Solving a Dynamic User Equilibrium model based on splitting rates with Gradient Projection algorithms

•It is shown how gradient projection algorithms can effectively be applied also to sequential local choices at nodes for users directed toward a given destination.•It is proved that, in the case of deterministic route choices, the Dynamic User Equilibrium based on arc conditional probabilities formulated as a Variational Inequality problem is equivalent to that based on path probabilities.•The flow propagation of demand flows travelling towards a given destination based on given travel times and arc conditional probabilities is formulated and solved as a sequence of square linear systems, one for each temporal layer, without introducing bushes of efficient arcs.•In the proposed framework, it is also possible to consider large time interval of several minutes, which is an extremely relevant feature if computing times are an issue, like in operation.•Convergence measured by the relative gap is reached in an acceptable number of iterations (e.g. 100) to a good level (e.g. 10−4) for moderate congestion (without spillback) and to a fair level (e.g. 10−2) for high congestion (with spillback).

This article shows how Gradient Projection (GP) algorithms are capable of solving with high precision a Dynamic User Equilibrium (UE) model based on Splitting Rates, i.e. turning movements fractions by destination.Dynamic Traffic Assignment (DTA) is formulated as a Variational Inequality problem defined on temporal profiles of arc conditional probabilities that express a sequence of deterministic route choices taken at nodes by road users directed toward each destination.Congestion is represented through a macroscopic traffic model capable to reproduce a range of phenomena having increasing complexity, from links with bottleneck to intersections with spillback. Different time discretizations, from few seconds to few minutes, are also possible, which allows a range of applications from planning to operation.This assignment model, which is fully link based, is proved to be equivalent to a path based formulation. It also allows for the computation of a handy gap function for analyzing convergence to equilibrium.Numerical experiments on test networks are presented, showing that the proposed GP algorithms converge to dynamic equilibrium in a reasonable number of iterations, outperforming the Method of Successive Averages (MSA).