Flow exchanges in multi-reservoir systems with spillbacks

Abstract Large-scale traffic flow models based on the Network Macroscopic Fundamental Diagram (MFD) are usually grounded on the bathtub analogy and a conservation equation for vehicle accumulation inside a given urban area. Recent studies have proposed a different approach where the MFD defines the spatial mean speed that is shared by all vehicles in a region while their traveling distance is tracked individually. The former approach is also referred to as “accumulation-based” while the latter is usually named “trip-based”. While extensive studies of both model properties have been carried out for the single reservoir case (a unique region), the multi-reservoir setting still requires some research effort in particular to clearly understand how inflow merge at a reservoir entry and outflow diverge at exit should be managed. These two components play a significant role in the evolution of the whole system, when flows are exchanged between multiple reservoirs. One of the crucial questions is to ensure that congestion properly propagates backwards through a succession of reservoirs when oversaturated situations are observed. In this paper, we propose a thorough analysis of how to handle congestion propagation in the accumulation-based framework with several trip lengths or categories, e.g. internal and external trips. This allows to derive a congestion propagation model for the trip-based approach in a multi-reservoir setting. Based on theoretical considerations and simulation studies, we develop a consistent framework to restrict the inflow and adapt to oversaturated traffic conditions in a reservoir including several trip lengths. Two inflow merging schemes are investigated. The first one is inspired from the existing literature and shares the available supply based on the demand flow ratio at the entry boundary. It is called “exogeneous” in contrast to the second “endogenous” scheme, which shares the supply with respect to the internal accumulation ratio on the different routes. At the reservoir exit, a new outflow diverging scheme is also introduced to better reproduce the effect of queuing vehicles that are prevented from exiting the reservoir when congestion spills back from neighboring reservoirs. Compared to the conventional outflow model, our new approach proves to avoid unrealistic gridlocks when the reservoir becomes oversaturated. Both entry and exit flow models are investigated in details considering the accumulation-based and trip-based frameworks. Finally, the most consistent approach is compared with two other existing MFD models for multiple reservoirs. This demonstrates the importance of properly handling entry and exit flows at boundaries.