CALIBRATION PROCEDURE FOR GIPPS CAR-FOLLOWING MODEL

Transcription

1 CALIBRATION PROCEDURE FOR GIPPS CAR-FOLLOWING MODEL Hesham Rakha (Corresponding Author) Charles Via Department of Civil and Environmental Engineering Virginia Tech Transportation Institute 35 Transportation Research Plaza (536) Blacksurg, VA 2461 Phone: (54) Fax: (54) Caroline Cavagni Pecker Laoratório de Sistemas de Transportes Universidade Federal do Rio Grande do Sul Praça Argentina, no. 9 sala 42 Porto Alegre, Brazil 94-2 Phone: Fax: carol@producao.ufrgs.r Helena Beatriz Bettella Cyis Laoratório de Sistemas de Transportes Universidade Federal do Rio Grande do Sul helenac@producao.ufrgs.r Word count: 432 (text) (tales and figures) = 7,552 (total) Revised paper sumitted for pulication in the Transportation Research Record. March, 3 th 27

2 Rakha, Pecker, and Cyis 2 ABSTRACT This paper presents a methodology for calirating the car-following model proposed y Gipps. This caliration procedure is concerned with steady-state conditions. Steady-state occurs when the leader and follower vehicles travel at very similar and near constant speeds, maintaining similar space headways etween each other. The steady-state caliration is important ecause it determines the roadway capacity, the speed-at-capacity, and jam density (spatial extent of queues when fully stopped). The caliration process allows the identification of adequate values for deceleration rates, (maximum deceleration rate the driver is willing to use), (the maximum deceleration rate estimated for the leader), and driver reaction time (T). Two different ehavior assumptions were developed concerning the deceleration rates: (i) equal to, and (ii) different from. The first hypothesis assumes the leader will e as aggressive as the follower. The second assumes that the driver considers his/her leader to have a different maximum deceleration rate. The results are presented in tales and graphs that correlate the car-following parameters to the fundamental traffic stream variales for the different ehavior assumptions. The procedures are then tested on sample field data to demonstrate the adequacy of the caliration procedures.

3 Rakha, Pecker, and Cyis 3 INTRODUCTION Traffic stream motion can e formulated either as discrete entities or as continuous compressile fluid. Microscopic car-following models, which form the discrete entity approach, characterize the relationship etween a vehicle s desired speed and the distance headway (h) to its preceding vehicle in the same lane. Since the first car-following theory was presented half a century ago, many models have een developed. Among others, the most studied are the Gazis-Herman- Rothery (GM) model and the collision avoidance formulation (1). A detailed description of the various car-following model formulations is eyond the scope of this paper ut can e found elsewhere in the literature (1). Alternatively, macroscopic traffic models descrie the motion of the traffic stream y approximating the flow to a continuous compressile fluid. These models relate three traffic stream variales, flow rate (q), density (k), and space-mean-speed (ū). The early models assumed flow condition as a single regime phenomenon. Some of these models were developed y Greenshields, in 1935, Greenerg, in 1958, and Underwood, in Also in 1961, Edie proposed the first multiregime model using Underwood and Greenerg formulations for free-flow and congested conditions, respectively (2). Macroscopic and microscopic traffic models can e related to each other, ased on the relationship etween the respective approach macroscopic (flow, density, and space-mean-speed) and microscopic (headway, spacing, and individual vehicle speeds) parameters. In such case, the sum of successive time headways (s n ) within a long analysis period can e approximated to its duration (T ), as demonstrated in Equation [1]. Therefore, the flow rate (q) can e expressed as the inverse of the average vehicle time headway. In the same way, the traffic density (k) can e approximated for the inverse of the average distance headway (h n ) for all vehicles within a section of roadway of length (L), as demonstrated in Equation [2]. T N sn n= 1 N 1 T 1 1 n N N n= 1 q = s q s L N hn n= 1 N 1 L 1 1 n N N n= 1 k = h k h Microscopic traffic simulation uses car-following models to capture the interaction among vehicles traveling in the same lane. The car-following process can e modeled y an equation of motion for steady-state conditions and two additional constraints that govern the ehavior of vehicles moving from one steady-state to another (decelerating and accelerating). The first constraint governs the vehicle acceleration ehavior, which is typically a function of the vehicle dynamics. The second constraint ensures that vehicles maintain a safe position relative to the lead vehicle. The caliration of car-following ehavior within a microscopic simulation model can e viewed as a two-step process. The first step calirates the steady-state condition. The steady-state caliration has a crucial importance ecause it imposes the roadway capacity and determines the speed-at-capacity and the jam density (spatial extent of queues when fully stopped). [1] [2]

