ML20149E714

Applicant Exhibit A-20,consisting of Undated Article, Further Evaluation of Single & Two-Regime Traffic Flow Models

ML20149E714
Person / Time
Site:	Seabrook
Issue date:	11/05/1987
From:	Ceder A, May A CALIFORNIA, UNIV. OF, BERKELEY, CA
To:
References
OL-A-020, OL-A-20, NUDOCS 8802110247
Download: ML20149E714 (15)

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,I FURTHER, EVALUATION OF SIN,GLE A.,ND TWO-RElld e

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' j ; jifN ! u; L Avishal Chderf and %dolf D. May, Institute of Transportation an pN ll' ) !l il

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University of California, Berkeley

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Because many road facilities operate under high-density conditions, it is important to consider more accurate interrelationships among the basic traffic flow variables. Previous papers by May and Keller concerned with

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the evaluation of traffic flow models have examined the macroscopic rela-tionships derived from the generalized car-following model designed by Gazis, lierman, and Rothery. Their results form the basis for considera-j tion of other data sets that could be subjected to similar evaluation pro-cedures. This paper presents an investigation of single-regime traffic flow models in which 32 sets of speed concentration measurements were used.

Those 32 data sets are also used to investigate two-regime t raffic flow models.

Then 13 newsets of data are evaluated based on predictions fromthe investi-gatlons of the single-and two-regime models. Procedures developed by May and Keller are used as a guide to investigate single-regime traffic flow models g

in an m, t matrix format in order to study the variability of those ex-i, 4

ponents of the sensitivity component that belong to the generalized car-following equation. The deficiencies of the various models are identified, and the need to investigate two-regime models is stressed. Two-regime g

traffic flow models are investigatedin an m,1 matrix format that is derived g

from the generalized car-following equation. Both the single-and two-a regime models show consistency in the m,1 matrix,which makes it pos-sible to predict the results of a new data set. The results of the additional 13 sets of data confirm the predictions. The overall analysis of the 45 data sets emphasizes the most appropriate m,1 values for the single-and two-regime approaches, particularly those concerned with traffic flow models for freeway lanes.

• Tile NEED to consider more accurate interrelationships among the basic traffic w

flow variables has become imperative as the number of road facilities operating at near-capacity has increased. Development of flow control and ramp-metering tech-j niques and design of new roadways must be based on the relationships among speed, flow, and concentration, particularly under high-concentration conditions.

j In recent years a number of steady-state flow equations for the interrelationships g

among traffic flow variables have been suggested.

Previous papers (1,2) show that the microscopic and macroscopic theories of traffic flow can be reduced to the equation of the general car-following model formulated by Gazis, !!erman, and Rothery (3):

"'(*

(1)

E,.i(t + T) = a [X,(t) - X,.i(t)), [X,(t) - X,.i(t))

• When the research was performed, Mr. Ceder was on leave from the Road Safety Center, Technion, Haifa. Israel Publication of this paper sponsored by Committee on Traffic Flow Theory and Characteristics.

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where the single and double dots represent speed and acceleration (deceleration) and t

X., X,.i = positions of the leading car and the following car, respectively, T = time lag of response to stimulus, and m,t,and c = constant parameters.

The steady-state flow formulation of this equation can be obtained by integrating the above equation; it is given by Gazis et al. as f.(u) = cf,(s) + c '

(2) 1 where

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u = steady-state speed of the traffic stream, s = constant average spacing, and c' and c = some appropriate constants consistent with physical restrictions.

By selecting proper combinations of the exponents m and iin equations 1 and 2, known microscopic and macroscopic traffic flow models can be obtained.

In previous papers (1_,2), an evaluation process was used to determine appropriate values of m and I; it was applied to two sets of typical data-namely, freeway and tunnel data.

Evaluations of the m and f coordinates in a matrix format for the single-regime models were rather surprising inasmuch as the selected m and I coordinates for the freeway data were quite similar to those found in the tunnel data Q). !!owever, the two-regime models indicate differences between the selected freeway and tunnel models in the free-flow regin.e, although identical results were found in the congested regime (2_).

These results form the basis for consideration of other data sets that can be evaluated with similar procedures.

This paper presents an investigation of single-and two-regime traffic flow models based on equations 1 and 2 and an evaluation of new sets of data based on the predic-tions made.

First, flow relationship equations are determined for the single-regime models and for parameters such as free-flow speed, optimum speed, optimum concentration, maxi-mum flow, and jam concentration for each set of data and for each m, Icombination.

The results are summarized in a two-dimensional matrix.

Second, two regime traffic flow models concerned with free-flow and congested-flow regimes are investigated by using an evaluation process similar to that used for single-regime models.

After the characteristics of the single-and two-regime traffic flow models are identified, new sets of data are evaluated by using the same evaluation procedure used for the single-and two-regime models.

DATA SELECTION Before we proceed into the three major parts of this work, a brief description of the actual traffic data is given.

To ensure appropriate speed-concentration relationships requires that traffic flow variables be sampled over the range of all possible concentrations. The two groups of data sets evaluated in this paper are based on data collected during a fixed time period.

The first group of 32 data sets, based on speed-concentration measurements, was collected at the following locations:

1. Eisenhower Expressway, Chicago:

2. Ifolland Tunnel, New York; q

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3 teeleration) and

3. !!ollywood Freeway, Los Angeles (10 data sets, five locations for 2 days each);

4. Pasadena Freeway, Los Angeles (eight data sets, four locations for 2 days each);

pectively,

5. Penn-Lincoln Parkway, Pittsburgh (sLx data sets, three locations for 2 days each);

6. U.S. highway in Virginia (two data sets, one for median Lane and one for shoulder lane); and

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7. Atunich-Salzburg Autobahn, Germany (four data sets, one for median lane and one for shoulder lane and for both directions).

