Dynamic User Equilibrium with Side Constraints: Interpretation and Application to Traffic Management

In managing a traffic network, it is desirable in some case to control the traffic such that the traffic volume on a certain link does not exceed a predefined threshold (e.g. its capacity). One method is to introduce physical capacity constraints for those links. Another method is to use an asymptotic travel time function that forces the travel time of the link to approach infinity when the flow approaches the link’s threshold. The main focus in this paper is on the latter approach, i.e. introducing explicit capacity constraint on the link volume. This type of constraint is widely referred to as ‘side constraint’ in traffic assignment field. It has been shown that, under the static user equilibrium (UE), the traffic assignment problem with side-constraints generates a flow pattern which satisfied the modified Wardrop UE principle in the sense that the equilibrium condition holds in terms of generalized travel costs: the original Wardrop equilibrium travel costs plus link queuing delays. In addition, the Lagrange multiplier as derived from the side-constraint of the assignment problem can be viewed as the traffic control parameter (delay or adjusted travel cost function) on that link to ensure that the UE link flow satisfy the side-constraint. This paper aims to extend the static side-constraint traffic assignment to the dynamic framework. The motivation is based on the application of this approach to the real-world traffic management scheme which involves a highly dynamical system with temporal demand and supply interaction. The dynamic framework allows a time-dependent control scheme to be derived from the proposed model. In this paper, we focus on problem formulation and analysis of such side-constraint problem with the Dynamic User Equilibrium (DUE) using linear exit-flow dynamic link model. Under the dynamic framework, the inflow and outflow of each link are implicitly restricted by the dynamic link model already. Thus, the side-constraint will be related to the ‘length’ of queue to ensure the desired congestion level on different links. In this paper, we formulate this problem as a continuous time optimal control problem with state dependent time delays and saturation, where the queue length is taken as a state variable and the inflow to a link is taken as a control variable. The state dependent formulation avoids the pitfalls of improper flow propagation and violation of causality of the original exit-flow dynamic link model. However, the induced state dependent time delays by such formulation complicates the control problem. The Pontryagin Minimum Principle is applied to our analysis to obtain the necessary condition. The Lagrange multipliers is obtained as the costate variable. After obtaining the optimality condition, we provide the interpretation of the costate variable as compared to the Lagrange multiplier in the static case.