An Extended Traffic Flow Model for Inner Urban Freeways

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AN EXTENDED TRAFFIC FLOW MODEL FOR INNER URBAN FREEWAYS M. Cremer* and A. D. May** *A rbl'i/ lbl'l"l'ic!t AUlulIIll lilierullgsIPc!tll ik. T l'cil 11 , [ 'lI il'l'niliil // all/burg- H arburg, 2050 I/all/b urg 80, FRG ** / lII /illlll' fur T rampullaliulI SludipI . Cll i"I'l"I i/y of Caliju ,."ia. BerkPip)', CA 9-1720, [ 'SA

Abstract . Convent i onal macroscop i c mode l s for t raffic flow in t he pas t ha ve s hown a serious drawback when app l ied to inner urban freeways : traffi c phenome na were not s imul ated realistically at high demand situations when t r affi c flow gets congested in front of a bottleneck or i n the merging area of the numerous on- ramps . In th i s paper an extended macroscopic model is pr esented whi ch ove r comes these malfunct i ons by s i mp le but effective modifications of the mode l equat i ons . While the computational effort is i ncreased on l y sl i ghtly , the model ' s performance is imp r oved i mp r ess i ve ly when a pplied to urban freeways wi th many on- and off- ramps a nd bott l enecks . A ver ifica tio n of t he model with real data f rom Santa Mon i ca Freeway underlines the e f f i ci enc y of the mo de l' s extensions _ Keywords . Tr affic f l ow; modelling ; model ve r ifi cation ; topologica l inhomogenities . I NTRODUCTION

f l ow i n an excellent manner provid ed that the free way ' s topo l ogy i s homogeneous and has no bott l enecks . However , i n cases of i nhomoge neous topo l ogies wi th lane drops and hi gh on- and off-ramp vo lumes as given at i nner urba n fr eeway s the mode l has shown unsatisfactor y and unr ealistic behaviour.

Tr affic flow mode l s are an i mportant tool for many pur poses i n d,s i gning and operating roads and freeways . These inc l ude ana l ysis , design of traffi c fac i liti es , design of automatic su r vei l lance systems , cost benefit analys i s and development of cont r ol strategi es . Depending on the f i e l d of ap pl i cation and on the degree of refinement , a l a r ge numbe r of models for the process of traffic fl ow may be found i n the li terature (for an overvi ew see May , 1981 and Gibson , 1981) . Accord i ng to what states of traffic flow are descr i bed , the mode l s mi ght be di vi ded into two classes : static models wh i ch represent only steady state flow dis t ri but i ons and d ynami c models which describe i nstat i ona r y transitions too . Another way of cl as sification discernes microscopic models where each i nd i vi dual vehicle ' s motion is modelled and macroscopic models where aggregate variables are used to def i ne traffic state .

It is the purpose of th i s pa pe r t o propos e some mod ifications of t he mode l equations by wh i ch t hese shortcomi ngs a r e ove rcome . In another r ece nt repor t by Babcock et a l . (1982) t hes e difficu lties wer e avoided by an adaptive scheme of repeated subdivision of the spati a l and t emporal step wi dth whereby the computation t i me becomes state dependent and may inc r ease dr as ti call y in c ritical cases . In contrast to that ap proach the tempor a l a nd spatial step width i s kept fixed he re . This l eads for a given freeway configura ti on with a prope rl y prespecified segmentat i on to a fi xed system ' s order with much less computat i onal requirements which is particularly use f ul f or rea l time appli cations .

According to still grow i ng traffic demand , design and operati on of traffic facilities have increasingly to deal with large , interconnected systems rather than wi th isolated subsystems . Moreover , the necessity to improve the efficiency and utili zation of traffic facilities by operational means requires that instationary situations and dynami c phenomena of traffic flow are taken into conside r ation . For these problems a dynamic , macroscopic model appears to be most suitable since it is able to handle instationary transitions wh i le keeping the computat i onal effort at a moderate level even when large systems have to be modelled .

