Optimal Control of Urban Traffic Signals by Means of Band Transformations

OPTIMAL CONTROL OF URBAN TRAFFIC SIGNALS BY MEANS OF BAND TRANSFORMATIONS Edward S. K. Chien David W. C. Shen Department of Systems Engineering University of Pennsylvania Philadelphia, Pennsylvania

ABSTRACT In general, the objective of urban traffic control is to synchronize traffic signals for the minimization of either total delay or total number of stops or their weighted combination. In this paper, an adaptive optimization procedure for traffic signal systems via multi-band modeling is proposed. The urban traffic behavior at an intersection is described in terms of oversaturated, saturated and free-flowing bands. Only three attributes are needed to describe a band at an intersection approach. Optimal traffic signal control via band transformations is much simpler than using the conventional queue model for road traffic flow. The adaptive optimization procedure predicts a surface, whose shape is generally unknown, via an approximate-analytic method and incorporates learning from experience concerning the locations of local minima from the predicted performance characteristics along unidirectional searches . The fixedtime coordination is readily updated in response to slow traffic variations by computing the sensitivities of optimal signal settings with respect to changes of band parameters.

INTRODUCTION Recent major research effort in urban traffic control can roughly be classified into two broad classes: (a) A traffic signal system is first simplified by means of serial-parallel combination of links. Dynamic programming technique is then used to optimize the total delay time. (b) Optimization techniques are directly applied to traffic signal systems without network simplification. Class (a) assumes that the delay to traffic in one direction along any link in a network solely depends upon the relative offset between the signals at each end of the link. This assumption is valid only when the traffic is close to saturation. Class (b) is more accurate in the description of general traffic behavior, both saturated and free flow patterns are considered. However, a sophiscated model that provides accurate information for optimization is likely to be more complex. This complexity of the model usually increases the difficulty of optimization. Morgan and Little(I,2,3) developed an algorithm

153

using mixed-integer linear programming for synchronizing traffic signals on an artery to achieve maximum bandwidth. They define bandwidth as the portion of the common cycle time of signals, during which period a vehicle can pass through the artery from one end to the other without stopping if traveling at predetermined speed on each section of the road and not impeded by other traffic. However, maximum bandwidth does not necessarily mean minimum delays or stops to the main street traffic. Newell et al(4,S,6) assumed that traffic is close to saturation and without turning on a two-way street. Traffic arrives at any approach can be treated as one platoon with length approximately equivalent to the green time duration of the upstream approach. Hence, delays or stops to traffic on any link solely depend upon the relative offset between signals at each end of the link. Hillier(7) orignated a serial-parallel combination method to simplify tree-type traffic signal networks. He assumed that delay to traffic on any link in a network is a function only of offset between signals at each end of the link. Hence, minimizing total delays for the entire system is converted to a problem of scheduling offsets and dynamic programming technique is used to solve this problem. A1lsop(8,9) and Gartner(IO) extended Hillier's method to gridtype traffic signal networks. No simulation model is needed to calculate delay and stops in this combination approach. This simplicity is the result of the assumed simple relation between delay and offset . However, the val·idity of this assumption is questionable for unsaturated traffic and/or with turning. SIGOP (~nal ~timization)(11,12), developed in U.S.A. by the Traffic Research Corporation, first computes "ideal" offset for each link of a network using a simple simulation model. The control strategy of SIGOP is to minimize a weighted sum of squares of the differences between the "ideal" and the optimal offsets for each link. Gradient technique is used to minimize the least square deviation from "ideal" offsets. A huge number of estimations may be needed to terminate the optimization process unless a good starting point is chosen. An additional shortcoming of SIGOP is the simulation model used to determine the "ideal" offsets does not adequately describe the traffic behavior. TRANSYT (traffic network study tool)(13,14) was developed in England at Road Reaearch Laboratory. TRANSYT uses experimentally derived 50-part histograms

to simulate downstream platoon shapes, with considerations of the random fluctuations of the platoons. However, it neglects the effect on the queue length by the propagation rate of "starting wave". A hillclimbing procedure is used to find the minimum of the linear combination of delays and stops for the system. The step size in the search process is varied in each iteration and is multiples of 1/50 of common cycle time. However, the number of vehicles in each unit of 1/50 of cycle time must be calculated twice for each approach. The overall computation effort increases in the order of square of the number of intersections. Recognizing that the effectiveness of optimization of urban traffic signal system is closely interrelated with the modeling of traffic behavior, there is a justifiable need for developing efficient optimal traffic control techniques and constructing a traffic behavior model capable of providing accurate information for optimization in short simulation time. It is hoped that the method presented in this paper may finally lead to on-line adaptive control ' of urban traffic networks.

