Séminaire LICIT-ECO7 (présentiel - ENTPE) : Space in the Tub

Professor Erif Verhoef
Full professor in Spatial Economics at VU Amsterdam and research fellow at the Tinbergen Institute

"Amphi du haut", ENTPE, 3 rue Maurice Audin, Vaulx-en-Velin

Résumé + biographie de l'orateur au format pdf

There is a vivid and rapidly growing line of research deploying macroscopic, single-facility models to represent congestion in urban areas. Often referred to as “Macroscopic Fundamental Diagram” (MFD) models, these models empirically identify, and/or utilize, instantaneous relations between (space-)averaged density and speed in an urban area. This approach typically produces relations that mimic patterns found with the conventional fundamental diagram of traffic congestion, which relates instantaneous and strictly local (rather than space-averaged) measures of flow, speed and density on a single facility. Some key references include Geroliminis and Dagonzo (2007, 2008); and Mariotte et al. (2017).
An important feature in these models is the occurrence of so-called hypercongestion: the range of relatively heavily congested traffic conditions where a further increase in traffic density depresses speed by so much that the product of speed and density, flow, decreases. In the corresponding hypercongested range of the backward-bending speed-flow relation, speed thus increases with traffic flow. This might even be taken to suggest a counterintuitive naïve recommendation that to reduce congestion and increase speed, an increase in traffic flow should be encouraged through subsidies, rather than applying the much more intuitive road toll. Hypercongestion is generally considered to be especially inefficient in that the same traffic flow can be served at a higher speed, lower density combination; while the traffic system is so much congested that its output has fallen below the maximum capacity.
Closely related to these MFD models are economic dynamic equilibrium models that have become known as “bathtub models”, which describe traffic dynamics in a single-facility representing a downtown area, where traffic conditions are assumed to be spatially homogeneous but to vary in continuous time. The spatially uniform speed then responds to the continuously evolving space- averaged traffic density, following an aggregate space-averaged density-speed relation of the same nature as those employed in MFD models (Arnott, 2013; Fosgerau, 2015). Again, hypercongestion is an important phenomenon of interest, and will occur in these models whenever density exceeds the critical value for which flow is at its maximum.
This paper investigates the implications of the perhaps seemingly relatively innocent assumption of spatially homogeneous traffic conditions on the insights and policy recommendations that can be derived from MFD and bathtub models. To that end, we use a structural continuous-time – continuous-space model of traffic congestion based on car-following theory, used earlier in Verhoef (2001, 2003, 2005). The model contains three links forming a Y-shaped network, on which the merging of traffic where the two upstream links feed into the single downstream link causes hypercongested queuing on the upstream links in the dynamic equilibrium. From this dynamic equilibrium we can use the time varying space-averaged measures of aggregate traffic density and speed, to reproduce observations that would feed into an MFD or bathtub model calibrated to represent the same network. By comparing these space-averaged measures to the underlying spatially differentiated variables, we can draw a number of rather far-reaching conclusions that warn against a naïve use of MFD and bathtub models for the evaluation of traffic congestion, and policies to address them.
First, there is a difference between the true relation between traffic density and speed as implied by the car-following equation employed (which applies throughout the entire network), and the relation that is suggested by the set of combinations of network averaged measures of aggregate density and speed that can be observed during the peak. Therefore, even if the same speed-flow relation applies everywhere on a network, it certainly need not be correctly reflected by repeated observations of space-averaged traffic characteristics for that same network.
Second, a network-wide speed-flow relation derived from the aggregate measures therefore naturally deviates from the true speed-flow relation that follows from the car-following equation, and that – as discussed in Verhoef (2003) – is the relevant function to base optimal dynamic congestion tolls on. The MFD speed-flow relation is therefore also a biased basis for the determination of optimal congestion tolls.
And third, this study underlines the importance of distinguishing between two types of hypercongestion: speed-hypercongestion, where a speed below the speed corresponding with the maximum flow is observed, and throughput-hypercongestion, where the performance of the network in terms of producing completed trips has fallen below its capacity because of heavy congestion. While for non-spatial single facility models these two types of hypercongestion perfectly coincide in occurence, this is not true for continuous-time - continuous-space models. For the present exercise, the MFD or bathtub model derived from the space-varying structural model suggests that hypercongestion drives the performance of the network below its capacity during the most heavily congested part of the peak. In reality, this hypercongested queuing takes place upstream of the network’s exit and does not cause the ouflow to fall below the exit capacity.
This study thus underlines the importance of maintaining spatial differentiation of traffic variables in the modelling of congestion and congestion policies whenever analytically or numerically feasible.