基于基本图和三相交通流理论的离散建模方法研究
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摘要
科学解决交通问题的首要任务是深入研究交通流的演化机理,掌握交通流的运行规律。现阶段的交通流理论已发展成以传统基本图方法与新兴三相交通流理论为核心的科学体系。基本图方法假设稳态交通流的流量和密度具有一一对应的函数关系,并将交通流分成自由流和拥挤流两类。三相交通流理论进一步将拥挤流细分成宽运动堵塞和同步流,并假设稳态同步流在流量密度平面上呈现二维分布。合理解决基本图方法和三相交通流理论的争议问题一直是交通流理论研究的热点和难点。遵循相互包容、协同发展的科学原则,本文立足于基本图方法,解决了三相交通流理论对基本图方法的部分批判问题,在深入研究格子流体力学和元胞自动机两类离散模型的基础上提出了更加符合实测现象的模型,通过理论分析和数值模拟研究了交通流的各种非线性现象,探讨了不同类型模型之间的内在联系。研究工作如下：
1)在Nagatani格子流体力学模型基础上,提出了考虑前后格子密度差效应的格子模型(DDLM模型)。通过线性稳定性分析得到了DDLM模型的中性稳定性曲线,并发现考虑密度差效应可以提高交通流的稳定性。通过非线性稳定性分析得到了稳定区域、亚稳定区域和不稳定区域存在的以Burgers方程、KdV方程和mKdV方程描述的三角激波、孤立波和扭结波。随后提出了匝道系统的格子流体力学模型,并设计了随机型和确定型匝道两种数值模拟方法。随机型匝道的模拟结果和基本图方法的实测结果一致,确定型匝道的模拟结果发现了四种新的组合拥挤模式,发现DDLM模型的堵塞可以产生于同步流,因此三相交通流理论对基本图方法模型堵塞产生于自由流的批判不适用于DDLM模型。
2)在BL刹车灯模型的基础上提出了期望时间间距刹车灯模型(DTGBLM模型)。DTGBLM模型不仅可以产生符合实测的同步流以及三相交通流理论的扩展同步流模式和全面模式,而且过大减速度现象出现的概率可忽略不计,因此DTGBLM模型解决了三相交通流理论对BL模型的批判问题。其次在FMCD和BLDAD刹车灯模型的基础上构建了提前慢化刹车灯模型(ARBLM模型)。ARBLM模型需借助外界扰动才能形成同步流,这和实测通行能力陡降现象借助瓶颈产生的机理一致,因此ARBLM模型不仅可以解释通行能力陡降现象的产生机理,而且比三相交通流理论的FMCD模型更加符合实际。
3)从传统的多车期望效应跟驰模型思想出发,将多车平均间距作为车辆的期望速度引入随机慢化过程,提出了平均车辆间距元胞自动机模型(ASGM模型)。ASGM模型不仅可以模拟自由流到同步流以及同步流到宽运动堵塞的一阶相变,而且可以模拟三相交通流理论的各种实测拥挤模式。由于ASGM模型和大多数三相交通流理论模型的模拟结果没有本质区别,稳态拥挤交通流在流量密度平面上呈现二维分布不是模型符合实测的基本条件。深入分析多车期望效应跟驰模型,提出了扩展平均车辆间距元胞自动机模型(EASGM模型)。EASGM模型能够模拟宽运动堵塞的同步出流,并且证实了车辆期望时间间距、慢启动的随机慢化概率和安全控制参数都可影响宽运动堵塞的出流状态。ASGM模型和EASGM模型不仅建立了连续的跟驰模型和离散的元胞自动机模型之间的联系,而且解决了三相交通流理论批判基本图方法不能模拟宽运动堵塞同步出流的问题。
4)提出了三阶段标定法,并对上述新模型和几个不同类型、不同理论体系的模型进行了标定。三阶段标定法首先挑出可以根据缺省值和实测数据直接给定的参数,然后采用试差法或启发式算法标定模型的其他参数,最后从第一步中挑选出能够直接影响车速的参数,并再次采用试差法或启发式算法对其进行标定。标定结果表明新模型的标定误差均小于原模型;三相交通流理论模型和基本图方法模型之间误差相当,没有优劣之分。
Engineering practice proves that the first principle of relieving congestion is investigating and exploring the mechanisms of traffic flow. Traffic flow theory was developed under this background, whose main branches are the traditional fundamental diagram approach and the novel three phase traffic flow theory. Fundamental diagram approach assumes the unique relationship between flow and density, while three phase traffic flow theory denies this assumption. The controversies between the fundamental diagram approach and the three phase traffic flow theory are one of the hotspots in the traffic flow theory. This dissertation is based on these theories:
1) Based on the Nagatani lattice hydrodynamic model, the density difference lattice model (DDLM) is proposed. DDLM not only considered the leading lattice but also the following lattice. Linear stability curve are obtained by the linear stability analysis, which demonstrates that the density difference can enhance the stability. Burgers, KdV, mKdV equations and the coexistence curve are derived by the nonlinear stability analysis. Simulations show that the triangular shock wave, soliton and kink and anti-kink wave described by these equations are simulated. Then the lattice hydrodynamic model with the onramp system is proposed and the stochastic and deterministic onramp simulation methods are designed. Onramp simulations are in accordance with the empirical findings. Moreover four new kinds of congested patterns are founded, two of which are very similar to the GP and Dissolving GP in the three phase theory. Thus DDLM not only confirms the empirical findings of the fundamental diagram approach, but also put forward new findings which need to be consolidated and proves that wide moving jams may not emerges in the free flow in the models of fundamental diagram approach.
2) The synchronized flow of the brake light cellular automaton model (BLM) and the traffic breakdown of FMCD and BLDAD brake light cellular automaton models are not in accordance with the real traffic. Based on BLM, the Desired Time Gap Brake Light Model (DTGBLM) is established by incorporating the desired time gap. Simulations show that the synchronized flow of DTGBLM is in accordance with the empirical results, such as the WSP and GP founded by the three phase traffic flow theory. Deceleration analysis demonstrates that although there are unrealistic decelerations, its occurrence probability is very small and can be neglected. Based on FMCD and BLDAD, the Advance Randomization Brake Light Model (ARBLM) is proposed. The mechanisms to reproduce the synchronized flow and related traffic breakdown phenomenon of ARBLM are investigated. All results show that the criticisms of break light model that the three phase theory imposed are solved by DTGBLM and ARBLM.
3) Combining the classical car following model which considers of multiple leading vehicles with the NaSch model, the Average Space-Gaps cellular automaton Model (ASGM) within the fundamental diagram approach is proposed. Simulations show that the ASGM produces the same spatiotemporal dynamics as many of the more complex three-phase models. Besides many aspects that are consistent with traffic data. It seems that the absence of a unique fundamental diagram is not an essential requirement for this model class, at least with respect to the spatiotemporal dynamics. With the further consideration of the classical car following model which considers of multiple leading vehicles, the Extended ASGM (EASGM) is proposed, which can reproduce the synchronized outflow of wide moving jams. ASGM and EASGM not only bridge the cellular automaton models with car following models, but also invalidate the criticism that models within the fundamental diagram approach cannot reproduce the synchronized outflow of wide moving jams.
4) In order to verify the validity of the proposed models, some models under different background are calibrated and validated with the new models proposed in this paper. The three stage calibration method is explored. In the first stage, the parameters that can be determined by the default or empirical values are picked out. In the second stage, other parameters are calibrated by the traditional calibration methods. In the last stage, one should calibrate the parameters that influence the velocity directly, which are chosen from the parameters in the first stage by traditional calibration methods. Calibration results show that all proposed models in this paper are better than their original models. And models in the three phase traffic flow theory or in fundamental diagram approach has nothing to do with the calibration results.
