交通流元胞自动机模型的解析和模拟研究
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摘要
伴随着社会经济的不断发展,交通需求不断增长。因此,交通问题日益成为制约经济发展、影响人类生活的一个突出的世界性难题。为了有效地指导交通规划、设计与控制,缓解失衡的交通供求关系,现代交通流理论研究在上世纪三十年代应运而生。八十多年来,交通科学家和物理学家们提出过上百个模型。从上世纪三十到四十年代的概率论模型,到五、六十年代的运动学模型和车辆跟驰模型,再到七、八十年代流体力学模型,都为揭示交通中复杂的物理现象起了非常重要的作用。九十年代以来,交通流元胞自动机模型开始异军突起,以其规则简单、意义明晰、易于扩展以及较高的计算效率而为越来越多的交通学者和工程师所青睐。本文从解析和模拟两个方面对交通流元胞自动机模型进行研究。一方面将简单完全非对称排他过程TASEP这一最简单的元胞自动机模型扩展应用到基本道路形式,运用平均场分析及畴壁(domain wall)理论等方法对模型进行数学解析,从数学解析的角度揭示简单系统中的复杂非平衡态物理现象,尝试建立起交通流与非平衡态统计力学的联系,以推动相关学科的发展;另一方面通过比较分析几个能够模拟同步流的元胞自动机模型,挑选出能较好符合实测结果的MCD(或FMCD)模型,将其应用于模拟双道复杂交通系统,得到更符合交通实际的时空特性,为交通工程实际提供一定的理论参考。本文的主要工作如下:
1.将TASEP扩展应用到含捷径道路和双道交叉系统两种基本道路形式的交通流研究,以数学解析这一全新视角深入理解交通系统中的非平衡态现象。
·在TASEP原型的精确解基础上,采用平均场分析和定量畴壁(domainwall)理论得到了TASEP在含捷径道路的扩展模型的数学解析解。模拟显示,相图可区分为三个稳态相。对于三个稳态相,可忽略格点间的相关性,采用平均场分析的方法求解得到了三相所对应的系统密度分布,所得结果与模拟结果较好地符合一致。对于相边界,由于存在强相关性,平均场分析不再适用,采用定量畴壁(domain wall)理论方法进行求解,所得相边界上的系统密度分布除在捷径起、终点间主道路段与模拟结果有小偏差之外均较好地符合模拟结果。
·在双道交叉系统中惊奇地发现了自发对称破缺现象。我们利用定性畴壁(domain wall)理论对自发对称破缺现象的成因进行了解释。此外,采用平均场分析方法对系统相图的相边界值进行了解析,所得结果与模拟结果一致。最后,考察了车辆在交叉点上的换道效应对自发对称破缺现象的影响,结果发现当换道概率大于某一临界值时,自发对称破缺现象消失。
2.运用模拟细致研究了同步流这一复杂相态。首先,通过对同步流元胞自动机模型的模拟结果进行比较分析,得出元胞自动机模型较好地模拟同步流的一个必要条件为定态解在流量密度平面占据二维区域。实现这一条件的不同模型机制对模拟所得交通相态的微观特性有很大影响。KKW模型和Lee模型由于显含或隐含的均匀系统车速机制使得模拟得到的同步流过于均匀,微观特性与实测结果有较大偏差,而MCD(或FMCD)模型利用刹车灯作用使车辆在某段距离内能够维持车速不变,所得交通相态的微观特性相对更加符合实测结果。
其次,将MCD模型应用于模拟双道复杂交通系统,解决了以往双道元胞自动机模型大多数采用基本图方法体系下不能模拟同步流的模型,而难以反映系统真实的时空特性这一问题,得到双道复杂系统的真实时空特性。
·对于双道均匀系统,相比于单道MCD模型,由于换道效应,能够模拟得到非致密的宽运动堵塞,且其堵塞出流既可为自由流又能处于同步流,更符合交通实测。此外,以往双道模型模拟所得换道概率与实测结果有较大偏差,而我们所用对称换道规则模拟得到的换道概率随密度变化与交通实测结果一致。最后,为改进对称换道规则,令速度为0车辆不能换道,提升了系统通行能力,还首次在对称换道规则下模拟得到了两道非对称的时空特性。
·考虑到公车系统对整个交通系统时空特性有重要影响这一事实,综合考察公车小车混合双道系统。模拟显示,由于公车系统的作用,密度被区分为四个区间,即系统可以呈现四种不同的时空特性。由于小车的换道效应,在高小车密度下,公车集簇因有小车进入而逐渐消失。我们还固定公车数目在不同公车站数目下考察了系统通行能力,给出了一个建议的公车站数目设置。
3.对含自适应巡航控制车辆的智能交通系统进行了初探。首次采用元胞自动机和车辆跟驰模型混合建模的方法研究了自适应巡航控制车辆及手动驾驶车辆混合交通系统的交通流特性。既消除了以往全用元胞自动机模拟由元胞自动机内在随机引入的虚假速度扰动,又能够模拟跟驰模型所不能模拟出的三相之间的相变概率特性。另外,我们还提出了一种改进的自适应巡航控制系统的距离控制策略,不仅能够保持恒定时间车头距策略和非线性策略的优点,而且能够克服这两种策略的缺点。从而为系统设计人员提供建议。
论文不仅将TASEP这一简单的元胞自动机模型扩展应用于基本道路形式,从数学解析角度揭示了系统复杂的非平衡态物理现象,以推动非平衡态统计力学的发展,具有重要的理论意义,还将同步流元胞自动机模型应用于模拟复杂交通系统,揭示交通系统的真实时空特性,为工程界提供参考,具有一定的实用价值。
With the rapid development of society economy,traffic demand continuously increases.Therefore,transportation problem gradually becomes an emergent world issue,which limits economy development and influences human life greatly.In order to guide the traffic planning,designing,controlling effectively and alleviate the unbalanced supply and demand,the modern traffic flow theory emerged in the 1930s as the times require.Since the birth of the modern traffic flow,more than 100 theoretical models have been proposed by traffic scientists and physicists,from probabilistic theory models in the 1930s-1940s,to kinematic models and car-following models in the 1950s-1960s,and then to fluid-dynamical models in the 