MODELING TRAFFIC BEHAVIOR BY BAND TRANSFORMATIONS Traditionally, the formulation of traffic flow is in terms of queue. The number of vehicles in a queue at an approach at any time can be expressed in terms of its initial value and the time integration of the difference between the departure and arrival flows at that approach. Furthermore, the total delay at this approach in a cycle time again requires a time integration of queue. Clearly, the queue length is determined by the position of the last vehicle in queue. This vehicle does not start to move until the "starting wave" propagates from the front of the queue to the rear. In order to simplify the calculation, it is usually assumed that the last vehicle and the first vehicle in a queue move at the same time. As a result, the queue model tends to underestimate the delays and stops. To overcome the shortcomings of the queue model the band concept originated by Little et al(1,2,1,15) is developed into a useful tool for optimization of signal offsets in coordinated traffic control. A band, or a platoon, is a group of vehicles of which traffic behavior can be represented by some common characteristics. Three important factors pertaining to a band are the flow index (y), the bandwidth (BW), and the head arrival time (tha), or head departure time (thd)' The bandwidth is the time length of the band, i.e., the time needed for the band to pass a point on the road. The flow index is defined as the ratio of the flow rate of a band to the saturation flow rate at an approach. The head arrival time is the ideal arrival time of the band at an approach if there were no vehicles preceding the band. The head departure time is the actual departure time of a band from an approach. Note that term "approach" used here can be thought of as the stop line at the intersection for that directioned traffic. Let y(i,k) be the flow index of band k at approach i. By definition i k _ flow rate of band k y( , ) - saturation flow rate at approach i

or y(i,k)

no. of veh. in band k/bandwidth of band k saturation flow rate at approach i (2)

In terms of the flow index, band k is classified as (a) oversaturated if y(i,k»l; (b) saturated if y(i,k)=l; or (c) free-flowing if y(i,k)
1. Assumptions The following reasonable assumptions are used in multi-band modeling of traffic behavior: (1) Cycle length and green splits are fixed for a given period of time. (2) Within each band, vehicles arrive at an approach separated by the same time interval. This assumption is a good approximation since the variations of spacing are already taken into account by the multibands. (3) All vehicles travel down the entire link at the same speed. This assumption is valid since the published standard deviation is known to be small for normally distributed urban journey speeds. (4) Queue is completely discharged during the green period of that approach. (5) Arrival traffic pattern at any source approach repeats itself from cycle to cycle. The assumptions (4) and (5) are valid provided the inputs are low enough to preclude congestions at the approach. The advantage of these two assumptions is that the steady state solution is readily obtained, which is independent of the initial conditions in the network.

2. Transformation of An Arrival Band into Departure Band(s) at An Approach If none of the vehicles in the arrival band is stopped at the approach before it departs, this arrival band is called an undelayed band. A delayed band is the arrival band in which every vehicle is stopped at least once at the approach before it can depart from the approach. Clearly, a partially delayed arrival band can be decomposed into a delayed portion and an undelayed portion. Since lis is the minimum time headway for the departure traffic, where s is the saturation flow rate of that approach, for any departure band the traffic flow index of a departure band is either equal to 1, or equal to the flow index of its corresponding arrival band, if the latter is not greater than 1. Hence, the departure band for a delayed arrival band is always a saturation band. Let us define the tail arrival time, tta(i,k), be the ideal arrival time of the tail of the k-th band at approach i, if all the vehicles in the k-th band are undelayed. So, tta(i,k) = tha(i,k)

+ BWa(i,k)

(3)

where BWa(i,k) is the bandwidth of the k-th arrival band at approach i.

