Splitting of Traffic Flows to Control Congestion in Special Events

We deal with the optimization of traffic flows distribution at road junctions with an incoming road and two outgoing ones, in order to manage special events which determine congestion phenomena. Using a fluid-dynamic model for the description of the car densities evolution, the attention is focused on a decentralized approach. Two cost functionals, measuring the kinetic energy and the average travelling times, weighted with the number of cars moving on roads, are considered. The first one is maximized with respect to the distribution coefficient, and the second is minimized with respect to the same control parameter. The obtained results have been tested by simulations of urban networks. Decongestion effects are also confirmed estimating the time a car needs to cross a fixed route on the network.


Introduction
The vehicles congestion is one of the most important problem of modern cities, challenging many researchers to find techniques to control it.A solution to the problem is represented by the use of more lanes and the construction of crossings, but in many areas the solution is not feasible, and moreover the building and expanding of roads to accommodate the increase of vehicles is more expensive.In particular, the presence of unexpected heavy traffic in situations such as accidents leads to delay in the arrival of the emergency services and supplies to where they are needed.In the case of special events, escorts, closures of roads, traffic directions, and control functions can be performed, when necessary, to ensure the safe and efficient movement of vehicles, splitting the traffic flows at intersections in such way to improve the viability.An example is in Figure 1, where policemen are involved to manage traffic at junctions.In this context, using a fluid dynamic model able to foresee the traffic density evolution on road networks see 1-4 , we propose a strategy to redistribute in an optimal way flows at junctions.According to the adopted model, the car densities on each road follow a conservation law see 5 , while dynamics at junctions is uniquely solved using the following rules: A the incoming traffic at a node is distributed to outgoing roads according to some distribution coefficients; B drivers behave so as to maximize the flux through the junction.
If a junction J is of 1 × 2 type namely, one incoming road, 1, and two outgoing ones, 2 and 3 , rule A is expressed by a distribution parameter α, indicating the percentage of cars going from road 1 to road 2. Assigning initial densities for incoming and outgoing roads and using rule B , we finally compute the asymptotic solution as function of α.
Here, considering the distribution coefficient as control parameter, we aim to redirect traffic at junctions of 1×2 type in order to improve urban traffic and face emergency situations.In particular, we analyze two optimization problems over a fixed time horizon: minimizing an objective function W 1 , estimating the kinetic energy; maximizing a functional W 2 , measuring the average travelling time of drivers, weighted with the number of cars moving on roads.Indeed, we prove that both functionals are optimized for the same value of α.
Some control strategies for the right of way parameters and distribution coefficients have already been treated in 6, 7 , where three cost functionals, related to average velocity, average travelling time, and flux, have been introduced for 1 × 2 and 2 × 1 junctions.Cost functionals W 1 and W 2 have been studied in 8 for the optimal control of green and red phases of traffic lights, while in 9 parameters of 2 × 2 junctions have been optimized for the fast transit of emergency vehicles along an assigned path in case of car accidents.
The analysis of the functionals W 1 and W 2 on a whole network is a very hard task, so we follow a decentralized approach: an exact solution is found for single 1 × 2 junctions and asymptotic W 1 and W 2 .The global sub optimal solution for networks is obtained by localization: the exact optimal solution is applied locally for each time at each junction of 1 × 2 type.
The analytical optimization results are then tested by simulations for numerics, see 10-12 , analyzing optimal and random distribution coefficients.The first ones are given by the optimization algorithm; the second ones consider, at the beginning of the simulation process, random values of α, kept constant during the simulation.Then effects of the decentralized approach on the global performance of two networks have been analyzed.
Simulation results for a symmetric topology show that, assuming random distribution coefficients, the congestion of one road can determine high traffic densities on the whole network, while decongestion phenomena occur when optimal α values are used.In the case study of a portion of the Salerno urban network in Italy, characterized by an asymmetric topology, with 1 × 2 and 2 × 1 junctions, some interesting aspects arise: random coefficients frequently provoke hard congestions, as expected; optimal distribution coefficients allow a local redistribution of traffic flows.While random simulation curves of the cost functional W 1 are always lower than the optimal one, the optimal curve of W 2 is higher than some random ones.This is not surprising because, at 2×1 junctions, traffic densities can remain high.Hence, for such a network, locally optimal solutions alleviate critical traffic situations, but the aim of the global optimization of W 1 and W 2 is not achieved.Moreover, using an algorithm see 13 for tracing car trajectories on a network, some simulations are run to test how the total travelling time of a driver is influenced by distribution coefficients.As intuition suggests, the time for covering a path of a single driver decreases when optimal α values are used.
The paper is organized as follows.Section 2 is devoted to the description of the model for road networks and to the construction of solutions to Riemann Problems 1 × 2 junctions.In Section 3, we define the cost functionals W 1 and W 2 and optimize them with respect to the distribution coefficients at a single junction.Simulation results for complex networks are presented in Section 4. Section 5 ends the paper through conclusions.

