On Convexity and Stability of Multicommodity Dynamical Flow Networks

The Traffic Assignment Problem is a central problem in transportation systems science and has been widely studied throughout the years. The objective of the TAP objective is to find an optimal allocation of traffic so that an aggregate cost function (e.g., the total travel time) of all users in the transportation network is minimized (social optimum). In real-life transportation networks the flow that travels across a road changes over time and this leads to dynamic formulations of the TAP. In particular, the Dynamic Traffic Assignment (DTA) problem is a optimal control problem that entails optimizing dynamic routing and network flows over given transportation network and time horizon. A related problem is the Freeway Network Control (FNC) problem, differing from the DTA problem in that the routing of the users is a known exogenous input, instead of being part of the optimization process. The control must be carried out in a distributed way, so that each road has only access to information about the "neighbouring" ones. However, both these problems are nonconvex, but it has been shown that it is possible to relax them through the use of the Cell Transmission Model (CTM). This model gives a better representation of real life traffic networks which provides a robust tool for modeling and analyzing traffic dynamics. Traffic flow is governed by fundamental principles of mass conservation and the flow-density relationship, identified by supply and demand functions, that limit the flow based on the current density. By relaxing the supply and demand functions into linear constraints the problems become convex and the flow can be controlled through variable speed limit, ramp metering, and routing. In other words the main objective of the FNC and the DTA is to minimize a given cost function in order to find an optimal dynamic allocation of flow across the network. In this thesis it is studied how those problem can be interpreted and solved in a multi commodity scenario which provides a more accurate representation of real-life traffic networks. For instance, consider a scenario where electric vehicles are permitted to traverse the entire network freely, while petrol cars are restricted from accessing certain roads. In this multi-commodity scenario it is important to control both flows in a separate way, which is achieved by considering two different demand functions and a shared supply function. This can be interpreted as limiting the amount of flow of a single commodity based on the traffic volume of that commodity in each cell. Moreover, the joint flow entering a cell is also restricted, based on the sum of the traffic volumes of both commodities. The achieved control can then be thought as assigning different variable speed limits to each kind of vehicles based on their density in a certain road. In order to solve both the multi-commodity DTA and FNC, it has been used the Alternating Direction Method of Multipliers (ADMM) introduced by Stephen Boyd as it has already been demonstrated to perform effectively in the single-commodity scenario. The results and simulations show that the selected algorithm is suitable for solving both problems. An optimal value of traffic allocation is reached, which consider different speed limits for the two types of commodities. The thesis also provides a stability analysis of multi-commodity dynamical flow networks, extending recent results based on the theory of contractive dynamical systems.