Three-phase traffic theory and highway capacity

Abstract

Hypotheses and some results of the three-phase traffic theory by the author are compared with results of the fundamental diagram approach to traffic flow theory. A critical discussion of model results about congested pattern features which have been derived within the fundamental diagram approach to traffic flow theory and modeling is made. The empirical basis of the three-phase traffic theory is discussed and some new spatial–temporal features of the traffic phase “synchronized flow” are considered. A probabilistic theory of highway capacity is presented which is based on the three-phase traffic theory. In the frame of this theory, the probabilistic nature of highway capacity in free flow is linked to an occurrence of the first order local phase transition from the traffic phase “free flow” to the traffic phase “synchronized flow”. A numerical study of congested pattern highway capacity based on simulations of a KKW cellular automata model within the three-phase traffic theory is presented. A congested pattern highway capacity which depends on features of congested spatial–temporal patterns upstream of a bottleneck is studied.

Introduction

Real traffic is a dynamical process which occurs both in space and time. This spatial–temporal process shows very complex dynamical behavior. In particular, highway traffic can be either free or congested. Congested traffic states can be defined as the traffic states where the average vehicle speed is lower than the minimum possible average speed in free flow (e.g., Ref. [1]). It is well known that in contrast to free traffic flow, in congested traffic a collective behavior of vehicles plays an important role (the collective flow by Prigogine and Herman [2]), and a synchronization of vehicle speeds across different highway lanes usually occurs [1]. In congested traffic, complex spatial–temporal patterns are observed, in particular a sequence of moving traffic jams, the so-called “stop-and-go” phenomenon (e.g., the classical works by Treiterer [3] and Koshi et al. [1]). Recall that a moving jam is a moving localized structure. The moving jam is spatially restricted by two upstream moving jam fronts where the vehicle speed and the density change sharply. The vehicle speed is low (sometimes low as zero) and the density is high inside the moving jam.

Congested traffic usually occurs at a highway bottleneck, e.g., at the bottleneck due to an on-ramp. In empirical investigations, the onset of congested traffic is accompanied by the breakdown phenomenon, i.e., by a sharp decrease in the vehicle speed at the bottleneck (see e.g., papers by Athol and Bullen [4], Banks [5], [6], Hall et al. [7], [8], Elefteriadou et al. [9], Kerner and Rehborn [10] and by Persaud et al. [11]). It has been found that the breakdown phenomenon has a probabilistic nature, i.e., the probability of the speed breakdown is an increasing function of the flow rate in free flow at the bottleneck [11]. Besides, it has been found that the capacity of congested bottleneck, i.e., highway capacity after the breakdown phenomenon at the bottleneck has occurred is often lower than the capacity in free flow before—the so-called phenomenon “capacity drop” [5], [6], [7], [8].

Concerning the important role of highway bottlenecks it should be noted that although congested traffic can occur away from bottlenecks [12], they have an important impact just like defects in physical systems which can play an important role for the phase transitions and for the formation of spatial–temporal patterns. The role of the bottlenecks in traffic flow is as follows: Congested traffic occurs most frequently at highway bottlenecks (e.g., [13], [14]). The bottlenecks can result from for example due to road works, on and off ramps, a decrease in the number of highway lanes, road curves and road gradients.

Although the complexity of traffic is linked to the occurrence of spatial–temporal patterns, some of the traffic features can be understood if average traffic characteristics are considered. Thus, important empirical methods in traffic science are empirical flow–density and speed–density relationships which are related to measurements of some average traffic variables at a highway location, in particular at a highway bottleneck. The empirical relationship of the average vehicle speed on the vehicle density must be related to an obvious result observed in real traffic flow: the higher the vehicle density, the lower the average vehicle speed. When the flow rate, which is the product of the vehicle density and the average vehicle speed, is plotted as a function of the vehicle density one gets what is known as the empirical fundamental diagram. It must be noted that the empirical fundamental diagram is successfully used for different important applications where some average traffic flow characteristics should be determined (e.g., [14], [15]).

