二维连续型动态交通分配模型及其数值模拟
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摘要
动态交通分配模型及其算法构成了智能交通系统中交通流诱导的理论基础。满足Wardrop第一或第二平衡原理的交通分配模型统称为平衡模型,否则,称为非平衡模型。基于交通系统的不同抽象方式,交通分配模型又可分为离散型(或微观)模型和连续型(或宏观)模型。其中,连续型模型是路网的一种较为抽象的表达,主要应用于大范围区域交通研究的初期规划和建模阶段,从宏观角度关注区域总体趋势、格局分布以及用户群体路径选择行为等。与传统的离散型模型相比较,连续型模型在国内外的研究并不多见；但较之前者,后者有诸多优势,这主要表现在它们易于描述规模庞大/路段稠密的城市交通系统,便于刻画庞大用户群体的路径选择行为,且能更好地诠释动态交通系统的宏观交通特性(如交通密度、速度、流量),从而为城市交通系统的规划与设计提供有用的信息。鉴于此,本文采用连续方法研究动态交通分配模型及其数值模拟。
本文的主要研究内容包括数学建模、算法设计和模拟应用三部分。
第一部分(第二章)(1)将连续型单用户等级和多用户等级静态交通平衡分配模型推广为满足动态用户最优原则(即反应型和预测型动态用户最优条件)的动态交通平衡分配模型, (2)将一维各向同性和各向异性车辆交通流动力学模型推广为描述动态网络交通流的非平衡分配模型,从而得到一系列连续型动态交通平衡与非平衡分配模型(即流体力学中非线性偏微分方程组的耦合系统),并对这些连续型模型作相关的理论分析,讨论模型粘性解存在性和模型线性稳定性条件等。
第二部分(第三章)构造求解动态交通平衡与非平衡分配模型的混合数值方法,主要包括非结构网格上交通流量守恒方程空间离散的高分辨率有限体积方法,用户最优路径选择模型(即动态或静态Hamilton-Jacobi方程)空间离散的迎风差分方法和有限元方法,以及半离散方程的显式TVD Runge-Kutta时间离散方法等。
第三部分将所推广的连续型动态交通平衡与非平衡分配模型及其算法用于研究一些典型的城市交通问题,包括(1) (第四章)将单用户等级预测型动态交通平衡分配模型及其算法用于研究拥有单一吸收点(如城市商业中心)的城市交通系统内预测型单向路网交通流的平衡分配问题; (2)(第五章)将多用户等级反应型动态交通平衡分配模型及其算法用于研究拥有两对出入口的地铁站台内反应型双向行人流的平衡分配问题；(3)(第六章)将单用户等级反应型动态交通非平衡分配模型及其算法用于研究设置有障碍物的步行设施内反应型单向行人流的非平衡分配问题。数值模拟结果表明,连续型动态交通分配模型可以短时预测动态网络交通流(包括路网交通流和行人交通流)的宏观交通特性,交通拥挤形成与消散过程以及用户路径选择行为等。这些定性或定量的结果将对交通系统的规划与设计、交通控制与管理等提供有用的信息。
综上所述,本文发展了一系列描述动态网络交通流的连续型模型,并构造了模型求解的混合数值方法,较为系统地研究了规模庞大/路段稠密的城市交通系统中动态交通分配问题。论文中涉及到的模型及其数值模拟结果为交通科学的理论研究和应用实践提供了新的参考。
The dynamic traffic assignment (DTA) models together with their solution algo-rithms constitute the theoretical basis of intelligent transportation systems (ITS) for the guidance of traffic flow. These satisfying Wardrop's first or second principles of equi-librium are classified as the traffic equilibrium assignment models, and the others as the traffic non-equilibrium assignment models. By the abstraction of a traffic system, the modeling approaches can be either discrete or continuous. The resultant continuum models concisely depict traffic networks with the complicatedly intersected roads and are mainly used for the initial planning and modeling phase in broad-scale regional traffic studies. The focuses hereby are on the general trend and the distribution pattern of area transportation networks as well as path-choice behavior of any user group at the macroscopic level. Compared with the discrete modeling approach, there are many advantages for the continuum modeling approach. First, it is particularly suited for de-picting a large/dense transportation system. Second, it conveniently depicts a collective path-choice behavior of large amounts of travelers. Third, it provides full information with the quantities of important macroscopic variables (the density, average velocity and flow rate) for the construction and management of transportation networks.
