A Hybrid Model of Traffic Flow: \Impacts of Roadway Geometry on Capacity

A Hybrid Model of Traffic Flow:
Impacts of Roadway Geometry on Capacity

Jorge A. Laval and Carlos F. Daganzo
Department of Civil and Environmental Engineering
Transportation Group
University of California, Berkeley

Abstract. This paper introduces a continuum-discrete procedure for the simulation of vehicular traffic flow and some initial results. The approach combines a continuum stream of fast vehicles with discrete trajectories for slower ones. This allows us to estimate the effects of road geometry on traffic flow, recognizing that each vehicular class is affected differently and that the classes may interact. An application of the procedure to predict the capacity of uphill grades leads to a simple formula that provides an efficient tool for the analysis and management of these bottlenecks. Unfortunately, predictions with the procedure disagree significantly with the recommendations in the latest version of the Highway Capacity Manual, which were obtained with microsimulation. This deserves further study.
Jorge Laval ©2004

Contents

1  Introduction
2  The numerical method
3  Capacity of upgrades
    3.1  Simulation
    3.2  Analytical formulae
        3.2.1  Analytical solution of the upgrade bottleneck
4  Comparison with HCM
5  Discussion

1  Introduction

This paper presents preliminary results of a hybrid model for the simulation of traffic flow composed by two vehicle classes: A continuum class of fast vehicles (cars) modelled with kinematic wave (KW) model theory [9,[12], and a discrete class of slower vehicles (trucks) treated as boundary conditions for the continuum problem. Similar work but more limited in scope has been presented in [5] (following [8].)
The proposed model is Markovian in that the state of each user class in the future (ie, densities for cars and speeds and positions for trucks) can be obtained with conditions in the present, as shown in Fig. 1. The arrows in the figure depict the interactions: Trucks affect cars as per the KW theory of moving bottlenecks (KW-MB) [11]. Cars affect trucks just as they affect other cars; ie, they preclude trucks from travelling faster than the KW stream as predicted by a constrained free-motion (CFM) model. To implement the procedure numerically, we use the approach in [3] for the KW-MB sub-model and a free-motion simulation based on [4] for the CFM sub-model.
f-1.gif
Figure 1: Interactions of the hybrid model
We are confident that the hybrid model of the figure will make robust predictions because its most complicated component, the KW-MB theory, is macroscopic in nature and has been improved and validated in the field [10]. The hybrid model also exhibits important practical advantages steaming from its simplicity and mathematical stability -the model does not amplify errors in the input data. Most notably, the model can (i) be calibrated by measuring a few easily observable parameters, (ii) make reasonable predictions with imprecise input estimates; and (iii) be validated by parts.
The proposed procedure has merit because these features are not generally shared by microsimulation models. Interestingly, we find that the predictions of the effect of trucks on uphills by the proposed procedure and by the Highway Capacity Manual [13] (HCM), which is based on microsimulation, differ significantly.
The paper also derives an approximate expression for the capacity of an upgrade. This simple formula turns out to match simulation results accurately under all conditions. Since the formula is analytical, it provides an efficient tool for the analysis and management of this common type of bottlenecks. A set of charts is also provided as an illustration, but more testing of the results is needed.
The paper is organized as follows: § 2 develops the numerical method and demonstrates its convergence. § 3 presents the example of the upgrade bottleneck and § 4 a comparison with the HCM. A brief discussion is given in § 5.

2  The numerical method

Let x=f(t) be the position, x, of a moving bottleneck (truck) at time t. Its desired acceleration a(v,t,x) is assumed to be given by a free-motion model; ie, a model based on Newtonian mechanics that gives the trajectory of a vehicle as a function its characteristics, current speed and the set of geometrical parameters at its current position (eg, grade, curvature, roughness, altitude, etc.) The free-motion model incorporated in [4] is used in this paper.
We assume that the speed, v(t), in the time-interval [t,t+Dt) is given by the CFM relation:
v(t)= min
{vdwn(t),vdes(t)},
(1)
where vdwn(t) is a numerical estimate for the speed of the KW stream immediately downstream of the truck in [t,t+Dt) and vdes(t) is the "desired" truck speed. This speed is obtained with the relation
vdes(t) .
=
 