These 32 data sets were based on samples taken at 1-min time intervals, and mean (2) speeds and mean concentrations are calculated for each interval.

A procedure similar to that developed by Drake et al. (4) was used to systematically reduce data points on the 32 data sets. That is, the number of measurements falling in the most sparse 5-vehicle-per-mile concentration range was determined, and a like l

number of measurements were randomly sampled from each of the other 5-vehicle-l per-mile ranges. This statistical procedure provides uniform distribution of the data l

points over the available concentration range.

fictions.

The second group of 13 data sets is based on data collected on the 42-mile (68-km) i Los Angeles Freeway surveillance and control system. This second group of data sets i I and 2, known was collected on the Santa Monica Freeway at 11 stations (SM-12 to SM-22) along 5 miles (8 km) of a four-and five-lane directional freeway. In addition, two data sets e appropriate were obtaired from a collector-dutributor road and an on-ramp within the 5-mile free-eway and tunnel way section. Data were collected on the same day for all stations duing the morning I

peak and were based on 5-min roadway occupancy and volume measurt ments. Accord-e-Mgime ng to Athol (5), there is a linear relationship between occupancy and cancentration in nates for the which three tiines the occupancy can be associated with the concentration value. As owever, the two-will be seen in equation 3, the relationship between speed and concentration depends nel models in on normalized concentration, and therefore the exact linear transformation from oc-y ed regime (2).

cupancy to concentration is not of major importance. However, for consideration of t

n be evaluated absolute values of concentration and speed in the second group of data sets, the linear transformation should be taken into account.

II w models A systematic procedure for uniformity of data points over the concentration range the predic-was performed on the second group of data sets. This procedure was similar to that (e models and used with the first data sets, but, instead of reducing the number of data points, it in-g creased the number of observations by weighting them. In each 5-vehicle-per-mile ytration, maxi-range, the number of observations was increased up to the number of observations ombination' falling in the densest 5-vehicle-per-mile concentration range. In addition, in each range, equal consideration has been given to individual data points. This procedure ongested-flow makes it possible to have approximately 100 data points in each set as was used in the ed for single-first group of data sets. It is worth mentioning here that data collected during a fixed time period represent the traffic flow variables during that period. However, from the

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comparison of 1-min and 5-min senples, it appears that no difference in the magnitudes

$cedure used of the traffic flow characteristics between the two samples is evident. This latter point will be shown later. Thus, the traffic flow models can be evaluated (at least with the data in this paper) with 5-min samples as well as with 1-min samples.

l SINGLE-REGIME MODELS pion of the The objective is to select single-regime models for 32 speed-concentration data sets 2raffic flow that satisfy preselected statistical and traffic flow criteria. The evaluation procedure I

%o groups of was initially developed in earlier papers (1,2) and will be briefly summarized here.

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, time period.

In the evaluation procedure, an m, 2 matrix is used in which the various microscopic fs,was and macroscopic theories of traffic flow can be positioned. Each m and I combination represents a specific model that can be axpressed matherratically by equations I and 2.

The selected model is cne that satisfies preselected statistical and traffic flow criteria.

For the single-recime mo icl, only models with an x-intercept (jam concentration) and a y-intercept (free-flow speed) were considered. This limited the investigation of

,--__x_____

4 the m, f matrLx to the region where m < 1 and I > 1.

Further, it was required that in equation 1 the speed function and the spacing function of the sensitivity component re-(

main in the numerator and denominator respectively. This limited the investigation of 1

the m, I matrix to the region where m a 0 and i a 0.

The combination of these two re-l quirements restricted the investigation of the m, I matrix to the region where O s m I

s 1 and I > 1 An upper limit was placed on I such that I s 3.1 becau.se this limit on i I

covers all the previous macroscopic models, as will be shown later. For this range of m and i values, the following macroscopic equation can be derived from equation 1:

i u'" = u'"

1-(3) k

\\ s/

where u, u, = steady-state and free-flow speeds ar.d k, kj = concentration and jam concentration.

In addition, the constant a of equation 1 can be determined for the restricted m, f region as

=

5 (4) a=f[

x m and I as a Function of the Traffic Flow Characteristics From equations 1 and 2 we see that m and I are the basis for evaluating driver behavior at both the microscopic and macroscopic levels. When the above-mentioned require-ments for m and i values are considered, a dependency of m and 1 on traffic flow char-acteristics can be ottained. Such a dependency will include k,, u,, and optimum pars:n-eters u, k. of speed and concentration respectively.

Equation 3 has the following form at maximum flow:

=1-(5)

Rearrangement of equation 5 gives

.b.. t" In 1-m=1-k, (6)

The steady-state flow equation is q = u x k, where q is the flow, and for optimum conditions (maximum flow) dq/dk = 0.

When the optimum parameters are substituted in the optimum condition [after the first derivative with respect to k in the equation q = f(k)), the following equation is ottained:

L

O 5

0 required that in My component re-M..

1 - m (7)

=

\\ 3f A-m l

k lhe investigation of i

@n of these two re-t ton where O s m me this limit on i Substituting equation 6 into equation 7 gives For this range

] from equation 1:

(1 - 1)fn ei

+1

=1 (8) k/

-1p e-c (3) 3 In t

ks The nonlinear fluctuations of I can be estimated from equation 8 as a function of u,,

i k, u,, and k,, and thereafter the fluctuations of m can be determined from equation 6.

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Evaluation Procedure and Results For the single-regime model, four criteria were used to select the best model:

mean deviation, jam concentration, free-flow speed, and maximum flow. A model was ac-cepted if all of the following preselected criteria were met: (a) the mean deviation (4) within 10 percent of the minimum mean deviation: (b) the jam concentration between 185 and 250 vehicles per mile: (c) the free-flow speedwithinan 8-mph (13-km/h) accept-able range; and (d) the maximum flow within a 300-vehicle-per-hour acceptable range.