The paper is organized as fo l lows . I n the next section the basic traffic flow mode l i s introduced and its malfunctions at bott l enecks a r e demonst r a ted . In the following section an anal ys i s is performed to identify the reasons f or th e models fail ures . On the bas i s of th i s ana l ys i s several improvements are proposed for the mode l s equati ons to overcome these failu r es . The pa r ame ters of the extended model t hen are calibra t ed us ing real da t a which were collected du r ing a t hree hour s period at a section of Santa Mon i ca Fr eeway i n Los Angeles . It will be shown by a compari son of r eal measure ments and model predict i ons that wi t h t he proposed and calibrated extensions the mode l is able to re produce critical situations hi gh l y r ea li s tically . Finally , a conclusion wi th a short ou tl ook fo r future work is given .

This paper deals with improvements of a macroscopic model wh i ch was primarily designed to describe the dynami cs of traffic flow on homogeneous free ways as given in rural areas . The model was first proposed by Payne i n 1973 and later improved acco r ding to practical experiences and to further i nvestigations by several authors (Payne , 1973 and 1979 , Cremer 1976 and 1979) . It has been shown by a thor ough va l idation using several sets of data fr om a Ge r man Autobahn (Cremer and Papageorgiou , 1981) that this model reproduces real traffi c phenomena including trans i t i ons from free flow to congest i on as well as from congesti on back to free

BASIC TRAFFIC FLOW MODEL In th i s section the basic dynami c model is pr esented and its shor t comi ngs a r e disc ussed when applyi ng it to urban f r eeways .

239

240

1\1. Cremer an d A. D. 1\la)'

Consider the section of an urban freeway shown in figure 1. The section may have on- and off- ramps and may contain a change of number of lanes .

Taking into account these three influences , an empirical equation can be set up for the transition of the mean speed within a subsection v . (k+1) J

vj(k) +

f [ V(C j ) -

v j ] (k)

+ ;j . [ v j (V j _ 1- v j ) ] (k)

Fig . 1.

(2)

Freeway section

The section is now considered to be formally subdivided in subsections each with a length 6j between 300m and 1000m . (The subsections should be chosen in such a way that ramps or lane drops are located at the boundaries of the subsections . ) With respect to this configuration the following discrete time macroscopic variables are introduced: c . (k) J

traffic density within subsection j at time k . T (veh/km) (the definition of density here does not involve the number of lanes lj)'

v. (k) J

mean speed of the vehicles within subsection j at time k·T (km/h)

q. (k)

traffic volume leaving subsection j and entering subsection j+1 within time interval k · T ~ t < (k +1) · T (veh/h)

J

r. (k) J

on- ramp volume entering subsection j within time - interval k · T < t < (k+1) · T (veh/h)

s. (k) J

off- ramp volume leaving subsection j within time - interval k·T < t < (k+1) . T (veh/h) -

Here T is the temporal step width which has to be chosen such that the ratio 6j/T always exceeds the maximum value of speed . Here the choice of T = 5 sec would be reasonable . k is the running integer time index . To proceed , we formulate equations which describe how the defined variables are interacting and how transition from time instant k·T to time instant (k+ 1)·T is effected. For the density such a relation is easily obtained by a simple balance of entering and leaving vehi cles within a subsection: Cj (k+1)

cj (k ) + ;j [ qj _1(k) - qj (k)

+( rj(k) - Sj(k) ) ]

( 1)

Following the ideas of Payne (1971), the mean speed within a subsection is essentially deter mined by three phenomena: - dynamic adapt ion of the mean speed to the stationary speed- density characteristic V(c) accor ding to a time constant T representing driver ' s inertia (this time constant is not the driver ' s reaction time in an unforeseen situation which is much less), - convection of a speed gradient Vj _1 - Vj in downstream direction during time interval T, - driver ' s anticipation of a density gradient Cj +1 - Cj as seen in downstream direction .