4 Rakha, Pecker, and Cyis 4 The second step is the non-steady-state caliration. Non-steady-state ehavior influences how vehicles move from one steady-state to another. It captures the capacity reduction that results from traffic reakdown and the loss of capacity during the first few seconds as vehicles depart at traffic lights, for instance, usually known as the start loss. Under certain circumstances, the nonsteady-state ehavior can influence steady-state ehavior. For example, vehicle dynamics may prevent a vehicle from attaining steady-state conditions (a desired free-flow speed). A typical example of such a case is the motion of a truck along a significant upgrade section when the truck is unale to attain its desired free-flow speed. This paper presents a methodology for calirating the car-following model proposed y Gipps (3). Gipps formulation is an important collision avoidance model which has een incorporated to several microscopic simulation models, such as AIMSUN (4), SISTM (5) and DRACULA (6). This caliration process is concerned with the steady-state condition. Steady-state occurs when the leader and follower vehicles travel at very similar and constant speeds, maintaining similar space headways etween each other (7). The process is performed for a single lane of unidirectional traffic, assuming that all vehicles have similar parameters and all drivers have similar characteristics. Section 2 reviews the details of Gipps model, descriing the constraints (acceleration/deceleration) and parameters. In section 3, an analysis of the Gipps car-following model for steady-state conditions is presented. The collision avoidance equation is descried as a macroscopic traffic stream model and the analysis of steady-state is performed. The caliration of two different hypotheses of steady-state ehavior is presented in Sections 4 and 5. In Section 6 the caliration procedures that were developed in Sections 4 and 5 are applied to field data. Finally, Section 7 presents the conclusions of the study and recommendations for further work. OVERVIEW OF GIPPS CAR FOLLOWING MODEL This section descries the Gipps car-following formulation. Prior to descriing the model formulation, the set of notations and their definitions are summarized in Figure 1 that illustrates two vehicles moving from left to right. Vehicle n-1 is the leader with a length of L n-1 while vehicle n is the follower. The vehicles positions, speeds, and acceleration rates at time t are denoted as x(t), u(t), and a(t), respectively. The distance headway etween vehicles n and n-1 is denoted as h n (t). According to Gipps, the speed of the follower vehicle is controlled y three conditions. The first condition ensures that the vehicle does not exceed its desired speed (U n ). The second condition ensures that the vehicle accelerates to its desired speed with an acceleration rate that initially increases with speed and then decreases to zero as the vehicle approaches its desired speed. The comination of these conditions results in Equation [3] which controls the vehicle acceleration while vehicles are distant from each other (free-flow ehavior). The equation coefficients were otained from fitting a curve to field data collected on a road of moderate traffic. un() t un() t un( t + T) = un( t) ant(1 ).25 U + U [3] n n where u n (t) is the speed of vehicle n at time t (m/s); a n is the maximum desired acceleration rate of vehicle n (m/s 2 ); T it the driver s reaction time (s); and U n is the desired speed of vehicle n (m/s). In a constrained traffic situation, when vehicles are traveling close to each other, the third condition ecomes dominant and controls the ehavior of the follower vehicle while decelerating. The speed

