BECS Collaborative Research: Modeling the Dynamics of Traffic User Equilibria Using Differential Variational Inequalities

本项目的目标是采用一种新的数学范式，即微分变分不等式（DVI）来研究交通分析中的两个基本问题，即具有流量传播和延迟的连续时间动态用户平衡问题，以及面对网络中断的交通不平衡问题。我们的目标是发展一种非线性动力系统理论，将经典的常微分方程（ODE）方法与当代数学规划的进展相结合，研究短时（如日内动力学）和长时间（如瞬态动力学）交通流的平衡过程，回答以下问题：如何计算动态用户平衡，由于网络中断，流量如何从不平衡状态演变到平衡状态，以及当前流量状态离平衡状态有多近。本研究的结果将导致一个新的范式，将交通网络分析方法与数学科学的最新进展相结合，并弥合当前交通分析实践与整体考虑短期和长期交通动态的需求之间的差距。这项研究将是开发交通动态分析数学框架的第一步，从长远来看，它有可能彻底改变传统的交通分析方式，如网络信号优化、动态拥堵定价、交通中断后的应急管理等。由于平衡分析和系统动力学在其他科学和工程领域的广泛应用，本研究中发展的概念和方法也可以为这些领域的基于平衡的分析提供新的视角。研究团队计划将研究成果融入本科和研究生课程，并让多学科背景的本科生和研究生参与本研究。该小组还计划与各种运输管理机构的决策者和从业人员接触，对他们来说，拟议研究的结果将使他们能够更好地管理运输系统。

The goal of this project is to employ a novel mathematical paradigm, known as a differential variational inequality (DVI), to study two fundamental problems in transportation analysis, namely, the continue-time dynamic user equilibrium with flow propagation and delays, and the problem of traffic disequilibrium in the face of network disruption. Our aim is to develop a nonlinear dynamical system theory that combines classical ordinary differential equation (ODE) methods with contemporary mathematical programming advances to study the equilibrium of short-time (e.g. within-day dynamics) and the equilibration process of longer-time (e.g. transient dynamics) traffic flows, answering questions such as: how to compute dynamic user equilibria, how traffic evolves from a disequilibrium state, due to network disruptions, toward an equilibrium, and how close the current traffic state is to an equilibrium. Results from this study will lead to a new paradigm integrating traffic network analysis methods with recent advances in mathematical science, and bridge the gaps between current practice of traffic analysis and the needs to consider short-term and long-term traffic dynamics in a holistic manner. The research will be the first step to develop a mathematical framework for traffic dynamic analysis that, in a long term, has the potential to revolutionize the traditional way of conducting transportation analyses such as network signal optimization, dynamic congestion pricing, emergency management after disruptions, among others. Since equilibrium analysis and system dynamics are widely used in other science and engineering fields, the concepts and methodologies developed in this research can also provide new perspectives for equilibrium based analysis in these fields. The research team plans to integrate research findings to undergraduate and graduate courses, and involve undergraduate and graduate students with multidisciplinary backgrounds in this research. The team also plans to reach out to policy makers and practitioners at various transportation management agencies for whom findings from the proposed research will enable them to better manage transportation systems.