1970s-1980s,all of which play a vital role in describing the complex phenomena in traffic.Since the 1990s, cellular automaton(CA) models,with simple rules,clear definition,good expansibility and high computational efficiency,have risen in traffic flow theory as a new force and gradually win the good graces of the traffic scholars and engineers.This paper studies the traffic CA models through both analytics and simulation.On one hand, Totally Asymmetric Simple Exclusion Process(TASEP),the simplest CA model in traffic,is extended to simple road sets.The mean-field analysis and domain-wall theory have been applied to obtain analytic model solutions in order to disclose rich interesting non-equilibrium phenomena analytically and bridge traffic science and non-equilibrium statistics,and thus promote the development of relative disciplines. On the other hand,we select MCD(or FMCD) model that can describe the microscopic structure of the synchronized flow better after comparing the simulation results of three CA models that can reproduce the synchronized flow.Then,MCD model is applied to simulate complex two-lane traffic systems in order to reproduce their real spatio-temporal characteristics,which provides theoretical references for traffic engineering.The contents of the paper are as follows.
1.TASEP is applied in two basic road sets:the road with a shortcut and a two-lane intersected road system.The non-equilibrium phenomena in traffic systems are studied with a new in-depth analytic perspective.
●Based on the exact solution of the origin single-lane TASEP,the analytic solution of TASEP in the road with a shortcut is obtained with mean-field analysis and domain wall theory.The simulation shows that the phase diagram can be classified into three stationary phases.Because the correlation between the neighboring sites can be neglected,the density profiles corresponding to the three phases are obtained by mean-field analysis,which are in good agreement with the simulation results.For phase boundaries,the mean-field analysis cannot be adopted due to strong correlation.The quantitative domain wall theory is adopted instead to calculate the density profiles on the phase boundaries.The obtained analytic results are in good agreement with the simulation results except slight deviation in the middle main road segment divided by the shortcut.
●In the two-lane intersected system,the spontaneous symmetry breaking phenomena occurs surprisingly.The qualitative domain-wall theory is adopted to explain the reason why it occurs.Furthermore,the value of the phase boundary is obtained by mean-field analysis,which is in good agreement with the simulation results.Finally,we investigate the lane-changing effect at the intersection on spontaneous symmetry breaking.It is found that the spontaneous symmetry breaking phenomena disappears if the lane-changing probability exceeds a critical value.
