TRANSYT derivatives for area traffic control optimisation with network equilibrium flows

A bi-level mathematical programming technique has previously been proposed, in which an area traffic control problem is dealt with as the upper level problem whilst the users' equilibrium traffic assignment is dealt with as the lower level problem. The performance index of the upper level problem is defined as the sum of a weighted linear combination of the rate of delay and number of stops per unit time for all traffic streams, which is evaluated by the traffic model from TRANSYT. Approximate mathematical expressions for various components of the performance index and the average delay to a vehicle at the downstream junction in the TRANSYT model for both undersaturated and oversaturated links are considered. As a step towards heuristics for finding good solutions to this form of the bi-level problem, in this paper, explicit expressions have been obtained for the partial derivatives of all these quantities with respect to the signal control variables. Numerical calculations have been shown on example test network and good agreement of the results with these derivatives determined by numerical differentiation is obtained. (C) 2003 Elsevier Science Ltd. All rights reserved.