The acceptable ranges in free-flow speed and maximum flow were estimated from each data set and differed from one data set to another.

The results of this investigation of single-regime models using the 32 data sets are 3 driver behavior given in Table 1. The results are discussed for (a) models considering minimum mean

,loned require-deviation only, (b) models considering all criteria, and (c) models considering previously raffic flow char-identified macroscopic models.

optimum param-The models having the smallest mean deviation for each of the 32 data sets are given in Table 1. Almost all of these models lie along the m = 0.8 or 0.9 axis with I values between 1.6 and 3.0.

However, no models are acceptable when the traffic flow criteria are also considered. The most consistent undesirable characteristic of these minimum mean deviation models is the extremely large values for jam concentration (Figure 1).

The selected models considering all criteria are also given in Table 1. The models (5) selected for 24 of the 32 data sets meet all criteria. Seven of the selected models do not meet the maximum flow criterion, and two do not meet the free-flow speed criterion.

These selected models are shown on the m, I matrix in Figure 2.

The selected models generally follow a diagonal line extending from m = 0, A = 2 (Greenshields' model, 7) to m = 1, t = 3 (Drake, Schoefer, and May's model, 4.).

To emphasize the zone of the re-sults in the m, i matrix, an envelope line marking the area that contains all celected models is drawn (Figure 2). One interesting thing shown in Figure 2 is that the selected m, I combinations for freeway shoulder lanes and the tunnel lane tend to te located along the upper right edge of this envelope area; i.e., there is a tendency toward rela-(6) tively lower 1 and higher m values.

The Greenberg (6), Greenshields (7), Underwood (8_), and Drake et al. (4) macroscopic models are shown in the m, I matrix in Figure 2 in relation to the selected models.

None appears to be superior to the other macroscopic integer models. It should be noted that the Greenshields model (7) results in a linear speed-concentration relation-f optimum ship and usually exhibits the undesirable characteristic of an extremely low jam con-fatter the centration. The Greenberg model (6_) results in a concave-shaped speed-concentration tion is relationship and does not have a y-intercept (free-flow speed of infinity). The Under-wood model (8_) results in a concave-shaped speed-concentration relationship and l

usually exhibits the undesirable characteristic of an extremely high free-flow speed I

Table 1. Selected models for single regime.

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Figure 2. Location of selected smgle-reg;me models (32 c'ata setsL O

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h and does not have an x-intercept (jam ccncentration of infinity). The Drake et al. (4_)

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model results in a concave-shaped speed-concentration relationship in the low concen-tration range and a convex-shaped relationship in the high concentration range. It has the undesirable characteristic of not having anx-intercept (jam concentrationofinfinity).

2 l$ 'll 3 l.sg Consequently, the advaritage of the noninteger m, I models is to minimize or eliminate the undesirable features of the integer m, i rnodels.

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TWO-REGIME MODELS

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1. l 3 lg The initial work on single-regime models was extended to an investigation of two-i regime models to obtam improved representation of the data sets, particularly at near-l 3 2a se un

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capacity levels of flow. Edie (9) first proposed the two-regime approach, and the in-j

["j 2p spection of the 32 sets of speed-concentration measurements supported such an ap-m 54

a. o..

proach.

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The procedures used in the two-regime model evaluation were identical to those

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used in the single-regime model evaluation with two exceptions. For the congested-i flow regime, only data points with concentration values of more than 50 vehicles per a su sa m.

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mile were included, and the free-flow speed and maximum flow criteria were removed.

I For the free-flow regime, only data points with concentration values of less than 60

!?,8j y lyi vehicles per mile were included, and the jam concentration criterion was removed,

rn is s y 'ly y llg;ur The selection of 50 to 60 vehicles per mile as the possible discontinuity range between Ny l,y the congested-flow and free-flow regimes was based on the inspection of the speed-d concentration data sets.

i Concested-Flow Recime Xted single reg,me The two criteria used in selecting the congested-flow regime models were mean devia-tion and jam concentration. A model was acceped if its mean deviation was within 10 percent of the minimum mean deviation and if its jam concentration was between 185 15 s.o i

and 250 vehicles per mile. The best selected model has the smallest mean deviation of several models (m, I combinations) that meet the jam concentration criterion. The boundaries of the m, I combinations :nvestigated were 0 s m < 1 and 0 s i s 3.1. The boundaries are based on previous investigations to determine the proper range for m i

u and I.

The extended region (over the region of the single-regime models) in the m, I matrix for the congested-flow regime is 0 s m < 1 and 0 s I < 1.

This region has the undesirable characteristic of not having a y-intercept (free-flow speed of infinity),

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which is not of major importance for congested-flow models. Ilowever, this extended region requires a different macroscopic equation than equation 3, which can be deter-

"y mined from equation 2 as j,

l*

u' ' ' = a 1*

k"'-k (9)

+T

- a h a 4

[

As mentioned earlier, only data points with concentration values of more than 50 ve-i hicles per mile were included in this analysis.

[^

.y GW c el The results of this investigation of the congested-flow regime using the 32 data sets are given in Table 2. These results are discussed for (a) models considering minimum

'0 mean deviation only, (b) models considering all criteria, and (c) initial (m = 1 = 0) and extended (m = 0, L = 1) car-following models.

The models having the smallest mean deviation for each of the 32 data sets are given in Table 2.

Almost all of these models lie either along the m - O axis with i values between 0 and 1 or along the t = 0 axis with m values between 0 and 1.

Eight of the models are represented by m = 0, t = 0.9, which is very close to the extended car-following model or Greenberg's model (6_) (m = 0, t = 1). liowever, only eight of the

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Table 2. Selected models for two-regime measureme:;ts (congested flow).

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6. u.

' ??

