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Autor(in)

Datum

2018Typ

- Doctoral Thesis

ETH Bibliographie

yes
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Abstract

This thesis addresses the optimal control of traffic networks. In road traffic control, the objective is to reduce congestion and minimize the total delay incurred by all drivers, for example by controlling traffic lights, adjusting variable speed limits, and by ramp metering for freeways. If a model of the traffic dynamics is available, then an optimal control problem can be posed as a mathematical optimization problem. However, such optimization problems are typically non-convex, due to the nonlinear relationship between traffic density and traffic flow. Computing the global optimizer of non-convex problems is computationally intractable in most cases, even for problems of only moderate size. In addition, problem data in traffic models are often not known with certainty. For example, predictions of future traffic demand are never exact. Furthermore, the flow-density relationship ultimately depends on the behavior of individual drivers and therefore, significant variance is observed in practice, in particular during congestion. The main objective of this thesis is to identify a class of traffic control problems that can be reduced to convex optimization problems, and therefore solved efficiently, despite the nonlinear traffic dynamics and potential uncertainty in the model.
In particular, we consider a special case of the dynamic traffic assignment problem, in which the aim is to optimize the operation of a traffic network by controlling flows, e.g. via traffic lights or variables speed limits, but without re-routing traffic. We consider a first-order, compartmental model based on the celebrated cell transmission model, with diverging junctions described by first-in, first-out (FIFO) dynamics. If the fundamental diagram is relaxed, then a convex problem is obtained. Prior work has established conditions under which solutions of the relaxed problem are feasible in the original system dynamics, or can be mapped to a feasible solution with equal objective value. In this thesis, we generalize these conditions, for the case when demand functions are concave and non-decreasing and supply functions are concave and non-increasing. In this case, we prove that if the objective is to minimize the total time spent in traffic, controlling solely the flows into merging junctions is sufficient to achieve the optimal objective value of the relaxed problem. We derive this result by introducing an alternative system representation, obtained via a state transformation. In the new representation, the system dynamics are convex and state-monotone. Notably, this result also implies that the non-transformed FIFO diverging dynamics are monotone with respect to a particular, polyhedral cone.
Subsequently, we consider the uncertainty inherent in traffic models, which poses a challenge for model- and optimization-based control approaches. We propose a robust counterpart to the deterministic traffic control problem, in which we introduce uncertainty sets for the flow-density relationship and future traffic demand. We seek to minimize the worst-case, cumulative delay incurred by all drivers, for any uncertainty realization. For a network with controlled merging junctions, and uncertainty sets that satisfy certain technical conditions, we show that the robust counterpart can be reduced to a finite-dimensional, convex, and deterministic optimization problem, whose numerical solution is tractable.
Finally, we show that monotonicity of the traffic dynamics in our setting can be leveraged to analyze an important special case, the problem of (optimal) freeway ramp metering. Empirical studies have shown that in freeway ramp metering, decentralized feedback laws often achieve performance comparable to optimization-based, centralized control approaches. While heuristic explanations for this observation have been suggested, a theoretical analysis of this phenomenon has been lacking. We use our prior results to derive sufficient optimality conditions for a particular, decentralized and non-anticipative ramp metering policy, for a freeway with monotone dynamics. Notably, we demonstrate numerically that this policy shows comparable closed-loop behavior to the successful Alinea ramp metering policy. We therefore conclude that if optimization-based or coordinated ramp metering policies seek to improve performance substantially over decentralized policies such as Alinea, then they need to specifically target non-monotone effects, in particular the capacity drop of a congested bottleneck. Mehr anzeigen

Persistenter Link

https://doi.org/10.3929/ethz-b-000278557Publikationsstatus

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Verlag

ETH ZurichThema

Traffic control; Optimal control; Cell transmission model; Convex optimization; Robust optimizationOrganisationseinheit

03751 - Lygeros, John / Lygeros, John
ETH Bibliographie

yes
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