1)在Nagatani格子流体力学模型基础上,提出了考虑前后格子密度差效应的格子模型(DDLM模型)。通过线性稳定性分析得到了DDLM模型的中性稳定性曲线,并发现考虑密度差效应可以提高交通流的稳定性。通过非线性稳定性分析得到了稳定区域、亚稳定区域和不稳定区域存在的以Burgers方程、KdV方程和mKdV方程描述的三角激波、孤立波和扭结波。随后提出了匝道系统的格子流体力学模型,并设计了随机型和确定型匝道两种数值模拟方法。随机型匝道的模拟结果和基本图方法的实测结果一致,确定型匝道的模拟结果发现了四种新的组合拥挤模式,发现DDLM模型的堵塞可以产生于同步流,因此三相交通流理论对基本图方法模型堵塞产生于自由流的批判不适用于DDLM模型。
2)在BL刹车灯模型的基础上提出了期望时间间距刹车灯模型(DTGBLM模型)。DTGBLM模型不仅可以产生符合实测的同步流以及三相交通流理论的扩展同步流模式和全面模式,而且过大减速度现象出现的概率可忽略不计,因此DTGBLM模型解决了三相交通流理论对BL模型的批判问题。其次在FMCD和BLDAD刹车灯模型的基础上构建了提前慢化刹车灯模型(ARBLM模型)。ARBLM模型需借助外界扰动才能形成同步流,这和实测通行能力陡降现象借助瓶颈产生的机理一致,因此ARBLM模型不仅可以解释通行能力陡降现象的产生机理,而且比三相交通流理论的FMCD模型更加符合实际。
3)从传统的多车期望效应跟驰模型思想出发,将多车平均间距作为车辆的期望速度引入随机慢化过程,提出了平均车辆间距元胞自动机模型(ASGM模型)。ASGM模型不仅可以模拟自由流到同步流以及同步流到宽运动堵塞的一阶相变,而且可以模拟三相交通流理论的各种实测拥挤模式。由于ASGM模型和大多数三相交通流理论模型的模拟结果没有本质区别,稳态拥挤交通流在流量密度平面上呈现二维分布不是模型符合实测的基本条件。深入分析多车期望效应跟驰模型,提出了扩展平均车辆间距元胞自动机模型(EASGM模型)。EASGM模型能够模拟宽运动堵塞的同步出流,并且证实了车辆期望时间间距、慢启动的随机慢化概率和安全控制参数都可影响宽运动堵塞的出流状态。ASGM模型和EASGM模型不仅建立了连续的跟驰模型和离散的元胞自动机模型之间的联系,而且解决了三相交通流理论批判基本图方法不能模拟宽运动堵塞同步出流的问题。
4)提出了三阶段标定法,并对上述新模型和几个不同类型、不同理论体系的模型进行了标定。三阶段标定法首先挑出可以根据缺省值和实测数据直接给定的参数,然后采用试差法或启发式算法标定模型的其他参数,最后从第一步中挑选出能够直接影响车速的参数,并再次采用试差法或启发式算法对其进行标定。标定结果表明新模型的标定误差均小于原模型;三相交通流理论模型和基本图方法模型之间误差相当,没有优劣之分。
Engineering practice proves that the first principle of relieving congestion is investigating and exploring the mechanisms of traffic flow. Traffic flow theory was developed under this background, whose main branches are the traditional fundamental diagram approach and the novel three phase traffic flow theory. Fundamental diagram approach assumes the unique relationship between flow and density, while three phase traffic flow theory denies this assumption. The controversies between the fundamental diagram approach and the three phase traffic flow theory are one of the hotspots in the traffic flow theory. This dissertation is based on these theories:
1) Based on the Nagatani lattice hydrodynamic model, the density difference lattice model (DDLM) is proposed. DDLM not only considered the leading lattice but also the following lattice. Linear stability curve are obtained by the linear stability analysis, which demonstrates that the density difference can enhance the stability. Burgers, KdV, mKdV equations and the coexistence curve are derived by the nonlinear stability analysis. Simulations show that the triangular shock wave, soliton and kink and anti-kink wave described by these equations are simulated. Then the lattice hydrodynamic model with the onramp system is proposed and the stochastic and deterministic onramp simulation methods are designed. Onramp simulations are in accordance with the empirical findings. Moreover four new kinds of congested patterns are founded, two of which are very similar to the GP and Dissolving GP in the three phase theory. Thus DDLM not only confirms the empirical findings of the fundamental diagram approach, but also put forward new findings which need to be consolidated and proves that wide moving jams may not emerges in the free flow in the models of fundamental diagram approach.
2) The synchronized flow of the brake light cellular automaton model (BLM) and the traffic breakdown of FMCD and BLDAD brake light cellular automaton models are not in accordance with the real traffic. Based on BLM, the Desired Time Gap Brake Light Model (DTGBLM) is established by incorporating the desired time gap. Simulations show that the synchronized flow of DTGBLM is in accordance with the empirical results, such as the WSP and GP founded by the three phase traffic flow theory. Deceleration analysis demonstrates that although there are unrealistic decelerations, its occurrence probability is very small and can be neglected. Based on FMCD and BLDAD, the Advance Randomization Brake Light Model (ARBLM) is proposed. The mechanisms to reproduce the synchronized flow and related traffic breakdown phenomenon of ARBLM are investigated. All results show that the criticisms of break light model that the three phase theory imposed are solved by DTGBLM and ARBLM.
3) Combining the classical car following model which considers of multiple leading vehicles with the NaSch model, the Average Space-Gaps cellular automaton Model (ASGM) within the fundamental diagram approach is proposed. Simulations show that the ASGM produces the same spatiotemporal dynamics as many of the more complex three-phase models. Besides many aspects that are consistent with traffic data. It seems that the absence of a unique fundamental diagram is not an essential requirement for this model class, at least with respect to the spatiotemporal dynamics. With the further consideration of the classical car following model which considers of multiple leading vehicles, the Extended ASGM (EASGM) is proposed, which can reproduce the synchronized outflow of wide moving jams. ASGM and EASGM not only bridge the cellular automaton models with car following models, but also invalidate the criticism that models within the fundamental diagram approach cannot reproduce the synchronized outflow of wide moving jams.
4) In order to verify the validity of the proposed models, some models under different background are calibrated and validated with the new models proposed in this paper. The three stage calibration method is explored. In the first stage, the parameters that can be determined by the default or empirical values are picked out. In the second stage, other parameters are calibrated by the traditional calibration methods. In the last stage, one should calibrate the parameters that influence the velocity directly, which are chosen from the parameters in the first stage by traditional calibration methods. Calibration results show that all proposed models in this paper are better than their original models. And models in the three phase traffic flow theory or in fundamental diagram approach has nothing to do with the calibration results.
引文
[1]翟忠民.道路交通组织优化[M].人民交通出版社,2004.
[2]梅超群.冲突区域的交通流建模与非线性特征分析[D].北京航空航天大学,2009.
[3]戴世强,冯苏苇,顾国庆.交通流动力学：它的内容,方法和意义[J].自然杂志,1997,19(4)：196-201.
[4]Chowdhury D, Santen L, Schadschneider A. Statistical physics of vehicular traffic and some related systems[J]. Physics Reports,2000,329(4):199-329.
[5]Nagel K, Rickert M. Parallel implementation of the TRANSIMS micro-simulation[J]. Parallel Computing,2001,27(12):1611-1639.
[6]Helbing D, Hennecke A, Shvetsov V, et al. MASTER:macroscopic traffic simulation based on a gas-kinetic, non-local traffic model[J]. Transportation Research Part B:Methodological, 2001,35(2):183-211.
[7]Kerner B S. The physics of traffic:empirical freeway pattern features, engineering applications, and theory[M]. Springer,2004.
[8]http://www.matsim.org/.MATISM.
[9]Treiber M, Kesting A, Helbing D. Three-phase traffic theory and two-phase models with a fundamental diagram in the light of empirical stylized facts[J]. Transportation Research Part B:Methodological,2010,44(8):983-1000.
[10]Schonhof M, Helbing D. Criticism of three-phase traffic theory[J]. Transportation Research Part B:Methodological,2009,43(7):784-797.
[11]Schadschneider A, Chowdhury D, Nishinari K. Stochastic transport in complex systems: From molecules to vehicles[M]. Access Online via Elsevier,2010..
[12]Eubank S, Guclu H, Kumar V S A, et al. Modelling disease outbreaks in realistic urban social networks[J]. Nature,2004,429(6988):180-184.
[13]Greenshields B D, Bibbins J R, Channing W S, et al. A study of traffic capacity[C]. Highway Research Board Proceedings.1935.
[14]Lighthill M J, Whitham G B. On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads[J]. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences,1955,229(1178):317-345.
[15]Pipes L A. An operational analysis of traffic dynamics[J]. Journal of applied physics,1953, 24(3):274-281.
[16]Prigogine I, Herman R. Kinetic theory of vehicular traffic[R].1971.
[17]Kesting A. Traffic Flow Dynamics[M]. Springer,2013.
[18]Kesting A. Microscopic modeling of human and automated driving:Towards traffic-adaptive cruise control[M]. VDM Publishing,2008.
[19]Schonhof M, Helbing D. Empirical features of congested traffic states and their implications for Traffic Modeling[J]. Transportation Science,2007,41(2):135-166.
[20]Helbing D, Treiber M, Kesting A, et al. Theoretical vs. empirical classification and prediction of congested traffic states[J]. The European Physical Journal B,2009,69(4):583-598.
