The well-known Lighthill–Whitham–Richards (LWR) theory is the fundamental pillar for most macroscopic traffic models. In the past, many methods were developed to numerically derive solutions for LWR problems. Examples for such numerical solution schemes are the cell transmission model, the link transmission model, and the variational theory (VT) of traffic flow. So far, the eulerian formulation of VT found applications in the fields of traffic modelling, macroscopic fundamental diagram estimation, multi-modal traffic analyses, and data fusion. However, these studies apply VT only at the link or corridor level. To the best of our knowledge, there is no methodology yet to apply VT at the network level. We address this gap by developing a VT-based framework applicable to networks. Our model allows us to account for source terms (e.g. inflows and outflows at intersections) and the propagation of spillbacks between adjacent corridors consistent with kinematic wave theory (KWT). We show that the trajectories extracted from a microscopic simulation fit the predicted traffic states from our model for a simple intersection with both source terms and spillbacks. We also use this simple example to illustrate the accuracy of the proposed model, and the ability to model complex bottlenecks. Additionally, we apply our model to the Sioux Falls network and again compare the results to those from a microscopic KWT simulation. Our results indicate a close fit of traffic states, but with substantially lower computational cost. The developed methodology is useful for extending existing VT applications to the network level, for network-wide traffic state estimations in real-time, or other applications within a model-based optimization framework.

著名的 Lighthill-Whitham-Richards (LWR) 理论是大多数宏观交通模型的基本支柱。过去，开发了许多方法来数值推导 LWR 问题的解决方案。此类数值求解方案的示例是单元传输模型、链路传输模型和交通流的变分理论 (VT)。到目前为止，VT 的欧拉公式在交通建模、宏观基本图估计、多模态交通分析和数据融合等领域得到了应用。然而，这些研究仅在连接或走廊级别应用 VT。据我们所知，目前还没有在网络级别应用 VT 的方法。我们通过开发适用于网络的基于 VT 的框架来解决这一差距。我们的模型允许我们考虑源术语（例如 交叉口的流入和流出）以及与运动波理论 (KWT) 一致的相邻走廊之间的溢流传播。我们表明，从微观模拟中提取的轨迹符合我们模型中预测的交通状态，用于具有源项和溢出的简单交叉路口。我们还使用这个简单的例子来说明所提出模型的准确性，以及对复杂瓶颈进行建模的能力。此外，我们将我们的模型应用于 Sioux Falls 网络，并再次将结果与微观 KWT 模拟的结果进行比较。我们的结果表明交通状态非常接近，但计算成本大大降低。所开发的方法可用于将现有的 VT 应用程序扩展到网络级别，用于实时网络范围的流量状态估计，