(1)

154

Similarly, the tail departure time, ttd(i,k), is the actual departure time of the tail of the k-th band from approach i, i.e., (4)

where BWd(i,k) is the bandwidth of the k-th departure band from approach i. At an approach, the number of vehicles in a departure band must be equal to the number of vehicles in its corresponding arrival band. By the definition of flow index y, we easily obtain Ya(i,k) BWa(i,k)

This is a transformation of a partially delayed arrival band into two departure bands: a saturated departure band followed by a free-flowing band. A simple example illustrating the transformations of bands is shown in Figure 2. There are four arrival bands at approach i: one saturated band and three free-flowing bands with different values of y. Due to delay, five departure bands are formed. Three of them are saturated and two free-flowing. The first free-flowing departure band is the undelayed portion of the second free-flowing arrival band. Essentially, those three saturated departure bands can be combined into a single saturated band.

(5)

3. Band Transformation in a Grid Network where subscript a and d denote arrival band and departure band, respectively. In terms of the relationships between thd and tha' ttd and tta' there are five basic arrival-departure band transformations as shown in Figure 1. For simplicity, we omit subscript (i,k) in the following discussion, assuming that thd is known. Case 1

thd = tha,

ttd = tta

(Figure la)

The arrival band is obviously undelayed at an approach. This case is possible only when YaSl and YdaYa Case 2

thd = tha,

ttd > tta

(Figure lb)

The arrival band is delayed because the departure bandwidth is greater than the arrival bandwidth. Hence, Yd=l and Ya=YdBWd/BWa >1. This is a transformation from an over saturated arrival band to a saturated departure band. Case 3

thd > tha'

ttd > tta

(Figure lc)

The arrival band is again a delayed band. This is apparent when Ya~l. For Ya
Band transformation in an artery is quite simple since no loops are formed by the links. Loops not only exist but also are coupled together in a grid network. At an interior approach along a loop, the departure bands can only be determined when arrival bands are completely known. These arrival bands can not be determined unless the departure bands from its adjacent upstream approaches are known. Since the latter involves turning traffic, band transformation in a grid network becomes quite complicated due to the presence of "vicious circles". A general solution of this problem can only be achieved by a cut-and-try procedure. Two methods have been attempted to break the "vicious circle". One method assumes the departure band from one of the approaches along a loop being a uniformly distributed free-flowing band; the other method assumes a saturated departure band. The second method is preferred because of the rapid convergence. It often happens that, at some approaches, the unknown arrival bands are indeed transformed into saturated departure bands. The characteristics of this saturated departure band are completely determined. This information accelerates the convergence.

4. Delays, Stops and Queue Length

This case is possible only if Ya
For a pair of arrival band and departure band, the average waiting time for every vehicle in the arrival band before it can depart from the approach is the time lapse between the arrival and the departure times of the mid-points of those bands. So the average delay per vehicle, d(i,k), is

(a) If BWd=YaBWa' the arrival band is a delayed band (Figure Id). This is a transformation from a free-flowing arrival band to a saturated departure band while the queue is completely discharged at the end. In this case, BWa-(thd-tha)/(l-Ya) •

Therefore, the delay to the k-th band at approach i, D(i,k) is

Case 4

thd > tha'

ttd

=

tta (Figures Id & le)

(b) If BWd>YaBWa, the arrival band is a partially delayed band (Figure le). It is evident that BWa>(thd-tha)/(l-Ya) • The BWa can be decomposed into two parts as follows: BWal

D(i, k)

d(i,k) BWd(i,k) s(i)

(9)

The number of stops to the k-th band at approach i, M(i,k), is equal to the number of vehicles in the k-th band, if it is delayed, i.e.,

(6)

M(i,k) = {

(7)

The first part corresponds to a delayed portion with completely discharged queue. Consequently, the remaining portion of the arrival band is undelayed.

