Published Apr 12, 2017
Oussama Derbel Tamás Péter Benjamin Mourllion Michel Basset


In case of the Intelligent Driver Model (IDM) the actual Velocity–Density law V(D) applied by this dynamic system is not defined, only the dynamic behaviour of the vehicles/drivers is determined. Therefore, the logical question is whether the related investigations enhance an existing and known law or reveal a new connection. Specifically, which function class/type is enhanced by the IDM? The publication presents a model analysis, the goal of which was the exploration of a feature of the IDM, which, as yet, ‘remained hidden’. The theoretical model results are useful, this analysis important in the practice in the field of hybrid control as well. The transfer of the IDM groups through large-scale networks has special practical significance. For example, in convoys, groups of special vehicle, safety measures with delegations. In this case, the large-scale network traffic characteristics and the IDM traffic characteristics should be taken into account simultaneously. Important characteristics are the speed–density laws. In case of effective modelling of large networks macroscopic models are used, however the IDMs are microscopic. With careful modelling, we cannot be in contradiction with the application of speed–density law, where there IDM convoy passes. Therefore, in terms of practical applications, it is important to recognize what kind of speed–density law is applied by the IDM convoys in traffic. Therefore, in our case the goal was not the validation of the model, but the exploration of a further feature of the validated model. The separate validation of the model was not necessary, since many validated applications for this model have been demonstrated in practice. In our calculations, also the applied model parameter values remained in the range of the model parameters used in the literature. This paper presents a new approach for Velocity–Density Model (VDM) synthesis. It consists in modelling separately each of the density and the velocity (macroscopic parameter). From this study, safety time headway (microscopic parameter) can be identified from macroscopic data by mean of interpolation method in the developed map of velocity–density. By combining the density and the velocity models, a generalized new VDM is developed. It is shown that from this one, some literature VDMs, as well as their properties, can be derived by fixing some of its parameters.


Download data is not yet available.



velocity–density model, microscopic traffic simulation, adaptive cruise control

Ardekani, A. S.; Ghandehari, M.; Nepal, M. S. 2011. Macroscopic speed-flow models for characterization of freeway and managed lanes, Bulletin of the Polytechnic Institute of Iasi 7: 149–160.

Bando, M.; Hasebe, K.; Nakanishi, K.; Nakayama, N. 1998. Analysis of optimal velocity model with explicit delay, Physical Review E: Covering Statistical, Nonlinear, Biological, and Soft Matter Physics 58(5): 5429–5435. https://doi.org/10.1103/PhysRevE.58.5429

Bede, Z.; Péter, T. 2014. Optimal control with the dynamic change of the structure of the road network, Transport 29(1): 36–42. https://doi.org/10.3846/16484142.2014.895959

Bede, Z.; Péter, T.; Szauter, F. 2013. Variable network model, IFAC Proceedings Volumes 46(25): 173–177. https://doi.org/10.3182/20130916-2-TR-4042.00026

Del Castillo, J. M. 2012. Three new models for the flow–density relationship: derivation and testing for freeway and urban data, Transportmetrica 8(6): 443–465. https://doi.org/10.1080/18128602.2011.556680

Del Castillo, J. M.; Benítez, F. G. 1995. On the functional form of the speed-density relationship – I: general theory, Transportation Research Part B: Methodological 29(5): 373–389. https://doi.org/10.1016/0191-2615(95)00008-2

Derbel, O.; Mourllion, B.; Basset, M. 2012a. Extended safety descriptor measurements for relative safety assessment in mixed road traffic, in 2012 15th International IEEE Conference on Intelligent Transportation Systems (ITSC), 16–19 September 2012, Anchorage, Alaska, US, 752–757. https://doi.org/10.1109/ITSC.2012.6338774

Derbel, O.; Péter, T.; Zebiri, H.; Mourllion, B.; Basset, M. 2012b. Modified intelligent driver model, Periodica Polytechnica Transportation Engineering 40(2): 53–60. https://doi.org/10.3311/pp.tr.2012-2.02

Dömötörfi, Á.; Péter, T.; Szabó, K. 2016. Mathematical modeling of automotive supply chain networks, Periodica Polytechnica Transportation Engineering 44(3): 181–186. https://doi.org/10.3311/PPtr.9544

Derbel, O.; Peter, T.; Zebiri, H.; Mourllion, B.; Basset, M. 2013. Modified intelligent driver model for driver safety and traffic stability improvement, IFAC Proceedings Volumes 46(21): 744–749. https://doi.org/10.3182/20130904-4-JP-2042.00132

Drake, J. S.; Schofer, J. L.; May, A. D. 1967. A statistical analysis of speed-density hypotheses in vehicular traffic science, Highway Research Record 154: 112–117.

Drew, D. R. 1968. Traffic Flow Theory and Control. McGraw-Hill Inc. 467 p.

Ge, H. X.; Cheng, R. J.; Li, Z. P. 2008. Two velocity difference model for a car following theory, Physica A: Statistical Mechanics and its Applications 387(21): 5239–5245. https://doi.org/10.1016/j.physa.2008.02.081

Greenberg, H. 1959. An analysis of traffic flow, Operations Research 7(1): 79–85. https://doi.org/10.1287/opre.7.1.79

Greenshields, B. D. 1935. A study of traffic capacity, Highway Research Board Proceedings 14: 448–477.

