Microscopic Density Characteristics Microscopic density are represented by
Microscopic Density Characteristics • Microscopic density are represented by the longitudinal spacing characteristics • Chapter focus is spacing • Particularly important to safety, capacity, and LOS – Minimum spacing for safety – Spacing v. s. capacity (do not forget speed) – Larger spacing better LOS
Spacing=Distance Headway • The longitudinal space occupied by individual vehicles consists of: – (1) the physical space occupied by the vehicle (vehicle length), and – (2) the distance gap (clearance) between the vehicle and either the vehicle immediately ahead or immediately behind the subject vehicle. – dn+1(t) = distance headway of vehicle n+1 at time t (feet), – Ln = physical length of vehicle n (feet), and – gn+1(t) = distance between vehicle n and vehicle n+1 at time t (feet) Note: Spacing and clearance are more commonly used in newer literature
Distance Headway vs. Time Headway d n+1 = hn+1 x v n • hn+1 = time headway of vehicle n+1 at point p (seconds), and • vn = speed of vehicle n during the time period hn+1 (fps) • The time headway is easier to measure by a roadside observer, and the distance headway can then be calculated
Density vs. Spacing • Density is defined as the number of vehicles occupying a single lane of a length of roadway over a distance of one mile • Unit is vehicles per mile (per lane) • dn: individual distance headway (spacing) • N: number of observed distance headways
Spacing and Car-following • Car-following leads to spacing characteristics • Car-following considerations: – Spacing in front – Safety – Desired speed – Separation w. r. t. speed and flow
Car-Following • Follow-the-leader model started in 1950 s • Currently 3 categories of models: – safe-distance models – stimulus-response models – psycho-spacing models
Car-Following Notations
Car-Following Model Basics • Acceleration/deceleration of the trailing vehicle is specified at time t + Δt (not t). The Δt term is the interval of time required for the driver of the trailing vehicle to decide that acceleration/deceleration is necessary. This element of time can be termed as a reaction time. • The distance headway is simply the distance between the lead and trailing vehicles at the same point in time, • The relative velocity of the lead and trailing vehicles is simply the difference of vehicle velocities at the same point in time. A positive relative velocity indicates that the lead vehicle is pulling away from the trailing vehicle, and a negative relative velocity indicates that the trailing vehicle is gaining on the lead vehicle, and • Acceleration can be positive or negative (deceleration).
Safe-Distance Car-Following Models • Safe-distance car-following models describe the dynamics of a single vehicle in relation to its predecessor. – Pipes’ Model – Leutzbach Model – Forbes’ theory
Pipes’ Model • Pipes’ Model (1953): “A good rule for following another vehicle at a safe distance is to allow yourself at least the length of a car between you and the vehicle ahead for every ten miles an hour (16. 1 km/hr) of speed at which you are traveling” For a vehicle length of 20 ft:
Pipes’ Model • Min. spacing changes with speed • Min headway does not change as much • Is hmin too large?
Leutzbach Model • Leutzbach (1988) discusses a more refined model describing the spacing of constrained vehicles in the traffic flow. • He states that the overall reaction time T consists of: – perception time (time needed by the driver to recognize that there is an obstacle); – decision time (time needed to make decision to decelerate), and; – braking time (needed to apply the brakes). T: reaction time
Forbes’ Theory • Consider two successive vehicles: if the first vehicle stops, the second vehicle only needs the distance it covers during the overall reaction time T with unreduced speed, yielding Forbes’ model. For a vehicle length of 20 ft and a reaction time of 1. 5 sec: Dt: reaction time
Forbes’ Theory • Min. spacing changes with speed • Min headway does not change as much • Is hmin too large?
Smaller hmin?
Stimulus-Response Car-Following Models • In general, the response is the braking, deceleration, or the acceleration of the following vehicle, delayed by an overall reaction time T. • Stimuli – Spacing – Speed Difference – etc
Chandler (1958), Gazis (1961) Chandler et al : where vn (t) and an(t+T) respectively denote velocity and acceleration of vehicle n at t and t+T, and g denotes the driver’s sensitivity. Thus, the stimulus is defined by the velocity difference between leader and follower. Gazis et al. : Thus, the following vehicle adjusts its velocity vn(t) proportionally to both speed and distances differences with delay T. The extent to which this occurs depends on the values of c, l and m.
GM Models- 1 st Generation 1 st model only considers relative speed
GM Models- 1 st& 2 nd Generations • 2 nd model proposed 2 states for sensitivity factor, higher value for low spacing (Fig 6. 5)
GM Models- 3 rd Generation 3 rd model considers spacing
GM Models- 4 th Generation • Improving the sensitivity by introducing the speed of the following vehicle.
