Traffic_Flow_Models09

Traffic_Flow_Models09 - Introduction to Transportation...

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Unformatted text preview: Introduction to Transportation Engineering Traffic Flow Models Dr. Antonio A. Trani Professor of Civil and Environmental Engineering Virginia Polytechnic Institute and State University Blacksburg, Virginia Fall 2009 Virginia Tech 1 Topics for Discussion • Why modeling traffic? • Approaches to model traffic • Parameters connected with traffic flow • What role do vehicle dynamic/kinematic equations play? • Examples Virginia Tech 2 Why Traffic Modeling? Required to estimate capacity of any transportation facility Highway capacity - how many cars per hour? Railway capacity - how many rail cars per hour Airport capacity - how many aircraft can land per hour? Required to estimate level of service of transportation facilities a) Level of service is connected with delays imposed by the system on vehicles and people To study impacts of our own actions (building infrastructure) Virginia Tech 3 A Difficult Problem to Understand • Traffic phenomena is complex • Traffic phenomena is usually a stochastic process (described by random variables) • Los Alamos (New Mexico) statement: “Modeling traffic phenomena has proven to be more difficult than predicting and modeling sub-atomic level reactions inside the atom - for nuclear warhead simulations” • Statement after four years of work developing the newest traffic and transportation planning software package called TRANSIMS Virginia Tech 4 Approaches to Modeling Traffic Microscopic Attempts to look into individual driving behaviors Vehicle-following models Macroscopic Looks at the traffic as a fluid-flow or heat-transfer phenomena Vehicles are not identified individually but as a group of entities moving on the system Technically, both microscopic and macroscopic models consider the human in their solutions Virginia Tech 5 Parameters Connected with Traffic Models • Speed • Volume and Rate of Flow • Density or Concentration • Spacing and Headway • Clearance and Gap Virginia Tech 6 How Do We Collect Traffic Data? • External devices Road traffic counters (loop detectors) Traffic data collectors Radar guns Weight-in-Motion • Internal devices (in-vehicle technology) GPS data collection devices Car chip collectors Speed transducer collectors Virginia Tech 7 Loop Detectors • Hardware/software application • Measures traffic volume, time stamp, speed, gap source: Sensource source: Jamar Technologies Virginia Tech 8 Traffic Data Collectors • Measure traffic data at intersections (turning movements), vehicle delays, queue lengths, saturation flows • Can be connected to software to expedite the analysis TDC-12 source: Jamar Technologies Virginia Tech 9 RADAR/LIDAR Technology • Measures spot speed (instantaneous speed) • Used in law-enforcement and also in traffic studies RADAR System LIDAR System source: Stalker Radar Systems Virginia Tech 10 Video Traffic Monitoring • Used in incident detection (hardware/software) • Can measure real-time (or stored) traffic data; including volume, occupancy, speed and vehicle class over time source: Autoscope Systems Virginia Tech 11 Vehicle Collection (Car Chip) • Measures up to five vehicle parameters • Good to monitor driver behavior (or traffic analysis) • Downloads data to a PC source: http://www.thecarchip.net/ Virginia Tech 12 Weight-in-Motion (WIM) Devices • Measure vehicle weight • Good to measure road infrastructure use and deterioration source: Virginia Tech Transportation Institute Virginia Tech 13 GPS Data Collectors • Measure vehicle performance parameters • Good for traffic behavioral studies • Used in vehicle tracking (fleet applications) source: cybergraphy Virginia Tech 14 Sample GPS Data • Sample data collected in Phoenix, AZ • Using Global Positioning System technology • Smoothed speed is sometimes necessary to remove data outliers Time (s) 0 2 4 6 8 10 12 14 16 18 Smoothed Speed (km/h) Speed (km/h) 41.43 41.43 41.43 41.43 52.11 45.17 55.60 48.82 58.92 52.35 60.76 55.30 61.86 57.59 66.28 60.63 65.73 62.42 65.91 63.64 Acceleration (m/s2) 0.00 0.00 0.52 0.51 0.49 0.41 0.32 0.42 0.25 0.17 Virginia Tech Fuel (l/s) HC (mg/s) CO (mg/s) NO 0.00000 0.00000 0.00000 0.00268 2.36032 37.23742 0.00344 3.12970 56.08933 0.00363 3.40516 63.20015 0.00381 3.69682 70.53934 0.00386 3.83389 73.50874 0.00385 3.89856 74.39976 0.00421 4.46872 88.83619 0.00401 4.29287 82.74030 0.00395 4.27326 81.33448 15 Sample GPS Car Data Data collected in Phoenix driving on arterial roads 100 Raw Speed 90 Smoothed Speed 80 Speed (km/h) 70 60 50 40 30 20 10 0 0 100 200 300 400 500 600 700 800 Time (s) Virginia Tech 16 Phoenix Car Data (Detail) Data collected