Recently dependencies of the empirical fundamental diagram on the type of the congested pattern at a freeway bottleneck and on the freeway location where the empirical fundamental diagram is measured have been found [16].

Over the past 80 years scientists have developed a wide range of different mathematical models of traffic flow to understand these complex non-linear traffic phenomena (see the classical papers by Lighthill and Whitham [17], Richards [18], Herman et al. [19], Gazis et al. [20], Kometani and Sasaki [21], Newell [22], Prigogine [23], Payne [24], Gipps [25], the books by Leuzbach [26], May [14], Daganzo [13], Prigogine and Herman [2], Wiedemann [27], Whitham [28], Cremer [29], Newell [30], the reviews by Chowdhury et al. [31], Helbing [32], Nagatani [33], Nagel et al. [34] and the conference proceedings [35], [36], [37], [38], [39], [40], [41]). Clearly these models must be based on the real behavior of drivers in traffic, and their solutions should show phenomena observed in real traffic.

Up to now by a development of a mathematical traffic flow model which should explain empirical spatial–temporal congested patterns, it has been self-evident that hypothetical steady-state solutions of the model should belong to a curve in the flow–density plane (see, e.g., [17], [20], [22], [23], [2], [38], [39], [40], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [28], [29], [26], [58], [59], [60], [61], [62], [63], [64], [65], [66], [67], [68], [69], [70] and the recent reviews [31], [32], [33], [34]). The above term steady state designates the hypothetical model solution where vehicles move at the same distances to one another with the same time-independent vehicle speed. Therefore, steady states are hypothetical spatially homogeneous and time-independent traffic states (steady states are also often called “homogeneous” or “equilibrium” model solutions; we will use in the article for these hypothetical model traffic states the term “steady” states or “steady speed” states). The curve in the flow–density plane for steady-state model solutions goes through the origin and has at least one maximum. This curve is called the fundamental diagram for traffic flow.

The postulate about the fundamental diagram underlies almost all traffic flow modeling approaches up to now (see reviews [31], [32], [33], [34]) in the sense that the models are constructed such that in the unperturbed, noiseless limit they have a fundamental diagram of steady states, i.e., the steady states form a curve in the flow–density plane. The fundamental diagram is either a result of the model (e.g., for the models by Gazis et al. [20], Gipps [25] Nagel–Schreckenberg cellular automata (CA for short) [42], [43], [44], [52], [69], Krauß et al. [51], Helbing, Treiber and co-workers [56], [66], [71], Tomer, Halvin and co-worker [68], [72]) or the fundamental diagram is hypothesized in a model (e.g., the optimal velocity models by Newell [22], Whitham [45], Bando, Sugiyama and co-workers [47], [48] and the macroscopic model by Payne [24]). Moving jams which are calculated from these models for a homogeneous road (i.e., a road without bottlenecks) are due to the instability of steady states of the fundamental diagram within some range of vehicle densities (see reviews [31], [32], [33], [34]). This is one of the reasons why we find it helpful to classify these models as belonging to what we call the ”fundamental diagram approach”.

In 1955 Lighthill and Whitham [17] wrote in their classical work (see p. 319 in [17]): “... The fundamental hypothesis of the theory is that at any point of the road the flow (vehicles per hour) is a function of the concentration (vehicles per mile)...”. Apparently the empirical fundamental diagram was the reason that the fundamental diagram approach has already been introduced in the first traffic flow models derived by Lighthill and Whitham [17], by Gasis et al. [20], and by Newell [22].

Concerning the theoretical fundamental diagram, it must be noted that in real congested traffic complex spatial–temporal traffic patterns are observed (e.g., [1], [3]). These patterns are spatially non-homogeneous. This spatial behavior of congested patterns is a complex function of time. An averaging of traffic variables related to congested patterns over long enough time intervals gives a relation between different averaged vehicle speeds and densities. Thus, the empirical fundamental diagram is related to averaged characteristics of spatial–temporal congested patterns measured at a highway location rather than to features of the hypothetical steady states of congested traffic on the theoretical fundamental diagram. For this reason, the existence of the theoretical fundamental diagram is only a hypothesis. This is also confirmed by a recent empirical study where it has been found that the empirical fundamental diagram strongly depends on the congested pattern type and on the freeway location [16].