[21]Kerner B S, Konhauser P. Cluster effect in initially homogeneous traffic flow[J]. Physical Review E,1993,48(4):R2335.
[22]Bando M, Hasebe K, Nakayama A, et al. Dynamical model of traffic congestion and numerical simulation[J]. Physical Review E,1995,51:1035-1042.
[23]Helbing D, Hennecke A, Treiber M. Phase diagram of traffic states in the presence of inhomogeneities[J]. Physical Review Letters,1999,82(21):4360.
[24]Lee H Y, Lee H W, Kim D. Dynamic states of a continuum traffic equation with on-ramp[J]. Physical Review E,1999,59(5):5101.
[25]Treiber M, Hennecke A, Helbing D. Congested traffic states in empirical observations and microscopic simulations[J]. Physical Review E,2000,62(2):1805.
[26]Gupta A K, Katiyar V K. Phase transition of traffic states with on-ramp[J]. Physica A: Statistical Mechanics and its Applications,2006,371(2):674-682.
[27]Lee H Y, Lee H W, Kim D. Phase diagram of congested traffic flow:An empirical study[J]. Physical Review E,2000,62(4):4737.
[28]Richards P I. Shock waves on the highway[J]. Operations research,1956,4(1):42-51.
[29]Payne H J. Models of freeway traffic and control [J]. Mathematical models of public systems,1971,1(1):51-61.
[30]Payne H J. FREFLO:A macroscopic simulation model of freeway traffic[J]. Transportation Research Record,1979 (722):68-77.
[31]Ross P. Traffic dynamics[J]. Transportation Research Part B:Methodological,1988, 22(6):421-435.
[32]Papageorgiou M. A hierarchical control system for freeway traffic[J]. Transportation Research Part B:Methodological,1983,17(3):251-261.
[33]Michalopoulos P G, Yi P, Lyrintzis A S. Continuum modelling of traffic dynamics for congested freeways[J]. Transportation Research Part B:Methodological,1993,27(4): 315-332.
[34]Kuhne R D. Macroscopic freeway model for dense traffic-stop-start waves and incident detection[C], Proceedings of the 9th International Symposium on Transportation and Traffic Theory (ISTTT9).1984:21-42.
[35]Kuhne R. Freeway control and incident detection using a stochastic continuum theory of traffic flow[C], International Conference on Applications of Advanced Technologies in Transportation Engineering (1st:1989:San Diego, Calif.).1989.287-292.
[36]Zhang H M. A theory of nonequilibrium traffic flow[J]. Transportation Research Part B: Methodological,1998,32(7):485-498.
[37]Berg P, Woods A. Traveling waves in an optimal velocity model of freeway traffic[J]. Physical Review E,2001,63(3):036107.
[38]Daganzo C F. Requiem for second-order fluid approximations of traffic flow[J]. Transportation Research Part B:Methodological,1995,29(4):277-286.
[39]Aw A, Rascle M. Resurrection of second order" models of traffic flow[J]. SIAM journal on applied mathematics,2000,60(3):916-938.
[40]Zhang H M. A non-equilibrium traffic model devoid of gas-like behavior[J]. Transportation Research Part B:Methodological,2002,36(3):275-290.
[41]Jiang R, Wu Q S, Zhu Z J. A new continuum model for traffic flow and numerical tests[J]. Transportation Research Part B:Methodological,2002,36(5):405-419.
[42]Xue Y, Dai S Q. Continuum traffic model with the consideration of two delay time scales[J]. Physical Review E,2003,68(6):066123.
[43]Zhang P, Wong S C, Dai S Q. A conserved higher-order anisotropic traffic flow model: description of equilibrium and non-equilibrium flows[J]. Transportation Research Part B: Methodological,2009,43(5):562-574.
[44]Helbing D, Hennecke A, Shvetsov V, et al. Micro-and macro-simulation of freeway traffic[J]. Mathematical and computer modelling,2002,35(5):517-547.
[45]Daganzo C F. The cell transmission model:A dynamic representation of highway traffic consistent with the hydrodynamic theory [J]. Transportation Research Part B: Methodological,1994,28(4):269-287.
[46]Daganzo C F. The cell transmission model, part Ⅱ:network traffic[J]. Transportation Research Part B:Methodological,1995,29(2):79-93.
[47]Reuschel A. Fahrzeugbewegungen in der Kolonne bei gleichformig beschleunigtem oder verzogertem, Leitfahrzeug, Zeitschrift des Oesterreichischen Ingenieurund Architekten-Vereines,1950,95(9):59-62.
[48]Chandler R E, Herman R, Montroll E W. Traffic dynamics:studies in car following[J]. Operations Research,1958,6(2):165-184.
[49]Gazis D C, Herman R, Potts R B. Car-following theory of steady-state traffic flow[J]. Operations Research,1959,7(4):499-505.
[50]Greenberg H. An analysis of traffic flow[J]. Operations Research,1959,7(1):79-85.
[51]Gazis D C, Herman R, Rothery R W. Nonlinear follow-the-leader models of traffic flow[J]. Operations Research,1961,9(4):545-567.
[52]Newell G F. Nonlinear effects in the dynamics of car following[J]. Operations Research, 1961,9(2):209-229.
[53]Helbing D, Tilch B. Generalized force model of traffic dynamics[J]. Physical Review E, 1998,58(1):133.
[54]Jiang R, Wu Q, Zhu Z. Full velocity difference model for a car-following theory[J]. Physical Review E,2001,64(1):017101.
[55]Lenz H, Wagner C K, Sollacher R. Multi-anticipative car-following model[J]. The European Physical Journal B-Condensed Matter and Complex Systems,1999,7(2): 331-335.
[56]Nagatani T. Stabilization and enhancement of traffic flow by the next-nearest-neighbor interaction^]. Physical Review E,1999,60(6):6395.
[57]Nakayama A, Sugiyama Y, Hasebe K. Effect of looking at the car that follows in an optimal velocity model of traffic flow[J]. Physical Review E, Statistical, nonlinear, and soft matter physics,2002,65(1 Pt 2):016112-016112.
[58]Ge H X, Dai S Q, Dong L Y, et al. Stabilization effect of traffic flow in an extended car-following model based on an intelligent transportation system application[J]. Physical Review E,2004,70(6):066134.
[59]Li Z P, Liu Y C. Analysis of stability and density waves of traffic flow model in an ITS environment[J]. The European Physical Journal B-Condensed Matter and Complex Systems,2006,53(3):367-374.
[60]Tomer E, Safonov L, Havlin S. Presence of many stable nonhomogeneous states in an inertial car-following model[J]. Physical Review Letters,2000,84(2):382-385.
[61]Helbing D, Batic D, Schonhof M, et al. Modelling widely scattered states in 'synchronized'traffic flow and possible relevance for stock market dynamics[J]. Physica A:Statistical Mechanics and its Applications,2002,303(1):251-260.
[62]薛郁.随机计及相对速度的交通流跟驰模型[J].物理学报,2003,52(11)：2750-2756.
[63]Bonsall P, Liu R, Young W. Modelling safety-related driving behaviour-impact of parameter values[J]. Transportation research part A:policy and practice,2005,39(5): 425-444.
[64]Gipps P G A behavioural car-following model for computer simulation[J]. Transportation Research Part B:Methodological,1981,15(2):105-111.
[65]Nagel K, Schreckenberg M. A cellular automaton model for freeway traffic[J]. Journal de Physique I,1992,2(12):2221-2229.
[66]Takayasu M, Takayasu H.1/f noise in a traffic model[J]. Fractals,1993,1(04):860-866.
[67]Benjamin S C, Johnson N F, Hui P M. Cellular automata models of traffic flow along a highway containing a junction[J]. Journal of Physics A:Mathematical and General,1996, 29(12):3119.
[68]Barlovic R, Santen L, Schadschneider A, et al. Metastable states in cellular automata for traffic flow[J]. The European Physical Journal B-Condensed Matter and Complex Systems,1998,5(3):793-800.
[69]Li X, Wu Q, Jiang R. Cellular automaton model considering the velocity effect of a car on the successive car[J]. Physical Review E,2001,64(6):066128.
[70]Fukui M, Ishibashi Y. Traffic flow in 1D cellular automaton model including cars moving with high speed[J]. Journal of the Physical Society of Japan,1996,65(6): 1868-1870.
[71]Helbing D, Schreckenberg M. Cellular automata simulating experimental properties of traffic flow[J]. Physical review E,1999,59(3):R2505-2508.
[72]Knospe W, Santen L, Schadschneider A, et al. Towards a realistic microscopic description of highway traffic[J]. Journal of Physics A:Mathematical and general,2000, 33(48):L477.