Since the driver reacts to the density per lane rather than to the density as defined above , the density gradient term (last term on the right hand side) contains the factor lj/lj+1 which will be effective only in case the number of l anes lj in subsection j is different from the number lj+1 i n subsection j +1 . The constant K (density per lane) in the denominator effects that this term has less influence i n low density situations . The influence of this term is weighted by the sensitivity fac tor v . Due to its empirical character , this equation holds only approximately . For the stationary speed- density characteristi c a highly general expression has been chosen : (May and Keller , 1967) (3)

(It should be pointed out that the parameters a and b are related to the parameters 1 and m used in (May and Keller , 1967) by a = 1- 1 and b = 1- 1/m . ) In analogy to mass transport in hydromechanics , the volumes from one subsection into another are expressed as a product of density and speed . qj(k) = a , cj(k) . Vj(k) +

(l - a)

, c j + 1 (k) . v j +1 (k) (4 )

The weighting factor a (0.5 < a < 1. 0) reflects that the volumes are defined-topologically i n between two neighbouring subsections the states of which both may affect the actual f l ow . Inserting eqn . (3) into eqn . (2) and eqn. (4) into eqn . (1), the traffic state within the whole sec tion at each time instant k · T is represented by the set of density and mean speed values {cj(k), v .(k); j = 1, .. . . n}. These variables are therefore c~lled state variables in the sequel. The transition of the state within a section of n subsections is described by the 2n nonlinear recur sive equations ( 1) and (2) for j = 1 , ... n where the main stream flow qo(k) entering subsection 1 and the on- and off- ramp flows act as exte r nal inputs . As is evident by inspection of the modular structure of the model and its equations, the model can be used to simulate traffic flow on free ways of any length and also on freeway networks by linking together sections appropriately . A more detailed discussions of the model ' s equations can be found in (Cremer , 1979) or (Cremer and May , 1984) . The model has been verified and validated using several data sets from a section of a German Auto bahn . For this a parameter optimization rout i ne was used to identify the model parameter v f ' c , a , b , a, K, v and T in such a way that best co~ incidence between the model behaviour and real observations was obtained . The measurements were taken from a uniform two lane freeway section with a length of 2 . 7 km and with no ramps and no l ane drop bottlenecks . The results of the model verifi cation in this case were excellent and have shown

An Extended Traffic Flow Model that the calibrated phenomena including congestion and back ly . (See Cremer and details. )

model reproduces real traffic transition from free flow into to free flow highly accurate Papageorgiou (1981) for more

In view of the excellent performance of the model at homogeneous topological conditions it was surprising that the model gave unrealistic results when applied to sections with a lane drop bottleneck or to sections with high ramp volumes as may often be found at inner urban freeways . This mal function was also observed by Hauer and Hurdle (1979) using a similar , slightly less elaborated model and also by Babcock et al. (1982) . It was all the more surprising as the influence of a bottleneck seemed to be satisfactorily be represented in a change of the speed density characteristic V(c) and in the factor lj/lj+l of the density gradient term . To clarify the models shortcomings consider the following simulation example . A freeway section with 12 subsections and a lane drop behind subsec tion 5 according to figure 2 is first fed with a mainstream input volume qo = 3000 veh/h which is below the two lane capacity. Then the entrance volume is increased instantaneously to 3750 veh/h , a value which now exceeds the capacity of the bottleneck by 10 % (the V(c) - characteristic was that of a German Autobahn) .

241

has moved in front of the bottleneck , the velocity gradient seems to be rather low within the first two subsections of the bottleneck in contrast to real observations . By this way a nor mal velocity level is reached only two kilome ters behind the congestion at the entrance of the bottleneck . 4 . The volumes at the end of the two lane part are 10 % below capacity . Similar unrealistic congestion patterns were obtained when merging of higher on- ramp flows were simulated under high demand conditions. In (Babcock et al . , 1982) these malfunctions have been overcome by an adaptive scheme which repea tedly subdivides the subsections together with the temporal step width whenever the traffic state becomes critical and shows considerable spatial gra dients . As a disadvantage of this procedure the system ' s order (i . e. the number of state vari ables Cj' Vj) and by that way the computation time increases drastically in critical situations and may even reach the time required by microscopic models . It was the intention of the presented approach to look for extensions of the basic model by which the model ' s malfunctions at bottlenecks and higher ramp volumes would be eliminated . The extensions should be simple enough to make sure that the ad ditional computational requirements remain fairly low .

Bollleneck

b :: l.--l~::I:::: l-::--1 - --f---1-- --1---- 1----1- -- -1----1 Fig . 2 .

