Dynamic traffic assignment approximating the kinematic wave model: System optimum, marginal costs, externalities and tolls

System marginal costs, externalities and optimal congestion tolls for traffic networks are generally derived from system optimising (SO) traffic assignment models and when they are treated as varying over time they are referred to as dynamic. In dynamic system optimum (DSO) models the link flows and travel times or costs are generally modelled using so-called ‘whole link’ models. Here we instead develop an SO model that more closely reflects traffic flow theory and derive the marginal costs and externalities from that. The most widely accepted traffic flow model appears to be the LWR (Lighthill, Whitham and Richards) model and a tractable discrete implementation or approximation to that is provided by the cell transmission model (CTM) or a finite difference approximation (FDA). These handle spillbacks, traffic controls and moving queues in a way that is consistent with the LWR model and hence with the kinematic wave model and fluid flow model. An SO formulation using the CTM is already available, assuming a single destination and a trapezoidal flow-density function. We extend the formulation to allow more general nonlinear flow density functions and derive and interpret system marginal costs and externalities. We show that if tolls computed from the DSO solution are imposed on users then the DSO solution would also satisfy the criteria for a dynamic user equilibrium (DUE). We extend the analysis to allow for physical or behavioural constraints on the link outflow proportions at merges and inflow proportions at diverges. We also extend the model to elastic demands and establish connections between the present DSO model and earlier DSO models.