Macroscopic fundamental diagram (MFD) has been receiving increasing attention recently due to its potential to describe traffic dynamics and guide the design of traffic control schemes at the network level. Perimeter control and route guidance are two main MFD-based traffic control approaches. However, current MFD-based perimeter control seldom considers travelers’ route choice behavior, while MFD-based route guidance studies usually assume directly that travelers would follow the guidance and neglect the effects of traffic control. This paper aims to integrate the MFD-based perimeter control (i.e., the behavior of a system manager) and the dynamic user equilibrium based route choice behavior (i.e., the behavior of travelers) into one rigorous mathematical framework. Given a traffic network that has been divided into multiple homogeneous regions, we use MFD to describe the dynamics of each region, and use point queue model to capture the dynamics of queues formed at the boundaries. Besides, we model travelers' route choice behavior by the instantaneous dynamic user equilibrium (IDUE) principle, and design an efficient range perimeter control method from the system perspective. We model the interactions between the system manager and the travelers as a non-zero sum, non-cooperative differential game, where the system manager aims to improve the system performance while travelers try to minimize their own travel times. Meanwhile, they share the common constraints (i.e., MFD dynamics and point queue dynamics at boundaries). Mathematically, this leads to a differential complementarity system (DCS). We propose a time-stepping approach to discretize and solve the DCS model, based on which the solution existence and convergence are also established. Numerical results show that the proposed method can limit the vehicle accumulations within the efficient range of each region, which helps improve the network performance. Compared with the condition without perimeter control, the proposed control method can improve network-wide traffic performance up to 14.18%.

宏观基本图（MFD）由于具有描述流量动态并指导网络级别流量控制方案设计的潜力，因此最近受到越来越多的关注。周界控制和路线引导是基于MFD的两种主要交通控制方法。但是，当前基于MFD的周边控制很少考虑旅行者的路线选择行为，而基于MFD的路线指南研究通常直接假设旅行者会遵循该指南并忽略交通管制的影响。本文旨在将基于MFD的边界控制（即系统管理员的行为）和基于动态用户平衡的路线选择行为（即旅行者的行为）整合到一个严格的数学框架中。给定一个已划分为多个同质区域的交通网络，我们使用MFD来描述每个区域的动态，并使用点队列模型来捕获在边界处形成的队列的动态。此外，我们利用瞬时动态用户平衡（IDUE）原理对旅行者的路线选择行为进行建模，并从系统角度设计一种有效的距离范围控制方法。我们将系统管理员与旅行者之间的互动建模为一个非零和，非合作的差分博弈，其中系统管理员旨在提高系统性能，而旅行者则试图尽量减少自己的旅行时间。同时，它们共享公共约束（即，MFD动态和边界处的点队列动态）。从数学上讲，这导致了差分互补系统（DCS）。我们提出了一种时间步长方法来离散化和求解DCS模型，在此基础上，还建立了解的存在性和收敛性。数值结果表明，该方法可以将车辆的累积量限制在每个区域的有效范围内，从而有助于提高网络性能。与没有边界控制的情况相比，该控制方法可以将整个网络的流量性能提高多达14.18％。