v(t-Dt)+a(v,t-Dt,x)Dt.
(2)
The position is then updated with
f(t+Dt)=f(t)+v(t)Dt.
(3)
Equations (1)-(3) are the essence of the CFM module at the top of Fig. 1. The KW-MB module, which is used to model cars and calculate vdwn(t), is based on [3]. The overall procedure is numerically stable and therefore tends to the continuum solution as Dt 0. This is illustrated below by means of a simple example that could be solved exactly by hand.

Numerical demonstration

Consider a homogenous freeway with n=2 lanes and the idealized flow (q) - density (k) diagram of Fig. 2a. (We assume an isosceles triangle in this illustration to avoid all numerical errors except for those introduced by the hybrid procedure.) Traffic is flowing at capacity (state C) when, at (t,x) = (0,0), a truck with vdes=30 mph enters the segment. This implies that the truck possesses infinite acceleration, but allows us to obtain the exact solution by hand. The truck trajectory turns out to be the piecewise linear dark line of Fig. 2b. The flow downstream of the truck when it holds back a queue (called the capacity of a moving bottleneck in [10], and denoted by QD in Fig. 2b) is assumed to be equal to the capacity of the unblocked lanes. In other words,
QD .
=
 
Q n- 1

n
.
(4)
At the same time but at x=.4 a traffic jam J begins to propagate upstream imposing a speed vJ=12 mph to incoming traffic. This traffic jam forces the truck to decelerate, as shown in Fig. 2b. This figure displays the exact continuum solution of the problem.
f-2.gif
Figure 2: (a) (k,q) diagram for the numerical demonstration (b) (t,x) diagram of the exact solution (c) (t,x) density map of numerical solution with Dt = 1 sec (d) (t,x) density map of numerical solution with Dt = .5 sec.
Parts (c) and (d) of the figure present two (t,x) density maps produced by the numerical method. Different shades of gray indicate different traffic states. Note how the solution converges to the exact solution of part (b) as the time-step is reduced.

3  Capacity of upgrades

Here we use the model to predict the maximum steady state flow that can be sustained without control in a long level freeway with n identical lanes and an uphill in the middle. Each lane has a triangular fundamental diagram with free-flow speed u = 60 mph, wave velocity w = 15 mph and jam density for one lane k = 150 vpmpl, in rough agreement with empirical evidence [6] [1] [2]. Thus, the maximum flow of cars on all lanes is Q=n[(w u)/(w + u)]k. This is the capacity of the flat segments. The uphill has a G percent grade for L mi. There are two user types, cars and trucks, in proportions 1-r and r, where r << .5. Cars are able to maintain the free-flow speed u on the uphill, but trucks cannot. When they slow down they create queues.
In keeping with simplicity, we assume that (4) holds. Although [10] observed that QD increases with v, (4) can be used as a rough approximation. This assumption is needed to develop an analytical formula, but more elaborate ways of defining QD may be used with the simulation.

3.1  Simulation

We applied the hybrid model for all possible combinations of the following: n={1,2,3} lanes, L={.1,.4,.6,.9,1.3} mi, G={2.5,4.5,6.5} % grade, and r=1,25 %. Two truck types were considered separately: Heavy (average) trucks with weight-to-power ratios of 228 (140) lb/hp and weight-to-frontal-area ratios of 682 (312) lb/ft2. The uphill capacity was estimated as the ratio of the number of vehicles crossing an arbitrary location to the simulation period.
Fig. 3 shows a time-space density map from a simulation. Truck arrivals to the uphill are depicted by white circles. Notice how trucks change speed as they climb and reach the flat road again. The trajectory of the second truck is slower because it reaches the bottom of the hill at a slower speed than the first truck. The third truck is inactive while it is in a queue, but activates its own queue when it cannot keep up with traffic acceleration ahead of it.
f-3.gif
Figure 3: (t,x) density map of a sample simulation.
The figure is interesting because it shows that when trucks cannot keep up they create a void with less traffic directly in front. These voids reduce capacity. Obviously, the magnitude of the effect depends on L, G and the percentage and characteristics of the trucks.