5 Rakha, Pecker, and Cyis 5 of the follower vehicle (see Equation [4]) is affected y the driver reaction time, the speed and distance difference etween leader and follower, and the deceleration rates they are willing to employ. Gipps pointed out that a safety margin should e added to the driver s reaction time. The safety margin would assure the vehicle s aility to stop even when there is a delay to initiate its reaction for some reason. For the purpose of this study, the safety margin is constant and equivalent to T/2 (reaction time divided y two). This safety value is implicit in Equation [4]. 2 2 u un( t + T) = T + T + 2 [ xn 1( t) Ln 1 xn( t) ] un( t) T + 2 () t where and are deceleration parameters of vehicle n (m/s 2 ); is the actual most severe deceleration rate the vehicle is willing to employ in order to avoid a collision; and is the estimated most severe deceleration rate the leader vehicle is willing to employ. It is an estimated value ecause it is impossile for the follower to evaluate the real intention of his/her leader; L n-1 is the effective length of vehicle n-1 (the actual length plus a safety margin); x n (t) is the position of vehicle n at time t; x n-1 (t) is the position of the preceding vehicle at time t; and u n-1 (t) is the speed of the preceding vehicle (m/s). The parameters related to deceleration rates ( and ) are very important for the raking process modeling. These parameters influence the distance headway etween follower and leader vehicles and thus affect the lane capacity. Assuming the vehicles will travel as close to their desired speed as possile and considering the dynamics limitations, the speed of vehicle n at time t + T can e computed as un() t un() t un( t) ant(1 ).25 +, Un U n un ( t + T) = min [5] un 1() t - T + T + {2 [ xn 1( t) Ln 1 xn( t) ] un( t) T + } According to the aove formulation, once the road is unconstrained and the space headways etween the vehicles are large enough to allow them to travel at their desired speed, the first argument of Equation [5] is applied. In this case, the following vehicle is ale to accelerate according to the empirical equation of vehicle dynamics. Alternatively, in congested conditions, where short headways are typical, the second argument of Equation [5] is applied. In such a case, the speed is limited y the leader vehicle performance. Each vehicle estalishes its speed in order to avoid a collision ased on the assumption that the leader deceleration rate will not exceed. The following section presents the proposed caliration procedures for the Gipps carfollowing model for steady-state conditions. GIPPS CAR-FOLLOWING MODEL IN STATIONARY STATE (STEADY-STATE) A detailed mathematical analysis of Gipps car-following model under steady-state conditions is presented in the literature (8). Consequently, the paper will only summarize the major findings of the Wilson study and present a caliration procedure of the model. In his study, Wilson (8) presented a mathematical analysis of simplified scenarios and identified parameter regimes that deserve further investigation. The paper also showed the derivation of uniform flow solutions (steady-state) and speed-headway functions under simplifying conditions concerning parameters,, and T, and an analysis of the linear staility of the uniform flow, identifying stale and nonstale flow regimes. n 1 [4]

