Stochastic macroscopic traffic flow modeling for highways with heterogeneous drivers

Abstract

Traffic flow modeling is a fundamental element in describing and predicting the characteristics of vehicular movements and an important component in various fields of transportation. Stochastic phenomena are commonly found in traffic dynamics. For example, the traffic flow can vary with traffic density, and the travel time over a given period may fluctuate within a certain range across different days. This stochasticity has received increasing attention. Although numerous studies have focused on fundamental diagrams and microscopic traffic modeling, few studies have taken macroscopic approaches.

The Lighthill–Whitham–Richards (LWR) model is popular for macroscopic–level modeling owing to its simplicity and ability to effectively explain shock formation and propagation. However, this model is deterministic and describes traffic dynamics as equilibrium values over the long term. Therefore, in the current study, a new framework is proposed to account for the uncertainties associated with traffic dynamics. Driving behavior is treated as an endogenous source of stochasticity that should be consistent for a single driver and heterogeneous across drivers. Accordingly, several types of stochastic LWR (SLWR) models are developed. The governing equation of the SLWR models is a conservation law formulated as a time-dependent stochastic partial differential equation, which can consider different traffic stream models. First, the free-flow speed is assumed to be a random parameter, and a linear speed–density relationship is applied. Next, a nonlinear speed–density relationship is discussed. Finally, the jam density, being correlated with the free-flow speed, is included as another random parameter, and a nonlinear speed–density relationship is used. For solving the SLWR models, the Monte Carlo (MC) method is robust and appropriate for obtaining benchmark results. However, the convergence rate is slow, and efficient solution methods need to be explored.

A dynamically bi-orthogonal (DyBO) method based on the spatial basis and stochastic basis is then applied. The DyBO method enables the spatial basis and stochastic basis to evolve over time while maintaining dynamic bi-orthogonality, which substantially reduces the costs of forming a covariance matrix and solving the eigen-problems. The DyBO formulation for each SLWR model is specific. First, a DyBO formulation is derived for the SLWR model with a linear speed–density relationship. Second, a Taylor series expansion is used to handle the exponential term in the DyBO formulation for the SLWR model with a nonlinear speed–density relationship. Third, multivariate Hermite polynomials are used to represent the correlated stochastic parameters calibrated empirically for the SLWR model with a correlated speed–density relationship. Based on simulation experiments with a temporal or geometric bottleneck, the SLWR models can effectively describe stochastic dynamic traffic evolutions, and shocks and propagations due to the bottlenecks are observed. Some typical traffic phenomena such as a capacity drop can also be reproduced. Compared with the MC method, the DyBO method can achieve an acceptable level of accuracy while substantially reducing computation costs. Furthermore, increasing the number of terms of the spatial basis and Hermite polynomials improves the accuracy, and the Taylor series expansion can be coupled successfully with the DyBO method.