A Riemann Solver for Road Networks
A road network is described by a couple I, J , where I represents the set of roads, modelled by intervals a i , b i ⊂ Ê , i 1, . . ., N, and J is the collection of junctions.
Indicating by ρ ρ t, x ∈ 0, ρ max the density of cars, ρ max the maximal density, f ρ ρv ρ the flux with v ρ the average velocity, the traffic dynamics is described on each road by the conservation law Lighthill-Whitham-Richards model, 3, 4 : We assume that: F f is a strictly concave C 2 function such that f 0 f ρ max 0. Choosing ρ max 1 and v ρ 1 − ρ, a flux function ensuring F is which has a unique maximum σ 1/2.In order to capture the dynamics at a junction, we solve Riemann Problems RPs , Cauchy Problems with a constant initial datum for each incoming and outgoing road, the basic ingredient for the solution of Cauchy Problems by Wave-Front-Tracking algorithms.
If m ≥ n, a possible RS at J is defined by the following rules see 1 : A traffic is distributed at J according to some coefficients, collected in a traffic distribution matrix A α j,i , i 1, . . ., n, j n 1, . . ., n m, 0 < α j,i < 1, n m j n 1 α j,i 1.The ith column of A indicates the percentages of traffic that, from the incoming road I i , distribute to the outgoing roads; B fulfilling A , drivers maximize the flux through J.
Focus on a 1 × 2 junction J.We indicate the cars density on the incoming road 1 by and on the outgoing roads ψ, ψ 2, 3, by Consider the flux function 2.2 and let ρ 1,0 , ρ 2,0 , ρ 3,0 be the initial densities at J. The maximal flux values on roads are defined by

Distribution Parameters Optimization
Fix a 1 × 2 junction J and an initial datum ρ 1,0 , ρ 2,0 , ρ 3,0 .We define the cost functional W 1 t and W 2 t , which measure, respectively, the kinetic energy and the average travelling time weighted with the number of cars moving on roads:

3.1
For a fixed time horizon 0, T , with T sufficiently big, consider the traffic distribution coefficient α as control.We aim to maximize W 1 T and to minimize W 2 T separately.The functionals assume the form: where s 1 and s ψ , ψ 2, 3, are given by International Journal of Mathematics and Mathematical Sciences with

3.4
According to the solution of the RP at J, we have where γ 1 is given by 2.5 .The values of α, which optimize W 1 T and W 2 T , are reported in the following theorem for the sketch of the proof, see the appendix .
Theorem 3.1.Fix a 1×2 junction J. Assuming T sufficiently big, the cost functionals W 1 T W 2 T is maximized (minimized) for α 1/2, with the exception of the following cases (for some of them, the optimal control does not exist but it is approximated): , we distinguish three subcases: , we distinguish two subcases: and ε is small and positive.

In case A , we get
where with α 1 0.6.For T sufficiently big, W 1 T and W 2 T have one discontinuity point at α α 1 , as shown in Figure 2. The optimal control does not exist, but one can choose α α 1 ε.
In case B , we have that holds.Then, the solution to the RP at J is where with α 2 0.7575.For T sufficiently big, the cost functionals W 1 T and W 2 T have one discontinuity point at α α 2 , as shown in Figure 3.The optimal control exists, and it is α 1/2, for which Case B : behaviour of W 1 and W 2 for T sufficiently big. is satisfied, and we obtain where

3.15
with α 1 0.238 and α 2 0.428.The cost functional W 1 T and W 2 T , reported in Figure 4 for T sufficiently big, have two discontinuity points at α α 1 and α α 2 .Hence, an optimal value for α does not exist, but we can choose α α 2 − ε.

Road Traffic Simulation
We present some simulation results in order to test the optimization algorithm for the cost functionals.In particular, we analyze the effects of different control procedures, applied locally at each junction, on the global performances of networks and compute the travelling time of a car on assigned paths.For simplicity, from now on we drop the dependence on T from W 1 and W 2 .

