Urban tranport phenomena in the street canyon Maciej

A field deterministic model of the vehicular dynamics in a generic urban street canyon with two neighboring canyons is considered. The assumed hydrodynamical model of vehicular movement is coupled to the gasdynamical model of the air and emitted pollutants. The vehicles are assumed to move on distinct left lanes and right ones. At the upstream and downstream ends there are coordinated traffic lights which introduce control parameters to the field model. The model of optimal control of street canyon dynamics is based on the two optimal multi-criteria control problems. The problems consist of minimization of the dimensionless functionals of the cumulative total travel time, global emissions of pollutants, and global concentrations of pollutants, both in the studied street canyon, as well as together with the two nearest neighbor substitute canyons, respectively. 47.10.+g, 47.62.+q, 89.60.-k, 89.40.-a Typeset using REVTEX ∗Electronic address: mduras @ riad.usk.pk.edu.pl 1 I. THE FIELD MODELS OF VEHICLES AND POLLUTANTS. In the present article we consider two coupled deterministic field model of the urban street canyon. The vehicular fields are one-dimensional in spatial variable x and they depend on time variable t. The considered fields are the vehicular number density k l,vt(x, t), vehicular velocity w l,vt(x, t), emissivity of exhaust gases e s l,ct,vt(x, t), and vehicular heat emissivity σ l,vt(x, t), where vt is emission type’s number, ct is number of the constituent of emitted exhaust gases. It is assumed that there are n1 = nL left lanes s = 1 and l is the left lane’s number, l = 1, ..., nL, whereas for n2 = nR right lanes s = 2 and l is the right lane’s number, l = 1, ..., nR, vt = 1, ..., V T , ct = 1, ..., CT . V T is total number of types of vehicular emissions, and CT is total number of emitted exhaust gases. The abovementioned model consists of the equations of balances of vehicular numbers [5] and vehicular equations of state (Greenshields’ equilibrium speed-density u-k model [6]) as well as exhaust gas emissivities and heat emissivities. The vehicular flow in the canyon is multilane bidirectional one-level rectilinear, and it is considered with two systems of coordinated signalized junctions at the upstream and downstream ends of lanes [1], [2], [3], [4]. The considered vehicles belong to distinct vehicular classes: passenger cars, and trucks. Emissions from the vehicles are based on technical measurements of the following types of pollutants: carbon monoxide CO, hydrocarbons HC, nitrogen oxides NOx. The field model of pollutants is gasdynamical. The gases are Newtonian viscous perfect and noninteracting. The considered fields are three-dimensional in spatial variables (x, y, z) and one-dimensional in temporal variable t and they are mixture’s density ρ(x, y, z, t), velocity v(x, y, z, t), temperature T (x, y, z, t), pressure p(x, y, z, t), as well as constituents’ mass concentrations ci(x, y, z, t), and partial pressures pi(x, y, z, t), i = 1, ..., N , where N is total number of mixture’s elements. (N = NE − 1 + NA). The first NE − 1 = 3 gases are the exhaust gases emitted by vehicle engines during combustion CO,CH,NOx. The remaining NA = 9 gases are the constituents of air: O2,N2,Ar,CO2,Ne,He,Kr,Xe,H2. The field equations are the balances of mixture’s mass, linear momentum, energy, and equation of state (Clapeyron’s law) as well as the balances of masses of constituents and constituents’ equations of states (Dalton’s law). The studied sources are mixture mass source S(x, y, z, t), mixture energy source σ(x, y, z, t), and constituents’ mass sources S i (x, y, z, t) (exhaust gases and consumed oxygen). The boundary-initial problems are of mixed types (Dirichlet’s and von Neumann’s types). The equations of dynamics are solved by the finite difference scheme. The two separate multi-criteria optimization problems are posed by defining the dimensionless functionals of cumulative total travel time, global emissions of pollutants, and global concentrations of pollutants, either in the studied street canyon or in the canyon and its two nearest neighbor substitute canyons. The vector of control is a five-tuple composed of two cycle times, two green times, and one offset time between the traffic lights. The optimal control problems consists of minimization of the two functionals over the admissible control domain manifold. II. THE GOVERNING EQUATIONS. Under the abovementioned specifications, the set of governing equations is formulated as follows [7], [8]: