- Email: [email protected]
Optimization of highway networks and traffic forecasting
ARTICLE IN PRESS
Physica A 346 (2005) 165–173 www.elsevier.com/locate/physa
Optimization of highway networks and traffic forecasting Andreas Schadschneidera,, Wolfgang Knospeb, Ludger Santenc, Michael Schreckenbergb a
Institut fu¨r Theoretische Physik, Universita¨t zu Ko¨ln, 50923 Ko¨ln, Germany Theoretische Physik, Universita¨t Duisburg-Essen, 47048 Duisburg, Germany c Theoretische Physik, Universita¨t des Saarlandes, 66041 Saarbru¨cken, Germany b
Available online 17 September 2004
Abstract The understanding of traffic dynamics on highway networks is essential for traffic control. Nowadays realistic microscopic models exist that allow highly efficient (faster-than-real-time) simulations of large networks. These simulations can be used not only to determine the state of the network from incomplete information, but also for traffic forecasting. We also discuss an analysis of the highway network in Germany to identify bottlenecks and possible optimization strategies. r 2004 Elsevier B.V. All rights reserved. Keywords: Cellular automata; Modelling of traffic flow; Optimization
1. Introduction The effective capacity of highway networks is of great importance in our daily life as it influences e.g. travel times or transport costs. In the last few years, the increasing amount of vehicular traffic has led more and more to their saturation and an everincreasing number of traffic jams. In contrast to the past, it is no longer possible to react to this by construction of new highways. Therefore, in order to use the existing Corresponding author. Tel.: +49-221-470-4312; fax: +49-221-470-5159.
E-mail address: [email protected] (A. Schadschneider). 0378-4371/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2004.08.063
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network more efficiently, several control strategies like signal control, variable message signs, route guidance, and ramp metering have been proposed for the optimization of the highway’s performance. In recent years important progress has been made in understanding empirical and theoretical aspects of traffic flow, also due to contribution by physicists (see e.g. the reviews [1–3]). Based on the analysis of empirical data [4,5] improved models have been suggested that yield quite realistic results. Among those, cellular automata models are especially attractive because their simplicity allows for an efficient and realistic simulation even of very large networks [6]. Since these simulations are fasterthan-realtime also traffic forecasting is possible. We will briefly describe this in Section 3. Another aspect concerns a possible optimization of existing networks. By analyzing the empirical traffic data of a network we will identify and characterize in Section 4 the sections of the network which limit the performance, i.e., the bottlenecks. It is clarified whether the bottlenecks are of topological nature or if they are constituted by on-ramps. This allows to judge possible optimization mechanisms (e.g. ramp metering control systems) and reveals in which areas of the network they have to be applied.
2. Traffic flow: empirical results and modelling approaches Sophisticated applications in traffic planning and forecasting require not only an accurate representation of the network structure but also simple and at the same time accurate models that are able to reproduce the empirically observed behavior even microscopically. Before we discuss applications and the analysis of the highway network in Germany in more detail, we briefly review the essential empirical results for highway traffic and their implications for the modelling of traffic dynamics. For the networks that we discuss later, empirical data are obtained from inductive loops. These provide local information about the traffic state by measuring the traffic flow J, the (average) velocity v, the density r as well as other interesting quantities like the time-headway between consecutive cars. The function JðrÞ is usually called fundamental diagram. It is the most important quantitative information about the traffic state. Nowadays it is believed that three different phases of traffic flow can be distinguished [4]. In free flow all drivers can move with their desired velocities vmax and interactions between the vehicles can be neglected. In this phase the flow increases linearly with increasing density and the fundamental diagram is given by J rvmax : Another phase is characterized by (wide) moving jams. Here jams exist where the density is large and the velocity small. The jam front moves opposite to the driving direction of the cars with an almost universal velocity of vjam ¼ 15 km=h: The fundamental diagram in this phase is given by J ðr rmax Þvjam ; i.e., the flow decreases with increasing density. Apart from the jam phase another congested phase exists which is called synchronized flow. Here the average velocity is significantly smaller than in free flow. The main characteristics is the absence of a functional
