Continuous and timed Petri nets for the macroscopic and microscopic traffic flow modelling
Introduction
The traffic flow is often considered as a heterogeneous system. It takes a continuous form in motorway circulation and a discrete one in intersection ways. On one hand, the continuous form of traffic flow is mainly modelled from a macroscopic point of view, which has been inspired from the hydrodynamic theory [1], [12], [18], [19]. In this case, the traffic flow is described by global variables as the flow rate, the flow density and the flow average velocity. On the other hand, the discrete form of traffic flow is modelled from a microscopic point of view that focuses on the individual vehicles behaviours in road. This approach describes the traffic in a finer way: vehicles are individually represented by considering the interactions between them. Generally, this theory represents each vehicle by its acceleration, its velocity and its position in road [2]. The literature on this topic mentions two famous models: car following models [4], [9], [14] and cellular automata models [8], [17].
Petri nets have been used extensively for modelling, analysis and synthesis of dynamic systems, particularly discrete-event systems. A major step in the effort to enhance the modelling power of Petri nets concerns the works about continuous Petri nets [3]. The basic idea is to consider the marking of the places as a positive real number instead of an integer one and the firing of transitions as a continuous process instead of a discrete event one. As a consequence, Petri nets are useful for modelling, analysis and control of hybrid systems.
Concerning the domain of transportation, some works have been done with discrete and also continuous Petri nets (PN) either to represent the traffic flow in motorway corridors or in complex road junctions. A lot of works in transportation area inspired from PN are based on simulation tools. Wang has investigated simple intersections and network control using timed Petri nets and performances evaluation based on SimNet [23]. DiCesare has proposed a model of six signalized intersections in order to implement traffic signal control strategies and to evaluate their impact on the traffic flow using POSES Petri nets simulator [5]. Some analytical works have been also developed with Petri nets either to provide a realistic representation of the continuous flow in interurban traffic networks [6], [21], or to take into account the discrete behaviours in intersections [10], [13].
In order to complete the referenced works and to give an original tool for the performances evaluation of traffic flow, this paper provides a mathematical framework based on Petri nets for the traffic flow modelling. Two particular classes of Petri nets called continuous Petri nets with variable speed (VCPN) and timed Petri nets (TPN) are investigated [3]. The variables of traffic flow are expressed by those of VCPN and TPN. As a consequence, an adaptive spatial discretization of PN traffic flow models is proposed that results in two limit behaviours: a macroscopic one and a microscopic one. The main advantage of the proposed approach lies in the coexistence of two description levels in the same PN representation.
The basic characteristics of PN are reported in Section 2. Macroscopic models based on VCPN are described in Section 3, whereas in Section 4, the TPN are introduced in order to provide microscopic models of the vehicles behaviour. Section 5 shows the portability and application of the proposed traffic PN model on the large scale networks. The complexity of the resulting representation in terms of basic elements and computational requirements is also discussed. Some conclusions and future developments are discussed in Section 6.
Section snippets
Basic Petri nets definitions
Among the set of tools devoted to the study and analysis of discrete-event systems, PN models offer the best compromise between graphic representation and analytic description through mathematical equations that describe the evolution of the systems. Petri nets have been used in a wide range of applications: performance evaluation [15], [20], flexible manufacturing systems [16], [22] and so on. In order to investigate new fields of applications, many extensions of PN have been developed. These
Model description
The proposed VCPN models are inspired from the hydrodynamic theory [12], [19] with an adaptive spatial discretization. In VCPN models each place Pi corresponds to the road segment i, i = 1, … , L, and each transition Ti stands for the separation between the segments Pi and Pi+1. The marking mi(t) of place Pi stands for the number of vehicles ni(t) in the considered segment and the transition firing speed vi(t) stands for the average flow rate qi(t).
The average flow density ρi(t) and average speed Si(
Timed Petri nets models of the traffic
From a macroscopic point of view, VCPN models present the same results as CA ones, whereas in microscopic approach the CA and VCPN models result in different STD. Therefore, it is not possible, with VCPN models, to follow the individual trajectories of the vehicles on the road. In order to overcome this drawback, we will introduce the TPN for the traffic modelling.
Modelling of large scale networks
Another difficulty with Petri net models of the traffic is the complexity and size required to represent large scale networks including a lot of road junctions and also some intersections. In this section, a modular and portable model design is proposed for this purpose. In that case the first step is to define the elementary nets that are suitable to model the basic elements of the traffic network. In this work two basic elements are considered: the road junctions and the simple road
Conclusions
This paper focuses on Petri nets approaches for the modelling of traffic flow. On one hand, the continuous form of traffic flow is modelled by VCPN. An analogy between the parameters of traffic flow and those of VCPN is found. On the other hand, TPN are investigated in order to present the discrete behaviour of vehicles. Furthermore, it could be interesting to study the stochastic behaviour of vehicle drivers by using stochastic distributions not only as input conditions but also as model
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