31 09 6 03 207 6 10 434*

6 04 180*

23 53 So lo 100 01 00 44 A*

J2 03 4 47 201 4 47 1 09*

4 49 380' 34 44 50 to 104 04 00 4 01 t n*

05 01 4 03 2M 4 02

$27' 4 06 ISS*

25 53 So lo 116 04 00 3t' 42*

0.- 00 3 49 227 3 39 327 3 96 Sif as 33 So to 112 00 09 2 Se 14-02 01 2 79 234 3 59 32f 2 49 229 27 59 M to 113 01 0 ', 2!

+=

01

0. 9 2 SS 239 3 S3 4.2 tf 2 SS 193 28 SG So to 105 02 03 3 S4

-4*

OO O2 3 $4 tot 3.54 34P 3 $9 167 29 6

$0 to 44 09 00 2 44 d'

31 00 3 09 234 3 10 179*

3 21 121*

30 8

60 to 60 00 09

3. 4 ~

• 4*

JO at 3 41 las 3.45 498*

3.45 lef 31 4

So to 67 00 09 15-

• f' 31 09 1 12 233 1 S4 921*

1.53 163*

32 17

• 1. 0 to 67 C0 00 7 C.

T v.7 02 7.04 197 7 03 158' 7.04 276

• >es r.on meee uwee Figure 3. Characterestics of congelted-flow 3ime models.

Figure 4. Location of selected two-regime models (32 data sets).

6 I'

'i

'3' i -

WTAca

~

-Selected models

~

trouceiwo d

-min deviofion reoc 0

0 e

m: 6 8

d

[e e

T a

u 2

l(z ?( no.t.D.s.Ce a

~

e goue

~

(G Rt (= B( DG k

~

t g,

f.O 4oc -

,t l

l1

!T o

e i

'e E

fl \\

\\\\

l h

N M

~'s

,a, f

I i

I,

• ~

fu%%EL SHotu( A u= E 200 IG"II'S'

( *'Ecl

\\

i, c

5, l 'y N_

o

_ 2.0

)

"I o

o

"! i t :

i.

t: F

-o 0

80 0 iiiI i1

!1 i o

5 to 15 20 25 30 2.5 2^

2.5 CATA SET b

s.c g D

- a

> ^

w..,,,o 0

45

,,o 1

m-s~ cogsted-fice reges o-free flee regee I

..AW At}

O l

9

}

f models are acceptable when the jam concentration and the mean deviation criterla are L

considered.

Nm N' ' * '

The selected models considering all criteria are also given in Table 2.

The models ki selected for all 32 data sets meet both the mean deviation and jam concentration cri-Firure 3 shows the two characteristics of the congested-flow models of both the

!N selected a?d minimum deviations models. This figure and Table 2 indicate that neither

)

I teria.

R 8 8t' m and J values nor the mean deviation is sensitive to changes in the jam concentration 7 El This effect can be anticipated from equations 8 and 6 for changes in m and I.

values.

I

'.3 On the other hand, the lack of data points under extremely high concentration conditions

' av may explain the nonsensitivity property of the mean deviation with respect to the jam i

ll$

concentration.

The selected models are shown on the m,1 matrix in Figure 4. Almost all of these

' son@

models lie along the m = 0 axis with i values between 0 and 1. This is the region of y

the m,1 matrix that lies between the initial (m = 0, L = 0) and the extended (m = 0, E

l 1 = 1) car-following models. These two models as they relate to the data set results 8o4 8

are discussed below.

y The mean deviation and jam concentration for the initial and the extended car-arr i

l following models for each of the 32 data sets are also given in Table 2.

Although the resulting mean deviations are all within 10 percent of the minimum mean deviation, the 8$ '

l models are generally not acceptable because the jam concentrations lie outside the O!.

specified range. Although the initial car-following model generally has smaller mean g

j deviations, the extended car-following model fulfills the jam concentration criteria in These results give significant support to the earlier work on car-most cases.

sw f

following theory (3, 9).

Free-Flow Recime t**rme modus The three criteria used in s*lecting the free-flow regime models were mean deviation, free-flow speed, and maximam flow. A model was accepted if the mean deviation was within 10 percent of the minimum mean deviation, if the free-flow speed was within an 8-mph (13-km/h) acceptable range, and if the maximum flow was within a 300-vehicle-

"5','

- O per-hour acceptable rance. The acceptable ranges in free-flow speed and maximum i.e The flow were estimated from each data set and differed f rom one data set to another.

f

_~

As boundaries of the m, I combinations investigated were O s m < 1 and 0 < t < 3.1.

mentioned earlie*, only data points with concentration values of less than 60 vehicles as per mile were included in this analysis.

The results of this investigation of the free-flow regime using the 32 data sets are given in Table 3. These results are discussed for (a) models considering minimum

-Lo mean deviation only, (b) models considering all criteria, and (c) models considering other previously identified macroscopic models.

Although half of the models having the minimum mean deviation for each of the 32 data sets lie in the vicinity of m = 0 and i = 3, the remaining models are scattered in ts

%m% :

the matrix from m = 0 to m = 0.9 and from i = 1.1 to i = 3.1.

However,15 of the models b' h. w are acceptable whet.the minimum mean deviation and the free-flow speed and ;naximum flow criteria are considered.

M ~

The selected models considering all criteria are also given in Table 3.

By selecting g

models that slightly increase the minimum mean deviation, the free-flow speed criterion I

is fulfilled for all selected models and 23 models fulfill the maximum flow criteria.

t3 These selected models are graphically represented on the m,1 matrix shown in Fig-These models lie either along the m = 0 axis with 1 values between 1.7 and 3.1 ure 4.

'^

an.

,y or along the m = 0.8 axis with I values between 1.9 and 3.1 An interesting point about t.o Figure 4 is that the m and 1 free-flow regime models associated with measurements

~

taken in tunnel and shoulder lanes are somewhat scattered away from most of the free-

% =,..

