MIXED TRAFFIC DENSITIES

Transcription

1 em feature Thinkstock Photo Estimation of MIXED TRAFFIC DENSITIES in Congested Roads Using Monte Carlo Analysis by Brian Freeman, Bahram Gharabaghi, and Jesse Thé Introducing a novel stochastic Monte Carlo method to estimate the number and type of vehicles on congested road sections. Brian Freeman, P.E., PMP, QEP, is team leader of air regulatory management for the United Nations Development Program s Kuwait Integrated Management Project, and a Ph.D. candidate with the University of Guelph, Ontario, Canada; Bahram Gharabaghi, Ph.D., P.Eng, is an associate professor of engineering at the University of Guelph, Ontario, Canada; and Jesse Thé, Ph.D., P.Eng, is president and founder of Lakes Environmental Software Inc. and a professor at the University of Waterloo, Canada. jesse. the@weblakes.com. Vehicle congestion is a serious problem throughout the world that impacts transport, health, and communication infrastructure. Estimating the number and type of vehicles in a traffi c jam can greatly assist planners and modelers interested in specifi c areas of emission impacts and evaluating secondary congestion outcomes, such as lost time/ productivity, cellular bandwidth requirements, and emergency response to large accidents. A Novel Approach The authors of this article developed a novel stochastic Monte Carlo approach to estimate the number and type of vehicles on congested road sections. This method assumes that each vehicle occupies road space based on its length and inter-vehicle gap during congested traffi c. The inter-vehicle gap is subject to variability due to driver behavior and average speed. By assigning vehicle spaces on a road based on the average speed, each vehicle can be treated as an independent variable using Monte Carlo simulation to identify ranges of possible outcomes. Under multiple sampling, the most likely number of mixed vehicles in a 1-km unit road length can be represented by the mode or median of the distribution of results, which is normal due to the Central Limit Theorem. Once the modes of different speeds are calculated, vehicle density curves can be estimated for combined traffi c and individual vehicle types. Various models have been used to estimate traffi c on roads, including statistical models, 1 Kalman fi l t e r s, 2,3 and neural networks. 4 These models looked at how traffi c fl owed over time to assess traffi c management strategies and required complexcomputations and historical data to calibrate 8 em april 2015 Copyright 2015 Air & Waste Management Association

2 th ANNIVERSARY istock.com/pgiam the necessary equations for a specifi c stretch of road. Monte Carlo methods have been used to validate results of traffi c fl ow models, 5 but not to generate results. These models look at traffi c fl ow under various conditions and not at the extreme condition of grid lock traffi c. Under this state, a simpler model can be used. L RS = L + IVG IVG (1) Assumptions Vehicles in congested roads move at homogenous speeds, due to the lack of options available to individual drivers. Assuming that all vehicles follow an average speed in slow moving, grid-locked traffi c is fundamental to this model. Fast moving traffi c (i.e., > 40 km per hour, or KPH) is assumed to be free fl owing and not applicable to this model. The model accepts different road sections with different mixes of vehicles. For example, a highway near residential areas would have more sedans and sport utility vehicles (SUVs), while a road near a port or industrial zone would have more heavy goods vehicles and multi-axle rigs. Methodology The number of vehicles on a given unit road length depends on the length of the vehicle (L) and the space cushion a driver keeps from the car in front, or the inter-vehicle gap (IVG). The recommended IVG is around 2 3 seconds at the vehicle speed. 6 For example, at 120 KPH, this RS (5 KPH) = 7.8 m (5 KPH * 2 sec) RS (40 KPH) = 27.4 m (40 KPH * 2 sec) (2) Figure 1. Required road space for a vehicle. Figure 2. Two-second vehicle spacing at 5 and 40 KPH. Copyright 2015 Air & Waste Management Association april 2015 em 9