155

BWd(i,k) s(i)

if D(i,k)#O

o

if D(i,k)=O

(10)

The total delay, Ds(i), and total number of stops, Ms(i) to all traffic at approach i are

1. Optimization Criteria ~D(i.k)

(11)

k

DM(i.k)

k

In a traffic signal system, let m be the number of unidirectional links. A unidirectional link is defined a s one direction on a roadway between two adjacent signalized intersections. For given common cycle time and splits at all intersections, let 8(i) be the offset of link i, i=1.2 •.. . •m. The total delay and total stops to the k-th band a t approach i are already given in equations (9) & (10). Note that D(i,k) and M(i,k) are nonlinear functions of 8 (i) via the head departure time.

(12)

Note that the above summations are performed on all the bands at approach i. Then the queue length at approach i. Z(i). is Z(i) = Ms(i)/K(i)

(13)

where IlK (i) is the mean queueing gap at approach i.

The total delay D in a traffic signal system is

5. Logic Flow Diagram of Multi-band Model

(14)

A simplified logic flow diagram for the multi-band mode ling is shown in Figure 3 for the determination of the band characteristics at approaches in a traffic signal s ystem. The input data includes number of intersections and indices of source approaches. input traffic volume at source approaches. red and green durations. signal offsets. lengthes of approaches. designed traveling speed. saturation flow rate at each approach, through and turning traffic ratios, etc. The arrival bands are known at source approaches. The arrival bands at an interior approach can be calculated from the departure bands of its adjacent upstream approaches. The band transformations at every approach can be readily computed thus providing the necessary informations . such as total delay and total stops. for system optimization.

where ui is the wei ghting factor for link i. The total number of stops M in a traffic signal s ystem is

M

(15)

where vi is the weighting factor for link i. For control of the traffic signal system, we adjust the offsets to minimize the above two criteria, either singly or their weighted combination.

2. Cyclic Behavior of Performance Function

The multi-band model is simple in the sense that only three attributes are involved. Nevertheless, it offers the capability of describing the realistic traffic behavior. Phenomena such as platoon dispersion. vehicles entering or leaving the street between signalized intersections (e.g .• the presence of a garage). and left-hand turns of vehicles at intersections can be easily incorporated.

ADAPTIVE OPTIMIZATION OF COORDINATED TRAFFIC SIGNALS The optimal control of an urban traffic signal system is essentially a nonlinear high-dimensional multimodal optimization problem. SIGOP performs a large number of trials with different initial points in order to find the global optimum. TRANSYT uses hill-climbing technique which is basically a unimodal coordinate search. In general. all of the previous methods requires extremely long computing time because past experience has not been utilized for recognizing the performance pattern over the control space. Indeed. a special characteristic of urban trafficsignal optimization is the cyclic behavior of the performance function over the n-dimensional signaloffset space. This fact. which turns out to be very helpful in locating the global minimum by the adaptive optimization procedure, unfortunately has escaped the attention of previous investigators.

The signal-offset space is an n-dimensional cyclic sapce. This is easily seen because any signal-offset whose effect on the traffic is equivalent to its value within a cycle time. It follows that the performance function along any search direction in an n-dimensional signal-offset space is cyclic. no matter what the period may be. This cyclic property transforms a unidirectional search into segments which cross different regions in a unit-cycle space. For example, consider a two-dimensional offset space. A multimodal performance function along a search dir ection ~(k) between points A4 & A3 in Figure 4 is mapped into three separated segments in the unitcycle space (Figure 5). The usefulness of this cyclic property is evidenced by the fact that a unidirectional search can locate widely dispersed local minimum regions. This provides the basis for the development of an adaptive optimization procedure.

3. Adaptive Optimization Procedure An adaptive optimization procedure is proposed here to synchronize traffic signals for the minimization of the performance function. This procedure includes the following features: (1) It predicts a surface. whose shape is generally unknown. via an approximate-analytic method. (2) It incorporates learning from experience concerning the locations of local minima from the predicted performance characteristics along unidirectional searches. For the prediction of the performance behavior along a chosen search ~. an n-th order orthogonal polynomial

156

of step size p is fitted to m sampled performance values. where n~(m-2). n is chosen for best-order fit which minimizes a variance term of the errors between the fitted polynomial and the actual data. The polynomial thus fitted adapts itself to the performance function having irregular contour along the search direction. If it is multimodal. several minima are located simultaneously by finding the real roots of the derivative of the polynomial with respect to the step size p. Note that these minima are scattered in the unit-cycle sapce. The promising regions of local minima are determined by an adaptive strategy. The strategy is based on learning the formation of clustering. thereby unwanted regions are eliminated.

