CN103761871A - Macroscopic traffic flow model branch-and-bound analysis and control method for blocking road - Google Patents

Abstract

The invention relates to a system theory analysis and control method, in particular to a macroscopic traffic flow model branch-and-bound analysis and control method for blocking a road, and belongs to the field of road traffic flow analysis and control. According to the method, two equivalent variable substitutions are introduced to transform a traffic flow problem into a system stability problem, and therefore whether the traffic is blocked up or other abnormal phenomena are caused is judged macroscopically from the point of overall stability. Branch-and-bound analysis is carried out on the model after equivalence transformation, the balance point of the system is solved uniformly, and branch points causing the change of the character of the system are found out. Corresponding control schemes are designed according to the different branch points, the unstable branch points of the system are changed into the stable branch points, or the unstable branch points are removed to be changed into the stable area of the system, namely, the stable running state of the traffic flow, and the actual operation can be achieved by controlling the opening and closing time of an entrance ramp adjusting rod and setting a variable information display board.

Description

The macroscopic traffic flow branch-and-bound control and analysis of blocked road

Technical field

The present invention relates to a kind of macroscopic traffic flow branch-and-bound control and analysis of Systems Theory control and analysis, particularly a kind of blocked road.

Background technology

Modern Traffic is when promoting socio-economic development, and the traffic jam issue thereupon bringing is more and more serious, is embodied in to block up more and more frequently, blocks the duration more and more longer, even at blocked road, also often occurs the phenomenon of traffic congestion.Research blocked road traffic problems, replace traditional manual shift to control by systemtheoretical analytical approach design control strategy, are to alleviate a solution of traffic jam issue now.

The complexity of traffic system is than the also complexity in people's imagination, it has related to the Comprehensive Control of urban traffic network, the synthetical collection of transport information and network transmission technology, traffic intelligent information fusion and treatment technology, traffic flow inductive technology, and vehicle transport intelligent dispatching method, municipal intelligent traffic planing method, traffic safety detects, many-sided contents such as traffic environment overall evaluation system, and influence each other between each factor, restriction mutually, the synthesis that correlativity is extremely strong, be difficult to adopt unified description form to portray this challenge.Early stage traffic flow research mostly adopts that to take theory of probability and mathematical statistics be the basic description magnitude of traffic flow and the modeling method of density relationship, many traffic behaviors, are all described out such as the phenomenons such as dissipation on a large scale of traffic collapse, lag-effect and flow-density plane.The development pole the earth of non-free traffic flow research has enriched the content of traffic flow theory, mainly comprises vehicle-following theory, and vehicle queue is theoretical, fluid mechanics model etc.Vehicle-following theory is described is the interaction between adjacent two cars in fleet of travelling on the bicycle road of overtaking other vehicles in restriction, use dynamic (dynamical) method to study front guide-car's motion state and change cause with the respective behavior of the car of speeding, by analyzing each vehicle, with the mode of speeding, understand single-way traffic properties of flow one by one; Congested in traffic typical case's performance is exactly vehicle queue, vehicle queue's phenomenon is a dynamic process changing along with the time, can reflect that traffic flow is from unimpeded to crowded, Zhongdao is stopped up such change procedure, for this process, a lot of scholars attempt to explain the congested in traffic inherent mechanism producing by setting up some models; In the macroscopic traffic flow based on fluid mechanics model, traffic flow is regarded as the compressible continuous fluid medium being comprised of a large amount of vehicles, the average behavior of research vehicle collective, and the individual character of single unit vehicle does not highlight.Macroscopic traffic flow is portrayed traffic flow with average density ρ, average velocity v and the flow q of vehicle, studies their satisfied equations.Compare with microvisual model, macromodel can be portrayed the collective behavior of traffic flow better.The research of existing traffic flow is mostly, on the basis of mechanism research, some traffic behaviors are carried out to approximate description and simplicity of explanation, or on the basis of forefathers' research, increase and consider some influence factors, adopt the method for souning out, by changing the value of some parameters, make it approach corresponding traffic behavior, to producing the reason of this phenomenon, from systemtheoretical angle, do not carry out essence explanation, can not directly provide traffic congestion condition, the integral body that particularly directly provides traffic jam issue when various traffic parameters change is described, make traffic system research worker be not easy to direct use, a lot of control methods can not effectively be applied.

