Journal of Advanced Transportation

Volume 2017, Article ID 4629792, 10 pages

https://doi.org/10.1155/2017/4629792

## Estimating Macroscopic Volume Delay Functions with the Traffic Density Derived from Measured Speeds and Flows

Department of Transportation Systems, Cracow University of Technology, Ul. Warszawska 24, 31-155 Kraków, Poland

Correspondence should be addressed to Rafał Kucharski; lp.ude.kp@iksrahcukr

Received 25 July 2016; Revised 12 January 2017; Accepted 5 February 2017; Published 26 February 2017

Academic Editor: Alexandre G. De Barros

Copyright © 2017 Rafał Kucharski and Arkadiusz Drabicki. 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

This paper proposes a new method to estimate the macroscopic volume delay function (VDF) from the point speed-flow measures. Contrary to typical VDF estimation methods it allows estimating speeds also for hypercritical traffic conditions, when both speeds and flow drop due to congestion (high density of traffic flow). We employ the well-known hydrodynamic relation of fundamental diagram to derive the so-called quasi-density from measured time-mean speeds and flows. This allows formulating the VDF estimation problem with a speed being monotonically decreasing function of quasi-density with a shape resembling the typical VDF like BPR. This way we can use the actually observed speeds and propose the macroscopic VDF realistically reproducing actual speeds also for hypercritical conditions. The proposed method is illustrated with half-year measurements from the induction loop system in city of Warsaw, which measured traffic flows and instantaneous speeds of over 5 million vehicles. Although the proposed method does not overcome the fundamental limitations of static macroscopic traffic models, which cannot represent dynamic traffic phenomena like queue, spillback, wave propagation, capacity drop, and so forth, we managed to improve the VDF goodness-of-fit from of 27% to 72% most importantly also for hypercritical conditions. Thanks to this traffic congestion in macroscopic traffic models can be reproduced more realistically in line with empirical observations.

#### 1. Introduction

In this paper we will solve the estimation problem where traffic speed is a function of the traffic flow, generically expressed as and further called volume delay function (VDF) or link-congestion function. The solution of the problem is a function which reproduces traffic speeds observed in field measurements. The VDF is commonly applied in static macroscopic traffic assignment to describe the resultant link travel times, as a function of flow (result of assignment) and capacity and free-flow travel time (constant parameters of the link). The purposes of this function are to reproduce congestion effects in the macroscopic model and to serve as an objective function in the assignment problem where the travel times are minimized [1]. The VDF is usually formulated in an easily integrable and differentiable form, since the assignment algorithm searches for the solution by using the integrals of VDF [2]. Unlike the physical representations of the traffic flow, the VDF allows the flow to exceed the capacity (which is by definition impossible within the traffic flow definitions). As a result, the flow volumes used in macroscopic assignment (and in turn in VDF) are not strictly related to the physically measured flows. The macroscopic flow (as we will denote it) is treated more like a demand flow which becomes delayed if it exceeds capacity. We will exploit this distinction in the proposed method.

The VDF shall reproduce both travel times and traffic flows realistically. Usually, the focus is to reproduce the actually observed flow pattern in the network, and it is well known that travel times in macroscopic model are a rough approximation neglecting fundamental traffic phenomena (such as bottlenecks, spillbacks, capacity drop, and gridlocks) which can be handled with dynamic traffic flow models [3]. The relationship between travel delay and flow volume used in macroscopic traffic flow models is extremely simplified and behaviorally unrealistic, yet it is commonly applied in big-scale traffic demand models [4]. Nevertheless, to improve the representativeness of resulting travel times, the VDF are usually estimated to match the observed traffic speeds/travel times, which raises following estimation issue, which we address in the paper.

In VDF the flow can become greater than capacity, while in field data measurements the vehicle flow, by definition, cannot exceed physical capacity. This raises an issue while estimating the shape of VDF. Namely, the macroscopic static traffic flow models represent the congestion with functional formulation which cannot be empirically observed and, in turn, cannot be estimated to reproduce the actual traffic speeds. Usually practitioners overcame this and estimate only the hypocritical part only for which the problem does not raise since the observed speed can be expressed with unique monotonically decreasing function of flow. The hypercritical part (when the flow starts decreasing) is usually neglected and arbitrary parameterizations are used [5].

The contribution of the paper is introducing a practical method to estimate the VDF from time-mean speed and flow overcoming issue of estimating the speeds for flows exceeding capacity. It is achieved by using the hydrodynamic relation of the fundamental diagram and more specifically by extending measured traffic flows and instantaneous speed with the proposed quasi-density. This allowed reformulating the volume delay estimation problem into the density-delay estimation problem and thus obtaining an improved goodness-of-fit with available measurements. Finally, by expressing the macroscopic flow with a quasi-density, the estimated VDF can be used in macroscopic traffic assignment where densities are not available. This way the VDF becomes not only estimated with the empirical data, but also coherent with principal traffic flow relations.

The paper is organized as follows. The following part reviews the literature, followed by Section 2 when the method is formally introduced. It starts with introducing the estimation problem for VDF, followed with dataset description and revealing problems while estimating VDF from field data. Subsequently we introduce idea of extending measured speed and flows to point densities and reformulate the VDF estimation using them. In Section 3 we present the results of the proposed method and in Section 4 we discuss them and conclude the paper.

