The Optimisation of Traffic Signal Control Over a Rolling Horizon

Copyright e IFAC TransportatiOD Systems.

TiaojiD. PRC. 1994

THE OPTIMISATION OF TRAFFIC SIGNAL CONTROL OVER A ROILING HORIZON M G H Bell and D W Brookes· ·Transport Operations Research Group, Department of Civil Engineering. University of Newcastle upon Tyne, NE1 7RU, UK

Abstract. Discrete time forms of adaptive traffic signal control which optimise over a rolling horizon are focused on. Markov chain techniques to compute the evolution of expected queues require prohibitively large amounts of memory and processing. Expressions are developed for the propagation forward in time of the first and second moments of queue size. A bulk service queuing model with random arrivals and departures is used. Particular forms of distribution are not required, but binomial arrivals and departures were used in simulation. Initial results suggest that a short rolling horizon is adequate in the absence of public transport priority. Key words. Traffic control; queuing theory; delays; Markov processes; computer simulation; feedforward control

1. INTRODUCTION 45degreeRne

TIme

Vehicle responsive traffic signal control where decisions are taken at regular intervals of time is the subject of considerable attention at present because of the potential fleXIbility offered. Associated with this is the notion of the rolling horizon, pioneered by Gartner and Kaltenbach, where the best sequence of decisions over the horizon is sought, the first part of the sequence is implemented, the horizon is rolled forward and then the process is repeated with the benefit of the latest detector data (see Fig. 1). This approach has been adopted by a number of signal control systems, specifically OPAC (Gartner, 1983) in the United States, and PRODYN (Henry, Farges and Tuffal, 1983) and UTOPIA (Mauro and Di Taranto, 1989) in Europe. Both PRODYN and U10PIA have featured in various ways in the European research and development programme DRIVE. These systems endeavour to minimise delay, or a weighted sum of delay and stops, where these quantities have been calculated on the basis of non-random arrivals and departures at each intersection. The assumption of non-random arrivals may be adequate for network control

Implemented

plan

Time

Figure 1: The rolling horizon approach

where platoon dispersion is less significant, but will not be adequate for the control of more isolated intersections. This paper concerns the calculation of expected delay and stops for random arrival headways and service times. The discussion in this paper refers to stage-based control, although the methods described apply also

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an efficient way to solve the problem (see Bell, 1992, for a description and evaluation of the algorithm), and is more efficient than either forward or backward dynamic programming.

to group-based control. Streams of traffic, each controlled by a group of traffic signals, are assigned to a limited number of stages, where all groups in a stage receive green simultaneously. The streams included in a stage must be compatible with each other. For safety reasons, clearance times must be allowed for when switching from one stage to the next. Since the clearance time between each incompatible pair of streams will depend on the geometry of the intersection, clearance times will not in general be equal to each other. The time allowed for the transition from one stage to the next must be no less than the largest clearance time involved. While modem microprocessor controllers, like the M32 controller from Siemens, allow for the control of the signal groups individually, the problem of fmding an optimum policy is considerably more complex for group-based control than for stage-based control because of the larger number of control variables. The OPAC, PRODYN and UTOPIA systems, as well as the work presented in this paper, therefore relate to stage-based control.

Figure 2: A decision tree for two-stage signal control As mentioned earlier, OPAC, PRODYN and UTOPIA consider only the deterministic component of delay. While under network control the random component may be less significant, this will not be the case for the more isolated intersection. Bell (1990) has descn"bed how expected delay may be calculated by Markov chain techniques. However, Markov chain techniques require the storage of large probability vectors representing the queue size distributions at the end of each interval, where the number of elements in the vector equals the size of the maximum queue that could have accumulated to that time. The calculation this way of the expected delay and stops over the rolling horizon for a given sequence of decisions requires prohibitively large amounts of both memory to store the probability vectors and computation for their convolution.

The discrete time approach to adaptive signal control requires the specification of a rolling horizon over which optimal stage durations (and potentially also the stage sequence) are sought. A rolling horizon of up to 2 minutes is required because of the short-term costs incurred in switching from one stage to the next and possibly also because of the need to give priority to public transport in a non-disruptive way (as in the case of the UTOPIA system). Effects beyond the rolling horizon can be allowed for in less detail through the inclusion of a terminal cost function. This assigns a cost to the state of the system at the end of the horizon. Decisions are made at regular intervals of the order of 5 seconds (this is the value adopted in OPAC and PRODYN; UTOPIA uses a decision interval of 3 seconds). Expected delay over the horizon is approximated by the sum of the expected queues at the end of each decision interval within the horizon. Since decisions are only made at intervals, it is possible to represent all sequences of decisions over the rolling horizon as a decision tree (see Fig. 2). By valuing each arc according to the expected cost of the decision represented, and by adding arcs to represent the terminal costs, a decision network can be formed (see Fig. 3). Finding the sequence of decisions that minimises expected cost is equivalent to finding the shortest path through the network. A modified form of Dijkstra's shortest path finding algorithm offers