3 (3) (4) % CI 90% CI 5% CI Meters Probability (P) Probability (P) Vehicle Class (5) (6) % CI 90% CI 5% CI # of Vehicles Seconds Figure 3. Possible SUV road space at 5 KPH. Figure 4. Probability distribution of observed vehicles. Figure 5. Probability distribution of IVG timing. Figure 6. Total number of vehicles estimated on 1-km road at 5 KPH. Tab le 1. Vehicle types. represents 67 m, while at 5 KPH, this represents 2.8 m. For an SUV with a length of 5 m traveling at 5 KPH, the most likely road space (RS) required to operate the vehicle is L + IVG = 7.8 m, as shown in Figure 1. Figure 2 shows the difference between vehicles spacing at different speeds assuming a 2-sec IVG, at 5 KPH and 40 KPH. The total number of vehicles (n) on 1-km road moving at the same speed can be estimated by summing the number of individual vehicle lengths (L i ) and individual IVG (IVG i ): n # of Vehicles (n), where (RS ι ) 1,000 meters =1 Both IVG and L are independent variables subject to a wide range of values. A vehicle s length may average from 1.8 m for a sedan, and up to Vehicle Class Vehicle Type Company Model Year Most Likely Length (m) Gross Vehicle Mass (kg) Fuel Type Probability (P) 1 Sedan Honda Civic LX ,650 Petrol SUV Toyota Prado VX ,990 Petrol Bus, Midsize Toyota Coaster ,180 Diesel Bus, Large Tata Starbus ,860 Diesel em april 2015 Copyright 2015 Air & Waste Management Association

4 th ANNIVERSARY (7) 725# (8) 7E5# 745# % CI 90% CI 5% CI # of Vehicles 765# 755# 25# E5# 45# 65# 5# 5 KPH 10 KPH 15 KPH 20 KPH 40 KPH Buses, Large Buses, Medium SUVs Sedans # of Sedans per 1,000 m (9) (10) KPH Total Number of Vehicles y = 509x R² = Total Number of Sedans y = x R² = Average Speed (KPH) Average Speed (KPH) Figure 7. Number of vehicles at different speeds. Figure 8. Number of sedans estimated on 1-km road at 5 KPH. Figure 9. Total mixed vehicles on 1-km road based on average speed. Figure 10. Total number of sedans on 1-km road based on average speed. 9.7 m for a large bus. IVGs are independent of the vehicle due to driver behavior and changes in speed due to the vehicle traveling ahead. At 5 KPH, IVGs range from 0.5 to 4 m. The range of possible road space used by a 5-m long SUV may vary, as shown in Figure 3. This is especially apparent at lower speeds (i.e., < 20 KPH) due to stop-and-go driving patterns. Specifi c road use is important when estimating the types of vehicles in a sample population of vehicles. During model development, four classes of vehicles were used. Vehicle classes were selected to represent existing traffi c based on observations in our case study city, Kuwait City, as shown in Table 1. Initial probabilities of occurrence (P) were assigned to a discrete probability distribution, as shown in Figure 4, such that the total probability to select a vehicle was 1. The number and type of vehicle classes can be expanded to account for better classifi cation, such as engine size, weight, fuel types, and age. A discrete algorithm was set up in a spreadsheet for multiple speeds ranging from 5 to 40 KPH. Table 2 shows speed sets and conversion to meters KPH m/sec Table 2. Modeled speeds. Copyright 2015 Air & Waste Management Association april 2015 em 11

5 Class Type Road Space #Vehicles #Sedans #SUVs #Buses, Medium #Buses, Large Vehicle 1 1 Sedan Vehicle 2 2 SUV Vehicle 3 1 Sedan Vehicle 4 3 Bus, Medium Vehicle 5 1 Sedan Vehicle 6 2 SUV Vehicle 7 1 Sedan Vehicle 8 1 Sedan Table 3. Sample of an iteration showing vehicle class and road space selection. Table 4. Expected number of vehicles by class in 1-km for profile. Vehicle Class per second. We modeled speeds less than 5 KPH as 5 KPH due to the IVG maintained by drivers at lower speeds. We assigned vehicle spaces based on a maximum number of 401 vehicles possible on a 1-km road moving at 5 KPH. This maximum value assumes that only sedans are on the road driving at the smallest possible safe distance. During modeling, however, the number of vehicles in the same stretch of road never exceeded 170. Safe IVG timing values were assumed to range from 0.5 seconds to 2 seconds and 4 seconds (maximum) in a continuous triangle distribution, as shown in Figure 5. Each vehicle length was assigned its own probability distribution using a pert distribution and vehicle manufacturer data. We ran our stochastic model using 5,000 iterations on each variable. During each iteration, a vehicle class was randomly selected from the 4 classes for each space. The vehicle length was then selected based on the class of vehicle. The safe distance was added to the vehicle length by randomly selecting a time spacing and multiplying it by the average speed to get the safe distance. If the cumulative length was less than 1,000 m, the class was assigned a value of 1 Expected Number of Vehicles in 1-km Road Sedan # of Sedans = Integer (324.32x ) SUV # of SUVs = Integer (179.11x ) Bus, Medium # of Medium Buses = Integer (12.55x ) Bus, Large 1 to allow tallying and grouping. Vehicle classes at the end of the list that exceeded the 1-km length were assigned a zero and not counted. Table 3 shows a portion of an iteration at 5 KPH. Results Palisade Software Version 6.2 Industrial Edition 7 was used to provide the Monte Carlo analysis using a Latin Hypercube sample generator. A total of 5,000 iterations were run at 5, 10, 15, 20, and 40 KPH at the same time. Our model captures the total number and class of vehicles in 1 km of road. Lane changing was not considered. Figure 6 shows the results of all vehicles traveling at an average of 5 KPH, including 90% confidence intervals. The expected number, and average, of total vehicles is 155 with a standard deviation of 4 vehicles. The calculated mean of different types of vehicles at different speeds are shown in Figure 7. Figure 8 shows the distribution of sedans traveling at 5 KPH. Graphing the statistical mean for the total number of vehicles over different average speeds yields a power curve, as shown in Figure 9. Fitting the curve with a power series trend line provides very high correlation (R 2 ) that can approximate the expected value at each speed. Similar curves (mean of each speed) for each vehicle class were prepared and summarized in Table 4. A curve for large buses was not included because the expected value at each speed is 1. Table 4 summarizes the expected number for vehicles by class in 1 km. This is the integer value of the equation, where x equals the average traffic speed in KPH. Figure 10 shows a power curve for sedans. 12 em april 2015 Copyright 2015 Air & Waste Management Association