Summary of the invention

The deficiency that is difficult to directly process traffic jam issue in order to overcome the existing macroscopic traffic flow of blocked road, the present invention is by introducing two substitution of variable of equal value, set up new traffic model, obtained the relation of traffic jam issue and system stability, as expanded the stability in discrete system unit circle to whole complex plane, traffic flow problem is converted into system stability problem, thereby can adopt global stability angle to judge from macroscopic view whether traffic there will be, blocks up or other abnormal occurrence; Model after converting by parity price carries out branch-and-bound analysis, after the equilibrium solution of Unified Solution system, from equilibrium point, finds out the take-off point that causes that the property of system changes; Due to traffic non-linear phenomena corresponding to different take-off points, for different take-off points, design corresponding control program, make the unstable take-off point of system become Stable Branch point, or make unstable take-off point disappearance become the stabilized zone of system, the i.e. steady operational status of traffic flow; The Systems Theory analytical approach of branch-and-bound can be from the reason of explaining that in essence traffic non-linear phenomena produces, for traffic control, the decision-making of blocked road provides basic foundation, can directly apply to processing traffic jam issue.

The technical solution adopted for the present invention to solve the technical problems is: a kind of macroscopic traffic flow branch-and-bound control and analysis, is characterized in adopting following steps:

1, the macroscopic traffic flow of setting up according to hydrokinematics model can be unified to be described below:

$\left\{\begin{array}{c}\frac{\∂\mathrm{\ρ}}{\∂t}+\frac{\∂\left(\mathrm{\ρv}\right)}{\∂x}=\mathrm{\π}[r(x,t),s(x,t)]\\ \frac{\∂v}{\∂t}+v\frac{\∂v}{\∂x}=f(\frac{\∂v}{\∂x},\frac{\∂\mathrm{\ρ}}{\∂x},\frac{{\∂}^{2}v}{\∂x^2},\frac{{\∂}^2\mathrm{\ρ}}{\∂x^2},V_e\left(\mathrm{\ρ}\right))\end{array}\right.$

Wherein, the average density that ρ is vehicle, unit is veh/m, the average velocity that v is vehicle, unit is m/s, and t is the time, and unit is s, and x is and the distance of road starting point, and unit is m, and ρ, v are the function about x, t, V _e(ρ) be equivalent speed function, f be about

v _e(ρ) function, the rate of change of the density that the vehicle flowrate that π [r (x, t), s (x, t)] is entered or rolled away from by ring road for the t moment, x section causes, unit is veh/ms ^-1, r (x, t)=r _o(x, t)+r _qthe vehicle flowrate that (x, t) entered by ring road for the t moment, x section, s (x, t)=s _o(x, t)+s _qthe vehicle flowrate that (x, t) rolled away from by ring road for the t moment, x section, r _o(x, t), s _o(x, t) normal vehicle flowrate for sailing into, roll away from by ring road, r _qthe flow reducing amount that (x, t) causes for the ring road highway of controlling that No entry, s _qthe flow recruitment that (x, t) forces outgoing vehicles to cause for information display board, flux unit is veh/s;

2, introducing state variable is:

$\left\{\begin{array}{c}\mathrm{\η}(x,t)=\frac{1}{{\mathrm{\ρ}}_m-\mathrm{\ρ}(x,t)}\\ \mathrm{\σ}(x,t)=\frac{1}{v(x,t)}\end{array}\right.$

Wherein, ρ _mtraffic flow density while occurring blocking for traffic, unit is veh/m, ρ (x, t) be the average density of vehicle, unit is veh/m, v (x, t) be the average velocity of vehicle, unit is m/s, as long as state variable η (x, t) >0 or σ (x, t) value of >0 increases, just illustrate that density increases or speed reduces, whole traffic system trends towards unstable, as state variable η (x, when t) >0 is tending towards infinite, that represent traffic density is tending towards saturated traffic density, produce traffic congestion, as state variable σ (x, when t) >0 is tending towards infinite, represent that vehicle average velocity goes to zero, produce traffic congestion,

Relation between η (x, t), σ (x, t) density and speed is brought in the unified macroscopic traffic flow of describing, and macroscopic traffic flow new after substitution of variable is described as:

