交通流的随机因素分析和相变研究
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摘要
交通拥堵和交通污染带来了巨大的社会压力和经济损失。如何有效地利用交通资源,改善交通供求关系,以科学的理论来指导交通建设和交通运营,成为国际社会广泛关注的重大问题。近几十年来,不同领域的研究者从不同的角度对交通流的特性进行了分析,建立了许多交通流模型,取得了大量成果。本文在现有交通流模型的基础上,针对诱发相变等复杂交通动力学现象的关键因素如随机噪声、空间非均匀性等进行着重分析,提出若干改进的模型,并进行相应的理论分析和数值模拟。此外,还探讨了格子流体力学模型在考虑某些实际因素时的相变问题。全文的主要工作如下:
一、综合考虑实际中影响驾驶员加速行为的各种随机性因素提出了一种加权概率元胞自动机模型。
在经典的NaSch模型和FI模型的基础上,综合考虑驾驶员和车辆的差异性及其外部噪声等随机因素的影响,基于概率论的思想提出了一种可广泛应用于不同道路条件(诸如限速与无限速高速公路)的加权概率元胞自动机模型(简称WP模型)。模型中,引入了一般的随机加速概率分布函数,更好地刻画了由各种随机因素所决定的驾驶特征。数值模拟表明:限速WP模型所得的结果与实测数据符合得相当好,时空演化斑图显示出一种新的均匀流(由高速和低速两类),并为区分两类均匀流给出了一种判据。最后,通过分析车间距的时间演化分布,进一步证实了该新均匀流的存在;对于无限速的WP模型,我们分别从基本图、时空演化斑图、车间距演化分布、速度分布四个角度将其与NaSch模型和FI模型进行了对比,结果表明,WP模型给出的流量与实测结果更为接近。所有这些结论充分表明了该模型形式简单,物理意义清晰,可以较好地描述实际交通现象。
二、从新的概率论主方程的角度,结合三相流理论,研究了开放的空间不均匀的道路缩减瓶颈中引起的交通崩溃现象。
将概率论主方程应用于描述在开放的空间不均匀的道路缩减瓶颈中引起的交通崩溃现象。出发点是Kerner的三相流理论,将交通崩溃与从自由流到同步流的一阶相变联系起来,同时将这种空间不均匀的道路结构假设为一个始终位于瓶颈附近的确定性的局部扰动。通过建立描写车辆集簇演化的主方程及相应的理论分析,我们得到了描写交通崩溃的平均延迟时间和成核率的表达式。最后通过一个具体的例子详细讨论了离开概率、势函数、势垒、平均延迟时间及相应的成核率与道路缩减程度及平均来流流率之间的关系。
三、将优化速度模型应用于描述实际交通中由道路几何形状的变化——上(下)坡引起的交通拥堵的特征。
通过引入计及空间位置、坡度以及可变安全间距的广义优化速度函数,拓展了经典的Bando优化速度模型,针对具有上(下)坡路段的限速交通流进行了研究。理论分析和数值模拟结果表明,具有上(下)坡路段的交通流基本图中,整条道路的流率随着密度单调递增,在某个临界点达到饱和,其后随着密度的进一步增加,流率维持其饱和值,直到超过另一临界密度后,流率随着密度的增加逐渐下降。在忽略驾驶员敏感度的轻微影响后,饱和流率的大小等于上(下)坡路段的最大流率,而且基本图与敏感度、坡度和坡长紧密相关。时空演化斑图给出了不同初始车辆数下由稀疏波和激波导致的不同交通相的分离。另外,通过对上(下)坡路段的模拟发现,在相同敏感度、坡度和坡长下,车辆在下坡路段对应的饱和流率和最大平均速度均大于上坡路段,这与实际道路交通情况一致。
四、基子ITS的应用,提出考虑后车效应的格子流体力学模型,研究了前方任意辆车和后方一辆车对交通流的影响,并进一步探讨了后车效应的引入导致的交通流的各向异性问题。
根据实际交通中前后车辆对当前车辆作用的不同,分别定义后向和前向优化速度函数,通过线性稳定性分析得到中性稳定性曲线,并在临界点附近采用约化摄动法导出了不稳定区域密度波演化的mKdV方程。研究结果表明:选用合适的后向优化速度函数,考虑后车效应将会提高交通系统的稳定性,相反,选取不恰当的后向优化速度函数则会增大交通拥堵。另外,通过对扭结—反扭结波传播速度的计算,我们发现:在不稳定区域内,一定权重下考虑后车因素会导致交通流密度波的传播速度为负值。进一步,我们详细讨论了此现象蕴含的交通流各向异性问题并给出合理的物理解释。
五、从格子流体力学的角度分析了考虑车辆逐步加速特性时交通流的特征,重点讨论了优化速度函数的改变对系统稳定性的影响。
基于实际交通中的车辆驾驶特性,提出了两种改进的格子流体力学模型。在模型中,通过引入计及车辆逐步加速效应,且与实测数据符合较好的新的优化速度函数,更合理地刻画了单车道交通流的动力学相变特性。线性稳定性分析表明:在考虑驾驶员逐步加速特性的情况下,交通流呈现出更为复杂的多重相变,且临界密度依赖于驾驶员敏感度和车辆密度,相变的复杂程度与优化速度函数的拐点数紧密相关。数值仿真结果与理论分析一致,证实了改进模型的正确性和有效性。
最后,对全文进行了总结,并指出需要进一步研究的问题。
Recently, traffic has become a global exasperating problem, and the increasing traffic jams and traffic pollution are exerting great pressure on society and causing enormous economic losses. How to make effective use of existing transportation resources so as to improve the relationship between traffic demand and supply, and how to guide transport development and operation with the aid of scientific theories are of great importance, drawing general concern of the international community. Over the past decades, a lot of scholars in different fields have been conducting extensive research on this subject and proposing a variety of traffic flow theories, yielding an enormous amount of results. In this dissertation, based on the previous work, several improved mathematical models are presented through analyzing emphatically the stochastic factors and spatial inhomogeneity in real transportation systems, and the corresponding theoretical analysis and numerical simulation are performed. Moreover, the phase transtion in lattice hydrodynamic model with some real traffic effects considered is investigated. The main contents are as follows.
Ⅰ. From a global viewpoint of the stochastic acceleration behavior of drivers, a weighted probabilistic cellular automaton model (the WP model, for short) is established by introducing a random acceleration probabilistic distribution function.