[73]邝华,孔令江,刘慕仁.多速混合车辆单车道元胞自动机交通流模型的研究[J].物理学报,2004,53(9)：2894-2898.
[74]董力耘,薛郁,戴世强.基于跟车思想的一维元胞自动机交通流模型[J].应用数学和力学,2002,23(4)：331-337.
[75]雷丽,薛郁,戴世强.交通流的一维元胞自动机敏感驾驶模刑[J].物理学报,2003,52(9)：2121-2126.
[76]Helbing D. Derivation of non-local macroscopic traffic equations and consistent traffic pressures from microscopic car-following models[M], Modelling and Optimisation of Flows on Networks. Springer Berlin Heidelberg,2013:247-269.
[77]陈建阳.交通流微观模型与宏观模型的统一[J].同济大学学报：自然科学版,1997, 25(1)：46-49.
[78]Helbing D. Traffic and related self-driven many-particle systems[J]. Reviews of modern physics,2001,73(4):1067.
[79]Kerner B S. Introduction to modern traffic flow theory and control:the long road to three-phase traffic theory[M]. Springer,2009.
[80]Kerner B S. Empirical macroscopic features of spatial-temporal traffic patterns at highway bottlenecks[J]. Physical Review E,2002,65(4):046138.
[81]关伟,何蜀燕,马继辉.交通流现象与模型评述[J].交通运输系统工程与信息,2012,12(3)：90-97.
[82]Kerner B S. Criticism of generally accepted fundamentals and methodologies of traffic and transportation theory:A brief review[J]. Physica A:Statistical Mechanics and its Applications,2013.
[83]Kerner B S, Klenov S L. A microscopic model for phase transitions in traffic flow[J]. Journal of Physics A:Mathematical and General,2002,35(3):L31.
[84]Kerner B S, Klenov S L, Wolf D E. Cellular automata approach to three-phase traffic theory[J]. Journal of Physics A:Mathematical and General,2002,35(47):9971.
[85]Kerner B S, Klenov S L, Schreckenberg M. Simple cellular automaton model for traffic breakdown, highway capacity, and synchronized flow[J]. Physical Review E,2011,84(4): 046110.
[86]Knospe W. Synchronized traffic:Microscopic modeling and empirical observations[D]. Universitat Duisburg-Essen, Fakultat fur Physik,2002.
[87]Jiang R, Wu Q S. Cellular automata models for synchronized traffic flow[J]. Journal of Physics A:Mathematical and General,2003,36(2):381.1.
[88]Jiang R, Wu Q S. Spatial-temporal patterns at an isolated on-ramp in a new cellular automata model based on three-phase traffic theory[J]. Journal of Physics A: Mathematical and General,2004,37(34):8197.
[89]Jiang R, Wu Q S. First order phase transition from free flow to synchronized flow in a cellular automata model[J]. The European Physical Journal B-Condensed Matter and Complex Systems,2005,46(4):581-584.
[90]Lee H Y, Lee H W, Kim D. Origin of synchronized traffic flow on highways and its dynamic phase transitions[J]. Physical Review Letters,1998,81(5):1130-1133.
[91]Gao K, Jiang R, Hu S X, et al. Cellular-automaton model with velocity adaptation in the framework of Kerner's three-phase traffic theory[J]. Physical Review E,2007,76(2): 026105.
[92]Gao K, Jiang R, Wang B H, et al. Discontinuous transition from free flow to synchronized flow induced by short-range interaction between vehicles in a three-phase traffic flow model[J]. Physica A:Statistical Mechanics and its Applications,2009, 388(15):3233-3243.
[93]Kerner B S, Klenov S L. Deterministic microscopic three-phase traffic flow models[J]. Journal of Physics A:mathematical and general,2006,39(8):1775.
[94]He S, Guan W, Song L. Explaining traffic patterns at on-ramp vicinity by a driver perception model in the framework of three-phase traffic theory[J]. Physica A:Statistical Mechanics and its Applications,2010,389(4):825-836.
[95]Zhao B H, Jiang R, Yuan Y M, et al. Anticipation effect explaining features of synchronized traffic flow[C]. Transportation Research Board 88th Annual Meeting.2009 (09-1802).
[96]Davis L C. Multilane simulations of traffic phases[J]. Physical Review E,2004,69(1): 016108.
[97]Yang H, Lu J, Hu X, et al. Cellular automaton model based on empirical driver's oscillation behavior reproducing the findings of Kerner's three-phase traffic theory[J]. Physica A:Statistical Mechanics and its Applications,2013,392(18):4009-4018.
[98]Jin C J, Wang W, Jiang R, et al. On the first-order phase transition in a cellular automaton traffic flow model without a slow-to-start effect[J]. Journal of Statistical Mechanics:Theory and Experiment,2010,2010(03):P03018.
[99]Knorr F. Applicability and Application of Microscopic Traffic Simulations [D]. Universitat Duisburg-Essen, Fakultat fur Physik》 Theoretische Physik,2013.
[100]Pottmeier A. Realistic Cellular Automaton Model for Synchronized Two-Lane Traffic-Simulation, Validation, and Applications[D]. Universitat Duisburg-Essen, Fakultat fur Physik Theoretische Physik,2008.
[101]Nagatani T. Modified KdV equation for jamming transition in the continuum models of traffic [J]. Physica A:Statistical Mechanics and its Applications,1998,261(3):599-607.
[102]Nagatani T. Jamming transition in a two-dimensional traffic flow model[J]. Physical Review E,1999,59(5):4857.
[103]Nagatani T. Jamming transition of high-dimensional traffic dynamics[J]. Physica A: Statistical Mechanics and its Applications,1999,272(3):592-611.
[104]Nagatani T. Jamming transition in traffic flow on triangular lattice[J]. Physica A: Statistical Mechanics and its Applications,1999,271(1):200-221.
[105]Nagatani T. Jamming transitions and the modified Korteweg-de Vries equation in a two-lane traffic flow[J]. Physica A:Statistical Mechanics and its Applications,1999, 265(1):297-310.
[106]Ge H X, Dai S Q, Xue Y, et al. Stabilization analysis and modified Korteweg-de Vries equation in a cooperative driving system[J]. Physical Review E,2005,71(6):066119.
[107]Zhu W X, Chi E X. Analysis of generalized optimal current lattice model for traffic flow[J]. International Journal of Modern Physics C,2008,19(05):727-739.
[108]Di-Hua S, Chuan T, Wei-Ning L. A traffic flow lattice model considering relative current influence and its numerical simulation[J]. Chinese Physics B,2010,19(8):080514.
[109]Ge H X, Cheng R J. The "backward looking" effect in the lattice hydrodynamic model[J]. Physica A:Statistical Mechanics and its Applications,2008,387(28):6952-6958.
[110]Li X, Li Z, Han X, et al. Effect of the optimal velocity function on traffic phase transitions in lattice hydrodynamic models[J]. Communications in Nonlinear Science and Numerical Simulation,2009,14(5):2171-2177.
[111]Zhu H B Lattice models of traffic flow considering drivers'delay in response[J]. Chinese Physics B,2009,18(4):1322.
[112]Peng G H, Cai X H, Liu C Q, et al. A new lattice model of traffic flow with the consideration of the driver's forecast effects[J]. Physics Letters A,2011,375(22): 2153-2157.
[113]Peng G, Nie F, Cao B, et al. A driver's memory lattice model of traffic flow and its numerical simulation[J]. Nonlinear Dynamics,2012,67(3):1811-1815.
[114]Peng G H, Cai X H, Liu C Q, et al. A new lattice model of traffic flow with the anticipation effect of potential lane changing[J]. Physics Letters A,2012,376(4): 447-451.
[115]Xue Y, Tian H H, He H D, et al. Exploring jamming transitions and density waves in bidirectional pedestrian traffic[J]. The European Physical Journal B,2009,69(2): 289-295.
[116]Tian H, He H, Wei Y, et al. Lattice hydrodynamic model with bidirectional pedestrian flow[J]. Physica A:Statistical Mechanics and its Applications,2009,388(14): 2895-2902.
[117]Tatsumi T, Kida S. Statistical mechanics of the Burgers model of turbulence[J]. Journal of Fluid Mechanics,1972,55:659-675.
[118]Nagatani T. Jamming transitions and the modified Korteweg-de Vries equation in a two-lane traffic flow[J]. Physica A:Statistical Mechanics and its Applications,1999, 265(1):297-310.
[119]Tang C F, Jiang R, Wu Q S. Phase diagram of speed gradient model with an on-ramp[J]. Physica A:Statistical Mechanics and its Applications,2007,377(2):641-650.
