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).