Multi-class kinematic wave theory of traffic flow

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Transportation Research Part B 42 (2008) 523–541 www.elsevier.com/locate/trb

Multi-class kinematic wave theory of traffic flow S. Logghe a,*, L.H. Immers b,c,1,2 a

b

Elzenhoek 4, 8210 Zedelgem, Belgium Katholieke Universiteit Leuven, Department of Mechanical Engineering, Traffic, Logistics and Infrastructure, Celestijnenlaan 300A, 3001 Leuven, Belgium c TNO Business Unit Mobility and Logistics, P.O. Box 49, 2600 AA Delft, The Netherlands Received 12 September 2007; received in revised form 16 November 2007; accepted 18 November 2007

Abstract The kinematic wave theory of traffic flow was independently developed by Lighthill and Whitham [Lighthill, M.J., Whitham, G.B., 1955. On kinematic waves. II. A theory of traffic flow on long crowded roads. Procedings of Royal Society A 229, 281–345] and Richards [Richards, P.I., 1956. Shockwaves on the highway. Operations Research 4, 42–51]. The original LWR model was extended in different directions to incorporate more and realistic details. The distinction of classes in traffic flow has received considerable attention recently. This paper proposes a framework for the different existing multi-class extensions of the kinematic wave theory. It turns out that the difference between all models lies in the assumption on how several classes interact. A new model is proposed where classes interact on a non-cooperative way. Slow vehicles act as moving bottlenecks for the fast vehicles, while fast vehicles maximize their speed without influencing slower vehicles. This leads to anisotropic behaviour of the traffic stream. This means that vehicles only react on stimuli in front of them. The new multi-class model is presented and illustrated in the moving bottleneck example of Newell [Newell, G.F., 1998. A moving bottleneck. Transportation Research Part B 32(8), 531–537]. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: LWR; Continuum traffic model

1. Introduction Lighthill and Whitham (1955) and Richards (1956) independently developed the first dynamic traffic flow model. This LWR model describes traffic on a link using a conservation law and an equilibrium relation between density and flow, better known as the fundamental diagram.

*

Corresponding author. Tel.: +32 50 39 32 10. E-mail addresses: [email protected] (S. Logghe), [email protected], [email protected] (L.H. Immers). URLs: http://www.kuleuven.be/traffic, http://www.tno.nl (L.H. Immers). 1 Tel.: +32 16 32 16 69; fax: +32 16 32 29 86. 2 Tel.: +31 15 269 68 11; fax: +31 15 269 68 54. 0191-2615/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.trb.2007.11.001

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S. Logghe, L.H. Immers / Transportation Research Part B 42 (2008) 523–541

Some properties of observed traffic patterns cannot be described by the kinematic wave theory e.g. the start–stop phenomena in congested traffic and the heterogeneous traffic composition. Numerous extensions are based on the adding of a higher order equation. Another research direction focuses on the non-homogeneous properties of traffic flow. The modelling of heterogeneous properties can be done in two ways. A first approach is to make a distinction between lanes. For each traffic lane, a separate LWR model is formulated with separate fundamental diagrams. Exchange terms are provided between these parallel models, based on the driving conditions on the lanes. This approach was proposed by Munjal and Pipes (1971) and Munjal et al. (1971) and further extended by Holland and Woods (1997) and recently by Laval and Daganzo (2005). A second approach subdivides the flow of vehicles into classes. Each class encompasses a group of driver– vehicle entities that share the same properties. Thus the heterogeneous characteristics of the modelled traffic flow are the result of interactions between homogeneous subgroups within the traffic flow. In this paper we elaborate on the multi-class approach. This subdivision into classes has the advantage that the heterogeneous properties are allocated to the vehicles and drivers, not to the infrastructure. This multi-class approach is also developed for other types of dynamic traffic operations models. A first attempt of a higher order multi-class model was set up by Colombo (2002) and Bagnerini and Rascle (2003). Helbing (1997), Hoogendoorn and Bovy (1999) and Tampe`re et al. (2003) used this approach in kinetic models. Heterogeneous properties are also present in almost all micro-simulation packages. In the following section we develop a framework for multi-class LWR models. The interaction of classes is considered as an assignment of road space to classes. Subsequently all the existing multi-class LWR models are reformulated within this framework and new approaches are proposed. A multi-class model where the classes interact as non-cooperative equilibrium is constructed in the following section. The moving bottleneck of Newell (1998) is treated as case study of this new heterogeneous model. 2. A framework for multi-class kinematic wave theory Within this section we develop a framework for multi-class kinematic wave models. First we formulate the general multi-class LWR model. Afterwards we interpret this multi-class model as an assignment of road space to classes. 2.1. Formulation of the multi-class kinematic wave model In the following derivations we examine traffic on a link with unchanging road characteristics. Here, traffic flow is composed of vehicles that belong to n different classes. For each class, density ki, flow qi and average speed ui can be defined. For the road overall, indicated with the tot, the following relations apply X ki k tot ¼ X qtot ¼ qi ð1Þ qtot X k i utot ¼ ¼ ui k tot k tot By this definition of the average speed of the heterogeneous flow, the relation q = k u continues to apply both for each separate class and for the total flow. In a k–q diagram the total heterogeneous traffic flow can be seen as a vector–sum of the joint homogeneous classes. The traffic conservation law, that forms the basis of the LWR model, applies both to each class i apart and to the overall traffic flow, ok i oqi þ ¼0 ot ox

ð2Þ

As in the homogeneous kinematic wave theory, an equilibrium relation is used that expresses the flow of a class in function of the density. However, in a multi-class model this equilibrium flow is a function of all class densities,


qi ¼ Qei ðk 1 ; . . . ; k n Þ

525

ð3Þ

Introducing these equilibrium relations into the traffic conservation law leads to a system of n partial differential equations: oK oK þ Q0e ¼0 ot ox with 2

3 k1 6 . 7 7 K¼6 4 .. 5 and kn

ð4Þ 2 oQ

e1

ok 1

6 6 Q0e ¼ 6 ... 4

oQen ok 1

..