[120]Meng Q, Weng J. An improved cellular automata model for heterogeneous work zone traffic[J]. Transportation research part C:Emerging Technologies,2011,19(6): 1263-1275.
[121]Appert-Rolland C, Du Boisberranger J. Macroscopic relaxation after on-ramps in real data and in cellular automata simulations[J]. Transportation Research Part C:Emerging Technologies,2011,34(0):162-175.
[122]Knorr F, Schreckenberg M. Influence of inter-vehicle communication on peak hour traffic flow[J]. Physica A:Statistical Mechanics and its Applications,2012,391(6): 2225-2231.
[123]Li X G, Gao Z Y, Li K P, et al. Relationship between microscopic dynamics in traffic flow and complexity in networks[J]. Physical Review E,2007,76(1):016110.
[124]Yuan Y M, Jiang R, Wu Q S, et al. Synchronized flow and phase separation in a two-lane cellular automaton Model[J]. International Journal of Modern Physics C,2007,18(02): 267-279.
[125]Yuan Y M, Jiang R, Wu Q S, et al. Traffic behavior in a two-lane system consisting of a mixture of buses and cars[J]. International Journal of Modern Physics C,2007,18(12): 1925-1938.
[126]Zheng L, Ma S, Zhong S. Analysis of honk effect on the traffic flow in a cellular automaton model[J]. Physica A:Statistical Mechanics and its Applications,2011,390(6): 1072-1084.
[127]Wang R, Jiang R, Wu Q S, et al. Synchronized flow and phase separations in single-lane mixed traffic flow[J]. Physica A:Statistical Mechanics and its Applications,2007,378(2): 475-484.
[128]Xiang Z T, Li Y J, Chen Y F, et al. Simulating synchronized traffic flow and wide moving jam based on the brake light rule[J]. Physica A:Statistical Mechanics and its Applications,2013,392(21):5399-5413.
[129]Hafstein S F, Chrobok R, Pottmeier A, et al. Cellular automaton modeling of the autobahn traffic in north rhine-westphalia[J]. Proc. of the 4th MATHMOD,2003: 1322-1331.
[130]Nicklin R C. Kinematics of tailgating[J]. The Physics Teacher,1997,35:78-79.
[131]Bexelius S. An extended model for car-following[J]. Transportation Research,1968,2(1): 13-21.
[132]Sawada S. Generalized optimal velocity model for traffic flow[J]. International Journal of Modern Physics C,2002,13(01):1-12.
[133]Hasebe K, Nakayama A, Sugiyama Y. Dynamical model of a cooperative driving system for freeway traffic[J]. Physical Review E, Statistical, nonlinear, and soft matter physics, 2003,68(2 Pt 2):026102-026102.
[134]Neubert L, Santen L, Schadschneider A, et al. Single-vehicle data of highway traffic:A statistical analysis[J]. Physical Review E,1999,60(6):6480.
[135]Jia B, Li X G, Chen T, et al. Cellular automaton model with time gap dependent randomisation under Kerner's three-phase traffic theory[J]. Transportmetrica,2011,7(2): 127-140.
[136]Gardes Y, May A D, Dahlgren J, et al. Freeway calibration and application of the PARAMICS model[C],81st Annual Meeting of the Transportation Research Board, Washington, DC.2002.
[137]Cheu R L, Recker W W and Ritchie S G Calibration of INTRAS for Simulation of 30-Sec Loop Detector Output, Transportation Research Record,1994,1457:108-116
[138]Cheu R L, Jin X, Ng K C, et al. Genetic algorithm approach in FRESIM calibration[C], Fifth International Conference on Applications of Advanced Technologies in Transportation Engineering.1998.
[139]Ma J, Dong H, Zhang H M. Calibration of microsimulation with heuristic optimization methods[J]. Transportation Research Record:Journal of the Transportation Research Board,2007,1999(1):208-217.
[140]Ben-Akiva M E, Davol A, Toledo T, et al. Calibration and evaluation of MITSIMLab in Stockholm[C] 82nd Annual Meeting Preprint CDROM, Transportation Research Board. 2002.
[141]Ngoduy D, Maher M J. Calibration of second order traffic models using continuous cross entropy method[J]. Transportation Research Part C:Emerging Technologies,2012,24: 102-121.
[142]Ossen S, Hoogendoorn S P, Gorte B G H. Interdriver differences in car-following:a vehicle trajectory-based study[J]. Transportation Research Record:Journal of the Transportation Research Board,2006,1965(1):121-129.
[143]Ossen S, Hoogendoorn S P. Car-following behavior analysis from microscopic trajectory data[J]. Transportation Research Record:Journal of the Transportation Research Board, 2005,1934(1):13-21.
[144]Toledo T, Koutsopoulos H N, Ahmed K I. Estimation of vehicle trajectories with locally weighted regression[J]. Transportation Research Record:Journal of the Transportation Research Board,2007,1999(1):161-169.
[145]Chundury S, Wolshon B. Evaluation of CORSIM car-following model by using Global Positioning System field data[J]. Transportation Research Record:Journal of the Transportation Research Board,2000,1710(1):114-121.
[146]Treiber M, Kesting A. Microscopic Calibration and validation of car-following models-a systematic approach[J]. Procedia-Social and Behavioral Sciences,2013,80:922-939.
[147]Brockfeld E, Kuhne R D, Wagner P. Calibration and validation of microscopic models of traffic flow[J]. Transportation Research Record:Journal of the Transportation Research Board,2005,1934(1):179-187.
[148]Wagner P. Fluid-dynamical and microscopic description of traffic flow:a data-driven comparison[J]. Philosophical Transactions of the Royal Society A:Mathematical, Physical and Engineering Sciences,2010,368(1928):4481-4495.
[149]Treiber M, Kesting A, Thiemann C. How much does traffic congestion increase fuel consumption and emissions? Applying a fuel consumption model to the NGSIM trajectory data[C] 87th Annual Meeting of the Transportation Research Board, Washington, DC.2008.
[150]Thiemann C, Treiber M, Kesting A. Estimating acceleration and lane-changing dynamics from next generation simulation trajectory data[J]. Transportation Research Record: Journal of the Transportation Research Board,2008,2088(1):90-101.
[151]Baumann M. A neural network model for driver's lane-changing trajectory prediction in urban traffic flow[J]. Mathematical Problems in Engineering,2013,2013.
[152]Toledo T, Zohar D. Modeling duration of lane changes[J]. Transportation Research Record:Journal of the Transportation Research Board,2007,1999(1):71-78.
[153]Castillo J M D, Benitez F G. On the functional form of the speed-density relationship—Ⅰ: general theory[J]. Transportation Research Part B:Methodological,1995,29(5): 373-389.
[154]Castillo J M D, Benitez F G. On the functional form of the speed-density relationship—Ⅱ: empirical investigation[J]. Transportation Research Part B:Methodological,1995,29(5): 391-406.
[155]姜锐,吴清松.交通流复杂动态特性的微观和宏观模式研究[D].中国科学院研究生院学报,2002.
[156]Lakouari N, Jetto K, Ez-Zahraouy H, et al. VELOCITY CORRELATIONS IN THE NAGEL-SCHRECKENBERG MODEL[J]. International Journal of Modern Physics C, 2013.
[2]梅超群.冲突区域的交通流建模与非线性特征分析[D].北京航空航天大学,2009.
[3]戴世强,冯苏苇,顾国庆.交通流动力学：它的内容,方法和意义[J].自然杂志,1997,19(4)：196-201.
[4]Chowdhury D, Santen L, Schadschneider A. Statistical physics of vehicular traffic and some related systems[J]. Physics Reports,2000,329(4):199-329.
[5]Nagel K, Rickert M. Parallel implementation of the TRANSIMS micro-simulation[J]. Parallel Computing,2001,27(12):1611-1639.
[6]Helbing D, Hennecke A, Shvetsov V, et al. MASTER:macroscopic traffic simulation based on a gas-kinetic, non-local traffic model[J]. Transportation Research Part B:Methodological, 2001,35(2):183-211.
[7]Kerner B S. The physics of traffic:empirical freeway pattern features, engineering applications, and theory[M]. Springer,2004.
[8]http://www.matsim.org/.MATISM.
[9]Treiber M, Kesting A, Helbing D. Three-phase traffic theory and two-phase models with a fundamental diagram in the light of empirical stylized facts[J]. Transportation Research Part B:Methodological,2010,44(8):983-1000.
[10]Schonhof M, Helbing D. Criticism of three-phase traffic theory[J]. Transportation Research Part B:Methodological,2009,43(7):784-797.
