Journal of Advanced Transportation

Volume 2017 (2017), Article ID 3127398, 12 pages

https://doi.org/10.1155/2017/3127398

## A Road Pricing Model for Congested Highways Based on Link Densities

^{1}School of Industrial Engineering, Universidad Diego Portales, Ejército 441, Santiago de Chile, Chile^{2}School of Government, Universidad del Desarrollo, Av. Plaza 680, Santiago de Chile, Chile

Correspondence should be addressed to Louis de Grange

Received 7 June 2017; Revised 27 July 2017; Accepted 9 August 2017; Published 17 October 2017

Academic Editor: Sara Moridpour

Copyright © 2017 Louis de Grange et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A road pricing model is presented that determines tolls for congested highways. The main contribution of this paper is to include density explicitly in the pricing scheme and not just flow and time. The methodology solves a nonlinear constrained optimization problem whose objective function maximizes toll revenue or highway use (2 scenarios). The results show that the optimal tolls depend on highway design and the level of congestion. The model parameters are estimated from a Chile’s highway data. Significant differences were found between the highway’s observed tolls and the optimal toll levels for the two scenarios. The proposed approach could be applied to either planned highway concessions with recovery of capital costs or the extension or retendering of existing concessions.

#### 1. Introduction

This study presents an analytic road pricing model based on macroscopic traffic models to determine the tolls for congested highways. The main motivation and contribution of this paper are to study fare design on urban highways, but considering the link density instead the link flow. The main advantage of this use of density data is that it captures the relationship between flow and cost (trip time or speed) on each link more accurately [1]. We develop two scenarios for our model, one in which the revenue is maximized and another in which the use of the road is maximized; however, other scenarios or criteria can be developed within the framework we propose. In both cases, the proposed formulation posits a constrained optimization problem. In the maximum revenue scenario (MR) the objective function maximizes toll revenue while in the maximum infrastructure use scenario (MIU) it maximizes highway use. The constraints for both scenarios are derived from the fundamental traffic equation relating flow, speed, and density on each highway segment. The parameters of the equation depend on highway design. The model’s optimality conditions give optimal tolls for the MR and MIU scenarios, for each segment and operating period.

The model was implemented using data on the main controlled-access highway in the Santiago region of Chile. The parameters of the model were estimated using regression analysis, taking particular care to address possible problems of collinearity and endogeneity of tolls, traffic speed, and traffic density. The results of this application are that the optimal tolls in the MR scenario are significantly higher than those for the MIU scenario. They also indicate that once a highway’s infrastructure capital costs have been recouped, the MIU optimal tolls will be significantly different from those designed to include capital cost recovery and should be modified accordingly. In some time periods (and highway segments, if tolls vary by segment) this will mean a major decrease in the tolls; in others it may mean an increase.

Two extensions to the proposed model are also presented. One adds a toll ceiling constraint to the MR scenario while the other incorporates financing constraints in the MIU scenario to recover the capital costs.

The remainder of this article is organized into five sections. Section 2 reviews the literature on optimal highway tolls under traffic congestion. Section 3 sets out our proposed optimization model for the MR and MIU optimal scenarios and its main properties. Section 4 applies the model to an urban highway in Chile, discusses the regression model used to estimate the parameters, and compares the results for the two scenarios with each other and with the observed situation reflected in the highway data. Section 5 develops two extensions to the optimization model incorporating a toll ceiling constraint for the MR scenario and financing constraints for the MIU scenario. Finally, Section 6 summarizes and discusses the main conclusions.

#### 2. Literature Review

Road pricing is implemented for two main reasons [2]. The first is to mitigate road network congestion by increasing the cost of trips along certain routes in order to shift traffic flows from peak to off-peak periods, from congested to less congested routes, or from private vehicles to public transport. The second reason is to recover the cost of building and maintain a road network when the cost is financed totally or partially by private investment.

However, according to the same author the greater part of the published research has in fact been concerned with the congestion pricing issue. Indeed, it has been widely recognized as an issue of primary importance from both theoretical and a public policy standpoints. Successful application of congestion charges in cities such as London, Singapore, and Stockholm has stimulated interest among researchers around the world.

The origins of congestion pricing theory go back to Pigou [3] and Knight [4], who used the example of a congested highway to address the issue of externalities and optimal congestion costs. Since then, the notion of marginal cost pricing, that is, imposing a charge equal to the difference between the private cost and the social cost of a road in order to maximize net social benefit, has been extended to road networks in general [5–8]. Yang and Huang [9] review the marginal cost principle in the presence of queues and delays. Wu and Huang [10], considering body congestion in carriage and vehicle congestion in bottleneck queues in a competitive highway/transit system, investigate the departure patterns of commuters through analysing the equilibria under three road-use pricing strategies. An up-to-date overview of road pricing may be found in Rouhani [11].

