Bounded-METANET: A new discrete-time second-order macroscopic traffic flow model for bounded speed

Abstract

Macroscopic traffic flow models are essential for the analysis and control of large-scale transport networks. While second-order models like METANET capture non-equilibrium traffic dynamics, they can produce unrealistic speeds, such as negative values or those exceeding the free-flow limit, leading to unreliable simulations. This paper introduces Bounded-METANET, an enhanced second-order model designed to inherently produce physically consistent outputs. The formulation eliminates the convection term and incorporates anticipation and merging influences within the relaxation term through a mathematically bounded “virtual density” approach. Consequently, simulated speeds are confined to the range [0, $v_{\text{free}}$] without requiring saturation functions, improving model stability and calibration efficiency. The model was evaluated against the original METANET and the first-order Cell Transmission Model (CTM) in two case studies: one involving synthetic data from the SUMO simulator and another using empirical loop detector data from a German motorway. Bounded-METANET consistently outperformed both benchmarks by maintaining physical consistency in traffic flow dynamics. In the synthetic scenario, it achieved the lowest root mean square error for speed and density (9.97% and 17.11% reductions respectively vs. METANET), while in the real-world case it produced superior flow estimates with enhanced shockwave representation. Pareto analysis shows Bounded-METANET’s frontier dominates METANET across all speed-flow weightings. By enforcing physical bounds on traffic variables, Bounded-METANET provides a more reliable framework for traffic simulation, prediction, and control.