$\left\{\begin{array}{c}\frac{\∂\mathrm{\η}}{\∂t}=-\frac{1}{\mathrm{\σ}}\frac{\∂\mathrm{\η}}{\∂x}+\frac{{\mathrm{\ρ}}_m{\mathrm{\η}}^2-\mathrm{\η}}{{\mathrm{\σ}}^2}\frac{\∂\mathrm{\σ}}{\∂x}+{\mathrm{\η}}^2\·\mathrm{\π}[r(x,t),s(x,t)]\\ \frac{\∂\mathrm{\σ}}{\∂t}=f(\frac{\∂\mathrm{\η}}{\∂x},\frac{\∂\mathrm{\σ}}{\∂x},\frac{{\∂}^2\mathrm{\η}}{\∂x^2},\frac{{\∂}^2\mathrm{\σ}}{\∂x^2},V_e({\mathrm{\ρ}}_m,\mathrm{\η}))\end{array}\right.$

Wherein, η >0 is state variable, and σ >0 is state variable, and x is and the position of road starting point, and unit is m, and t is the time, and unit is s, V _e(ρ _m, η) be equivalent speed function, ρ _msaturated traffic density during for traffic congestion, unit is veh/m, when state variable η is tending towards infinite, that represent traffic density is tending towards saturated traffic density, produces traffic congestion, when state variable σ is tending towards infinite, represent that vehicle average velocity goes to zero, produce traffic congestion;

3, according to the new macroscopic traffic flow after substitution of variable, the equilibrium solution of solving system:

When system reaches stable (balance) state, should have:

$\left\{\begin{array}{c}\frac{\∂\mathrm{\η}}{\∂t}=0\\ \frac{\∂\mathrm{\σ}}{\∂t}=0\end{array}\right.$

Suppose to enter with the vehicle flowrate rolling away from and equate by ring road, i.e. π [r (x, t), s (x, t)]=0, now, the equilibrium point of the model after substitution of variable meets following balance equation:

$\left\{\begin{array}{cc}(\mathrm{\η}-{\mathrm{\ρ}}_m{\mathrm{\η}}^2)\frac{\∂\mathrm{\σ}}{\∂x}+\mathrm{\σ}\frac{\∂\mathrm{\η}}{\∂x}=0& \left(a\right)\\ f(\frac{\∂\mathrm{\η}}{\∂x},\frac{\∂\mathrm{\σ}}{\∂x},\frac{{\∂}^2\mathrm{\η}}{\∂x^2},\frac{{\∂}^2\mathrm{\σ}}{\∂x^2},V_e({\mathrm{\ρ}}_m,\mathrm{\η}))=0& \left(b\right)\end{array}\right.$

To solving after (a) formula conversion in above-mentioned equation, have

And (b) formula of bringing into can obtain conversion after the equilibrium solution of model be:

Conventionally, equivalent speed function V _e(ρ) can choose the following expression-form being proposed by people such as Del Castillo and (see document Del Castillo J M, Bennitez F G.On functional form of the speed-density relationship-I: General theory; II: Empirical investigation.Transpn Res B, 1995,29:373～406):

$V_e\left(\mathrm{\ρ}\right)=V_f\{1-\exp [1-\exp \left(\frac{c_m}{V_f}(\frac{{\mathrm{\ρ}}_m}{\mathrm{\ρ}}-1)\right)\left]\right\}$

Wherein, V _ffor free stream velocity, unit is m/s, c _mspreading of disturbance during for traffic jam, unit is m/s, ρ _mtraffic flow density while occurring blocking for traffic, unit is veh/m, and η >0 is state variable, and it is V after substitution of variable _e(ρ _m, being expressed as η):

$V_e({\mathrm{\ρ}}_m,\mathrm{\η})=V_f\{1-\exp [1-\exp \left(\frac{c_m}{V_f({\mathrm{\ρ}}_m\mathrm{\η}-1)}\right)\left]\right\}$

Each parameter meaning is the same;

4, take equilibrium point as initial point, utilize MatCont software searching system take-off point, the system take-off point meeting within the scope of actual physics meaning is carried out to specificity analysis, be i.e. traffic non-linear phenomena corresponding to different take-off points:

The scope that meets actual physics meaning refers to that the selection of model parameters under different road conditions meets the condition of actual traffic operation, the corresponding actual density of scope of state variable η, σ, the boundary value of speed;

Owing to being for the stability problem of discussing system under long-time, when processing PDE model, regarding parameter or constant as with respect to the partial derivative of position x after speed, density equivalence transformation, discuss a lot of compared with the traffic flow non-linear phenomena in shorted segment, these a lot of are coupled together compared with shorted segment is exactly the overall section that we will study, it is processed thinking and is similar to Integral Thought, think under synchronization the small or steady change of the traffic flow in short space, certainly, this short space is relative; Selecting system equilibrium point, as system initial value, utilizes MatCont to paint individual software, by choosing different parameters as the continuous variable parameter of system, and then obtains system branch, as Hopf branch, An Jie branch, BT branch, GH branch etc.;