3.2  Analytical formulae

A set of formulas are currently been developed. The simplest of these, derived in section § 3.2.1, is based on the crawl speed of the trucks on the upgrade, vc. It expresses the uphill capacity as rQ, where r [0,1]. The result is
r = rmin

1-e-rrminQT (1-rmin)
, where
(5)

T = L w+vc

wvc
,
(6)

rmin=1- w(u-vc)

nu(w+vc)
.
(7)
The parameter T is the duration of the queueing episode caused by an isolated truck (see Fig. 5,) and rmin is the limit of r for L.
Consideration of (5) shows that it always behaves reasonably in limit cases. For example, as r increases, we see from (5) that r rmin. This is as expected for in this case every car enters the upgrade when a truck is still on it. We also see that rmin [0,1] if vc (0,u), as required. Note that even in the worst case scenario of our sample (heavy trucks on an infinite one-lane 6.5% upgrade) rmin > .75.
f-4.gif
Figure 4: Capacity charts showing agreement between simulation and approximation.
Extensive comparisons show that (5) matches simulation results accurately under all conditions studied here, see Fig. 4. In the entire data set the average absolute difference between the r values obtained with simulation and analytically is .01. The largest differences arose for short upgrades. The maximum difference were .05 for heavy and .03 for average trucks, both with n=1.

3.2.1  Analytical solution of the upgrade bottleneck

This section presents the derivation of equation () and may be skipped without loss of continuity.
Let the crawl speed vc be the steady state speed of a vehicle on an infinite upgrade, where the speed drops to a point where the engine lacks power to accelerate. For a particular truck type, vc is only a function of G. Using the free motion model in [4] we found that the following polynomials give an accurate estimate of vc for the relevant truck types if .01 G .09:
vc =



-.09 G 3 + 2G 2 - 18 G + 71
(heavy trucks,)
.44 G 2 - 9.4 G + 71
(average trucks.)
(8)
The approximate solution presented here assumes that all truck trajectories are identical and piecewise linear with slopes vc and u, as in Fig. 5. In this simple setting the disturbance caused by a truck at the bottom of the uphill is given by (6) and the corresponding queued flow by (see Fig. 2a)
QU = Q n- 1

n
+ wvc

w + vc
k.
(9)
The advantage of this crude approximation is that the truck arrival process at the bottom of the upgrade becomes a renewal process. It is clear from Fig. 5 that flows at the bottom of the uphill are either Q or QU.
f-6.gif
Figure 5: (a)(t,x) diagrams of a single truck on the uphill, (b) mean of the simplified process.
Let the headway between two consecutive trucks be the random variable h with mean ~h and probability density function f(h). Since every vehicle arrival at the bottom of the upgrade has a probability r of being a truck, we see that the truck headway (in continuous time) must have the following density function:
f(h) =



rQUe-rQUh
when h T,
e-Tr(Q-QU)rQe-rQh
when h > T.
(10)
Taking expectation and after manipulation,
~
h
 
=[ e-rQUT

Q
+ (1-e-rQUT)

QU
]/r.
(11)
Since the average number of vehicles (ie, cars between two consecutive trucks plus the leading truck) is 1/r, the capacity of the upgrade can be expressed as 1/(r~h), which in combination with (9), gives () as claimed.