It has recently been found that empirical features of the phase transitions in traffic flow and most of empirical spatial–temporal pattern features [74], [75], [76] are qualitatively different from those which follow from mathematical traffic flow models in the fundamental diagram approach which is considered in the reviews [31], [32], [33], [34].

For this reason in 1996–2000 the author, based on an empirical traffic flow analysis, introduced a concept called “synchronized flow” and the related three-phase traffic theory [12], [74], [75], [77], [78], [79], [80], [81], [82], [83], [84].

In the concept “synchronized flow”, there are two qualitatively different phases, the traffic phase called “synchronized flow” and the traffic phase called “wide moving jam”, which should be distinguished in congested traffic [74], [75], [77], [78], [79], [80]. This distinguishing is based on qualitatively different empirical spatial–temporal features of these phases. Traffic consists of free flow and congested traffic. Congested traffic consists of two traffic phases. Thus, there are three traffic phases:

(1) Free flow.

(2) Synchronized flow.

(3) Wide moving jam.

In the three-phase traffic theory, features of spatial–temporal congested patterns are explained based on the phase transitions between these three traffic phases.

Objective criteria to distinguish between the traffic phase “synchronized flow” and the traffic phase “wide moving jam” are based on qualitative different empirical spatial–temporal features of these two traffic phases. A wide moving jam is a moving jam which possesses the following characteristic feature. Let us consider the downstream front of the wide moving jam where vehicles accelerate escaping from a standstill inside the wide moving jam. This downstream jam front propagates on a highway keeping the mean velocity of this front. As long as a moving jam is a wide moving jam this characteristic effect—the keeping of the velocity of the downstream jam front—remains even if the wide moving jam propagates through any complex traffic states and through any highway bottlenecks. In contrast, the downstream front of the traffic phase “synchronized flow” (where vehicles accelerate escaping from synchronized flow to free flow) is usually fixed at the bottleneck. Corresponding to the definition of the traffic phase “wide moving jam” and to the concept “synchronized flow” where there are only two traffic phases in congested traffic, any state of congested traffic which does not possess the above characteristic feature of a wide moving jam is related to the traffic state “synchronized flow”. A more detailed consideration of the objective criteria of traffic phases and empirical examples of the application of these objective criteria for the determination of the phase “synchronized flow” and the phase “wide moving jam” in congested traffic can be found in Ref. [76]. It must be noted that there are at least two well-known empirical effects in congested traffic (e.g., [1]): (1) Synchronization of the average vehicle speed between different freeway lanes. (2) A wide spreading of empirical data in the flow-density plane. These effects can occur in both traffic phases of congested traffic, “synchronized flow” and “wide moving jam”. The distinguishing between these traffic phases is made only based on the above objective criteria for traffic phases rather than on the speed synchronization effect or on the effect of the wide spreading of empirical data in the flow-density plane.

It should be stressed that the concept “synchronized flow” and the related methodology of the congested pattern study, which has been used for the definition of the three traffic phases below, is based on an analysis of empirical spatial–temporal features of congested patterns [74], [77], [78] rather than on a dynamical analysis of data (e.g., in the flow–density plane) only. First, a spatial–temporal study of traffic must be made. Only after the traffic phases “synchronized flow” and “wide moving jam” have already been distinguished, based on the objective criteria for the traffic phases discussed above some of the pattern features can further be studied, e.g., in the flow–density plane. In particular, this procedure has already been used in Ref. [77]: At the first step, a spatial–temporal analysis of empirical data has been made and the phases “synchronized flow” and “wide moving jam” have been identified. At the next step, the traffic phase “synchronized flow” has been plotted in the flow–density plane without any wide moving jams and some of the features of “synchronized flow” have been studied. Measured data on sections of the highway A5 in Germany which have been used in [12], [74], [75], [76], [77], [78], [79], [80], [81], [82], [83], [84] also comprise the information about vehicle types (number of vehicles and long vehicles) and individual vehicle speeds passing the detector during each one minute interval of averaging. Using the latter information in addition to the spatial–temporal data analysis of 1 min averaged data, the determination of the type of synchronized flow in empirical studies of synchronized flow (the type (i), or (ii) or else (iii) [77]) have been made. Besides, the individual vehicle speeds allow us to answer the question of whether there are narrow moving jams in synchronized flow or not. All these steps of data analysis have been made in [12], [74], [75], [76], [77], [78], [79], [80], [81], [82], [83], [84]. We will illustrate this in Section 2 where an empirical example of a synchronized flow pattern will be studied.

