Estimation of urban network capacity with second-best constraints for multimodal transport systems

•Network capacity estimation problem with second-best constraints (NCSC).•Systematic modelling framework for the NCSC based on bi-level model.•Modified improved gradient projection (MIGP) algorithm for combined modal split and traffic assignment.•The capacity estimation method based on sensitivity analysis. Transport network capacity enhancement is a significant aspect of urban transport planning and demand management, and a suitable measurement of the network capacity is of considerable importance. In this paper, the network capacity with second-best constraints (NCSC) is investigated to meet some specific development requirements of urban transport networks. Herein, the network capacity is restricted to an inferior “second-best solution”, due to various concerns/constraints regarding the public transport mode share, serviceability, and emissions, etc. For the sake of presentation, these constraints are termed as second-best constraints, and the NCSC problem can also be referred as second-best network capacity (SBNC) problem. A bi-level model is formulated to analyse the NCSC problem. The upper-level model maximizes the total origin-destination (OD) demand, which incorporates the second-best constraints into consideration. The lower-level model is a transport network equilibrium model, which measures the network performance under a given OD demand pattern. To better investigate some important second-best constraints (e.g., public transport mode share) and also the demand elasticity, the modelling framework is extended to a multimodal transport network. An exact solution method is developed for the NCSC problem; wherein, a modified improved gradient projection (MIGP) algorithm is designed for the lower-level multimodal flow equilibrium problem, and a tailored sensitivity analysis-based (SAB) method is employed for solving the NCSC problem. The proposed models and solution methods are verified by numerical examples, demonstrating that NCSC can be an efficient tool for transport planning and management.