Mehdi DANECH-PAJOUH INRETS-GRETIA [email protected] TEC October 2001

Road traffic indicators as a performance guide I. Introduction Road traffic indicators provide a means of evaluating the performance of a road or a sub-network. They are computed from macroscopic or microscopic variables which are measured by sensors. They are similar to economic indicators such as the price index, the rate of inflation or the unemployment rate. Experts in both fields define their own indicators which may be of several different types. A traffic indicator may or may not have a physical meaning, the important thing is how it changes over time and in the course of a journey. It is important to distinguish between a qualitative or quantitative scale that is based on an indicator with the indicator itself. We shall now present a number of familiar traffic engineering indicators with their two computational variants and discuss successively: ¾ ¾ ¾ ¾

The fundamental aspects of measuring traffic variables. The indicators, the two computational variants and the difference between them. The representativeness of the times displayed on Variable Message Signs. The role of these indicators for operators.

II. The three macroscopic variables output by sensors In order to provide a general description of the vehicle flows on a section of road, sensors generally output the three main traffic variables, namely, flow (the number of vehicles counted in unit time), occupancy rate (from which an approximate assessment of the concentration can be obtained) and speed (distance travelled in unit time). Depending on the equipment installed on the road in question these variables are measured for each lane or each direction at regular intervals of, for example 20 seconds, 1 minute or 6 minutes).

III. Indicators (flow weighted) An indicator is based on the aggregation of macroscopic (or microscopic) variables. This aggregation can be performed temporally (for an hour, a day, or a month) or

1

spatially (for a route, a journey or a sub-network). To avoid confusion we shall assume that the three variables are measured by road section1, by direction and at the same instant for the entire journey (sub-network) with the same measurement interval (for example one minute). In other terms, each indicator presents a snapshot of the route at the instant of measurement. The length of road section i is shown by li the traffic flow by qi and the speed value by vi . The total length of the journey (or route or network) is equal to

L=

n

∑l i =1

i

A journey made up of n sections starting point

end

i=1

i=2

i=3

i= n

L = total length of journey III. 1 Total distance travelled

ID =

n

∑q i =1

i

⋅ li

This indicator measures the total distance covered by all the vehicles during the period in question. It is expressed in units of distance. III.2 Average flow n

IQ = ( ∑ q i ⋅ l i ) / L i =1

This is obtained by dividing the first indicator by the total length of the journey. It is expressed in number of vehicles per unit of time.

1

It is assumed that the measured speed is spatial (see 2)

2

III.3 Total time spent n

IT = ∑ qi .li / vi i =1

This indicator is expressed in units of time and measures the total time spent by all vehicles during the time period in question. It can also be interpreted as the number of vehicles making the journey. III.4 Average speed

IV = ID / IT By definition this is the distance travelled divided by the time spent. It is a harmonic mean which is weighted both by the lengths of the sections (which is a static factor) and the flows (a dynamic factor).

 n   n  IV =  ∑ q i ⋅ l i  /  ∑ q i .l i / v i    i =1   i =1 III.5 Average journey time

AJT = L / IV This depends on the length of the journey (L) and the average speed. n    n  AJT =  L .∑ q i .l i / v i  /  ∑ q i ⋅ l i   i =1   i =1 

III.6 Total delay

 n  1 1 r   IR = ∑ qi ⋅ li ⋅  − r  vi < vi   vi vi   i =1  IR = [0 vi ≥ vir ]

3

The total delay in a given period is the difference between the time spent at actual speeds and that computed using free speeds (design speeds). The same design speed can be used for all the sections. It is expressed in units of time. III.7 Fluidity

qi ⋅ li ∑ r i =1 vi IF = 10 ⋅ n qi ⋅ li ∑ i =1 vi n

One way of measuring fluidity is to divide the time spent at design speed by the time spent at actual speeds. This indicator varies over quite a small range (between 0 and 10) and therefore readily lends itself to use as a qualitative variable that describes the level of fluidity using a number of categories. In addition, the reciprocal of this indicator provides a scale of congestion. This indicator has no units. III.8 The density of congestion2 n2  n1  n c IC = ∑ qi ⋅ li / vi + ∑ q j l j / v ⋅ /(∑ qk lk / v c ) j =1  i=1  k =1

n sections which comprise the route or journey are broken down into n = n1 + n 2 . The n1 sections are those on which the operating speed is below a critical speed (for example 60 km/h). The speed measured on the n2

For this indicator the

sections is higher than this critical speed. This indicator therefore measures the density of congestion as determined with reference to a critical speed. When no operating speed is below this critical speed the value of the indicator is unity. In other cases it is greater than unity. The contribution of each section depends on the flow measured on it. This indicator also has no units.

