Robust control of traffic networks under uncertain conditions

Journal of Advanced Transportation, Fall, 2008 by S.P. Hoogendoorn, V.L. Knoop, H.J. van Zuylen

Uncertainty of traffic network operations has been a subject of lively debate in the last decade. However, little effort has been put in developing control frameworks that are not only aimed at improving the average performance of the system, but also at improving the system robustness and reliability. In fact, it can be argued that most of the current control approaches are only aimed at improving the efficiency, which can even be counterproductive from a robustness point of view.

The main contributions of this article is the proposition of a new control framework based on the notion of controlled Markov processes, which explicitly takes into account the uncertainty in predicted traffic conditions and system performance. Furthermore, in contrast to traditional optimal control approaches, the objective function can include general statistic of the random system performance, such as the mean, standard deviation or 95-percentile.

The contribution aims to make clear how different performance function specifications yield different control strategies. This is shown for a relatively simple case study.

Keywords: Stochastic control, reliability, robust networks

I. Introduction

Dutch transportation policy is not only aimed at achieving the largest throughput, but also at increasing transport reliability. In fact, current policy aims to achieve that 95% of all motorway trips in the peak hour will be on time. From the perspective of the direct users of the transportation users, this will mean a considerable improvement, because behavioral studies have already shown that besides mean travel time, travel time reliability plays an considerable role in the valuation of a trip (Bogers et al, 2006).

Despite of this, for the deployment of most of the Dynamic Traffic Management (DTM) measures, we generally only take consider efficiency impacts (in terms of maximizing throughput, reducing emissions and noise, etc.). Little research has been done how DTM can be put to use to increase reliability. Amongst the few examples is the work of Liu (2006), showing how tolling can be used to improve reliability. Also, in Akamatsu and Nagae (2007) a stochastic optimal control framework is put forward, in which an optimal system optimum assignment is performed in case of stochastic system dynamics. The research shows how the problem can be formulated it as a dynamic programming problem. The paper focuses on minimizing the expected value of the objective function given the uncertain system dynamics. In Hoogendoorn et al (2007), a similar problem was tackled but for generic objective functions.

In this contribution we put forward a new control methodology showing how to control for reliability for generic control inputs and objectives. Instead of using a deterministic prediction model, which is done in traditional optimal control theory, a stochastic model is used instead. The uncertain system dynamics imply that the predicted performance is a random variable as well, rather than a single deterministic value. The control objective which is optimized can then be specified using a specific stochastic distribution, enabling not only the consideration of the average system performance, but any statistic of the random system performance or combinations thereof. The concepts are illustrated by a number of relatively simple examples.

2. Formulation of the Traffic Control Problem

Dynamic Traffic Management offers many possibilities to influence traffic flow operations in networks. Examples are providing route information or guidance, ramp-metering, mainline metering, tidal flow, dynamics speed limits, intersection control, etc.

We assume that the status or control settings of these measures can be represented by some control input u. This vector can include all kinds of control settings, such as the green-time, whether or not a specific lane is closed, the route advice people receive via the VMS, etc. Furthermore, the control will be dynamic, i.e. u = u(t). The control u(t) influences the (current and future) state x(t) of the system (this is formulated mathematically in the next section). We generally assume that the next state (say, x(t+dt)) is determined by the current state x(t), the control u(t), and any 'disturbances' that may be applied (including the boundary conditions, such as traffic flowing into the considered network). This implies that the state captures the entire history of the system.

In the remainder, we set out to determine the optimal control settings [u.sup.*], i.e. the control that steers the state of the traffic network in some optimal way, according to some performance criteria chosen. In case the system dynamics are deterministic, optimal control theory can be effectively used to find the optimal control settings minimizing the choice performance criteria. We refer to Hegyi (2004) for (one of many) thorough overviews of application of optimal control applications in ITS. Rather that considering deterministic models to predict network flow operations, a stochastic model is put forward. In doing so, we can explicitly consider the impact of uncertainty in determining the optimal control laws, which yield for robust system operations.