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relation between flow and density. The corresponding time-series shows an irregular behavior and the data points are spread out on a two-dimensional area instead of well-defined lines, like for the other two phases. With regard to traffic control it is important to notice that typically large values flow are observed in synchronized states and that there is a high probability in synchronized traffic that fluctuations cause jams. Therefore, it is reasonable to start ramp metering if synchronized states are observed. Another interesting observation with relevance for the modelling of traffic dynamics concerns the time-headway distributions. The time-headway is the time interval between the passing of the detector location by consecutive cars. Surprisingly the observed distributions show a large contribution at small timeheadways about 1 s [5]. This is much shorter than the recommended safe headway which (in Germany) corresponds to 1.8 s. The occurrence of such short headways means that the drivers not only adjust their driving style according to the distance to the preceding vehicle but also take into account the situation in front of that car. Therefore anticipation is an important ingredient for any realistic model. In the last 50 years many different models have been suggested to describe the dynamics on highways [1–3]. They range from macroscopic approaches based on fluid dynamics or kinetic gas theory to car-following approaches where equations of motion for each car are given based on the traffic situation (headway, velocity, etc.). In recent years approaches based on cellular automata (CA) have become more and more popular. In CA space, time and state variables are discrete. Therefore they are ideally suited for highly efficient simulation even of large networks. Their dynamics is usually given in terms of intuitive rules that have stochastic components. The prototype model has been developed by Nagel and Schreckenberg [7]. Its update rules are given by four simple steps (acceleration, crash avoidance, randomization and motion). Although the basic structure of traffic dynamics is correctly reproduced by this model, it is not sufficient for a realistic microscopic description of traffic flow as needed for traffic forecasting. Therefore several modifications have been suggested over the years. The brake-light model [8] is able to reproduce traffic dynamics even on a microscopic scale. It takes into account anticipation through an interaction horizon where the future velocity of the preceding car is estimated. Also information about velocity changes is obtained through brake-lights. This is important for synchronized traffic.
3. Online simulations and traffic forecasting In the following, we will discuss results and applications for the highway network of North Rhine-Westphalia, the most populated state of Germany with 18 Million inhabitants. The network has a total length of about 2250 km and includes 67 highway intersections and 830 on- and off-ramps. About 4000 inductive loops provide minute aggregated data of flow, occupancy and velocity online via traffic control centers.
ARTICLE IN PRESS 168
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Before a forecast can be made one has to know the current state of the network. This sounds simpler than it actually is! The detectors are not distributed evenly over the whole network. Therefore for certain parts one has very accurate information about the traffic situation, whereas for other parts almost nothing is known. Further complications arise through the sources and sinks in the form of on- and off-ramps. In order to obtain information about the state of the whole network, the local data from the inductive loops are used as an input for a simulation of the network which then (after some iterations) provides a sort of interpolation for the parts of the network without detectors. This is called online simulation [9]. The information is then made available on the internet (www.autobahn.nrw.de); see Fig. 1. The online simulation approach can be extended to an actual forecast. This is possible due to the efficiency and accuracy of the models used. CA allow for a simulation of the whole network several times faster than real time. The online data are used as input for 30 and 60 min forecasts that are made available on the internet. These forecasts have a good accuracy and are used by more than 200,000 drivers everyday. However, future research has to incorporate the feedback of the forecasts on the decisions of the drivers [10]. They might change their plans, e.g., the travel route or travel time or even try to avoid the trip or use public transport.
4. Analysis and optimization of traffic networks In the following, we summarize the main results of [11]. The highway network of North Rhine-Westphalia has an average traffic load of about 30,000 veh/24 h per
Fig. 1. The current state of the highway network is displayed on the internet at www. autobahn.nrw.de.
