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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

In this paper we will solve the estimation problem where traffic speed is a function of the traffic flow, generically expressed as

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 [

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 [

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

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 [

The common formulations of the volume delay functions are based on the standard BPR function (Bureau of Public Roads [

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 [

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 [

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 [

To formulate the problem, the following notation will be used:

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

Figure

Travel time multiplier and speed in the BPR function.

By assuming constant length the relative speed drop can be derived from BPR function (

In the paper, we discuss the estimation problem of the BPR function where we look for optimal values of parameters

We do not analyze the solution of problem (

We illustrate the proposed method with field measurement data from the Warsaw count locations where the vehicle flows and speeds are measured. Vehicle flow is continuously measured in over hundred locations in crucial points across the Warsaw road network, some of which are equipped with the double induction loops, capturing also instantaneous vehicle speeds [

We used measurements collected from one loop placed at the main road site at the boundary of Warsaw. It measures traffic flows towards the city centre on the three-lane arterial road with a posted speed limit of 70 km/h which becomes highly congested during peak hours. For all the approaching vehicles, instantaneous speeds were measured and the datasets were aggregated over 15-minute time intervals. We analyzed data collected for six months was used, which covered 15 thousand records and almost 6 million vehicles (for further references see [

Figure

Average weighted instantaneous speed, given with

where

Capacity, initially assumed to be the 95th percentile of the measured vehicle flows and further estimated within VDF estimation.

Free-flow speed initially assumed to be the 85th percentile of the measured speed and further estimated within VDF estimation.

Hourly vehicle flow: for 15-minute flow aggregation, we estimate the hourly flow rate by multiplying the 15-minute flow rate by four; this enables us to trace short-term flow variations, which could otherwise disappear if, for example, the moving sum is used.

Dummy instantaneous travel time multiplier, computed from the speed drop rate with

Flow and speed percentiles.

Percentile | Speed | Flow |
---|---|---|

80 | 60 | 2316 |

85 | 61 | 2376 |

90 | 62 | 2476 |

95 | 63 | 2640 |

99 | 66 | 2900 |

99,99 | 70 | 3128 |

Measurement site, Al. Krakowska, Warsaw, Poland, (c) Google Satellite.

Traffic speed and flow histograms.

Data shown in Figure

The directly observed measures allow us to plot the correlation between observed instantaneous speeds against the traffic flows

Average instantaneous speed [km/h] against hourly flow [veh/h].

Travel time multiplier [—] against hourly flow [veh/h] and the BPR function estimated for this data (red) with output parameters equal to

Estimating function fitted the data of Figure

To overcome the problem stated above, let us utilize the fundamental hydrodynamic relation between speed and flow and introduce the third measure of the fundamental diagram: density, denoted by

Mind that the above holds true only under certain conditions (stationary homogeneous traffic [

Thus, being aware of the limitations and possible biases, for the purposes of the proposed method, we will derive density from measured time-mean speed and time-mean flow with (

This can be deemed appropriate and sufficient for the method proposed in the paper, since it is used only as unobserved intermediate variable and the actual density (i.e., number of vehicles per kilometer)—as it will be seen in further sections—is not required. Quasi-density computed with (

The fundamental diagram depicting traffic flow against quasi-density

Fundamental diagram; hourly flow (veh/h) against quasi-density (veh/km).

Figure

Speed (km/h) against quasi-density (veh/km).

Travel time multiplier against quasi-density (veh/km).

Once we stated that the BPR function shape is observed against density, not a flow, we will propose the way to obtain densities in the macroscopic assignment where they are not available. To this end, let us first further exploit distinction between the measured (physical) flow

We postulate to consider macroscopic flow in terms of the demand flow and state that the best approximation possible of macroscopic flow can be achieved in relation to the density, not the flow. Similarly to the macroscopic flow, density grows with the increasing demand volume but reaches its maximum value way above the capacity rate, at the “traffic-jam” density conditions. If so, let us assume that the macroscopic flow shall be understood in terms of density rather than the measured flow and reformulate the traditional VDF flow-speed relation into the density-speed relation.

Since density is not available in the macroscopic static models it cannot be used directly in the VDF formulas. To overcome this problem, let us propose the following mapping from observed quasi-densities to flows of the macroscopic assignment. Such linear mapping is defined with (

Density-at-capacity

We stated that the BPR function can be observed empirically, yet not as a function of flow, but as a function of quasi-density

Quasi-density is used here as a substitute of the flow, while the density-at-capacity

In this formulation, modelled speed is directly expressed with the observed traffic flow and speed values along with estimated parameters

Once the BPR parameters have been estimated, we need to reformulate the BPR function for the purposes of macroscopic assignment algorithm. To this end, we use the inverse mapping from densities to macroscopic flow (

The output BPR function formulated with (

Let us apply the above reformulations and propose the final form of the estimation problems (

BPR function (red) fitted to

Finally, let us further relax problem (

Speed versus density with the estimated BPR function,

The classical VDF assumes that the flow can exceed capacity, which is somehow acceptable for the macroscopic assignment problem, yet it is hardly consistent with actual field measurements (since, by definition, the capacity is the maximum flow which can be possibly observed). Consequently, the VDF cannot be estimated directly from the measured flows (at least for the hypercritical part: see Figure

This research can be further extended to cover another data sources like floating car data or space-mean densities. Also the exposed speed-density relations can advocate new mathematical formulations of VDF which will yield better fit. The naïve least-squares formulation of the estimation problem might be revised to weight more the hypercritical observations (which are crucial to reproduce congestion). Finally the proposed method shall be applied on the real-size strategic model to show its applicability.

The authors declare that there is no conflict of interests regarding the publication of this paper.