in Phoenix driving on arterial roads 100 Raw Speed Smoothed Speed 90 80 Speed (km/h) 70 60 50 40 30 20 10 0 100 150 200 250 Time (s) Virginia Tech 17 Sample GPS Car Data Data collected in Phoenix driving on arterial roads 100 Raw Speed 90 Smoothed Speed 80 Speed (km/h) 70 60 50 40 30 20 10 0 0 100 200 300 400 500 600 700 800 Time (s) Virginia Tech 18 Sample Traffic Data Data collected at various locations Holland (Beltway) Germany (Autobahn) U.S. (I-4) These plots demonstrate how speed (u), density (k) and flow (q) are related Virginia Tech 19 Highway 401 Data (U.S.) The data shows the basic relationship between speed (u) and flow (q) Data courtesy of Dr. H. Rakha (VTTI) Virginia Tech 20 Highway 401 Data (U.S.) The plot shows the basic form of the density-speed relationship Data courtesy of Dr. H. Rakha (VTTI) Virginia Tech 21 Highway 401 Data (U.S.) • Basic Density vs. Flow Relationship Data Data courtesy of Dr. H. Rakha (VTTI) Virginia Tech 22 Autobahn Data (Germany) The data shows the basic relationship between speed (u) and flow (q) Data Data courtesy of Dr. H. Rakha (VTTI) Virginia Tech 23 Autobahn Data (Germany) The plot shows the basic form of the density-speed relationship Data courtesy of Dr. H. Rakha (VTTI) Virginia Tech 24 Autobahn Data (Germany) • Basic density (k) vs. flow (q) relationship Data Data courtesy of Dr. H. Rakha (VTTI) Virginia Tech 25 Macroscopic Traffic Flow Models We cover two basic models: • Greenshield • Greenberg In all traffic flow models, the following fundamental traffic flow equation applies, q = u⋅k (1) where: q is the traffic flow (vehicles/hr per lane), u is the flow speed (km/hr) and k is the flow density (vehicles per lane-km) Virginia Tech 26 Greenshield’s Model (circa 1936) Assumes a linear relationship between flow speed (u) and flow density (k) u Speed[km/h] Free-flow speed uf k u = u f ⋅ 1 – -- k j Jam density kj k Density [veh/km-lane] k q = u ⋅ k = u f ⋅ 1 – --- ⋅ k k j Virginia Tech (2) 27 Greenshield’s Model 2 k q = u f ⋅ k – ----- kj (3) Maximum flow Flow[veh/h] k2 k – --- q = uf ⋅ kj qm Jam density Density [veh/km] 0 km kj k where: k m is the density for maximum flow and q m is the maximum flow. Virginia Tech 28 Greenshield’s Model The condition for maximum flow (qm) is achieved when, km um = kj -2 = uf ---- (4) 2 Then qm = uf kj --------4 (5) Prove this relationship using calculus. Virginia Tech 29 Greenberg’s Model (circa 1959) Assumes a nonlinear relationship between flow speed (u) and flow density (k) k ---j u = c ⋅ ln k (6) kj q = u ⋅ k = c k ⋅ ln --- k (7) where: k j is the jam density, c is a model constant (later to be proven the speed for maximum flow), u is the space mean speed (just like in other models) and k is the flow density. (8) Virginia Tech 30 Greenberg’s Model Use calculus to prove that, km = kj --e (9) is the density for maximum flow. You can also prove that the speed for maximum flow occurs at, um = c (10) Virginia Tech 31 Greenberg’s Model Since the relationship q = u ⋅ k for all traffic flow conditions, the condition for maximum flow (qm) is, ck j qm = ------ (11) e Prove this relationship using calculus. Virginia Tech 32 Example 1 The Blacksburg Middle school board hires you as a transportation engineer to ease complaints from parents driving vehicles and making a left turn to the school entrance during the peak hour in the morning (see Figure 1). The road is divided and has a left turn queueing island allowing cars to stop before making the turn. Measurements at the road by the town engineer indicate that traffic flow in this section has a jam density of 70 veh/km-lane and the free flow speed of 50 km/hr (restricted by the speed limit). Assume Greenshield’s model traffic flow conditions hold true. Virginia Tech 33 School Entrance To Blacksburg R = 12 m. Spacing = S Car Traffic Counters Car To Radford FIGURE 1. Blacksburg Middle School Traffic Situation. The typical acceleration model for a car is known to be: a = 4.0 – 0.1 V where: a is the acceleration of the car (in m/s2) and V is the vehicle speed in m/s. During the morning peak period, traffic counters at the site measure an average of 20 Virginia Tech 34 vehicles per kilometer per lane traveling from Radford to Blacksburg (see Figure 1). a) Find the typical spacing (S) and the average headway (h) between vehicles traveling from Radford to Blacksburg during the peak morning period. b) Find if the average headway (h) allows a typical driver to make a left turn if the driver has a perception/reaction time of 0.5 seconds. The radius of the curve to make a left turn is 12 meters. According to AASHTO standards, the critical vehicle length is 5.8 meters. Virginia Tech 35 Solution to Part (a) Find the Spacing (Sp) between