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measurement section for each driving direction. However, there are large differences between the highways (Fig. 2). Obviously, only a few sections with very large traffic volumes exist which concentrate on the main urban areas and have an average load of 40,000–80,000 veh/24 h. Rather generally, the traffic volume decreases with increasing distance from these main urban areas. About 15% of the vehicles are trucks. The share of trucks on the traffic volume increases with the distance to the conurbations, while the absolute number of trucks does not change significantly on the main highways. Thus, the long-distance traffic is mainly dominated by trucks, while at short distances commuter traffic leads to large traffic loads of the network. 4.1. Determination of bottlenecks In order to allow an evaluation of the traffic load, the probability to find a jam is calculated. Traffic volumes larger than the highway capacity can be related directly to a large jam probability, and thus bottlenecks of the highway network can clearly be identified. Due to the size of the analyzed data set we introduce a simple criterion for the identification of jams: If the density of at least one minute is larger than 50% (the density is given as occupancy which is the percentage of time a detector is covered by a vehicle), a jam is supposed. This criterion was motivated by former empirical studies [5] that indicated that densities higher than 50% are typically neither observed in free-flow nor in synchronized traffic. The jam probability is determined as the fraction of the number of days a jam was found during the observation period (265 days). The analysis of the data shows that
0-10,000 10,000-20,000 20,000-30,000 30,000-40,000 40,000-60,000 60,000-80,000 Fig. 2. Number of vehicles per measurement location per 24 h and driving direction.
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large jam probabilities can be found in four regions of the inner part of the network, related to the conurbations (Fig. 3). As expected, the jam probability shows temporal variations. During the weekend there are less jam than during the rest of the week. From Monday to Thursday one can clearly identify two rush-hour peaks in the morning and the afternoon. On Fridays the afternoon peak is more pronounced. The jam probability in the four regions shows that consecutive measurement sections are often jammed even at the same time, which may be a consequence of jams that have a large spatial extension. However, a sequence of jammed detectors is always restricted by highway intersections, not by on- and off-ramps. More importantly, a jam which branches from one highway to another highway via an intersection has been observed only in one case, namely for the Westhofener Kreuz. There are rather strong correlations between jams here and on the surrounding highways. Therefore, it can be considered as the dominating bottleneck of that region. For the other intersections strong correlations only exist with detectors on one highway. This indicates that the intersections in these cases are not the bottlenecks, but rather the on- and off-ramps. Only one example for a topological bottleneck has been observed. It was generated by a reduction from three to two lanes in both driving directions. Jams regularly emerged in the south of the bottleneck. These jams could have a large extension passing some intersections undisturbed. After the road construction has been finished, the jam probability was reduced drastically. A systematical impact on other parts of the highway network could not be observed.
2
1 3 4
10-20% 20%-30% 30%-40% 40%-50% 50%-70% 70%-100% Fig. 3. Jam probability for the driving direction south/west. The encircled regions are the Ruhrgebiet (region 1), the area of Dortmund (region 2), the area of Krefeld and Du¨sseldorf (region 3) and the area of Cologne (region 4).
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4.2. Braess paradox In order to avoid or at least reduce congestion, why not just build new roads? First of all, in many densely populated areas this is simply not possible. Secondly, additional roads do not necessarily lead to an improvement! This surprising result is known as Braess paradox [12]. A simple example is shown in Fig. 4. Suppose that 6 drivers want to move from A to D at the same time. They have the choice between two different routes ABD and ACD. The travel time on each road segment depends on the number f of cars using it. Explicit expressions are given in Fig. 4. In the situation of Fig. 4a the travel time of all cars is minimized if three cars choose the route ABD and the other three cars ACD. The travel time for each vehicle is T a ¼ 83: Now an additional road is built connecting C with B directly. This new road is fast with travel time function f þ 10: Some drivers decide to use the new route. If each of the now possible route choices is taken by two drivers the new travel time is T b ¼ 92 for everybody. It is minimal in the sense that if just one driver takes a different route, his/her travel time will increase to T~ b ¼ 93 or 103. Thus there exists no better choice for individual drivers, i.e., the solution corresponds to a user optimum. The Braess paradox is not a true paradox. It results from a situation where user optimum and system optimum are different, e.g., as in Fig. 4b, whereas they agree in Fig. 4a. Although the choice of travel times in the present example is somewhat artificial, Braess paradox is of some relevance for real traffic (and probably other!) networks. It has also its correspondence in real physical systems, like coupled springs [13].