[11]Schadschneider A, Chowdhury D, Nishinari K. Stochastic transport in complex systems: From molecules to vehicles[M]. Access Online via Elsevier,2010..
[12]Eubank S, Guclu H, Kumar V S A, et al. Modelling disease outbreaks in realistic urban social networks[J]. Nature,2004,429(6988):180-184.
[13]Greenshields B D, Bibbins J R, Channing W S, et al. A study of traffic capacity[C]. Highway Research Board Proceedings.1935.
[14]Lighthill M J, Whitham G B. On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads[J]. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences,1955,229(1178):317-345.
[15]Pipes L A. An operational analysis of traffic dynamics[J]. Journal of applied physics,1953, 24(3):274-281.
[16]Prigogine I, Herman R. Kinetic theory of vehicular traffic[R].1971.
[17]Kesting A. Traffic Flow Dynamics[M]. Springer,2013.
[18]Kesting A. Microscopic modeling of human and automated driving:Towards traffic-adaptive cruise control[M]. VDM Publishing,2008.
[19]Schonhof M, Helbing D. Empirical features of congested traffic states and their implications for Traffic Modeling[J]. Transportation Science,2007,41(2):135-166.
[20]Helbing D, Treiber M, Kesting A, et al. Theoretical vs. empirical classification and prediction of congested traffic states[J]. The European Physical Journal B,2009,69(4):583-598.
[21]Kerner B S, Konhauser P. Cluster effect in initially homogeneous traffic flow[J]. Physical Review E,1993,48(4):R2335.
[22]Bando M, Hasebe K, Nakayama A, et al. Dynamical model of traffic congestion and numerical simulation[J]. Physical Review E,1995,51:1035-1042.
[23]Helbing D, Hennecke A, Treiber M. Phase diagram of traffic states in the presence of inhomogeneities[J]. Physical Review Letters,1999,82(21):4360.
[24]Lee H Y, Lee H W, Kim D. Dynamic states of a continuum traffic equation with on-ramp[J]. Physical Review E,1999,59(5):5101.
[25]Treiber M, Hennecke A, Helbing D. Congested traffic states in empirical observations and microscopic simulations[J]. Physical Review E,2000,62(2):1805.
[26]Gupta A K, Katiyar V K. Phase transition of traffic states with on-ramp[J]. Physica A: Statistical Mechanics and its Applications,2006,371(2):674-682.
[27]Lee H Y, Lee H W, Kim D. Phase diagram of congested traffic flow:An empirical study[J]. Physical Review E,2000,62(4):4737.
[28]Richards P I. Shock waves on the highway[J]. Operations research,1956,4(1):42-51.
[29]Payne H J. Models of freeway traffic and control [J]. Mathematical models of public systems,1971,1(1):51-61.
[30]Payne H J. FREFLO:A macroscopic simulation model of freeway traffic[J]. Transportation Research Record,1979 (722):68-77.
[31]Ross P. Traffic dynamics[J]. Transportation Research Part B:Methodological,1988, 22(6):421-435.
[32]Papageorgiou M. A hierarchical control system for freeway traffic[J]. Transportation Research Part B:Methodological,1983,17(3):251-261.
[33]Michalopoulos P G, Yi P, Lyrintzis A S. Continuum modelling of traffic dynamics for congested freeways[J]. Transportation Research Part B:Methodological,1993,27(4): 315-332.
[34]Kuhne R D. Macroscopic freeway model for dense traffic-stop-start waves and incident detection[C], Proceedings of the 9th International Symposium on Transportation and Traffic Theory (ISTTT9).1984:21-42.
[35]Kuhne R. Freeway control and incident detection using a stochastic continuum theory of traffic flow[C], International Conference on Applications of Advanced Technologies in Transportation Engineering (1st:1989:San Diego, Calif.).1989.287-292.
[36]Zhang H M. A theory of nonequilibrium traffic flow[J]. Transportation Research Part B: Methodological,1998,32(7):485-498.
[37]Berg P, Woods A. Traveling waves in an optimal velocity model of freeway traffic[J]. Physical Review E,2001,63(3):036107.
[38]Daganzo C F. Requiem for second-order fluid approximations of traffic flow[J]. Transportation Research Part B:Methodological,1995,29(4):277-286.
[39]Aw A, Rascle M. Resurrection of second order" models of traffic flow[J]. SIAM journal on applied mathematics,2000,60(3):916-938.
[40]Zhang H M. A non-equilibrium traffic model devoid of gas-like behavior[J]. Transportation Research Part B:Methodological,2002,36(3):275-290.
[41]Jiang R, Wu Q S, Zhu Z J. A new continuum model for traffic flow and numerical tests[J]. Transportation Research Part B:Methodological,2002,36(5):405-419.
[42]Xue Y, Dai S Q. Continuum traffic model with the consideration of two delay time scales[J]. Physical Review E,2003,68(6):066123.
[43]Zhang P, Wong S C, Dai S Q. A conserved higher-order anisotropic traffic flow model: description of equilibrium and non-equilibrium flows[J]. Transportation Research Part B: Methodological,2009,43(5):562-574.
[44]Helbing D, Hennecke A, Shvetsov V, et al. Micro-and macro-simulation of freeway traffic[J]. Mathematical and computer modelling,2002,35(5):517-547.
[45]Daganzo C F. The cell transmission model:A dynamic representation of highway traffic consistent with the hydrodynamic theory [J]. Transportation Research Part B: Methodological,1994,28(4):269-287.
[46]Daganzo C F. The cell transmission model, part Ⅱ:network traffic[J]. Transportation Research Part B:Methodological,1995,29(2):79-93.
[47]Reuschel A. Fahrzeugbewegungen in der Kolonne bei gleichformig beschleunigtem oder verzogertem, Leitfahrzeug, Zeitschrift des Oesterreichischen Ingenieurund Architekten-Vereines,1950,95(9):59-62.
[48]Chandler R E, Herman R, Montroll E W. Traffic dynamics:studies in car following[J]. Operations Research,1958,6(2):165-184.
[49]Gazis D C, Herman R, Potts R B. Car-following theory of steady-state traffic flow[J]. Operations Research,1959,7(4):499-505.
[50]Greenberg H. An analysis of traffic flow[J]. Operations Research,1959,7(1):79-85.
[51]Gazis D C, Herman R, Rothery R W. Nonlinear follow-the-leader models of traffic flow[J]. Operations Research,1961,9(4):545-567.
[52]Newell G F. Nonlinear effects in the dynamics of car following[J]. Operations Research, 1961,9(2):209-229.
[53]Helbing D, Tilch B. Generalized force model of traffic dynamics[J]. Physical Review E, 1998,58(1):133.
[54]Jiang R, Wu Q, Zhu Z. Full velocity difference model for a car-following theory[J]. Physical Review E,2001,64(1):017101.
[55]Lenz H, Wagner C K, Sollacher R. Multi-anticipative car-following model[J]. The European Physical Journal B-Condensed Matter and Complex Systems,1999,7(2): 331-335.
[56]Nagatani T. Stabilization and enhancement of traffic flow by the next-nearest-neighbor interaction^]. Physical Review E,1999,60(6):6395.
[57]Nakayama A, Sugiyama Y, Hasebe K. Effect of looking at the car that follows in an optimal velocity model of traffic flow[J]. Physical Review E, Statistical, nonlinear, and soft matter physics,2002,65(1 Pt 2):016112-016112.
[58]Ge H X, Dai S Q, Dong L Y, et al. Stabilization effect of traffic flow in an extended car-following model based on an intelligent transportation system application[J]. Physical Review E,2004,70(6):066134.
[59]Li Z P, Liu Y C. Analysis of stability and density waves of traffic flow model in an ITS environment[J]. The European Physical Journal B-Condensed Matter and Complex Systems,2006,53(3):367-374.
[60]Tomer E, Safonov L, Havlin S. Presence of many stable nonhomogeneous states in an inertial car-following model[J]. Physical Review Letters,2000,84(2):382-385.
[61]Helbing D, Batic D, Schonhof M, et al. Modelling widely scattered states in 'synchronized'traffic flow and possible relevance for stock market dynamics[J]. Physica A:Statistical Mechanics and its Applications,2002,303(1):251-260.
[62]薛郁.随机计及相对速度的交通流跟驰模型[J].物理学报,2003,52(11)：2750-2756.
[63]Bonsall P, Liu R, Young W. Modelling safety-related driving behaviour-impact of parameter values[J]. Transportation research part A:policy and practice,2005,39(5): 425-444.
[64]Gipps P G A behavioural car-following model for computer simulation[J]. Transportation Research Part B:Methodological,1981,15(2):105-111.
