## Static Traffic Assignment with Junction Modelling

### Authors

H Muijlwijk, Omnitrans International bv; K Zantema, Goudappel Coffeng, NL

### Description

Junction delays contribute significantly to travel times in urban areas. With junction modelling, the Traffic Assignment Problem becomes asymmetric and is no longer convex. Multiple equilibria are shown and suggestions for solving methods are given.

### Abstract

The last decade, a number of new solution methods for the user equilibrium-based Traffic Assignment Problem have been presented, for example, bush-based algorithms. These new methods are not necessarily able to cope with traffic models with extensive junction modelling. Due to the asymmetry of the cost function on turns, the problem can no longer be formulated as an optimization program. Therefore, even the conventional Frank-Wolfe cannot be applied properly on models with junction modelling. The effect of junction modelling on the assignment is discussed in this paper, and new suggestions for solving methods are suggested.

In urban areas, a significant portion of the travel time is incurred at junctions. In order to get realistic traffic flows, the costs on all alternative routes need to be realistic, which means that the effect of junction delays cannot be ignored. A common way to model junctions is to define cost functions on all turning movements. This turn delay will be part of the total route travel time, making sure that junctions that cause great delay will attract less traffic.

The turn delay naturally depends on the load on the turn itself, but also on the load on the conflicting turns, which makes the cost function non-separable. The turns become dependent of each other, and this dependency has implications for the Traffic Assignment Problem and its solution methods. Actually, some conditions for uniqueness and convergence are no longer met.

The first implication, caused by the asymmetry of the cost function, is that the objective function of the Beckmann formulation is not explicitly known. This means that the Traffic Assignment Problem can no longer be formulated as an optimization problem. This is of great influence to most solving methods. The second implication of the addition of junction modelling is that the uniqueness of the solution is no longer guaranteed, so there may exist multiple user equilibria. In an accurate estimation of the junction delay, the influence of the load on the conflicting turns may be larger than the influence of the load on the turn itself, for example at priority junctions. This results in a great influence of off-diagonal elements, so the monotonicity condition of the cost function is not guaranteed. Consequently the problem is no longer convex, and several local minima may appear. This results in a trade-off, either the junction delays are realistic but there may exist multiple solutions, or a unique solution is guaranteed but the junction delays are not realistic.

These issues were mostly known as theoretical statements. This study shows these statements are not only theoretically, they appear to hold in realistic networks. We found that the Frank-Wolfe algorithm is indeed not solving the Traffic Assignment Problem properly, and returns solutions that not meet the Wardrop equilibrium conditions. Also, we found that in realistic model networks of some large Dutch cities (100.000+ inhabitants), multiple user equilibria exist and are easily obtained using different initializations. These equilibria were obtained by Method of Successive Averages. The relative differences between these solutions found in the case studies are up to 14% of the total load on local roads, the absolute differences are up to 68 vehicles per hour, on a total load of 722 vehicles per hour.

In the paper, mathematical proofs and some case studies will be presented. Also possible solution methods are presented, that are able to cope with models with junction modelling. These methods use for example a ´diagonalized´ cost function, so the objective function of the Beckmann formulation becomes explicitly known, or a Variational Inequality formulation is used. Furthermore, suggestions are given for dealing with the existence of multiple user equilibria, some heuristics are presented.

This paper discusses an issue which is usually neglected when developing new and faster solutions methods for the Traffic Assignment Problem, namely the effect of junction modelling on the obtained user equilibrium and the solving methods. Junction modelling is of great importance for an accurate calculation of total route travel times, especially in urban areas. In the paper the effects of junction modelling on the assignment are listed and illustrated in realistic networks, and solutions are given.

#### Publisher

Association for European Transport