In Bell and Brookes (1992), approximate ways to calculate expected delay and stops which do not require the retention of probability vectors are described, and results are compared with those obtained by the Markov chain approach. The best approximation to the Markov chain approach considered was one where the first and second moment of the queue length distnbutions are propagated forward in time. This paper extends this method to allow for some random variation in the service times. The approach adopted is based on a bulk service model of the queuing process. The model assumes that a queue never reforms during green, an assumption that would require the maximum service time to be less than the

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random variable in this paper. If there is no queue on green, and if the maximum service time is less than the minimum arrival headway, vehicles will cross the stop line as soon as they arrive. If, however, the maximum service time is greater than the minimum arrival headway it will be poSSIble for a queue to form during green. The bulk service model adopted here excludes this poSSIbility .

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It is assumed in the simulation experiments that

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the likelihood of an arrival or a departure of a vehicle at a particular instant is independent of the history of previous arrivals and departures. The arrival and departure processes are then binomial over any time interval, simplifying the expected queue and stop calculation. However, the following analysis is more general and applies to a wide range of traffic flow processes, not just binomial processes.

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Figure 3: A decision network with terminal costs

Expressions for expected delay and stops presupposing a deterministic departure process appear in Bell and Brookes (1992). This paper relaxes the assumption of deterministic departures. minimum arrival headway. 3. RECURSIVE ES1lMA1l0N OF QUEUE

2. lHE TRAFFIC MODEL Let time be measured in units equal to the minimum service time (taken to be about 1 s). Consider a period starting at time u and ending at time v. The initial queue will be denoted by q(u), and the fmal queue by q(v). Consider a red period during which the modelled queue would build up, followed by a green interval during which it would partly or completely discharge. Let the length of the green period be g and the length of the red period be r. Further, let c = r + g. The maximum number of arrivals during the red and green periods is c , and the maximum number of departures is g.

In order to calculate the expected delay and stops to the vehicles encountering the junction, it is first necessary to describe the traffic model. Vehicles are presumed to travel undelayed to the stop line, either joining a notional vertical queue or being serviced (crossing the stop line). As a consequence of this assumption, detectors upstream of the intersection give advance information on vehicle arrivals at the stop line, the degree of forewarning depending on the undelayed travel time from the detector to the stop line. In practice, the extent of forewarning is unlikely to exceed 15 seconds.

Let Pi be the probability that i vehicles arrive over the cycle and s; be the probability that j vehicles depart during the green interval. Given the initial queue q(u), the cycle time c and the green time g, as well as the arrival and departure distnbutions, it is possible to calculate the fIrst and second moments of the fmal queue length probability distribution as follows:

The detection period (the average undelayed travel time from the upstream detector to the stop line) is assumed to be known. Outside this period it is assumed that no advance information on arrivals is available, so arrivals and departures must be estimated. Where signal controlled junctions are feeding the intersection in question, arrivals outside the detection period will be known with more precision. However, this paper focuses on the more isolated intersection. Vehicles can only depart when the traffic signal is effectively green. If on green there is a queue, the rate at which the vehicles can leave the stop line will depend on the service time, treated as a

where k = max(O,j-q(u» is the number of new arrivals that can be processed.

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It is assumed that the minimum arrival headway is greater than the maximum service time, so that the possibility of a queue reforming during green having once discharged is excluded.

(12)

(13) S'(~')

Generally the initial queue will not be known with certainty and therefore the first and second moment of the final queue may be written as

= Ij_o~j ..J
S"(~') = 2I j- o

(14) 2F('~')

(15)

where Eq(y) {q(v)}

= Eq(u) {Eq(y){q(v) I q(u)}}

(3)

Eq(y){q(V)2}

= Eq(u){Eq(y){q(V)2 I q(u)}}

(4)

r

I ~=

F(q(u» ==

=

F(~')

+

(q(u)-~')F(~')

~')2F'(~')

14=

Mj

for j ~ ~'

I

0

for j < ~'

I

Ph>'

for j ~ ~'

2Rj

for j <

2Rj + 0.5 Ph>'

for j

(17)

I = I

~'

(18) ~ ~'

The foregoing argument has left open the vehicle arrival and departure distnbutions. An appropriate distribution would be the binomial. Denote the probability of a vehicle arriving during an interval of time equal to the smallest conceivable headway, say 1 s, by Tr. Then p;

Similarly the expected value of the square of the final queue is given by

= B(i,Tr,c) = c! ~ (I_Try"; / (i!