GM Models- 5 th Generation 5 th model considers nonlinear responses
Psycho-Spacing (Psycho-Physical) models • The basic behavioral rules of such so-called psycho-spacing models are: At large spacings, the following driver is not influenced by velocity differences. • At small spacings, some combinations of relative velocities and distance headways do not yield a response of the following driver, because the relative motion is too small.
Psycho-Physical Models • Models based on driver’s perception • Driver switch from one model to another when a certain threshold is reached • Wiedemann (1974) developed a Psycho. Physical model, that was used in VISSIM later
Other Car-Following Models • Optimal Velocity Model – Follower’s acc/dec depends on his/her speed and the optimal speed he/she can attain • Fuzzy logic models – Based on human perception of environment, rules such as “if distance divergence is too far and relative speed is closing then no action” • Neural Network Models
Traffic Stability • The stability of the traffic stream can be defined by the reaction time being used. • High reaction times (sluggish behavior) generally result in exaggerated responses: acceleration/deceleration rates. This would define regions of "unstable" behavior, characterized by high values of reaction time and sensitivity response. • "Stable" behavior is further characterized by moderate values of both reaction time and sensitivity response. • If the product (C) of these two parameters (Δt and α) is high, unstable conditions are likely to occur, while if the product is low, stable traffic conditions will occur. • Related Paper: Influence of Reaction Times and Anticipation on the Stability of Vehicular Traffic Flow, Treiber, Martin ; Kesting, Arne ; Helbing, Dirk, Transportation Research Board 86 th Annual Meeting, 2007
Traffic Stability • The GM research defined two regions of traffic stability: – Local stability: involved with the car-following behavior of just two vehicles. – Asymptotic stability: concerned with the car-following behavior of a line of vehicles. • In order to define boundary conditions for these two levels of stability, the product (C) of reaction time (Δt) and sensitivity (α) was calculated and tested in the field. • The resulting values of C are provided in Table 6 -4 (page 180).
Traffic Stability
Stability Illustrated • http: //www. youtube. com/watch? v=Suugnp 5 C 1 M
Car-Following Data • “Wired vehicles” in 1950
Car-Following Data Single Loop Detector
Car-Following Data Double Loop Detector
Data from Controlled Experiments (Example) • Single Lane, 1080 m • No Overtaking • No Considering of Curvature • 85 vehicle gradually Enter/ Exit every 20 s • Detectors at 270 m, 540 m, 810 m • Increased demand stage (1700 s) • Constant demand stage (200 s) • Decreased demand stage (1700 s) Source: J. Wang, University of Leeds
Model Development • Data, data • Calibration of parameters – Minimizing errors based on a selected variable (micro/macro) – MLE, GA, etc. • Calibration vs. validation
Other Car-following Studies • Other data collection devices, such as GPSs • Other models, such as NNs, Fuzzy Logic, – Car-Following Model Based on Artificial Neural Networks in Urban Expressway Sections, Liu, Xiaoming; Wang, Li; Zhong, Xiaoming , Transportation Research Board 85 th Annual Meeting, 2006 – Study on Chaos Model of Expected Space Headway in Urban Expressway, Wang, Li; Yang, Xiaokuan; Zhang, Zhiyong; Du, Zhencai , Transportation Research Board 85 th Annual Meeting, 2006 – P. Chakroborty and S. Kikuchi. Calibrating the membership functions of the fuzzy inference system: instantiated by carfollowing data. Transportation Research Part C, 11: 91– 119, 2003. – C. Kikuchi and P. Chakroborty. Car following model based on a fuzzy inference system. – Transportation Research Record, 1365: 82– 91, 1992.
Lane Change • Lane change also affects headway • Car following and lane change are core logic of microscopic simulation • Car-following model may consider lane change; lane change should consider carfollowing – “ghost vehicle” during lane changing process in simulation
Lane-changing?
New Logic for Autonomous Vehicles • ACC: adaptive cruise control – automatically adjusts the vehicle speed to maintain a safe distance from vehicles ahead – On-board sensors (Radar or Lidar) • CACC: corporative adaptive cruise control – the preceding vehicle's acceleration is used – obtained from the Cooperative Awareness Messages it transmits using DSRC or WAVE • Lane changes • Truck platooning
Other New Developments • Speed, position data from Connected Vehicle (CV) technologies • New data, new model, new parameters (i. e. sensitivity, reaction time, etc. ) • Calibration and validation?
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