10f
D
0
f+1
f+50
C
f+50
D
f+50
10f
f+50
C
A
(a)
10f
B
A
B 10f
(b)
Fig. 4. An example for Braess’ paradox. The travel times for each segment depend on the number f of cars using it (f ¼ 0; 1; . . . ; 6). In (b) the situation where all three routes are chosen by two drivers (f ABD ¼ f ACD ¼ f ABCD ¼ 2) corresponds to a user optimum. The corresponding travel times are larger than that in situation (a) where each route is chosen by three drivers (f ABD ¼ f ACD ¼ 3).
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5. Conclusion The modelling of traffic dynamics has now reached a state where the quality of the models allows for very efficient large-scale computer simulations of highway networks. Data from detectors on the highways are used as an input for simulations that not only allow to determine the present state of the full network but also to make reliable forecasts about the behavior within the next hour. This information is then made available on the internet. Through a statistical analysis of empirical data we have identified the main bottlenecks that lead to congestions. These are predominantly on- and off-ramps rather than topological peculiarities of the highway network. It is the large in- or outflow at ramps that perturbs the stream of vehicles on the main highway. It is expected that ramp metering systems are able to counteract this destabilization of the flow and reduce the formation of jams [14,15], especially in the presence of synchronized traffic. In this way a restricted flow on the ramps may lead to a significant increase of the capacity of the main highways. Of special interest for a capacity optimization is the existence of ‘‘hot spots’’ in the network: The highest jam probabilities are spatially and temporally well localized. These are the parts of the network, where the flow has to be optimized. Presently these sections self-organize in a congested state, leading fluxes that are far below their capacity. The situation can be improved a lot by controlling the number of entering cars and by optimizing the traffic stream at the on- and off-ramps at this section. The small number of bottlenecks that are present in the network shows that it is possible to improve the capacity with a reasonable technical effort. Of particular interest are sections where the main input is from other, less crowded, highways. In these cases a restricted input does not lead to a collapse of the urban traffic.
Acknowledgements We are grateful to the Landesbetrieb StraXenbau NRW for data support and to the Ministry of Transport, Energy, and Spatial Planning as well as to the Federal Ministry of Education and Research of Germany for financial support (the latter within the BMBF project ‘‘DAISY’’). L.S. acknowledges support from the DFG under Grant No. SA864/2-1.
References [1] [2] [3] [4] [5] [6]
D. Chowdhury, L. Santen, A. Schadschneider, Phys. Rep. 329 (2000) 199. D. Helbing, Rev. Mod. Phys. 73 (2001) 1067. A. Schadschneider, Physica A 313 (2002) 153. B.S. Kerner, Netw. Spat. Econ. 1 (2001) 35. W. Knospe, L. Santen, A. Schadschneider, M. Schreckenberg, Phys. Rev. E 65 (2002) 056133. K. Nagel, J. Esser, M. Rickert, in: D. Stauffer (Ed.) Annu. Rev. Comp. Phys. 7, p. 151, World Scientific, 2000.