[65]Nagel K, Schreckenberg M. A cellular automaton model for freeway traffic[J]. Journal de Physique I,1992,2(12):2221-2229.
[66]Takayasu M, Takayasu H.1/f noise in a traffic model[J]. Fractals,1993,1(04):860-866.
[67]Benjamin S C, Johnson N F, Hui P M. Cellular automata models of traffic flow along a highway containing a junction[J]. Journal of Physics A:Mathematical and General,1996, 29(12):3119.
[68]Barlovic R, Santen L, Schadschneider A, et al. Metastable states in cellular automata for traffic flow[J]. The European Physical Journal B-Condensed Matter and Complex Systems,1998,5(3):793-800.
[69]Li X, Wu Q, Jiang R. Cellular automaton model considering the velocity effect of a car on the successive car[J]. Physical Review E,2001,64(6):066128.
[70]Fukui M, Ishibashi Y. Traffic flow in 1D cellular automaton model including cars moving with high speed[J]. Journal of the Physical Society of Japan,1996,65(6): 1868-1870.
[71]Helbing D, Schreckenberg M. Cellular automata simulating experimental properties of traffic flow[J]. Physical review E,1999,59(3):R2505-2508.
[72]Knospe W, Santen L, Schadschneider A, et al. Towards a realistic microscopic description of highway traffic[J]. Journal of Physics A:Mathematical and general,2000, 33(48):L477.
[73]邝华,孔令江,刘慕仁.多速混合车辆单车道元胞自动机交通流模型的研究[J].物理学报,2004,53(9)：2894-2898.
[74]董力耘,薛郁,戴世强.基于跟车思想的一维元胞自动机交通流模型[J].应用数学和力学,2002,23(4)：331-337.
[75]雷丽,薛郁,戴世强.交通流的一维元胞自动机敏感驾驶模刑[J].物理学报,2003,52(9)：2121-2126.
[76]Helbing D. Derivation of non-local macroscopic traffic equations and consistent traffic pressures from microscopic car-following models[M], Modelling and Optimisation of Flows on Networks. Springer Berlin Heidelberg,2013:247-269.
[77]陈建阳.交通流微观模型与宏观模型的统一[J].同济大学学报：自然科学版,1997, 25(1)：46-49.
[78]Helbing D. Traffic and related self-driven many-particle systems[J]. Reviews of modern physics,2001,73(4):1067.
[79]Kerner B S. Introduction to modern traffic flow theory and control:the long road to three-phase traffic theory[M]. Springer,2009.
[80]Kerner B S. Empirical macroscopic features of spatial-temporal traffic patterns at highway bottlenecks[J]. Physical Review E,2002,65(4):046138.
[81]关伟,何蜀燕,马继辉.交通流现象与模型评述[J].交通运输系统工程与信息,2012,12(3)：90-97.
[82]Kerner B S. Criticism of generally accepted fundamentals and methodologies of traffic and transportation theory:A brief review[J]. Physica A:Statistical Mechanics and its Applications,2013.
[83]Kerner B S, Klenov S L. A microscopic model for phase transitions in traffic flow[J]. Journal of Physics A:Mathematical and General,2002,35(3):L31.
[84]Kerner B S, Klenov S L, Wolf D E. Cellular automata approach to three-phase traffic theory[J]. Journal of Physics A:Mathematical and General,2002,35(47):9971.
[85]Kerner B S, Klenov S L, Schreckenberg M. Simple cellular automaton model for traffic breakdown, highway capacity, and synchronized flow[J]. Physical Review E,2011,84(4): 046110.
[86]Knospe W. Synchronized traffic:Microscopic modeling and empirical observations[D]. Universitat Duisburg-Essen, Fakultat fur Physik,2002.
[87]Jiang R, Wu Q S. Cellular automata models for synchronized traffic flow[J]. Journal of Physics A:Mathematical and General,2003,36(2):381.1.
[88]Jiang R, Wu Q S. Spatial-temporal patterns at an isolated on-ramp in a new cellular automata model based on three-phase traffic theory[J]. Journal of Physics A: Mathematical and General,2004,37(34):8197.
[89]Jiang R, Wu Q S. First order phase transition from free flow to synchronized flow in a cellular automata model[J]. The European Physical Journal B-Condensed Matter and Complex Systems,2005,46(4):581-584.
[90]Lee H Y, Lee H W, Kim D. Origin of synchronized traffic flow on highways and its dynamic phase transitions[J]. Physical Review Letters,1998,81(5):1130-1133.
[91]Gao K, Jiang R, Hu S X, et al. Cellular-automaton model with velocity adaptation in the framework of Kerner's three-phase traffic theory[J]. Physical Review E,2007,76(2): 026105.
[92]Gao K, Jiang R, Wang B H, et al. Discontinuous transition from free flow to synchronized flow induced by short-range interaction between vehicles in a three-phase traffic flow model[J]. Physica A:Statistical Mechanics and its Applications,2009, 388(15):3233-3243.
[93]Kerner B S, Klenov S L. Deterministic microscopic three-phase traffic flow models[J]. Journal of Physics A:mathematical and general,2006,39(8):1775.
[94]He S, Guan W, Song L. Explaining traffic patterns at on-ramp vicinity by a driver perception model in the framework of three-phase traffic theory[J]. Physica A:Statistical Mechanics and its Applications,2010,389(4):825-836.
[95]Zhao B H, Jiang R, Yuan Y M, et al. Anticipation effect explaining features of synchronized traffic flow[C]. Transportation Research Board 88th Annual Meeting.2009 (09-1802).
[96]Davis L C. Multilane simulations of traffic phases[J]. Physical Review E,2004,69(1): 016108.
[97]Yang H, Lu J, Hu X, et al. Cellular automaton model based on empirical driver's oscillation behavior reproducing the findings of Kerner's three-phase traffic theory[J]. Physica A:Statistical Mechanics and its Applications,2013,392(18):4009-4018.
[98]Jin C J, Wang W, Jiang R, et al. On the first-order phase transition in a cellular automaton traffic flow model without a slow-to-start effect[J]. Journal of Statistical Mechanics:Theory and Experiment,2010,2010(03):P03018.
[99]Knorr F. Applicability and Application of Microscopic Traffic Simulations [D]. Universitat Duisburg-Essen, Fakultat fur Physik》 Theoretische Physik,2013.
[100]Pottmeier A. Realistic Cellular Automaton Model for Synchronized Two-Lane Traffic-Simulation, Validation, and Applications[D]. Universitat Duisburg-Essen, Fakultat fur Physik Theoretische Physik,2008.
[101]Nagatani T. Modified KdV equation for jamming transition in the continuum models of traffic [J]. Physica A:Statistical Mechanics and its Applications,1998,261(3):599-607.
[102]Nagatani T. Jamming transition in a two-dimensional traffic flow model[J]. Physical Review E,1999,59(5):4857.
[103]Nagatani T. Jamming transition of high-dimensional traffic dynamics[J]. Physica A: Statistical Mechanics and its Applications,1999,272(3):592-611.
[104]Nagatani T. Jamming transition in traffic flow on triangular lattice[J]. Physica A: Statistical Mechanics and its Applications,1999,271(1):200-221.
[105]Nagatani T. Jamming transitions and the modified Korteweg-de Vries equation in a two-lane traffic flow[J]. Physica A:Statistical Mechanics and its Applications,1999, 265(1):297-310.
[106]Ge H X, Dai S Q, Xue Y, et al. Stabilization analysis and modified Korteweg-de Vries equation in a cooperative driving system[J]. Physical Review E,2005,71(6):066119.
[107]Zhu W X, Chi E X. Analysis of generalized optimal current lattice model for traffic flow[J]. International Journal of Modern Physics C,2008,19(05):727-739.
[108]Di-Hua S, Chuan T, Wei-Ning L. A traffic flow lattice model considering relative current influence and its numerical simulation[J]. Chinese Physics B,2010,19(8):080514.
[109]Ge H X, Cheng R J. The "backward looking" effect in the lattice hydrodynamic model[J]. Physica A:Statistical Mechanics and its Applications,2008,387(28):6952-6958.
[110]Li X, Li Z, Han X, et al. Effect of the optimal velocity function on traffic phase transitions in lattice hydrodynamic models[J]. Communications in Nonlinear Science and Numerical Simulation,2009,14(5):2171-2177.
[111]Zhu H B Lattice models of traffic flow considering drivers'delay in response[J]. Chinese Physics B,2009,18(4):1322.
[112]Peng G H, Cai X H, Liu C Q, et al. A new lattice model of traffic flow with the consideration of the driver's forecast effects[J]. Physics Letters A,2011,375(22): 2153-2157.