2.The synchronized flow is investigated in detail through simulation.Firstly, through comparing and analyzing the simulation results of three CA models that can reproduce the synchronized flow,we conclude that the steady model solutions occupy a two-dimension region in the flow-density plane is a necessary condition for a CA model to reproduce the synchronized flow well.The different mechanism of different models to satisfy the condition has great impact on the microscopic structure of the traffic states.For example,the explicit and implicit mechanism of the KKW model and Lee model to homogenize the traffic flow makes the obtained synchronized flow so homogenous that its microscopic structure deviates from the empirical results.However,the mechanism of MCD (or FMCD) model is that a vehicle can keep its velocity unchanged in certain distance determined by its brake light.Therefore,the microscopic structure of the synchronized flow in MCD model is relatively in better agreement with the empirical results.
Then,MCD model is applied to describe two-lane complex traffic system,which can solve the problem that most previous two-lane CA models,in the frame-work of the fundamental diagram approach,cannot reproduce the real spatio-temporal features.
●For two-lane homogeneous system,due to the lane-changing effect,the incompact wide moving jams,whose outflow can be either free flow or the synchronized flow,are obtained.It agrees with the empirical results but cannot be obtained in single-lane MCD model.Moreover,the lane-changing frequency against density in previous work deviates from the empirical results,however,in two-lane MCD model,the obtained lane-changing frequency agrees well with the empirical results.Finally,we improve the symmetric lane-changing rules by restricting the lane change of the stopped vehicles,which results in higher road capacity.It is the first time that asymmetric traffic behavior can be maintained under symmetric lane changing rules.
●Considering the great impact of the bus system on the spatio-temporal characteristics of the whole traffic system,we investigate a heterogeneous two-lane traffic system consisting of the mixture of buses and cars.Due to the influence of the buses,the whole density range can be classified into four regions,i.e.,the system can be in four states.It is also found that due to the interaction of the cars,the bus clusters gradually disappear with the increase of the car density.Moreover,we study the road capacity under different sets of the number of bus stops with fixed number of buses and an optimal number is suggested.
3.A preliminary study on the intelligent transportation systems has been conducted. It is the first time that a hybrid model of the CA model and car-following model is proposed to study the characteristics of the mixed traffic of the adaptive cruise control vehicles and the manual vehicles,which not only removes the artificial velocity fluctuation in the CA model for ACC vehicles, but also maintains the ability to simulate the probabilistic characteristics of the phase transitions among the three traffic states.Moreover,We propose a new range policy for ACC vehicles,which not only maintains the advantages of the constant time headway policy and the nonlinear range policy,but also overcomes their shortcomings,for better design in ACC vehicles.
1.将TASEP扩展应用到含捷径道路和双道交叉系统两种基本道路形式的交通流研究,以数学解析这一全新视角深入理解交通系统中的非平衡态现象。