IV. The indicators (unweighted variant) If the variable flow is not used when formulating the indicators that have been described above, new quantities will be obtained. Some of these will have no statistical meaning, and others will only be of interest because they are easy to

2

This is a variant of the indicator developed by the SIER (Service Interdépartemental d’Exploitation

Routière d’Île-de-France) with v = 60km / h c

4

compute. In what follows the two variants will be distinguished by use of capital letters for the weighted variant and lower case letters for the unweighted variant. The "total distance travelled" indicator becomes the total length of the journey:

id =

n

∑l i =1

i

= L

"Total time spent" and "total delay" will lose their meaning as they will no longer express total times. The indicators "average speed", "average journey time" and "density of congestion" merit further consideration because they are used in France by some fairly important operators. This is the subject of the rest of this paper. •

Average speed :

iv =

L n

∑

i =1

li vi

Here this is the harmonic mean of speeds weighted by a static factor (the length of the road sections). Whatever the number of vehicles (one or more) on every section, the average speed for the journey is the same: this is obviously not the case with the weighted variant.

•

The average journey time :

ajt = L / IV =

n

∑

i =1

li vi

This is the ratio between the total length and the average speed. It is also the sum of the "average journey times" on all the sections. In this formulation, the "average journey time on each link" and the individual journey time on each link" become indistinguishable. However, summing averages without taking into account the number of individuals with each value is contrary to the fundamental principles of statistical calculation.

•

The density of congestion n2  n1  c ic = ∑ li / vi + ∑ l j / v ⋅ /( L / v c ) j =1  i=1 

This is the indicator used by the SIER ( v c = 60km / h ) in order to describe the congestion on the sub-network it manages. It is equal to unity when no measured 5

speed is below the critical speed. In other situations its value exceeds unity. Apart from the influence of length, all sections contribute in the same manner, irrespective of variations in flow.

V. A comparison between the two variants In the following situations the two variants of the "average journey time" and "density of congestion" indicator will give the same results: 1. The trivial case where the journey consists of a single section of road which gives rise to the expression:

AJT = ajt = l / v

Ic = ic = v c / v 2. In the case where the two flows are identical we will have: q1 = q 2 = .....q n = q : n

AJT = ajt = ∑ li / vi i =1

n2  n1  IC = ic = ∑ li / vi + ∑ l j / v c ⋅ /( L / v c ) j =1  i=1 

3. In the case where the measured speeds are equal and v1 = v 2 = .....v n = v : we will have:

TP = tp = L / v

{ Ic = ic = v c / v v < v c } { Ic = ic = 1v ≥ v c }

The two variants may give different results in all other situations, which leads us to identify some shortcomings in the unweighted variant: 1 : Producing an artificial equal distribution of demand The previous section shows clearly that the unweighted average journey time variant also gives an average rather than an individual journey time. But the effect of the distribution over the different road sections of the vehicles making the journey at time t is not considered. Put another way, it is assumed that the number of vehicles on each section is the same throughout the journey (a false equalization of demand). 2 : Accentuating an overestimation The average journey time that is obtained from the unweighted variant is too sensitive to low speeds. It only needs one of the measured speeds to be relatively low for the journey time to become abnormally high. When the journey time is to be displayed on a VMS, a reasonable upper limit is imposed on the computed values. In the weighted procedure this low value is only of minor importance so a single disruption will not degrade the quality of the estimate. By their nature, weighted

6

indicators are smoother than unweighted ones and avoid excessively high values. This comment also holds true for the congestion indicator. 3 : Removing the interaction between supply and demand The use of unweighted variants is partly justified by their ease of application. Their validity is determined only by the validity of the variable speed. However, the weighted variant is based on speed and flow, which means that one must be more cautious about its validity. In addition, it must not be forgotten that at any moment the flow expresses at least a portion of demand. Including it in the average journey time calculation provides a straightforward way of taking account of the impact of the interaction between supply and demand. Ignoring this effect when calculating congestion levels is likely to result in larger differences than for journey times.