Owing to the variety of vehicle acceleration behavior from a synthetic viewpoint, a weighted probabilistic cellular automaton model (the WP model, for short) for traffic flow is constructed on the basis of the classical NaSch model and FI model. A probabilistic distribution function for random acceleration, which can be widely used in treating general traffic situations, e.g., those with or without speed limit, is introduced. The numerical simulations show that the speed limit WP model leads to the results consistent with the empirical data rather well, and a new kind of traffic phenomenon called neo-uniform flow, consisting of the high-speed and low-speed ones, is resulted. Furthermore, we give the criterion for distinguishing the high-speed and low-speed neo-uniform flows and elucidate the mechanism of this kind of transportation characteristics. As for the non-speed-limit WP model, we compare the numerical results with those obtained with the NaSch model and FI model from four different aspects, and show that the maximum traffic flux is closer to the observed data. All these are helpful in understanding and depicting the realistic traffic behavior.
Ⅱ. The traffic breakdown caused by the reduction of lanes at a highway bottleneck in the spatially inhomogeneous open road system is examined via a proposed stochastic master-equation model with the three-phase theory.
Based on the Markov processes, a new stochastic approach to the spatially inhomogeneous open traffic system at a highway bottleneck caused by the reduction of lanes is developed. The model is in the context of three-phase traffic theory, where the breakdown phenomenon is associated with a first-order phase transition from the free flow phase to the synchronized flow phase. In addition, a permanent and motionless non-homogeneity that can be considered a deterministic vehicle cluster localized in a neighborhood of the bottleneck is assumed. An appropriate master equation for the car cluster evolution is derived. The mean time delay and the associated nucleation rate of traffic breakdown are found and the nucleation rate of traffic breakdown as a function of the reduction of lane numbers and the coming flow rate per lane is studied. The studied example further verifies the analytical results.
Ⅲ. A generalized optimal velocity model is presented and applied to investigate the traffic charactersitics under the changed road conditions, i.e., the speed limit traffic for the roads with upgrade (or downgrade).
Through introducing a generalized optimal velocity function to consider the spatial position, slope grade and variable safety headway, the effect of slope on a single-lane highway is investigated with a generalized optimal velocity model. The theoretical analysis and simulation results show that the flux of the whole road with the upgrade (or downgrade) increases monotonously with density, saturates at a critical density, then maintains this saturated value in a certain density range and finally decreases with density. The value of saturated flux is equal to the maximum flux of the upgrade (or downgrade) without considering the slight influence of the driver's sensitivity. And the fundamental diagrams also depend on the sensitivity, slope grade and slope length. The spatio-temporal pattern gives the segregation of different traffic phases caused by the rarefaction wave and shock wave for a certain initial vehicle number. The comparison between the upgrade and the downgrade indicates that the value of saturated flux of the downgrade is larger than that of the upgrade under the same condition. This result is in accordance with the real traffic.