·在TASEP原型的精确解基础上,采用平均场分析和定量畴壁(domainwall)理论得到了TASEP在含捷径道路的扩展模型的数学解析解。模拟显示,相图可区分为三个稳态相。对于三个稳态相,可忽略格点间的相关性,采用平均场分析的方法求解得到了三相所对应的系统密度分布,所得结果与模拟结果较好地符合一致。对于相边界,由于存在强相关性,平均场分析不再适用,采用定量畴壁(domain wall)理论方法进行求解,所得相边界上的系统密度分布除在捷径起、终点间主道路段与模拟结果有小偏差之外均较好地符合模拟结果。
·在双道交叉系统中惊奇地发现了自发对称破缺现象。我们利用定性畴壁(domain wall)理论对自发对称破缺现象的成因进行了解释。此外,采用平均场分析方法对系统相图的相边界值进行了解析,所得结果与模拟结果一致。最后,考察了车辆在交叉点上的换道效应对自发对称破缺现象的影响,结果发现当换道概率大于某一临界值时,自发对称破缺现象消失。
2.运用模拟细致研究了同步流这一复杂相态。首先,通过对同步流元胞自动机模型的模拟结果进行比较分析,得出元胞自动机模型较好地模拟同步流的一个必要条件为定态解在流量密度平面占据二维区域。实现这一条件的不同模型机制对模拟所得交通相态的微观特性有很大影响。KKW模型和Lee模型由于显含或隐含的均匀系统车速机制使得模拟得到的同步流过于均匀,微观特性与实测结果有较大偏差,而MCD(或FMCD)模型利用刹车灯作用使车辆在某段距离内能够维持车速不变,所得交通相态的微观特性相对更加符合实测结果。
其次,将MCD模型应用于模拟双道复杂交通系统,解决了以往双道元胞自动机模型大多数采用基本图方法体系下不能模拟同步流的模型,而难以反映系统真实的时空特性这一问题,得到双道复杂系统的真实时空特性。
·对于双道均匀系统,相比于单道MCD模型,由于换道效应,能够模拟得到非致密的宽运动堵塞,且其堵塞出流既可为自由流又能处于同步流,更符合交通实测。此外,以往双道模型模拟所得换道概率与实测结果有较大偏差,而我们所用对称换道规则模拟得到的换道概率随密度变化与交通实测结果一致。最后,为改进对称换道规则,令速度为0车辆不能换道,提升了系统通行能力,还首次在对称换道规则下模拟得到了两道非对称的时空特性。
·考虑到公车系统对整个交通系统时空特性有重要影响这一事实,综合考察公车小车混合双道系统。模拟显示,由于公车系统的作用,密度被区分为四个区间,即系统可以呈现四种不同的时空特性。由于小车的换道效应,在高小车密度下,公车集簇因有小车进入而逐渐消失。我们还固定公车数目在不同公车站数目下考察了系统通行能力,给出了一个建议的公车站数目设置。
3.对含自适应巡航控制车辆的智能交通系统进行了初探。首次采用元胞自动机和车辆跟驰模型混合建模的方法研究了自适应巡航控制车辆及手动驾驶车辆混合交通系统的交通流特性。既消除了以往全用元胞自动机模拟由元胞自动机内在随机引入的虚假速度扰动,又能够模拟跟驰模型所不能模拟出的三相之间的相变概率特性。另外,我们还提出了一种改进的自适应巡航控制系统的距离控制策略,不仅能够保持恒定时间车头距策略和非线性策略的优点,而且能够克服这两种策略的缺点。从而为系统设计人员提供建议。
论文不仅将TASEP这一简单的元胞自动机模型扩展应用于基本道路形式,从数学解析角度揭示了系统复杂的非平衡态物理现象,以推动非平衡态统计力学的发展,具有重要的理论意义,还将同步流元胞自动机模型应用于模拟复杂交通系统,揭示交通系统的真实时空特性,为工程界提供参考,具有一定的实用价值。
With the rapid development of society economy,traffic demand continuously increases.Therefore,transportation problem gradually becomes an emergent world issue,which limits economy development and influences human life greatly.In order to guide the traffic planning,designing,controlling effectively and alleviate the unbalanced supply and demand,the modern traffic flow theory emerged in the 1930s as the times require.Since the birth of the modern traffic flow,more than 100 theoretical models have been proposed by traffic scientists and physicists,from probabilistic theory models in the 1930s-1940s,to kinematic models and car-following models in the 1950s-1960s,and then to fluid-dynamical models in the 1970s-1980s,all of which play a vital role in describing the complex phenomena in traffic.Since the 1990s, cellular automaton(CA) models,with simple rules,clear definition,good expansibility and high computational efficiency,have risen in traffic flow theory as a new force and gradually win the good graces of the traffic scholars and engineers.This paper studies the traffic CA models through both analytics and simulation.On one hand, Totally Asymmetric Simple Exclusion Process(TASEP),the simplest CA model in traffic,is extended to simple road sets.The mean-field analysis and domain-wall theory have been applied to obtain analytic model solutions in order to disclose rich interesting non-equilibrium phenomena analytically and bridge traffic science and non-equilibrium statistics,and thus promote the development of relative disciplines. On the other hand,we select MCD(or FMCD) model that can describe the microscopic structure of the synchronized flow better after comparing the simulation results of three CA models that can reproduce the synchronized flow.Then,MCD model is applied to simulate complex two-lane traffic systems in order to reproduce their real spatio-temporal characteristics,which provides theoretical references for traffic engineering.The contents of the paper are as follows.
