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## Study of the shock wave induced by closing partial road in traffic flow

Sun Xiao-Yan, Zhu Jun-Fang
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• #### 摘要

本文根据实际交通中经常遇到的交通事故或部分道路施工等情况, 建立了部分道路关闭的交通流模型. 采用平均场理论分析和确定性NS元胞自动机规则分别对模型进行解析和数值模拟, 结果表明, 系统存在三种稳定的物理状态:低密度相、激波相和高密度相, 并找到了系统发生相变的临界密度. 理论分析和数值模拟能很好地符合.

#### Abstract

There often occurs traffic accident or road construction in real traffic, which leads to partial road closure. In this paper, we set up a traffic model for the partial road closure. According to the Nagel-Schreckenberg (NS) cellular automata update rules, the road can be separated into cells with the same length of 7.5 m. L = 4000 (corresponding to 30 km) is set to the road length in the simulations. For a larger system size, our simulations show that the results are the same with those presented in the following. In our model, vmax rtial road is closed (for convenience, we define the road length as L1), vmax 2= 2 (corresponding to 54 km/h) in the section of normal road (we define the road length as L2). In our simulations, let L1= L2 = 2000. We would like to mention that changing these parameter values does not have a qualitative influence on the simulation results. The simulation results demonstrate that three stationary phases exist, that is, low density (LD), high density (HD) and shock wave (SW). Two critical average densities are found:the critical point ρcr 1= 3/8 separates the LD phase from the SW phase, and ρcr 2= 1/2 separates the SW phase from the HD phase. We also analyze the relationship between the average flux J and average density ρ. In the LD phase J = 4/3ρ, in the HD phase J= 1 -ρ and J is 0.5 in the SW phase. We investigate the dependence of J on ρ. It is shown that with the increase of ρ, J first increases, at this stage J corresponds to the LD phase. Then J remains to be a constant 0.5 when the critical average density ρcr 1 is reached, and J corresponds to the SW phase (this time,J reaches the maximum value 0.5). One goal of traffic-management strategies is to maximize the flow. We find that the optimal choice of the average density is 3/8 ρρcr 2 is reached, J decreases with the increase of average density, which corresponds to the HD phase. We also obtain the relationship between the shock wave position and the average density by theoretical calculations, i.e. Si = i+4-8ρ, which is in agreement with simulations.

#### 作者及机构信息

###### 1. 北京师范大学系统科学学院, 北京 100875; 2. 广西师范学院物理与电子工程学院, 南宁 530023; 3. 西南科技大学理学院, 绵阳 621010
• 基金项目: 国家自然科学基金(批准号:71461002,11402058,11202175)、广西壮族自治区自然科学基金(批准号:2014GXNSFAA118012)和广西高等学校优秀中青年骨干教师培养工程(第一期)资助的课题.

#### Authors and contacts

###### 1. School of systems Science, Beijing Normal University, Beijing 100875, China; 2. College of Physics and Electronic Engineering, Guangxi Teacher Education University, Nanning 530023, China; 3. School of Science, Southwest University of Science and Technology, Mianyang 621010, China
• Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 71461002, 11402058, 11202175), the Natural Science Foundation of Guangxi Province, China (Grant No. 2014GXNSFAA118012), and the project of outstanding young teachers’ training in higher education institutions of Guangxi.

#### 参考文献

 [1] Chowdhury D 2000 Phys. Rep. 329 199 [2] Gao H W, Gao Z Y, Xie D F 2011 Acta Phys. Sin. 60 058902 (in Chinese) [郭宏伟, 高自友, 谢东繁 2011 物理学报 60 058902] [3] He H D, Lu W Z, Dong L Y 2011 Chin. Phys. B 20 040514 [4] Lakouari N, Ez-Zahraouy H, Benyoussef A 2014 Phys. Lett. A 378 3169 [5] Nagatani T 2014 Physica A 413 352 [6] Tang T Q, Huang H J, Shang H Y 2010 Chin. Phys. B 19 050517 [7] Jia B, Jiang R, Wu Q S 2003 Int. J Mod. Phys. C 14 1295 [8] Zhang L, Du W 2012 J. Wuhan Univ. Tech. 36 886 (in Chinese) [张邻, 杜文 2012 武汉理工大学学报 36 886] [9] Zhang Ao M H, Gao Z Y 2012 J. Trans. Sys. Engin. Inf. Tech. 12 46 (in Chinese) [张敖木翰, 高自友 2012 交通运输系统工程与信息 12 46] [10] Qian Y S, Zeng J W, Du J W, Liu Y F, Wang M, Wei J 2011 Acta Phys. Sin. 60 060505 (in Chinese) [钱勇生, 曾俊伟, 杜加, 刘宇斐, 王敏, 魏军 2011 物理学报 60 060505] [11] Kanai M Phys. Rev. E 2005 72 035102(R) [12] Yamauchi A, Tanimoto J, Hagishima A, Sagara H 2009 Phys. Rev. E 79 036104 [13] Nakata M, Yamauchi A, Tanimoto J, Hagishima A 2010 , Physica A 389 5353 [14] Jia B, Gao Z Y, Li K P, Li X G 2007 Models and Simulations of Traffic System Based on the Theory of Cellular Automaton (Beijing:Science Press) (in Chinese) [贾斌, 高自友, 李克平, 李新刚 2007 基于元胞自动机的交通系统建模与模拟 (北京:科学出版社)] [15] Li L, Jiang R, Jia B, Zhao X M 2011 T heory and Application of Modern Traffic flow(Vol. 1)-freeway traffic flow (Beijing:Tsinghua University Press) [李力, 姜锐, 贾斌, 赵小梅 2011 现代交通流理论与应用卷I-高速公路交通流 (北京:清华大学出版社)] [16] Sun D 2011 Ph. D. Dissertation (Hefei:University of Science and Technology of Chian) (in Chinese) [孙舵 2011 博士学位论文 (合肥:中国科学技术大学)] [17] Nagel K, Schreckenberg M 1992 J. Phys. I (France) 2 2221 [18] Sun X Y, Xie Y B, He Z W, Wang B H 2011 Phys. Lett. A 375 2699 [19] Derrida B, Evans M R, Hakim V 1993 J. Phys. A:Math. Gen. 26 1493