.

oQe1 ok n

3

7 .. 7 . 7 5

oQen ok n

The eigenvalues of Q0e represent the speeds of the characteristics of the solution. These kinematic waves show how the properties of traffic propagate. Also shock waves will occur in the analytical solution. These are lines in the t–x solution plane where class densities change discontinuously. The Rankine–Hugoniot condition shows how the changing class densities and flows across the shock wave are related to the speed z of the wave: z¼

dx Dqi ¼ dt Dk i

ð5Þ

The speed of the shock wave is, therefore, related to changes in density and flow. Note that this shock wave condition applies both to the separate class variables and to the total traffic flow. 2.2. Assigning road space to classes In this section, the general multi-class formulation of (4) will be interpreted. To this end we first define a homogeneous fundamental diagram of a class. When traffic consists of vehicles of only one class, the original LWR model is valid. In that case, the equilibrium relation (3) is valid with zero values for all other class densities. This relation will be referred to as the homogeneous fundamental diagram of a class. A superscript h is used in the notation. qi ¼ Qhei ðk i Þ ¼ Qei ð0; . . . ; 0; k i ; 0 . . . ; 0Þ

ð6Þ

Just as we do with the vehicle population, we divide the road space into different fractions. Each class can then make exclusive use of a fraction of road space assigned to it. We define the fraction of the total road that is assigned to class i as ai. The road fractions are always positive and the sum of all fractions can not exceed 1: 0 6 ai X ai 6 1

ð7Þ

When the road is completely used, the sum of the fractions equals 1. When we assume that a class behaves on an assigned fraction as being subject to a scaled version of its homogeneous fundamental diagram, the flow equals ki qi ¼ ai Qhei ð8Þ ai In this case we can consider the road to be divided into parallel spaces used by each class separately. The assignment of fractions of road space to the various classes derives from the interaction between the vehicles. The equilibrium flows for each class are, therefore, the result of the homogeneous fundamental diagrams Qhei ðk i Þ and of the assumed interactions between the classes. A microscopic interpretation can also be given to the concept of road fractions. We define a vehicle’s space s as the space between the rear bumper of the vehicle ahead and the rear bumper of the vehicle under

526


consideration. This space s equals the distance gap d for the vehicle plus the length of the vehicle L. In the case of a homogeneous flow of vehicles, the space equals the inverse of density k ¼ 1=s

ð9Þ

We can also rewrite this relation in a different way to clarify that the number of vehicles multiplied by the space per vehicle equals the total road space, or 1¼sk

ð10Þ

This relation can be formulated for each class separately if the vehicles of class i move on a fraction ai of the total road space. The space si of the class i vehicles and the density ki of class i vehicles are then related as follows: ai ¼ si k i Now, we can rewrite class flow (8) with (11) to 1 qi ¼ si k i Qhei si In this case the speed of class i is a function of the space: 1 h ui ¼ si Qei si

ð11Þ

ð12Þ

ð13Þ

Instead of assigning a fraction of the road per class, we can also interpret this as a choice for the different spaces si per class. In all the stationary multi-class LWR models developed so far, the equilibrium flow from (3) can be rewritten using road fractions and the scaled version of the homogeneous fundamental diagram according to (8). Formulating a multi-class fundamental diagram then comes down to determining the way in which the road fractions are assigned to the different classes. Section 3 will review the existing multi-class kinematic wave models based on this formulation. 2.3. Anisotropic traffic It is assumed that traffic flow is anisotropic (Daganzo, 1995a). This means that a driver reacts only to stimuli ahead of him and that the traffic situation behind the vehicle does not influence driver behaviour. Exceptions, such as bumper tailing or an advancing police car are, therefore, not taken into consideration. In a homogeneous model this anisotropic condition has been met when the speed of the characteristics and shock waves is smaller than or equal to the speed of the vehicles. It is also known that this condition cannot always be satisfied when considering the total traffic stream on a multi-lane road (Zhang, 2003). On the other hand, we can assume that the anisotropic property remains valid for each individual vehicle. When considering several classes of homogeneous traffic, the anisotropic property must remain valid, though only per class. Thus, the speed of a shock wave and characteristics needs only to be smaller than the vehicle speeds of those classes that are influenced by the particular shock wave. When a shock wave does not affect a slow class, the speed of this wave may exceed the speed of this vehicle class. By incorporating this anisotropic property condition per class, we prevent distortions by averaging over total traffic flow. 3. Multi-class kinematic wave models revisited Within the framework of formula (8) all multi-class LWR models developed so far can be described. In this section, the existing models will be reformulated and discussed using this formula. In addition, some new possible assumptions will be formulated that can lead to other multi-class LWR models. For illustration purposes we work with two classes.