(c-i)!)

(19)

Denote the probability of a vehicle departure during the same unit of time by cP. Then

Eq(V){q(V)2} == Eq(u){S(q(u»}

= S(~') + (~-~')S'(~') +

O.5Ph>'

L

0.5(Eq(u){q(U)2}(6)

2~~'+~'2)F'(~')

-

r

+ 0.5(q(u)(5)

(~-~')F(~')

+

Pj

L

== Eq(u){F(q(u)}

= F(~') +

I r

where F(~') and F'(~') are the first and second derivatives of F(~') evaluated at ~'. The expected fmal queue is then given by Eq(v){q(v)}

(16) L

Denote the conditional first and second moments by F(q(u» Eq(y){q(v) I q(u)} and S(q(u» Eq(V){q(V)2 I q(u)} respectively. Further, let the expected initial queue be ~ = Eq(u){q(u)} , and let ~. be the nearest integer to ~. Then a truncated Taylor series expansion about ~. gives

=

for j < ~'

1

Sj

= BG,cP,g) = g! cJ>i (1-4> )&.j /

G! (g-j)!)

(20)

0.5(Eq(u){q(U)2}(7)

2~~'+~'2)S"(~')

4. RECURSIVE ESTIMATION OF STOPS where S'(~') and S"(~') are the first and second derivatives evaluated at ~'. Both F(~') and S(~') are given by expressions (1) and (2) respectively, and their derivatives at ~. can be approximated using the finite difference expressions (8) to (11) below F'(Jl') == O.5(F(Jl'+I) F'(~')

==

F(~'+I)

S'(~')

==

0.5(S(~'+I)

S"(~')

==

S(~'+I)

-

-

F(~'-I»

2F(~')

-

+

F(~'-I)

S(~'-I»

2S(~')

+

S(~'-I)

A vehicle is said to have stopped if it joins the vertical queue at the stop line. During a red period all the arriving vehicles will stop, but during a green period vehicles which encounter a queue will have to stop while those that do not will not stop. Given the traffic flow model described earlier, the probability of a vehicle stopping during a unit of time during which the signal indication is green will be equal to the probability of a non-zero queue at the end of the previous unit of time multiplied by the probability of a vehicle arrival during the current unit of time. This is also equal to the expected number of stops during the current unit of time. The total expected number of stops over a period is then simply the sum of the expected number of stops during each unit of time within the period.

(8) (9) (10) (11)

Letting Pj = Ii_t..cPi and Rj = Ii_k..c(i + ~' -j)Pi , where k is defined as before, then the derivatives are given by

Let the number of stops that occur between the 1016

vehicle rules out giving absolute priority to another.

times u and v be denoted by H(u.v). Then

The alternative to absolute pnonty is relative priority. whereby the delay or stop imposed on a

Let T(q(U»

= ~(u.y){H(u.v) I q(u)}

priority vehicle is weighted in some way. This resolves conflicts between priority vehicles and does not neglect the delay imposed on nonpriority vehicles. Weights may be determined in various ways. For example. the degree of priority could depend on the lateness of the bus or tram or on the number of passengers on board.

(22)

As in the case of expected delay. a Taylor series approximation can be utilised. Then

= Eq(u) {T(q(U»} T(U') + (U-U')T'(U') +O.5(Eq(u){q(u)2} - uu' + U'2)T"(U') (23)

~(u.y){H(u.v)} =:

6. THE EFFECT OF HORIZON LENGTH If the arrival and departure processes are binomial. then the expected number of stops conditional on the initial queue is

An important issue is the length of the rolling horizon. On the one hand, the horizon should be long enough to capture the benefits of any stage change. If this is not the case, the control will be biassed against a stage change, as this incurs lost time and therefore carries a short-term cost. On the other hand. the computation required grows exponentially with the length of the horizon because of the way the decision tree branches as time advances through the horizon. Furthermore. decisions taken later in the horizon are based on current data, and so are subject to considerable uncertainty. As more data will become available before these decisions are imminent. there may be little benefit in evaluating them.

T(q(U» ::::: '7T( C - Im_o_Q-Ili_o..mlj_o..i BG.'7T.r+gQ+m+1)B(g-Q+i+1,4>.g-Q+m+1) ) (24) where Q = max(O.g-q(u», and B( .•.•. ). '7T and 4> are as defined in Section 3. Note that the calculation of the derivatives T' and T" by fmite differencing as before requires much more computational effort than in calculating derivatives used in the delay calculations.

