In relation to the lower level problem in the bi-level problem (3.0), q = g*(\|/) , the
user equilibrium traffic assignment problem can be form ulated as a variational inequality problem in the following way.
To find a value ^*(Y) G Q (\j/) of ^(\|/) such that
c(g*(V)) ( 9(V) - ^ 0
V ^(\|/) E ^ (Y) = { q{^)'. 9 (V )^ = S / ( V ) A /(x]/)^ = D ^ , /(\|/) > 0 }
(3.40) 3.5.1 Link travel time function
Follow ing the mathematical expressions for the indicators of traffic conditions, by adopting the traffic model from TRA N SY T as an assessm ent tool for the estim ation of the conditions to the performance index for each link, we suppose that the corresponding link travel time function for a particular time period can be expressed as the sum of the un delayed travel time along the link and the estim ated average delay, which is incurred by vehicles on the link leading to the dow nstream signal- controlled junction in that time period.
• V) = (3.41)
Since d ^ ( t ‘ + ) = + d'^ ° ( t ‘ +')
the link travel time function for that time period can be expressed as
. V) = c ° + d ^ + d ' * ° ( t ‘ + ') (3.42) where the relevant mathematical expressions are (3.22), (3.29), (3.34), (3.39), and the
link travel time is a separable function of .
In relation to the uniqueness of the solution for equilibrium link flows, the follow ing inequalities are assumed throughout this work.
8c ,
> 0 , M a e L (3.43)
dc d d ^
where --- = + --- , \/ a e L according to expression (3.42).
D etailed expressions for the derivatives of average delay in terms of the uniform and random plus oversaturation delay with respect to the average flow will be discussed in Chapter 4.
3.6 Conclusions
In this Chapter the bi-level program m ing approach has been used in form ulating the optim isation problem of area traffic control for equilibrium flows. The fundam ental term inology and the corresponding notation throughout this thesis for the area traffic control optimisation have been respectively given in Section 3.2 and 3.3. The upper level and the lower level problem s of the bi-level form ulation have been given respectively in Section 3.4 and 3.5, in w hich the corresponding objective function and the constraint set have been respectively stated. In the following Chapters, the way of searching for the feasible solutions to this bi-level problem of area traffic control optimisation is discussed on the basis of the results provided by this Chapter.
Chapter 4 Sensitivity Analysis
4.0 Introduction
The bi-level formulation for the optim isation of area traffic control leads to a non- convex problem in Chapter 3 due to the non-convexity of the objective function and the non-linearity of the condition for equilibrium network flows. Therefore, only local optimal values for the signal timings can be found for the solution to the bi-level formulation. In this Chapter, the sensitivity analysis approach will be used as an appropriate tool to obtain inform ation about the derivatives o f the rate o f delay and the number of stops with respect to the current signal timings. These derivatives will be used as a basis for the solution m ethod developed later in this work for the bi-level formulation of the optimisation of area traffic control for equilibrium netw ork flows. The arrangement of this Chapter can be described as follows. In the next section, an introduction for the sensitivity analysis approach to the bi-level form ulation will be given. The specification of each elem ent in the derivatives for the bi-level form ulation will be discussed respectively in the follow ing sections. In Section 4.2, the derivatives for all links with respect to the com mon cycle time at current signal timings will be discussed. In Section 4.3, the derivatives for the upstream links which are controlled by a particular signal group at a particular junction with respect to the start and duration of green will be discussed. In Chapter 3, the objective function adopted was the performance index in terms of a w eighted linear com bination of the rate of delay and the number of stops, which was used in the traffic model from TRA N SY T, in which cyclic flow profiles are used to represent vehicle platooning. A ccordingly, for a change in the signal timings of one particular signal group at the corresponding junction, not only the primary effects will be caused on the upstream links by the change in the signal timings but consequent effects will also be caused on the dow nstream and further dow nstream links since the IN profiles for the dow nstream and further downstream links are affected by the OUT profiles for the upstream links which are controlled by that signal group. Therefore, in Section 4.4, the derivatives for the downstream links with respect to the start and duration of green for that signal group will be discussed, where the dow nstream links are the im mediate ones away
from the corresponding signal group. Furtherm ore, in Section 4.5, the derivatives for the further dow nstream links which are the im m ediate links away from the dow nstream links with respect to the start and duration of green for that signal group will be discussed accordingly. Conclusions for this Chapter will be given in Section 4.6.