Freeway section with a bottleneck

The spatial profiles as predicted by the model are shown for the density Cj ' the mean speed Vj and the volumes qj for a time instant twenty mlnutes after the increase of entrance volume qo .

bottle neck / 200 100 -- .. .. C Vlkm / hl -·-' .. [veh / km] 1----..-;;.,

/v

100 50

q [veh/hl 4000

2000

\ __ L c 0 ~0--~--~-~~3--~--~5~-~6~m

Fig. 3 .

Basic model's response to mainstream traffic demand exceeding the bottleneck capacity

From this and other simulations the following points of unrealistic models behaviour were found : 1. Congestion starts about 1 to 2 kilometers beyond the lane drop and moves slowly in upstream direction afterwards . 2. In the beginning, there are over capacity flows within the bottleneck over a spatial range of nearly 3 kilometers. 3. After about thirty minutes, when the congestion

MODEL EXTENSIONS To direct the intended modifications to the right points of the equations, a thorough analysis of the model ' s mechanisms was performed first to identify the reasons for the unrealistic behaviour in the above mentioned situations . From this investigation the following observations were obtained . 1. At the beginning of a congestion due to over capacity flow at a bottleneck the convection term (second term on the right hand side of eqn. (2)) leads to unrealistic high speed values in the first subsection behind the lane drop which then are transferred into the fol lowing subsections by the same term . Unreal istic high speed values together with high density values result in over capacity flows . 2. Later when congestion has moved to the front of the bottleneck, the same term now pulls down the speed values in the bottleneck leading to flows considerably below capacity. 3 . The density gradient term which was calibrated for uniform geometrical conditions causes too strong an interaction between the last three lane subsection and the first two lane subsection at the entrance of the bottleneck. 4 . Merging of a high on- ramp flow impedes the mainstream with too high a retardation and spatial delay. In contrast to the situation of traffic flow on a uniform freeway section, a lane drop obviously re duces driver's inertia to adapt his velocity to a foreseen new situation . From these observations the following extensions were inferred to improve the model . 1. limitation of the volumes entering the bottleneck to avoid over capacity flow rates, i. e .

242

~1.

q.(k) < 1. 1 . Cap J

-

~ I ay

C rcme r a nd A. D.

(5 )

J+

where Cap is capacity per lane (e .g . 1700 2000 veh/h/lane) . bottlene c k 2 . modification of the density gradient term in f r ont of a bottleneck:

Q (ve h/ h J 4 00 0

200

c

l veh/km]

-..:~ -..:.

(6)

13 > 0

100

where 13 is a dimensionless new parameter.

2000

3. weakening of the convection term for the first subsection of the bottleneck

y.;

j

C

0

[ v.

J+ 1

(k) · (v.(k) - v. l(k)) ] J

..:..

3

2

6 km

5

(7)

J+

Fig . 4 .

where y < 1 is another dimensionless parameter which has to be calibrated . All these extensions were tested individually by simul ations to investigate the specific effect of each one with respect to the above listed malfunc tions . It turned out that the models shortcomings could be removed not by a single extension alone but only by a combination of all three extensions . The i mprovement when i nsert i ng these modifications into the model equations can be seen clearly from f i gure 4 which shows the behaviour of the extended model fo r the simulation example of figure 3 . It can be seen from comparison of both figures that the extended model shows a fairly realistic behaviour now though the parameters have not yet been calibrated .

Extended model ' s response to mainstream traffic demand exceeding bottleneck capacity

- c. (k) J

v ·T c.(k) + 1. . K

/',. ·T

J

J

J

(8)

Similarly , the relieving effect of higher off- ramp volumes is taken into account by replacing the term 6·T//', .· r. 1(k) by - 6 . T//',j . S.(k) . J

J+

J

In this case it has turned out that only one modi fication was sufficient to obtain realistic behaviour of the model .

As it was mentioned already above the basic model did not reproduce congestions caused by higher onramp flows correctly . In this case too congestion is modelled as starting too late and at the wrong place about half a kilometer downstream from the mergi ng point . Here it appeared to be the most reasonable and natural way to reflect the impact of higher entrance volumes in the driver ' s antici pation represented by the density gradient term , since the driver reacts by decelerating and lane changes already when he approaches the merging point . Therefore the following simple modification of the denSity gradient term was introduced for the subsection in front of an on- ramp :

Until now the investigations of the effect of the individual modifications of the models equations were only done by qualitative studies where the new introduced parameters were not calibrated. It will be the subject of the next section to cal i brate the add i tionally introduced parameters 13 , y and 6 together with the parameter set of the basic model (the extensions may lead to new parameter values of the basic model too) using real data . MODEL

VERIFICATIO~

To calibrate the parameters of the extended model

Sub sec ti on tengt h (m e te r) 451

J 0

41 9

tt

81

~

Oe tec to r No Fig. 5 .

