Optimal freeway ramp control without origin–destination information

Abstract

This paper develops an analytical framework for ramp metering, under which various ramp control strategies can be viewed as ramifications of the same most-efficient control logic with different threshold values, control methods, and equity considerations. The most-efficient control logic only meters the entrance ramps nearest critical freeway mainline sections so as to eliminate freeway internal queues, which is derived from a new formulation of the optimal ramp control problem. Instead of assuming the availability of real-time origin–destination information, the new formulation takes advantages of the stability and predictability of off-ramp exit percentages. Those properties of the off-ramp exit percentages are supported by empirical data, and allow us to formulate the optimal ramp control problem as a linear program whose input variables are all directly measurable by detectors in real-time. The solution is also tested on a real-world freeway section in a microscopic traffic simulator for demonstration. Time-dependent origin–destination tables and off-ramp exit percentages are compared as two alternative ways to represent the true real-time demand patterns that are important to freeway ramp metering.

Introduction

The first attempt to solve the ramp metering control problem via optimization at the freeway system level goes back to Wattleworth in two papers (Wattleworth, 1963, Wattleworth, 1967). The linear program proposed in these papers and its followers (see Lovell, 1997 for a review) are essentially time-invariant optimization minimizing the total travel time in the freeway system. Some distinct attributes of these models include: (a) They all incorporate a constraint equation which ensures that freeways operate under free-flowing conditions. Hence, the difficulties of dealing with freeway mainline dynamics are avoided. (b) Time-dependent origin–destination (OD) demand information is assumed to be available. (c) They assume that there are no diversions from freeways to surface arterial streets. Recently, Lovell and Daganzo (2000) extended Wattleworth's steady-state model to include time-dependency, and developed a computationally-efficient greedy heuristic solution. However, the heuristic is only appropriate for small-scale networks, and OD information is still a required input.

There is also a body of literature combining optimal control theory and macroscopic traffic flow models (Chang and Li, 2002; Kotsialos et al., 2002; Papageorgiou, 1995; Zhang and Recker, 1999). The freeway mainline dynamics therein are described by a set of time-discrete equations based on finite difference approximations of specific macroscopic traffic flow models. Control strategies developed along this line also confront the difficulty of getting accurate OD information in real-time. Unclear reliability of the estimated OD information and computational complexity are two drawbacks preventing these strategies from being implemented widely. The models themselves are usually complicated, which makes it hard to solve to global optimality.

On the other hand, numerous operational ramp metering algorithms have been developed in practice. Over the years, ramp metering systems have extended to many urban areas around the globe¹ after its debut in Chicago, IL, in the early 1960s. Surprisingly, every city has its own strategy, some of which are summarized by Bogenberger and May (1999). Simulation evaluation studies on various ramp control strategies become more and more popular. Although many of these practical algorithms are limited in many aspects and based on extensive engineering judgment instead of optimization models, they are successful in reducing total delay as demonstrated by some field experiments (Levinson et al., 2002) and many simulation studies (Hourdakis and Michalopoulos, 2003; Kwon, 2000; Lomax and Schrank, 2000). Local traffic responsive metering algorithms, which base control decisions on real-time traffic data collected in the vicinity of individual on-ramps, have also been very successful without even touching the issue of system-level optimization, no matter what type of local controller is used (Linear: Papageorgiou et al., 1991; Artificial Neural Network: Zhang, 1997; or Fuzzy-logic: Taylor et al., 1998).

This brief retrospective examination of both the research and practice sides of ramp metering brings our attention to several interesting phenomena and questions. First, there is a gap between the state-of-the-art and the state-of-the-practice, and it is not clear which is behind. Researchers have been working on formal optimization problems with assumptions about data availability and complex mathematical models. In contrast, practitioners have developed various ad hoc but operational strategies. In our conversation with several engineers in the Minnesota Department of Transportation who oversee the Twin Cities freeway management system, we were told that researchers in the field of ramp metering are far behind practitioners. It is probably true that practitioners developed more real-world congestion-mitigating ramp control strategies on their own. But what is it that makes those operational strategies successful while few formal theories or models are adopted by practitioners? People working with real-world metering systems long ago drew the conclusion empirically that the most benefits of ramp metering were from the decision to have a metering system, but not from the sophistication of the algorithm (Newman et al., 1969). Is that true and why?

The second observation is that in theoretical development of optimal ramp control solutions, researchers tend to use time-dependent OD tables to represent true freeway demand patterns, which help formulate the problem mathematically. Many researchers have considered the optimal ramp control problem as a two-stage problem implicitly or explicitly: an accurate and efficient real-time OD estimation procedure should be developed, and then that information can be used as input to the following optimization stage. However, in practice, few operational control strategies use OD tables. There must be some way other than OD tables that the practitioners take care of the true demand patterns. Their success implies that the alternative method is somewhat reasonable. What is it?

Finally, we have seen the following trend in ramp metering studies. The mathematical models become more and more complicated as the scope is expanded from local to coordinated and integrated control. However, the only validation step taken seems to be simulating the final product of all research efforts––the resulting ramp control strategy. If the simulation shows positive results, the theoretical model or procedure is considered as acceptable. However, this reasoning process could be dangerous because sometimes very crude metering algorithms (e.g. a pre-timed) can also significantly reduce total delay. Simulation studies can show whether one strategy outperforms another, but do not shed light on how and why that is the case. If satisfactory efficiency performance is obtained in one control strategy, is it because the traffic flow model successfully predicts real traffic conditions, or because an OD estimation procedure is incorporated, or because equity is put on a low priority, or something else? If we do not pursue answers to the how and why questions, successful simulation, even field evaluation results, do not necessarily imply that the underlying theory is superior. Therefore, an analytical framework under which those questions can be explored is clearly in order.

This paper, as a small step to address the questions and research needs identified in the above discussion, develops an analytical framework for ramp metering studies, with the hope of leading towards a more unified and generic ramp control theory. Under this framework, various individual elements that constitute a complete ramp control strategy can be easily decomposed and studied separately. We formulate the optimal ramp control problem without using the time-dependent OD tables to represent true demand patterns. Instead, the stability of off-ramp exit percentages is studied and used in the analysis. The solution to the new formulation reveals that the most-efficient ramp control logic is actually a very simple one, which to some extent explains the success of many operational ramp control strategies. In this regard, we hope that the paper can also help bridge some of the gaps between research and practice in both directions. A simulation experiment is executed only to demonstrate that the core ramp control logic in the analytical framework can be implemented in real-time. The findings also reveal that the most efficient control strategy is also the least equitable one. Considering the enormous political and public interests on balancing efficiency and equity of ramp meters, this topic is also briefly discussed.

The remainder of the paper is organized as follows. The next section (Section 2) proposes an analytical framework for ramp metering studies. The following section (Section 3) is devoted exclusively to two alternative ways of considering real-time freeway demand pattern in ramp metering––OD tables vs. off-ramp exit percentages. In order to complete the construction of the analytical framework, a ramp metering logic is required. Therefore, Section 4 details our formulation of the optimal ramp control problem, the solution of which can serve as the control logic. Thanks to the stability and predictability properties of off-ramp exit percentages, it is able to formulate the problem as a linear program. The simulation experiment on the solution to the linear program is described in Section 5, followed by a discussion on equity considerations in Section 6. Conclusions and suggestions for future studies are delivered at the end of the paper.