6 Rakha, Pecker, and Cyis 6 However, in order to enhance the practice of traffic modeling it is desirale to relate the model parameters to some measurale traffic stream parameters. The present work aims to estalish mathematical relationships etween Gipps parameters (, and T) and the fundamental traffic variales (free-flow speed, speed-at-capacity, capacity, and jam density). The main assumptions considered to validate the mathematical relationships are (i) traffic flows on a single lane highway (multilane scenarios with lane-changing effects are not considered) and (ii) characteristics of all drivers and vehicles are similar. Once steady-state conditions are achieved (characterized y similar headways etween vehicles traveling at similar speeds), the following equity is true u ( t + T) = u () t = u () t = u, [6] n n n 1 where ū is defined as the space-mean-speed of the traffic stream of vehicles. This equation holds ecause all vehicles are traveling at near equal constant speeds, and thus the time lag T has no impact on the model. Wilson applied the aove statement into Equation [4] to yield the following quadratic equation: ( 1) u 2 3Tu + 2( xn 1() t Ln 1 xn()) t = [7] The microscopic headway etween vehicles n and n-1, h n (t) is no longer time dependent and thus ecomes constant, and denoted y h. Furthermore, Gipps formulation considers that a safety margin is included in the vehicle length (L n-1 ) in order to ensure that vehicles do not invade other vehicle space. Consequently, L n-1 can e defined as the minimum space headway etween two vehicles (h j ) and, according to Equation [2], is the inverse of the jam density. Equation [7] then reverts to ( 1) u 2 3Tu 2 ( h hj ) + = [8] Equation [8] can then e simplified to the following equation if we assume that the traffic stream speed u is in km/h ( 1 ) h = hj + Tu + u = a1 + a2u + a3u [9] where: a = h = ; a = T; a = ( 1 k ) 1 j 2 3 j Wilson demonstrated that using Equation [9] it is possile to develop two different hypotheses concerning and. Section 4 presents the solutions for equal to. This condition underlines the hypothesis that the driver assumes his leader will e identical in the level of aggressiveness. Section 5, on the other hand, shows the solutions when is different from. In this case, the driver considers his leader may have a different maximum deceleration rate. HYPOTHESIS OF EQUAL TO When is assumed to e equal to, the constant a3 reverts to zero and thus Equation [9] is simplified to 2.4 u = ( h h j ). [1] T

7 Rakha, Pecker, and Cyis 7 Under these conditions, the Gipps model presented in Equation [9] reverts to the Pipes model (9), which is a stimulus-response model. The vehicle speed is the result of a stimulus weighed y a sensitivity factor. An earlier study (7) demonstrated that the sensitivity parameter can e related to the traffic variales (capacity, jam density, and free-flow speed), as 1 1 u = ( h hj ), [11] q k u c j f where q c is the roadway capacity (veh/h/lane), k j is the jam density (veh/km/lane), and u f is the traffic stream free-flow speed. The space headway acts as the stimulus and the fundamental traffic variales (q c, k j and u f ) are used to compute the sensitivity factor. Considering the analogy etween the simplified version of the Gipps model (Equation [1]) and Pipes (Equation [11]) the reaction time in units of seconds (T) can e computed as 1 1 T = 24 q k u. [12] c j f Equation [12] provides a unique formula that can e used to calirate the driver reaction time using desired macroscopic traffic stream parameters without the need to gather microscopic carfollowing data. This formula is the first contriution of the paper. For example, considering a freeway facility with a lane capacity of 24 veh/h/lane, a free-flow speed of 1 km/h, and a jam density of 1 veh/km/lane, this would correspond to a driver reaction time of.76 s. Alternatively, the modeler would need to code a driver reaction time of.76 s in order to model the aove identified macroscopic parameters. Tale 1 demonstrates the variation in roadway capacity (veh/h/lane) as a function of the driver reaction time parameter for different roadway free-flow speeds and jam densities using Equation [12]. The capacities that exceed 23 veh/h/lane are highlighted to demonstrate the various trends. First, as the driver reaction time increases, the roadway capacity decreases. Furthermore, as the roadway free-flow speed and jam density parameters increase, the roadway capacity increases. Figures 2 and 3 demonstrate the influence of the equation parameters on the model ehavior. Figure 2 presents the model results for the following set of parameters: driver reaction time (T) equal to 1. s, a free-flow speed (u f ) equal to 8 km/h, and a minimum vehicle spacing of 7.14 m. Alternatively, Figure 3 illustrates the speed-flow-density curves considering different reaction times: a) T =.8s and ) T = 1.2s. Figure 2 shows that the Gipps model ( = ) reverts to the Pipes model shape. In the uncongested regime, speed is insensitive to traffic flow and density (see Figure 2a and 2) as is the case with the model states. The speed-headway relationship is linear (see Figure 2c) and the flowdensity curve has an inverted V shape (see Figure 2d). HYPOTHESIS OF DIFFERENT FROM Another possile assumption is the assignment of different values to the vehicle deceleration parameters ( ). The driver may estimate that his/her leader is either more aggressive or less aggressive than him/herself. The equation that corresponds to this situation is the resultant speed y solving Equation [8] to derive

