A comparison of the accumulation-based, trip-based and time delay macroscopic fundamental diagram models

Abstract

Macroscopic fundamental diagram (MFD) is widely applied in network modelling and management, such as route guidance and vehicle relocation, which are formulated as generalised dynamic traffic assignment (DTA) problems. MFD can effectively reduce the spatial dimension thus making the generalised DTA problems computationally efficient. In the literature, three MFD models, the accumulation-based model, the trip-based model, and the time delay model, were proposed to capture the traffic flow propagation under different traffic conditions and demand scenarios. However, no consensus has been reached on their computational efficiency and which model should be chosen under certain traffic conditions and demand scenarios. In this paper, we revisit these models regarding two important theoretical properties regarding flow propagation in the DTA, i.e. the first-in-first-out (FIFO) principle and causality. Corresponding dynamic network loading algorithms are designed to compare their numerical accuracy and computational efficiency. Numerical comparisons with Lighthill-Whitham-Richards (LWR) model and a micro simulator confirm that the accumulation-based model is valid in saturation, the trip-based model is valid in free-flow, while the time delay model provides a good approximation in both free-flow and saturation scenarios. On the other hand, violation of strict causality is observed in the accumulation-based and trip-based models, rendering it hard to pursue analytical DTA. This issue is not observed in the time delay model. Overall, the time delay model is a promising alternative for dynamic network loading in large-scale network applications.

Keywords:

• Macroscopic fundamental diagram
• dynamic network loading
• accumulation-based model
• trip-based model
• time delay model

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The MFD mapping network accumulation to the trip completion rate denoted by $G(n)$ is the most commonly used form in the literature. Here, we use G-MFD for this specific case while MFD as a general relationship for all kinds of traffic variables throughout the paper.

2 Since estimating the initial network condition is never an easy task, zero initial network condition is commonly assumed for convenience of network loading in the DTA literature but not compulsory.

Additional information

Funding

The work in this paper was jointly supported by research grants from the National Natural Science Foundation of China [grant number 72071214], the National Key R&D Program of China [grant number 2018YFB1600500], and the Research Grants Council of Hong Kong [grant number R5029-18 & 15211518E].