Another claim of the concept of “synchronized flow” is the hypothesis about steady states of synchronized flow. This is the fundamental hypothesis of the three-phase traffic theory. The fundamental hypothesis of the three-phase traffic theory reads as follows [74], [78], [79], [80]:

Hypothetical steady states of synchronized flow cover a two-dimensional region in the flow–density plane (Fig. 1). This means that in these hypothetical steady states of synchronized flow, where all vehicles move at the same distance to one another and with the same time-independent speed, a given steady vehicle speed is related to an infinite multitude of different vehicle densities and a given vehicle density is related to an infinite multitude of different steady vehicle speeds. This hypothesis means that there is no fundamental diagram for hypothetical steady speed states of synchronized flow. This hypothesis has recently been used in a microscopic three-phase traffic flow theory [85], [86], [87]. It occurs that this theory, which is also based on other hypotheses of the three-phase traffic theory (Section 4), explains and predicts main features of empirical phase transitions and spatial–temporal congested patterns found in [74], [76].

The fundamental hypothesis of the three-phase traffic theory is therefore in contradiction with the hypothesis about the existence of the fundamental diagram for hypothetical steady states of mathematical models and theories in the fundamental diagram approach. An explanation of the fundamental hypothesis of the three-phase traffic theory will be done in Section 4.1.1.

The term “synchronized flow” should reflect the following features of this traffic phase: (i) It is a non-interrupted traffic flow rather than a long enough standstill as it usually occurs inside a wide moving jam. The word “flow” should reflect this feature. (ii) There is a tendency to a synchronization of vehicle speeds across different lanes on a multi-lane road in this flow. Besides, there is a tendency to a synchronization of vehicle speeds on each of the road lanes (a bunching of the vehicles) in synchronized flow due to a relatively low mean probability of passing in synchronized flow. The word “synchronized” should reflect these speed synchronization effects.

The term “wide moving jam” should reflect the characteristic feature of the jam to propagate through any other states of traffic flow and through any bottlenecks keeping the velocity of the downstream jam front. The word combination “moving jam” should reflect the feature of the jam propagation as a whole localized structure on a road. If the width of a moving jam is considerably higher than the widths of the jam fronts and the speed inside the jam is zero then the moving jam possesses this characteristic feature. The word “wide” (the jam width in the longitudinal direction) should reflect this characteristic feature of the jam propagation keeping the velocity of the downstream jam front. However, the distinguishing between the traffic phases “synchronized flow” and “wide moving jam” is made only based on the objective criteria for traffic phases which have been considered in Section 1.2.1.

This article is organized as follows. In Section 2, firstly an overview of known features of synchronized flow is made (Section 2.1) and then new empirical results about spatial–temporal features of synchronized flow are presented. These empirical results should help to understand some of the hypotheses to the three-phase traffic theory. A critical analysis of the application of the fundamental diagram approach for a description of phase transitions and of spatial–temporal features of congested patterns will be made in Section 3. In Section 4, a comparison of already known hypotheses to the author's three-phase traffic theory with results of the fundamental diagram approach to traffic flow theory will be considered. In this section, we will also consider some new hypotheses about a Z- and double Z-shaped characteristics of traffic flow. The double Z-characteristic should explain phase transitions which are responsible for the wide moving jam emergence in real traffic flow. Other new results are presented in Section 5 where a probabilistic theory of highway capacity which is based on the three-phase traffic theory is considered. This general theory will be illustrated and confirmed by new numerical results of a study of a KKW cellular automata model within the three-phase traffic theory.