where Yi (k) =

{~/m

l/N

if Si(k-l)Si(k)~O if Si(k-l)Si(k»O if m = 0

i=1.2 •..• N

SiCk) is the i-th component of the k-th search direction S(k). OsmSN is the number of components of ~ that did-not change sign at the last step. The above described algorithm is activated only in the presence of a sharp "ridge" by detecting the successive overshooting of the projected local minimum in the direction of one or more of the coordinate axes. The elements that change directions are penalized while the others are reinforced in such a way that a direction of search will approach the direct ion of a "ridge".

The optimization procedure is roughly divided into the following two stages:

4. Cluster Seeking Algorithm Stage I: A best possible minimum is located from unidirectional searches. This is achieved by using the lowest minimum in one unidirectional search as a starting point for the next search. All local minima along unidirectional searches are classified into cluster regions.

Along a unidirectional search. all local minima on a segment in the unit-cycle signal-offset space belong to different cluster regions. The well-known maxrnin-distance procedure is adopted to identify the cluster regions.

Stage 11: Local optimization is performed in each of the cluster regions. During this process. a polynomial fitting is still used for each unidirectional search in order to locate minimum points which may update existing clusters or even form new cluster regions. This process is continued until no more new cluster regions are formed. Due to the cyclic behavior of the performance function. in each local optimization. one can choose the point with the minimum performance value in the same updated cluster region as the starting point for each iteration in order to accelerate the convergence. Global minimum is finally chosen from all local minima. In case the performance function possesses a sharp "ridge". the course of the search will follow a zig-zag path along the "ridge" requiring a considerable amount of steps before the estimate of the signal-offset vector e reaches sufficiently close to the optimum value. A-learning algorithm that introduces the weighting matrix P(k) is adopted to modify the gradient search algorithm as follows. ~(k)

=

~(k-l)

-

pP(k)~(k)

(16)

The matrix P(k) is defined by p (k) 0 1 0 P2 (k)

P(k)

=

N: [

o

:

Due to the cyclic behavior. the Euclidean distance between two offset vectors 8 and ~ in an N-dimensional signal-offset space is defined-as

II~-!II

=

([118(j)-CPU) 112)!

(19)

j

118 (j) -cP (j ) 11

where

b (j) =

(20)

if if if

{~

-1

(8(j)-CP(j»<-C/2 -C/2S(8(j)-CP(j»SC/2 (8U)-CP(j»>C/2

(21)

8(j) and CP(j) are the j-th components of offset vectors e and~. C is the cycle length. Let {~(1).~(2) •..• ~(k) •.. } be the set of cluster centers; T~(1).~(2) •..• ~(i) •.. } be the set of local minima to be assigned to cluster regions. For a local minimum 8(i) we save the minimum distance among those from ~(i) to each cluster center. If the maximum distance among those minimum distances of all the local minima is an appreciable fraction of the "typical" maximum distance among those existing cluster regions. then this local minimum forms the center of a new cluster region. The preceding procedure is repeated until no new cluster region is identified. Then local minimum is assigned to the closest cluster region.

(17)

After all the local minima are assigned to cluster regions. the cluster cent er in each cluster region is updated such that the sum of the squared distances from all points in one region to the new cluster center is minimized. In other words. the new cluster center ~(k) in cluster region k is computed so that the clustering performance index

0

where N is the dimension of the signal-offset space. o ~ PiCk) ~ 1. Pi(O) = l/N. i = 1.2 •.•.• N N[Pi(k) = trP(k) = N. k =1.2 •... i

A linear reinforcement learning algorithm is used to update at every step the components of matrix P(k) and alter the direction of the search according to O
Jc = [

e

II~-!(k)

112

e

£

cluster region k

(22)

is minimized. The j-th component of ~(k) which minimizes this clustering performance index is given by

(18)

157

(23)

Ijl(j) = (E9i(j»/L + (Ebi(j»C/L i

i

minimizing variables for (1) minimum total delay. and (2) minimum total stops. Conjugate gradient is used to generate search directions. Forsythe's(18) three-term recurrence relation is utilized for polynomial curving fitting of a performance function along each search direction. Up to fifty points are sampled for this purpose. The transformation of arrival bands to departure bands for each interior approach along a loop converges in the order of 3 to 4 loop-iterations for a given offset vector. Table 11 lists the optimum offsets for both performance criteria.

where 9i(j) is the j-th component of §! in cluster region k. L is the number of points in cluster region k.

