system to train and test the new neuro-fuzzy model for estimating the train delays and to establish the connection between the train parameters and the corresponding delay. We tested the Petri net (PN) model on a part of the Belgrade railway node of Serbian Railways. The FPN module in the PN model generated the primary delays that were combined with timetable data to produce the arrival times of the trains. Analysis of the traffic conditions and data collected during the research suggested that, for the FPN module, the following parameters should be used as inputs for fuzzy logic train delay models: the train category, the time of arrival at the station, the distance travelled, and the infrastructure influence. The time of arrival, the train data and the infrastructure data were inputs to the Petri net simulation model. Results of the simulation model were verified by animating track sections occupied by trains and using a train time-distance graph. The FPN model presented in this paper was used as a new method in Serbian Railways research studies for estimating train delays.

This paper is structured as follows. Section 2 starts with a definition of fuzzy Petri nets. The FPN model to simulate train delays, and basic rules for train movements are described in Section 3. Section 4 presents the hierarchy and modules of the FPN. The steps for constructing a model of the railway system are provided in Section 5. Section 6 demonstrates the application and the results of the FPN model of the part of Belgrade railway node. Conclusions are provided in the final section.

Section snippets

Fuzzy Petri nets

A fuzzy Petri net is a Petri net that uses fuzzy logic rather than Boolean logic [13]. The fuzziness concept can be incorporated in Petri nets by applying a fuzzy reasoning mechanism over the Petri nets structure.

The fuzzy Petri net model for simulating train delays

The simulation model of the railway system was created using the ExSpect version 6.41 software [11], in which the HLPN extends the PN to data, time and hierarchy [17]. Tokens have types and values (colours). In addition, each place in the net also has a type, restricting the type of tokens allowed in that place. ExSpect specification provides a special kind of place, a store. A store always contains exactly one token, and it might be considered a variable. Another extension of ExSpect is the

Fuzzy Petri nets modules and the model hierarchy

The basic subsystem or module used for model creation is the track section module. There are similar modules that use the same concepts to model interlocking principles, safety and signalling systems. However, these modules differ in the number of connecting input and output pins used (for modules of the switch) and in the additional rules for train movement within a station area. Modules for generating trains use the timetable data imported from an external database used to generate tokens

Creating the FPN model of the railway system

Previously defined modules can be used to create an FPN simulation model of various railway systems. Creating the simulation model required connecting and arranging the modules according to the layout plan of the track sections [16].

When creating a model, one must take into account all of the rules and operating conditions of a railway system. The following algorithm is used to create the model.

  • The railway system is analysed, and definitions are prepared for the fuzzy logic model and the data

The application of a model

The model was tested on a part of the Belgrade railway node and was defined using the infrastructure data of the sections, the railway signal layout plan, and the timetable data from the year 2010. The boundaries of the model consisted of the Beograd Centar station, the Topcider station, the Rakovica station and the Karadjordjev Park station (Fig. 8). Three train categories were present in the model: freight, regional (including regional and suburban) and passenger (including long distance

Conclusion

An FPN can be used to model systems characterised by ambiguity and uncertainty. Train delays can be modelled intuitively or using the ANFIS model trained by historical data. Systems with unknown behaviour and without known data can be modelled as fuzzy systems defined by the experience of experts (in this case, the train dispatchers and operating staff). Such a model depends on expert knowledge along with the timetable and infrastructure data to create suitable FPN parameters to determine train

Acknowledgments

The authors acknowledge the support of research Project TR 36012, funded by the Ministry of Education, Science and Technological Development of the Republic of Serbia.

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