vehicles. Since the density of the traffic flow is known to be 20 veh/km-la we compute the spacing as the reciprocal of the density 1 1 S p = -- = ----- = 0.05 kilometers k 20 Sp = 50 meters To find the headway we need to figure out how fast the cars are traveling on the road. We use Greenshield’s model to estimate the speed when k = 20 veh/km-la. Virginia Tech 36 Space Mean Speed vs. Density Diagram u (km/hr) uf = 50 km/hr (13.89 m/s) kj = 70 veh/km-la 50 37.5 0 70 20 k (veh/km-la) u 50 u = u f – ---f k = 50 – ----- ( 20 ) = 35.71 km/hr kj 70 Virginia Tech 37 Solution to Part (a) Traveling at 35.71 km/hr (9.92 m/s) the headway (h) between successive cars is, 50 h = --------- = 5.04 seconds 9.92 Virginia Tech 38 Solution to Part (b) To check if the turning vehicle can make a safe maneuver, check the time to turn against the headway (h) calculated in part (a). Account for the reaction time of the turning vehicle. The time available to execute a safe turn is (h) - 0.5 seconds to account for reaction time, t available = 5.04 – 0.5 = 4.54 seconds Technically we should use the gap between two successive vehicles to estimate the time to turn left. In this case we have to subtract the time traveled by the oncoming vehicle to cover its car length at 9.92 m/s Virginia Tech 39 t gap 5.8 = 5.04 – 0.5 – --------- = 3.96 seconds 9.92 The distance traveled by a vehicle with a linearly-varying acceleration model is, –k t –k t k1 v0 k 1 t ---S = ----- – - ( 1 – e ) + --- ( 1 – e ) 2 k2 k2 k2 2 2 Note that t is either 3.96 or 4.54 seconds (depending on your assumption on when the stopped vehicle starts the left turn). Using values of k 1 , k 2 of 4 and 0.1, respectively, the left turning vehicle travels 35.6 meters in 4.54 seconds and 28 meters in 3.96 seconds. Virginia Tech 40 A plot of distance traveled vs. time is shown in the following diagram. The total distance to be traveled in the left turn maneuver to reach a safe point is, 2πR 2 π ( 12 ) d = --------- + L = ----------------- + 5.8 = 24.65 meters 4 4 The vehicle can execute the turn safely. Virginia Tech 41 Distance vs. Time Profile (Turning Car) The car reaches 24.65 m in 3.73 seconds The car travels 35.65 m in 4.54 seconds Virginia Tech 42 Differentiation of Speeds Used in Traffic Analysis Two types of speed sused in traffic analyses: • time-mean speed • space-mean speed The time-mean speed u is defined in the following way: t N 1 u t = --- ⋅ N ∑u (12) i i=1 where u represents recorded speed of the i-th vehicle. i We see that the time-mean speed can be calculated by calculating the arithmetic time mean speed. Virginia Tech 43 Space Mean Speed The space-mean speed is the average speed that has been used in the majority of traffic models. Let us note a section of the highway whose length equals D. We denote by t the time needed by the i-th vehicle to travel along this highway section. The space-mean speed u is defined in the following way: i s D u s = ------------------- = D --N t 1 --- ⋅ ti N (13) ∑ i=1 Virginia Tech 44 N 1 The expression --- ⋅ ∑ t represents the average travel time t i N i=1 of the vehicles traveling along the observed highway section. Virginia Tech 45 Example 2 Measurement points are located at the beginning and at the end of the highway section whose length equals 1 km (see figure below). The recorded speeds and travel times are shown in the Table. 1 km A Measurement points Virginia Tech B 46 Example 2 Table 1. Recorded speeds and travel times. Vehicle number Speed at point A [km/h] Travel time between point A and point B [sec] 1 80 45 2 75 50 3 62 56 4 90 39 5 70 53 Speeds of the five vehicles are recorded at the beginning of the section (point A). The vehicle appearance at point A and point B were also recorded. a) Calculate the time-mean speed and the space-mean speed. Virginia Tech 47 Solution to Example 2 The time-mean speed u at point A is: t N 1 u t = --- ⋅ N ∑u i i=1 1 1 km u t = -- ⋅ ( 80 + 75 + 62 + 90 + 70 ) = -- ⋅ 377 = 75.4 -----5 5 h Virginia Tech 48 Solution to Example 2 The space-mean speed represents measure of the average traffic speed along the observed highway section. The space-mean speed is: D u s = ------------------N 1 --- ⋅ ti N ∑ i=1 The total travel time for all five vehicles is: 243 45 + 50 + 56 + 39 + 53 = 243 [ sec onds ] = ----------- [ h ] = 0.0675 [ h ] . 3600 Virginia Tech 49 The space-mean speed is calculated as: 1 km - km u s = ---------------------------- ------ = 74.07 -----1 h h -- ⋅ ( 0.0675 ) 5 Virginia Tech 50 ...
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This note was uploaded on 12/31/2011 for the course CEE 3604 taught by Professor Katz during the Fall '08 term at Virginia Tech.

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