5. Sensitivity Analysis Within the framework of overall fixed-time control. a small traffic variation can easily be handled by computing the sensitivities of optimal signal settings with respect to input flow rates. V. where v

=

(24)

y s

The sensitivity matrix a9(1)/aV(1) a9(2)/aV(1) §:J.V =

~.V

For total delay minimization, 5 cluster regions are found after 40 iterations; whereas for total stops minimization, 7 cluster regions are found after 66 iterations. The average number of vehicles in the network per cycle is 287.23. The total delay per cycle is 2414 . 93 veh-sec under the minimum-delay criterion and 2480.48 veh-sec under the minimum-stop criterion, the difference is 2.7%. On the other hand. the total number of stops per cycle is 157.74 under the minimum-stop criterion and 180.57 under the minimum-delay criterion. the difference is 14.5%. It should be noted that total number of stops and total delay to vehicles at source approaches are not controllable under the assumption of uniform arrival. In fact, they constitute an appreciable portion of the total number of stops and total delay. The total number of stops occurred at source approaches is 60.90 and the total delay is 895.58 veh-sec. If we exclude them and only consider the interior approaches. the difference in total delay between two criteria becomes 4.3% whereas the difference in total number of stops increases to 23.6%.

is defined as

a9(1)/aV(2) a9(2)/aV(2)

a9(1)/aV(m)j a9(2)/aV(m)

a9(N)/aV(2)

a9(N)/aV(m)

•

[ a9(N)/aV(1)

(25) where V(l). V(2) •...• V(m) are input flow rates. 9(1). 9(2) •...• 9(N) are optimum signal offsets. Due to small perturbation of input flow rate V+~V. the new optimum setting §+~2 can be calculated from (26) The sensitivity coefficients are readily computed using the information provided by the multi-band model. Although the sensitivity analysis is only valid for small and slow traffic perturbation. it is anticipated that our approach may lead to the possibility of a multi-level control strategy for on-line generation of optimal signal offsets from sensors when traffic pattern changes. Conceptually. the control strategy may consist of two major hierarchies. namely. a short-term adaptive-learning hierarchy (the method presented in this paper) which is the main control hierarchy. and a long-term trafficsituation-recognition hierarchy which serves basically as a supporting hierarchy. The situationrecognition level is employed mainly to learn the traffic parameters to be used in the adaptive optimization level in order to accelerate its convergence. The situation-recognition procedure will be based on general pattern recognition techniques and is a subject for future investigation.

CONCLUSIONS A multi-band model and an interwoven optimization procedure have been developed for urban traffic control. An adaptive learning scheme has been proposed which provides the memory to the multi-dimensional algorithms in order to utilize the accumulated past experience of the search for accelerating the convergence of the optimum signal setting. The multiband model is simple in structure but capable of refinement.

ACKNOWLEDGEMENTS The authors wish to thank Drs. Samuel D. Bedrosian and Kenneth A. Fegley for the computer facilities and for their interest and encouragement.

AN ILLUSTRATIVE EXAMPLE LIST OF SYMBOLS For the purpose of showing the application of the method. we consider a grid network with 9 intersections as shown in Figure 6. The input traffic flow to source approaches are assumed to be uniformly distributed. The input data for 36 approaches are listed in Table I. In this Table. Pth and Ptu are the portion of through traffic and the portion of turning traffic~respectively. The common cycle time is 60 sec. The duration of green light for each approach is computed according to Webster's rule(16.l7). Approach pairs (1.3). (5.7). (9.11). (13.15). (17.19). (21.23). (25.27). (29.31) and (33.35) are chosen as intersection green onset references. The offsets of eight intersections are

s(i)