A Symmetric Network
In this subsection, we analyze a symmetric network with three simple junctions of 1 × 2 type, labelled by 1, 2, and 3, see Figure 5.In particular, the network consists of two inner roads, b and c, and five roads, that connect the inner roads to outside: a, d, e, f, and g.The conservation law with flux function 2.2 is approximated using the Godunov scheme, with space step Δx 0.0125, and time step, determined by the CFL condition see 10, 11 , equal to 0.5.We assume initial conditions zero for all roads at the starting instant of simulation t 0 , a 0.3 Dirichlet boundary datum for roads a, d, e, f, a 0.9 Dirichlet boundary condition for road g and a time interval of simulation 0, T , where T 30 min.Two different choices of the distribution coefficients are considered: locally optimal parameters at each junction, given by analytical results optimal case ; random parameters random case , that is, the distribution coefficients are taken randomly for each road junction when the simulation starts and then are kept constant.
The evolution of W 1 and of W 2 are depicted in Figures 6 and 7, reporting with a continuous line the optimal case and with dashed lines various random cases.In some random simulations, the α values are such that a lower traffic density goes to road g, with a consequent natural improvement of the network performances.This justifies the fact that some dashed W 1 and W 2 curves approach the optimal ones.In other cases, W 1 rapidly decreases and W 2 tends to infinity, indicating that the random choice of α provokes congestions on all network roads.However, in any case, the optimal case is better than the others.In fact, it describes the natural situation that happens on congested real urban networks in which the traffic is redirected to less congested roads.

A Real Urban Network
This subsection is devoted to the simulation on a portion of the urban network of Salerno, Italy.The network topology, depicted in Figure 8 Via Leonino Vinciprova segments d and e , Via Settimio Mobilio segments f, g, h, and i , and Via Guercio segment l .We distinguish inner roads segments, b, e, f, g, and h, and external ones, a, c, d, i, l.Junctions indicated by numbers 1, 3 and 5 are of 1 × 2 type,while 2 and 4 are of 2 × 1 types.The evolution of traffic flows is simulated by the Godunov method with Δx 0.0125, Δt Δx/2 in a time interval 0, T , with T 120 min.Initial conditions and boundary data for densities are in Table 1 and have been taken in order to simulate a congestion scenario.Notice that, for junctions 2 and 4, right of way parameters are chosen according to measures on the real network.
In Figure 9, we report the behaviour of W 1 and W 2 , where optimal simulations are indicated again by continuous lines, while random cases by dashed ones.Random simulations curves of W 1 are always lower than the optimal ones.In fact, when optimal parameters are used, a flows redistribution occurs on roads, with consequent reduction of congestions at junctions of 1 × 2 type.Focus now on W 2 , for which the optimal curve is higher than some random ones.This is not surprising as we deal with the simulation of a high congested asymmetric network.The traffic redirection at congested 1 × 2 junctions is of local type and, as expected, benefits occur only on roads and at junctions where the optimization procedure is applied.This is easy deducible considering Via Torrione, which presents a 2 × 1 junction, labelled by 2, where traffic high densities cannot be redirected.distribution coefficients are used, high traffic densities affect some roads and this justifies that the optimal curve is not the lowest for W 2 .
Suppose that a car travels along a path in a network, whose traffic evolution is modeled by 2.1 .The position of the driver x x t is obtained solving the Cauchy problem: where x 0 is the initial position at the initial time t 0 , while v 1 − ρ.Using numerical methods, described in 13 , we aim to estimate the driver travelling time and to prove the goodness of the optimization results.First, we compute the car trajectory along road g and the time needed for covering it in optimal case and random cases; then, we fix a car path within the Salerno network and study the exit time evolution versus the initial travel time t 0 the time in which the car enters into the network .
In Figure 10, we assume that the car starts its own travel at the beginning of road g at the initial times t 0 20 a and t 0 60 b and compute the trajectories x t along road g, in optimal case continuous line and random cases dashed lines .Although initial times t 0 are different, the evolution x t in the optimal case has always a higher slope with respect to trajectories in random cases because traffic levels are low.When random choices of parameters α are used, higher boundary conditions for roads c, i, and l cause an increase in density values by shocks propagating backwards.Hence, car velocities are reduced, travel times become longer and so exit times from road g.
In Table 2, we collect the exit times T out from road g the times needed to go out from road g , for the optimal choice of distribution coefficients opt and random choices r i , i 1, . . ., 9 , assuming t 0 20.
The exit time T out t from road g versus the initial time t 0 , assuming that the car starts its path from the beginning of the road is shown in Figure 11 a .Because of the decongestion effects, the choice of optimal coefficients continuous lines allows to obtain an exit time lower than the other cases dashed lines .Notice that the exit time becomes stable after a certain initial time value t 0 18.5 for the optimal distribution choice, unlike the random cases, for which t 0 35.5 and t 0 42 .Finally, in Figure 11 b , we consider the exit time T out t from a fixed route, a, g, h, l , versus the initial times t 0 .Precisely, we study the temporal variation of T out when a car, starting its trip at the beginning of road a at time t 0 , crosses roads a, g, h, l in order to exit from the network.Also in this case, different choices of distribution coefficients the optimal one is indicated by a continuous line, unlike the others affect the time for covering the path.When optimal parameters are not used, T out tends to infinity at some critical times t c t c 13 and t c 27 for the random case represented by dashed line, and t c 23 and t c 36 for the random case with dot-dashed line .This occurs because the car cannot enter road g or road h, since the traffic within them is blocked.For times greater than critical ones, traffic densities become stable a light decongestion allows the car to reach the destination and exit times reach a steady value T out 31.5 and T out 30.35 at times, respectively, t 0 42 and t 0 46.On the contrary, when optimal parameters are used, the exit time does not tend to infinity as the network is never congested and reaches the steady value T out 28.75 at time t 0 38 lower, as expected, than steady-state times of simulation with random parameters .

