Revisiting the traffic flow observability problem: A matrix-based model for traffic networks with or without centroid nodes

• Published in: Transportation research. Part B: methodological 2024-12, Vol.190, p.103099, Article 103099
• Main Authors: Zhuo, Yue, Shao, Hu, Lam, William H.K., Tam, Mei Lam, Cao, Shuhan
• Format: Article
• Language: English
• Subjects: Flow observability problem; Graph theory and matrix analysis; Node flow conservation; Planar and non-planar networks
• ISSN: 0191-2615
• DOI: 10.1016/j.trb.2024.103099

•Introduces a graph theory model for optimizing passive sensor deployment in traffic networks.•Creates a virtual network and uses an isomorphic graph to maintain node and link consistency.•Proposes two formulas for calculating the minimum observable links in different networks.•Develops a matrix framework to handle flow conservation, analyzing chords, cut sets, and loops.•Proposes an optimization model to quantify the impact of budget constraints. This study introduces a graph theory-based model that addresses the link flow observability problem in traffic networks by optimizing passive sensor deployment. The model aims to determine the minimal number of sensors and their optimal placement. It constructs a virtual network and uses isomorphic graph theory to map between the original and virtual networks, ensuring consistency in nodes, links, and link directions. Two formulas are proposed to calculate the minimum number of observable links required across different networks, factoring in links, ordinary nodes, centroid nodes, and added links. Key concepts such as chords, cut sets, and loops, along with their matrices, are analyzed. A matrix-based framework is developed to consider flow conservation conditions. Results show that solving the full link flow observability problem using node flow conservation equations yields a fixed number of sensors with non-unique deployment schemes, Additionally, a resource-constrained sensor network optimization (RSNO) model is presented, employing null space projection (NSP) as an objective function to quantify the impact of budget constraints particularly under the condition if all the link flows cannot be observed. Numerical examples demonstrate the RSNO model's applications.