On the other hand, that is not the case in the congested flow way m, I combinations.

models in which the m, f combinations of tunnel and shoulder lanes are among the other freeway m, I combinations. The free-flow regime model characteristics are shown in Figure 5 for both the selected and minimum deviation models, in addition, Figure 5 A.

~~

1 i

sm O

l Tabte 3. Sdected models for y_,,

o,. u i 3,w u i r

Data k

tWO4egiff%f rfteasuremefits E

toc ahosi Point s Rage m

MD se q.

m 4

MD u,

q.

(freef flow).

M i

57 14 to H fG 31 4 26 12 1.792*

0. 0 27 4 43 54

1. r2 M

2 e6 6 to 60

0. 0 14 3.01 47 1.324 0.0 ta 3 01 47 1.324 3

31 1S t3 23 00 11 4.70 48 1.072*

0.0 25 6.75 51 1.te4' 4

38 15 to M 00 3.1 4 11 53 1.752 00 3.1 4 Il 53 1.7 52 5

24 22 to t0 0.0 3.1 6.67 44 1.MT

0. 0 25 6 73 $1 1.919' t

t 42 13to 4 00 31 3 44 48 2.297

0. ft 3.1 3 44 48 2 297 e

e 1

17 le to 60

0. 0 3.1 3.23 47 1.575' O. 0 2.9 3 34 40 1,5tpf l

6 24 29 to FO O6 11 3.14 232' 9.637 00 31 3.30 48 2.951 9

9 19 18 to 84

0. 4 1.1 3 16 321*

2.0e4' 08 23 3 30 50 t $7f l l 30 40 9 t s d'O 30 20 5 04 45

2. 37 f 00 20 $M 45 2.373' e

11 15 12 to ta 0a 27 3 43 46 1.933 03 2.7 3 63 44 1 919 12 37 18 to #4 00 3i S.46 43 1 844 00 34 5 46 49 t.e44 1

13 19 15 t > M

9. 7 30 1 42 49 2.242' 04 30 1.42 4#

2.244 1

14 50 14 t. 50 06 31 1.60 52 2.188' O7 30 1 60 52 2.217 1$

21 I4 to 63 08 31 2 64 43

1. e 30 08 33 2 e4 43

1. a3J 14 22 14 ts M 04 24 1 54 44 2.194 08 24 1 54 4a 2.194 1

15 22 t s E3 00 31 2 29 en

1. f 4 4' O.2 22 2.44 55 1349' l

is 10 131.> M 00 13 3 12 72' 3.122"

0. 0 19 31C 50 2.134

'I!

19 19 18to*O OO 31 2 00 41 t. 9 61 '

00 27 2 04 41 1,7M 23 31 t$ M N

0. t 24 1.75 4%

1.912 01 24 I 78 45 1.212 ji 21 3s 7 es M 0.0 11 4 61 43

1. 52 C 00 31 4 El 43 2.28" 22 34 7 13 *)

00 t.4 8.10 4.330' 04 3.1 6 25 47

2. 2 t ?

I 23 31 Il to td 00 30 5 98 10 1.702 00 3.0

$ 98 M

1 '.32 24 32 12 ts (4 00 31 3 94 53

1. e* 2 00 31 3 94 13

1. ** 2 25 30 f t to M 00 31 3 22 19

1. %l 00 31 3 22 39 1.501 24 26 23 to M 00 11 2 34 33 1.537 02 23 2 49 42 1Al q

27 62 11 to M 0a 30 3 34 St 1.817 08 29 1 14 60 1.819

!s 71 It tn 80 02 31 S.44

• 1

2. ! ! S' 00 3.1 5 45 70 2.132 29 91 3 to 61 00 27 5 47 41 1.400 08 17 % 47 81 1.44 30 97 1 to 60 01 25 6 80 77 1.446 01 25 6 a0 77 1.t 44 31 89 1 ts M 09 19 4 71 64 1.617 09 19 4 15 46 tAl?

32 114 t to 60 00 1.7

$.74 76 1.823 0.0 1.-

5.74 to 1.623

• tame ec==e= cewea Figure 5. Characteristics of

i i

i

i:

8

-sexted models free-flow regime models.

-- - -- m m devotion models 2

6 a

6 4

o 3

v 2

O T T 70 1 I

1 2

e

,o.

c i

a 6C e

E l '.

So r f,

5 3

40 A

g 31 (i

'TT l !.

i 85*-

c p2OOOs j,'

5

[1500 h j

I I-

^'

=

100C

=

0 5

10 15 20 25 30 CATA SET I

Table 4. Selected models for single regime (13 data sets).

htama Dmanon M *!

Sciected W xiel Dat a k

P an-e hwe4r Poess Pane m

4 MD k,

e, g.

m t

MD k.

s.

4.

$M t!

P4 3 to 132 09 27 3 Il 317" SS 2.032 07 24 3 76 23'

$9 2.099 su-13 e*

6 to 93 08 27 3 $7 226 et 2.0v1 01 27 3 ST 224 41 2.091 EW 14 107 3 ts !!S 09 26 3 $4 195'

$7 2,104 07 2S 3 el 127 ST 2.141 S W.15 33 6 to 114 00 26 2 60 3( 2' e0 2.013 08 26 2.48 245 60 2.037 e'.9*

119 S t$ t21 at 74 3 44 4t t' 83

) 988*

07 23 3 57 ft) 63 1.924

!M 17 9?

6 t o it6 99 23 2 se 4 H' et

1. e #4

0. 7 23 3 21 241 42 2,cs4

$U*tt 12 3131's 09 25 26#

3 o*

17

1. e4 6*

0.7 26 3 is 194 St 1.916

• M 19 93 3 to IM 39 27 3 03 329' 81 2.02 0e 24 3 04 23e at 2 070 SW 23 94 4 to r4 09 29 3 34 267' 56 2.043 07 27 3 40 189 t4 2.N9 su-ft 94 313 104 09 23 23?