527

3.1. Multi-commodity classes When traffic is subdivided into classes where driver behaviour does not differ between vehicles of the various classes, we will call these multi-commodity classes. Modelling networks requires commodity classes for each OD-relation (e.g. Daganzo, 1995b; Lebacque, 1996). Characteristic for multi-commodity classes is that the total flow does not depend on the composition of traffic flow. Consequently, the homogeneous fundamental diagrams for a class are identical to the fundamental diagram Qe(k) for the total traffic flow. By equating the road fractions ai with the relative class density, the equilibrium flow of a class also becomes a fraction of the total flow. Here, the fractions are defined as ai ¼

ki k tot

ð14Þ

Therefore, the flow per class can be described as a fraction of total flow qi ¼

ki Q ðk tot Þ ¼ ai qtot k tot e

ð15Þ

By choosing these fractions the speed of the traffic flow for all vehicles from all classes is identical. This amounts to a first in first out (FIFO) rule where the traffic composition is unable to change faster or slower than vehicle speeds. 3.2. Special lanes Daganzo (1997) extended the LWR model for two classes, both with the same fundamental diagram Qe(k). But a set of special lanes was taken that are accessible to one of the two classes only. This model is used to model diverges as outlined in Fig. 1. Exiting traffic is then confined to a few traffic lanes, while the remaining traffic can use all traffic lanes. Non-exiting traffic, therefore, has exclusive access to ‘special’ lanes and spreads itself in such a way as to allow for the largest possible flow. This model can also be written using the general fraction equation (8). The exiting class then has a fraction that is limited due to the road section that it can use. Suppose that the class 1 vehicles are confined to md lanes of the total m lanes of the road. We can then write the following fraction equations: a1 ¼ min

k 1 md ; k tot m

a2 ¼ 1 a1 ¼ max

ð16Þ

k2 md ;1 k tot m

ð17Þ

In addition to an analytical computation of this model for a triangular fundamental diagram, the numerical framework was also adapted and interpreted (Daganzo et al., 1997).

Fig. 1. The type of freeway diverge that can be modeled using the special lanes method.

528


3.3. Equal space Wong and Wong (2002) recently formulated a heterogeneous model with various vehicle classes. A homogeneous fundamental diagram Qhei ðkÞ applies per vehicle class and each class is supposed to proceed on a fraction of the total road. These fractions are equated to relative class density: ki ð18Þ ai ¼ k tot Flow per class then equals: qi ¼ k i

Qhei ðk tot Þ k tot

ð19Þ

In this approach flow is calculated per vehicle type using the homogeneous fundamental diagram, at total density. In this way the intrinsic properties of a class remain valid while total traffic flow is taken into account. The model of Zhu et al. (2003) and Zhu and Wu (2003) also operates according to this principle. Here, the homogeneous fundamental diagrams, however have a more specific formulation. In a microscopic interpretation of (18) according to (11), the space si of class i vehicles turns out to be computed as follows: si ¼

1 k tot

ð20Þ

This shows that the space si of all vehicles is identical. This model assumes, therefore, that each vehicle, independent of class, maintains a speed that leads to a fixed user space. It is questionable whether vehicles in heterogeneous traffic will search for identical spaces. The assumption to take relative density as class fraction limits the workable homogeneous fundamental diagrams Qhei ðkÞ. Since the space is identical for all classes, minimum space for all vehicles must also be identical. This is why this model requires identical jam densities for all classes. Within this model, the anisotropic condition cannot be met. Slow vehicles are also influenced by the behaviour of faster vehicles. This implies that slow vehicles are possibly influenced by the properties of vehicles coming from behind. 3.4. Equal distance gap Analogous to the homogeneous space model, Benzoni-Gavage and Colombo (2002) formulated a homogeneous gap model. Here it is assumed that the distance gap for all vehicles in a heterogeneous traffic flow is identical. From the combination with the microscopic interpretation of the fractions (11) with the distance gap definition we know that a i ¼ d k i þ Li k i

ð21Þ

Here Li is the length of the vehicles from class i and d is the gap that is equal for all vehicles. If we take account of a fully utilised road, then the gap becomes equal to P 1 Li k i d¼ ð22Þ k tot This implies that the equilibrium flow for a class is given by X ki ki k tot P qi ¼ ai Qhei : 1 k i Li þ k tot Li Qhei ¼ ai k tot 1 k i Li þ k tot Li

ð23Þ

The properties of this model agree fairly well with those of the homogeneous space model. When the length of all vehicle classes is equal, both models are identical. Using the gap rather than the space compensates for the difference in vehicle length. This allows classes to be modelled with various jam densities. The restraint imposed on the homogeneous fundamental diagrams is, therefore, no longer valid.


529

The anisotropy is also a weak point in this model. Faster vehicles help determine the distance gap of slower vehicles and thus lead to an influence by upstream traffic. In addition, it remains doubtful whether striving towards equal space gaps is the driving force of real vehicles. 3.5. Distance gap proportional to vehicle length The free-flow model of Chanut and Buisson (2003) can also be worked out within the framework of (8). This model distinguishes two classes with different free-flow speeds ufi. If the speed is below a fixed capacity speed uM, the free-flow regime applies. The homogeneous fundamental diagrams for both classes are then given by ðufi uM Þ k i Qhei ðk i Þ ¼ k i ufi ð24Þ b:k Ji Here the expression N/Li from the original publication, with N for the number of traffic lanes of the road, and Li for the vehicle-length of vehicles in class i was equated to the jam density kJi of class i. In the model of Chanut and Buisson (2003) the flow of a class in a heterogeneous flow is described as ðufi uM Þ k 1 k2 Qei ðk 1 ; k 2 Þ ¼ k i ufi þ ð25Þ b k J1 k J2 Applying (8)–(24) and equating to (25) gives an expression for the fraction of the road space that is assigned to class i: ai ¼

k1 k J1 k1 k J1

þ kkJ22

ð26Þ

The microscopic interpretation of this expression, using (11) leads to si ¼ Li þ Li

