Macroscopic Traffic Dynamics Under Fast-Varying Demand

Abstract

Aggregated dynamic traffic models based on the Network Macroscopic Fundamental Diagram (MFD) have been founded on stationary or steady-state approximations. Theoretically, they are only qualified with slow-varying demand profiles as inputs. In practice, MFD models are often fed with fast-varying demand patterns classically observed during peak hours. This paper investigates the accuracy of existing MFD model outputs in such a case. It also provides a fresh look at hysteresis loops that may arise on MFD shapes and show how they are connected to the demand profile. The exact solutions of the LWR model for a simple, but meaningful test case corresponding to the loading and the recovery of an arterial with a single bottleneck are first derived. The spatial integration of these solutions permits to derive the average flow-and the outflow-MFD patterns. The average flow-MFD exhibits a clockwise hysteresis loop with higher flows during the network loading than during the recovery. The outflow MFD shows an opposite trend with a counterclockwise hysteresis loop. The detailed analysis of the LWR solution highlights that these patterns are fully driven by the congestion dynamics and different wave propagation during both periods. This complements existing studies that tend to explain hysteresis by heterogeneous network loading due to unbalanced flows at intersections. It should be noted that the size of the production-MFD hysteresis loop is influenced by the demand loading profile, which means that the MFD is more sensitive to the demand than initially expected. The second part of the paper is about the implications of these observations on the MFD simulation. The main conclusion is that the accumulation-based formulation captures properly the aggregate dynamics of the LWR model during saturation but not in free-flow. It should be calibrated with an average flow-MFD that does not account for the hysteresis pattern to properly estimate the outflow. On the contrary, the original formulation of the trip-based model is only valid in free-flow. During saturation, it should be patched to modify the outflow values. The best approach is a hybrid model that combines the tripbased model in free-flow and the accumulation-based model during saturation. Note that in free-flow the trip-based model is better fed by the average flow-MFD with hysteresis as it gives better estimation for the mean speed. Finally, some evidences are presented and tend to confirm that all the conclusions remain valid for more complex urban settings.

Validation of Macroscopic Fundamental Diagrams-Based Models with Microscopic Simulations on Real Networks: Importance of Production Hysteresis and Trip Lengths Estimation

“…As proposed in Leclercq and Paipuri, and Mariotte and Leclercq(18,26), the outflow (or trip completion rate) is bounded by the maximum capacity and to sustain the outflow to maximum capacity when network reaches saturation to avoid the causality effect (see e.g., Merchant and Nemhauser, Friesz et al…”

“…Mahmassani et al showed that higher demand during congestion period results in the larger hysteresis loop in the MFD ( 17 ). Recently, Leclercq and Paipuri proposed a deeper investigation of the clockwise hysteresis phenomenon in the p-MFD by deriving the Lighthill–Whitham–Richards (LWR) solutions to an arterial case with internal bottleneck ( 18 ). They showed that when the network state is close to saturation, the congestion dynamics caused by bottlenecks with unequal shockwave speeds triggers the hysteresis shape in p-MFD.…”

“…Their work proposed a novel way to model the congestion spill-backs in the trip-based formulation. However, Leclercq and Paipuri showed that no model is perfect and the trip-based model exhibits inconsistent outflow patterns close to saturation ( 18 ). This can be avoided by monitoring the outflow, but the travel times in the reservoir are no longer consistent with trip lengths and mean speed.…”

Empirical Validation of Bimodal MFD Models

“…Another critical factor to consider while defining the functional fit is the scatter in the data. It was already shown the existence of the hysteresis phenomenon in the case of 2D-MFD due to various factors like network heterogeneity, demand pattern (Buisson and Ladier, 2009;Leclercq and Paipuri, 2019). A framework to account for this hysteresis phenomenon in the MFD modeling framework was proposed in Paipuri et al (2019).…”

On the application of variational theory on networks

“…The method of cuts introduced by Daganzo and Geroliminis (2008) estimates the MFD for a homogeneous ring road based on VT. Accordingly, related extensions and studies also refer and exploit VT (Boyaci and Geroliminis, 2011;Leclercq and Geroliminis, 2013;Laval and Castrillón, 2015;Tilg et al, 2020;Daganzo and Knoop, 2016;Leclercq and Paipuri, 2019;Girault et al, 2016;Ambühl et al, 2018;Loder et al, 2019). Again, these approximation methods apply VT only at the corridor level, even when trying to estimate traffic conditions at the network level.…”