W

• 2 1.993 06 2l 2.44 223

!4 1.979 5422 87 3 ts 78 09 25 2 93 If 2*

41 1.53 1 00 25 2 94 23$

61 1,I34*

Lorca %.raep i 10e 9 t o 144 0t t6 2 01 294*

il 2.153 00 16 2 02 241

$2 2.152 g emce % D.m.

91 3 to les 09 17 3 20

1. E s? ??

1.303 09 ie 3 24 1.313' SI 1.333 too w m es c+en

• f-

s.w um 11

}

_= a wo,,,,

N EIN shows that the acceptable values of the parameters u, and q, can be obtained by only "3i in U [U[

slightly increasing the mean deviation.

O Il '!]l U jjjf The Greenberg (6,), Greenshields (7), Underwood (8), and Drake et al. (4) models are shown on the m, i matrix in Figure 4 in relation to the selected models. None of these jj 'l 3 y " lOav macroscopic models appears to be appropriate for the various data sets. There is no 0 lUj.

ilj '

83 s ao justification for expecting the free-flow regime data sets to be represented by micro-I U " 2f.,r scopic (car-following) theories. Ilowever, it is interesting to note that the selected s

j] ll l0 ll [NI free-flow models have the characteristic of a large i value and a small m value. This

• • u 04 j'

se i-causes the sensitivity component of the car-following equation to be numerically small, li l" ll j:gl which would be expected in situatbns where vehicles are not in a car-following mode.

(

i ea" ' " s 4r

'n n us:

,, 31 4 61 el 2 *2r EXTENDED DATA ll N lU U ltj

8 47 uer "N O l,l Em As has been mentioned earlier, the second group of data sets consists of 13 sets of data M 'l lg uljl '[is U$

11 of which were taken from freeway stations and two from on-ramp ano co!!ector-distributor road within the freeway section. This second group of data sets is based ll ! Uf," M"i Based on the the research on single-and two-regime models, an attempt was made 87 in n on 5-min time interval samples and is averaged across the total directional roadway.

os 3o u u*

n in to predict the results of m, I combinations for both the minimum mean deviation models and selected models. These predictions and their verifications are discussed for (a) models considering single-regime approach, (b) models considering congested-flow m

regime only, and (c) models considering free-flow regime only.

Sincle-Regime Models The single-regime model characteristics of the first group of data sets are shown in j

Figures 1,2, and 3 and given in Table 1.

When the m, I combinations of the selected models are co'tsidered, it appears that the m and i values of most of the data sets are

/

(

within the region of 0.5 s m < 1 and 2 s t s 3 and tend to fall within the envelope of re-sults shown in Figure 2 and extending from m = 0, f = 2 to m = 1, t = 3.

Furthe rmore, l

all the m, I combinations associated with models of non-inner freeway lanes are located along the upper right edge of the envelope area. Therefore, this envelope of results will be the basis for predicting the m, I combinations of other data sets for freeway lanes, The results of the investigation of single-regime models using the second data sets o

are given in Table 4.

In addition, the m, I combinations of these data sets are shown in Figure 6 for the selected models. It should be noted that the evaluation procedure and preselected criteria were used in the same way for both groups of data sets.

Consequently, from the new selected m, A combinations the above prediction is in-deed verified by the second group of data sets. This conclusion is shown in Figure 6 where the selected m, i cf the freeway models are within the predicted envelope area in the m, i matrix.

Congested-Flow Models The conge: ted-flow regime model characteristics for the first group of data sets are H

shown in Figures 3 and 4 and given in Table 2.

From Figure 4 and Table 2 it appears d' -

that the m and 2 values of most of the data sets are within the region of 0 $ m 5 0.$

[jll and 0 < 1 s 1 and the m values tend to approach zero. This observation is the basis

[jy for predicting m, I combinations for other freeway data sets.

The results of the investigation of congested-flow regime models using the second ur.

.[j" group of data sets are given in Table 5 In addition, the selected m, t combinations of these data sets were located in the a, i matrix shown in Figure 7 Comparison of go Figures 4 and 7 emphasizes the identical tendency of m to approach zero, but 1 of the 6n ll[

second data set has a slight tendency toward values greater than 1.0.

.333

O

=W.'

12 i@ t

..J.- M Figure 6. Location of telected lingle-regirne models Figure 7. Location of telected two<egime rnodels p,

T ' tj (13 data sets).

(13 data set 3).

[GAE E mat %

O

• Tiat (An-e3 mto rottewims n',.

, i 0

tc g

,0S to

t. ':

o

..,3

-e e

m g

O

~

~

0.5 t_S LS e

O 0.5 turtNDf0Cami2 CD - eca0 2

F0L tC wi46, -

\\

Coi-m a W8 CD-Roa0 Get c hst mo I-8 LO 1.C-t Icat r =s-t tos:

u.a-xw I

2D a

,2D 8

gg 4

{

~

g3

_ F5EDCTNT t

N e

- EWELOFT o

08e # Atop a

M

)

l

~

$ (LI.%

^

b%;t e mol a

(cat t *S 2.5 6-4 2.5 2.o-i O

a.

o 8

-2.0 l

j 2.5 2.S

- so O -

!Ca a. E e+ a ]

i 2

g 1

0 0S ID E@

h"g

• 2 m ---

n a3 m

e-- eemswe.#ie. rega o-h e e f e n9 +=

Table 5. Selected rnodels for congested regime (13 data lets).

Mmewm Denerson M met SelectNI M

• net Dat a k

Salton Wester Pow e R ais e m

WD k,

m t

WD k.