5, for unsettled system take-off point design control program, make the unstable take-off point of system become Stable Branch point, or make unstable take-off point disappearance become the stabilized zone of system, i.e. the steady operational status of traffic flow:

Supposing the system traffic flow stable operation is at equilibrium point (η ₀, σ ₀) locate, at equilibrium point place, to model, this carries out linearization process, has:

$\left[\begin{array}{c}\frac{\∂\mathrm{\η}}{\mathrm{\σt}}\\ \frac{\∂\mathrm{\σ}}{\∂t}\end{array}\right]={\left[\begin{array}{cc}{f}_{11}(\mathrm{\η},\mathrm{\σ},{\mathrm{\ρ}}_m)& f_{12}(\mathrm{\η},\mathrm{\σ},{\mathrm{\ρ}}_m)\\ f_{21}(\mathrm{\η},\mathrm{\σ},{\mathrm{\ρ}}_m)& f_{22}(\mathrm{\η},\mathrm{\σ},{\mathrm{\ρ}}_m)\end{array}\right]}_{\begin{array}{c}\mathrm{\η}={\mathrm{\η}}_0\\ \mathrm{\σ}={\mathrm{\σ}}_0\end{array}}\left[\begin{array}{c}\mathrm{\η}\\ \mathrm{\σ}\end{array}\right]+\left[\begin{array}{c}{\mathrm{\Δ}}_{\mathrm{\η}}(\mathrm{\η},\mathrm{\σ},{\mathrm{\ρ}}_m,\frac{\∂\mathrm{\η}}{\∂x},\frac{\∂\mathrm{\σ}}{\∂x},\frac{{\∂}^2\mathrm{\η}}{\∂x^2},\frac{{\∂}^2\mathrm{\σ}}{\∂x^2})\\ {\mathrm{\Δ}}_{\mathrm{\σ}}(\mathrm{\η},\mathrm{\σ},{\mathrm{\ρ}}_m,\frac{\∂\mathrm{\σ}}{\∂x},\frac{\∂\mathrm{\η}}{\∂x},\frac{{\∂}^2\mathrm{\η}}{\∂x^2},\frac{{\∂}^2\mathrm{\σ}}{\∂x^2})\end{array}\right]$

Wherein, $\left\{\begin{array}{c}{f}_{11}(\mathrm{\η},\mathrm{\σ},{\mathrm{\ρ}}_m)=2\mathrm{\η}\·\mathrm{\π}[r(x,t),s(x,t)]\\ f_{12}(\mathrm{\η},\mathrm{\σ},{\mathrm{\ρ}}_m)=0\end{array}\right.,$ F ₂₁(η, σ, ρ _m), f ₂₂(η, σ, ρ _m) the expression of value second equation in model decide, and relevant to the size of equivalent speed, with

all about η, σ, ρ _m,

high-order infinitely small, note matrix ${\left[\begin{array}{cc}{f}_{11}(\mathrm{\η},\mathrm{\σ},{\mathrm{\ρ}}_m)& f_{12}(\mathrm{\η},\mathrm{\σ},{\mathrm{\ρ}}_m)\\ f_{21}(\mathrm{\η},\mathrm{\σ},{\mathrm{\ρ}}_m)& f_{22}(\mathrm{\η},\mathrm{\σ},{\mathrm{\ρ}}_m)\end{array}\right]}_{\begin{array}{c}\mathrm{\η}={\mathrm{\η}}_0\\ \mathrm{\σ}={\mathrm{\σ}}_0\end{array}}$ For A, when the eigenwert of A matrix is all containing negative real part, while dropping on S Left half-plane, model is at equilibrium point (η ₀, σ ₀) compared with stable in small neighbourhood;

From the expression formula of matrix A, can find out, the rate of change of the density that the size of the eigenwert of A and equivalent speed and the vehicle flowrate that is entered or rolled away from by ring road cause is relevant, by regulating its size can make traffic flow stable operation, therefore, by controlling the opening/closing time of ramp metering bar, regulate r (x, t) size, thereby change π [r (x, t), s (x, t)] size, regulates the size of equivalent speed by variable information display board is set, can make eigenwert that system meets matrix A all containing the condition of negative real part;

In actual control operation process, in requiring the section of controlling, according to actual traffic state, by control Entrance ramp adjuster bar opening/closing time, in the upper section in the larger section of flow or more front section, set the size of variable information display board speed, the ring road of usining enters blocked road flow as mode input, variable information display board is forced the display device of output regulated quantity as pressure speed and ring road, when regulating equivalent speed size, vehicles passing in and out number to ring road is controlled, thus realize traffic flow steadily, effectively operation.