8 Rakha, Pecker, and Cyis 8 u 8 1 3T 1 1 = ± + 2. [13] 2 9 ( 1 T ) ( h hj )( ) Considering different and, two separate cases are possile: (i) > or (ii) <. Based on the mathematical analysis from (8), the formulation for each case is derived. > When is greater than, the third parameter a 3 of the speed-headway (see Equation [9]) function is negative, which implies that the speed-headway curve is convex as opposed to concave. This ehavior is inconsistent with field data, as illustrated in Figure 4. Furthermore, Wilson (8) demonstrated that the speed-headway relationship may ecome unphysical and produce multiple solutions for some sets of parameters. In order to guarantee feasiility, the user must choose parameters that satisfy u f < 2.4 T ( 1). [14] In order to satisfy the condition of Equation [14] the ratio of to must e very close to 1. (e.g., 1.1), which means that the speed-headway function is near linear and thus would e similar to a Pipes model. Given the unrealistic ehavior of the model for such conditions and that these are not recommended within the commercial simulation software, this case is not discussed any further. < When is smaller than (which implies the leader willingness to decelerate is overestimated or that the vehicle characteristics are different), the model yields 8 1 3T u 1 1 = [15] 2 9 ( 1 T ) ( h hj )( ) In this equation we have removed the negative solution generated y Equation [13] and only included the positive solution. Comining the uncongested and congested regimes the general formulation for the speed-headway relationship can e cast as T u min uf, 1 1 = ( 1 T. [16] ) ( h hj )( ) Equation [16] is adjusted to generate the speed in units of km/h instead of m/s as is the case in Equation [15]. The final speed-headway is illustrated in Figure 5. The vehicle speed is zero for headways smaller than the minimum headway (h j ), and is constant after it reaches the free-flow speed (u f ). The curvature of the curve is convex as was discussed earlier in the paper.

9 Rakha, Pecker, and Cyis 9 Using the speed-headway relationship (Equation [16]), the speed-density relationship can e derived as ( ) T k kj u = min uf, ( 1 T. [17] ) Equation [17] considers the units conversions to ensure that estimated speed is in units of km/h. Alternatively, the density-speed relationship can e cast as 1 k =, [18] 2 a + a u + a u where a 1, a 2, and a 3 are as defined earlier. The speed-flow relationship can then e cast as 1u q =. [19] 2 a + a u + a u Recognizing the convex nature of the congested regime of the speed-flow relationship, if speed-atcapacity occurs prior to the desired speed it can e estimated y taking the derivative flow with respect to speed and setting it to zero as q 1 u( a2 a3u) + = = u u c a1 a2u a3u + + ( a1 + a2u + a3u ). [2] c1 2hj uc =± = 3.6 c3 ( 1 ) Given that the model is a dual-regime model, the speed-at-capacity may exceed the free-flow speed and thus should e constrained y the uncongested regime desired speed as u c 2 = min uf, 3.6. [21] kj ( 1 ) Equation [21] demonstrates that the speed-at-capacity is not impacted y the driver reaction time (T), ut instead is impacted y the,, and k j parameters, as demonstrated in Tale 2. Tale 2 demonstrates an increase in the speed-at-capacity as the / ratio approaches 1. (Pipes model) as demonstrated y the highlighted cells. Furthermore, as the driver deceleration rate increases, the speed-at-capacity increases. Using Equation [21] the roadway capacity can e estimated as 1uc qc =. [22] 2 1 Tuc uc ( ) k j Unlike the speed-at-capacity, the roadway capacity is impacted y the driver reaction time. Specifically, as the driver reaction time increases, the roadway capacity decreases, as demonstrated y comparing Tale 3, Tale 4, and Tale 5. The roadway capacity was calculated considering the free-flow speed (u f ) ranged from 6 to 12 km/h, the jam density (k j ) ranged etween 1 and 16 veh/km/lane, for three reaction times (T) of.5,.6, and.7s, respectively. The deceleration parameter () ranged from 2 to 8 m/s 2, while was set y the / ratio which varied from.45 to

10 Rakha, Pecker, and Cyis The tales demonstrate that the roadway capacity increases as the jam density increases, decreases as increases, and increases as the / ratio increases. SAMPLE APPLICATIONS OF PROPOSED CALIBRATION PROCEDURES This section presents some sample applications of the proposed caliration procedure using field data from two freeways and a single arterial roadway facility. The caliration tool was tested on data provided in the literature (1). These data included a freeway with a speed limit of 88 km/h (55 mph) (Figure 6), the Holland tunnel data, and data gathered from an arterial street that was monitored using the Split Cycle and Offset Optimization Tool (SCOOT) (Figure 7). Other data sources included a sample dataset of 5-min. data from Highway 41 in Toronto, Canada (Figure 4). Tale 6 summarizes the estimates of the four traffic stream parameters for different traffic stream models, as taulated in the literature (1), and as computed using the Gipps and Van Aerde (7, 1, and 11) functional forms. In addition, the literature provides independent estimates of the valid ranges of oserved values for each of the four parameters of interest. These oserved validity ranges serve as an independent measure of the quality of fit of the various models to the suject data. The results of Tale 6 demonstrate the efficiency of the Van Aerde functional form in fitting the data and highlight some of the deficiencies of the Gipps functional form in matching the field data. The caliration of the macroscopic traffic stream parameters (u f, u c, q c, and k j ) was achieved using a heuristic automated tool (SPD_CAL) for calirating steady-state traffic stream models (12,13). These parameters are summarized in Figure 6. Once the macroscopic parameters are derived, the next step is to use the caliration procedures developed in this paper to compute the Gipps car-following model input parameters. By considering the most severe deceleration rate to e employed to avoid a collision to e -5 m/s 2, the / ratio is computed to e.798 (i.e., = 6.27 m/s 2 ) y applying the Excel Solver to Equation [21] considering a desired speed-at-capacity of 74 km/h. The driver reaction time is then estimated to e.856 s y applying the Excel Solver to Equation [22] considering a desired capacity of 1699 veh/h. It should e noted that y changing the parameter from -5 m/s 2 to -4 m/s 2 another set of parameters that produce the same macroscopic ehavior could e achieved. The proposed caliration procedures were also applied to a sample arterial dataset that was extracted from the literature (1). The first step was to calirate the macroscopic traffic stream parameters, as illustrated in Figure 7 using the SPD_CAL heuristic. The figure illustrates a more paraolic functional form for the speed-flow relationship in comparison to the freeway fit that was presented earlier. The Gipps model is ale to predict a speed-at-capacity and capacity that is consistent with field data. In this case the speed-at-capacity is less than the desired speed (free-flow speed). The next step was to input the four macroscopic parameters into Equations [21] and [22] to compute the microscopic parameters,, and T. Using the Excel Solver the / ratio was set at.3 and the parameter was estimated to e 1.44 m/s 2 to produce a speed-at-capacity of 19 km/h. The driver reaction time (T) was computed to e s to produce a capacity of 55 veh/h. It should e noted that there are an infinite numer of cominations of the microscopic input parameters that would produce the same macroscopic ehavior. For example, the following solutions produce the same macroscopic ehavior (/ =.4, = 1.21 m/s 2, = 3.3 m/s 2, T = 2.65 s and / =.5, = 1.2 m/s 2, = 2.3 m/s 2, T = 2.66 s).

