There has recently been a growing interest in analysing road pricing schemes in urban areas using dynamic traffic assignment (DTA) tools. The motivation behind this development is the problem for static transportation models to accurately predict travel time savings, from introducing road pricing, in networks with severe congestion. Finding optimal toll levels and locations in urban road traffic networks has so far mainly been studied using either derivative-free heuristics (e.g. genetic algorithms and simulated annealing) or ascent methods. Both approaches rely on fast computations of the road users response (traffic flows, travel times and demands), given the road pricing scheme, and for the case of ascent methods, the methods also rely on fast computations (or rather approximation) of derivatives. Using DTA tools for evaluating the road users’ response to a pricing scheme is, however, very computationally expensive. Previously developed methods are therefore not suitable to use together with DTA.
Surrogate models, e.g. in terms of response surfaces, are commonly used for optimisation problems with expensive-to-evaluate objective functions. The surrogate model is used for approximating the expensive-to-evaluate objective function, and the optimisation is then done on the surrogate model instead. The performances of optimisation methods based on surrogate models are, however, dependent on experimental design, infill strategy and choice of surrogate model itself. The experimental design will give the initial set of toll levels, for which the DTA needs to be evaluated, the infill strategy determined additional toll levels to be evaluated by the DTA, and the choice of surrogate model will give the functional form to be fitted to the sampled toll levels.
We apply a surrogate model framework for optimising toll levels in a multiple cordon pricing scheme. In the first stage we evaluate the experimental design, infill strategy and choice of surrogate model, using a static macroscopic traffic model. This allows a large number of experiments to be carried out, which would not be possible with a DTA tool. It also allows us to compare the performance of the surrogate modelling approach with other global optimisation methods. In the second stage, the insight which has been gained from the experiments with the static model is used when applying the surrogate modelling approach to a DTA model of Stockholm.
Computational results are presented for a Stockholm network with three cordons, each with differentiated toll level in both directions, resulting in a total of six toll level variables. Surrogate models in the form of Radial Basis Functions and Kriging models are evaluated with a static model of Stockholm, for different initial experimental designs, infill strategies and choice of surrogate models. In comparison with previously developed derivative based methods for static models, our results show that the surrogate based optimisation approach performs better, since it allows for metaheuristic methods to search for global optimal solutions efficiently.
Nationella konferensen i transportforskning, 21-22 oktober 2014, Linköpings universitet, Campus Norrköping