An application of shock wave theory to urban traffic control via dynamic speed advisory

Abstract

In this work, shock wave theory is used to model traffic evolution within urban road segments between signalized intersections. The shock wave evolution depends algebraically on the boundary conditions at the signalized intersections, and on the fundamental diagram (FD) of each road segment, whereby e.g. the FD free flow speed can be seen as a control variable. Thus, the vehicular velocities, densities and queue evolutions can be described without the need for differential equations and their solution, noting that the traffic is characterized by cells, each with constant velocity and density, whose boundaries are the shock waves. The free flow speeds may be set by dynamic speed advisory control with the aim to optimize vehicles energy consumption and total time spent in the road network. In future work other variables such as green splits and phase differences between the intersection signals will be used for control and optimization purposes.

State of the Art The principal theoretical work on kinematic waves was done by Lighthill and Whitham [1955]. Given flow and density upstream and downstream

from the shock wave, its propagation velocity was calculated analytically. Furthermore, it was stated that such a velocity represents the slope of the chord joining the two points on the flow-density curve (i.e. fundamental diagram) which correspond to the states ahead of and behind the shock wave. The importance of this finding is that if the traffic states are known, then their future evolution can be easily predicted by describing the boundary (i.e. shock wave speed) between them. This insight was used by Richards [1956] for the study of shock waves on highways, and by Daganzo [1994] in the Cell Transmission Model (CTM) derivation and analysis. The first attempt to use shock wave theory for traffic control was presented by Hegyi et al. [2008] in order to propose a quick control scheme to resolve jams on highways by means of variable speed limits. Shock waves principles were also used for alternative highway traffic modeling approaches. In Canudas de Wit [2011] the proposed variable-length CTM-like model aims to capture the traffic conditions on an arbitrarily long road segment by describing the dynamics of the shock wave and its upstream and downstream traffic states. An adaptation of this model to the energy optimization by variable speed advisories in an urban traffic framework, with boundary flows enabled by cyclic traffic signals, was proposed by De Nunzio et al. [2014]. However only the evolution of upstream congestion boundaries was considered, and downstream rarefaction waves were neglected. In Canudas de Wit and Ferrara [2016] an improved variable-length model with downstream rarefaction was used for speed control on a ring road.

Contribution of this work An algebraic model of the traffic evolution within urban road sections, i.e. modeling of traffic light induced phenomena, is proposed. The model is able to provide a solution of traffic density and flow distributions without solving differential equations, and therefore the computational burden is drastically reduced. This is particularly desirable for large-scale traffic optimization. An energy consumption model and a travel time model have been adapted to the proposed traffic model and used for performance optimization in an urban scenario.

Figure 1: Urban road section and model notation.

Model Description In urban traffic, it is reasonable to assume that flows are generated by traffic lights or intersections, and hence inflows and outflows of a road segment are pulses starting at discrete time instants. Then, under such conditions, with the road segment decomposed into cells, see Fig. 1, during a finite time interval, each cell state will be defined by one point on the fundamental diagram (FD), see Fig. 2. Hence the slope between the FD points of two neighboring cells will be constant over the said finite time interval. According to the shock wave theory, this slope gives the propagation speed of the front or boundary between the cells. Hence, there is no need to solve differential equations to find neither the value of cell states which remain constant during the existence of the specific cell, nor the evolution of the cell boundaries which move at a constant speed determined algebraically. Let us consider an urban road section as in Fig. 1. It is reasonable to assume that the considered road section represents an elementary segment of the traffic network, meaning that no exogenous flows are allowed within the segment. The inflow and outflow are only enabled by the two traffic lights at the two ends of the section. The road section may be intuitively split up into cells, j = 1, . . . , k, of variable length lj [m] corresponding to different traffic states ρj [veh/m]. Each traffic state (or density) remains constant within its spatial domain,

Figure 2: Fundamental diagram. ρm is the stand-still or jam density, ρcr is the critical, or sweet-spot density, where the state changes from free flow to congested flow. ϕmax is the sweet-spot flow. v [m/s] stands for free flow velocity, and w [m/s] is the backward wave front propagation speed.

delimited by moving boundaries, e.g. boundary or front pj [m] in Fig. 1. According to the shock wave theory, over an infinitesimal time step dt [s] the front pj will move by

ρj+1 v(ρj+1 ) − ρj v(ρj ) dpj = dt (1) ρj+1 − ρj

where the symbols are defined in the caption of Fig. 2. Hence, the length lj of each cell is calculated by the difference of continuously moving fronts. In Fig. 2, the fundamental relations between speed, flow and density are shown. Note that the speed of the front in equation (1) depends only on the upstream and downstream traffic densities which remain constant. Therefore equation (1) may be rewritten in its algebraic form as:

ρj+1 v(ρj+1 ) − ρj v(ρj ) pj (t) = pj0 + · (t − tj0 ) (2) ρj+1 − ρj

where pj0 and tj0 are the starting position and time of the moving front pj , respectively. Typically, traffic dynamics and behavior present several shock waves generated for instance by traffic lights turning green or red, or by the intersection of two other shock waves. In order for these events to be predictable and for traffic dynamics to be described algebraically, the following assumptions must hold:

• the FD of the road section is known;

• the initial traffic conditions are known; • the traffic signals timings are known. Note that signalized intersections may present very complex phases and movements during a traffic light cycle, therefore the inflow to a particular road section may be determined by several simultaneous and/or consecutive movements. Thus, the green light duration enabling boundary flows may be thought of as an equivalent duration which comprises all the movements entering or leaving the section.

Performance metrics Let us introduce two optimization criteria of interest: mean vehicle energy consumption and travel time. Based on the modeled traffic conditions, vehicle trajectories can be determined by taking into account: traffic signals timings which enable boundary flows, and traffic densities and cells boundaries which determine travel speed and acceleration. The performance metrics will be evaluated on the trajectory of the vehicles completing the trip in the considered road section. If the traffic conditions are at an equilibrium (i.e. equal boundary flows), the performance may be evaluated solely on the vehicles entering the section during a traffic light cycle time. Let us consider the following simple expression for the power request P (t) [W] at the wheels P (t) = Ft (t)v(t) (3) where v(t) [m/s] is the vehicle speed. The traction force Ft (t) [N] at the wheels is dv(t) Ft (t) = m + a2 v(t)2 + a1 v(t) + a0 + mg sin(α) (4) dt where m [kg] is the vehicle mass, ai , i = 0, 1, 2 are given constants, g [m/s/s] is the acceleration due to gravity, and α [rad] is the slope of the road. In each cell j, traffic density and speed are constant, therefore the power request within each cell becomes a constant P̄ (v(ρj )), with no acceleration term. At the interface between adjacent cells, the speed difference triggers additional energy consumption which can be expressed as: Z v(ρj+1 )−v(ρj ) a Etrans = P (τ ) dτ (5)