1 ðk 1 L1 þ k 2 L2 Þ k 1 L1 þ k 2 L2

ð27Þ

Since k1 L1 + k2 L2 equals the physical space of all vehicles on the road under consideration, (27) demonstrates that the distance gap of the vehicles is proportional to the vehicle-length. The fraction assigned to a class is, therefore, proportional to the physical space occupied by the vehicles of a class on the road. In this assumption, headway is not determined by speed, but by vehicle length. The anisotropic condition is also not met. During the capacity regime, all classes in the model share the same speed uM. Therefore the boundary with the congestion regime, that is modeled according to a different assumption, is unequivocally fixed. In general, however, the assumption of proportionality of distance gap to vehicle length imposes no restrictions on the shape of the fundamental diagram. 3.6. Equal speed Another way to model multi-class traffic is to assume that the speed of all vehicles is equal. The fractions ai can be determined by equating the speed of all classes. If we look at a flow of traffic that consists of two classes only and that occupies the road completely, then equating the speeds leads to a1 h k 1 1 a1 h k2 Qe1 Qe2 ¼ ð28Þ k1 a1 k2 1 a1 Solving this equation gives fraction ai in function of class densities k1 and k2. Both Zhang and Jin (2002) and Chanut and Buisson (2003) use this equal speed assumption during the congestion regime.

530


In the model of Zhang and Jin (2002) we can derive the homogeneous fundamental diagrams as follows: 1 k i Li si si Using the equal speed assumption (28) we find the following fraction for class 1: k 1 s1 þ k 1 k 2 ðL1 s2 L2 s1 Þ a1 ¼ k 1 s1 þ k 2 s2 This fraction results in the speed that, for all classes, equals Qhei ðk i Þ ¼

u1;2 ¼

Qe1 ðk 1 Þ Qe2 ðk 2 Þ 1 k 1 L1 k 2 L2 ¼ ¼ k1 k2 k 1 s1 þ k 2 s2

ð29Þ

ð30Þ

ð31Þ

A homogeneous fundamental diagram can also be derived in the congestion model of Chanut and Buisson (2003). To this end, we use the same notation as we used in Section 3.5, b uM k Ji ki h 1 Qei ðk i Þ ¼ ð32Þ 1b k Ji Applying equality (28) yields the following fraction for class 1: a1 ¼

k1 k J1 k1 k J1

ð33Þ

þ kkJ22

This means that the speed for both classes equals the postulated formulation: Pk Qe1 ðk 1 Þ Qe2 ðk 2 Þ b uM 1 kJii u1;2 ¼ P ki ¼ ¼ k1 k2 1b k

ð34Þ

Ji

Logghe and Immers (2003) worked out the equal speed assumption for two classes with equal shaped homogeneous fundamental diagrams. This results in a scaling of classes using passenger car equivalents. During the free-flow regime, the equal speed assumptions restrict the homogeneous fundamental diagrams. Because the speed of all vehicles is the same, the free-flow speed of the various classes must also be the same. An equal speed for all vehicles corresponds with a perfect compliance with the FIFO rule. All vehicles follow one another at the same speed and to that end use, depending on the class to which they belong, the necessary space. 3.7. User-equilibrium In this section we develop another multi-class model that is based on the user-equilibrium. Here, as in the user-equilibrium of Wardrop (1952) all vehicles try to minimise their travel time, which amounts to maximising their speed. During a user-optimum the classes divide themselves across the total road space in such a way that a vehicle is unable to increase its speed without decreasing the speed of slower vehicles. This definition explicitly takes account of the anisotropic property. The speed of a vehicle depends solely on vehicles with an equal or lower speed. Faster vehicles can never decrease the speed of slower vehicles which means that vehicles approaching from behind can have no influence. In a further specification of this user-optimum we assume that slow vehicles do not occupy more space than is strictly necessary. This efficient road use also appears in European rules on overtaking on multi-lane roads. Slow vehicles cut in early so as to not take more space than is strictly necessary. Assuming a fully occupied road of two classes, the speed of the slowest class is given as a1 h k 1 ð1 a1 Þ h k2 uslowest ¼ min Qe1 Qe2 ; ð35Þ k2 k1 a1 ð1 a1 Þ We now look for the smallest possible road fraction for the slowest class while this slowest class speed is still maximal. In that case, the slowest class has maximised its speed and it is unable to increase its speed by taking extra road space.


531

Here, the following scenarios are possible The speed of both classes is equal. Increasing the speed of one class causes a reduction in speed for the other class. Increasing the road fraction of the slowest class does not affect its speed in any way. This means that the derivative of the homogeneous fundamental equilibrium relation of the slowest class equals zero: dQhei ðk i =ai Þ ¼0 dai

ð36Þ

A difference in speed between the two classes only applies in the second scenario. According to (36) this is only possible during free-flow and when there is a (local) straight free-flow branch in the homogeneous fundamental diagram of the slowest class. During congested traffic the user-optimum always gives equal speeds for both classes. This multi-class assumption founds the new model in Section 4. 4. Class interaction according to an user-equilibrium This section presents a new multi-class LWR model based on the user-equilibrium assumption of Section 3.7. The vehicles in a traffic flow are divided into two classes. Each class is described by a triangular fundamental k–q diagram. The vehicles conform to these homogeneous fundamental diagrams, even when the composition of the traffic flow is heterogeneous. This means that a scaled version of this diagram is valid when only a part of the road is assigned to that class. The two classes are assumed to interact based on a user-equilibrium. This states that no vehicle can increase its speed any further without influencing the speed of slower vehicles. Furthermore, vehicles are assumed to capture no more space than strictly necessary. The impact of the latter assumption is treated in Section 6. All possible stationary traffic states will now be examined and subsequently some transitions will be discussed. 4.1. Stationary states Each of the two vehicle classes on the link under consideration is described on the basis of a triangular fundamental diagram. When a flow of traffic is composed of vehicles belonging exclusively to one class, the LWR model with this homogeneous fundamental diagram applies. In the remaining description of the interactions between the two classes, the triangular diagrams are defined as q qM1 r.qM2 qM2 w1 w2

uf2 k

uf1 kM1 kM2 r.kM2

kJ2 kJ1

Fig. 2. The two triangular homogeneous fundamental diagrams.