Ya(i.k) BWa(i,k) tha(i.k) Yd(i.k) BWd(i,k) thd(i.k)

158

saturation flow at approach i traffic flow index of the k-th arrival band at approach i bandwidth of the k-th arrival band at approach i "ideal" arrival time of the head of the k-th band at approach i traffic flow index of the k-th departure band from approach i bandwidth of the k-th departure band from approach i departure time of the head of the k-th band from approach i

d(i,k) D(i,k)

Ds (i) M(i,k) Ms(i)

l/K(i) Z(i) ~(k)

o P(k) !(k)

~,V

average delay per vehicle of the k-th band at approach i total delay to the k-th band at approach i total delay to all bands at approach i total stops to the k-th band at approach i total stops to all bands at approach i mean queueing gap at approach i total queue length at approach i linear search direction at the k-th iteration signal-offset vector weighting matrix for modifying gradient search algorithm at the k-th iteration cluster center in cluster region k sensitivity matrix

11.

"SIGOP, Traffic Signal Optimization Program", Traffic Research Corp., N.Y., 1966 .

12.

"SIGOP, Traffic Signal Optimization Program, User's Manual", Peat, Marwick, Livingston & Co., N. Y. , 1968.

13.

Robertson, D. 1., "TRANSYT: Traffic Network Study Tool", Proc. Fourth International Symposium on Traffic Flow, Karls r uhe, Germany, 1968, pp. 134-144.

14.

Robertson, D. I., "TRANSYT: A Traffic Network Study Tool", Road Research Laboratory Report LR 253, Crowthorne, Berkshire, England , 1969.

15.

Dick, A. C., "A Method for Calculating Vehicular Delay at Linked Traffic Signals", Traffic Engineering and Control, Vol. 7, No. 3 , 1965, pp. 224-229.

16.

Webster, F. V., "Traffic Signal Settings", Road Research Technical Paper No. 39, Road Research Lab . , London, England , 1961.

17.

Webster, F. V., and Cobbe, B. M. , "Traffic Signals", Road Research Technical Paper No . 56, Road Research Lab . , London, England, 1966.

18.

Forsythe, G. E., "Generation and Use of Orthogonal Polynomials for Data-Fitting wi th a Digital Computer", SIAM Journal of Applied Mathematics , Vol. 5, No. 2, 1957, pp. 74-88.

REFERENCES 1.

2.

Morgan, J. T., and Little, J. D. C., "Synchronizing Traffic Signals for Maximal Bandwidth", Operations Research, Vol. 12, 1964, pp. 896-912. Little, J. D. C. , Martin, B. V., and Morgan, J. T., "Synchronizing Traffic Signals for Maximal Bandwidth", Highway Research Record , No. 118, pp . 21-24.

3.

Little, J. D. C., "The Synchronization of Traffic Signals by Mixed-Integer Linear Progranuning", Operations Research, Vol . 14, 1966, pp. 568-594.

4.

Newell, G. F., "Synchronization of Traffic Lights for High Flow", Quarterly Applied Math., Vol. 21, No. 4, 1964, pp. 315-324 .

5.

Bavarez, E., and Newell, G. F., "Traffic Signal Synchronization on a One-Way Street", Transportation Science, Vol. 1, No. 2, 1967, pp. 57-73.

TABLE I

Approaches in the Sample Problem

Intersect ion

Approach

Up st r eam

Input

Number

Number

Approach

Tra f f ie veh/hr

sou r c e

6.

7.

8.

9.

10.

Allsop, R. E. , "Selection of Offsets to Minimize Delay to Traffic in a Network Controlled by Fixed-Time Signals", Transportation Science, Vol. 2, No. 1, 1968, pp. 1-13. Allsop, R. E., "Optimization Techniques for Reducing Delay to Traffic in Signalized Road Networks", Ph.D. Thesis, University College, London, England, 1970 .