5. PRIORI1Y STRATEGIES There are a wide range of possible forms of priority. At one extreme, public transport vehicles can be given absolute priority. Given sufficient and accurate forewarning of vehicle arrivals (distinguishing priority from non-priority vehicles), the signals can sometimes be set so that the priority vehicle experiences no delay. In the case of non-segregated traffic. this requires ensuring that no queue is present on the relevant approach when the priority vehicle arrives at the junction. In practice, granting absolute priority would imply setting signals to minimise expected delay to the priority vehicle.

Such considerations tend to suggest the existence of an optimal horizon length. Simulation experiments have been carried out to examine the effect of horizon length on the performance of signal control.

7. THE SIMULATION MODEL The SIGSIM microscopic traffic simulation model was used. Each vehicle has its own properties. such as type. length. maximum acceleration. desired speed. and so on. and the motion of each vehicle is determined by Gipps's car following model (Gipps. 1981). Vehicles are generated at intervals according to a shifted negative exponential distribution. the mean arrival rate being set by the user.

An attractive aspect of absolute priority in the context of discrete time control is that it will in general reduce the number of sequences of decisions over the rolling horizon. In other words. it leads to a pruning of the decision tree. This should speed up optimisation.

Vehicle detectors can be positioned on each approach to the junction. and the information gathered is used by the optimisation program to estimate the formation of the queues at the junction over the detection period. For the investigation of the approach descnbed above. vehicle detectors were placed at the stop line (to monitor vehicles leaving the system) and at a

As the number of priority vehicles passing the junction per unit of time increases, problems can arise. The delays imposed on some non-priority vehicles. if not already unacceptable. may become unacceptable. Moreover. conflicts may arise where by giving absolute priority to one

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junction design and signal control, which is funded by the Science and Engineering Research Council, currently through grant GR/J19122.

position 110 m from the stop line (to estimate the time at which arriving vehicles join the notional vertical queue). Various quantities, such as the saturation flow for each lane, the signal state transitions, and the delay to simulated vehicles are written to output files. For rolling horizons of different lengths, the total simulated delay and stops can be measured.

11. REFERENCES Bell, M.G.H. & Brookes, D.W. (1992) Discrete time adaptive traffic signal control: The calculation of expected delay and stops. Transportation Research B, In Press.

8. PRELIMINARY SIMULATION RESULTS

Bell, M.G.H. (1992) Vehicle responsive traffic signal control. In: Proceedings of the IMA Conference on Mathematics in Transport Planning and Control (Ed. J. Griffiths), Oxford University Press, 253-265.

Preliminary simulation results for a four-arm intersection with five streams (one separately signalled right-turning stream) suggest that the benefits of looking forward tend to tail off after about 15 to 20 s. There is some evidence that extending the horizon beyond this might even be counter-productive. Further simulations are planned to study the impact of public transport priority on the optimal length of the horizon.

Bell, M.G.H. (1990) A probabilistic approach to the optimisation of traffic signal settings in discrete time. Proceedings of the 11 th International Symposium on Transportation and Traffic Theory, 619-632. Gartner, N.H. (1983) OPAC: A demandresponsive strategy for traffic signal control. Transport Research Record 906, Transportation Research Board.

9. CONCLUSIONS This paper proposes a method to estimate expected delay and stops in signal control algorithms where decisions are taken at regular intervals on the basis of optimisation over a rolling horizon. Previous operational algorithms, such as OPAC, PRODYN and UTOPIA. have dealt with deterministic arrivals and departures. Bell (1990) indicated the importance of considering the random component of delay, particular for junctions that are relatively isolated, and put forward a Markov chain approach that allowed for random arrivals but assumed deterministic departures. Bell and Brookes (1992) put forward a good approximation to the Markov chain approach which involved propagating the first and second moment of the queue forward in time. This paper extends the first and second moment method to allow for some random variation in the service times. There is a discussion on the optimum size of the rolling horizon. Initial simulation results suggested that a short horizon of 15-20 will suffice. Further research is planned to look at the effect of public transport priority on the optimum length of horizon.

Gipps, P.G. (1981) A Behaviourial Car Following Model for Computer Simulation. Transportation Research B, Vol. 15B, 105-111. Henry, J.J., Farges, J .L. & Tuffal, J. (1983) The PRODYN real time traffic algorithm. 4th IFACIFIP-IFORS Conference on Control in Transportation Systems, Baden-Baden. Mauro, V. & Di Taranto, C. (1989) UTOPIA. 6th IFAC-IFIP-IFORS Conference on Control. Computers, Communications in Transport, Paris. Silcock, P. and Sang, A. (1990) SIGSIGN: A phase-based optimisation program for isolated signal-controlled junctions. Traffic Engineering and Control, May.

10. ACKNOWLEDGEMENTS The authors have benefitted greatly from the continual support and advice of Richard Allsop, Tony Ridley, Ben Heydecker and Paul Silcock through a joint programme of research on 1018