SM It is 53 t s 131 00 14 0 61 1 90 00 14 0 91 190 BM 13 35 SS to 99 08 01 1.93 1:r 06 01 I 97 247

$W 14 47 50 to 135 0t

1. 0 t is 6tf 90 04 2 27 105 SM Il 44

!J h llo 08 0.3 I CI 367 90 of 1 04

!!8 EM 14 66 10 to 129 06 16 0 89 1.f >9*

O3 06 0 60 34e SM 17 43 50 to fit 03 01 0 el (95*

01 01 1 v4 27 $'

SW 18 it la te 10e 60 ie 1 31 llr

0. 4 II I 21 199

$W 19 46 50 to 105

0. 7 06 1.77 f.t r 00 03 1 41 til SW 20 le t o i t >9 00 03 2.76 157 00 03 3 76 233 SM fl 51 M t o 100 00 0t i 33 SJ 3" 01 0$ 1 33 tot SW 22 43 50 tz TO 00 09 l ie 15 # 04 16 1 Il 181 LaBren h-rampt M

M i s 144 00 14 1 91

! ?'f

0. 0 16 I 91 147 Yeelee LCD an#

53 53 t3 Ila 00 01 1 34 515' O4 00 1 39 238 o-.

Table 6. Selected rnodels for free flow regirne (13 data lets).

Wtaseem Denm Mwl se:ectet M asei Dar a k

p ar ton h ea tie r Pow s Rar se m

WD o.

e MD e,

S.

EW 13

' la 3 to H 09 18 3.44 64

2. 0*3 09 f.8 3 84 58

2. Oil

$W 13 el e to O OO 31 3.le M

1.007 00 31 3 es to IE1 SW 14 64 3 to 60 00 31 3 43 le 1.055 00 31 3 63 le 3.05 6 SM-ll 47 6 t3 60 00 31 3f4 57 2.031 00 31 3.66 51

2. C21 0

SW il 42 3 tom 00 31 3 25 83

1. s T9 0e 3.1 3 25 43

1. l? D f

SM li 68 8 to M 09 30 2 to el 1.090 09 30 2 60 41 1.090

$W-ll le 3 t s 60 00 31 3 74 16 1.731' 80 31 2 74 56 1.711*

S W-19 52 3 ts 4 00 31 3 97 to 3.033 00 31 3 97 60 a.C33 SW 20 60 8 is 80 00 31 3 77 54 1, t e4 0.0 30 2.77 54

1. D et sW 31 le 3 t s f4 00 24 2 64 el 1 r71*

0. 9 23 3 ft es

1. M4 SW 23 55 3 to 60 08 84 3 60 51 1.7tf 0t 3.1 3 si el

1. M 3 LaBros W rarrpi 43 9 to 60 00 13

3. I 8 ff 1.895' 0l 18 3 33 44 3.160 Vest-e ICD-an i 43 3 to M 0.9

1. 7 4 le ll 1.463 00 1.7 4 50 ll 1,453

• >= w esse men f

)

6 N%.;;

l

'm O

13

"* 'n dels Figure 8. Typical single and two4egime models.

i.

i ;i4 ; ;.

i>ia -

~

25oo sut ato.ut woctts

,og gi.a o.smet.,,. m.E jed# AGE-sMDtm LANE :

l

1. 13 g

Tyh,N(( (&g{

[

f

,O isoo j

\\)

I

~

E

-o.s

( co -noso :

soo : -

f f

7 o

to

\\

j tsoov

{

}

g,ano{g reatoiwt vocets i.s g_

[

e t

-J Of * *<XJOI

=

, s_

^

1 O

[

[

'0 icoe 2.s Soo j

i o

i c a 30

,00

,33

,00

,30 1>

,4g com.ct%taanos (vom) j5 b ei,,,,,,

> regee l

Free-Flow Recime Models The free-flow regime model aracteristics from the first group of data sets are shown in Figures 4 and 5 and given Table 3.

From Figure 4 and Table 3 it appears that the m and f values oi most of tb.itasets arewithinthe regionof 0 s m s 1 and 2.5 s 2 s 3.0 and tend to be centered ar, d m = 0,1 = 3.0. This tendency will be the basis for predicting m and 1 values :. cther freeway data sets.

The results of the invesucation of free-flow regime models using the second data sets are given in Table 6.

In addition, the selected m. t combinations are on the matrix shown in Fleure 7.

As can be seen from Figure 7, the above prediction is verified in which seven of 11 m, I combinations (of freeway lanes data) are centered around m = 0, L = 3.0.

To visualize the differences among the various models with respect to type of road facility, three groups of models were identified for nonshoulder freeway lanes, shoulder freeway lanes, and a tunnel lane. Thece average models are shown in Figure 8 for the flow-concentration relationship. The average m,1 values for the nonshoulder freeway lanes are m = 0.6, A = 2.4; m = 0.2, t = 0.5: and m = 0.2, L = 2.9 for the single regime, congested-flow regime, and free-flow regime respectively. The average m, f values for the shoulder lanes are m = 0.7, L = 2.2: m = 0.1, L = 0.6; and m = 0.8, L = 2.5 for the single, congested flow, and free-flow regimes respectively. The m, I combinations for the tunnel data are given in Tables 1,2, and 3 (data set 2).

The consideration of road facilities other than nonshoulder freeway lanes is focused on tunnel, shoulder lanes, on-ramp, and CD road data sets. In single-regime models, although it is possible to distinguish between m, 4 values for on-ramp and CD road data, l

In the it is unlikely that this distinction can be made for tunnel and shoulder lane data.

This is congested-flow models no distinction can be maae for the various data sets.

as expected because under high concentration conditions traffic behavior is similar on all types of road facilities. In the free-flow models, m, I combinations of tunnel, shoulder lane, on-ramp, and CD road are scattered away from most of the m, i freeway models. it is reasonable to assume that different driver behavior is reflected under toe concentration conditions on differ?nt types of road facilities (e.g., in a tunnel there are lower speeds and more cautious driving than on an open freeway lane).