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[7] K. Nagel, M. Schreckenberg, J. Physique I 2 (1992) 2221. [8] W. Knospe, L. Santen, A. Schadschneider, M. Schreckenberg, J. Phys. A 33 (2000) L 477 A 35 (2002) 3369; Phys. Rev. E 65 (2002) 015101; Phys. Rev. E 70 (2004) 016115. [9] J. Wahle, L. Neubert, J. Esser, M. Schreckenberg, Parallel Comp. 27 (2001) 719. [10] M. Schreckenberg, R. Selten (Eds.), Human Behaviour and Traffic Networks, Springer, 2004. [11] W. Knospe, L. Santen, A. Schadschneider, M. Schreckenberg, Traffic and Granular Flow ’03, to appear. [12] D. Braess, Unternehmensforschung 12 (1968) 258. [13] J.E. Cohen, P. Horowitz, Nature 352 (1991) 699; C.M. Penchina, L.J. Penchina, Am. J. Phys. 71 (5) (2003) 479. [14] H.H. Salem, M. Papageorgiou, Transp. Res. A 29 (1995) 303. [15] B.A. Huberman, D. Helbing, Europhys. Lett. 47 (1999) 196.
Physica A 346 (2005) 165–173 www.elsevier.com/locate/physa
Optimization of highway networks and traffic forecasting Andreas Schadschneidera,, Wolfgang Knospeb, Ludger Santenc, Michael Schreckenbergb a
Institut fu¨r Theoretische Physik, Universita¨t zu Ko¨ln, 50923 Ko¨ln, Germany Theoretische Physik, Universita¨t Duisburg-Essen, 47048 Duisburg, Germany c Theoretische Physik, Universita¨t des Saarlandes, 66041 Saarbru¨cken, Germany b
Available online 17 September 2004
Abstract The understanding of traffic dynamics on highway networks is essential for traffic control. Nowadays realistic microscopic models exist that allow highly efficient (faster-than-real-time) simulations of large networks. These simulations can be used not only to determine the state of the network from incomplete information, but also for traffic forecasting. We also discuss an analysis of the highway network in Germany to identify bottlenecks and possible optimization strategies. r 2004 Elsevier B.V. All rights reserved. Keywords: Cellular automata; Modelling of traffic flow; Optimization
1. Introduction The effective capacity of highway networks is of great importance in our daily life as it influences e.g. travel times or transport costs. In the last few years, the increasing amount of vehicular traffic has led more and more to their saturation and an everincreasing number of traffic jams. In contrast to the past, it is no longer possible to react to this by construction of new highways. Therefore, in order to use the existing Corresponding author. Tel.: +49-221-470-4312; fax: +49-221-470-5159.
E-mail address: [email protected] (A. Schadschneider). 0378-4371/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2004.08.063
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network more efficiently, several control strategies like signal control, variable message signs, route guidance, and ramp metering have been proposed for the optimization of the highway’s performance. In recent years important progress has been made in understanding empirical and theoretical aspects of traffic flow, also due to contribution by physicists (see e.g. the reviews [1–3]). Based on the analysis of empirical data [4,5] improved models have been suggested that yield quite realistic results. Among those, cellular automata models are especially attractive because their simplicity allows for an efficient and realistic simulation even of very large networks [6]. Since these simulations are fasterthan-realtime also traffic forecasting is possible. We will briefly describe this in Section 3. Another aspect concerns a possible optimization of existing networks. By analyzing the empirical traffic data of a network we will identify and characterize in Section 4 the sections of the network which limit the performance, i.e., the bottlenecks. It is clarified whether the bottlenecks are of topological nature or if they are constituted by on-ramps. This allows to judge possible optimization mechanisms (e.g. ramp metering control systems) and reveals in which areas of the network they have to be applied.