1.TASEP is applied in two basic road sets:the road with a shortcut and a two-lane intersected road system.The non-equilibrium phenomena in traffic systems are studied with a new in-depth analytic perspective.
●Based on the exact solution of the origin single-lane TASEP,the analytic solution of TASEP in the road with a shortcut is obtained with mean-field analysis and domain wall theory.The simulation shows that the phase diagram can be classified into three stationary phases.Because the correlation between the neighboring sites can be neglected,the density profiles corresponding to the three phases are obtained by mean-field analysis,which are in good agreement with the simulation results.For phase boundaries,the mean-field analysis cannot be adopted due to strong correlation.The quantitative domain wall theory is adopted instead to calculate the density profiles on the phase boundaries.The obtained analytic results are in good agreement with the simulation results except slight deviation in the middle main road segment divided by the shortcut.
●In the two-lane intersected system,the spontaneous symmetry breaking phenomena occurs surprisingly.The qualitative domain-wall theory is adopted to explain the reason why it occurs.Furthermore,the value of the phase boundary is obtained by mean-field analysis,which is in good agreement with the simulation results.Finally,we investigate the lane-changing effect at the intersection on spontaneous symmetry breaking.It is found that the spontaneous symmetry breaking phenomena disappears if the lane-changing probability exceeds a critical value.
2.The synchronized flow is investigated in detail through simulation.Firstly, through comparing and analyzing the simulation results of three CA models that can reproduce the synchronized flow,we conclude that the steady model solutions occupy a two-dimension region in the flow-density plane is a necessary condition for a CA model to reproduce the synchronized flow well.The different mechanism of different models to satisfy the condition has great impact on the microscopic structure of the traffic states.For example,the explicit and implicit mechanism of the KKW model and Lee model to homogenize the traffic flow makes the obtained synchronized flow so homogenous that its microscopic structure deviates from the empirical results.However,the mechanism of MCD (or FMCD) model is that a vehicle can keep its velocity unchanged in certain distance determined by its brake light.Therefore,the microscopic structure of the synchronized flow in MCD model is relatively in better agreement with the empirical results.
Then,MCD model is applied to describe two-lane complex traffic system,which can solve the problem that most previous two-lane CA models,in the frame-work of the fundamental diagram approach,cannot reproduce the real spatio-temporal features.
●For two-lane homogeneous system,due to the lane-changing effect,the incompact wide moving jams,whose outflow can be either free flow or the synchronized flow,are obtained.It agrees with the empirical results but cannot be obtained in single-lane MCD model.Moreover,the lane-changing frequency against density in previous work deviates from the empirical results,however,in two-lane MCD model,the obtained lane-changing frequency agrees well with the empirical results.Finally,we improve the symmetric lane-changing rules by restricting the lane change of the stopped vehicles,which results in higher road capacity.It is the first time that asymmetric traffic behavior can be maintained under symmetric lane changing rules.
●Considering the great impact of the bus system on the spatio-temporal characteristics of the whole traffic system,we investigate a heterogeneous two-lane traffic system consisting of the mixture of buses and cars.Due to the influence of the buses,the whole density range can be classified into four regions,i.e.,the system can be in four states.It is also found that due to the interaction of the cars,the bus clusters gradually disappear with the increase of the car density.Moreover,we study the road capacity under different sets of the number of bus stops with fixed number of buses and an optimal number is suggested.
3.A preliminary study on the intelligent transportation systems has been conducted. It is the first time that a hybrid model of the CA model and car-following model is proposed to study the characteristics of the mixed traffic of the adaptive cruise control vehicles and the manual vehicles,which not only removes the artificial velocity fluctuation in the CA model for ACC vehicles, but also maintains the ability to simulate the probabilistic characteristics of the phase transitions among the three traffic states.Moreover,We propose a new range policy for ACC vehicles,which not only maintains the advantages of the constant time headway policy and the nonlinear range policy,but also overcomes their shortcomings,for better design in ACC vehicles.
引文
References
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