532


Qhei ðkÞ

k ufi ¼ kMi ufi ðkMi kJi Þ ðk k Ji Þ

k 6 k Mi k > k Mi

ð37Þ

Besides the free flow speed ufi, the critical density kMi and the jam density kJi, the congestion-wave speed wi and the capacity flow qMi can also be defined as dependent parameters: qMi ¼ k Mi ufi qMi wi ¼ k Mi k Ji

ð38Þ ð39Þ

In the remaining computation the fastest class is indicated by the index 1. Fig. 2 sketches two homogeneous fundamental k–q diagrams. Without loss of generality, a smaller jam density, a smaller capacity and a less negative congestion-wave speed were assumed for class 2. We define a scale factor r to relate both diagrams. The capacity point for class 2 lies on the class-1 congestion branch when class 2 is scaled by a factor r. The value of r then becomes r¼

w1 k J1 k M2 ðw1 uf2 Þ

ð40Þ

We can now work out the proposed non-cooperative interactions of both classes. Three regimes or phases can be distinguished according to the class densities: the free-flow regime, the semi-congestion regime and the congestion regime. Fig. 3 gives the three regimes in a k1–k2 phase diagram. 4.1.1. Regime A: free-flow Traffic states where both vehicle classes can maintain their maximum speed are classified under the free-flow regime. The speed of a class is given by ai h k i ui ¼ Qei ð41Þ ¼ ufi ki ai From (37) it follows that this speed equals the free-flow speed ufi when the following condition is met

kJ2

k2

kJtot

k2/k1

kM2

C

kMtot

A

B k1 kM1

r.kM2

Fig. 3. The k1–k2 phase diagram with the three traffic regimes.

kJ1


ki 6 k Mi ai

533

ð42Þ

Since the sum of the fractions equals 1, the following boundary for this phase is derived k1 k2 þ 61 k M1 k M2 Working out Q0e in formula (4) leads to uf1 0 Q0e ¼ 0 uf2

ð43Þ

ð44Þ

The characteristics of this traffic stream are therefore equal to uf1 and uf2. In Fig. 3, this regime (A) is shown graphically on the phase-diagram. The vehicles of the different classes behave completely independent of each other and can overtake without problem. The classes do not influence each other and the user-equilibrium condition is met. 4.1.2. Regime B: semi-congestion At a larger density, the class 1 vehicles will be the first to decrease speed. Those traffic states in which the speed of the class 1 vehicles is less than uf1 but larger than – or equal to – the class 2 free-flow speed, are classified as the semi-congestion regime. The term ‘semi-congestion’ was taken from Daganzo (2002). It indicates that intrinsically slow vehicles can function in a free-flow regime, although faster vehicles are already experiencing congestion. The speed of class 2 remains equal to the maximum speed uf2, respecting the user-equilibrium. The minimum fraction for class 2 whereby this speed is maintained is a2 ¼

k2 k M2

ð45Þ

This gives class 1 the largest possible fraction, which in turn, maximises the speed of class 1 vehicles. a1 k1 k M1 uf1 ðk 1 k M2 k M2 k J1 þ k 2 k J1 Þ u1 ¼ Qhe1 ð46Þ ¼ k 1 k M2 k M1 k 1 k M2 k J1 Þ k1 a1 This regime remains valid as long as the speed of class 1 exceeds or equals the class 2 free-flow speed, or k1 k2 þ 61 r k M2 k M2

ð47Þ

Together with the inversed inequality of (43) this defines the semi-congestion regime. In Fig. 3 this regime is indicated with the index B in the phases-diagram. Working out Q0e in formula (4) leads to " # w1 k J1 w 1 k M2 Q0e ¼ ð48Þ 0 uf2 The characteristics of this traffic stream are therefore equal to w1 and uf2. The semi-congestion regime complies with the user-equilibrium. The slowest class 2 vehicles are able to maintain their maximum speed, while the class 1 vehicles can maximise their speed without influencing the class 2 vehicles. The slowest class 2 vehicles function as moving bottlenecks for the fast class 1 vehicles. This corresponds to the moving bottleneck theory of Newell (1998). 4.1.3. Regime C: congestion In the third regime, the speed of both vehicle classes is less than the free-flow speed of the slowest class. In that case the speed of both classes is identical and traffic operates according to a FIFO regime.

