Optimising traffic signal settings for periods of time-varying demand

Transpn. Res.-A, Vol. 30, No. 3, PP. 2077230, 1996

Conyriaht Q 1996 Elsevier Science Ltd Printed~in &eat Britain. All rights reserved 0965-8X4/96 $15.00+0.00

Pergamon O!XS-8564(95)00018-6

OPTIMISING

TRAFFIC SIGNAL SETTINGS FOR PERIODS OF TIME-VARYING DEMAND BIN HAN*

Transport

Studies Group+,

University

College London,

Gower

Street, London

WCIE 6BT, England

(Received 12 November 1993; in revisedform 13 March 1995) Abstract - When traffic demands are different in successive time periods, the signal settings that are optimal for each individual period are only local solutions to the problem. These settings may be readjusted and the changes between them shifted in time so that the overall performance for those periods taken together is improved. A sequential optimisation technique has been developed to minim& the total junction delay over the successive periods taken together by searching for the optimal signal timings and the time-shifts subject to certain queue length considerations, and each traffic stream in the junction can be undersaturated or oversaturated during a time period. Some example calculations are given and the results show that such a technique can provide modest improvements in the junction control performance and give less delay than the existing methods that optimise only for individual periods. Copyright 0 1996 Elsevier Science Ltd

INTRODUCTION

The conventional methods of calculating traffic signal settings (beginning with those of Webster, 1958; Miller, 1963) always assume that the demand flow is time-stationary, and usually 1h is the basic period for calculations. However, practical observations show that traffic demand is time-varying; it can change substantially even within lh, particularly during peak periods. The way in which flow varies in time has a great influence on the vehicular delays in practical operations and this is reflected in theoretical analyses. Hence it is important to consider the flow change pattern. In most attempts to do so the 1h peak period is divided into a sequence of sub-periods (usually 5- 15 min) during each of which the demand flow for each stream is assumed to be constant. The use of time periods shorter than 5 min is undesirable because it is difficult to model the flow profile to that detailed level and to evaluate the true demand trends; on the other hand, longer time periods than 15 min should be avoided otherwise real flow changes could be missed. Outside peak periods, however, these sub-periods may be longer since the traffic flows are then usually changing less quickly. The literature on setting traffic signals for periods of time-varying demand is rather limited. An early example is the computer program OSCADY (Burrow, 1987) which uses time-dependent queueing theory (Kimber & Hollis, 1979) to model the growth and decay of queues. This is achieved by considering the modelled period as a sequence of short time segments. During each segment the demand flow, capacity and signal timings are considered constant. The program allows for the calculation of the queue length at the end of each segment and the delay occurring during the segment. The optimisation techniques employed to derive the best signal timings for each time period are based on methods developed by Allsop (1971a, b; 1972). For each time period, if the junction is undersaturated, the delay is minimised; if the junction is oversaturated, then the capacity is maximised. The timings are then left as they are, and the switching timings *Present address: Dunn Engineering Associates, 66 Main Street, Westhampton Beach, NY 11978. t The Transport Studies Group now forms part of the University of London Centre for Transport the same London address. 207

Studies with

208

Bin Han

from one time period to the next are the same as the times at which the flow changes, no further step is taken to re-adjust the switching timings and the signal timings for each period. Hence it gives the signal settings which are in certain cases only local optimal solutions, and optimises with respect to capacity instead of delay during periods of oversaturation. A more recent investigation was reported in a paper by Kuwahara and Koshi (1990). The optimal signal timings for each individual time period are calculated by employing a convex programming approach. The switching timings (i.e. the times at which the signal timings are altered) are then determined via formulating the problem within the dynamic programming framework, so that the total delay for all the time periods can be minimised. However, there are two major limitations: (i) only undersaturated intersections are considered, so that the cases where the intersections are oversaturated cannot be accommodated; and (ii) when searching for the switching timings, the signal settings for each individual time period are not re-adjusted, so that the resulting switching timings and the signal settings for each time period may only be local solutions. In this paper, some basic definitions and notation are first introduced, followed by mathematical formulation of the problem. A sequential optimisation method, using the total delay over the whole time period as the performance index, is then established, which adapts the OPTIM subroutine in SIGSET(Allsop, 197lb) into the process of optimising the effective green times and cycle times in different periods. To prevent longer queue lengths at the end of the last period which will cause unnecessarily bigger delays to the subsequent periods that may follow, the optimiser can re-optimise the objective function by extending the length of the last period so that the effects of the final queues are taken into account. The example calculations show that the new approach can give somewhat better results than OSCADY. STATEMENT OF THE PROBLEM

Fixed-time signal control methods are based on the historical data such as arrival rates, saturation flow rates obtained from traffic surveys and traffic counts. A typical pattern of changing flow in one stream over a peak period at a junction is described in Fig. 1, where T is the length of the peak period, and T,, T2, . .. . T, = (Tk) are the time segments during each of which the flow can be regarded as constant. A straightforward method of dealing with this kind of problem is to set the traffic signals correspondingly with the variation of traffic flow levels, i.e. to find the optimal signal settings that give the best value of a performance index for each time period. The expected random and oversaturation queue length of each stream at the end of each time period is treated as the initial random and oversaturation queue length for that stream in the next period. An optimisation technique can then be employed to optimise the traffic

Fig. 1. Typical traffic flow pattern for a traffic stream over a peak period

209

Optimisingtrafficsignalsettings cl T

,__...._. ,.. .__.............

1

.

T

TP Al’

j

Tl’

i I

_!

hz’ I

0

t1

. ..

..... . ... . .............. ... --A

T2’ I

TP’ I tp-1

t2

I )

t

tp

Fig. 2. Signal timing re-adjustment and time shifts.

signal settings for each single time period, thus a sequence of signal settings specified by vectors A,, h2, .... hP to be defined in the next section can be obtained. The program OSCADY is an example of such a method. However, due to the substantial changes in the traffic characteristics at different times of day, the control strategy should be such that the signal settings can give the best junction performance index taking into account the effects of timings in any one period upon conditions in subsequent periods. Since the random and oversaturation queue length for each stream at the end of one segment will be the initial random and oversaturation queue length for that stream in the next segment, the signal settings for each segment are not independent of each other. A control policy that treats them as such will not be optimal when considering different time segments all together and can only be locally optimal in each time interval. There are some ways of making improvements such as vehicle-actuated control methods that will change the traffic signal settings according to the change of flow conditions. However, to investigate the potential use of fixed-time traffic signals, improvements may be made if the signal settings are re-adjusted and subject to time shifts, i.e. if the cycle time and green splits are re-adjusted for each segment but the times at which signal settings are changed for each segment are different from those at which the flow changes, as illustrated in Fig. 2. Suppose that (AI!, hzl, .... Ati) = {hk,} are the new signal settings after re-adjustment that are in force for each period (T]J, T21, . . . . T,r) = { Tp} respectively. Now we have two sequence of time-periods: the one corresponding to the flow changes {T/J = (Tl, T2r . . . . T,) and the one corresponding to the signal settings {Tkf} = (T,!, T2#, ...) T,!). For the purpose of practical operations let p’ = p, i.e. the number of changes in signal settings is the same as that in flows. The optimisation problem now becomes how to find the signal settings (&I, AZ).... Ad) = {hkf} and the sequence of time-periods {Tp} that gives an optimal performance index subject to suitable constraints.

NOTATION

Since considerable number of mathematical expressions and operations are involved in this paper, it is necessary to clarify the notation before further discussion. The following notation will be used General notation m = the number of stages in a signal cycle n = the number of traffic streams in a junction Y = the flow ratio of a stream (=q/s).

210

Bin Han

The following subscripts will be used unless otherwise specified i j r u e o 0 f K

denotes denotes denotes denotes denotes denotes denotes denotes denotes

in stage i (i = 1, 2, . ... m) in streamj(j = 1, 2, . ... n) the random and oversaturation component the uniform component equilibrium oversaturation initial final total lost time for a signal cycle (K = 2 Ki) i=l

aoj denotes the proportion of the lost time that is effectively green for stream j. (j = 1, 2, . ... n) A denotes the stage matrix, which is defined A = (QjJ

(j=

1,2, .’

where if stream j has green in stage i

Time periods

(1) Whole time: T (to, tP) (T is divided into p time periods). (2) The end of the kth time period: tk (k = 1, 2, . ... p). (3) The length of the kth time period: Tk. The symbol Tk will be used to denote both time period and its duration, so that Tk = tk t&I (Fig. 3). Tk

I

I

d-1

;k Fig. 3. The relationship between tk, tk_ I, and Tk.

(4) Array of p time periods array of p time periods of demand T1, Tz, .,,, TP: { Tk} array of p time periods of signal settings T,l, Tzf, .%.,TPl: {Tkl) {Tk} = (TI,

T2, .* * , Tp)

(Tkt} = (T,!, T2t, . . . . T,r) . Time-shifts (1) The shift of time Tk : &tk (Fig. 4).

6tk is defined as the difference between the end of Tk and that of Te, i.e. the amount by which the kth signal settings hkf end earlier or later than hk. 20,

if tkr 3 lk

< 0,

if tk! < tk .

btk = tk’ - tk

211

Optimising traffic signal settings

Tk

I

I

-

Ak

I

I

tk-1

tk

Tk'

I

I

hk' I

I

I tk'

tk-1'

Fig. 4. Definition of 64.

(2) Array for time-shifts in p time periods: (Stk} where

{stk} = (6t0,6t1, . . . , St/J,

St0 = St, = 0 .

Flow in a stream Flow in stream j: qy Flow in stream j during Tk Saturationflow

: qjk.

in a stream

Saturation flow in stream j : sj Saturation flow in stream j during Tk

: sjk.

TrafJic signal settings (1) Cycle Time: c time during Tk: ck (defined when 6tk-, = 6tk = 0) cycle time during Tkl : ck’ Then &,k = Kick and &,k’ = K/Q.

Cyde

(2) Effective green time for a stage proportion of effective green time for stage i: Xi effective green time for stage i in period Tk : xik’ck i = 1,2, ....m (defined when C?tk-l= 6tk = 0) effective green time for stage i in period Tkt : xiv.ck’ i = 1,2, ....m. (3) Effective green time for a stream proportion of effective green time for stream j: effective green time for stream j during Tk: effective green time for stream j during Te

Aj

hjk.ck

(defined when 6tk-1 = 6tk = 0)

: Ajp’ck’.

(4) Array of traffic signal settings vector for traffic signal settings for a single period Tk : hk (defined when 6t,+, = btk = 0) vector for traffic Signal Settings for a Single period Tk, : hk’ (defined when 6tk-t = btk = 0)

array for traffic signal settings for p time periods {Tkj : {hk) (defined when &k-r = 6tk = 0) array for traffic signal settings forp time periods {Tp} : {hk’} (defined when btk_t = 6fk = 0) {hk}

=

(Al, Al>

” ‘, &)

{hk’} = (ii,,, A,(, . . . , i$,‘) .

Bin Han

212

The relationship between Ajk and Xjkis Ajk

=

2

(j=1,2,*..,n;i=0,1,2;..,m).

aijXik

i=O

Queue lengths for stream j : Lj

(1) Uniform Queue Length: L, average uniform queue length during the time period Tk : Lujk average uniform queue length during the time period Tk’ : Lujk. (2) Random Queue Length: Lrj random queue length at time tk : Lrjk random queue length at time tp : Lrjpm Then is the initial random queue length for period Tk+ 1 and the final random queue length for period Tk, and similarly Lrjp for Tk+ 1’and TV. (3) Total Queue Length for Stream j at tk : Ljke Total Queue Length for Stream j at tk’: Ljp Lrjk

Ljk = Lrjk f Lujk .

Therefore is the initial total queue length for period Tk+ I and the final total queue length for period Tk, and similarly Ljp for Tk+ 1’and Tkl. Ljk

Delays

(1) Delay rate: D delay rate in stream j: Di. If the signal settings are the same throughout Tk it is useful to define delay rate during Tk in Stream i : Djkm And if the arrival rates are the same throughout Tp it is useful to define delay rate during TV in stream j : Djp. (2) Total delay at the junction: W delay at the junction over Tk: wk.

total

If the signal settings are the same throughout

Tk, then

wk = Tk . 2 Djk /=I

total delay at the junction over TK : wki. If the arrival rates are the same throughout wk’

Tk!, then

= Tk/* 2

Djk’ .

j=l

where Tk and TV

OVerlap,

total

delay at the junction over Tk fl TK= Wkk’.

(3) Total delay at the junction over the whole time @ time periods)

w=f:wk=e

wk’,

k=l

k=l

Optimisingtrafficsignalsettings

213

MATHEMATICALFORMULATION Basic assumptions on signal settings

In this paper the following assumptions are made: 1. there are p time periods; 2. the traffic conditions (e.g. flow and saturation flow) in each time period are constant; 3. there are n streams in the junction and m stages in the signal cycle; 4. the constraints on the signal timings (i.e. the stage order, the stage matrix and the minimum green time for each stage, etc) are the same in all the time periods. Under these assumptions, traffic signal settings for an isolated junction are to be calculated by a stage-based method. The calculation of the Performance Index

In the following discussions we use the total delay over all the time periods for the junction as a Performance Index, and the problem can then be expressed as: To find a signal sequence (AIt, hzl , . .,, Ai) = {hp} and a series of time periods (T1j, T2f, . .. . T,f) = { Tkt} to minimise the total delay W for the whole time period T, and W is defined by

where Wkf = total delay in time period Tkj (k = 1,2) . . ..p>. subject to the constraints 6to = btp = 0

-Tk f & < Tk+l

(k=1,2,yp-I)

(2) (3)

and other constraints for optimising signal timings for a single time period, e.g. minimum green time constraint. The constraints (3) are introduced to make sure that a certain set of signal settings hk will be implemented in no more than 3 time periods, i.e. only in extreme cases will hk be implemented in Tk_ 1, Tk and T, + I. Under such constraints the following relationship exists Tkr = Tk - 6tk-, + btk

(k = 1, 2, *. . , p) .

(4)

By the relationship (4), the problem can be restated as follows: To find the succession of signal settings (&I, hzf, . ... h,+ ) = {hkf} and time-shifts (do, &I, bt,, ..*, St,) = {bt,> to minimise W as defined in (l), subject to the constraints (2), (3) and other constraints for optimising signal timings for a single time period. The initial settings (hl, hz, . ... hp) = (hk} are such that { Tk} = { Tk), i.e. the initial signal settings correspond to the flow level in each time segment. The signal settings obtained through the method using the new sheared delay formula introduced in another paper by the author (Han, 1995) will be used as such settings. The performance index W corresponding to (hk}, when { Tkt} = (Tk}, can be easily calculated by

where Djk

=

&jk

+ hjk

Bin Han

214

and &jk

=

&jk(Cljk,

sjk, Ajk,

ck)

and according to the sheared delay formula (Kimber & Hollis, 1979) Drjk = drjk [ qjkr

sjk,

Ajk,

hj(k-l)]

.

Suppose, that, through some optimisation techniques, we have found the signal settings @If, h2’, . . . . Q) = {&I and a series of time periods (Tt!, T~I, . . . . T,,) = {Tko, then the next problem is how to evaluate the performance index W when {hk,} are implemented. Obviously W cannot be evaluated according to (5) since there can be more than one set of signal settings in each time period in which the flow is constant. The calculation of the performance index must therefore be conducted in more detail, even within one period. In fact, each time period can be sub-divided into up to 3 time intervals, in each of which there is only one combination of flow levels and signal settings. If the total number of time periods is p (p 2 2), then the way the delay in each of these time periods is estimated can be shown by the following example. Consider an intermediate period Tk. No period of this type exists if p < 3, and period k is this type for if p = 3.

k = 2, k=2,...,p-

if p > 3 .

1,

Consider the case where Stk-i < 0, &k < 0 (Fig. 5): There are two sets of signal timings during Tk : hk’ and h&l’, which are in force for the intervals Tk fl Tk! and Tk II Tk+t, respectively. Total delay in the interval Tk n Tkl is Wkk =

2 i

D,(q, 1 jk,

Sjk, hjk’,

Ck’, J%j(k-1)r

‘(Tk

Tk + 6tk)

j=l

f

6tk)

.

1

The random queue length for stream j at the end of the interval Tk fl Tp is Lrjk’

=

Lrjk’(qjk,

sjky njki,

Lrj(k-I),

Tk + fitk)

.

Then the total delay in the interval Tk fl Tk+p is n

Wk(k+l)

c j=l

=

Dj(Cljk,

Sjk,

Ajj(k+l)'>

Ck+l'r

Lrjk’,

-6tk)

’ (--6tk)

.

Hence the total delay in period Tk is wk

=

wkk

+

Wk(k+l)

Tk-1

Tk+l

hk-1

-hk -6tk-1 i-i

hk+l -6tk /Vi Tk+l'

Tk-1' I Ak-1'

-hk' Fig. 5. Cakdation

of bV, in case where 6fk_ I < 0, bfk < 0.

Ak+l'

215

Optimising traffic signal settings

The random queue length for stream j at the end of period Tk is Lrjk =

Lrjk(qjk,

sjk,

Aj(k+l)'r

Lrjk’r

-6tk)

.

Other situations can be considered in a similar manner. Evaluation of the effects of {&I} and {Stk} on the Peformance Index After the two performance indices Wand ?V corresponding to two successions of signal settings {hk} and {hk~}have been calculated, it is important to compare Wand lV so that the effects of re-adjusting the signal timings and making time-shifts can be evaluated. The criterion that will be used for this purpose is the percentage reduction in total delay: SW/W=

(W-

W’)/W.

(6)

The criterion SW/W will be used in example calculations. OPTIMISATION OF {hk,} FOR FIXED {6t,}

The discussions in this paper have not yet considered the problem of how to find {&} and {bt,}, but only the evaluation of the Performance Index once {hkr) and {btk} are found. In fact, it is difficult to solve the problem of simultaneous optimisation of {hk’} and {6tk}, since the number of variables involved is very large and no existing technique is available except exhaustive grid search. However, the problem can be solved by the sequential optimisation of {hk,} and {6tk} iteratively. This section introduces the optimisation of {hkr} without changing {&}, and the next section discusses the problem of optimising {btk} under fixed {&I}. An iterative algorithm of sequentially optimising {hk~}and {btk} is given later. When {btk} are fixed, to optimise {&I}, we must first consider the factors affecting the choice of {h&t}. First consider the problem of optimising a particular hk’ = 1, 2, .... p. Since the random queue lengths at the end of TV will be the initial random queue lengths at the beginning of Tk+ 1~,any change in hk’ will directly influence the delay in the subsequent periods after Tkf. On the other hand, such a change will also indirectly influence the delay in the periods before Tkl, since the change in hkl will eventually cause changes in h,,f and hence W,I, h = 1, 2, . ... k-l. Therefore the signal settings in one period will be determined not only by the traffic conditions in this period, but also by the traffic conditions and signal settings in the periods before and after this period, i.e. hk’ must be treated as global variables, and such influences must be considered in order to get the globally optimal signal settings. The calculation of the derivatives with respect to hkt To optimise hk’ (k=l, 2, .... p), it is important to evaluate the derivatives of W with respect to hk’, i.e. aw/axjkl (i=O,1,2, .... m; k= 1,2, ....p) should be known. At each step in the iterative process of optimisation to be introduced in this paper, we optimise hk’ under existing hi, h= 1, 2, .... k-l, k+l, .... p. In doing so we only consider the direct effect of changing hk’ on the delay in periods Tp and after, and the indirect effect of changing hk’ on the delay in periods before Tp is taken into account later in the iterative optimisation process. Now (7)

where Wj/+ is the delay in stream j during Thl, and the change in hk’ will only have direct effect on W,I, h = k, k+ 1, .... p, WV itself is influenced through changes in the

216 Ajp

Bin Han G=

132, .. ..n) (in accordance with the relationship Aj = 5

aij.& ( j = 1,2,. . . , n) and

i=O

the W,f (h=k+ 1, k+2, . ... p) are influenced through changes in the initial queue lengths Lri(h_,Jt (j = 1, 2, ,,., n ; h = k+ 1, k+ 2, .. .. p). Hence

d

wjh'

“rj(h-1)

The derivatives a hj@ , a wj,I$/C!?&kf and 8 are multiples of the corresponding derivatives of The derivatives aLrjkj/dAjkl and can also easily be calculated. When {&k} # {0}, by analogy with the earlier discussion on the calculation of the performance index, there are further steps in the calculation, i.e. the period TV may be sub-divided into up to 3 time-intervals according to the number of different sets of demand flow levels. For example, when k= 1, to calculate ~3Wjll/aAj,‘: wj,

/a

Wjk'/aLrj(k_l)t

Djkm

8Afdrjkt/aLrj(k_,)f

(a) If 6tl < 0, since there is no change in the flow qjl during Tlf, d Wjll/aAjlt calculated directly by

awj,l/anjll = -?aA,,,Wjl'14jlr

(

sj17

hjl'? c1’,

can be

Lrjo, T,j ))

(b) If St, > 0, since T,l = T1 + 6tl, the flow qj1 is changed to qj2 at the end of T1, hence 8 Wjll/aAj,, consists of 2 components corresponding to periods T1 and T, fl TIT, and given by

and

where

wjk/,$

is the delay in stream j during Tk il Tk/,

Hence d W/~&J (i=O,l, 2, ,.., m; k= 1, 2, . ... p) can be evaluated according to the above formulation using the new sheared delay formula (Han, 1996). The sequential optirnisation of {hi > To optimise hk’, k = 1, 2, ,*., p, the SICSET (Allsop, 1971b) optimisation subroutine OPTIM that was used to calculate the optimal signal settings for a single period can stil1 be used here, since the signal timing requirements (e.g. the definition of the stage matrix)

Optimisingtrafficsignalsettings

217

and the constraints for optimising signal timings for a single time period are unchanged and the variables are the same. The only difference is in the objective function, which is reflected in the difference in the derivatives. Hence {hk,) can be optimised for given {&,} by the following sequential algorithm: ALGORITHM 0 (initial signal settings): Step 0: Find the initial solution {hkt} = {hk}, where {hk} is the sequence of traffic signal settings that give the optimal performance index for each single time period Tk, k= 1, 2, ...) P. ALGORITHM 1 (sequential optimisation of {hkt}: Choose a small cl (e.g. cl = 0.0001) for use in the criterion for stopping the algorithm. Step 1: Let the current Performance Index for the whole period T be WC. For given optimise (hlf, h21, . . ., ApI) = {hkf}.

{&},

Step 1.1: Optimise A,!. Step 1.2: Optimise hZl for the previously optimised A,t. ......

Step 1.~: Optimise hd for the previously optimised h,~, h21,h,,_,. Steps 1.l- 1.p are realised by implementing OPTIM, which uses the derivatives given by (8) and (9). Then a new value of the Performance Index is obtained, which is denoted by W new. Step 2: If (WC-W,,,)

> cl. WC, go to Step 1.1. Otherwise stop.

Hence {hkt} is optimised with respect to the total delay W in the whole period T rather than each hk, being optimised individually with respect to the total delay wkl in period TV, i.e. the effects of each settings hk’ on Was a whole are considered. OPTIMISATION OF {&} FOR FIXED {At,}

In the above algorithm, the total delay for the whole period can be decreased by re-optimising {&} even if {bt,) = {0}, but it may be decreased further if the signal settings are subject to time-shift, i.e. if the change of signal settings from one period to the next is not necessarily simultaneous with the change of the flows but can be earlier or later. To investigate this possibility, simulation testing were made for the same two simple intersections as described by Han (1996), where 2 time periods were considered, which are both 10 min long. The initial signal settings {hk} and the re-optimised settings {hk’} (through the sequential optimisation of {hk’} by using Algorithm 1 are both tested, i.e. by continuing A1 (or Alt) into the second period (in which case 6tl > 0), or by starting AZ(or A*?) before the end of the first period (in which case St, < 0). Delay as a function of the time-shift An example of the effects on the total delay for the whole periods of making the timeshift to the initial settings {hk} and the re-optimised settings {hk’} can be seen in Fig. 6(a) and Fig. 7(a), where the delays are estimated by the new delay expression, and in Fig. 6(b) and Fig. 7(b), where the delays are estimated by the OSCADY formula. From these curves it can be seen that, for the initial signal settings {hk} the time-shift for minimum total delay is 6tl = 36 s, i.e. W is minimised if the signal settings are changed 36 s later than the change of flows; and for the re-optimised signal settings {&} the time-shift which yields minimum total delay is 6t, = 72 s. Diagrams similar to Fig. 6 were also plotted for a number of cases (Han, 1990) and the following characteristics were found: 1. The curve for the total delay W as a function of the time shift 6tk (k = 1, 2, . ... p), W= W(&k), is non-convex, and non-differentiable at &k = 0 (k = 1, 2,... , p-l) under fixed (hk’}.

Bin Han Delay as a function of time-shift with initial signal settings

(a) Delay (pcu-minutes)

5

10 Time Shift (minutes)

Fig. 6(a). Delay estimated by the new delay formula.

Delay as a function of time-shift with initial signal settings

(b) Delay (pen-minutes)

600

400

200

0 -10

-5

0

5

10 Time-shift (minutes)

Fig. 6(b). Delay estimated by the OSCADY delay formula.

2. W = W&J has only one turning value, a minimum, in a neighbourhood of zero which is very often as large as the largest range (-Tk, Tk+ 1) of vaIues of Stk that need to be considered within the constraints (3). In least favourable cases like this example, there are other turning values towards the extremities of the range. In such cases, as 6tk increases from -Tk, W(&) may pass through another turning value before decreasing into the neighbourhood of zero that contains a minimum. Again as &k increases towards Tk+ 1, W(&) may pass through another turning value. In the example shown here, there is one local maximum point for W(6tk) near 6tl = -T, = -9 min, and another near 6t1 = 8 min. 3. Where there are other local maximum or local minimum points for W(St,) in the cases studied the 6tk such that W = W(&) attains its lowest value is always the one minimum point in the neighbourhood of zero.

219

Optimising traffic signal settings Delay as a function of time-shift with re-optimised signal settings

(a)

Delay (pewminutes) 720 tk

-10

0

-5

5

IO Time-shift (minutes)

Fig. 7(a). Delay estimated by the new delay formula. Delay as a function of time-shift with re-optimised signal settings

(b) Delay (pcu-minutes)

400

200 c(Wu)osc

-I- Total Delay Wosc

-10

-5

0

5

10 Time-shift (minutes)

Fig. 7(b). Delay estimated by the OSCADY delay formula.

These findings suggest that it is difficult to apply analytical methods (methods that require the derivatives 8 W(Gt~)/&$J to search the optimal ark. The fact that any change in the signal settings or time-shift for one period will result in changes in the delays in the subsequent periods implies that it is also difficult to optimise the elements of {&,J simultaneously, especially when the number of time periods p is large, and the amount of computation tequired to evaluate any changes in the 6~ is big. Hence it is only practical to optimise {StjJ sequentially. The optimisation method -

Golden Section method

To optimise (&J sequentially under fixed {h&j, first we consider the optimisation of

Bin Han

220 6tk,

k= 1, 2, , . ... p-l. The initial feasible region for 6tk is defined according to the current and the constraints (3), and will be denoted by the interval (ak, bk).

6tk_1, Stktl

ak =

Then

b

Then

=

k

--Tic

if&k_] GO

-Tk + btk-,

if &k-i > 0

T/c+1+ 6tk+l Tk+l

ifbtk+l

VW

<0

(lob)

if&k+1 >O.

If w(&k) has only one turning value, a minimum, in (ak, bk), then the optimal 6tk can be found by using the Golden Section method. However, since it is possible that there are other local minimum points in this interval for 6tk, local optimisation in this interval may give a suboptimal 6tk. Based on the assumption that the optimal 6t.k should be in the neighbourhood of the current value 6tkO, say, the optimal Stk can be found by iterative reduction of the interval while optimising 6tk using the Golden Section method. Such an algorithm for optimising {bt,} can be described as follows: ALGORITHM

2 (sequential optimisation of {c!Yt,)):

Step 0: Define the initial feasible region (ok, bk) for btk (k = 1, 2, .,., p-l). Step 1: Optimise {btk} (k = 1, 2, . ... p-l). Step 1.1: Choose a small value e2 for use in a stopping criterion. Step 1.k: Optimise 6tk (k = 1, 2, . ... p-l). Step I.k.0: Evaluate the initial performance index Wo = W(&&. Step I.k.1: Let the current feasible region be (akF b& wherefis the number of times that (&, bk) is reduced (f = 0, 1, . . . ), and aklo and bklo are determined by eqn (10). Step l.k.2: Let the current search interval for btk be (akh, b&, where h is the current iteration number (h = 1, 2, ...). and ski = akrand bkl = b,+ The length of the interval is crh = bkh - akh. Then find 2 trial points &i and 5h2,which are given by thl

=

bkh

-

Tffh

th2

=

akh

+ Tab

M 0.618 is the factor by which the interval is reduced.

Where 7 = w

Step i.k.3: Compare W(&) and W(&). If

w(
<

W&2),

<(/z+ 1)2 =

then
t(h+

let ak(h+ ])I

=

1) = bk(h+

akh, bk(h+ I)-TQ(h+

the current solution as btkh =
1) =

th2;

I). Take

TQ(h+

1)~ and

take

Step l.k.4: If )W(&,l) - W(&Q)l/W(&l) > c2, go t0 Step l.k.3. Otherwise compare W. and W, = W(&h). If wh > Wo, reduce the feasible region as follows: If 6tkh > b&o, let akf+ 1 = akV, bklf+ 1 = btkh ; If b&t, < bt/& let &,f+ 1 = Sikh, bkv+ 1 = bkp Then go to Step l.k.1 to search 6tk again in the revised feasible region. Otherwise stop and take &kh as the solution. Such an algorithm will generate a sequence of feasible region (ak(O,bklo), (akIl, bkll), (ak(2, b/c/2),...y which will converge to the neighbourhood of 6tk, in which the only turning value is a minimum, and hence a performance index which is no worse than W. can be found.

Optimising traffic signal settings A SEQUENTIAL OPTIMISATION

221

METHOD

The discussion so far on the optimisation of traffic signal settings for periods of timevarying demand has been confined to optimising {hkf} without changing {&}, or optimising {btk} without changing {&I}. However, since {hk’} and {htk} are interrelated, the signal settings given by either Algorithm 1 or Algorithm 2 are only local optima and can be improved if the 2 algorithms are combined together. Hence the following sequential optimisation method is proposed. The algorithm ALGORITHM

3 (sequential optimisation of {hk’} and {fit,}):

Step 0: Find the initial solution {hk’} = (hk} and {&} = {0}, where {hk} is the succession of traffic signal settings that give the optimal performance index for each single time period Tk, k = 1, 2, .... p. Choose a small value t3 for use in a stopping criterion. Step 1: For current (6s}, optimise {hk’} using Algorithm 1. Step 2: Let the current Performance Index be WC. For the current {hk’}, optimise {btk}. Let the resulting Performance Index be W,,,. If (WC - W,,,) > t3. WC, go to Step 1. Otherwise stop. As seen in Figs 7(a, b) since the curve for the total delay W as a function of the time shift Stk (k= 1, 2, .... p) is non-convex, the above algorithm does not necessarily always lead to the global optimum, however in most cases it should, since the Golden Section method can usually find the optimum for curves like those in Figs 7(a, b). TAKING INTO ACCOUNT THE SUBSEQUENT CONDITIONS

Although in mathematical terms, the traffic signal settings can be calculated for periods of time-varying demand for any given flow pattern and junction layout by using Algorithm 3, some practical factors must also be considered in the optimisation process, one of which is the queues left at the end of the last period TP. These queues will influence the delays in the period after T,, if they are not small enough to be neglected. However, Algorithm 3 takes no account of the final queues and the delays in the subsequent periods after T,, and it is possible that unnecessarily long final queues will occur as a result of such an algorithm, especially when the degree of saturation in TP is high. This will cause bigger delays than necessary after TP. Hence it is necessary to make further improvement to Algorithm 3 so that, whilst keeping the total delay W as the Performance Index, some consideration can also be given to the delays in the subsequent periods after TP. In the calculation of signal settings for a succession of periods in practice, two requirements of the flow data are that the data should be specified for as many time periods as is practicable, and the flows in the last period should be low enough for the junction to be undersaturated in T,, and this period should be long enough for equilibrium to be approached during it. This is so that there will not be big delays in the subsequent periods that could be influenced by the choice of timings for periods T, to TP. Hence a good knowledge about the arrival pattern is preferable and it is reasonable to assume that the last period is always undersaturated when setting tra#ic signals for periods of time-varying demand. Under this assumption, the queue length for each stream will approach an equilibrium value in TP if TP is long enough. The delay-minimising algorithm can then be applied to a series of time periods at the end of which all the streams are in equilibrium. The resulting signal settings will usually be the optimal solution which can give the minimum estimated total delay over such a series of time periods. However, in practice, the engineer may have difficulty in specifying the length of the last period TP so that the queues in each stream are in equilibrium at the end of T,,, or

222

Bin Han

may give a shorter T, than is required for this. In the latter case it may be reasonable to assume that the flow levels for all the streams are the same after TP as they were during r,. Hence when optimising the signal settings for p time periods, if it happens that at the end of TP, some streams still have longer queues than their equilibrium queue lengths, then T, can be extended to, say, T,,, and the signal settings re-optimised for T,, T2 .... TP_,, TF. This process can be repeated until all the queues are in equilibrium before or at the end of Tpe. In this way some account is taken of the subsequent delays. To sum up, the following two assumptions are made to find the signal settings that can take account of queues remaining at the end of TP. (1) In the last period T, the junction is undersaturated; (2) The flows in TP may last longer than TP. Based on these two assumptions, an improved algorithm is now proposed. A further algorithm To accommodate the delays in the subsequent periods after TP, the following algorithm can be implemented. ALGORITHM delays) :

4 (sequential optimisation of {&I} and {&} with allowance for subsequent

Step 0: Let TPe = TP. Choose a small number e4 (e.g. e4 = 0.01) for use in a stopping criterion. Step 1: Implement Algorithm 3 to find the solution (hk’} and {bt,} for the time periods Ti, 7-2, . . . . Tpe. Step 2: Calculate the final random queue length Lrjp at the end of TPe for each stream j 0’=1,2 > ..., n), and the equilibrium random queue length L,ejp. Step 3: If for all j 0 = 1,2, .... n), (Lrjp - Lrejp) < Ed.Lrejp,then the current signal settings are the final solution and go to Step 4. Otherwise extend T,, by Step 3.1: Let TPe = TPe + St,,, where St,, is an increment step (e.g. St,, = 5 min). Step 3.2: Same as Step 2. Step 3.3: If (L,p - L,ejp) < Eq.Lrejpholds for allj (j = I, 2, .... n), then go to Step 1; otherwise go to Step 3.1. Step 4: evaluate W for periods T,, T2, .... TPe and stop. EXAMPLE CALCULATIONS

To demonstrate further the optimisation method described in this paper, two example calculations were conducted. Example 1 is for a simple intersection, whereas Example 2 is Table 1. Additional data for example Junction 1

Length of time period Flow ratio in stream 1 Flow ratio in stream 2 Initial queue length in stream 1 Initial queue length in stream 2

Period T,

Period T,

T, = IOmin

Tz = 10min

Y,, = 0.6

Y,z = 0.45

Y,, = 0.4

Y*z = 0.3

Lr,lJ = 0.0

Lrzo = 0.0

Optimising Table 2. Results for example

T2c

=

T2

Cl

~“,IL,I,

120.00

120.0

’

0.5631

0.5600

2

0.3502

0.3702

0.3733

’

0.5583

0.5631

0.5600

0.3502

0.3702

2

j =;

D/r, j =

’

2

’

2

c2 (sf

Xi2, i = :

A,2 i =

i

=

0,~ j = min)

’

2

’

2

New formula

New formula OSCADY formula

’

2

OSCADY formula % reduction in total delay estimated by

OSCADY

87.49

X,I, j =

X,2,

Present method Optimised settings

0.5583

Ajl,j =

Total delay (Pcu estimated by

I optimised over Tr and T, only

(s)

Xi,, i =

i-2

Junction

Initial settings

61, = 106.3 s

223

traffic signal settings

0.3733

5.99

19.33

8.20

18.13

8.21

18.91

5.53

19.91

7.77

14.62

7.80

13.86

1.0746

1.0655

1.0714

1.1421

1.0805

1.0715

173.80

190.38

194.13

166.12

164.85

161.67

85.38

73.48

70.4

0.5291

0.5298

0.5256

0.3772

0.3614

0.3551

0.5291

0.5298

0.5256

0.3772

0.3614

0.3551

4.30

6.61

3.69

5.05

3.60

6.85

3.94

5.51

3.57

4.76

3.49

5.76

0.8505

0.8494

0.8562

0.7953

0.8301

0.8448

159.43

134.23

151.30

154.07

126.27

124.78

660.15

621.88

633.91

654.00

615.16

631.78

5.80

1.90

5.93

2.63

a much more complicated junction. In both cases, initial settings are obtained optimising traffic signal settings for each individual time period only.

by

Example 1: Crossroads

The first example calculation was made for the example Junction 1 discussed in a preceding paper (Han, 1995, Fig. 6). The additional data for this junction are given in Table 1. The results of sequential optimisation in the case T, = T2 = 10 min are summarised in Table 2. hi and h2 are obtained by Algorithm 0, through minimising the total delay in T, and T2 separately. Algorithm 4 is used for the sequential optimisation of hit, h21 and fit,, but to reduce the period over which calculation is needed, the error limit e4 is set to 0.2, and no extension is made if the final random queues are shorter than 120% of their equilibrium values. However, if an extension for T2 is needed, a maximum value for Tz, is imposed, which is (T&ax = 50 min. This constraint is introduced to make sure that the sum of TI and Tze is not going to exceed 1 h, since it is not likely that the traffic flows will stay unchanged for more than 1 h. After the optimisation, the queues and delays are estimated by both the OSCADY formulae and the new delay expression. If an extension for TZ is made, then the total delay are evaluated over (T, + Tze) as well as over Ti + Tz.

Bin Han

224

Table 3. Results for example Junction

Initial settings

br, = 0.0 s T2,

=

T2

Cl

120.0

l

0.5634

0.5600

0.3502

0.3700

0.3733

0.5583

0.5634

0.5600

0.3502

0.3700

2

2

j =:

’

2

c2 6)

Ar2, i = :

l

A,,, i = 2 .i

=

X,2, j = 4, Total delay (Pcu estimated by

min)

.i =

:

0.3733

5.99

19.33

8.20

18.07

8.21

5.53

19.91

7.77

14.67

7.80

18.91 13.86

1.0746

1.0655

1.0714

1.1421

I .0805

1.0715

173.80

190.04

194.09

166.72

165.07

161.66

78.48

76.18

70.4

0.5298

0.5312

0.5256

0.3682

0.3638

0.3551

0.5298

0.5312

0.5256

0.3682

0.3638

0.3551

3.94

3.40

3.80

3.29

3.60

3.63

3.73

2.71

3.67

2.75

3.44

2.97

’

0.8494

0.8471

0.8562

2

0.8146

0.8246

0.8448

’

408.42

390.64

397.94

368.85

352.70

365.35

1141.27

II 16.47

1126.99

1117.71

1098.64

1110.55

2.17

0.93

1.71

1.07

2

New formula OSCADY formula

% reduction in total delay estimated by

OSCADY

120.00

D,l,.i=’ 2

L&,2,

Present method Optimised settings

87.49

X,I, j =

T2

T2 to 50 min

0.5583

Aji, j=’

L,IlLjl,

after extending

(s)

xi,, i =

Tl

1 optimised

New formula OSCADY formula

The optimised signal settings suggested that bti = 106.3 s (i.e. the first set of signal settings should be extended 106.3 s into T, before being switched to the second set of signal settings), so that the random queue lengths can be reduced much quicker, and eventually, the total delay can be reduced. It can be seen from Table 2 that compared with the initial settings, the sequentially optimised settings can reduce the total delay by about 6%, which is about 2% less than that given by OSCADY. For the case of extending T, to T,, = 50 min, the results are summarised in Table 3. In this case, the reduction in total delay becomes smaller (about 2% compared with 6%) which is similar to what is given by OSCADY. Example

2: the Chapel Hill Junction

This junction has been used elsewhere in the traffic signal literature, e.g. by Heydecker and Dudgeon (1987) and it is a real junction in Huddersfield, England, which has 8 vehicular streams and a pedestrian stream (stream 9). However, the delay to the pedestrians is not considered in the optimisation process in this paper. The data for saturation flows, arrival rates in the second period T,, the junction diagram [Fig. 8(a)] and stage diagram [Fig. 8(b)] are based on that in Heydecker and Dudgeon (1987). However, the

Optimising

225

traffic signal settings s9 =9ooopcuh

9A

3

s3 = 2622 peti

2 1

I

92= 3997 pcu/h

k

SI= 3763pm/h

s4

f

=

4

3494pcu/ll

so = 2965pcu/h Fig. 8(a). The Chapel

Hill Junction.

Stage 1

Stage 2

Stage 3

Stage 4 Fig. 8(b). The stage

diagram.

flow rates in the first period T1 (where the junction is oversaturated) are artificial. The stream data are given in Table 4. The following parameters were used: maximum cycle time: 120 s minimum green time for each stage: 6s lost time after each stage: 5 s for Stage 1,3,4; 1.5 s for Stage 2.

Bin Han

226

Calculations were also made for the program OSCADY and the results are compared by using the criterion 6 W/W. For T1 = T2 = 10 min, the results are given by Table 5. Table 4 Stream 4

number S

950 250 2622

IS00 633 3494

0.00

0.00

4

3

4

3

I

8

9

1600 871 2978

1250 122 1835

1000 925 3360

1000 655 2965

1* 1* 9000

6.50

0.00

0.00

1.30

0.50

0.00

4

2

4

4

2

I

3

2

2

1

2

3

1

3

2

3

Flow in T,(pcu/h) Flow in T&w/h) Sat. flow (pcu/h)

369 123 3163

900 260 3997

Extra effective green time (s)

7.00

First stage to receive green Last stage to receive green

*This small arrival rate is introduced

to exclude the influence of the pedestrian

Table 5. Results for Chapel Hill Junction

optimised Present

61, = 0.0 s T2,

=

T2

Cl

&I, i=

(s) 1 2 3 4 1 2 3

Ajl,

j = Z 6 I 8 9 1 2 3 4

LujlILr,I, j =

S 6 I 8 9

Xj,,

j=

6

1

1 2 3 4 j = 5 6 7 8 9

1 2 3 4 41, j = 5 6 I 8 9

stream on the optimisation

process.

over r, and Tz only

Initial settings

method Optimised settings

OSCADY

12.29 0.3449 0.2583 0.0856 0.0830

100.48 0.3403 0.2756 0.1601 0.0597

120.00 0.3225 0.2450 0.2450 0.0500

0.1798 0.0856 0.8245 0.3482 0.4971 0.8245 0.3826 0.3518 0.0856

0.1294 0.1601 0.7752 0.3403 0.4500 0.7752 0.4636 0.3454 0.1601

0.1083 0.2450 0.7008 0.2992 0.4142 0.7008 0.5133 0.3267 0.2450

2.76 3.14 0.46 7.96 7.48 1.21 5.45 6.37 0.01

0.38 93.34 0.21 49.64 25.50 2.20 I.55 6.48 0.00

4.33 7.50 1.05 to.95 10.29 2.76 5.72 9.02 0.01

1.28 44.14 0.24 53.99 46.22 13.32 0.68 1.64 0.00

5.42 11.04 2.22 12.21 12.04 5.85 5.62 10.87 0.01

3.14 4.42 0.33 77.12 63.02 8.01 0.48 12.38 0.00

0.5453 2.6307 0.4394 1.2331

0.7579 1.4064 0.4674

1.0809 0.8262 0.7779 0.9586 0.0013

1.1943 0.8787 0.6420 0.9766 0.0007

0.9052 0.9191 0.5170 1.4350 1.2972 0.9720 0.5798 1.0325 0.0005

31.40 499.9 1 6.69 338.43 221.30 32.76 69.31 113.20 0.10

55.12 305.03 12.87 389.08 345.77 51.52 63.88 146.12 0.10

79.00 146.81 25.47 514.02 444.3 1 118.38 60.93 188.09 0.10

1.2616

-

conrinued oppsire

Optimising

continued

Table 5 -

Present Initial settings

6t, = 0.0 s 7-2,

=

T2

c2 6)

X,2, i = 2 3 4 2 3 I$~, j = Z 6 7 8 9

1 2 3 4 .L,2l-L,2,

j

=

5

6 7 8 9

I 2 3 X,2, j = Z 6 7 8 9 1 2 3 D,2? j = ;’ 6 I 8 9 Total delay (Pcumin) estimated by

New formula OSCADY

% reduction in total delay estimated by

New formula OSCADY formula

227

traffic signal settings

method Optimised settings

OSCADY

73.94

65.59

60.10

0.2366 0.1670 0.2920 0.0811

0.2718 0.2157 0.1694 0.0915

0.2962 0.2246 0.1048 0.0998

0.1758 0.2920 0.6201 0.2550 0.3854 0.6201 0.4969 0.2434 0.2920

0.1982 0.1694 0.7315 0.3 148 0.4395 0.7315 0.4278 0.2795 0.1694

0.2163 0.1048 0.7870 0.3328 0.4792 0.7870 0.3760 0.3045 0. IO48

0.89 1.48 0.41 6.82 4.78 1.76 3.32 4.94 0.01

0.03 0.97 0.02 12.97 3.61 0.69 0.42 5.78 0.00

0.74 1.75 0.18 3.31 3.52 0.78 3.81 3.98 0.01

0.02 1.51 0.01 2.99 3.24 0.41 0.70 2.40 0.00

0.65 1.86 0.10 2.87 2.79 0.45 4.15 3.39 0.01

0.02 0.83 0.01 5.73 3.22 0.36 1.17 1.86 0.00

0.1859 0.2228 0.1538 0.7106 0.7589 0.6345 0.5540 0.9076 0.0004

0.1649 0.3839 0.5956 0.5755 0.6654 0.5379 0.6435 0.7905 0.0007

0.1511 0.6205 0.1211 0.5444 0.6103 0.4999 0.7321 0.7255 0.0011

9.10 262.21 4.32 364. I I 144.92 25.48 37.50 109.21 0.10

8.30 162.77 2.01 235.08 193.28 18.17 45.00 73.35 0.10

6.82 29.37 1.1 I 409.03 253.27 8.88 53.02 68.21 0.10

2347.1 1

2197.53

2482.26

2211.42

2113.41

2406.55

6.31

I 1.47

6.96

12.18

It can be seen from Table 5 that for this complex situation, a greater improvement is obtained by sequential optimisation of the signal settings even though no time-shift results. The cycle time in T, is increased (from 72.3 to 100.5 s); but the cycle time in T2 is decreased (from 73.9 to 65.6 s). Although the uniform delay in T1 is increased, the total random queue length at the end of T1 is decreased (from 179.3 pcu to 168.1 pcu), which therefore causes less delay in T2. However, since at the end of T2, the queue lengths for some streams are still greater than 120% of their equilibrium values, the second period T2 is extended to Tze = 50 min. The results are given in Table 6.