For the aforementioned advantages, this dissertation focuses on the DTA mod-els and algorithms based upon the continuum assumption. The contents are arranged to include three major subjects:(1) mathematical modeling; (2) design of numerical schemes; and (3) the simulation applications, which are indicated as follows.
In Chapter 2, the continuum single and multiple user-class static traffic equilib-rium assignment models are extended to the dynamic ones which satisfies the dynamic user-optimal principles (i.e. the reactive and predictive user-optimal conditions). Fur-thermore, both one dimensional (1D) isotropic and anisotropic vehicle traffic flow mod-els are extended to deal with the assumed two dimensional (2D) transportation systems. The resultant models substantially enhance the nonlinearity and the dynamic property of traffic flow on road networks and might truly reflect non-equilibrium transportation systems. Theoretical properties including solution existence and linear stability of these models are also investigated.
In Chapter 3, some hybrid numerical methods are constructed to solve the devel- oped models. The solution procedure includes three major steps in the following order. First, the unstructured high-resolution finite volume methods are designed for the space discretization of the flow conservation equations. Second, the upwind difference and finite element schemes are used for the space discretization of the user-optimal path choice models, which are essentially the dynamic or static Hamilton-Jacobi equations. Finally, the explicit TVD Runge-Kutta methods are applied for the time discretization of the resultant ordinary differential equations.
In Chapters 4-6, the proposed equilibrium and non-equilibrium models and their solution algorithms are applied to study several typical urban DTA problems. Chapter 4 investigates the single user-class predictive dynamic traffic equilibrium assignment pat-tern of traffic flow on a road network with a single absorption point (e.g., a central busi-ness district); Chapter 5 deals with the multiple user-class reactive dynamic traffic equi-librium assignment pattern of traffic flow on a dense network (e.g., bi-direction pedes-trian flow moving in a railway platform with two pairs of entrances and exits); Chapter 6 applies the non-equilibrium isotropic model to simulate the single user-class reactive dynamic traffic non-equilibrium assignment pattern of unidirectional pedestrian flow passing a walking facility with an obstruction. All these numerical simulations indicate that macroscopic traffic characteristics of dynamic traffic flow on networks, the forma-tion and dissipation process of traffic congestion, and travelers' path-choice behaviors can be predicted in short term by the proposed continuum DTA models. Both qualita-tive and quantitative results will provide useful information for planning and design of an urban transportation system as well as for traffic control and management.
In summary, this dissertation systematically studies the DTA problem for a large/dense transportation system through the development of 2D continuum models and the construction of hybrid numerical schemes. The contents together with the sim-ulation results as a whole will surely provide new references for theoretical researches and practical applications in the transportation science.
本文的主要研究内容包括数学建模、算法设计和模拟应用三部分。
第一部分(第二章)(1)将连续型单用户等级和多用户等级静态交通平衡分配模型推广为满足动态用户最优原则(即反应型和预测型动态用户最优条件)的动态交通平衡分配模型, (2)将一维各向同性和各向异性车辆交通流动力学模型推广为描述动态网络交通流的非平衡分配模型,从而得到一系列连续型动态交通平衡与非平衡分配模型(即流体力学中非线性偏微分方程组的耦合系统),并对这些连续型模型作相关的理论分析,讨论模型粘性解存在性和模型线性稳定性条件等。
第二部分(第三章)构造求解动态交通平衡与非平衡分配模型的混合数值方法,主要包括非结构网格上交通流量守恒方程空间离散的高分辨率有限体积方法,用户最优路径选择模型(即动态或静态Hamilton-Jacobi方程)空间离散的迎风差分方法和有限元方法,以及半离散方程的显式TVD Runge-Kutta时间离散方法等。
第三部分将所推广的连续型动态交通平衡与非平衡分配模型及其算法用于研究一些典型的城市交通问题,包括(1) (第四章)将单用户等级预测型动态交通平衡分配模型及其算法用于研究拥有单一吸收点(如城市商业中心)的城市交通系统内预测型单向路网交通流的平衡分配问题; (2)(第五章)将多用户等级反应型动态交通平衡分配模型及其算法用于研究拥有两对出入口的地铁站台内反应型双向行人流的平衡分配问题；(3)(第六章)将单用户等级反应型动态交通非平衡分配模型及其算法用于研究设置有障碍物的步行设施内反应型单向行人流的非平衡分配问题。数值模拟结果表明,连续型动态交通分配模型可以短时预测动态网络交通流(包括路网交通流和行人交通流)的宏观交通特性,交通拥挤形成与消散过程以及用户路径选择行为等。这些定性或定量的结果将对交通系统的规划与设计、交通控制与管理等提供有用的信息。
综上所述,本文发展了一系列描述动态网络交通流的连续型模型,并构造了模型求解的混合数值方法,较为系统地研究了规模庞大/路段稠密的城市交通系统中动态交通分配问题。论文中涉及到的模型及其数值模拟结果为交通科学的理论研究和应用实践提供了新的参考。