534


The fractions can be extracted by equating the class speeds a1 k M1 uf1 k1 1 a1 k M2 uf2 k2 k J1 ¼ k J2 k 1 ðk M1 k J1 Þ a1 k2 ðk M2 k J2 Þ 1 a1 This gives the same speed for both classes u¼

k1 : k J1

þ kkJ22 1

k1 k J1 w1

2 þ kJ1kw 2

ð49Þ

ð50Þ

Working out Q0e of formula (4) leads to 2 2 2 k1 k 1 :k 2 k2 k2 1 w þ 2 w w þ w 2 1 1 1 2 k J1 k J1 k J2 k J2 k J2 6N Q0e ¼ 6 4 k2 1 2 k2kw k2kw þ w2 N 2 ...k J2 J2 J1

1 N2

2 k1 k J1

k1 N 2 ...k

J2

:

k 1 w2 k J1

1 k1kw þ w1 J1

3

w2 þ 2 kkJ11 kk2J2 w2 kkJ11 w2 þ

2 k2 k J2

w1

7 7 5

ð51Þ

with N2 ¼

2 1 k 1 w2 k 2 w1 þ w1 w2 k J1 k J1

The eigenvalues of this matrix are the kinematic wave speeds during traffic congestion k1 ¼

k1 k J1

þ kkJ22 1

k1 k J1 w1 k1 k J1

2 þ kJ1kw 2

k1 k J1 w1

2 þ kJ1kw 2

k2 ¼

þ kkJ22

¼u ð52Þ

The first characteristic corresponds with the traffic speed. The second expression can be interpreted as the speed of a weighted congestion branch wtot. To this end we study the traffic stream for the given traffic composition. The line through the origin of the phase diagram as in Fig. 3, represents all traffic states with the same composition. The place where it crosses the boundary with the semi-congestion regime gives a weighted density kMtot: k Mtot ¼

r k M2 ðk 1 þ k 2 Þ k1 þ r k2

ð53Þ

Analogously, the weighted jam density kJtot can be expressed as k Jtot ¼

ðk 1 þ k 2 Þ k1 þ kkJ22 k J1

ð54Þ

The second eigenvalue k2 can be interpreted as the inclination of the weighted congested branch. This second eigenvalue equals the following expression: uf2 k Mtot k2 ¼ wtot ¼ ð55Þ k Mtot k Jtot Traffic composition does not change along this second wave. Congestion is therefore characterized by a homogeneous speed with FIFO properties. The composition can only change along a trajectory and the class flows are proportional to the class densities. 4.2. Transitions This section gives an overview of the different transitions in the new model using Riemann problems. In a Riemann problem we consider an infinite road, with, as initial condition, two traffic states. Upstream, one traffic state applies for x < x0. At x0 this changes discontinuously to the second downstream traffic state.


535

The transitions can comprise different types of shock waves and rarefraction fans. Also new intermediate states can be introduced in the solutions. In a first step, we consider the Rankine–Hugoniot conditions between an upstream traffic state U and a downstream traffic state D. These conditions (5) for the Riemann problem give D qU 1 q1 U D k1 k1 qU qD 2 z ¼ 2U k2 kD 2

z¼

ð56Þ

The speed of the shock z must be the same for the two classes. If only one class is involved, then the density and flows of the other class stay unaffected. To have physical significant solutions, the entropy conditions must hold (Ansorge, 1990). In concave fundamental diagrams, this means that the speed of the shock must lay between the speeds of the respective characteristics: D kU i P z P ki

ð57Þ

Based on these conditions, we first show the possible shocks for transitions within the three defined traffic phases. Afterwards the possible transitions between regimes are discussed. 4.2.1. Regime A: free-flow In this case, both the upstream and downstream traffic state belong to the free flow regimes. Only two types of shocks can occur within the free flow phase based on (57) and the characteristics in the free flow regime (48). (1) Shock wave with speed uf1 This shock can only affects class 1 while class 2 stays unaffected by this shock. Furthermore the shock runs parallel with the vehicle trajectories. We will call this contact discontinuity a class 1 slip. (2) Shock wave with speed uf2 Analogously, there can be a class 2 slip that does not affect class 1. 4.2.2. Regime B: semi-congestion Two types of shocks can occur when both the upstream and downstream traffic states are semi-congested. Again, the equality of the entropy condition (57) holds due to the non-strict concavity of the fundamental diagram. (1) A shock with speed w1 This shock runs in parallel with the congestion branch of the class 1 fundamental diagram and does not affect class 2. Traffic density of class 1 can both compress are decompress over this shock. This is related to the straight shape of the fundamental diagram. (2) A shock wave with speed uf2 This shock affects both classes. An increase in density of one class leads to a corresponding decrease of the other class. 4.2.3. Regime C: congestion (1) A shock with speed u This is a slip that runs parallel with the trajectories. Only traffic composition changes. (2) A shock wave with speed wtot Traffic composition is maintained over this shock wave. Vehicular speeds are changed. Both an increase or decrease of density is possible. All the intraregime shocks must end within the boundaries of the considered regime. Transitions between two states of different regimes are more complex and can lead to intermediate states. We refer to Logghe (2003) for details of the different possible transitions.

536


4.2.4. From regime A to regime B Compression shock waves can exist based on conditions (56) and (57). Based on the characteristics within the regimes and condition (57) we get two possibilities (1) A shock wave for class 1 traffic The shock wave can have a speed between uf1 and w1. (2) Shock wave for class 2 The shave wave has always a speed of uf2. 4.2.5. From regime A or B to regime C Also here compression shock waves exist that agrees with conditions (56) and (57). Both class 1 and class 2 traffic will have the same shock wave speed. Free flow or semi-congested traffic will compress to a state with the speed of the downstream congested traffic condition. A slip with that speed will separate the change in traffic composition. 4.2.6. From regime C to regime A or B The characteristics wtot in regime C and w1 in regime B are partly related. Based on (55) we see that jwtot j P jw1 j if jw2 j P jw1 j