In a specifically urban context, de Grange and Troncoso [12] survey the literature emphasizing the relevant conceptual and practical considerations for achieving a successful design, evaluation, and implementation of road pricing measures. They note, for example, that evaluating the social welfare implications of a Pigovian tax on commuter trips must take into account the charge’s effect on not only congestion but also the labour market, already distorted by preexisting income taxes. In a study of the redistributive impact of road pricing, Foster [13] and Arnott et al. [14] show that it is regressive, an important finding for gauging the political cost of implementing such a policy. Other interesting works that take account of road pricing’s redistributive effects are Gehlert et al. [15] and Linn et al. [16]. The cases of Sweden and Norway are explained in detail in the work of Börjesson et al. [17] and Ieromonachou et al. [18], respectively.

For multiple reasons, first-best optimal marginal cost prices cannot be implemented in practice [12]. To begin with, such measures tend to encounter strong political and public resistance. Also, the large additional administrative, personnel, and equipment expenditures involved in collecting tolls across an entire road network are typically not fully covered by the revenue generated. Many studies have therefore sought alternative or second-best pricing solutions that are technically inferior but can be feasibly implemented.

One such second-best or suboptimal solution is a bilevel optimization model suggested by Yan and Lam [19] that assumes fixed demand between origin-destination pairs. The upper level minimizes the road network’s total trip time while the lower level defines network user behaviour (traffic assignment or route choice model). In a similar vein, Yang and Zhang [20] propose bilevel models that explicitly incorporate social and spatial equity conditions for various classes of vehicle drivers with different time valuations.

Zhang and Yang [21] study cordon-based road pricing models for delimited areas within a road network. Zhang and van Wee [22] propose the maximization of road network reserve capacity as the toll design criterion. Other works consider the time dimension of optimal congestion charges [14, 23–28]. Liu et al. [29] address the toll pricing framework for the first-best pricing with logit-based stochastic user equilibrium (SUE) constraints. Daganzo and Lehe [30] introduce trip-time heterogeneity and propose a usage-based, time-varying toll that alleviates congestion while prioritizing shorter trips.

Another approach aimed at avoiding the complexities of optimal toll schemes that are theoretically efficient but difficult to implement is based on the origins and destinations (O-D) of road network users [31]. In this system, users pay as a function of their destination. Thus, the same toll is levied on all routes connecting a given O-D pair regardless of their length.

Ahn [32] incorporates into optimal toll design the interactive effects of a decline in private vehicle traffic on congestion and a possible consequent improvement in bus services sharing the same roads. They conclude that road pricing has the potential to improve private welfare when there is congestion even without considering toll revenue use. Ekström et al. [33] evaluate the performance of a surrogate-based optimization method, when the number of pricing schemes are limited to between 20 and 40. A static traffic assignment model of Stockholm is used for evaluating a large number of different configurations of the surrogate-based optimization method. Their results show that the surrogate-based optimization method can indeed be used for designing a congestion charging scheme, which return a high social surplus.

On the issue of tolls as an instrument for financing road construction, optimal road pricing studies have also considered schemes that involve the building, operation, and transfer of road infrastructure (known as the BOT framework). Though this research is less abundant, recent decades have witnessed an increase worldwide in the supply of private toll highways in both developed and developing countries.

This public policy option has made it possible to undertake multiple large infrastructure projects simultaneously without putting pressure on public funds. For example, as of 2007 about one-third of the Western European highway network was under concession [34]. In Chile, private concessions have been let for Route 5 between La Serena and Puerto Montt, the country’s main north-south highway, and various urban highways in greater Santiago, the nation’s capital.

In this context, Verhoef and Rouwendal [35] study the relationship between optimal pricing, capacity choice, and road network financing. The authors report that optimal pricing and capacity choice policies may result in user charges for moderately congested areas of an already-built network that are only slightly less than those for highly congested areas. They also conclude that a fixed charge per kilometre may produce optimally efficient results.

Both Verhoef [34] and Ubbels and Verhoef [36] study the capacity and toll level choices of private bidders in a government-organized highway concession auction, considering and comparing different criteria for the award of the concession.

The involvement of the private sector in building and running highways may have advantages in terms of lower costs, greater innovation, and availability of funds. However, private objectives do not necessarily coincide with the maximization of social welfare [37, 38]. It is important, therefore, that regulatory authorities have the proper tools to design appropriate concession contracts.

With this caveat in mind, Verhoef [34] examines how concession award criteria impact route capacity and tolls. The author concludes that the socially optimal toll and capacity values are achieved when the award is based on the level of use. However, when second-best aspects are taken into account, maximizing level of use is no longer the optimal social solution but remains a second-best alternative.

Woensel and Cruz (2007) use queueing theory tools to incorporate dynamic and stochastic aspects of traffic behaviour into the calculation of marginal congestion cost and the consequent design of optimal congestion charges by the relevant authorities. In this way, Zhu and Ukkusuri [39] propose a dynamic tolling model based on distance and accounts for uncertain traffic demand and supply conditions. Their results show that the total travel time of tolling links reduces by 25% over simulation runs. Verhoef [40] presents dynamic extension of the economic model of traffic congestion, which predicts the average cost function for trips in stationary states.

