In this Chapter, the optimisation problem of area traffic control and user equilibrium traffic assignm ent has been dealt with as a com bined problem , i.e. the equilibrium network design problem. Various form ulations and the corresponding solution m ethods for this equilibrium network design problem for area traffic control optim isation have been investigated respectively. The bi-level program m ing technique applied to the form ulation of such problem has been considered as an appropriate approach in dealing with the dependence of equilibrium flow on signal timings. Furtherm ore, reviews of sensitivity analysis and the corresponding applications for the bi-level program m e have been conducted. As we have m entioned before, with the background of rigorous mathematical sensitivity analysis for equilibrium network flow s, the bi level program m e for the area traffic control optim isation problem regarded as a constrained optimisation problem can be solved by the techniques using derivatives in m athem atical program m ing for optim isation problems. In the following chapters, the bi-level formulation for the optim isation of area traffic control with equilibrium network flows will be proposed to w hich the solution m ethod in term s o f the derivatives for system performance values with respect to the decision variables of interest will be discussed accordingly.
Table 2.1 Approximate expressions for the derivatives o f the uniform com ponents for upstream links; i.e. V a g
c (sjz
^ 2- ^
- 4>j)
- C 4(4>j)(42 -
(
- 4 (t ) )
- c
note: c is com m on cy c le time; C - iz
c
0g , 4)^ are resp ectively the starts and duration o f effe c tiv e greens for link a controlled by signal
d d d d
controlled group j at junction m in the netw ork, i.e.
39.
30,„ ’
a|.
/^(•) i Qa ’ respectively the arrival rate as represented by the IN profile, average flo w and
saturation flo w for link a and , Z^2 resp ectively the starting and ending tim e in one c y c le for
queue form ed on link a.
Table 2.2 A pproximate expressions for the derivatives of the random plus oversaturation com ponents for upstream links; i.e. V a g Aj^
% ^j m
0 - ^ 1 0
note- ^ - '• ^ K''" ^ r c (2 n. 7)
4 j , ' 20>,I--2C)" ' a»,. ' (i^ .t-2q‘
w here , L^' are respectively the capacity, degree o f saturation and initial queue for link a over
T able 2.3 A pproxim ate expressions for the derivatives of the uniform com ponents for dow nstream links; i.e. \/ b e
dDu dSu
dx dx
d I M )... C i , ' ? d u t)
dt
j: e I I and /j(«) , are defin ed as in T able 2.1.
Table 2.4. The changes in the IN pattern on dow nstream links; i.e. V 6 E
64 67,
.C A (t)(# i4 ) - #3,h)
note: / o r p=J,2,3, Riy^, t) = — , / o r r + c such that Riy^, t +kc) = Riy^, t) ,
1 - g T
k is integer, 4 , is the proportion o f traffic on upstream link a into dow nstream link b, and
F = Ç F is a m od ified sm oothing factor used as a counterpart for F defined in T R A N S Y T ,
is the number o f tim e steps into w hich the c y c le tim e is divided for the evaluation o f perform ance
index. T he values o f , for p-1,2,3 are given as
c , y2=(0/ c + X, - 0,) mod c , y^=(Q^+2^2+ ^ ^
Table 2.5 Approximate expressions for the derivatives o f the random plus oversaturation com ponents for downstream links;i.e. \ / h e
a o r K °
a i » a i »
0 ,
0
if upstream link a undersaturated
^fai^a
.
, Otherwise0 ( exp (- (
%
d o r 1 + ACT 4C7(2I^+%, 11, n
2(11, r - 2 0 ’' dqi, 11» T - 2C
^ ’ V-b ^ » ^b are defin ed as in T able 2.2.
T a b l e 2 . 6 . A p p r o x i m a t e e x p r e s s i o n s f o r t h e d e r i v a t i v e s o f t h e u n i f o r m c o m p o n e n t s f o r fu r t h e r d o w n s t r e a m li n k s ; i .e . V c s C j^ 6D / dD^ a s" / a j / ^ jm 0 i f 0 A -
note: where //* ) , , z^j , z,2 are d efined as in T able 2.1 and 4. is; defined as in Table 2.4.
T a b l e 2 . 7 A p p r o x i m a t e e x p r e s s i o n s f o r t h e d e r i v a t i v e s o f t h e r a n d o m p l u s
o v e r s a t u r a t i o n c o m p o n e n t s f o r f u r t h e r d o w n s t r e a m l i n k s ; i . e . V c e Cj„
d S "”
4 . a ê » a i »
Chapter 3 Problem Formulation
3.0 Introduction
In Chapter 3, the bi-level programm ing is used to form ulate the optimisation problem of area traffic control for equilibrium netw ork flow s, in which the upper level problem deals with the optimisation of area traffic control with respect to the common cycle time, and the starts and durations of green times, w hilst the lower level problem deals with the equilibrium network flows which corresponds to solving the user equilibrium traffic assignment. In the following sections, firstly, a general description of the b i level formulation for equilibrium network flows is given in Section 3.1. Fundam ental term inology and notation for the optimisation o f area traffic control will be then given in Sections 3.2 and 3.3 respectively. Secondly, in relation to the form ulation of the upper level problem, in Section 3.4, the m athem atical expressions of the com ponents for the system perform ance which are in term s of a weighted linear com bination of m ean delay and num ber of stops per unit tim e adopted from TRANS YT will be discussed. Thirdly, in Section 3.5, the form ulation for user equilibrium traffic assignm ent in the low er level problem will be discussed, in which a separable link travel time function is determined by the sum of the undelayed travel time, i.e. travel tim e in prevailing traffic conditions if not delayed by dow nstream signals, and the average junction delay incurred by traffic at the downstream end of the link. Conclusions for this Chapter will be given in Section 3.6.