VI. The representativeness of the journey times displayed by VMS Before approaching the problem of representativeness, we shall consider the extent to which the two concepts of journey time (individual and average) differ from each other. Let us assume that the journey in question is made up of n sections (or n basic parts) of different lengths. The individual journey time on this journey is the exact time an individual takes to make it, and is the sum of the journey times on each section. This journey time is specific to this individual and depends on at least three factors: the individual, the vehicle used and the road. The average journey time of a group of individuals is the mean of the individual journey times. The implicit assumption here is that we are able to obtain the individual journey time on each section. However, in this case the time taken to make the journey is an additive function of the times taken to make the different basic parts of the journey. But it is well known that the journey times displayed on VMS are obtained from speed value, one of the three macroscopic traffic parameters. The problem is that it is not possible to compute an individual journey time from macroscopic variables. The unweighted average journey time variant used by some operators for the VMS displays is neither an individual journey time nor a true average journey time (in the statistical sense of the term). Does one of the two variants represent the average journey time more accurately than the other? It would be possible to conduct a measurement campaign using test vehicles which follow a stream of vehicles and produce average journey times for that vehicle stream. The resulting measurements would then be compared with the results given by the two variants. A campaign of this type would require extremely good synchronisation between the starting point (next to a VMS) and the display of a journey time. It would also be possible to perform a campaign using floating vehicles (each of which overtakes the same number of vehicles that overtake it, see 2); unfortunately, although in theory this procedure would measure average journey times it is almost impossible to apply. A third option exists which is to reconstruct average journey times from very detailed historical data and by following space-time diagrams (to take account of the

7

delays that affect measurements). There are two additional weighted and unweighted variants in order to compute this "fictitious" average journey time. n    n  AJT ( t ) =  L .∑ q i ( h i ) l i / v i ( h i )  /  ∑ q i ( h i ) ⋅ l i     i =1  i =1

ajt ( t ) =

n

∑

i =1

Where

li v i ( hi )

hi is the time of arrival on road section i for a vehicle leaving the VMS at q i ( hi ) is the flow measured on road section i at time hi and Vi ( hi ) is the

time t. speed measured at the same instant.

Even in this case the estimated times will be average times rather than individual times, and this must be borne in mind when comparing them with the times obtained using the two variants. In spite of the complexity of the computing task this third approach has been tested in INRETS laboratories (see 4 and 6). The results are summarized in the tables below. Porte d’Ivry - Porte de la Chapelle (Paris outer orbital motorway, 13200 metres) Traffic generally congested, comparison with weighted fictitious average journey times Date Time Number of Unweighted Weighted journey time Average observations journey time fictitious journey Relative Absolute Relative Absolute time error error error error (weighted) 11/01/96 15h-21h 325 13,0% 2m53s 9,1% 2m1s 21m57s 16/01/96

15h-21h

250

8,5%

1m33s

6,2%

1m8s

17m58s

17/01/96

6h-12h

176

22,8%

5m8s

12,4%

2m48s

22m3s

12/01/96

6h-12hh

261

34,7

8m9s

15,7

3m41s

22m49s

12/01/96

6h-12h

292

21,4

5m20s

11,2

2m47s

24m24s

1304

2,6%

5m5s

11,6%

2m36s

21m55s

Global

8

Porte d’Ivry - Porte de la Chapelle (Paris outer orbital motorway, 13200 metres) Traffic generally congested, comparison with unweighted fictitious average journey times Date Time Number of Unweighted Weighted journey time Average observations journey time fictitious journey time Relative Absolute Relative Absolute (unweighted) error error error error 11/01/96 15h-21h 325 9,0% 2m7s 8,3% 1m57s 23m11s 16/01/96

15h-21h

254

6,3%

1m11s

5,8%

1m6s

18m26s

17/01/96

6h-12h

199

15,3%

3m38s

12,9%

3m3s

22m38s

12/01/96

6h-12hh

264

23,4%

6m19s

18,8%

5m5s

25m43s

12/01/96

6h-12h

290

15,2%

4m7s

11,8%

3m11s

26m28s

1332

15,9

3m52s

12,9%

3m8s

23m17s

Global

SIER network (6 am to 12am) Traffic generally congested, comparison with weighted fictitious average journey times Unweighted Weighted journey Date journey length Number Average journey time time of obs. fictitious journey Relative Absolut Relative Absolute time error e error error (weighted) error 16/01/98 A6 Corbeil - 29200 181 35,9% 15m30s 26,4% 11m21s 42m16s BP via A6-a m 20/01/98 A6 Corbeil - 29200 163 26,7% 10m43s 20,3% 8m9s 39m40s BP via A66-a m 16/01/98 A6 Savigny – 15500 170 24,0% 7m0s 14,9% 4m20s 28m33s BP via A6-a m 20/01/98 A6 Savigny - 15500 161 16,9% 4m14s 12,5% 3m7s 24m42s BP via A6-a m 16/01/98 A6a Orly- BP 5100m 188 23,8% 4m8s 19,9% 3m27s 16m20s 20/01/98 A6a Orly- BP 5100m 178 20,5% 2m50s 17,9% 2m28s 13m9s