The dynamic traffic assignment (DTA) models together with their solution algo-rithms constitute the theoretical basis of intelligent transportation systems (ITS) for the guidance of traffic flow. These satisfying Wardrop's first or second principles of equi-librium are classified as the traffic equilibrium assignment models, and the others as the traffic non-equilibrium assignment models. By the abstraction of a traffic system, the modeling approaches can be either discrete or continuous. The resultant continuum models concisely depict traffic networks with the complicatedly intersected roads and are mainly used for the initial planning and modeling phase in broad-scale regional traffic studies. The focuses hereby are on the general trend and the distribution pattern of area transportation networks as well as path-choice behavior of any user group at the macroscopic level. Compared with the discrete modeling approach, there are many advantages for the continuum modeling approach. First, it is particularly suited for de-picting a large/dense transportation system. Second, it conveniently depicts a collective path-choice behavior of large amounts of travelers. Third, it provides full information with the quantities of important macroscopic variables (the density, average velocity and flow rate) for the construction and management of transportation networks.
For the aforementioned advantages, this dissertation focuses on the DTA mod-els and algorithms based upon the continuum assumption. The contents are arranged to include three major subjects:(1) mathematical modeling; (2) design of numerical schemes; and (3) the simulation applications, which are indicated as follows.
In Chapter 2, the continuum single and multiple user-class static traffic equilib-rium assignment models are extended to the dynamic ones which satisfies the dynamic user-optimal principles (i.e. the reactive and predictive user-optimal conditions). Fur-thermore, both one dimensional (1D) isotropic and anisotropic vehicle traffic flow mod-els are extended to deal with the assumed two dimensional (2D) transportation systems. The resultant models substantially enhance the nonlinearity and the dynamic property of traffic flow on road networks and might truly reflect non-equilibrium transportation systems. Theoretical properties including solution existence and linear stability of these models are also investigated.
In Chapter 3, some hybrid numerical methods are constructed to solve the devel- oped models. The solution procedure includes three major steps in the following order. First, the unstructured high-resolution finite volume methods are designed for the space discretization of the flow conservation equations. Second, the upwind difference and finite element schemes are used for the space discretization of the user-optimal path choice models, which are essentially the dynamic or static Hamilton-Jacobi equations. Finally, the explicit TVD Runge-Kutta methods are applied for the time discretization of the resultant ordinary differential equations.
In Chapters 4-6, the proposed equilibrium and non-equilibrium models and their solution algorithms are applied to study several typical urban DTA problems. Chapter 4 investigates the single user-class predictive dynamic traffic equilibrium assignment pat-tern of traffic flow on a road network with a single absorption point (e.g., a central busi-ness district); Chapter 5 deals with the multiple user-class reactive dynamic traffic equi-librium assignment pattern of traffic flow on a dense network (e.g., bi-direction pedes-trian flow moving in a railway platform with two pairs of entrances and exits); Chapter 6 applies the non-equilibrium isotropic model to simulate the single user-class reactive dynamic traffic non-equilibrium assignment pattern of unidirectional pedestrian flow passing a walking facility with an obstruction. All these numerical simulations indicate that macroscopic traffic characteristics of dynamic traffic flow on networks, the forma-tion and dissipation process of traffic congestion, and travelers' path-choice behaviors can be predicted in short term by the proposed continuum DTA models. Both qualita-tive and quantitative results will provide useful information for planning and design of an urban transportation system as well as for traffic control and management.
In summary, this dissertation systematically studies the DTA problem for a large/dense transportation system through the development of 2D continuum models and the construction of hybrid numerical schemes. The contents together with the sim-ulation results as a whole will surely provide new references for theoretical researches and practical applications in the transportation science.
引文
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