Ferrari [41] proposes a method of calculating tolls that partitions the cost burden between motorists and public financing in such a way as to optimize social welfare. The approach assumes deterministic road networks. The author applies the model to a real case and concludes that the optimal toll for a road segment is independent of its fixed costs but strongly depends on the marginal cost of public funds and motorists’ willingness to pay. Mun and Ahn [42] present a model of a transport system with two road links in a series that describes traffic patterns under various pricing regimes; this serial link approach is similar to our model representation.

Álvarez et al. [43] estimate optimal charges for the use of highways in Spain based on vehicle type. They compare these charges to the tolls actually imposed, finding that the latter are generally above marginal cost. In making their estimates they emphasize the difference between externalities on urban and interurban highways. Whereas the main external effect in the urban case is congestion, in the interurban case it may be accidents along the route.

On the other hand, there is extensive literature regarding the acceptance of payment by the road (recent analyses are presented in Hensher and Li [44], Hysing [45], and Grisolía et al. [46]). While it is beyond the scope of this article, this is a fundamental issue in the design of road pricing policies.

#### 3. Optimal Toll Model

The model we develop in what follows incorporates various characteristics similar to those discussed in the studies reviewed above but differs in that we make particular use of the flow-speed relation given by macroscopic traffic models. In this context, a given flow can occur with high or low speeds and the optimal price will be different in each case. The proposed formulation is easy to implement in practice and generates price recommendations for changing highway traffic conditions.

The use of link densities has a number of advantages over the traditional flow-based approach that allow it to achieve a higher degree of realism. It recognizes that link maximum flow is not fixed but rather is a function of density levels. These maximum flows are determined as a function of the speed and density on each link as given by the fundamental traffic equation. Finally, the density-based approach identifies whether a reduced flow level on a given link is due to low latent demand for its use (e.g., low density) or, on the contrary, to heavy congestion (e.g., high density) reducing the flow that can use the link, thereby generating traffic queues and longer delays.

In this context, Ohta [47] shows how flow can become a misleading variable if it is interpreted as a direct policy variable. This is because both equilibrium flow and optimal flow keep increasing with demand only up to a certain critical point, beyond which they start to decline. However, from equilibrium point of view (which is outside the scope of our model), there is an interesting debate with Verhoef [48]. Liu et al. [49], using a macroscopic approach and microsimulation, show that the incorrect use of performance curves to estimate demand can thus seriously underestimate equilibrium traffic levels and the costs of congestion.

We begin by letting be the density of vehicle flow on link or segment of a highway in period . As will be shown later, we consider density as a good proxy for demand that depends on the price or toll as well as temporal and spatial control variables . Now consider the following relation:where is the parameter of toll for segment in period and are the parameters of the control variables (e.g., dummies for time of day, day of the week, geographical zone) and is the model intercept. The sign of is negative () and thus consistent with microeconomic theory, meaning that the higher the charge, the lower the demand and therefore the density. For simplicity’s sake we define .

With these basic elements we can now present our two optimization model variants for generating optimal tolls in MR and MIU scenarios. The MR version is set out in Section 3.1 and the MIU version in Section 3.2.

##### 3.1. MR Optimal Toll Model

The optimality criterion for the MR scenario is the maximization of highway toll revenue. The optimization problem is then as follows:where is the toll, is the traffic flow, and is the traffic speed along segment in period . If the highway operating cost is constant, it will not affect the above model’s optimality conditions; if, on the other hand, it is directly proportional to flow, can simply be defined as the difference between revenue and operating cost per unit of flow. Either way, then, maintenance/operating cost does not need to be explicitly incorporated into the model [50].

The speed along a segment is a decreasing function of density (a good survey of such relations is found in Wang et al. [51]) and is defined in the following manner:where is the baseline parameter representing the speed of free-flowing traffic under normal highway conditions, are the parameters for the control variables (e.g., dummies for time of day, day of the week, geographical zone, and incidents), and is the density parameter for segment in period . The sign of consistent with traffic theory is . For simplicity we define .

The optimality condition for objective function (2) above is

Given constraint (3), that is, ,

Substituting (6) into (5), we have

This expression gives the optimal value of the toll for segment in period in the private benefit scenario.

Differentiating (1) and (4) we get

Plugging (8) into (7), we obtain

Finally, given that and that , we arrive at

This expression gives the optimal toll for segment in period in the private benefit scenario. Since, by (1), density depends on the toll and, by (4), depends on density , (10) can be easily represented as a nonlinear function in . More specifically, it is a quadratic function and therefore easy to solve.

The relationship between MP optimal toll and density is shown in Figure 1 for a simple baseline case of a highway 1 kilometre long under the following predefined relations: and .