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Journal of Advanced Transportation
Volume 2017, Article ID 4629792, 10 pages
Research Article

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;

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.


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.