2. Traffic flow: empirical results and modelling approaches Sophisticated applications in traffic planning and forecasting require not only an accurate representation of the network structure but also simple and at the same time accurate models that are able to reproduce the empirically observed behavior even microscopically. Before we discuss applications and the analysis of the highway network in Germany in more detail, we briefly review the essential empirical results for highway traffic and their implications for the modelling of traffic dynamics. For the networks that we discuss later, empirical data are obtained from inductive loops. These provide local information about the traffic state by measuring the traffic flow J, the (average) velocity v, the density r as well as other interesting quantities like the time-headway between consecutive cars. The function JðrÞ is usually called fundamental diagram. It is the most important quantitative information about the traffic state. Nowadays it is believed that three different phases of traffic flow can be distinguished [4]. In free flow all drivers can move with their desired velocities vmax and interactions between the vehicles can be neglected. In this phase the flow increases linearly with increasing density and the fundamental diagram is given by J rvmax : Another phase is characterized by (wide) moving jams. Here jams exist where the density is large and the velocity small. The jam front moves opposite to the driving direction of the cars with an almost universal velocity of vjam ¼ 15 km=h: The fundamental diagram in this phase is given by J ðr rmax Þvjam ; i.e., the flow decreases with increasing density. Apart from the jam phase another congested phase exists which is called synchronized flow. Here the average velocity is significantly smaller than in free flow. The main characteristics is the absence of a functional
ARTICLE IN PRESS A. Schadschneider et al. / Physica A 346 (2005) 165–173
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relation between flow and density. The corresponding time-series shows an irregular behavior and the data points are spread out on a two-dimensional area instead of well-defined lines, like for the other two phases. With regard to traffic control it is important to notice that typically large values flow are observed in synchronized states and that there is a high probability in synchronized traffic that fluctuations cause jams. Therefore, it is reasonable to start ramp metering if synchronized states are observed. Another interesting observation with relevance for the modelling of traffic dynamics concerns the time-headway distributions. The time-headway is the time interval between the passing of the detector location by consecutive cars. Surprisingly the observed distributions show a large contribution at small timeheadways about 1 s [5]. This is much shorter than the recommended safe headway which (in Germany) corresponds to 1.8 s. The occurrence of such short headways means that the drivers not only adjust their driving style according to the distance to the preceding vehicle but also take into account the situation in front of that car. Therefore anticipation is an important ingredient for any realistic model. In the last 50 years many different models have been suggested to describe the dynamics on highways [1–3]. They range from macroscopic approaches based on fluid dynamics or kinetic gas theory to car-following approaches where equations of motion for each car are given based on the traffic situation (headway, velocity, etc.). In recent years approaches based on cellular automata (CA) have become more and more popular. In CA space, time and state variables are discrete. Therefore they are ideally suited for highly efficient simulation even of large networks. Their dynamics is usually given in terms of intuitive rules that have stochastic components. The prototype model has been developed by Nagel and Schreckenberg [7]. Its update rules are given by four simple steps (acceleration, crash avoidance, randomization and motion). Although the basic structure of traffic dynamics is correctly reproduced by this model, it is not sufficient for a realistic microscopic description of traffic flow as needed for traffic forecasting. Therefore several modifications have been suggested over the years. The brake-light model [8] is able to reproduce traffic dynamics even on a microscopic scale. It takes into account anticipation through an interaction horizon where the future velocity of the preceding car is estimated. Also information about velocity changes is obtained through brake-lights. This is important for synchronized traffic.
3. Online simulations and traffic forecasting In the following, we will discuss results and applications for the highway network of North Rhine-Westphalia, the most populated state of Germany with 18 Million inhabitants. The network has a total length of about 2250 km and includes 67 highway intersections and 830 on- and off-ramps. About 4000 inductive loops provide minute aggregated data of flow, occupancy and velocity online via traffic control centers.