Gumz, F.; Török, Á. 2015. Investigation of cordon pricing in Budakeszi, Periodica Polytechnica Transportation Engineering 43(2): 92–97. https://doi.org/10.3311/PPtr.7579

Helbing, D.; Tilch, B. 1998. Generalized force model of traffic dynamics, Physical Review E: Covering Statistical, Nonlinear, Biological, and Soft Matter Physics 58(1): 133–138. https://doi.org/10.1103/PhysRevE.58.133

Herty, M.; Klar, A. 2003. Modeling, simulation, and optimization of traffic flow networks, SIAM Journal on Scientific Computing 25(3): 1066–1087. https://doi.org/10.1137/S106482750241459X

Holden, H.; Risebro, N. H. 1995. A mathematical model of traffic flow on a network of unidirectional roads, SIAM Journal on Mathematical Analysis 26(4): 999–1017. https://doi.org/10.1137/S0036141093243289

Kesting, A. 2008. Microscopic Modeling of Human and Automated Driving: Towards Traffic-Adaptive Cruise Control: Doctoral Thesis. Faculty of Traffic Sciences, Dresden University of Technology, Germany. 218 p.

Kesting, A.; Treiber, M.; Schönhof, M.; Helbing, D. 2008. Adaptive cruise control design for active congestion avoidance, Transportation Research Part C: Emerging Technologies 16(6): 668–683. https://doi.org/10.1016/j.trc.2007.12.004

Lakatos, I. 2015. Development of a new method for comparing the cold start- and the idling operation of internal combustion engines, Periodica Polytechnica Transportation Engineering 43(4) 225–231. https://doi.org/10.3311/PPtr.8087

MacNicholas, M. J. 2008. A simple and pragmatic representation of traffic flow, in Symposium on the Fundamental Diagram: 75 Years (Greenshields 75 Symposium), 8–10 July 2008, Woods Hole, MA, US, 1–17.

Mahnke, R.; Kaupužs, J. 1999. Stochastic theory of freeway traffic, Physical Review E: Covering Statistical, Nonlinear, Biological, and Soft Matter Physics 59(1): 117–125. https://doi.org/10.1103/PhysRevE.59.117

Molina, J. 2005. Commande de l’inter-distance entre deux véhicules: Thèse pour obtenir le grade de docteur. Institut National Polytechnique de Grenoble, France. 164 p. (in French).

Newell, G. F. 1961. Nonlinear effects in the dynamics of car following, Operations Research 9(2): 209–229. https://doi.org/10.1287/opre.9.2.209

Paveri-Fontana, S. L. 1975. On Boltzmann-like treatments for traffic flow: a critical review of the basic model and an alternative proposal for dilute traffic analysis, Transportation Research 9(4): 225–235. https://doi.org/10.1016/0041-1647(75)90063-5

Péter, T. 2012. Modeling nonlinear road traffic networks for junction control, International Journal of Applied Mathematics and Computer Science 22(3): 723–732. https://doi.org/10.2478/v10006-012-0054-1

Peter, T.; Bokor, J.; Strobl, A. 2013. Model for the analysis of traffic networks and traffic modelling of Győr, IFAC Proceedings Volumes 46(25): 167–172. https://doi.org/10.3182/20130916-2-TR-4042.00023

Péter, T.; Fazekas, S. 2014. Determination of vehicle density of inputs and outputs and model validation for the analysis of network traffic processes, Periodica Polytechnica Transportation Engineering 42(1): 53–61. https://doi.org/10.3311/PPtr.7282

Péter, T., Lakatos, I.; Szauter, F. 2015. Analysis of the complex environmental ımpact on urban trajectories, in ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 2–5 August 2015, Boston, Massachusetts, US, 9: 1–7 https://doi.org/10.1115/DETC2015-47077

Pipes, L. A. 1967. Car following models and the fundamental diagram of road traffic, Transportation Research 1(1): 21–29. https://doi.org/10.1016/0041-1647(67)90092-5

Prigogine, I.; Herman, R. C. 1971. Kinetic Theory of Vehicular Traffic. Elsevier. 100 p.

Rakha, H.; Gao, Y. 2010. Calibration of Steady-State Car-Following Models Using Macroscopic Loop Detector Data. Final Report VT-2008-01. Virginia Tech Transportation Institute, US. 24 p.

Tettamanti, T.; Milacski, Z. Á.; Lőrincz, A.; Varga, I. 2015. Iterative calibration method for microscopic road traffic simulators, Periodica Polytechnica Transportation Engineering 43(2): 87–91. https://doi.org/10.3311/PPtr.7685

Török, Á; Kiss, Á; Szendrő, G. 2015. Introduction to the road safety situation in Hungary, Periodica Polytechnica Transportation Engineering 43(1): 15–21. https://doi.org/10.3311/PPtr.7510

Treiber, M.; Hennecke, A.; Helbing, D. 2000a. Congested traffic states in empirical observations and microscopic simulations, Physical Review E 62(2): 1805–1824. https://doi.org/10.1103/PhysRevE.62.1805

Treiber, M.; Hennecke, A.; Helbing, D. 2000b. Microscopic simulation of congested traffic, in D. Helbing, H. J. Herrmann, M. Schreckenberg, D. E. Wolf (Eds.). Traffic and Granular Flow’99: Social, Traffic, and Granular Dynamics, 365–376. https://doi.org/10.1007/978-3-642-59751-0_36

Underwood, R. T. 1961. Speed, volume and density relationship, in B. D. Greenshields, H. P. George, N. S. Guerin, M. R. Palmer, R. T. Underwood (Eds.). Quality and Theory of Traffic Flow: a Symposium. Bureau Highway Traffic, Yale University, 141–188.

Van Aerde, M. W. 1995. Single regime speed-flow-density relationship for congested and uncongested highways, in 74th Annual Meeting of the Transportation Research Board, 22–28 January 1995, Washington, DC, US, 1–26.

Wang, H.; Ni, D.; Chen, Q.-Y.; Li, J. 2013. Stochastic modeling of the equilibrium speed–density relationship, Journal of Advanced Transportation 47(1): 126–150. https://doi.org/10.1002/atr.172