4  Comparison with HCM

The recommendations of HCM 2000 [13] for the capacity on upgrades are based on the microsimulation model FRESIM, developed by the Federal Highway Administration. Using the notation of this paper and neglecting recreational vehicles, the HCM capacity expression reads
r .
=
 
1

1+r(ET-1)
,
(12)
where ET is the passenger car equivalent of a truck, ie, the number of cars that would produce the same effect on capacity as a single truck. The values for ET are tabulated for different values of r, L and G in exhibit 23-9 of the HCM (not reproduced here) for the average truck, which has a weight-to-power ratio "between 125 and 150 lb/hp".
It was not possible to reconcile the HCM values with those presented here. For example, the HCM recommends r curves that tend to much smaller values than rmin as r increases. Moreover, there are cases were the value of r predicted by the HCM increases in some range of r. This is shown in Fig. 6 for G=2.5% and L=0.9 mi. This figure also includes the curves produced with (5) for the heavy and average truck.
f-5.gif
Figure 6: Comparison of HCM values and eq. (5) for G=2.5%, L=0.9 mi and n=2 lanes.
There may be several reasons for the disagreement. First, the validation of microsimulation models is a tremendous task; second, nothing guaranties that, if conditions change, a microsimulation output should be reasonable; third, the number of lanes is not considered in (12); finally, the free motion models in FRESIM may themselves need validation (as suggested in [14].) Of course, it is also possible that our model is unrealistic. Physical evidence is needed to resolve the discrepancy. The advantage of the proposed model is that it can be validated by parts, avoiding the expensive effort required to validate it as a whole.

5  Discussion

The hybrid model in this paper can be used in any situation where the road affects a small subset of vehicles, even if they are heterogenous; eg, to capture the joint effect of local and express buses at stops near traffic signals (generalizing the work in [5]) or the effect of light and heavy trucks on upgrades (generalizing the work in this paper.) A simulation with heterogeneous vehicles should allow for slow-vehicles to overtake one another, recognizing that QD changes during each passing interaction. The changes are trivial if interactions always occur pairs. Multi-vehicle interactions can also be modeled but if they are frequent and complex (with a high proportion of slow vehicles) the model may lose its stability properties.
The proposed simulation model can be applied to other geometrical bottlenecks, such as combinations of horizontal and vertical curves. All one needs is an estimate of the maximum vehicle acceleration as a function of x, which should be known given the road geometry. Analytic formulae can then be developed for simple bottlenecks following the logic of the paper.
The homogeneity assumption made in this paper prevents trucks from clustering. Clustering can have either a favorable or unfavorable effect, depending on the trucks' relative positions. For example, if two trucks hit the bottom of an upgrade on the shoulder lane right after they have passed each other, only the trailing truck will have a significant effect on the traffic stream. This is a best-case scenario since the pair acts as a single truck. A worst-case scenario arises if the two trucks are passing each other slowly, blocking two lanes, when they hit the upgrade. These limiting cases are now analyzed by changing the parameters of our formula. The results roughly indicate how sensitive the formula is to the homogeneity assumption. If we denote by r(r,n) the prediction of (5) for a fraction r of trucks and n lanes, relevant limiting cases for n = 2 are:
Case 2 is modeled as if trucks were evenly distributed on both lanes with no passing allowed; thus, we set n = 1. Case 3 is modeled as if only 50% of the trucks existed; thus, we reduce the truck proportion by a factor of 2. Consideration shows, as expected, that:
r(r,n) [r(r,1);r(r/2,2)].
(13)
Furthermore, the bounds in this relation are reasonably tight; see Fig. 7 for heavy trucks, L=0.2 mi, and G=6 %. The bounds should be even tighter for n > 2.
f-7.gif
Figure 7: Three cases for eq. (5) for G=6%, L=0.1 mi, heavy trucks.
Equation (5) can be used both, for control and design. For design, one should relate the traffic parameters of the formula -u,w and k- to road geometry. To put the formula on firmer ground, further study is needed to see whether the vc(G) functions of TWOPAS are good approximations for a system close to capacity - since it the slow vehicles may not act as if they were totally insulated from the traffic stream when flow is heavy.
The ideas in this paper are now being extended to situations where lanes and lane-changing are explicitly considered, using the framework in [7]. These extensions have the potential for reproducing the growth of disturbances caused by off-ramps and the behavior of freeway bottlenecks (eg, merges, diverges and slow vehicles) in more detail.
Acknowledgment- This research was supported in part by the University of California Transportation Center.

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