Second-Order Macroscopic Modelling of Stop-and-Go Traffic Under Fixed and Feedback-Based Signal Control

Urban traffic congestion is strongly influenced by signalized intersections. Analysing how different control strategies influence traffic flow at these intersections requires mathematical models that capture these interactions. This study develops a second-order macroscopic traffic flow model that incorporates both fixed-time and feedback-based adaptive signal control systems. Unlike most macroscopic models where signal control is treated as an external boundary condition, the proposed formulation embeds signal logic directly into the dynamic velocity equation. This allows signal timing to directly influence traffic dynamics and enables the analysis of stop-and-go waves near signalized intersections. Linear stability analysis identifies a critical relaxation threshold separating stable flow from stop-and-go oscillations. This threshold is governed by driver sensitivity, adaptation strength, and the slope of the equilibrium speed-density relation. Numerical simulations across free-flow, mild congestion, and heavy congestion conditions confirm the analytical findings. In free-flow conditions, both control strategies perform similarly with no measurable difference in speed or discharge flow. Under mild congestion, adaptive control improves average speed with marginal discharge flow gains, as the hysteresis switching logic effectively reduces unnecessary red-phase duration. Under heavy congestion, however, performance converges to that of fixed-time control strategies, indicating the limits of adaptivity when demand persistently exceeds capacity. Sensitivity analysis reveals a regime-dependent optimal detection span. An intermediate span delivers the best balance of speed and flow improvement under mild congestion, while wider spans introduce flow penalties under heavy congestion. These results suggest that adaptive signal control is most effective when demand exceeds the critical density but has not yet saturated capacity.