THE CELL TRANSMISSION MODEL: A DYNAMIC REPRESENTATION OF HIGHWAY TRAFFIC CONSISTENT WITH THE HYDRODYNAMIC THEORY

CARLOS F. DAGANZO Department of Civil Engineering and Institute of Transportation Studies,

University of California, Berkeley CA 94720, U.S.A.

(Received 23 October 1992; in revisedform 13 July 1993)

Abstract-This paper presents a simple representation of traffic on a highway with a single entrance and exit. The representation can be used to predict traffic’s evolution over time and space, including transient phenomena such as the building, propagation, and dissipation of queues. The easy-to-solve difference equations used to predict traffic’s evolution are shown to be the discrete analog of the differential equations arising from a special case of the hydrodynamic model of traffic flow. The proposed method automatically generates appropriate changes in density at locations where the hydrodynamic theory would call for a shockwave; i.e., a jump in density such as those typically seen at the end of every queue. The complex side calculations required by classical methods to keep track of shockwaves are thus eliminated. The paper also shows how the equations can mimic the real-life development of stop-and-go traffic within moving queues.

1. INTRODUCTION

Accurate descriptions of highway traffic flow over transportation networks, whether at the planning or operations level, must recognize that the vehicles traveling on any section of the network must be bound for specific destinations.

Static traffic assignment models used for transportation planning (see Sheffi, 1985, for example) achieve this goal by describing the flow on a link of the network by its components by final destination; e.g., by specifying a variable yid that represents the amount of flow on link i that is ultimately bound for destination d. Unfortunately, this is much more difficult to do for dynamic network flow problems (with time-dependent origin-destination (O-D) flows) because the functional dependence of the link flows at time t, yid(f), on the collection of all past flows is quite complex. This problem manifests itself both at the planning level, where networks are quite complex, and at the operations level, where networks are simpler, but more detail is sought about the system’s evolution.

Although dynamic traffic assignment models -planning level models involving large networks- typically recognize that traffic travels to many destinations, the models are based on simplistic flow relationships that are not perfectly consistent with the conserva- tion laws of traffic. A planned sequel to this paper will discuss this in more detail.

Traffic operations models can be microscopic or macroscopic. Microscopic simula- tions (e.g., Schwerdtfeger, 1984; Cremer and Ludwig, 1986; Nagel and Schreckenberg, 1992) assume that the behavior of an individual vehicle is a function of the traffic condi- tions in its environment. Although microscopic simulations usually keep track of each vehicle’s destination, their assumptions are difficult to validate because humans’ behavior in real traffic (not in contrived “car-following” experiments) is difficult to observe and measure. This is unfortunate because for a simulation to work the microscopic details have to be just right. (As is well known from nonlinear system theory, microscopic details have a way of affecting the macroscopic world unpredictably.) In this paper, thus, we focus on macroscopic models.

Macroscopic models assume that the aggregate behavior of sets of vehicles, easier to observe and validate, depends on the traffic conditions in their environment. The hydrodynamic theory of traffic flow (Lighthill and Whitham, 1955; Richards, 1956)

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underlies most of these models. While not perfect, its shortcomings are well understood. It is justifiably and routinely used in traffic engineering analyses. Unfortunately, most macroscopic models do not differentiate their component flows by destination; each stream is treated as the flow yi of a single commodity. Thus, when the flow reaches a fork in the road, or some other junction where there is a route choice, the models usually specify either a fixed proportion of turns or a fixed exit flow. In reality, though, neither is fixed; both depend on when vehicles bound for the various destinations reach the junction in question.

This deficiency was addressed by Vaughan, Hurdle, and Hauer (1984), Vaughan and Hurdle (1992), and Hurdle (1992), who formulated the problem for a simple network consisting of a series of links connected by junctions. Unfortunately, the solution to the differential equations resulting from the hydrodynamic model of traffic flow, using the classical method of characteristics, is tedious even for a single link, let alone a series of links. In an attempt to solve the multiple link problem, Newell (1993) proposed a shortcut solution method for the hydrodynamic model of one link. The new method predicts the variation of traffic flow at one end of the link from the behavior of traffic at the other end, without evaluating the behavior at intermediate points. (Based on the cumulative flow curves at the entrance and exit to the link, the method had also been proposed by Luke (1972) for the solution of a geological erosion problem.) This ability to “jump” from one end of the link to the other is exploited by Newell (1991) to solve the freeway traffic flow prediction problem formulated by Vaughan, Hurdle, and Hauer.

This paper presents an alternative way of predicting traffic behavior for one link by evaluating flow at a finite number of carefully selected intermediate points, including the entrance and exit. As in Newell (1993), the difference equations that form the basis for this procedure are shown to be discrete approximations to the differential equations of the hydrodynamic theory for a special form of the equation of state. Although it requires more computer memory than Newell’s method, the proposed procedure can be readily extended to complex networks such as those involving loops and diverging branches; this will be shown in a planned sequel.traffic engineering class speech draft.

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