ARTICLE IN PRESS 168
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Before a forecast can be made one has to know the current state of the network. This sounds simpler than it actually is! The detectors are not distributed evenly over the whole network. Therefore for certain parts one has very accurate information about the traffic situation, whereas for other parts almost nothing is known. Further complications arise through the sources and sinks in the form of on- and off-ramps. In order to obtain information about the state of the whole network, the local data from the inductive loops are used as an input for a simulation of the network which then (after some iterations) provides a sort of interpolation for the parts of the network without detectors. This is called online simulation [9]. The information is then made available on the internet (www.autobahn.nrw.de); see Fig. 1. The online simulation approach can be extended to an actual forecast. This is possible due to the efficiency and accuracy of the models used. CA allow for a simulation of the whole network several times faster than real time. The online data are used as input for 30 and 60 min forecasts that are made available on the internet. These forecasts have a good accuracy and are used by more than 200,000 drivers everyday. However, future research has to incorporate the feedback of the forecasts on the decisions of the drivers [10]. They might change their plans, e.g., the travel route or travel time or even try to avoid the trip or use public transport.
4. Analysis and optimization of traffic networks In the following, we summarize the main results of [11]. The highway network of North Rhine-Westphalia has an average traffic load of about 30,000 veh/24 h per
Fig. 1. The current state of the highway network is displayed on the internet at www. autobahn.nrw.de.
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measurement section for each driving direction. However, there are large differences between the highways (Fig. 2). Obviously, only a few sections with very large traffic volumes exist which concentrate on the main urban areas and have an average load of 40,000–80,000 veh/24 h. Rather generally, the traffic volume decreases with increasing distance from these main urban areas. About 15% of the vehicles are trucks. The share of trucks on the traffic volume increases with the distance to the conurbations, while the absolute number of trucks does not change significantly on the main highways. Thus, the long-distance traffic is mainly dominated by trucks, while at short distances commuter traffic leads to large traffic loads of the network. 4.1. Determination of bottlenecks In order to allow an evaluation of the traffic load, the probability to find a jam is calculated. Traffic volumes larger than the highway capacity can be related directly to a large jam probability, and thus bottlenecks of the highway network can clearly be identified. Due to the size of the analyzed data set we introduce a simple criterion for the identification of jams: If the density of at least one minute is larger than 50% (the density is given as occupancy which is the percentage of time a detector is covered by a vehicle), a jam is supposed. This criterion was motivated by former empirical studies [5] that indicated that densities higher than 50% are typically neither observed in free-flow nor in synchronized traffic. The jam probability is determined as the fraction of the number of days a jam was found during the observation period (265 days). The analysis of the data shows that
0-10,000 10,000-20,000 20,000-30,000 30,000-40,000 40,000-60,000 60,000-80,000 Fig. 2. Number of vehicles per measurement location per 24 h and driving direction.
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large jam probabilities can be found in four regions of the inner part of the network, related to the conurbations (Fig. 3). As expected, the jam probability shows temporal variations. During the weekend there are less jam than during the rest of the week. From Monday to Thursday one can clearly identify two rush-hour peaks in the morning and the afternoon. On Fridays the afternoon peak is more pronounced. The jam probability in the four regions shows that consecutive measurement sections are often jammed even at the same time, which may be a consequence of jams that have a large spatial extension. However, a sequence of jammed detectors is always restricted by highway intersections, not by on- and off-ramps. More importantly, a jam which branches from one highway to another highway via an intersection has been observed only in one case, namely for the Westhofener Kreuz. There are rather strong correlations between jams here and on the surrounding highways. Therefore, it can be considered as the dominating bottleneck of that region. For the other intersections strong correlations only exist with detectors on one highway. This indicates that the intersections in these cases are not the bottlenecks, but rather the on- and off-ramps. Only one example for a topological bottleneck has been observed. It was generated by a reduction from three to two lanes in both driving directions. Jams regularly emerged in the south of the bottleneck. These jams could have a large extension passing some intersections undisturbed. After the road construction has been finished, the jam probability was reduced drastically. A systematical impact on other parts of the highway network could not be observed.
2
1 3 4
10-20% 20%-30% 30%-40% 40%-50% 50%-70% 70%-100% Fig. 3. Jam probability for the driving direction south/west. The encircled regions are the Ruhrgebiet (region 1), the area of Dortmund (region 2), the area of Krefeld and Du¨sseldorf (region 3) and the area of Cologne (region 4).
