Modelling and simulation of traffic flow on highways has been investigated intensively during the last years. On the one hand models describing detailed traffic dynamics on single roads have been constantly developed and improved. To describe traffic flow on networks detailed dynamic models based on partial differential equations have been used. However, the number of roads which
can be treated with such an approach is restricted, in particular, if optimization problems have to be solved. On the other hand large traffic networks with strongly simplified dynamics or even static description of the flow have been widely investigated. In particular, optimal control problems for traffic flow on networks arising from traffic management, are a major focus of research in this field.
The purpose of our investigation is to derive and develop a hierachy of simplified dynamical models based on the 'correct' dynamics described by partial differential equations. These models should include reasonable dynamics and, at the same time, they should be solvable for large scale networks. Special focus is on optimal control problems and optimization techniques. We start with macroscopic models based on partial differential equations. In particular, dealing with optimal control questions for such large scale networks where the flow is described by partial differential equations is very expensive from a computational point of view. Therefore, we concentrate on the derivation of simplified dynamic models derived from the models based on partial differential equations. The resulting models are network models which are based on nonlinear algebraic equations or combinatorial models based on linear equations. In the simplest case well known static combinatorial problems like min-cost-flow models are obtained. For the different models we study optimal control problems and various optimization methods, i.e., combinatorial and continuous optimization techniques. Using strongly simplified models large scale networks can be optimized with combinatorial approaches in real-time. However, including more complex dynamics reduces the advantage of the combinatorial algorithms compared to continuous optimization procedures.
- TU Darmstadt
- Armin Fügenschuh, Michael Herty, Axel Klar, Alexander Martin, Combinatorial and Continuous Models and Optimization for Traffic Flow on Networks , SIAM Journal on Optimization, Vol. 16, No. 4, pp. 1155 – 1176, 2006.