Conclusions
In this paper, an optimization study has been presented to improve the urban traffic conditions in case of special events in which splitting of flows is needed.
The optimization has been made over traffic distribution coefficients at junctions, using two cost functionals, that measure, respectively, the kinetic energy and the average travelling times of drivers, weighted with the number of cars moving on roads.An exact solution has been found for simple junctions having one incoming road and two outgoing roads.The obtained analytical results have been tested through simulations, showing that in some cases a total decongestion effect is possible.This is also confirmed by evaluations of cars trajectories on some roads and fixed routes on the network: using optimal distribution coefficients, times needed to cover paths are the lowest.

Appendix
We report the proof of Theorem 3.1.Consider new functionals, W 1 and W 2 , in which the terms not depending on α are neglected.Since the solution to the RP at J depends on the value of the parameter α, we distinguish various cases.Here, for sake of brevity, we analyze some of them.
Assume see Figure 12 a that . As for 0 < α ≤ α 1 , s 1 1, s 2 −1, and for α 1 < α < 1, s 2 s 3 −1, W 1 and W 2 assume the form: Our aim is to maximize W 1 and to minimize W 2 separately with respect to the parameter α.
If α 1 < α < 1, we get where ii if α 1 ≥ 1/2, the optimal value for both W 1 and W 2 does not exist.One can choose α α 1 ε, with ε small and positive constant.

Figure 1 :
Figure 1: Example of car flows redistribution.

Figure 2 :
Figure 2: Case A : behaviour of W 1 and W 2 for sufficiently big.

Figure 4 :
Figure 4: Case C : behaviour of W 1 and W 2 for T sufficiently big.

Figure 5 :
Figure 5: Topology of the symmetric network.

Figure 6 :Figure 7 :
Figure 6: W 1 t evaluated for optimal distribution coefficients continuous line and random choices dashed lines .a evolution in 0; T .b zoom around the asymptotic optimal values.

Figure 8 :Figure 9 :
Figure 8: Topology of the portion of the real urban network of Salerno.

Figure 10 :
Figure10: Evolution of a car trajectory x t along road g with initial travel times t 0 20 a and t 0 60 b , evaluated for optimal distribution coefficients continuous lines and random choices dashed lines .

Figure 11 :
Figure 11:Exit time from road g versus initial travel time t 0 in 0, 80 a and exit time from the path a, g, h, l versus initial travel times t 0 in 0, 80 b , evaluated for optimal distribution coefficients continuous lines and random choices dashed and dot dashed lines .

Table 1 :
Initial conditions, boundary data and right of way parameters for roads of the network.
Even if the right of way parameters which characterize 2 × 1 junctions are optimized according to the values in 8 , traffic conditions almost remain the same.Hence, although optimal International Journal of Mathematics and Mathematical Sciences

Table 2 :
Initial conditions, boundary data and right of way parameters for roads of the network.