11 Rakha, Pecker, and Cyis 11 CONCLUSIONS The caliration of micro-simulation models is one of the major challenges that traffic modelers have to deal with. Furthermore, the model s ehavior is extremely sensitive to the parameters values input to the model. To make matters worse, it is extremely data intensive and difficult to gather microscopic driver ehavior. This paper presented a methodology for calirating the car-following model proposed y Gipps. The caliration procedure focuses on steady-state conditions, assuming that all drivers have similar ehavior and characteristics. The caliration procedure converts the car-following model to its associated macroscopic traffic stream model, and key macroscopic traffic stream parameters (free-flow speed, speed-at-capacity, capacity, and jam density) are calirated using loop detector data. The paper then develops procedures to estimate the model s microscopic parameters from the four macroscopic parameters. The procedure application was demonstrated through some example illustrations using data from two North American freeways and an arterial roadway. The Gipps model does have its limitations in representing traffic stream ehavior; however, the proposed caliration approach should assist modelers in identifying the optimum values of the microscopic car-following parameters to achieve desired macroscopic traffic stream ehavior.

12 REFERENCES 1. Brackstone, M., and M. McDonald. Car-following: A Historical Review. Transportation Research, 2F, 1999, pp Gartner, Nathan H., Carrol J. Messer, and Ajay Rathi. Traffic Flow Theory - A State-of-the-Art Report: Revised Monograph on Traffic Flow Theory. FHWA, Accessed July, Gipps, P.G. A Behavioral Car-following Model for Computer Simulation. Transportation Research, 15B, 1981, pp Barceló, J. Microscopic Traffic Simulation: a Tool for the Analysis and Assessment of ITS Systems. cn.com. Accessed July, Hardman, E. J. Motorway Speed Control Strategies Using SISTM. In Proceedings of the 8 th International Conference on Road Traffic Monitoring and Control. London, UK, 1996, pp Liu, R., D. Van Vliet, and D.P. Watling. DRACULA: Dynamic route assignment comining user learning and microsimulation. Paper presented at PTRC, Vol E, 1995, pp Rakha, H., and B. Crowther. Comparison and Caliration of FRESIM and INTEGRATION steadystate Car-following Behavior, Transportation Research, 37A, 23, pp Wilson, R.E. An analysis of Gipps' car-following model of highway traffic. In IMA Journal of Applied Mathematics 66, 21, pp Pipes, L.A. Car-following Models and Fundamental Diagram of Road Traffic. Transportation Research, Vol 1, 1967, pp May, A.D. Traffic flow fundamentals. 199: Prentice Hall. 11. Van Aerde, M., Single regime speed-flow-density relationship for congested and uncongested highways. Presented at the 74th TRB Annual Conference, Washington DC, Paper No. 9582, Van Aerde, M. and H. Rakha. Multivariate caliration of single regime speed-flow-density relationships. in Proceedings of the 6th 1995 Vehicle Navigation and Information Systems Conference Seattle, WA, USA: Vehicle Navigation and Information Systems Conference (VNIS) IEEE, Piscataway, NJ, USA, 95CH Rakha, H. and M. Arafeh. Tool for Calirating Steady-State Traffic Stream and Car-Following Models. Accepted for presentation at the 86 th Transportation Research Board Annual Meeting, Washington DC, 27. ACKNOWLEDGEMENTS The authors acknowledge the financial support of the Mid-Atlantic University Transportation Center (MAUTC) and the editorial help of Vikki Fitchett.

13 Rakha, Pecker, and Cyis 13 LIST OF TABLES TABLE 1 Variation in Roadway Capacity (veh/h/lane) for =...14 TABLE 2 Variation in Speed-at-Capacity (< )...15 TABLE 3 Variation in Capacity (< ) (T=.6s)...16 TABLE 4 Variation in Capacity (< ) (T=.7 s)...17 TABLE 5 Variation in Capacity (< ) (T=.8 s)...18 TABLE 6 Comparison of Flow Parameters for Single-Regime, Multiple-Regime Models, and Gipps Model...19 LIST OF FIGURES FIGURE 1 Car-following notations...2 FIGURE 2 Steady-state Gipps traffic stream models ( = )...21 FIGURE 3 Sensitivity of traffic stream models to driver reaction times, a) T =.8s e ) T = 1.2s...22 FIGURE 4 Sample fit to North American freeway data FIGURE 5 Sample speed-headway relationship ( < )...24 FIGURE 6 Calirated speed-flow-density fit to freeway data FIGURE 7 Calirated speed-flow-density fit to arterial data....26