Ⅳ. Two extended cooperative driving lattice hydrodynamic models are proposed by considering the backward looking effect. The influnce of more preceding vehicles and one following vehicle on the studied vehicle is investigated. And the anisotropy of traffic flow is further discussed through examining the negative propagation velocity as the effect of following vehicle is involved.
In light of the different influences of the preceding and following vehicles on the considered vehicle in real traffic, the proper forward and backward optimal velocities are defined, respectively. The neutral stability line is obtained by using the linear stability theory and it is found that considering the following vehicle effect and adopting the appropriate backward optimal velocity funciton could lead to the improvement of the traffic flow stability. The modified Korteweg-de Vries equations (the mKdV equation, for short) near the critical point are derived by using the nonlinear perturbation method to show that the traffic jam could be described by the kink-antikink soliton solutions for the mKdV equations. It is shown that in the unstable region, to a certain extent, the negative propagation velocity appears as the effect of following vehicle is involved. Moreover, the related anisotropy of traffic flow with the negative propagation velocity is discussed in detail and corresponding physical nature is explored.
Ⅴ. Based on two extended lattice hydrodynamic models, the traffic characteristics are anaylzed with considering the effect of stepwise acceleration of a vehicle, in which emphasis is particularly laid on the investigation of the influence of optimal velocity function on the system stability.
With the consideration of the behavior of stepwise vehicle acceleration, two extended lattice hydrodynamic models are proposed. A new optimal velocity function is introduced to take the stepwise acceleration into account and to fit the obtained results to observed data better than those with the original lattice hydrodynamic models, which leads to a better description of the dynamical phase transition for single-lane traffic flow. Linear stability analysis shows that the multiple phase transitions occur through considering the effect of stepwise acceleration, and their properties depend on the vehicle density and sensitivity. Moreover, it is concluded that the complexity degree of phase transition is closely related with the turning points of the introduced optimal velocity function. The validity and correctness of the presented model is confirmed by direct simulations.
The final chapter of this dissertation is devoted to a summary and prospect of further study of the road traffic flow.
一、综合考虑实际中影响驾驶员加速行为的各种随机性因素提出了一种加权概率元胞自动机模型。
在经典的NaSch模型和FI模型的基础上,综合考虑驾驶员和车辆的差异性及其外部噪声等随机因素的影响,基于概率论的思想提出了一种可广泛应用于不同道路条件(诸如限速与无限速高速公路)的加权概率元胞自动机模型(简称WP模型)。模型中,引入了一般的随机加速概率分布函数,更好地刻画了由各种随机因素所决定的驾驶特征。数值模拟表明:限速WP模型所得的结果与实测数据符合得相当好,时空演化斑图显示出一种新的均匀流(由高速和低速两类),并为区分两类均匀流给出了一种判据。最后,通过分析车间距的时间演化分布,进一步证实了该新均匀流的存在;对于无限速的WP模型,我们分别从基本图、时空演化斑图、车间距演化分布、速度分布四个角度将其与NaSch模型和FI模型进行了对比,结果表明,WP模型给出的流量与实测结果更为接近。所有这些结论充分表明了该模型形式简单,物理意义清晰,可以较好地描述实际交通现象。
二、从新的概率论主方程的角度,结合三相流理论,研究了开放的空间不均匀的道路缩减瓶颈中引起的交通崩溃现象。
将概率论主方程应用于描述在开放的空间不均匀的道路缩减瓶颈中引起的交通崩溃现象。出发点是Kerner的三相流理论,将交通崩溃与从自由流到同步流的一阶相变联系起来,同时将这种空间不均匀的道路结构假设为一个始终位于瓶颈附近的确定性的局部扰动。通过建立描写车辆集簇演化的主方程及相应的理论分析,我们得到了描写交通崩溃的平均延迟时间和成核率的表达式。最后通过一个具体的例子详细讨论了离开概率、势函数、势垒、平均延迟时间及相应的成核率与道路缩减程度及平均来流流率之间的关系。
三、将优化速度模型应用于描述实际交通中由道路几何形状的变化——上(下)坡引起的交通拥堵的特征。
