車流動力模式之構建與模擬-以波茲曼輸運方程為基礎

Abstract

本研究提出一由波茲曼方程 (Boltzmann equation) 推導出巨觀動態車流模式之完整架構。波茲曼方程由Prigogine和Herman (1971)引用於描述動態車流，許多後續研究延續其觀念探討不同之車流行為。相關研究著眼於車流鬆弛行為、調整行為與互動行為之模式構建。由Prigogine和Herman所發展之車流波茲曼方程僅考慮了隨著時間與空間變化的相空間分布，但忽略了加速度之影響。Paveri-Fontana (1975) 修正了他們的模式，研究中加入加速度與隨時間變化之期望速度分配之影響。Helbing(1995, 1997, 2000)、Hoogendoorn與Bovy (1998, 2000, 2001)等人則延伸Paveri-Fontana之研究，推導出一系列之巨觀動態模式。然而，其研究中引用的二階動差函數在交通上並無實質意義。因此，本研究首先修正二階動差函數，以個別車輛之速度變異數代入模式推導當中，推導巨觀車流動力模式。關於車輛加速度，本研究藉由描述車輛間互動之交通場 (traffic field) 構建模式，所構建出之車流擴散模式 (traffic dispersion model) 除了描述加速度、車輛間之互動之外，亦在系統趨向均衡之假設下，將密度按均衡狀態分配 (equilibrium distribution) 於多車道道路上。本研究於二維平面空間探討多車道車流，主要優點在於可簡化討論車輛超車與變換車道行為，針對車道內不滿足跟車行為之車種，如：機車，亦能藉著二維空間之車流擴散模式描述。將車流擴散模式與有限空間限制條件結合，模式可應用至多種駕駛行為車流。將車流擴散模式與巨觀車流動力模式結合，即為動態多駕駛行為多車道車流模式。本研究亦發展模式求解之數值方法，所解之模式為偏微分方程系統，因此本研究引用分解演算法 (decoupling scheme) 進行模擬，並以簡化之漂移擴散模式 (drift-diffusion model) 為例探討。隨機動態模式構建為另一探討重點，鑒於模式之複雜度，模式推導時以多項式近似非線性項，期簡化隨機動態模式之計算複雜度。最後，本研究以動態未來研究方向與自我評述作為結論，期望作為相關研究之參考與基礎。

This study proposes a whole derivation of dynamic traffic flow model from Boltzmann equation to macroscopic models. Boltzmann-like equation is employed to describe dynamic traffic flow by Prigogine and Herman (1971). Many researches follow their concept and discuss different traffic behavior. Mostly, they endeavor to formulate the relaxation, adjusting and interaction between vehicles. Boltzmann-like equation developed by Prigogine and Herman only discussed distribution with time and spatial evolution but without acceleration effect. Paveri-Fontana (1975) modified their model and considered the influence of acceleration and time dependent desired speed distribution. Helbing (1995, 1997, 2000), Hoogendoorn and Bovy (1998, 2000, 2001) developed a series of macroscopic systems from Paveri-Fontana’s model. However, the second order moment function they employed does not have physical meaning in traffic flow. In this study, individual velocity variance is employed as the second moment function to modify the model. Furthermore, this study introduces the concept of traffic field, which describes the acceleration effect interacting among vehicles so as to distribute density on a multilane road, into Boltzmann equation. The multilane domain is considered as a two-dimensional space. The reason we consider a multilane domain as a two-dimensional space is that the driving behavior of road users may not be restricted by lanes, especially motorcyclists. According to the traffic field and the equilibrium assumption, a traffic dispersion model is obtained. The finite-space requirement of multiclass users can also be described by the dispersion model. By coupling the gas-kinetic model and the dispersion model, a new self-consistent multiclass multilane traffic model is presented. Numerical simulation methods are also developed in this study. Since the model is a system of partial differential equations, a decoupling scheme is suggested and employed to simulate the drift-diffusion model, which is a simplified model of the self-consistent model. Stochastic dynamic traffic flow modeling is another research topic discussed in this study. A decomposed polynomial is employed to approximate the nonlinear term of the model because of the computing complexity. Conclusions of this study and the perspectives of traffic flow researches are discussed in the end of this dissertation.