sou r ce

17 18 19 20

13,14 6,7

0.7 0.6 0.8 5

0.3 0.4 0.2 0.5

21.79 36 . 21 23 . 79 36.21

35 35

0.7 09 0.5 0.7

0.3 0. 1 0.5 0.3

36.05 23,95

36.05 23.95

40

0.8 0.5 0.6 0.8

0.2 0.5 0. 4 0.2

31 17 28.83 31.17 28.83

40 35 40

0.' 0.7 0.5 o6

0. 4 0.3 0.5 0. 4

36 . 73 23.27 36 . 73 23.27

35 35 35 35

21

0.8 0.8 0, 4 0.5

0.2 0,2 0.6 0.5

31.00 29.00 31 . 00 29.00

35 40

350

0,7 0.8 0. 4 0.8

0.3 0.2 0.6 0, 2

32 .36 27.64 32.36 27.64

0.4 o2 0.3 0,3

31.82 28.18

350

0.6 0.8 0,7 0.7

400 400

0.5 0.8 o8 0.•

0.5 0.2 0,2 0.4

39.09 20.91

600

29,32 17 , 18 10,11

23

source

24

33,36

25 26 27 28

sou r ce 14 ,15 3 1 ,32

29 30 31

25,26 18,19

Saturati.on Flow Rate :

4 00 4 00

23,24

22

33 34 35 36

159

o

700

2, 3 19 ,20 25 ,28

500 500

35,36

32

Gartner, N., "Optimal Synchronization of Traffic Signal Networks by Dynamic Programming", Fifth International Symposium on Theory of Traffic Flow and Transportation, Univ. of California, Berkeley, 1971, pp. 281-295.

30 40

21,24

13 14 15 16

29, 30 22,23

1800 veh/hr

Spped mph

29.50

5,6

source

Travel inli';

JO.50 29 . 50 30.50

11 , 12 17 ,20

9 10 11 12

Hillier, J. A. , "Appendix to Glasgow's Experiment in Area Traffic Control", Traffic Engineering and Control, Vol. 7, No. 9, 1966, pp . 569-571.

Green Durat ion

0.2 0.2 0.3 0. 4

1,2

source

Ptu

0.8 0.8 0.7 0.6

450 500

7,8 13,16

Newell, G. F . , "Traffic Signal Synchronization for High Flows on a Two-Way Street", Proc. Fourth International Symposium on Tra~Flow, Karlsruhe, Germany, 1968, pp. 87-92.

Pth

28 .18

30

35

35 40 35

35 35 40

31,8 2 20.91 39.09

40 35

Optimum Offsets for the Grid Network

TABLE II

Offset Min-De1ay

Intersection Number

sec Min-St op

1 2 3 4 5

0.00 9.78 0.09 8.32 19.71

0.00 3.12 58.87 59.92 57.81

6 7 8 9

28.90 16.12 30.91 34.44

4.10 54 . 04 0.15 28.74

l NO

(.)

(b)

f·i/

START

(d)

T

NEED I NIT IAL DATA ?

YES

READ I NITI AL DATA

J

Figure 1

(e)

Five Basic Band Transformations

t

DETERM I NE CALCULAT I ON SEQUENCE FOR I NTER I OR AP PROACHES

t

t SOURCE APPROACH :

AT CAL CULAT E RED & GREEN ONSETS ; DEPARTURE BANDS; DELAYS , STOPS, QUEUE L ENGTH

t

NO /

\

SOURCE

~~ROACHES DONE?

Di sta nce

~

YES \

J

--,

t

t

L

App roach 1

AT I NTER I ON APPROACH : CALCULATE RED & GREEN ONSETS ; ARR I V AL BANDS

t

CA L CU L ATE DEPARTURE BANDS; DELAYS, STOPS, QUEUE LENGTH

t

NO /

INTERIORA~PROACHES DONE?

YES

t

END T ime

Figure 3

Logic Flow Diagram for Multi-Band Model Figure 2

160

Band Transformations in Time-Space Diagram

PI

--'~4--GL---------~-----'2~E--~L--L----------~~--------~D~A~3-- ~( k )

'2

Figure 4

Performance Function along Search Direction

~(k)

~1

0

15

J6 70 18190 J 2

30

14

600 ft

17

5

8

26

27[ 29

20

32

31 [

650 ft

21

9

12

24

33 36

22

750 f t

Figure 6 Figure 5

Performance Contour Map in Unit-Cycle Space

161

7 00 ft

Grid Network of Sample Problem