DE102016225404A1 - A method, apparatus, computer program and computer program product for determining a fundamental diagram of a traffic flow for a space-time domain - Google Patents

Abstract

A method for determining a fundamental diagram of a traffic flow for a space-time domain comprises the steps of: providing vehicle data representative of data of vehicles in the space-time domain, determining vehicle speeds depending on the vehicle data, determining depending on the speeds and a continuity equation a traffic flow, an assignment of density and speed and determine depending on the assignment of the fundamental diagram.

Description

The invention relates to a method for determining a fundamental diagram of a traffic flow for a space-time domain. The invention further relates to an apparatus for determining a fundamental diagram of a traffic flow for a space-time domain. The invention further relates to a computer program and computer program product for determining a fundamental diagram of a traffic flow for a space-time domain.

Traffic can be described macroscopically by the three state variables flow, density and velocity via the so-called fundamental diagram of the traffic flow. A well-estimated fundamental diagram makes it possible to draw conclusions about the overall state by measuring only one state variable of the traffic. Knowing the overall condition of the traffic is crucial for many applications such as traffic planning, traffic information and traffic control.

The object underlying the invention is to efficiently determine a well-estimated fundamental diagram.

The object is solved by the features of the independent claims. Advantageous embodiments are characterized in the subclaims.

The invention is characterized by a method for determining a fundamental diagram of a traffic flow for a space-time domain. The invention is further characterized by an apparatus for determining a fundamental diagram of a traffic flow for a space-time domain, the apparatus being adapted to carry out the method for determining a fundamental diagram of a traffic flow for a space-time domain or an optional embodiment of the method.

The method provides vehicle data representative of data of vehicles in the space-time domain. Depending on the vehicle data speeds of the vehicles are determined. Depending on the speeds and a continuity equation of a traffic flow, a correlation of density and velocity is determined. Depending on the assignment, the fundamental diagram is determined.

The method makes it possible to determine a fundamental diagram with the help of only one state variable, namely the velocity. As a result, the fundamental diagram can be determined more cost-effectively and more efficiently, since, for example, no sensor data which measures at least two state variables, such as loop detectors, are necessary. There are enough vehicle data, by means of which exactly one state variable (the speed) can be determined. Such vehicle data are, for example, position data which has been determined by means of a global navigation satellite system for position determination, for example by means of the Global Positioning System (GPS). Since the measurement of traffic data is associated with high costs, a reduction of the variables to be measured represents a high added value. Using the proposed method thus costs can be saved in the acquisition of the necessary data.

The determined fundamental diagram is then provided, for example, to an application and used in the application. Such applications are, for example, a traffic planning, a representation of traffic information and / or a traffic control.

According to an optional embodiment, the assignment of density and velocity is determined depending on the velocities and the continuity equation and the method of least squares.

Using the method of least squares, the fundamental diagram can be estimated in a very simple and thus fast and cost-effective way.

According to another optional embodiment, in the least squares method, the continuity equation replaces flux by density multiplied by velocity, and then performs discretization in time and location. The density is replaced by a given speed dependent function, and the least squares solution gives the average densities at a given number of speed nodes.

By means of this embodiment of the method of least squares, average densities can be determined in a very simple manner and thus the assignment of density and speed can be determined.

According to another optional embodiment, the assignment of density and velocity is determined depending on the speeds and the continuity equation and a Markov chain Monte Carlo method, MCMC method.

It can be taken into account by the MCMC method that a variance of the densities is not constant but is dependent on the speed. Thus, a very accurate fundamental diagram can be determined by the MCMC method.

According to a further optional embodiment, in the MCMC method, probability densities are determined as a function of the speeds. A simulation of the densities, flows and velocities is carried out with the help of a given discretization of the traffic flow model. Subsequently, a determination of the probability of the simulated densities, flows and velocities is derived by comparing the simulated density values with the probability densities. The probability density of the density values results, for example, from given velocities in the space-time domain and selected interpolation points of functions r (v) and σ (v).

By means of this embodiment of the MCMC method, a very accurate fundamental diagram can be determined in a very simple way.

According to a further optional embodiment, the predetermined discretization is a Godunov discretization.

By Godunov discretization can be done a very accurate yet simple discretization.

According to another optional embodiment, the determination of the probability is performed several times.

With this an average result over all speeds can be determined and thus the accuracy of the determined fundamental diagram can be increased.

According to a further optional embodiment, a predetermined heuristic optimization method is additionally performed.

As a result, the accuracy of the determined fundamental diagram can be further increased.

According to another optional embodiment, simulated annealing is the heuristic optimization method.

According to a further aspect, the invention is characterized by a computer program, wherein the computer program is designed to perform the method for determining a fundamental diagram of a traffic flow for a space-time area or an optional embodiment of the method.

