Solving the dynamic user equilibrium problem via sequential convex optimization for parallel horizontal queuing networks

1 This article considers the dynamic user equilibrium (DUE) problem for parallel networks. The network 2 dynamics are modeled using a Godunov discretization of the Lighthill-Williams-Richards partial differential 3 equation with a trapezoidal flux function. The model is augmented with an additional constraint that prevents 4 vehicle holding which is a flaw in the discretization. The departure rates are assumed to be fixed. Under 5 these assumptions, we show that the future allocation of the demand among the different paths at the origin 6 has no effect on the travel time of the vehicles already in the network. This enables us to show that the DUE 7 for a fixed time steps horizon can be decomposed into a series of static UE problems and solved sequentially. 8 Thus, the DUE problem can be solved as a sequence of convex optimization problems. 9 Jean-Baptiste Lespiau, Samitha Samaranayake, Alexandre M. Bayen 2 INTRODUCTION 10 Dynamic traffic assignment (DTA) models have been studied since the seminal works of Merchant and 11 Nemhauser in 1978 (1, 2). The principle of the user equilibrium (UE) (or Wardrop equilibrium) alloca12 tion, in which all travelers with the same origin-destination pair see the same travel time, was introduced 13 by Wardrop (3) in the context of static traffic assignment and has been expanded to dynamic models by 14 Beckman (4). The system optimal (SO), also introduced by Wardrop, corresponds to the minimization of 15 the total travel time of all agents. Both the UE and the SO have many applications in traffic planning and 16 intelligent transportation systems (ITS). The UE is used to represent the behavior of selfish agents and the 17 SO is an upper bound in terms of network efficiency. 18 The static user equilibrium has been studied extensively in game theory (5), since it is a particular 19 case of the Nash equilibrium concept. Several algorithms give arbitrarily good approximations of the static 20 UE. Blum (6) presents a class of no-regret algorithms that give a strategy that, if applied by all agents, 21 converges to a Nash equilibrium in static games, when latency functions are increasing, continuous and 22 have bounded slopes. Fischer (7) presents another algorithm based on a replication-exploration protocol. 23 The dynamic user equilibrium (DUE) has been formulated using different hypothesis for the de24 cision variables (route and/or departure time) and different models for the dynamics. Huang and Lam (8) 25 study the simultaneous route and departure time choice problem using vertical queues to model the traffic. 26 Lo and Stezo (9) formulate the DUE as a finite dimensional variational inequality problem and propose an 27 alternating direction method to solve it. They also extend their model to handle elastic travel demand (10). 28 Friesz and al. (11) present a continuous-time network loading procedure based on the Lighthill-Williams29 Richards (LWR) model, formulate the DUE as a variational inequality problem and solve it using a fixed 30 point algorithm. All of these methods are computationally complex even in simple networks. 31 We consider a macroscopic model (i.e. traffic flows and numbers of agents have continuous values. 32 It is based on the assumption that one agent represents a negligible fraction of the overall traffic) based on 33 the LWR partial differential equation (12, 13). Specifically, we use a Godunov discretization (14) of this 34 equation, also known as the Cell Transmission Model (CTM) (15, 16) in transportation literature. We assume 35 that the relationship between flow and density can be approximated to a first order using the trapezoidal 36 fundamental diagram as seen in empirical studies (17). 37 We focus on the single source, single destination DUE problem with parallel paths, where the desti38 nation is not capacity restricted. As depicted in Fig. 1, each path is composed of an initial buffer of infinite 39 capacity linked to a road network. Each path has its own buffer and the travel times of paths are independent 40 from each other. 41 Single origin Single destination