Dynamic urban transportation network models: theory and implications for intelligent vehicle-highway systems
The book deals with modelling of the dynamic processes performed by flows of vehicles in a road network. The flows form a dynamic system, input of which is given by a set of vehicle inflows at individual origin network nodes for each origin-destination pair. When the elastic departure time is considered, the input is described by the cumulative numbers of vehicles departing from the origin nodes for the given destination nodes during a given time period. The basic parameters, which are used here to describe the system behaviour, are time dependent values of the inflow and the exit flow for each network link and the number of vehicles on the link for each origin-destination pair at each instant of time of the given time period. The following common constraints are required for above values in any dynamic model presented in the book: a) relationship between state variables (the number of vehicles on the link) and control variables (the values of inflow and exit flow on the link); b) flow conservation constraints; c) flow propagation constraints; d) definitional constraints; e) nonnegativity constraints; f) boundary constraints. Besides the common constraints, which are required for usual case of a dynamic flow system, the vehicle drivers are supposed to comply with a general rule of behaviour. The rule of behaviour states that any vehicle driver chooses the time shortest way to his destination whenever he has a choice. The authors made an assumption that each vehicle driver is given the relevant information about the state of network in the considered system. Various cases are studied under these presumptions. There are completed mathematical programming models there for the instantaneous and the actual traversing times, which are known to drivers for their decision making. This way, the instantaneous dynamic user optimal route choice model and the ideal dynamic user optimal route choice model arise. It is proved that an optimal solution of the programs mentioned above also satisfies the travel-time-based instantaneous dynamic user optimal choice conditions and the travel-time-based ideal dynamic user optimal choice conditions, respectively. A method for solving the above programs is developed here. The method consists of the Frank-Wolfe method (the gradient method for the constrained range of feasible solutions) and diagonalization technique. The resulting method enables to reach the numerical solution of the problem by several iterations for simpler networks. Furthermore, the generalization of the models for the stochastic case has been done and the stochastic models are studied here, too. Besides the Pontryagin's Minimum Principle, which is broadly used as the first order conditions in the theoretical accounts over the book, the other principle: variational inequality is introduced and its equivalence with the first order conditions is proved for some instances. The final part of the book is devoted to studies of the link travel time functions and to the analysis of the problems which would be connected with possible implementations in the intelligent vehicle highway systems.
scientific article; zbMATH DE number 692286