##### 1.1. Literature Review

All of practically applied VDF formulations follow the basic principles of traffic flow theory; that is, the speed decreases with the increasing flow, or, equivalently, with the increasing saturation rate. Saturation rate is computed as the ratio between the flow and the capacity, with capacity being unknown and (as we show further) estimated internally within the proposed estimation problem. Reference [5] singles out mathematical and behavioral conditions which should be met by the VDF; that is, it should be described with a continuous, strictly increasing, and nonnegative function, which clearly contradicts the actual traffic flow [6]. The relation between the speed and saturation rate in VDF is highly nonlinear: initially, speed remains almost intact and rises slowly until some congestion is generated and then after reaching the assumed saturation level, speed starts to fall down significantly. The characteristic point of the VDF is when the flow equals capacity. In several VDF the formula is conditional and assumes different parameterization before and after the capacity threshold is reached. The VDF remains continuous, but the differential usually becomes much higher above this threshold, which allows separating two different regimes both in parameterization and in interpretation.

The common formulations of the volume delay functions are based on the standard BPR function (Bureau of Public Roads [7]), which has been adopted in the US guidelines and widely applied ever since in transportation planning practice [8]. A number of alternative VDF formulations have been proposed, so as to match better with observed operating conditions and road facility characteristics, which include most notably: Davidson [9], Conical [10], Akcelik [11], and Vatzek [5], as well as empirically modified BPR variations [12]. One of recurrent reasons for revisiting the standard VDF formulas in research works was identifying drawbacks in function performance for volumes approaching and exceeding the capacity rate. Studies observed that default BPR functions and their later modifications are likely either to overestimate travel times for the congested conditions [11] or lead to overestimated traffic volumes when capacity is exceeded and conversely underestimated traffic flows for relative free-flow conditions [13]. Reference [10] further noted that by adjusting the BPR parameters the travel times become more constant (“suppressed”) in uncongested traffic conditions, but at the same time much more sensitive for volumes approaching and exceeding the capacity. Empirical practice shows that steepness of the speed-volume curve in congested conditions is strongly related to road-design and is different for freeways than urban arterials [11]. Reference [10] reckons that the maximal steepness rate of a VDF should be limited to limit the risk of travel time overestimation.

The VDF can be empirically observed only below the capacity rate, which divides the estimation problem into two parts: realistic curve estimation for the hypocritical part and arbitrary formulation for the hypercritical part [5]. This leads to dual approach to VDF estimation: realistic, which can be supported with empirical data in the hypocritical part; and theoretical, as it cannot be observed, in the hypercritical part. The hypercritical part of the VDF is formulated solely to be used in the assignment model, while the hypocritical part is estimated to fit the empirical observations. Other authors [2] claim that the VDF is completely unrealistic and shall be treated only as part of the traffic assignment algorithm. Importantly, [3] claims that the static assignment cannot reproduce both traffic flows and travel times realistically and further argues that the purpose of the VDF is not to reproduce the actual travel speeds, but to guarantee the convergence of the Wardropian algorithm [14] towards the user-equilibrium. Consequently, the VDF can be both easily differentiated (to obtain the search direction) and integrated. The goal function of equilibrium assignment is usually formulated with the VDF formula [15] which yields the total link travel time from the first to the last vehicle. This may however result in conflict between the empirical representation of the traffic flow and the performance and convergence of the traffic assignment algorithm.

Reproducing the travel time delay solely from a direct speed-volume estimation can prove to be a complex and challenging issue in practice. Empirical works, such as [16, 17], aimed to calibrate the VDF for congested traffic regimes but observed major obstacles in collecting traffic flow data for traffic states where volumes exceeded capacity—in the end, simulation models were used to generate the necessary data instead. A different and perhaps more interesting approach is to extend the scope of analysis to include the speed-density relationship, thus exploiting a much wider extent of macroscopic fundamental diagram. Empirical estimations based on a speed-density relationship are likely to be more accurate when identifying distinct traffic conditions (free-flow, congested, and mixed traffic conditions) and the jam density limits [18, 19]. In general overview, the idea of utilizing the speed-density relationship to estimate the speed-flow curve remains (to the best of our knowledge) relatively not much investigated in traffic modelling research. Reference [20] discusses such transition, illustrated with an example of estimating a speed-flow curve from a calibrated speed-density one, which as a result underestimates the capacity rate; the limitation reason is the discontinuity between uncongested and congested flows (so-called capacity drop). Yet [21] compares two approaches in VDF curve estimation and argues that expressing the average speeds with volumes yields a low correlation which can be substantially improved when the speed-density relation is used instead.

#### 2. Method

In this paper we argue that the VDF relation can be observed over the broader domain, by utilizing flow densities instead of flow volumes to describe resultant speeds (travel times) on network. We will demonstrate that the VDF formula can be both algorithmically efficient and provide an improved goodness-of-fit with the field data, not just for flows in the hypocritical part, but also for the hypercritical part of the fundamental diagram. To illustrate the method proposed in the paper we will use the classic BPR formula [7], which has been ever since studied in numerous research works in its either original or modified form [12] and is discussed up to now [4].

To formulate the problem, the following notation will be used: : flow [veh]; : speed [km/h]; : quasi-density [veh/km]; : travel time [s]; : free-flow speed [km/h]; : capacity [veh/h]; : density-at-capacity [veh/km]; , : estimated parameters of VDF [—]; : theoretical speed computed with VDF [km/h]; : theoretical travel time computed with VDF [s].

To illustrate the method proposed in the paper, we will utilize the simplest BPR function, formulated with

It expresses travel time as a function of free-flow travel time, flow-to-capacity ratio, and the two parameters and .

Figure 1 depicts travel time multiplier and speed modelled with BPR function (parameters and )*, *which increases steadily from one until the capacity limit and becomes even steeper afterwards. BPR (1) can be formulated relatively to express the travel time delay: