Linked traffic signals for maximum capacity

Tnmpn Rcs.,Vol. 1I, PP.229-234. Per&mm Press 1977. Printedin Great Britain

LINKED TRAFFIC SIGNALS FOR MAXIMUM CAPACITY

w. BACON Oxford Polytechnic, Oxford, England (Received 25 August, 1975;in revisedjonn I October 1976)

Ah&a&-The capacity of a single route which passes througha chain of non-priorityjunctionscan sometimesbe increasedif such junctionsare providedwith linked signals. Circumstancescan arise in which the potentialfor the use of the route becomes so groatthat it is desirableto qaximise the capacity ratherthan minimisethe delays to traffic.The authorhas previously describeda method of calculatingthe proportionof green time for each phase which would maximise the capacity of a single chain of signals, providedthat the timingsof the signals could be satisfactorilyco-ordinated.This paper describes a method of determiningwhether such co-ordinationis feasible, firstly by computing the feasible range of offsets for each link, relative to the platoons of vehicles, and subsequentlyby searchingfor a coherent system of offsets relative to all the tratIicsignals in the open or closed chain. INTRODUCTION Most area W&c control systems, in which the timings of

the signals are co-ordinated, are intended to reduce the delays to tm!Iic. In general, this is achieved by reducing the lengths and waiting times of the queues at the signal stop lines, at all signals within the area. The author (1974) has suggested an alternative strategy in which the aim is to maximise the capacity of a selected route, ignoring delays to tralk. If this aim is paramount and if delays as such are to be ignored, then totally d8erent tic conditions can be envisaged as occurring along the route. In particular, queues on the approaches to the signals may actually be contrived by the control system, rather than reduced to a minimum. It is perhaps therefore desirable to summa&e the concept in more detail. In the previous paper, the need to maximise route capacity was considered in a situation where tic was to be dissuaded from passing through an area (such as a CBD) and would consequently overload the selected route. The route could form a closed ring enclosing the area, but this is not essential. Such situations are perhaps more lie to occur in smaller cities which may not yet have linked signal systems and may have few alternative routes for displaced traliic. A similar situation could occur on the approach route to a tunnel or bridge. In order to design the system for maximnm capacity, it is necessary to know the origins and destinations of the traflic using the single chain of tic signals-in terms of where the vehicles enter and leave the system, i.e. at which point and in which direction they arrive at and depart from the chain. From this information the technique of linear programming can be used to compute the optima1 capacity of the chain of signals by finding the optimum proportion of green time for each phase, at each signalised junction. However, this objective could be achieved only if saturation flow could be maintained during all the green phases at all the junctions. It was suggested that this could be done by co-ordinating the timings of the signals, but ways of doing this .were not described; the purpose of the present paper is to consider this problem.

The technique of linear programming allows the designer to specify constraints at wili, and it is thus always possible, for example, to stipulate that the whole of an existing tratlic stream, or streams, shall be accommodated. If the route is a tunnel approach, one could stipulate that the sum of the streams going to the tunnel shall not exceed the capacity of the tunnel. Such constraints may or may not reduce the achievable capacity, but if in fact they do so, they may make an infeasible system feasible, for it should be remembered that the linear programming solution will not necessarily represent a practicable system. This is because it may not be possible to co-ordinate the signals to maintain saturation flow. Clearly short distances between junctions, and uneven spacings, are more diEcult to accommodate in any such systems. The purpose of the present paper is thus to describe how a system which maintains saturation flow at signals during all the green phases can be. designed, where possible, and how to ascertain whether it is possible. It is assumed for design purposes that the tra5c volumes will be uniform, with no variations even in turning movements, and with permanent queues at all entry points. The dispersion of platoons leaving the signals is allowed for. A standard fixed-time plan would be calculated off-line on this basis, but the timings would have to be advanced or retarded to suit fluctuations in tra5c volumes, within the constraints of the system, from information provided by vehicle detectors as discussed in the previous paper. The paper is divided into two parts. In Part 1, a single link between junctions is considered in isolation, with the object of obtaining a range of possible timings, or offsets, for that link, which will maintain saturation flow. At this stage, a solution is usually possible, since the problem of co-or&mating the timings on all the various links is ignored--as are the timings in opposite directions on the same link. In Part 2, the problem of co-ordination is discussed; the constraints on a solution are described; and a strategy for searching for solutions is suggested.

229

230

W. BACON PART1

purpose of this part of the paper is to establish a range of feasible values for the offset between one junction and the next, if all other offsets are ignored. An offset is detined here as the time lag between the start of the upstream green phase and that at the next junction downstream in which the same platoon crosses the stop line. This ditIers from the normal dell&on in that the offset is relative to a platoon of vehicles; however, the platoon may be released by the downstream green either before or after the vehicles that have turned onto the link, depending on whether a maximum or a minimum offset is to be found. The offset must fultll two objectives: (1) it must ensure that at any time when the lights are green at the downstream junction, there is a queue (30 vehicles) to maintain the flow; and (2) it must ensure that the link street is not too congested, i.e. that there is su@cient space to accommodate vehicles leaving the upstream junction. The second objective requires that the offset must not be too long, whilst the hrst requires, in effect, that it must not be too short. The

The necessary constraints are therefore as follows: (1) The shortest possible offset, xmin, must be that which ensures that the last vehicle required to till the green phase to saturation will arrive just in time. See Fig. 1, which is a time-distance diagram of the kind used to describe the early “green wave” systems. (2) If the offset is longer than this, some vehicles will be left behind at the end of the green phase, because part of the previous platoon will be in the queue in front of the turning vehicles. Since the minimisation of delay is not a specitlc objective of the system, the number of vehicles left behind will not, in principle, be constrained by considerations of delay, although obviously the shorter offsets will cause less delay, (3) It is therefore possible, even if it is not desirable, to allow the queue to back right UP to the upstream junction, so long as the tail of the queue begins to move forward at the precise moment when more vehicles wish to join the queue. The maximum possible offset is that which results in the tail moving forward at the moment when additional vehicles are released from the upstream junction. See Fig. 2, but note that this moment could occur during any phase,

2nd Junction

Direction of Travel

Tim@

Fii.

1. Tie-distance diagramfor calculationof minimumoffset.

A Distance

2nd Junction

x-(seconds) k Fii.

2. Tiiedistance diagramfor calculationof maximumoffset.

Linked tratlic signalsfor maximumcapacity because of turning vehicles. This would depend on the length of the street and on the lengths of the platoons in the queue. Fiis 1 and 2 both show the existence of a queue when the downstream green phase begins. The queue in Fig. 1 is of the minimum length, whilst that in Fig. 2 is the maximum. The question therefore arises as to how this queue can be created in the tlrst place, since it will not exist during off-peak periods, and how it can be set to its correct length, since the number of vehicles entering the link in each cycle is the same as the number leaving. This can only be done by slightly changing the ratio between the phases, in favour of the transverse phases, until the correct queues have formed on each of the links, as indicated by detectors. It could be done whenever the queues on the external approaches to the system exceed a certain level, or alternatively by time clock if traUlc flows are suthciently predictable. Extreme care would however be necessary to avoid unstable fluctuations in the lengths of the external queues. It should perhaps be mentioned that it is unlikely that any benetlt will be gained in practice from forming queues of more than say two platoons in length, as suggested by Fig. 2, which is drawn thus merely to illustrate the concept. The minimum length, shown in Fig. 1, is slightly less that one platoon. Minimum offsei

Figure 1 shows the conditions appropriate for the minimum offset, from which it can be seen that x&=g,+y-f*

(1)

where g,, is the effective length of the green phase at the 1st junction, in seconds; y, is the journey time of the tail end of the platoon, in seconds, over the distance (L i t&); d, .bing the mean distanceheadway between stationary vehicles in the queue; and n2, being the number of vehicles to be released from the queue in each green phase; and t2 is an allowance for the time lag from the onset of the green to the movement of successive vehicles waiting in the stationary queue. If uniform acceleration and speed is assumed, then t*=An,-B

(2)

where A and B are constants which depend on the acceleration and speed. It would however be desirable to determine the constants by field measurement, using regression analysis, since accelerations and speeds will not be constant. The second constant is an allowance for acceleration from rest to uniform speed. The journey time of the tail end of the platoon depends on the degree to which the platoon will disperse. S&don (l972) has suggested that the mathematical model of dispersion which was devised by Robertson (1%9) gives better results than other known models, at least in the U.K. Some simulations of dispersion by the author, usiug the .Robertson model, indicate that a good estimate of y is given by

y=2.8t-4.4 where t is the journey times of the nose of the platoon, in seconds, over the same distance (L - n,d). (This formula applies so long as t is greater than 5.6 set, which would almost always be true: the mean journey time of all vehicles was assumed to be 1.25 t, and the constant F was taken as l/(1 +O.S t), as recommended by Robertson; these values are probably applicable only in the U.K.) It is important to note here that f2 depends only on n2 and moreover, since L and d are constants, y also depends only on n2. Furthermore, n2 is a function of g,, and if the signals operate on a common master cycle, both g, and g2 are functions of the cycle time, since the proportion of the cycle allotted to each phase can be computed by linear programming. Consequently x* depends solely on the length of the common cycle, It will be shown later that the maximum offset also depends solely on the length of the common cycle. (In both cases, this is strictly correct only if the lost time per cycle is known in advance. It should not .be diBicult to estimate this as a proportion of the cycle time, but if the assumed value turns out to be very inaccurate it would be necessary to begin again with a better estimate.) Maximumo&l There are two possible situations to consider when calculating the maximum offset; one or the other must occur, but not both. Under the conditions shown in’& 2, which shows one possible situ&ion, the last vehicle in thequeue when it backs up to the upstream junction arrives during a green phase. The alteruative is that it could be a turning vehicmg in a red phase. In the tirst case, a formula for the maximum .offset can be derived as follows: x,

= maximum offset = the length of time from A to C =AB+BC =AB+DE-(DE-EC)

Let N = the number of complete platoons for which space is available on the link; this is an integer number obtained by dividing L by n2d and rounding off downwards. Then, DE = NC (set) where c is the length of the cycle, and (DE - BC) = A&/d) - B (set) on the same basis as that used for calculating r2 previously, and Al?= (+qEsec

where p,, is the proportion of non-turning tratllc on the green phase at the 1st junction, and s;, is the saturation flow at the stop line at the 1st junction, in vehicles/hr. assuming a homogeneous mixture of turners and nonturners on the green phase; where this is not the case, an appropriate value of p, could be assumed. This would depend very much on site conditions.

232

W. BACON

It can be seen from eqns (4) and@) that the maximum offset is a function of N; c, n2 and g,. As with the minimum offset, nz and g, are functions of c. Also N depends on n2 and therefore on c. In effect, therefore, the maximum offset depends solely on the length of the signal cycle.

Rearranging, AB =;D-NnD where 3600

D=plb Therefore x”“=$D-Nn2D+Nc-A$+B. Siiplif

ying : x”“==,N(c-n,D)-$(A-D)+B

(4)

where A, B, D, L and d are constants for design purposes. Equation (4) applies to the sitution shown in Fig. 2, in which the space behind the last platoon is not sulllcient for all the vehicles arriving duriug the next green phase, i.e. where

L-Nnzd
3600’

In the alternative situation, which could occur almost as frequently, there is suthcient space for the green phase arrivals, but not for all the arrivals during the subsequent red phase, i.e. the turning vehicles. In this case, the formulae for DE and (DE - BC) remain the same, but AB is given by

where pf and st are the equivalent values of p1 and s1 for the opposite or transverse green phase, and k is the lost time per phase change (say, 3 set). Therefore AB=;E-NnrE+gl

(

1-i

>

+k

CoudiGons Equations (l), (4) and (5) are applicable to the following conditions: (a) l3oth junctions must work to a common cycle. (b) The flows crossing the stop lines at the upstream junction must be a homogeneous mixture of turning and non-turning vehicles, with the same arrival pattern in every cycle. (c) The downstream green time is exactly equal to that required to release the number of vehicles arriving in each cycle. In linear programming terms, this green time is one of the bmding constraints on capacity. (d) In the case of eqn (l), it is assumed that the tail of the dispersing platoon will not be overtaken by turning vehicles from the next transverse green phase. The value of y is therefore probably couservative, although the value of tl is not, since a mean value is proposed. (e) In practice, the formulae would need to take account of, the number of lanes available for the queues. With regard to the Ilrst condition, this is the easiest case to deal with and simplifies the presentation. If one junction has a cycle which is a multiple or sub-multiple of the other, the calculation of the minimum offset is more complicated, as it is subject to a condition which can lead to a more complex equation. The calculation of the maximum offset is also dot quite so straightforward since N has to be defined in a more elaborate way. However the basic principles are not altered iu either case. PART2

The purpose of the next stage of the procedure is to tind the feasible values, if any, for a common cycle time, c, for the system-together with the associated offsets on each liuk for each feasible value of c. It is necessaryto begin by establishing a basic rule for the creation of a feasible system, as follows. Since at any one junction the start of the green phase must coincide for the two opposing directions of travel, unless late starts are permitted, it follows that the sum of the offsets for the two directions on each link must be a multiple of the cycle time. Referring to Fig. 3, this can be stated in the form: xl2+ xzl = l12c (etc.)

E=(l-;s,.

where x12,is the time offset from junction 1 to junction 2; .x2,,is the time offset from 2 to 1; and ilz, is an integer number (b 1) associated with the link. Note that since x,z+x2, = x2,+ xl1 it follows that i,* must be equal to i2, (etc.). In other words, only one such number must be assigned to each link.

so p?=- f;E-NfiE+g,

(I-;)+k+Nc-A$+B.

simplifying: y=N(c-n,E)-i(A-E)+g,(I-;)+B+k.

(6)

(5)

Method As mentioned in Part 1, the minimum and maximum offsets are both functions of the cycle time, c, and

233

Liuked tratllcsignalsfor maximumcapacity

4th Junction

3rd Junction

2nd Junction

1stJunction

Fig.

3. J)ime&tence

diagram for CO-Ordii

consequently c81111ot be calculated in advance. Further, the “i” values can be any integer numbers, and are not necessarily the same as, or related to, one another. It therefore appears to the author that the existence of more than one range of feasible values cannot be ruled out, and that it is not possible to use a “programmmg” or other rigorous method of search for a feasible system. The suggested procedure is as follows: (1) Calculate, for each junction in the system, the length of the shortest cycle that will just pass all the tratllc, i.e. that for which lost time is at the allowable limit. The largest of these values would represent the lower limit of the range of search for a feasible value of C.

(2) Calculate, for each junction approach, the length of cycle which will result in the queue backing up to the adjacent junction, denoted by:

OffSetS

possible alternatives are further constrained by the restriction that, if the cycle time, c, is slightly less than that which causes the queue to back up to the adjacent upstream junction, xr, then there will still only be sufhcient space for one complete platoon (for this link and direction); only if the cycle time is equal to or less than half this value will there be room for two complete platoons, the size of the platoon being proportional to the length of the cycle. More formally, if c >f fl,

then il2 can only equal 1,

if C>$T,

then il2 can only equal 1 or 2, etc.

and if c > f W, then izl can only equal 1, if c > i fl,

then izt can only equal 1 or 2, etc. (7)

CT.

(For any given cycle length, the number of vehicles to be released during the downstream green phase can be calculated from the known flow on the link. For feasibility, there must be suthcient storage space for this number of vehicles,t because otherwise saturation flow cannot be maintained during the succeeding green phase.) The smallest of these values therefore represents the upper limit of the range of search. (3) Set up a step-by-step test of the values lying within this range, in increments of say 6 sec. Print out any feasible results at each step, and “warn@ flags against any cycle times which are almost feasible. The tests would be subject to a number of conditions and constraints as descrii below. Having established the overall range of search, the tThe queue can sometimes be “topped up” during the green

phase, after the tail of the queue has started to move and whilst thereis stilltimefor vehiclesto reachthe stoplint beforetheend of the green. Allowance for this possibility is both feasible and desirable.