9

SIER Network (6 am to 12am) Traffic generally congested, comparison with unweighted fictitious average journey times Date journey length Number Unweighted journey Weighted journey Average of obs. time time fictitious Relative Absolute Relative Absolute journey time (unweighted) error error error error 16/01/98 A6 Corbeil - 29200 185 27,1% 13m53s 25,3% 12m59s 50m2s orb. mot. via A6-a 20/01/98 A6 Corbeil – 29200 168 23,2% 10m52s 20,7% 9m28s 45m52s BP via A66-a 16/01/98 A6 Savigny - 15500 173 19,8% 6m23s 19,1% 6m9s 31m20s BP via A6-a 20/01/98 A6 Savigny - 15500 166 27,4% 7m14s 15,0% 3m58s 25m49s BP via A6-a 16/01/98 A6a Orly- BP 5100 187 21,5% 3m54s 19,0% 3m26s 17m6s 20/01/98 A6a Orly- BP

5100

178

16,8%

2m21s

14,5%

2m3s

13m15s

Under free-flow conditions (average speed more than 60 km/h) there is no significant difference between the average journey times given by the two variants, which is why the above tables only deal with periods of congestion. It can be seen that, whichever reference is used for comparison (weighted or unweighted fictitious average journey time), the error given by the proposed variant is always less than that given by the variant used for VMS displays. In addition, the Paris authorities and the SIER have conducted two separate "floating car" surveys to measure journey times. The SIER data have been exploited by the University of Delft as part of the DACCORD project. If we assume that the data was valid, this study concluded that the proposed variant is more accurate than the one in current use (see 8). For technical reasons, no conclusion has as yet been reached from the survey conducted by the Paris Authorities.

VII. Two numerical examples First example Journey : Paris outer orbital motorway between Porte de St Mandé (VMS) and Porte de la Chapelle Distance : 10,01 km Number of sections : 20 (one sensor per section) Data: smoothed average computed every 4 minutes Refresh rate: 1 minute

10

tuesday january 9 1996 BP St Mande-Chapelle (10,01 Km)

minutes

50 45 40

weighted AJT unweighted ajt

35 30 25 20 15 10

11h25

11h07

10h49

10h31

9h55

10h13

9h37

9h19

9h01

8h43

8h25

8h07

7h49

7h31

7h13

6h55

6h37

6h19

6h01

5 0

This diagram show that the largest differences between the two average journey time variants occur during congestion. At 8h02, the difference is about 10 minutes (JT=24 , jt=32 ). At this time the speeds measured on the 15 last sections (approaching the orbital motorway) were less than 20 km/h. This is also confirmed by the congestion index computed for the two variants.

11

Second example Journey : The A6 motorway between a VMS at Corbeil-Essones and the orbital motorway (Porte d’Italie) Distance : 29.2 km Number of sensors: 51 (one sensor per section) Critical speed : 60 km/h Data: smoothed average computed every 4 minutes Refresh rate: 1 minute

friday January 16 1998 Corbeil (A6)-BP(A6a) 29,2

minutes

70 60

unweighted ajt

50

weighted AJT

40 30 20 11h58

11h41

11h24

11h07

10h50

10h33

10h16

9h59

9h42

9h25

9h08

8h51

8h34

8h17

8h00

7h43

7h26

7h09

6h52

6h35

6h18

6h01

10

This second example also shows that the largest differences between the two average journey time variants are to be found during congestion. At 8.00 a.m. the difference is approximately 12 minutes (AJT=54, ajt=66 ). At this time the speeds measured on the 15 last sections (approaching the orbital motorway) were less than 20 km/h. This is also confirmed by the congestion index computed for the two variants.