6 th ANNIVERSARY Portable. Affordable. Reliable. Complete Air Monitoring System The Haz-Scanner measures and documents trace level (ppb) gas, particulates & meteorological parameters in real-time to US EPA & EU directives. Confgure up to 12 sensors with true simultaneous PM-2.5 & PM-10 readings. Custom Sensor Calibrations to Meet Your Needs Build your own system to your specifc a pplication(s). Wireless Networking Interface multiple systems 24/7 with cell phone alerts & remote global access to data without Cloud-based subscriptions. Battery, AC, or Solar Option Contact Environmental Devices or our distributor, SKC inc., for more information. MADE IN THE USA HazScanner.com DISTRIBUTED BY The results in Table 4 are representative only of that section of road with the traffi c profi le in Table 2. Different traffi c profi les will have different densities and results. Other factors that could affect the traffi c profi les include location of road section, type of road, season, time of day, weather conditions, and road construction activities. Future Investigations We developed a stochastic procedure to estimate the number of vehicles by class in a 1-km stretch of road at various speeds. This procedure can be linked to vehicle emission data to estimate realtime mobile source pollution on sections of road due to traffi c conditions (i.e., congested vs. free fl owing). Additional Monte Carlo analysis can be used to estimate the emissions from individual vehicles given expected vehicle emission rates. Validation of this procedure includes counting vehicles during various traffi c conditions and identifying vehicle classes. This will require capturing a section of road and identifying individual vehicles types and separation at various speeds to reduce IVG uncertainty. em References 1. Schreckenberg, M., et al. Discrete stochastic models for traffi c fl ow; Physical Review 1995, E 51 (4), Pourmoallem, N., et al. A Neural-Kalman fi ltering method for estimating traffi c states on freeways. In Proceedings Japan Society of Civil Engineers, Dotoku, Gakkai, Sun, X., et al. Mixture Kalman fi lter based highway congestion mode and vehicle density estimator and its application. In Proceedings of the IEEE 2004 American Control Conference, Boston MA. 4. Ghosh-Dastidar, S.; Adeli, H. Neural network-wavelet microsimulation model for delay and queue length estimation at freeway work zones; J. Transportation Engineering 2006, 132 (4), Mihaylova, L.; Boel, R. A particle fi lter for freeway traffi c estimation. In Proceedings of the 43rd IEEE Conference on Decision and Control, See for example, New York State Department of Motor Vehicles New Driver Study Guide; defensive-driving#all-spc; and California Driver Handbook, 7. Us ers Version 6.2 Industrial Edition, September 2013; ACKNOWLEDGMENT The State of Kuwait Supreme Council for Planning and Development funded this work through the United Nations Development Programme (UNDP) and the Kuwait Environment Public Authority (KEPA). Special thanks to Dr. Sami Al-Yakoob and Shah Faisal from UNDP and Meshal Abdullah from KEPA. Copyright 2015 Air & Waste Management Association april 2015 em 13