[113]Peng G, Nie F, Cao B, et al. A driver's memory lattice model of traffic flow and its numerical simulation[J]. Nonlinear Dynamics,2012,67(3):1811-1815.
[114]Peng G H, Cai X H, Liu C Q, et al. A new lattice model of traffic flow with the anticipation effect of potential lane changing[J]. Physics Letters A,2012,376(4): 447-451.
[115]Xue Y, Tian H H, He H D, et al. Exploring jamming transitions and density waves in bidirectional pedestrian traffic[J]. The European Physical Journal B,2009,69(2): 289-295.
[116]Tian H, He H, Wei Y, et al. Lattice hydrodynamic model with bidirectional pedestrian flow[J]. Physica A:Statistical Mechanics and its Applications,2009,388(14): 2895-2902.
[117]Tatsumi T, Kida S. Statistical mechanics of the Burgers model of turbulence[J]. Journal of Fluid Mechanics,1972,55:659-675.
[118]Nagatani T. Jamming transitions and the modified Korteweg-de Vries equation in a two-lane traffic flow[J]. Physica A:Statistical Mechanics and its Applications,1999, 265(1):297-310.
[119]Tang C F, Jiang R, Wu Q S. Phase diagram of speed gradient model with an on-ramp[J]. Physica A:Statistical Mechanics and its Applications,2007,377(2):641-650.
[120]Meng Q, Weng J. An improved cellular automata model for heterogeneous work zone traffic[J]. Transportation research part C:Emerging Technologies,2011,19(6): 1263-1275.
[121]Appert-Rolland C, Du Boisberranger J. Macroscopic relaxation after on-ramps in real data and in cellular automata simulations[J]. Transportation Research Part C:Emerging Technologies,2011,34(0):162-175.
[122]Knorr F, Schreckenberg M. Influence of inter-vehicle communication on peak hour traffic flow[J]. Physica A:Statistical Mechanics and its Applications,2012,391(6): 2225-2231.
[123]Li X G, Gao Z Y, Li K P, et al. Relationship between microscopic dynamics in traffic flow and complexity in networks[J]. Physical Review E,2007,76(1):016110.
[124]Yuan Y M, Jiang R, Wu Q S, et al. Synchronized flow and phase separation in a two-lane cellular automaton Model[J]. International Journal of Modern Physics C,2007,18(02): 267-279.
[125]Yuan Y M, Jiang R, Wu Q S, et al. Traffic behavior in a two-lane system consisting of a mixture of buses and cars[J]. International Journal of Modern Physics C,2007,18(12): 1925-1938.
[126]Zheng L, Ma S, Zhong S. Analysis of honk effect on the traffic flow in a cellular automaton model[J]. Physica A:Statistical Mechanics and its Applications,2011,390(6): 1072-1084.
[127]Wang R, Jiang R, Wu Q S, et al. Synchronized flow and phase separations in single-lane mixed traffic flow[J]. Physica A:Statistical Mechanics and its Applications,2007,378(2): 475-484.
[128]Xiang Z T, Li Y J, Chen Y F, et al. Simulating synchronized traffic flow and wide moving jam based on the brake light rule[J]. Physica A:Statistical Mechanics and its Applications,2013,392(21):5399-5413.
[129]Hafstein S F, Chrobok R, Pottmeier A, et al. Cellular automaton modeling of the autobahn traffic in north rhine-westphalia[J]. Proc. of the 4th MATHMOD,2003: 1322-1331.
[130]Nicklin R C. Kinematics of tailgating[J]. The Physics Teacher,1997,35:78-79.
[131]Bexelius S. An extended model for car-following[J]. Transportation Research,1968,2(1): 13-21.
[132]Sawada S. Generalized optimal velocity model for traffic flow[J]. International Journal of Modern Physics C,2002,13(01):1-12.
[133]Hasebe K, Nakayama A, Sugiyama Y. Dynamical model of a cooperative driving system for freeway traffic[J]. Physical Review E, Statistical, nonlinear, and soft matter physics, 2003,68(2 Pt 2):026102-026102.
[134]Neubert L, Santen L, Schadschneider A, et al. Single-vehicle data of highway traffic:A statistical analysis[J]. Physical Review E,1999,60(6):6480.
[135]Jia B, Li X G, Chen T, et al. Cellular automaton model with time gap dependent randomisation under Kerner's three-phase traffic theory[J]. Transportmetrica,2011,7(2): 127-140.
[136]Gardes Y, May A D, Dahlgren J, et al. Freeway calibration and application of the PARAMICS model[C],81st Annual Meeting of the Transportation Research Board, Washington, DC.2002.
[137]Cheu R L, Recker W W and Ritchie S G Calibration of INTRAS for Simulation of 30-Sec Loop Detector Output, Transportation Research Record,1994,1457:108-116
[138]Cheu R L, Jin X, Ng K C, et al. Genetic algorithm approach in FRESIM calibration[C], Fifth International Conference on Applications of Advanced Technologies in Transportation Engineering.1998.
[139]Ma J, Dong H, Zhang H M. Calibration of microsimulation with heuristic optimization methods[J]. Transportation Research Record:Journal of the Transportation Research Board,2007,1999(1):208-217.
[140]Ben-Akiva M E, Davol A, Toledo T, et al. Calibration and evaluation of MITSIMLab in Stockholm[C] 82nd Annual Meeting Preprint CDROM, Transportation Research Board. 2002.
[141]Ngoduy D, Maher M J. Calibration of second order traffic models using continuous cross entropy method[J]. Transportation Research Part C:Emerging Technologies,2012,24: 102-121.
[142]Ossen S, Hoogendoorn S P, Gorte B G H. Interdriver differences in car-following:a vehicle trajectory-based study[J]. Transportation Research Record:Journal of the Transportation Research Board,2006,1965(1):121-129.
[143]Ossen S, Hoogendoorn S P. Car-following behavior analysis from microscopic trajectory data[J]. Transportation Research Record:Journal of the Transportation Research Board, 2005,1934(1):13-21.
[144]Toledo T, Koutsopoulos H N, Ahmed K I. Estimation of vehicle trajectories with locally weighted regression[J]. Transportation Research Record:Journal of the Transportation Research Board,2007,1999(1):161-169.
[145]Chundury S, Wolshon B. Evaluation of CORSIM car-following model by using Global Positioning System field data[J]. Transportation Research Record:Journal of the Transportation Research Board,2000,1710(1):114-121.
[146]Treiber M, Kesting A. Microscopic Calibration and validation of car-following models-a systematic approach[J]. Procedia-Social and Behavioral Sciences,2013,80:922-939.
[147]Brockfeld E, Kuhne R D, Wagner P. Calibration and validation of microscopic models of traffic flow[J]. Transportation Research Record:Journal of the Transportation Research Board,2005,1934(1):179-187.
[148]Wagner P. Fluid-dynamical and microscopic description of traffic flow:a data-driven comparison[J]. Philosophical Transactions of the Royal Society A:Mathematical, Physical and Engineering Sciences,2010,368(1928):4481-4495.
[149]Treiber M, Kesting A, Thiemann C. How much does traffic congestion increase fuel consumption and emissions? Applying a fuel consumption model to the NGSIM trajectory data[C] 87th Annual Meeting of the Transportation Research Board, Washington, DC.2008.
[150]Thiemann C, Treiber M, Kesting A. Estimating acceleration and lane-changing dynamics from next generation simulation trajectory data[J]. Transportation Research Record: Journal of the Transportation Research Board,2008,2088(1):90-101.
[151]Baumann M. A neural network model for driver's lane-changing trajectory prediction in urban traffic flow[J]. Mathematical Problems in Engineering,2013,2013.
[152]Toledo T, Zohar D. Modeling duration of lane changes[J]. Transportation Research Record:Journal of the Transportation Research Board,2007,1999(1):71-78.
[153]Castillo J M D, Benitez F G. On the functional form of the speed-density relationship—Ⅰ: general theory[J]. Transportation Research Part B:Methodological,1995,29(5): 373-389.
[154]Castillo J M D, Benitez F G. On the functional form of the speed-density relationship—Ⅱ: empirical investigation[J]. Transportation Research Part B:Methodological,1995,29(5): 391-406.
[155]姜锐,吴清松.交通流复杂动态特性的微观和宏观模式研究[D].中国科学院研究生院学报,2002.
[156]Lakouari N, Jetto K, Ez-Zahraouy H, et al. VELOCITY CORRELATIONS IN THE NAGEL-SCHRECKENBERG MODEL[J]. International Journal of Modern Physics C, 2013.