Dynamic Optimization of Transportation Networks with Delays
Abstract: The topic of this thesis is the optimal control of transportation networks. The problem studied is a dynamical extension of a classical problem in economics, in which the objective is to distribute goods to maximize welfare, whilst satisfying constraints on production and consumption. The main contribution is to show that for a class of welfare functions and dynamics, the optimal control is highly structured, and can be implemented in a way that scales gracefully with network size. More specifically, it is shown that if the underlying transportation network is structured by a string graph with delays on the edges, an LQ optimal controller can be found by explicitly constructing the solution to a Riccati equation. Next the problem is studied from a user perspective. A method to compensate the users in the network, so that that their choices of levels are also the social optimum is derived. Finally the results are extended to handle directed tree graphs, more general cost functions, and variable production in the network. In all cases the optimal control can be found by sweeping through the graph once, calculating aggregate utilities and levels. This gives a serial implementation, that is suitable for systems were the is no need for high sample times, such as district heating systems and transportation networks.
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