ð58Þ

This means that the class 2 fundamental diagram has a steeper congestion branch when condition (58) holds. In that case the model behaves as the classical LWR model with a concave fundamental diagram. A fan between wtot and w1 will be part of the solution if traffic density decreases in the transition from congestion to a faster regime. If (58) does not hold, the characteristics in congestion wtot have a larger speed than the w1 characteristics in the semi-congestion phase. This is only possible when the class 2 congestion branch speed w2 is less negative than w1. In that case, the model behaves like a homogeneous LWR model with a non-concave fundamental diagram. Decompression shock waves can come in the solution in that way. Traffic composition changes over such wave and the fraction of class 2 vehicles increase because the speed of both classes differs after crossing the shock wave. The decompression shock wave is perfectly possible in this continuum model. However, many lane change manoeuvres must be executed when looking at the real traffic stream. The transition from a congested state comprises a perfect mixture of both classes and a homogeneous speed, while a free flow state comprises parallel separated classes with different speeds. In practice this decompression shock wave condition will probably not be met. Therefore, it is reasonable to assume that the discharging flow will not be optimal due to the detailed lane change processes over the shock. This can lead to a clarification of lower outflows and the ‘hysteresis’ effect purely based on heterogeneous vehicle properties and the incorporation of multi-lanes. 4.3. Numerical scheme The classical numerical scheme for the kinematic wave model (Daganzo, 1995c; Lebacque, 1996) is a cell method. The road under consideration is divided into cells with a length of Dx where the traffic situation is calculated per time interval Dt. The aim is to calculate the average cell density based on the traffic situation during earlier time intervals and the in- and outgoing flows in a cell. The flows over the cell boundaries can be calculated when solving the corresponding Riemann problem. An infinite road is considered where the density of the upstream cell applies upstream from a particular point. The traffic condition from the downstream cell applies downward from that point. Constructing the analytic solution of this problem is the corresponding Riemann problem. The flow over the cell boundary corresponds to the flow at the discontinuity in the Riemann problem. The basic principles of this method can also be applied to the multi-class model. Because all possible transitions are known, the numerical scheme is complete. A case study will illustrate it in the following section. However, the proposed scheme will introduce a lot of dispersion.


537

5. Case study: Newell’s moving bottleneck Newell (1998) studied the way in which a moving bottleneck influenced the traffic system in the LWR model. The traffic flow on a homogeneous road is studied where a fundamental diagram with lower capacity applies at the local level. Because this inhomogeneity moves at constant speed, we speak of a moving bottleneck. This original problem definition can also studied using a multi-class LWR model. The local presence of vehicles of the slowest class then acts as the moving bottleneck for the fastest class. Here, Newell’s case study is converted to a problem of two interacting classes. The use of triangular homogeneous fundamental diagrams, as illustrated in Fig. 5, simplifies the case study. We consider a homogeneous road with exclusively class 1 vehicles in the free-flow regime. As illustrated in Fig. 4, there is a feeder road at location x1, where a column of class 2 vehicles enters the road from t1 to t2. It is assumed that the vehicles that merge in at this point have absolute priority with regard to the advancing class 1 vehicles on the primary road itself. When the flow of the class 2 vehicles entering the road is too large, this convoy behaves like a static bottleneck for the upstream class 1 vehicles. All class 2 vehicles drive at free-flow speed uf2 until they leave the primary road at location x2. The effect of this column is shown using characteristics, shock waves, slips and fans in the t–x diagram of Fig. 4. In Fig. 5 the class 1 homogeneous initial state is indicated with A on the fundamental diagram. We assume that the convoy occupies half of the available road, so that the class 2 flow equals half the class 2 capacity qM2. When the convoy enters the road, only half the road space is available to class 1 vehicles. Upstream from the feeder, therefore, state C applies, where the flow equals half of the class 1 capacity qM1. The same class 1 flow will be processed near the convoy, but at the free-flow speed uf1. Since the speed of the class 1 vehicles exceeds that of the incoming convoy, namely the class 2 vehicles, this stream passes the head of the convoy. At that moment, more space is available to the class 1 vehicles, although the same state B continues to apply. This state B is separated from the initial state A by a uf1 slip. When the incoming column has fully entered the road, at t2, a semi-congestion state D applies upstream from the column’s tail. This state is separated from state C by a wave with speed w. In the example, the flow of state D exceeds the flow of the initial state A, hence the shock wave continues on from A to D. When all class 2 vehicles have left the primary road, we get a fan in which the class 1 capacity qM1 is attained. A homogeneous class 1 vehicle flow is simulated in the numerical computation for this case. The flows are adapted near the feeder road at location x1, between t1 and t2. A flow G2 = qM2/2 is used for the class 2

x

B M1

x2

A B B D

x1

C

A

A t t1

t2

Fig. 4. Analytical solution of the moving bottleneck in the t–x diagram.

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S. Logghe, L.H. Immers / Transportation Research Part B 42 (2008) 523–541 q M1 D

qM1/2+qM2/2 A

qA qM2 qM1/2 qM2/2

M2

B

C

convoy

k kJ2

kJ1

Fig. 5. Analytical solution to the moving bottleneck in the k–q diagram.

transition flow for the cell downstream from the feeder road. This limits the transition flow for class 1 to a maximum of qM1/2. The class 2 transition flow downstream from the exit road at location x2, is set at zero. This prohibits class 2 traffic downstream from the exit road. Class 1 and class 2 densities are illustrated in Figs. 6 and 7, using the same scale as in the analytical solution shown in Fig. 4. In each case, white signifies higher density levels and black lower density. In the numerical computation, the cell-interval and the time-interval were synchronised in order to comply uf1 = D x/Dt. Since this discretisation is optimal only for class 1, there is an increased dispersion for the class 2 vehicles. Fig. 7 clearly shows how the rigorously discontinuous properties of the convoy have dispersed near x2 . The dispersion of the numerical Godunov scheme limits its application somewhat. To anticipate this problem, modeling a rigorous discontinuous class pattern requires a much smaller cell-interval as compared to the physical length of the convoy. When this is not the case, the bottleneck is flattened which enables the processing of a higher class 1 flow than would seem possible from the analytical solution. The methods of Lebacque et al. (1998), Henn and Leclercq (2004) or Laval (2004) could be used for localised moving bottlenecks of smaller physical length.