14 Rakha, Pecker, and Cyis 14 TABLE 1 Variation in Roadway Capacity (veh/h/lane) for = uf (km/h) kj T (s) (veh/km)

15 Rakha, Pecker, and Cyis 15 TABLE 2 Variation in Speed-at-Capacity (< ) kj (veh/km/ln) and /' k j = 14 k j = 13 k j = 12 k j = 11 k j = 1 k j = 15 k j = 16 u f (km/h) and (m/s 2 ) u f = 6 km/h u f = 9 km/h u f = 12 km/h

16 Rakha, Pecker, and Cyis 16 TABLE 3 Variation in Capacity (< ) (T=.6s) kj (veh/km/ln) and /' k j = 14 k j = 13 k j = 12 k j = 11 k j = 1 k j = 15 k j = 16 u f (km/h) and (m/s 2 ) u f = 6 km/h u f = 9 km/h u f = 12 km/h

17 Rakha, Pecker, and Cyis 17 TABLE 4 Variation in Capacity (< ) (T=.7 s) kj (veh/km/ln) and /' k j = 14 k j = 13 k j = 12 k j = 11 k j = 1 k j = 15 k j = 16 u f (km/h) and (m/s 2 ) u f = 6 km/h u f = 9 km/h u f = 12 km/h

18 Rakha, Pecker, and Cyis 18 TABLE 5 Variation in Capacity (< ) (T=.8 s) kj (veh/km/ln) and /' k j = 14 k j = 13 k j = 12 k j = 11 k j = 1 k j = 15 k j = 16 u f (km/h) and (m/s 2 ) u f = 6 km/h u f = 9 km/h u f = 12 km/h

19 Rakha, Pecker, and Cyis 19 TABLE 6 Comparison of Flow Parameters for Single-Regime, Multiple-Regime Models, and Gipps Model Type of Model Model Free-speed (km/h) Speed-at- Cap. (km/h) Capacity (veh/h/lane) Jam density (veh/km/lane) Valid Data Range (1) Single- Regime Multi-Regime Greenshields Greenerg Underwood Northwestern Edie Regime Modified Greenerg 3-Regime Gipps Van Aerde Source: May, (1) pp. 3 and 33. Highlighted cells: Outside the valid data range for specified parameter.

20 Rakha, Pecker, and Cyis 2 L n-1 n n- 1 x n (t) x n-1 (t) h n (t) = x n-1 (t) - x n (t) FIGURE 1 Car-following notations.

21 Rakha, Pecker, and Cyis Flow (veh/h) Density (veh/km) a) ) Headway (m) Flow (veh/h) Density (veh/km) c) d) FIGURE 2 Steady-state Gipps traffic stream models ( = ).

22 Rakha, Pecker, and Cyis Flow (veh/h) Flow (veh/h) a) ) FIGURE 3 Sensitivity of traffic stream models to driver reaction times, a) T =.8s and ) T = 1.2s.

23 Rakha, Pecker, and Cyis Field Data Gipps Model Van Aerde Model Flow (veh/h/lane) Density (veh/km) Flow (veh/h) Headway (m) Density (veh/km) Gipps Model Van Aerde Model Free-Speed: 16 km/h Free-Speed: 16 km/h Capacity: 1892 veh/h/lane Capacity: 1888 veh/h/lane Speed-at-Capacity: 16 km/h Speed-at-Capacity: 85 km/h Jam Density: 1 veh/km/lane Jam Density: 1 veh/km/lane FIGURE 4 Sample fit to North American freeway data.