Note that this constraint is not the same as that implied by the maximum offset. It will usually be a more severe one, but so far as the writer can see, there may be instances where this is not so. The test to be applied to each trial value of c will depend also on the type of route on which the signals are located, as follows. Case 1 A single chain of signals not forming a closed ring: In this case, the offsets adjacent to one another in one direction of travel (x,~,xt,, . . . in Fi. 3) do not return to the same physical or geographical point, and therefore they do not form a closed set in their timings. They are thus independent of each other, apart from the interdependence of pairs of offsets on the same link, already specified by eqn (6). A systematic search can therefore be made in the following way. Set the test value of c at the lowest end of the range of search. Try each link in turn to ascertain whether any feasible value of i,, exists. If there is no

w. .gAcON

234

feasible value on any one link, the whole set must be discarded, since this cycle time is not feasible. If more than one feasible value exists, the lowest value can be selected, since it causes least delay, and the others can be discarded, since the liis are independent of each other. Increase the test value of c by say 6sec and repeat, regardless of whether a feasible solution has already been found. Continue thus through the range of search until the upper threshold is reached. Case 2

A single chain forming a closed circular ring: In this case, an additional constraint will apply. This is the requirement that the sum of the offsets, proceeding round the ring in one direction, must be a multiple of the cycle time, since they return to the same junction, i.e. (xl*+xu+x~++**

+ G.3 must equal ic

09

where i is any integer number ( B 1). If this requirement ia met, it follows automatically from eqn (6) that the sum of the offsets in the opposite direction will also be a multiple of the cycle time. Consequently either one direction or the other must be considered, but not both, ,A systematic search can therefore be made in the same way as for Case 1,:except that any feasible vahte of ie and not merely the lowest would be considered. Whenever a set of such values is found, the feasibility of “closing the ring” by satisfying condition (8) must be checked in the following manner. Find the range of (xl2+ xu + - * * + _x,_,) subject to the following: (ilzc - $3

congestion, such as that which might occur as the result of a new traflic management scheme. The question which can only be answered by experiment is whether additional capacity provided on the favoured route would actually be taken up, at least to the extent that driver behaviour matches the offsets, as here defined. TrafGc volumes might instead diminish to a lower level, as some people found journeys no longer worth making. Traffic studies have frequently demonstrated that trtic adjusts itself to the conditions provided for it; for example it is well known that trafhc volumes reflect road capacities to a considerable extent; but it is also well known that congestion is it’s own restraint; tra5c never rises above a certain level of congestion except on an occasional and irregular basis. It seems that extreme congestion will always be confined to a small number of critical sites, and will never occur over large areas. However, the author makes no apology for implicitly advocating a policy which would cause extreme congestion on a single route. With regard to the method described in Part 2, it see.ms unlikely that the range of search would be wide, or that may alternative solutions would be found, since in practice only long signal cycles would have the required capacity. Moreover there will usually be a shortage of space for the queue somewhere in the system which will impose an upper limit on the cycle time. Indeed it seems likely that in most real situations the target capacity would have to be set initially at a relatively high level and progressively reduced until a feasible system is found. It might then be necessary to reduce it still further until the delays reached a level judged to be realistic in the light of the choices open to drivers. After all some of them can stay in their homes.

S xl2d (Lc - x*)

x~rnx,*4x~ (iuc - x?) d xn G (&c - xd)

(9)

The type of system envisaged above would probably not be very successful except under conditions of severe

Bacon W. (1974) Some new concepts in area tr&lc control. PTRC Annual SummerMeetina,Univ. of Warwick.Julv. RobertsonD. I. (1%9)T,raw~py~: a-m@c nerwork~ti& tooi.RRL Report LX 253, Transport & Road Research J&oratory, Crowthorne,Berks. scddon P. A. (1972)The predictionof platoon dispersionin the combinationmethodsof link@ t&Ticsignals. TranspnRes. 6, 125.