Speed–density functional relationship for heterogeneous traffic data: a statistical and theoretical investigation

Gaddam, Hari Krishna; Rao, K. Ramachandra · 2018 · OpenAlex-citations

DOI: 10.1007/s40534-018-0177-7

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Summary

This study addresses the lack of accurate speed–density functional relationships for heterogeneous traffic conditions, specifically within the context of Indian urban arterials. Existing macroscopic traffic flow models, largely developed for homogeneous traffic, often fail to capture the complex dynamics of mixed-vehicle streams involving cars, two-wheelers, three-wheelers, and heavy vehicles. The authors aim to evaluate existing single-regime speed–density models for their statistical accuracy and adherence to static and dynamic traffic properties, and to propose new functional forms that overcome identified limitations. The researchers collected empirical data from two urban arterial sections in Delhi, India, using video image processing software to extract class-specific speeds and volumes. The study considered the entire road width as a single lane due to the absence of lane discipline. Key parameters, including free flow speed, jam density, and kinematic wave speed at jam ($C_j$), were estimated from field observations. Notably, $C_j$ was derived by establishing a linear relationship between driver reaction time and vehicle position in queues at signalized intersections, yielding a value of -12.42 km/h. Model parameters were calibrated using the Levenberg–Marquardt algorithm in R statistical software. The models were evaluated using root mean squared error (RMSE), average relative error (ARE), and cumulative residual (CURE) plots, while also being tested against static properties (e.g., speed approaching zero at jam density) and dynamic properties (e.g., convexity of the flow–density relationship near jam density). The analysis revealed that while some existing models, such as those by Wang et al. and Papageorgiou et al., demonstrated superior statistical fit compared to others, none of the existing functional forms satisfied all required static and dynamic properties of the fundamental diagram. Specifically, many models failed to accurately represent shock wave propagation or the convexity required for stable start waves in congested conditions. To address these shortcomings, the authors proposed two new speed–density functional forms. These new models were designed to simultaneously satisfy numerical accuracy metrics and the theoretical properties of traffic flow dynamics. The proposed forms incorporate the estimated kinematic wave speed and a saturation flow parameter, allowing them to better replicate the empirical behavior of heterogeneous traffic streams across both free-flow and congested regimes. The significance of this work lies in its contribution to macroscopic traffic modeling for developing economies with mixed traffic compositions. By establishing a functional relationship that adheres to both statistical accuracy and theoretical traffic dynamics, the study provides a more robust tool for dynamic traffic studies, such as modeling shock waves and queue lengths. The findings suggest that accurate modeling of heterogeneous traffic requires specific attention to kinematic wave speeds and saturation flows, offering a refined approach for traffic simulation and management in non-lane-based environments.

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• speed choice