Fig. 6. Numerical calculation of the class 1 flow in the t–x diagram.


539

Fig. 7. Numerical calculation of the class 2 flow in the t–x diagram.

6. Discussion The non-cooperative multi-class model describes slow traffic as an interaction of a slow class that acts as moving obstruction for the fast class. This leads to a traffic description where free flow speed and capacity depend on the traffic composition. A critical assumption in the proposed model is that vehicles capture no more space than strictly necessary. This means that vehicles always behave according to their fundamental diagram and that a road can be fully allocated to the different classes. We clarify both points in the context of the presented model. The validity of the fundamental diagram is a core element in KW modeling. However, we see two elements that can undermine it. First of all, the fundamental diagram represents stationary traffic. The development of higher order models explicitly incorporate non-stationary behavior to overcome this. Non-stationary behavior differs between classes and the need to apply higher order models increase when looking to multi-class traffic. By using the fundamental diagram, it is furthermore assumed that drivers optimize their driving behavior. In fact, it is for each individual driver always possible to hold a larger gap to the preceding vehicle. This nonoptimizing behavior leads to traffic states that can be represented below the curve of the fundamental diagram. Analogously, our critical assumption on the use of no more space than strictly necessary, coincide with the validity of the fundamental diagram in the homogeneous KW model. The assumptions in the user-optimal class interaction are a logical extension of the user-optimizing driving behavior of the different individual vehicles. The assumption that the road can be fully allocated to the different classes is only valid when both the physical vehicles and the road are perfectly divisible. Laval (2004) classifies these multi-class models therefore as one-pipe models. In reality, only fractions in multiples of lane numbers can, strictly speaking, occur. We discuss a multi-lane extension of the non-cooperative multi-class model to clarify the impact of the indivisible road. We assume a road with m traffic lanes that are numbered increasingly from the slowest to the fastest traffic lane. Two classes use this road of which class 2 is the slowest. Assuming a user-optimum during free-flow, the computed fraction for class 1 not necessarily is a multiple of 1/m. We assume that the computed speeds are, respectively, u1 and u2, where fraction a2 complies to j jþ1 < a2 6 m m

ð59Þ

540


It then follows that class 1 is limited by mj m ðj þ 1Þ > a1 P m m

ð60Þ

This is because the user-optimum applies, the fraction of class 2 cannot be decreased. Only the fraction of class 1 can be adjusted to take account of the number of lanes. In a first option, class 1 only utilises the m (j + 1) lanes. There are no class 2 vehicles on these lanes and class 1 can proceed on a parallel basis, at a different speed. This will only happen when the speed of class 1 on this limited number of lanes remains larger than the class 2 speed or mj1 h m k1 u2 6 u 1 ¼ Qe1 ð61Þ m k1 mj1 If the computed speed of class 1 at a fraction a1 = (m j 1)/m were to be smaller than u2, then the traffic flow acquires a homogeneous speed u2 and traffic becomes mixed. Thus in the interaction model according to the user-equilibrium the model can easily be adjusted to take into account the number of lanes. In this adjustment, slow vehicles are still not influenced by fast ones. The anisotropic condition remains valid. The behavioural multi-lane model of Daganzo (2002) is related to this approach. That model also focuses on two classes where lanes are chosen on a non-cooperative manner. The multi-lane extension also shows the critical point in continuum traffic variables: what implicit interval in space and time is traffic averaged over? When considering traffic on a lane during very short time and place intervals (e.g. 5 m, 0.5 s) traffic is always homogeneous. In that case most multi-class extensions of Section 3 become quite similar and traffic flow is modelled as a flow of successive packages of homogeneous traffic. Averaging over rather large intervals (e.g. 500 m, 20 s) leads to differences between the several multi-class approaches. It is clear that the spatial and temporal resolution affects traffic variables and model assumptions in a considerable way. Therefore, the kinematic wave modelling of multi-class and multi-lane traffic runs into the limits of describing discrete phenomena in a continuous way. The discrete properties of traffic in a continuum representation seem to come under pressure in different approaches. Helbing (2001) discussed how the observation of fluctuating traffic conditions becomes very difficult when using continuum traffic variables. Also in higher order continuum models the limitations of continuum traffic representations appear. Li (2005) showed how some oscillatory solitions appears in a numerical simulation of a payne like model. Meanwhile, Li showed that there are no oscillatory solutions of the corresponding continuum model. Therefore, there are limitations of describing discrete traffic phenomena in a continuous way. Based on the limits of the continuum models in the representation of discrete traffic, we expect a strong evolution of hybrid models that combine the continuum and a discrete approach. Laval (2004) already successfully started the development along this path. Hybrid models probably will lead to a unification of the historical separate evolution of continuum and microsimulation models. 7. Conclusion The proposed framework illustrates how multi-class traffic can be described as an interaction of homogeneous classes. Existing multi-class models were discussed and the underlying interaction assumptions were studied. A new model where classes interact according to a user-equilibrium was developed. This new model shows anisotropic behaviour: each vehicle is assumed to drive at maximum speed without influencing slower vehicles. The proposed model can also be interpreted as explicitly modelling slow vehicles as moving bottlenecks from the viewpoint of the fast vehicles. The revival of the multi-class approach during recent years will certainly lead to new and improved descriptions of traffic flow. However, a combination of this with a discrete and multi-lane description will be necessary to further improve and declare specific traffic flow phenomena.


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