I

O 14 CONCLUSIONS y

bd i 4

i O This paper has evaluated macroscopic and microscop!c models to determine which of I D[ 6.

cone them best represented observed sets of speed-concentration measurements. Single-that

}

4 and two-regime models of a free flow and congested flow were investigated. A total of 4

4 45 sets of measurements were analyzed; the results of the first 32 data sets were used b ?j to predict the results of the 13 remaining data sets.

ACI P

In regard to single-regime models the more significant findings were as follows:

a i

! The

1. The mean deviation of the selected models varied from 1.6 to 7.0 mph (2.6 to i Ang

')

11.3 km/h) with a mean value of 3.8 mph (6.1 km/h);

2. The traffic flow criteria for the selected models were satisfied in 35 of the 45 i

{

data sets; i

pI

3. All previously proposed m.1 intecer models had significant deficiencies in regard j

j to acceptable traffic flow parameter values and mean deviations

j 1.

4. The area of the m, i matrix in which the selected models are located is shown 1

i in Figures 2 and 6. and for inner freeway lanes the selected models tended toward m j

2.

and I of 0.6 and 2.4 respectively; and j

j

5. The major disadvantage of the single-regime approach was that the selected j

models did not represent the data sets at near-capacity conditions.

j 3.

l The most sienificant findincs with congested-flow two-regime models were that l

4.

1. The mean deviation of the selected models varies from 0.6 to 7.0 mph (1 to 11 km/ h) with a mean value of 2.9 mpn (4.7 km'h):

5.

2. The jam concentration criterion for the selected models was satisfied in 44 of the 45 data sets:

6.

s

3. Two previously proposed m, : integer models (m = 0, 4 = 1) were marginally 1

1' 7.

satisfactory but did not have the minimum mean deviations, and the jam concentration values were generally high:

4. The area of the m. I matrix in which the selected models are located is shown 8

In Figures 4 and 7. and the selected models tended toward m values approaching 0 and i values between 0 and 1: and 9

5. The two-regime approach did result in more models satisfying the jam concen-tration criterion but only a slight reduction in the mean deviation.

The most significant findings with the free-flow two-regime models are given below.

1. The mean deviation of the selected models varied from 1.4 to 6.8 mph (2.3 to 10.9 km'h) with a mean value of 3.7 mph (6.0 km/h).

2. The traffic flow parameter criteria for the selected models were satisfied in 35 of the 45 data sets.

3. The area of the m. I matrix in which the selected models are located is shown in Figures 4 and 7 The selected models are scattered over the lower portion of the m, i matrix: however, the largest cluster of selected models occurs at m = 0 and i = 3.

?

4. With the two-regime approach no more models satisited the maximum ihw criterion and there was no signtitcant reduction in the mean deviation.

In summary, 4

t'

1. Previously proposed macroscopic models did not accurately represent the speed-concentration data sets:

2. The use cf noninteger m, I macroscopic models for sir. ale-regime analysis pro-vided a significant improvement in accuracy and more realistic traffic parameter values but had the weakness of r.ct c.111 representing the data sets at near-capacity j

conditions:

3. The use of noninteger m, 2 macroscopic models combined with two regime analysts did support the visualappaarance of the two-regime phenomenon in the data sets h

N W

'%? W A

O O

15 but provided only slightly better re resentation of the data sets; and

4. For further improvement in electing macroscopic models to represent speed-1 to determine which conceaNn seu d masmmds a hM maM mW Ad M MW 3as urerhents' Sin I investigated. A total of 132 data sets were used j

A bgs were as follows:

~

The authors wish to express their appreciation to the Division of Ilighways at Los 3 to 7.0 mph (2'6 t

't

{

Angeles, District 7, for supplying the data sets.

isfied in 35 of the 45 I

1 at deficiencies in regard

^'

• ~

8'

• • I are located is h search '& cod M, N. po. M.

Dels tended toward m i

2. A. D. Af ay,.ir., and 11. E. A!. Keller. Evaluation of Single-and Two-Regime Traffic Y

3 that the selected Ka rls ruhe,1968.

a,

3. D. C. Gazis, R. Herman, and R. W. Rothery. Nonlinear Follow-the-Leader Alodels a e Row. Opuauons &sead, M 9,.W. 4, N. pp. WM.

' models were that

4. J. S. Drake. J. L. Schoefer and A. D. Alay, Jr. A Statistical Analysis of Speed y

. eses. Noc, N hnadonal 4mpmm on & Nory of hah to 7.0 mph (1 to 11 Flow, Elsevier. New York,1967 5, P. Athol. Interdependence of Certain Operational Characteristics Within a A!oving 2 satisfied in 44 of the Traffic Stream. Ilichway Research Record 72,196 5, pp. 58-87.

j

6. ii. Greenbere. An Analysis of Traffic Flow. Operations Research, Vol. 7, No. 4,

were marginally 19 59, pp. 4 93 -50 5.

e Jam concentration

7. B. D. Greenshields. A Study in Highway Capacity, lilU1 Proc., Vol. 14,1934, pp.

e located is sho'wn

8. R.

t nde rwood. Speed Volume and Density Relationships. Quality and Density s approaching 0 and of Traffic Flow. Yale Bureau of Traffic,1961, pp. 66-76,

9. L. C. Edie. Car-Following and Steady-State Theory for Non-Congested Traffic.

inR th' lam c neen-Operations Research, Vol. 9, No.1,1961, pp. 66-76.

f Wels are given below.

D 6.8 mph (2.3 to sere satisfied in 35

) located is shown in e portion of the m, 3 m = 0 and 4 3.

Examum flow criterion fepresent the speed-gime analysis pro-Ulc parameter

) near-capacity 1

h two-regime hon in the data sets I

~'

1

(