Traffic State Estimation in Heterogeneous Networks with Stochastic Demand and Supply: Mixed Lagrangian–Eulerian Approach:

A network fundamental diagram (NFD) represents the relationship between network-wide average flow and average density. Network traffic state estimation to observe NFD when congestion is heterogeneously distributed, as a result of a time-varying and asymmetric demand matrix, is a challenging problem. Recent studies have formulated the NFD estimation problem using both fixed measurements and probe trajectories. They are often based on a given ground-truth NFD for a single day demand. Stochastic variations in network demand and supply may significantly affect the approximation of an NFD. This study proposes a modified framework to estimate network traffic states to observe NFD while capturing the stochasticity in transportation networks. A mixed integer problem with non-linear constraints is formulated to address stochasticity in the NFD estimation problem. To solve this Nondeterministic Polynomial-hard (NP-hard) problem, a solution algorithm based on the simulated annealing method is applied. The problem is formulated and the solution algorithm is implemented to find an optimal configuration of loop detectors and probe vehicles to estimate the NFD of the Chicago downtown network and capture its day-to-day variations, considering a given available budget. Ground-truth NFDs and estimated NFDs based on a subset of loop detectors and probe vehicles are calculated using a simulation-based dynamic traffic assignment model, which is the best surrogate available to replicate real-world conditions. The main contribution of this study is to capture stochasticity in the demand and supply sides to find a more robust subset of links and trajectories to be acquired for the NFD estimation.