Real-time congestion pricing strategies for toll facilities

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Highlights

  • Total system delay can be minimized with many different pricing strategies.

  • Given an operator objective, the optimal pricing strategy can be reduced to finding a single parameter value.

  • Maximum revenue under minimum total delay implies toll facility at capacity with no queues for as long as possible.

  • The concept of alternative-specific marginal cost is clarified.

Abstract

This paper analyzes the dynamic traffic assignment problem on a two-alternative network with one alternative subject to a dynamic pricing that responds to real-time arrivals in a system optimal way. Analytical expressions for the assignment, revenue and total delay in each alternative are derived as a function of the pricing strategy. It is found that minimum total system delay can be achieved with many different pricing strategies. This gives flexibility to operators to allocate congestion to either alternative according to their specific objective while maintaining the same minimum total system delay. Given a specific objective, the optimal pricing strategy can be determined by finding a single parameter value in the case of HOT lanes. Maximum revenue is achieved by keeping the toll facility at capacity with no queues for as long as possible. Guidelines for implementation are discussed.

Introduction

There are currently more than a dozen cities around the world that implement zone- or cordon-based congestion pricing, and around 20 toll facilities in the United States subject to congestion pricing. The pricing strategies in these facilities are inspired by the “first-best toll” concept borrowed from the economics literature, which can be stated as “System Optimum (SO) will be equivalent to User Equilibrium (UE) with tolls derived from the SO solution” (see e.g. Carey and Watling, 2012). This concept has been adapted to the case of traffic flow rather directly, and it is our view that some important traffic dynamics properties may have been overlooked in doing so.

Although there are a number of studies examining the performance of High Occupancy Toll (HOT) lanes (see e.g. Supernak et al., 2003, Supernak et al., 2002a, Supernak et al., 2002b, Burris and Stockton, 2004, Zhang et al., 2009) and travelers’ willingness to pay (Li, 2001, Burris and Appiah, 2004, Podgorski and Kockelman, 2006, Zmud et al., 2007, Finkleman et al., 2011), SO tolling policies have received little attention. Existing studies focused on ad hoc objectives that the tolling agencies may seek to achieve, such as ensuring free-flow conditions on HOT lane. For example, Li and Govind (2002) developed a toll evaluation model to assess the optimal pricing strategies of the HOT lane that can accomplish different objectives such as ensuring a minimum speed on the HOT lane, or in the general-purpose lanes (GPL), or maximizing toll revenue. Zhang et al. (2008) proposed the logit model to estimate dynamic toll rates of the HOT lane after calculating the optimal flow ratios by using feedback-based algorithm on the basis of keeping the HOT lane speed higher than 45 mph. Yin and Lou (2009) explored two approaches including feedback and self-learning methods to determine dynamic pricing strategies for the HOT lane, and the comparative results showed that the self-learning controller is superior to the feedback controller in view of maintaining a free-flow traffic condition for managed lanes. Lou et al. (2011) further developed the self-learning approach in Yin and Lou (2009) to incorporate the effects of lane changing using the hybrid traffic flow model in Laval and Daganzo (2006). Burris et al. (2009) examined the potential impacts of different tolling strategies on carpools, which includes removing or reducing the preferential treatment for them in the HOV lane.

In our formulation the social cost to be minimized is total system delay, and does not include the effects that tolls may have on trip generation or departure-time choice. The proposed pricing strategies are real-time, in the sense that they respond to real-time traffic arrivals in a way that minimizes total system delay for that particular rush hour. Therefore, the underlying assumption is that demand is inelastic within the day, but it could very well be elastic from day to day. In this context, this paper proposes a real-time pricing mechanism that is consistent with known properties of marginal costs under inelastic demands, i.e.: the cost of adding an additional user to a specific alternative is given by the time until congestion clears, it is not well defined when capacity is reached, and the SO assignment is not unique (Muñoz and Laval, 2005, Kuwahara, 2007). Towards this end, this paper is organized as follows. Section 2 presents the problem formulation along with the SO and UE solutions. Section 3 summarizes the general properties of SO tolls, including expressions for delays and revenue. Section 4 examines the special case of HOT lanes, and finally Section 5 presents a discussion.

Section snippets

Problem formulation

Consider the equilibrium between two alternatives with finite capacity, one of which is priced. To fix ideas, we take the example of a Managed Lane (ML) competing with the general-purpose lanes (GPL), but the formulation to be developed also applies to other cases such as toll roads or zone-based pricing. Our focus is on real-time pricing strategies and therefore we do not assume that traffic demand is known in advance, but only as it realizes.

Let A(t) be the cumulative number of vehicles at

Properties of system optimum tolls

In this section we identify and examine the properties of the SO toll, π(t), that produces a SO assignment under UE. The goal of SO tolls is for every user to perceive the marginal cost it imposes to the system. This could be accomplished in our case by charging the externality in each alternative given by (5). Equivalently, since we want to maintain the GPL toll-free we will charge the difference in the externalities to the ML only. This is illustrated in Fig. 4a, which shows the marginal cost

HOT lanes under linear tolls

System optimum tolls on HOT lanes can be characterized within the proposed framework using Δ=0; typically μ1μ0 but we do not need this assumption. For simplicity and without loss of generality we neglect high occupancy vehicles (who do not pay the toll to use the HOT lane) in this analysis. The reader can verify using Fig. 3 that in this case w1(T0)=0, and therefore the boundary condition (10a) changes to:π(t0)=0,π(T0)=0.We now show that when the pricing strategy is linear, as defined

Discussion

As stated in the introduction, our definition of social costs does not include the effects that tolls may have on trip generation or departure-time choice. This does not mean that demand is considered inelastic here, at least in the traditional sense: demand is assumed inelastic for each particular rush hour (because users do not know the toll in advance), but next day demand may change due to pricing. In particular, we provided an answer to the question: given the current state of the system,

Acknowledgments

This research was supported by STRIDE/GDOT research project 2012–089S, and by CEDEUS, CONICYT/FONDAP 15110020. The authors would like to thank two anonymous referees and the Associate Editor for their valuable comments and suggestions, which greatly improved this paper.

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