Dynamic origin–destination demand flow estimation under congested traffic conditions

• Single-level OD demand estimation models considering various sources of measurements.
• A path flow-based OD demand estimation model for practical-sized network applications.
• A Lagrangian relaxation heuristic that dualizes dynamic user equilibrium constraint.
• Derive analytical gradients for the changes in link performances under congestion.

This paper presents a single-level nonlinear optimization model to estimate dynamic origin–destination (OD) demand. The model is a path flow-based optimization model, which incorporates heterogeneous sources of traffic measurements and does not require explicit dynamic link-path incidences. The objective is to minimize (i) the deviation between observed and estimated traffic states and (ii) the deviation between aggregated path flows and target OD flows, subject to the dynamic user equilibrium (DUE) constraint represented by a gap-function-based reformulation. A Lagrangian relaxation-based algorithm which dualizes the difficult DUE constraint to the objective function is proposed to solve the model. This algorithm integrates a gradient-projection-based path flow adjustment method within a column generation-based framework. Additionally, a dynamic network loading (DNL) model, based on Newell’s simplified kinematic wave theory, is employed in the DUE assignment process to realistically capture congestion phenomena and shock wave propagation. This research also derives analytical gradient formulas for the changes in link flow and density due to the unit change of time-dependent path inflow in a general network under congestion conditions. Numerical experiments conducted on three different networks illustrate the effectiveness and shed some light on the properties of the proposed OD demand estimation method.