通过引入计及空间位置、坡度以及可变安全间距的广义优化速度函数,拓展了经典的Bando优化速度模型,针对具有上(下)坡路段的限速交通流进行了研究。理论分析和数值模拟结果表明,具有上(下)坡路段的交通流基本图中,整条道路的流率随着密度单调递增,在某个临界点达到饱和,其后随着密度的进一步增加,流率维持其饱和值,直到超过另一临界密度后,流率随着密度的增加逐渐下降。在忽略驾驶员敏感度的轻微影响后,饱和流率的大小等于上(下)坡路段的最大流率,而且基本图与敏感度、坡度和坡长紧密相关。时空演化斑图给出了不同初始车辆数下由稀疏波和激波导致的不同交通相的分离。另外,通过对上(下)坡路段的模拟发现,在相同敏感度、坡度和坡长下,车辆在下坡路段对应的饱和流率和最大平均速度均大于上坡路段,这与实际道路交通情况一致。
四、基子ITS的应用,提出考虑后车效应的格子流体力学模型,研究了前方任意辆车和后方一辆车对交通流的影响,并进一步探讨了后车效应的引入导致的交通流的各向异性问题。
根据实际交通中前后车辆对当前车辆作用的不同,分别定义后向和前向优化速度函数,通过线性稳定性分析得到中性稳定性曲线,并在临界点附近采用约化摄动法导出了不稳定区域密度波演化的mKdV方程。研究结果表明:选用合适的后向优化速度函数,考虑后车效应将会提高交通系统的稳定性,相反,选取不恰当的后向优化速度函数则会增大交通拥堵。另外,通过对扭结—反扭结波传播速度的计算,我们发现:在不稳定区域内,一定权重下考虑后车因素会导致交通流密度波的传播速度为负值。进一步,我们详细讨论了此现象蕴含的交通流各向异性问题并给出合理的物理解释。
五、从格子流体力学的角度分析了考虑车辆逐步加速特性时交通流的特征,重点讨论了优化速度函数的改变对系统稳定性的影响。
基于实际交通中的车辆驾驶特性,提出了两种改进的格子流体力学模型。在模型中,通过引入计及车辆逐步加速效应,且与实测数据符合较好的新的优化速度函数,更合理地刻画了单车道交通流的动力学相变特性。线性稳定性分析表明:在考虑驾驶员逐步加速特性的情况下,交通流呈现出更为复杂的多重相变,且临界密度依赖于驾驶员敏感度和车辆密度,相变的复杂程度与优化速度函数的拐点数紧密相关。数值仿真结果与理论分析一致,证实了改进模型的正确性和有效性。
最后,对全文进行了总结,并指出需要进一步研究的问题。
Recently, traffic has become a global exasperating problem, and the increasing traffic jams and traffic pollution are exerting great pressure on society and causing enormous economic losses. How to make effective use of existing transportation resources so as to improve the relationship between traffic demand and supply, and how to guide transport development and operation with the aid of scientific theories are of great importance, drawing general concern of the international community. Over the past decades, a lot of scholars in different fields have been conducting extensive research on this subject and proposing a variety of traffic flow theories, yielding an enormous amount of results. In this dissertation, based on the previous work, several improved mathematical models are presented through analyzing emphatically the stochastic factors and spatial inhomogeneity in real transportation systems, and the corresponding theoretical analysis and numerical simulation are performed. Moreover, the phase transtion in lattice hydrodynamic model with some real traffic effects considered is investigated. The main contents are as follows.
Ⅰ. From a global viewpoint of the stochastic acceleration behavior of drivers, a weighted probabilistic cellular automaton model (the WP model, for short) is established by introducing a random acceleration probabilistic distribution function.
Owing to the variety of vehicle acceleration behavior from a synthetic viewpoint, a weighted probabilistic cellular automaton model (the WP model, for short) for traffic flow is constructed on the basis of the classical NaSch model and FI model. A probabilistic distribution function for random acceleration, which can be widely used in treating general traffic situations, e.g., those with or without speed limit, is introduced. The numerical simulations show that the speed limit WP model leads to the results consistent with the empirical data rather well, and a new kind of traffic phenomenon called neo-uniform flow, consisting of the high-speed and low-speed ones, is resulted. Furthermore, we give the criterion for distinguishing the high-speed and low-speed neo-uniform flows and elucidate the mechanism of this kind of transportation characteristics. As for the non-speed-limit WP model, we compare the numerical results with those obtained with the NaSch model and FI model from four different aspects, and show that the maximum traffic flux is closer to the observed data. All these are helpful in understanding and depicting the realistic traffic behavior.
Ⅱ. The traffic breakdown caused by the reduction of lanes at a highway bottleneck in the spatially inhomogeneous open road system is examined via a proposed stochastic master-equation model with the three-phase theory.
