Batista et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 2010 11(7):520-529 520 Optimal velocity functions for car-following models Milan BATISTA, Elen TWRDY (Faculty of Maritime Studies and Transport, University of Ljubljana, 6320 Portorož, Slovenia) E-mail: milan.batista@fpp.edu; elen.twrdy@fpp.uni-lj.si Received June 24, 2009; Revision accepted Nov. 27, 2009; Crosschecked Apr. 30, 2010 Abstract: The integral part of the optimal velocity car-following models is the optimal velocity function (OVF), which can be derived from measured velocity-spacing data. This paper discusses several characteristics of the OVF and presents regression analysis on two classical datasets, the Lincoln and Holland tunnels, with different possible OVFs. The numerical simulation of the formation of traffic congestion is conducted with three different heuristic OVFs, demonstrating that these functions give results similar to those of the famous Bando OVF (Bando et al., 1995). Also an alternative method is present for determining the sensi- tivity and model parameters based on a single car driving to a fixed barrier. Key words: Traffic flow, Car following, Optimal velocity function (OVF), Traffic congestion doi:10.1631/jzus.A0900370 Document code: A CLC number: U491.1 + 12 1 Introduction Car-following theory is focused on the study of single lane traffic with no passing where the driver in each following car is controlled by the car directly in front. For a review and historical development of the subject one should consult (Holland, 1998; Brack- stone and McDonald, 2000; Weng and Wu, 2002). Whilst today the study of car-following has practical applications in developing adaptive cruise control systems (Rajamani, 2006), the original theory mostly dealt with the stability analysis of driving with respect to velocity perturbations. The classical result is that instability leads to collision. However, as pointed out by Bando et al. (1995), the more likely phenomena resulting from instability is traffic congestion. To demonstrate this idea, Bando et al. (1995) proposed the optimal velocity model (OVM) in which the driver’s response is proportional to the difference between his optimum speed and his actual speed. The acceleration of a car is thus ( ) d ( ) , 1, 2, ..., , d n n n v Vh v n N t λ = − = (1) where 1 n n n h x x − = − is the distance between the nth car and its predecessor, n x and d d n n x v t = are the nth car position and velocity at time t, respectively, Ȝ is the sensitivity of the car-driver system, N is the total number of cars, and V(h) is the optimal velocity function (OVF). They then derived the general sta- bility criterion for the model in Eq. (1): * * d ( ) , d 2 h h V V h h λ = ′ = < (2) where * h is the car spacing of a steady state move- ment. To illustrate the possibility of spontaneous evaluation of traffic congestion at an unstable condi- tion, Bando et al. (1995) took Ȝ=1 and proposed the following OVF: () tanh ( 2) tanh 2, Vh h = − + (3) for which the stability condition is from Eq. (2) and becomes 2 1 tanh ( 2) 1 / 2. b − − < This condition di- vides the domain of OVF into three regions: a stable region near the origin followed by an unstable region, Journal of Zhejiang University-SCIENCE A (Applied Physics & Engineering) ISSN 1673-565X (Print); ISSN 1862-1775 (Online) www.zju.edu.cn/jzus; www.springerlink.com E-mail: jzus@zju.edu.cn © Zhejiang University and Springer-Verlag Berlin Heidelberg 2010