Physica A 392 (2013) 3537–3545 Contents lists available at SciVerse ScienceDirect Physica A journal homepage: www.elsevier.com/locate/physa An extended macro model for traffic flow with consideration of multi static bottlenecks T.Q. Tang a,b , P. Li b , X.B. Yang a,∗ a MOE Key Laboratory for Urban Transportation Complex Systems Theory and Technology, Beijing Jiaotong University, Beijing, 100044, China b School of Transportations Science and Engineering, Beihang University, Beijing 100191, China highlights • A macro model with consideration of multi static bottlenecks has been developed. • Impacts of the number of static bottlenecks have been explored. • Impacts of the distance between adjacent static bottlenecks have been explored. • These results can help us better understand and explain some traffic phenomena. article info Article history: Received 6 November 2012 Received in revised form 19 February 2013 Available online 17 April 2013 Keywords: Traffic flow Macro model, stability Static bottleneck abstract In this paper, we propose a macro model with consideration of multi static bottlenecks to study the impacts of multi static bottlenecks on traffic flow. The numerical results show that the influences are related to the number of static bottlenecks, the distance between two adjacent static bottlenecks and the initial density. © 2013 Elsevier B.V. All rights reserved. 1. Introduction Serious traffic problems have attracted researchers to develop many traffic flow models to explore various complex traf- fic phenomena [1–17]. However, the existing models cannot be used to directly study the effects of bottleneck on traffic flow since they do not consider this factor. In fact, bottlenecks (e.g., bus station, ramp, etc.) widely exist and produce some complex traffic phenomena (e.g., stop-and-go, queues, etc.). Hence, many models have been proposed to explore bottleneck. For example, Cassidy and Bertini [18] used empirical data to study the formation and discharge of queue near a bottleneck on freeway; Ishibashi and Fukui [19] explored the effects of a bottleneck on the high-speed vehicles; Ni and Leonard [20] proposed a kinematic model to study the priority that the vehicle enters a merging bottleneck; Daganzo and Laval [21] used the kinematic theory to study the impacts of a moving bottleneck on traffic flow; Yamamoto et al. [22] proposed a coupled map optimal velocity model to study the effects of a bottleneck on the car-following behavior; Kerner [23] used the three phase traffic flow theory to study the spatiotemporal features resulted by bottleneck; Lattanzio et al. [24] used the differ- ential equation theories to study the influences of a bottleneck on traffic flow from the mathematical perspective and Tang et al. [25] further extended the work. Nagatani and his/her cooperators explored the impacts of some special bottlenecks ∗ Corresponding author. Tel.: +86 10 82339327. E-mail addresses: tieqiaotang@126.com (T.Q. Tang), yangxb@bjtu.edu.cn (X.B. Yang). 0378-4371/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physa.2013.03.056