475

378

3 78

475

I

258

547

5 47

3 70

370

5 15

kJ J ty

411

411

322

~ 4

Section of Santa Monica Freeway with subsections and detectors

TABLE 1 List of Calibrated Model Parameters speed - density characteristic V

f

93 . 1 km/h

c

m

110 . veh/km /lane

a 1. 86

basic model b 4 .05

0:

0.95

K

9 .5 veh/km Ilane

V

23 . 9 km 2 /h

extensior.s T

20 . 4 sec.

f3 0 . 79

y

0 . 70

(\

1.35

5

243

An Extend ed Tra ffi c Fl u,,· :\I od cl and to investigate how the model ' s predictions match real observations , data from a section of eastbound Santa Monica Freeway in Los Angeles were used . The section has a total length of 6 . 33 kilometers (3 . 93 miles) which was divided into 15 subsections according to figure 5 . Data were col lected over a three hours period from 2 p .m. until 5 p .m. from six mainline detectors and from de tectors on all on- and off- ramps. The data contained volume measurements q . which were taken at 1 minute intervals from main~tream detectors and at five - minute (partly fifteen - minute) intervals from ramp detectors and occupancy measurements from mainstream detectors only . With respect to the sampling time of T = 5 sec of the model , the five - minute or even fifteen - minute data from the ramps may not contain the full information about the dynamic impact of the entering or leaving vol umes because they represent only accumulated and by that way smoothed measurements. The occupancy measurements from the mainstream de tectors were converted into a time sequence of mean speed values wri(k) which could be compared with corresponding predictions wmi(k) of the model .

by the model extensions at this bottleneck , the diagram of mean speed as predicted by the unextended mode l is included . While the extended model matches the real observations rather well, it is seen from the figure that the unextended model is not able to reproduce a half hour ' s congestion at this site .

W"

[km/hi 80

60

40 20

14 00

Fig . 6 . For the purpose of parameter calibration and model verification the following procedure was chosen . Traffic flow on the freeway section was considered to be a dynamic , causal process on which certain variables act as inputs . The process then reacts to these inputs by corresponding transi tions of its state variables . The system ' s re sponse to the inputs is observed by measurements which are taken as reference outputs or reac tions and which are either state variables themselves or are related to state variables by simple equations (e . g . measured volumes qj are related to state variables Cj ' Vj by eqn . (4)) . To be more specific , here the mainstream volume qo(k) entering the section at the upstream end and all entering and leaving ramp volumes were taken as input variables while all other mainstream measurements (detectors No . 1 to 5) of volume and speed were treated as output variables . The problem of parameter calibration now can be formulated more precisely : Given the traffic flow mpdel with eqn . ' s (1) to (4) together with extensions (5) to (8) ; find the set of model parameters which minimize the performance index

+ )J(w .(k) - w . (k))2 ! rl ml

(9)

where qri(k) , wri(k) are real measurements of reference outputs and qmi(k) , wmi(k) are the corresponding mode l outputs when the model is excited by the measured real input sequences . The weighting factors A and )J were chosen to be 10- 6 and 10- 2 , respectively, to balance the volume errors against the speed errors . This problem was solved by a suitable parameter optimization routine in an iterative manner where for each choice of a new parameter set a model simulat i on run had to be carried out to compute the actual value of the performance index PI. (For more details see Cremer and May (1985)) . Convergence was achieved with the set of optimal model parameters given in table 1. In figure 6 diagrams of the time sequences of measured and simulated mean speed at detector are shown . To underscore the improvement achieved

1500

1600

17 00

The diagram of mean speed at detector 1 real measurements extended model basic model without extensions

