## Optimisation of Area Traffic Control Subject to User Equilibrium Traffic Assignment

### Authors

CHSIOU S-W, University College London, UK

### Description

In relation to calculation of a signal timing plan for coordinated fixed-time control in a given road network in which an entry to exit matrix is given but the link flows are not, a combined technique has been used to formulate this problem by researchers

### Abstract

In relation to calculation of a signal timing plan for coordinated fixed-time control in a given road network in which an entry to exit matrix is given but the link flows are not, a combined technique has been used to formulate this problem by researchers: Allsop (1976), Allsop and Charlesworth (1977) and many others subsequently. There are two approaches in modelling the signal settings and link flows in this combined problem. In both approaches, it is recognized that the link flows are affected by changes in signal settings and thus the hnk flows and signal settings can not be treated independently; instead, the link flows are regarded as functions of signal settings. First, a mutually consistent approach is discussed by Charlesworth (1977), in which the signal settings and link flows are calculated alternately and obtained respectively by solving the signal setting optimisation for given link flows and by solving a user equilibrium traffic assignment for given signal settings. Second, a bi-level programming technique is adopted by Heydecker and Khoo (1990), Yang and Yagar (1995), in which the equilibrium link flows become a constraint set for the optimisation problem of signal settings, in which we may regard the signal setting optimisation as an upper level problem while the problem for a user equilibrium traffic assignment by which the link flows are obtained can be regarded as the lower level problem. In this bi-level programming approach, the solution obtained by minimising total delay for a signal controlled road traffic network subject to user equilibrium assignment can be expected to achieve a lower value than that obtained from the mutually consistent calculation approach (Heydecker and Khoo, 1990).

In this paper, we adopt the N-level programming approach in dealing with this combined problem, in which the optimisation of area uaffic control is dealt with as the upper level problem whilst a user equilibrium traffic assignment is dealt with as the lower level problem. In the upper level problem, opthnisation for an area traffic control is defined by the phase-based signal setting variables: the common cycle time, the starts and durations of green. Also, the performance index 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 in TRANSYT (Vincent 1980). In the' lower level problem, a general expression for a user equilibrium traffic assignment is formulated as a variational inequality problem. Although link flows are affected by the changes in signal settings, there is so far no explicit mathematical form in formulation of this bi-level program. In dealing with this difficulty, Friesz, Tobin, Cho and Metha (1990) used a sensitivity analysis based method, in which the first partial derivatives of the objective and link flows with respect to decision variables have been considered. By use of these derivatives, the perturbations around a current solutiori have been estimated and then the gradient-based solution method was used to identify the descent direction by which an improved solution may be achieved. In this paper, the gradient projection method is used to determine the descent direction at given signal settings and optimal choices of step length along the descent direction. Equal and simultaneous changes in the starts of green for all signal groups at any one junction, corresponding to changes in offsets are particularly considered. Furthermore, mixed search procedure is adopted in finding an improved local optimum. In the following sections, the bi-level program for optimisation of area traffic control subject to user equilibrium traffic assignment is formulated in Section 2. In Section 3, solution method and mixed search procedure to the bi-level program at any given signal settings including the identification of descent direction and optimal choices of step length are both discussed. In Section 4, Allsop and Charlesworth's trial road network is used for an illustrative numerical test of the proposed search procedure and comparison for effectiveness with mutually consistent calculations.

#### Publisher

Association for European Transport