Bin Han

228

Table 6. Results for Chapel Hill Junction optimised after extending T2 to 50 min

Initial settings

6r, = 0.0 s T2e =

T2

Cl 6) Xii,

i = : 3 4 1 2 3

.!ijl,

j = Z 6 7 8 9 1 2 3

LjlILjl,

i

=

i

6 I 8 9

1 2 3 4 X,,, j = 5 6 7 8 9

1 2 3 4 Dj,, j = 5 6 7 8 9

Present method Optimised settings

OSCADY

12.29

82.37

120.00

0.3449 0.2583 0.0856 0.0830

0.3438 0.2772 0.1558 0.0728

0.3225 0.2450 0.2450 0.0500

0.1798 0.0856 0.8245 0.3482 0.4971 0.8245 0.3826 0.3518 0.0856

0.1578 0.1558 0.7653 0.3062 0.4774 0.7653 0.4170 0.3500 0.1558

0.1083 0.2450 0.7008 0.2992 0.4142 0.7008 0.5133 0.3267 0.2450

2.16 3.14 0.46 7.96 7.48 1.21 5.45 6.37 0.01

0.38 93.34 0.21 49.64 25.50 2.20 1.55 6.48 0.00

3.32 6.01 0.94 8.49 8.50 2.47 5.54 7.30 0.01

0.59 47.50 0.25 73.16 33.80 3.66 I .04 6.80 0.00

5.42 11.04 2.22 12.21 12.04 5.85 5.62 10.87 0.01

3.14 4.42 0.33 77.12 63.02 8.01 0.48 12.38 0.00

0.5453 2.6301 0.4394 1.2331 1.0809 0.8262 0.7779 0.9586 0.0013

0.6214 1.4452 0.4734 I .4020 1.1254 0.8901 0.7137 0.9630 0.0007

0.9052 0.9191 0.5170 1.4350 1.2972 0.9720 0.5798 1.0325 0.0005

31.40 499.91 6.69 338.43 221.30 32.16 69.31 113.20 0.10

38.92 303.40 11.92 457.29 269.75 57.14 65.47 124.22 0.08

79.00 146.81 25.47 514.02 444.31 118.38 60.93 188.09 0.10

54.49

54.04

60.10

1 X,2, i = 2 3 4

0.2692 0.1581 0.1598 0.1101

0.2718 0.1827 0.1292 0.1110

0.2962 0.2246 0.1048 0.0998

1

0.2386 0.1598 0.7209 0.2774 0.4711 0.7209 0.3693 0.2784 0.1598

0.2406 0.1292 0.7505 0.3029 0.4753 0.7505 0.3637 0.2810 0.1292

0.2163 0.1048 0.7870 0.3328 0.4792 0.7870 0.3760 0.3045 0.1048

c2 6)

2 3 A,z, j = i 6 7 8 9

I

&&,z,

2 3 4 j = 5 6 I 8 9

0.56

1.49 0.16 3.06 2.61 0.70 3.84 3.31 0.01

0.01 0.48 0.01

1.07 0.69 0.40 1.31 1.92 0.00

0.55 1.58 0.13 2.82 2.54 0.56 3.88 3.26 0.01

0.01 0.58 0.01 0.87 0.70 0.35 1.41 1.83 0.00

0.02 0.65 1.86 0.65 0.10 0.01 2.87 0.62 2.79 0.78 0.31 0.45 4.15 1.20 1.27 3.39 0.01 0.00 _ continued opposite

Optimising

trafficsignalsettings

Table 6 -

T2,

=

T2

OSCADY

0.1359 0.5035 0.1270 0.5981 0.6154 0.5243 0.7569 0.7862 0.0009 28.38 154.16 9.04 241.38 179.40 46.35 264.3 I 261.36 0.28

0.1511 0.6205 0.1211 0.5444 0.6103 0.4999 0.7321 0.7255 0.0011 34.18 129.39 5.77 224.40 216.03 40.08 266.95 244.41 0.33

New formula

2635.23

2601.75

2844.05

OSCADY formula

2542.55

2511.08

2739.25

New formula

1.27

8.52

OSCADY formula

1.24

8.33

1 2 3 D,,, j = z

6 1 8 9

% reduction in total delay estimated by

Present method Optimised settings

0.1370 0.4071 0.1323 0.6531 0.6208 0.5458 0.7455 0.7935 0.0007 28.69 181.00 8.78 250.82 177.92 55.52 257.91 268.44 0.27

1

2 3 4 X,,, j = 5 6 I 8 9

Total delay (Pcu.min) estimated by

continued

Initial settings

6r, = 0.0 s

229

It can be seen from Table 6 that after the extension, the benefits arising from the sequential optimisation of signal settings are small (around 1.25%), but the results are still much better than that given by OSCADY. DISCUSSION

From the examples it can be seen that the new method of calculating signal settings for periods of time-varying demand can give somewhat better results than the program OSCADY. This can be explained by the fact that: (a) when the junction is oversaturated, OSCADY will maximise junction capacity rather than the total delay; (b) There is no interaction between the choices of signal settings in different periods in the optimisation process in OSCADY. The sequential optimisation method introduced in this paper, however, can take into account the interaction between successive time periods, by carrying forward the queue lengths left at the end of each time period, and by sequential re-optimisation and by shifting the times at which the settings change. In these ways an improved performance index can be obtained. The results of the example calculations are consistent with the possibility that the new method may offer greater improvements in more complicated cases. Therefore it can be advantageous to optimise the traffic signal settings for periods of time-varying demand globally. A simple period-by-period optimisation will give only locally optimal solutions, which can be appreciably inferior to the solutions found by the more global heuristics reported here. CONCLUSION

The program OSCADY can model periods of time-varying demand, but it has some limitations: (a) when the junction is oversaturated, the program optimises the signal

230

BinHan

settings with respect to maximum junction capacity rather then total delay; (b) furthermore, the signal settings given by OSCADY are only optimal in each single time period but not for the whole succession of time-periods, hence are only local solutions. In addition to using the author’s new delay expression, the problem of optimisation over periods of time-varying demand has been reformulated to allow successive re-optimisation of the signal settings for the various periods with respect to total delay for the whole succession of time periods, together with optimisation of the times at which the settings of the signal change. The optimisation method, based on the subroutine OPTIM in the earlier procedure SIGSET, is capable of improving the junction control performance. The total delay will be reduced to a greater or better extent compared with the local solutions such as those given by OSCADY. In addition, the algorithm can take some account of delays subsequent to the time periods considered. Hence progress has been made in solving the problem of optimising traffic signals for periods of time-varying demand for a junction by the work described in this paper. Acknowledgements ~ The author is very grateful to Richard Allsop, for his help, patience and careful comments, and for the British Council and the Chinese Government, for the financial support during the course of this research work. He would also like to thank Michael Bell of University of Newcastle-upon-Tyne and two anonymous referees for comments and suggestions,

REFERENCES Allsop R. E. (1971a) Delay-minimizing settings for fixed-time traffic signals at a single road junction. J. Inst. Maths Applies 8, 164-185.

Allsop R. E(l97lb) SIGSET: a computer program for calculating traffic signal settings. Traff. Engng Control 13, 58860. Allsop R. E. (1972) Estimating the traffic capacity of a signalized road junction. Transpn Res. 6, 245-255. Burrow I. J. (1987) OSCADY: a computer program to model capacities, queues and delays at isolated traffic signal junctions. Transport and Road Research Laboratory Report RRl05, Crowthorne. Han B. (1990) Some aspects of optimisation of traffic signal timings for time-varying demand. Ph.D. thesis, University of London. Han B. (1996) A new comprehensive sheared delay formula for traffic signal optimisation. Transpn Res. - A 30, 155-171.

Heydecker B. G. and Dudgeon I. W. (1987) Calculation of traffic signals to minimise delay at a junction. Proc. 10th Int Symp. on Transportation and Traffic Theory. Elsevier Science Publishing Co., N.Y. Kimber R. M and Hollis, E. M. (1979) Traffic queues and delays at road junctions. Transport and Road Research Laboratory Report LR909, Crowthome. Kuwahara M. and Koshi M. (1990) Decision of timings of signal program switching in pretime multi-program control. Proc. of the 11th Int. Symp. on Transportation and Trajic Theory. Elsevier Science Publishing Co., New York. Miller A. J. (1963) Settings for fixed-cycle traffic signals. Opl Res. Q. 14, 3733386. Webster F. V. (1958) Traffic signal settings. Road Research Technical Paper 39. HMSO, London.