#### 施引文献

•  [1] Chowdhury D 2000 Phys. Rep. 329 199 [2] Gao H W, Gao Z Y, Xie D F 2011 Acta Phys. Sin. 60 058902 (in Chinese) [郭宏伟, 高自友, 谢东繁 2011 物理学报 60 058902] [3] He H D, Lu W Z, Dong L Y 2011 Chin. Phys. B 20 040514 [4] Lakouari N, Ez-Zahraouy H, Benyoussef A 2014 Phys. Lett. A 378 3169 [5] Nagatani T 2014 Physica A 413 352 [6] Tang T Q, Huang H J, Shang H Y 2010 Chin. Phys. B 19 050517 [7] Jia B, Jiang R, Wu Q S 2003 Int. J Mod. Phys. C 14 1295 [8] Zhang L, Du W 2012 J. Wuhan Univ. Tech. 36 886 (in Chinese) [张邻, 杜文 2012 武汉理工大学学报 36 886] [9] Zhang Ao M H, Gao Z Y 2012 J. Trans. Sys. Engin. Inf. Tech. 12 46 (in Chinese) [张敖木翰, 高自友 2012 交通运输系统工程与信息 12 46] [10] Qian Y S, Zeng J W, Du J W, Liu Y F, Wang M, Wei J 2011 Acta Phys. Sin. 60 060505 (in Chinese) [钱勇生, 曾俊伟, 杜加, 刘宇斐, 王敏, 魏军 2011 物理学报 60 060505] [11] Kanai M Phys. Rev. E 2005 72 035102(R) [12] Yamauchi A, Tanimoto J, Hagishima A, Sagara H 2009 Phys. Rev. E 79 036104 [13] Nakata M, Yamauchi A, Tanimoto J, Hagishima A 2010 , Physica A 389 5353 [14] Jia B, Gao Z Y, Li K P, Li X G 2007 Models and Simulations of Traffic System Based on the Theory of Cellular Automaton (Beijing:Science Press) (in Chinese) [贾斌, 高自友, 李克平, 李新刚 2007 基于元胞自动机的交通系统建模与模拟 (北京:科学出版社)] [15] Li L, Jiang R, Jia B, Zhao X M 2011 T heory and Application of Modern Traffic flow(Vol. 1)-freeway traffic flow (Beijing:Tsinghua University Press) [李力, 姜锐, 贾斌, 赵小梅 2011 现代交通流理论与应用卷I-高速公路交通流 (北京:清华大学出版社)] [16] Sun D 2011 Ph. D. Dissertation (Hefei:University of Science and Technology of Chian) (in Chinese) [孙舵 2011 博士学位论文 (合肥:中国科学技术大学)] [17] Nagel K, Schreckenberg M 1992 J. Phys. I (France) 2 2221 [18] Sun X Y, Xie Y B, He Z W, Wang B H 2011 Phys. Lett. A 375 2699 [19] Derrida B, Evans M R, Hakim V 1993 J. Phys. A:Math. Gen. 26 1493
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##### 出版历程
• 收稿日期:  2014-09-09
• 修回日期:  2014-11-25
• 刊出日期:  2015-06-05

## 部分道路关闭引起的交通激波特性研究

• 1. 北京师范大学系统科学学院, 北京 100875;
• 2. 广西师范学院物理与电子工程学院, 南宁 530023;
• 3. 西南科技大学理学院, 绵阳 621010
基金项目: 国家自然科学基金(批准号:71461002,11402058,11202175)、广西壮族自治区自然科学基金(批准号:2014GXNSFAA118012)和广西高等学校优秀中青年骨干教师培养工程(第一期)资助的课题.

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