Based on the Markov processes, a new stochastic approach to the spatially inhomogeneous open traffic system at a highway bottleneck caused by the reduction of lanes is developed. The model is in the context of three-phase traffic theory, where the breakdown phenomenon is associated with a first-order phase transition from the free flow phase to the synchronized flow phase. In addition, a permanent and motionless non-homogeneity that can be considered a deterministic vehicle cluster localized in a neighborhood of the bottleneck is assumed. An appropriate master equation for the car cluster evolution is derived. The mean time delay and the associated nucleation rate of traffic breakdown are found and the nucleation rate of traffic breakdown as a function of the reduction of lane numbers and the coming flow rate per lane is studied. The studied example further verifies the analytical results.
Ⅲ. A generalized optimal velocity model is presented and applied to investigate the traffic charactersitics under the changed road conditions, i.e., the speed limit traffic for the roads with upgrade (or downgrade).
Through introducing a generalized optimal velocity function to consider the spatial position, slope grade and variable safety headway, the effect of slope on a single-lane highway is investigated with a generalized optimal velocity model. The theoretical analysis and simulation results show that the flux of the whole road with the upgrade (or downgrade) increases monotonously with density, saturates at a critical density, then maintains this saturated value in a certain density range and finally decreases with density. The value of saturated flux is equal to the maximum flux of the upgrade (or downgrade) without considering the slight influence of the driver's sensitivity. And the fundamental diagrams also depend on the sensitivity, slope grade and slope length. The spatio-temporal pattern gives the segregation of different traffic phases caused by the rarefaction wave and shock wave for a certain initial vehicle number. The comparison between the upgrade and the downgrade indicates that the value of saturated flux of the downgrade is larger than that of the upgrade under the same condition. This result is in accordance with the real traffic.
Ⅳ. Two extended cooperative driving lattice hydrodynamic models are proposed by considering the backward looking effect. The influnce of more preceding vehicles and one following vehicle on the studied vehicle is investigated. And the anisotropy of traffic flow is further discussed through examining the negative propagation velocity as the effect of following vehicle is involved.
In light of the different influences of the preceding and following vehicles on the considered vehicle in real traffic, the proper forward and backward optimal velocities are defined, respectively. The neutral stability line is obtained by using the linear stability theory and it is found that considering the following vehicle effect and adopting the appropriate backward optimal velocity funciton could lead to the improvement of the traffic flow stability. The modified Korteweg-de Vries equations (the mKdV equation, for short) near the critical point are derived by using the nonlinear perturbation method to show that the traffic jam could be described by the kink-antikink soliton solutions for the mKdV equations. It is shown that in the unstable region, to a certain extent, the negative propagation velocity appears as the effect of following vehicle is involved. Moreover, the related anisotropy of traffic flow with the negative propagation velocity is discussed in detail and corresponding physical nature is explored.
Ⅴ. Based on two extended lattice hydrodynamic models, the traffic characteristics are anaylzed with considering the effect of stepwise acceleration of a vehicle, in which emphasis is particularly laid on the investigation of the influence of optimal velocity function on the system stability.
With the consideration of the behavior of stepwise vehicle acceleration, two extended lattice hydrodynamic models are proposed. A new optimal velocity function is introduced to take the stepwise acceleration into account and to fit the obtained results to observed data better than those with the original lattice hydrodynamic models, which leads to a better description of the dynamical phase transition for single-lane traffic flow. Linear stability analysis shows that the multiple phase transitions occur through considering the effect of stepwise acceleration, and their properties depend on the vehicle density and sensitivity. Moreover, it is concluded that the complexity degree of phase transition is closely related with the turning points of the introduced optimal velocity function. The validity and correctness of the presented model is confirmed by direct simulations.
The final chapter of this dissertation is devoted to a summary and prospect of further study of the road traffic flow.
引文
[1].首部城市蓝皮书发布大城市交通拥堵问题突出,http://www.ce.cn/cysc/cysczh/200712/20/t20071220_13984389.shtml,2007-12-20.
[2].交通拥堵令美经济一年损失630亿,http://www.sina.com.cn http://news.sina.com.cn/w/2004-09-09/10053627222s.shtml,2004-9-9
[3].Helbing D.,Traffic and Related Self-driven Many Particle Systems.Reviews of Modern Physics,2001,73(4):1067-1141.
[4].国外公交运营管理面面观(上),http://www.china.com.cn/news/txt/2007-10/08/content_9013314_2.htm,2007-10-8
[5].京:公交IC卡面临二次搁浅,http://news.stock888.net/040615/101,1280,878518,00.shtml,2004-6-15
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