VIII. The role of the traffic indicators In principle, these indicators are intended for road traffic information purposes and information exchange between the operators of competing modes of transport. They may also have commercial importance for those disseminating information (for example via the radio, newspapers or the Internet). In the case of information which is directly disseminated by the operator (for example which is displayed on a roadside VMS) and perceived by drivers as relating to their convenience, the calculation procedure which is selected is of little importance for the following reasons: •

For drivers, journey time is a subjective entity, the displayed values only need to be within an acceptable range of accuracy (ref. 7).

12

•

In the case of relatively short journeys (less than 10 km) the fact that estimations reappear along a route can to an extent correct the inaccuracy of the unweighted procedure.

However, when these indicators are used for traffic management purposes, the choice of a calculation procedure becomes important: ¾ Sending vehicles onto alternative routes is one of the means available to road operators to improve traffic conditions during high demand. An indicator which does not take account of variations in demand is not suitable for use in this context. When a VMS is located upstream from a decision point on a motorway (an exit ramp) or on the adjacent non-motorway network, it will also perform the function of encouraging drivers to use alternative routes. Drivers' choice of route will depend on the journey times which are displayed for the existing alternatives. An overestimated journey time will have the opposite effect to that intended by the operator. ¾ In order to evaluate the effect of an incident an operator needs to make use of the variation in demand that is expressed (at least partially) by the flow and therefore needs indicators which are able to take account of this variation. ¾ The operator needs to evaluate the interaction between supply and demand on a continuous basis. A worksite reduces capacity, exceptional demand produced by a sporting or cultural event and a partial or total public transport strike are all examples of exogenous effects. In order to conduct real time traffic control or decide on future actions, operators not only need measurement instruments but also evaluation tools (indicators in this case) which are able to take account of the effect that such events have on the interaction between supply and demand (see 3 & 5). ¾ Bad weather reduces capacity. In order to assess its effect on flow, demand must be used to compute indicators. The choice of a computation procedure is also important with regard to the exchange of information between operators (whether in a commercial context or not) for the following reasons: ¾ Operators wish to be sure that the data they exchange with their partners is valid. This data includes sensor measurements and more complex calculated indicators. ¾ As the unweighted indicators are based on a single variable (speed), they have much less statistical validity. Those exchanging data prefer it to be more reliable. This last point raises an issue which relates to a stage before an indicator is calculated, i.e. the quality of the measured data. In order to guarantee high quality data, operators must use algorithms which detect outliers or missing data (see 2) and multidimensional algorithms for processing missing data. Such an approach is 13

essential because of the interaction between the measured variables, which are expressed by the "fundamental relationships" familiar to traffic engineers.

IX Conclusions This article has described a new variant for calculating a number of road traffic indicators. The value of this variant lies in: • the nature of the variables measured by the sensors, • the mathematical justification of the expressions, • its accordance with traffic theory, • its consistency with the way these indicators are used in road operation. In addition, the tests that have been performed in the laboratory or by means of surveys testify to the better performance of the proposed variant and the need to continue research in this area in order to improve our understanding or road traffic. Bibliography : 1. Probabilité Analyse des Données et Statistique, G. SAPORTA, Edition Technip, 1990 2.

Ingénierie du trafic routier. Eléments de théorie du trafic et applications, S. Cohen, Presses de l’Ecole Nationale des Ponts et Chaussées, nouvelle édition, Paris.1993

3.

Prévision du trafic à J+1 (J+2) une approche intermodale, S. Van Iseghem, M. Danech-Pajouh, RTS n° 65 Octobre-Décembre 1999

4.

DACCORD : On-Line Travel Time Predictions dr.ir. H.J.M. van Grol, Hague Consulting Group, The Netherlands, dr. M. Danech-Pajouh, INRETS, France, dr. S. Manfredi, CSST, Italy, dr. J. Whittaker, University of Lancaster, United Kingdom, 8th WCTR 1998

5.

Projet CAPITALS, Prévision du trafic à J+1 (J+2), rapport final, M. Danech-Pajouh, S. Van Iseghem, Rapport INRETS, juin 1998

6.

Projet DACCORD, prévision de temps de parcours sur le Bd périphérique et les voies rapides, M. Danech-Pajouh, S. Bercu Rapport INRETS, février 1998

7.

Méthode d’analyse de l’impact des informations dynamiques des panneaux à message variables sur le comportement des usagers franciliens. P. Jardin, J ; Laterrasse, Congrès International de l’ATEC 1998.

8.

DACCORD, Project TR 1017, Annex B, Travel time evaluation at the Paris test site, February 1999.

14