CONCLUSIONS An extended macroscopic model for traffi c flow on urban freeways with changing numbers of lanes and with many on- and off- ramps is presented . The work was motivated by the experience that a bas i c ver sion of this model which was developed for traffic flow on homogeneous cross- country freeways was not able to cope with traffic flow phenomena on non uniform urban freeways . Particularly , unrea listic traffic simulations were obtained from the former model at bottlenecks as given by a reduction of lanes or at on- ramps with higher entering flow rates . To derive simple but efficient extensions first an analysis was performed to find out what terms and what mechanisms in the basic model equations cause unrealistic behaviour . From this investigation conclusions were drawn how the model shoul d be modified by extensions to achieve correct simulation results . It was found that three mod i fica tions had to be included to accomplish this : limi tation of volumes entering a bottleneck to capacity , weakening the convection term at a change i n number of lanes and modification of the density gradient term (driver ' s anticipation term) when approaching a bottleneck or a merging area . It has been shown by a verification procedure ap plied to the extended model that the important problem of modelling bottleneck phenomena real istically has been solved by the introduced extensions . This makes it a promi sing tool for a wide range of applications like system analysis , traffic prediction , design of control laws and others . Some important features of the developed model de serve to be emphasized . First , the mode l uses a fixed temporal step width and fixed spat i al intervals no matter what the traffic state is. In that way the system order and the computational re quirements are fixed right from the beginning which might be an important advantage over models with state dependent adaptation of step width in real time applications. Moreover , model parameters relate to a longer freeway sect i on as long as the

244

M. Cremer and A. D. May

sUb- section roadway and geometrical or environmental conditions are fairly uniform . In this case the speed- density relationship and hence the capacity is only calibrated for the entire section and does not require calibration for each sub- sec tion. This greatly simplifies the calibration and application of the model. The model , however , will permit the user to select different speed - density relationships for different sub- sections when the section of freeway is not uniform. It has to be the subject of future work to test this extended model with more data sets from different traffic situations and from different sections of inner urban freeways. From these tests conclusions have to be drawn as to what extent the model can be applied to other freeways and which parameters have to be calibrated in other cases of applications . For a careful model validation procedure which includes a more general analysis on sensitivity and accuracy it would be highly desirable as pointed out above to have more accurate measurement data which are taken in ten seconds or at least half minute intervals to make them adequate to the model ' s dynamics. REFERENCES Babcock , P .S ., A.D. May , D.M. Auslander and M. Tomizuka (1982) . Freeway Simulation and Control . Research Report UCB- ITS- RR - 82 , 13. Inst . Transp. Studies , Un i v. of California . Cremer , M. (1976) . A New Scheme for Traffic Fl ow Estimation and Control. Proc . of the 3rd IFAC/IFIP/IFORS- Symp . on Control in Transp . Systems, Columbus , Ohio , pp. 29 - 37 .

Cremer , M. (1979). Der VerkehrsfluB auf Schnell straBen . Springer Verlag, Berlin, Heidelberg , New York. Cremer , M. and M. Papageorgiou (1981). Parameter Identification for a Traffic Flow Model . Automatica , Vol. 17 , No. 6. Cremer, M. and A.D . May (1985) . An Extended Traffic Model for Freeway Control . Research Report UCB- ITS- RR - 85- 7 . Inst . Transp. Studies , Univ . of California . Gibson, D. R.P. (1981). Available Computer Models for Traffic Operations AnalYS i s . National Academy of Sciences, Spec. Rep . 194, Washington D.C . Hauer , E. and V.F. Hurdle (1979). Discussion of FREFLO . Transp . Res. Rec. 722, pp. 75 - 76 . May, A.D . and H. Keller (1967). Non - Integer CarFollowing Models. Highway Res. Record 199, pp . 19 - 32 . May , A.D. (1981) . Models for Freeway Corridor Analysis . National Academy of Sciences , Spec . Rep . 194, Washington D. C. Payne, H.J . \ 1971) . Models of Freeway Traffic and Control. Simulation Council Proc., Vol . 1, pp . 51 - 61. Payne , H. J. (1973) . Freeway Traffic Control and Surveillance Model. Transp . Eng . Journal , Vol. TE4 , pp . 767 - 783 . Payne , H.J . (1979) . FREFLO : A Macroscopic Simu l a tion Model of Freeway Traffic. TRB Transp. Res. Record , No . 722 .