A Compartmental Model for Traffic Networks and Its Dynamical Behavior

Abstract

We propose a macroscopic traffic network flow model suitable for analysis as a dynamical system, and we qualitatively analyze equilibrium flows as well as convergence. Flows at a junction are determined by downstream supply of capacity as well as upstream demand of traffic wishing to flow through the junction. This approach is rooted in the celebrated Cell Transmission Model for freeway traffic flow. Unlike related results which rely on certain system cooperativity properties, our model generally does not possess these properties. We show that the lack of cooperativity is in fact a useful feature that allows traffic control methods, such as ramp metering, to be effective. Finally, we leverage the results of the paper to develop a linear program for optimal ramp metering.1 the few existing results on equilibria and convergence such as [9] and we present a simple linear program for obtaining a ramp metering control strategy that achieves the maximum possible steady-state network throughput.Our work is related to the dynamical flow networks recently proposed in [10,11] and further studied in [12]. In [10,11], downstream supply is not considered and thus downstream congestion does not affect upstream flow, an unrealistic assumption for traffic modeling. In [12], the authors allow flow to depend on the density of downstream links, but the paper focuses on throughput optimality of a particular class of routing policies that ensure the resulting dynamics are cooperative [13,14]. In contrast, the model proposed here is generally not cooperative. Furthermore, the adaptation to the CTM described briefly in [12, Section II.C] differs from our model in the following important respects: the model as discussed in [12, Section II.C] assumes a path graph network topology, requires identical links (i.e., identical supply and demand functions), and only considers trajectories in the region in which supply does not restrict flow (that is, α v (ρ) = 1 for all v ∈ V in our model), which is shown to be positively invariant given their assumptions. In this work, we generalize each of these restrictions.In a separate direction of research, many network models attempt to apply single road PDE models such as [15,16] directly to networks, see [17] for a thorough treatment. Recent results such as [18] and [19] provide analytical tools for traffic network estimation and modeling using PDE models. The CTM and related models, including our proposed model, can be considered to be a discretization of an appropriate PDE model [6]. Alternatively, these models and the model we propose in this work fit into the broad class of compartmental systems that model the flow of a substance among interconnected "compartments" [20,21,22].We first proposed a compartmental model of traffic flow in [23]. Here, we expand on the conference version by discussing the general lack of cooperativity for our proposed model, identifying how lack of cooperativity can be exploited to increase throughput via ramp metering, and providing an explicit optimization problem for...

On resilient control of dynamical flow networks

“…Example 5. In the continuous-time version of Daganzo's cell transmission model studied, e.g., in [33,34,35,36,37], the network flow dynamics reaḋ…”

Resilience of Traffic Networks with Partially Controlled Routing

“…where κ i (x) ∈ [0, 1] is a parameter that enforces the bounds (3) or, in other words, guarantees that every outgoing link has adequate supply to accommodate the demand of its incoming links. Different models for κ i (x) have been proposed in the literature, and prevalent roles have been played by FIFO policies [13] and proportional allocation rules [14]. We combine the link dynamical equations with (4) and (5) to derive the overall network dynamicṡ…”

“…We then rewrite F i (x k , r k (σ), λ k ) by taking its Taylor expansion for around σ 0 where δ σ = σ − σ 0 , and where we used the implicit differentiation rule to compute Ψ i (r k , x k , λ k , σ) = ∂Fi ∂σ + dFi dr c dr c dσ , with dr c /dσ = η 1 . By substituting into (13) and by rearranging the terms we obtain…”

An exact convex relaxation of the freeway network control problem with controlled merging junctions

“…The corresponding LP with 68684 variables and 137280 constraints is solved by Gurobi [15] in 40sec. 8 In the following, we will introduce uncertainty for external demands and fundamental diagrams and explore the impact on control performance.…”

“…The latter result relies on a monotone reformulation of the system dynamics. Even though it is known that the dynamics of FIFO-diverging junctions are not monotone if expressed in the densities [24,8], one can perform a state transformation to obtain a monotone model [32]. Flow control (or priority control) for merging junctions is necessary to retain monotonicity in the transformed system.…”