Adaptive Cruise Control: Towards higher traffic flows, at the cost of increased susceptibility to congestion

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3 x n 1( + ) α [x n 1( + )] [x n ( ) x n 1 ( )] (x n( ) x n 1( )) (4) where x n 1 ( ) denotes the position of the vehicle entering the cluster, x n ( ) denotes the position of the tailing vehicle in the cluster, and α denotes the sensitivity of the driver of the vehicle entering the cluster. The headway x n ( ) x n 1 ( ) is represented by h( ). The range of driver sensitivities is determined using typical traffic conditions and comfortable deceleration standards set by AASHTO. The typical traffic flow is assumed to have free flow velocity of about 25 m/s (about 55 miles/hour), free headway of about 100 m, and cluster velocity of about 0-2 m/s. Further, the maximum permissible deceleration is limited to 3.4 m/s 2, according to AASHTO standards. Using these values, the process of a vehicle joining a cluster is simulated to obtain the maximum observed deceleration. The simulation outputs are included in figure 2(a) for two different driver sensitivities. The simulation is repeated for various values of driver sensitivities and the acceptable driver sensitivities are determined to fall in the approximate range [0.4, 0.65] (Figure 2(b)). join v c { m=1 m mα (v c k ) 1 mα (h free 1 h mα cluster )} (5) where k is a driver dependent constant and v c is the velocity of the tailing vehicle in the cluster. However, it is observed that as an increasing number of terms is included in calculating join, the hypergeometric series converges quickly to the true solution obtained from numerical simulation (Figure 3(a)). An additional key insight of this paper is to recognize that the hypergeometric series is constrained by the range of permissible driver sensitivities. Considering the range of permissible driver sensitivities, it is observed that one can approximate a closed-form solution by using the first term of the series ( 1 ) and an appropriate truncation ratio ( T R ), such that join T R 1. For typical traffic conditions, the truncation ratio (T R ) is determined to be about 1.4 (Figure 3(b)). Fig. 2 (a) Maximum observed deceleration during simulation of a vehicle joining a cluster with typical traffic conditions. (b) Maximum observed deceleration with varying driver sensitivities. Next, the equation of the car-following model is solved to determine the time taken to join the cluster ( join ). The resulting expression for join is a hypergeometric series with no closed-form solutions: Fig. 3 (a) Time to join a cluster theoretical expression converges quickly to the numerical solution (b) Range of permissible driver sensitivities limits the variation of truncation ratio, T R join 1. The transition probability rate for joining the cluster is thus given by the inverse of time taken to join a cluster, as shown in equation (6). The growth dynamics for the cluster are determined as proposed in [13], but the

6 at which clusters begin to form. The results from the Monte Carlo simulation appear to match quite well with the theoretical analysis, as shown in Figure 7. Fig. 7 Results from the Monte Carlo simulation appear to agree with the theoretical analysis. Colormap indicates the total number of vehicles inside a cluster. 7. CONCLUSIONS The study has shown that as the percentage of ACC vehicles in the traffic system is increased the critical density also increases correspondingly. The increase in critical density implies that the density at which vehicle clusters begin to spontaneously appear is increased. This indicates that the traffic flow can operate at higher densities and consequently higher flow rates, since it is known from the fundamental diagram of traffic flow that, in the free flow regime, the flow increases as the density increases. Additionally, while increased ACC penetration may allow the traffic system to operate at increased densities and flows, it comes at a cost. As ACC penetration increases, a small percentage of drivers with low sensitivities are enough to cause a self-organized traffic jam. In other words, in a predominantly ACC traffic system, introduction of a small percentage of human drivers may cause a rapid reduction of critical density, resulting in a self-organized traffic jam. [6] P. Seiler, B. Song and J. K. Hedrick, Development of a collision avoidance system, SAE SPEC. PUBL., vol. 1332, pp , [7] S. Darbha and K. R. Rajagopal, Intelligent Cruise Control Systems and Traffic Flow Stability, California Partners for Advanced Transit and Highways, [8] J. Zhou and H. Peng, Range Policy for Adaptive Cruise Control Vehicles for Improved Flow Stability and String Stability, IEEE Transactions on Intelligent Transportation Systems, vol. 6, pp , [9] P. A. Ioannou and C. C. Chien, Autonomous intelligent cruise control, 1993, IEEE Transactions on Vehicular Technology,vol. 42, pp , [10] P. J. Zwaneveld and B. van Arem, Traffic effects of automated vehicle guidance systems, Department of Traffic and Transport, Netherlands Organization for Applied Scientific Research, [11] S. E. Shladover, Potential freeway capacity effects of Advanced Vehicle Control Systems, Proc. Of 2nd International Conference on Applications of Advanced Technologies in Transportation Engineering, Minneapolis, Minnesota, USA, [12] K. Nagel and M. Paczuski, Emergent traffic jams, Physical Review E,vol. 51, pp , [13] R. Mahnke and N. Pieret, Stochastic master-equation approach to aggregation in freeway traffic, Phys. Rev. E,vol. 56, pp , [14] R. Mahnke and J. Kaupužs, Stochastic theory of freeway traffic, Phys. Rev. E, vol. 59, pp , [15] J. Schmelzer, G Röpke, R. Mahnke, Aggregation Phenomenon in Complex Systems, Wiley-VCH, [16] A. Kesting et al. Jam-avoiding adaptive cruise control (ACC) and its impact on traffic dynamics [ed.] A Schadschneider, et al. Traffic and Granular Flow, pp , Berlin : Springer, [17] A. D. May, Traffic Flow Fundamentals, Prentice- Hall, Inc., REFERENCES [1] Federal Highway Administration (FHWA), Our nation's highways Washington D.C. : US DoT, [2] D.L. Schrank and T.J. Tomax, The 2007 Urban Mobility Report, Texas Transportation Institute, [3] B.S. Kerner and P Konhäuser, Cluster effect in initially homogeneous traffic flow, Phys. Rev. E, vol. 48, pp , [4] Y. Sugiyama et al., Traffic jams without bottlenecks - experimental evidence for the physical mechanism of the formation of a jam New Journal of Physics, vol. 10., [5] D.C. Gazis, R. Herman and R.B. Potts, Car-Following Theory of Steady-State Traffic Flow, Operations Research, vol. 7, pp , 1959.

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