According to a further aspect, the invention is characterized by a computer program product comprising executable program code, wherein the program code, when executed by a data processing device, executes the method for determining a fundamental diagram of a traffic flow for a space-time domain or an optional embodiment of the method.

In particular, the computer program product comprises a medium which can be read by the data processing device and on which the program code is stored.

• 1 ein Ablaufdiagramm eines Programms zum Ermitteln eines Fundamentaldiagramms eines Verkehrsflusses für einen Raumzeitbereich.

Embodiments of the invention are explained in more detail below with reference to the schematic drawing. It shows:

• 1 a flowchart of a program for determining a fundamental diagram of a traffic flow for a space-time area.

The 1 FIG. 12 is a flow chart of a program for determining a fundamental diagram of a traffic flow for a space-time domain. FIG.

A device 50 is designed to run the program. The device 50 has for this purpose in particular a computing unit, a program and data memory, as well as, for example, one or more communication interfaces. The program and data memory and / or the arithmetic unit and / or the communication interfaces can be formed in a structural unit and / or distributed over several structural units. The device 50 may be formed for example in a backend.

The device 50 Can also be used as a device 50 for determining a fundamental diagram of a traffic flow for a space-time domain.

The program is started in a step S1 in which variables can be initialized if necessary.

In a step S3, vehicle data representative of data of vehicles in the space-time area is provided. The vehicle data are, for example, position data of the vehicles, such as GPS data.

In a step S5, speeds of the vehicles are determined as a function of the vehicle data.

In a step S7, depending on the speeds and a continuity equation of a traffic flow, an assignment of density and velocity is determined and depending on the assignment, the fundamental diagram is determined.

In a step S9, the program is ended and may optionally be restarted in step S1.

• Bei einer ersten Ausführungsform wird die Methode der kleinsten Quadrate verwendet.

In the following, embodiments of step S7 are presented:

• In a first embodiment, the least squares method is used.

Starting point is the continuity equation of a traffic flow, in which a replacement of flow by density multiplied by speed takes place: $$\frac{\partial \rho }{\partial t}+\frac{\partial \left(\rho v\right)}{\partial x}=0$$

Here, ρ is the (traffic) density and v the speed.

If we now describe a rectangular space-time domain with D = [x _start , x _end ] x [t _start , t _end ] (where x is a position and t is a time) and possibly other boundary conditions, we can perform a discretization in time and place , for example using the "central differences": $$\frac{{\rho }_{x.t+1}-{\rho }_{x.t-1}}{2\Delta t}+\frac{{\rho }_{x+\mathrm{1,}t}{v}_{x+\mathrm{1,}t}-{\rho }_{x-\mathrm{1,}t}{v}_{x-\mathrm{1,}t}}{2\Delta x}=\mathrm{0th}$$

The equation with 0 is correct only in the ideal state.

The density can now be replaced by a given function r (v): $$\frac{r\left(v_{x.t+1}\right)-r\left(v_{x.t-1}\right)}{2\Delta t}+\frac{r\left(v_{x+\mathrm{1,}t}\right)v_{x+\mathrm{1,}t}-r\left(v_{x-\mathrm{1,}t}\right)v_{x-\mathrm{1,}t}}{2\Delta x}\ne 0$$

The equation is not equal to 0 in reality, since measurement errors, noise, etc. occur.

r (v) is a piecewise linear function that is based on n nodes (i = 1 ... n): $$r\left(v\right)=\left(1-\frac{v-v_i}{v_{i+1}-v_i}\right)r\left(v_i\right)+\left(\frac{v-v_i}{v_{i+1}-v_i}\right)r\left(v_{i+1}\right),$$

Subsequently, a formulation can be made as a problem of linear regression:

Here a large number of equations are generated from the given velocity information: $$\left[B_0|\widehat{B}\right]{\left[r_0|\widehat{r}\right]}^T=\mathrm{0th}$$

B is a matrix of coefficients which results from equation 3.

In particular, it is assumed that the macroscopic traffic speed is completely given in space and time.

The method requires in particular a "base". In the mentioned equation system ( 5 ) the interpolation point is marked as variable r0. The corresponding coefficients are described as column vector B0. The method makes it possible to specify each of the interpolation points or a multiplicity of random interpolation points. That is, one point must be given to the function r (v). The reason for the need for at least one vertex is that velocities and densities are computed relative to each other, but the method has no absolute reference to the traffic. A base that offers itself is one with low speed, where a maximum traffic density ρ _max occurs.

The solution of the least squares method then results from: $$\widehat{r}=-{\widehat{r}}_0{\left({\widehat{B}}^T\widehat{B}\right)}^{-1}{\widehat{B}}^T{B}_0,$$

The solution gives the average densities at the n nodes of the velocities.

In a second embodiment, a Markov chain Monte Carlo method, MCMC method, is used.

First, all velocities at the edge of the space-time domain are considered (t = 0, x = 0, t = T, x = L).

For each of these values, a value is chosen from a probability density resulting from functions r (v) and σ (v): $${\widehat{\rho }}_i\left(\Omega \right)~p\left(\widehat{\rho }\left(\Omega \right)|r.\sigma \right)$$

r (v) and σ (v) are piecewise linear functions which are formed from a vector of values as described above for r (v). ρ (Ω) is the density at the edge of the space-time domain.

This is followed by a simulation of the densities and flows and velocities, in particular with the help of the Godunov discretization of the traffic flow model and the velocities at the edge of the space-time domain: $${\rho }_{x.t+1}={\rho }_{x.t-1}-\frac{\Delta t}{\Delta x}\left({\rho }_{x+\mathrm{1,}t}{v}_{x+\mathrm{1,}t}-{\rho }_{x-\mathrm{1,}t}{v}_{x-\mathrm{1,}t}\right),$$

Now, the probability of the simulated state can be obtained by comparing the simulated density values with the probability density of the density values resulting from the given velocities in the space-time domain and the selected interpolation points of the functions r (v) and σ (v): $$p\left(equatiOns|\widehat{\rho }\left(\Omega \right).r.\sigma \right)={\displaystyle \underset{\left(x.t\right)\in \mathcal{D}}{\Pi }\mathcal{N}\left({\rho }_{x.t}|r\left(v_{x.t}\right).\sigma \left(v_{x.t}\right)\right),}$$

mit $$\underset{N\to \infty }{\text{lim}}I=p\left(\text{equations}|r,\sigma \right)\begin{array}{cc}& {}_{{}_{.}}\end{array}$$

To make a robust estimation of the probability of occurrence ( 9 ) of the randomly generated boundary values from the corresponding density functions, the calculation is repeated N times. As the number of calculations increases, the calculated probability of occurrence converges to the true value. $${\rm I}\approx \frac{1}{N}{\displaystyle \underset{i=1}{\overset{N}{\Sigma }}p\left(equatiOns|{\widehat{\rho }}_i\left(\Omega \right).r.\sigma \right)}.{\widehat{\rho }}_i\left(\Omega \right)~p\left(\widehat{\rho }\left(\Omega \right)|r.\sigma \right).$$

With $$\underset{N\to \infty }{\text{lim}}I=p\left(\text{equations}|r.\sigma \right)\begin{array}{cc}& {}_{{}_{,}}\end{array}$$

By maximizing the probability of occurrence as a function of the functions r (v) and σ (v), the best values for the underlying fundamental diagram can be determined by means of an optimization method such as simulated annealing.

The method makes it possible to determine a fundamental diagram with the help of only one state variable, namely the velocity. As a result, the fundamental diagram can be determined more cost-effectively and efficiently, since the measurement of traffic data is associated with high costs.

It can be taken into account by the MCMC method that a variance of the densities is not constant but is dependent on the speed. Thus, a very accurate fundamental diagram can be determined by the MCMC method.

Claims (12)

Method for determining a fundamental diagram of a traffic flow for a space-time area, in which Providing vehicle data representative of data of vehicles in the space-time domain, - depending on the vehicle data speeds of the vehicles are determined - Depending on the speeds and a continuity equation of a traffic flow, an assignment of density and speed is determined and the fundamental diagram is determined depending on the assignment. Method according to Claim 1 , which determines density and velocity as a function of velocities and the equation of continuity and the method of least squares. Method according to Claim 2 In the least squares method - in the continuity equation, a replacement of flow by density multiplied by velocity occurs and then a discretization in time and place is performed, - the density is replaced by a given function dependent on the velocity and the solution of the Least squares method gives the average densities at a given number of nodes of velocities. Method according to Claim 1 in which, depending on the velocities and the continuity equation and a Markov chain Monte Carlo method, MCMC method, the assignment of density and velocity is determined. Method according to Claim 4 in the MCMC method - determining probability densities depending on the speeds, - simulating the densities, flows and velocities using a given discretization of a traffic flow model and then determining the probability of the simulated densities, flows and velocities by comparing the derived density values with the probability densities. Method according to Claim 5 , where the given discretization is a Godunov discretization. Method according to Claim 5 or 6 , where the determination of the probability is performed several times. Method according to one of Claims 4 to 7 In addition, a predetermined heuristic optimization method is performed. Method according to Claim 8 , where simulated annealing is the heuristic optimization method. Computer program for determining a fundamental diagram of a traffic flow for a space-time area, wherein the computer program is formed by a method according to one of Claims 1 to 9 when executed on a data processing device. Computer program product comprising executable program code, wherein the program code when executed by a data processing device, the method according to one of Claims 1 to 9 performs. Apparatus (50) for determining a fundamental diagram of a traffic flow for a space-time domain, the apparatus being adapted to the method according to one of Claims 1 to 9 perform.

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