Complex Fundamental Diagram of Traffic Flow in the Deep Lefortovo Tunnel (Moscow)

The properties of traffic flow in long highway tunnels has been under individual consideration since the middle of the last century (see, e.g., Refs [1,2]). Interest to this problem is caused by several reasons. The first and, may be, main one is safety. Jams in long tunnels are rather dangerous and detecting the critical states of vehicle flow leading to the jam formation is of the prime importance for the tunnel operation. Second, the tunnel traffic in its own right is an attractive object for studying the basic properties of vehicle ensembles on highways. On one hand, it is due to the individual car motion being more controllable inside tunnels with respect to velocity limits and lane changing. On the other hand, long tunnels typically are equipped with a dense system of detectors, which provides a unique opportunity to receive a detailed information about the spacial-temporal structures of traffic flow.

The present work continues the investigation of tunnel traffic properties reported previously [3]. The analysis is based on empirical data collected during the last time in the Lefortovo tunnel located on the 3 rd circular highway of Moscow (Fig. 1). It comprises two branches and the upper one is a deep linear three lane tunnel of length about 3 km. Exactly in this branch the analyzed data were collected. The tunnel is equipped with a dense system of stationary radiodetetors (Remote Traffic Microwave Sensor, X model) distributed uniformly along it chequerwise at spacing of 60 m. Because of the technical features of the detectors traffic flow on the left and right lanes is measured at spacing of 120 m whereas on the middle lane the spacial resolution is 60 m. The data were averaged over 30 s. Each detector measures three characteristics of vehicle ensemble; the flow rate q, the car velocity v, and the occupancy k for three lanes individually. The occupancy is analog to the vehicle density and is defined as the total relative time during witch vehicles were visible in the view region of a given detector within the averaging interval. It is measured in percent. The detectors themselves and their records were analyzed initially to justify the reliability of the collected data.

The fundamental diagram under consideration was constructed as follows. The phase space {k, v, q} was divided into cells of size about 1 % × 1 km/h × 0.01 car/s. Each 30 seconds a detector contributes unity to one of the cells. Taking into account a certain rather long time interval of traffic flow observation, all the detectors, and then dividing the result by the total number of records we obtain the three-dimensional distribution P (k, v, q) of fixed traffic flow states over this phase space. In order to elucidate the obtained result we present the projection of P (k, v, q) on three phase planes {kq}, {kv}, and {vq}. Besides, in projecting onto the given phase planes some layers can be singled out, for example, the expression

specifies the projection of the layer DV = (vmin, vmax) onto the plane {kq} within a constant cofactor normalizing it to unity. Such distributions will be also referred to as slices of the fundamental diagram.

Figure 2 presents the projection of the whole fundamental diagram onto the plane {k, q} (the upper left frame) as well as its slices parallel to this plane. In the frame of the whole projection two branches are singled out by the relation v ≶ 21 km/h × k/kc2, where the critical occupancy kc2 = 31% according the results to be demonstrated further. The two branches with a small degree of overlap are Fig. 2. Projection of the fundamental diagram onto the plane {k, q} as well as its slices parallel to this plane. separated actually by the transition from light to heavy synchronized traffic (see below). The given slices of fixed velocity demonstrate the fact that, at least, three different states of heavy congested traffic were observed. It reflects in the existence of three branches visible well for v = 19, 13, 7 km/h. Their additional analysis demonstrated us that these branches are characterized individually by different mean lengths of vehicles. In particular, the higher is a branch in Fig. 2, the shorter, on the average, vehicles forming it. The distribution of the traffic flow states becomes rather uniform for very low velocities matching the jam formation. On the whole fundamental diagram the jammed traffic is described by the region looking like a certain “beak”.

Figure 3 Figure 4 exhibits the projection of the fundamental diagram onto the plane {q, v} and evolution of its slices for fixed values of the occupancy. In this figure the four different phase states of the analyzed tunnel traffic are visible. The free flow where the overtaking manoeuvres are most feasible corresponds to the three branches that can be related to trucks, passenger cars, and high-speed cars. As the traffic flow rate grows with the occupancy k the three branches terminate and are followed by a structureless two-dimensional

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