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4.2. Braess paradox In order to avoid or at least reduce congestion, why not just build new roads? First of all, in many densely populated areas this is simply not possible. Secondly, additional roads do not necessarily lead to an improvement! This surprising result is known as Braess paradox [12]. A simple example is shown in Fig. 4. Suppose that 6 drivers want to move from A to D at the same time. They have the choice between two different routes ABD and ACD. The travel time on each road segment depends on the number f of cars using it. Explicit expressions are given in Fig. 4. In the situation of Fig. 4a the travel time of all cars is minimized if three cars choose the route ABD and the other three cars ACD. The travel time for each vehicle is T a ¼ 83: Now an additional road is built connecting C with B directly. This new road is fast with travel time function f þ 10: Some drivers decide to use the new route. If each of the now possible route choices is taken by two drivers the new travel time is T b ¼ 92 for everybody. It is minimal in the sense that if just one driver takes a different route, his/her travel time will increase to T~ b ¼ 93 or 103. Thus there exists no better choice for individual drivers, i.e., the solution corresponds to a user optimum. The Braess paradox is not a true paradox. It results from a situation where user optimum and system optimum are different, e.g., as in Fig. 4b, whereas they agree in Fig. 4a. Although the choice of travel times in the present example is somewhat artificial, Braess paradox is of some relevance for real traffic (and probably other!) networks. It has also its correspondence in real physical systems, like coupled springs [13].
10f
D
0
f+1
f+50
C
f+50
D
f+50
10f
f+50
C
A
(a)
10f
B
A
B 10f
(b)
Fig. 4. An example for Braess’ paradox. The travel times for each segment depend on the number f of cars using it (f ¼ 0; 1; . . . ; 6). In (b) the situation where all three routes are chosen by two drivers (f ABD ¼ f ACD ¼ f ABCD ¼ 2) corresponds to a user optimum. The corresponding travel times are larger than that in situation (a) where each route is chosen by three drivers (f ABD ¼ f ACD ¼ 3).
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5. Conclusion The modelling of traffic dynamics has now reached a state where the quality of the models allows for very efficient large-scale computer simulations of highway networks. Data from detectors on the highways are used as an input for simulations that not only allow to determine the present state of the full network but also to make reliable forecasts about the behavior within the next hour. This information is then made available on the internet. Through a statistical analysis of empirical data we have identified the main bottlenecks that lead to congestions. These are predominantly on- and off-ramps rather than topological peculiarities of the highway network. It is the large in- or outflow at ramps that perturbs the stream of vehicles on the main highway. It is expected that ramp metering systems are able to counteract this destabilization of the flow and reduce the formation of jams [14,15], especially in the presence of synchronized traffic. In this way a restricted flow on the ramps may lead to a significant increase of the capacity of the main highways. Of special interest for a capacity optimization is the existence of ‘‘hot spots’’ in the network: The highest jam probabilities are spatially and temporally well localized. These are the parts of the network, where the flow has to be optimized. Presently these sections self-organize in a congested state, leading fluxes that are far below their capacity. The situation can be improved a lot by controlling the number of entering cars and by optimizing the traffic stream at the on- and off-ramps at this section. The small number of bottlenecks that are present in the network shows that it is possible to improve the capacity with a reasonable technical effort. Of particular interest are sections where the main input is from other, less crowded, highways. In these cases a restricted input does not lead to a collapse of the urban traffic.
Acknowledgements We are grateful to the Landesbetrieb StraXenbau NRW for data support and to the Ministry of Transport, Energy, and Spatial Planning as well as to the Federal Ministry of Education and Research of Germany for financial support (the latter within the BMBF project ‘‘DAISY’’). L.S. acknowledges support from the DFG under Grant No. SA864/2-1.
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