We consider a combined train control and scheduling problem involving multiple trains in a railway line with a predetermined departure/arrival sequence of the trains at stations and meeting points along the line. The problem is formulated as a multiphase optimal control problem while incorporating complex train running conditions (including undulating track, variable speed restrictions, running resistances, speed-dependent maximum tractive/braking forces) and practical train operation constraints on departure/arrival/running/dwell times.Two case studies are conducted. The first case illustrates the control and scheduling problem of two trains in a small artificial network with three nodes, where one train follows and overtakes the other. The second case optimizes the control and timetable of a single train in a subway line. The case studies demonstrate that the proposed framework can provide an effective approach in solving the combined train scheduling and control problem for reducing energy consumption in railway operations.
This paper extends the framework of "rational behavior adjustment process" (RBAP) to incorporating the "boundedly rational user equilibrium" (BRUE). The proportional-switch adjustment process (PSAP) and the network tatonnement process (NTP) are extended to the BRUE case, and their dynamical equations are shown to be Lipschitz continuous, which guarantees the global uniqueness of the classical solutions. A special group of the BRUE-RBAP is proposed, for which the path flows would increase if the paths are in the acceptable path set, and would decrease otherwise. Classical solutions to this special group of models may not exist. Stability of the BRUE-RBAP with classical solutions is proved with separable link travel cost functions. For non-separable link travel cost functions, the stability of the BRUE-PSAP is proved. Numerical examples are presented to demonstrate the evolution processes of BRUE-PSAP and BRUE-NTP under various bounded rationality thresholds and various initial states. The applicability of BRUE-PSAP in larger networks with asymmetric link travel cost functions is also illustrated.
Abstract. This paper offers a new look at the network flow dynamics from the viewpoint of physics by demonstrating that the traffic system, in terms of the aggregate effects of human behaviors, may exhibit like a physical system. Specifically, we look into the day-to-day evolution of network flows that arises from travelers' route choices and their learning behavior on perceived travel costs. We show that the flow dynamics is analogous to a damped oscillatory system. The concepts of energies are introduced, including the potential energy and the kinetic energy. The potential energy, stored in each link, increases with the traffic flow on that link; the kinetic energy, generated by travelers' day-to-day route swapping, is proportional to the square of the path flow changing speed. The potential and kinetic energies are converted to each other throughout the whole flow evolution, and the total system energy keeps decreasing owing to travelers' tendency to stay on their current routes, which is analogous to the damping of a physical system. Finally, the system will approach the equilibrium state with minimum total potential energy and zero kinetic energy. We prove the stability of the day-to-day dynamics and provide numerical experiments to elucidate the interesting findings.
This paper examines existing day-to-day models based on a virtual day-to-day route choice experiment using the latest mobile internet techniques. With the realized day-to-day path flows and path travel times in the experiment, we calibrate several welldesigned path-based day-to-day models who take the Wardrop's user equilibrium as (part of) their stationary states. The nonlinear effects of path flows and path time differences on the path swapping are then investigated. Participants' path preferences, time-varying sensitivity and learning behavior in the day-to-day process are also examined. The prediction power of various models with various settings (nonlinear effects, time-varying sensitivity, and learning) is compared. Assumption of rational behavior adjustment process in Yang and Zhang (2009) is further verified. Finally, evolutions of different Lyapunov functions used in the literature are plotted and no obvious diversity is observed.
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