Optimising toll prices based on a dynamic multi-region MFD SUE traffic model: Formulation and a case study of Zealand, Denmark

Road use tolling is an effective way of alleviating congestion. Although many tolling models have been developed, there is gap in the research for a model that: i) is dynamic, ii) accounts for the impacts of tolls on travel demand and departure time choice, iii) accounts for stochasticity in travellers’ route choices, iv) is well-behaved, producing continuous outputs, and v) is computationally feasible to apply to real-life large-scale networks. This paper fills this gap, by developing a tolling model based on the dynamic multi-region Macroscopic Fundamental Diagram (MFD) Stochastic User Equilibrium (SUE) traffic model introduced in Duncan et al. (2025). We begin by extending the model to account for elastic demand and departure time choice. Then, we integrate the model within a toll-price optimisation framework, where the tolling scheme is travel-time-based and the objective function maximises social welfare. We first test the model in a small-scale example multi-region MFD system, and then apply it to estimate an optimal toll-price in a real-life large-scale and detailed case study of Zealand, Denmark. Experiments find that the model is well-behaved and produces smooth objective function surfaces with a unique maximum. Travel behaviour implications of tolling are also realistic, where some travellers opt not to travel by car, some change their departure time, and some change their route. Results suggest that tolling could instigate a positive change in travel behaviour to benefit society.