The invention has the beneficial effects as follows: on the basis of existing model, by the new traffic model after substitution of variable, can access the relation of traffic jam issue and system stability, thereby adopting global stability angle to judge from macroscopic view whether traffic there will be blocks up or other abnormal occurrence, the Systems Theory analytical approach of branch-and-bound can be from the reason of explaining that in essence traffic non-linear phenomena produces, traffic control for blocked road, decision-making provides basic foundation, for concrete take-off point design control program, make traffic flow even running, can effectively alleviate blocked road traffic jam issue.

Below in conjunction with embodiment, the present invention is elaborated.

Embodiment

Take Jiang-Wu-Zhu macroscopic traffic flow as example, be specifically described:

1, Jiang-Wu-Zhu macroscopic traffic flow is described below:

$\left\{\begin{array}{c}\frac{\∂\mathrm{\ρ}}{\∂t}+\frac{\∂\left(\mathrm{\ρv}\right)}{\∂x}=\mathrm{\π}[r(x,t),s(x,t)]\\ \frac{\∂v}{\∂t}+v\frac{\∂v}{\∂x}=c_0\frac{\∂v}{\∂x}+\frac{V_e\left(\mathrm{\ρ}\right)-v}{\mathrm{\τ}}\end{array}\right.$

Wherein, the average density that ρ is vehicle, unit is veh/m, v is vehicle average velocity, and unit is m/s, and t is the time, unit is s, and x is and the distance of road starting point that unit is m, ρ, v are the function about x, t, π [r (x, t), s (x, t)] rate of change of the density that the vehicle flowrate that is entered or rolled away from by ring road for the t moment, x section causes, unit is veh/ms ^-1, r (x, t)=r _o(x, t)+r _qthe vehicle flowrate that (x, t) entered by ring road for the t moment, x section, s (x, t)=s _o(x, t)+s _qthe vehicle flowrate that (x, t) rolled away from by ring road for the t moment, x section, r _o(x, t), s _o(x, t) normal vehicle flowrate for sailing into, roll away from by ring road, r _qthe flow reducing amount that (x, t) causes for the ring road highway of controlling that No entry, s _qthe flow recruitment that (x, t) forces outgoing vehicles to cause for information display board, flux unit is veh/s, V _e(ρ) be equivalent speed function, c ₀for spreading of disturbance, unit is m/s, and τ is slack time, and unit is s, c ₀, τ is constant;

2, introducing state variable is:

$\left\{\begin{array}{c}\mathrm{\η}(x,t)=\frac{1}{{\mathrm{\ρ}}_m-\mathrm{\ρ}(x,t)}\\ \mathrm{\σ}(x,t)=\frac{1}{v(x,t)}\end{array}\right.$

Wherein, ρ _mtraffic flow density while occurring blocking for traffic, unit is veh/m, ρ (x, t) be the average density of vehicle, unit is veh/m, v (x, t) be the average velocity of vehicle, unit is m/s, as long as state variable η (x, t) >0 or σ (x, t) value of >0 increases, just illustrate that density increases or speed reduces, whole traffic system trends towards unstable, as state variable η (x, when t) >0 is tending towards infinite, that represent traffic density is tending towards saturated traffic density, produce traffic congestion, as state variable σ (x, when t) >0 is tending towards infinite, represent that vehicle average velocity goes to zero, produce traffic congestion,

Relation between η (x, t), σ (x, t) density and speed is brought in the unified macroscopic traffic flow of describing, and Jiang-Wu-Zhu macroscopic traffic flow new after substitution of variable is described as:

$\left\{\begin{array}{c}\frac{\∂\mathrm{\η}}{\∂t}=-\frac{1}{\mathrm{\σ}}\frac{\∂\mathrm{\η}}{\∂x}+\frac{{\mathrm{\ρ}}_m{\mathrm{\η}}^2-\mathrm{\η}}{{\mathrm{\σ}}^2}\frac{\∂\mathrm{\σ}}{\∂x}+{\mathrm{\η}}^2\·\mathrm{\π}[r(x,t),s(x,t)]\\ \frac{\∂\mathrm{\σ}}{\∂t}=(c_0-\frac{1}{\mathrm{\σ}})\frac{\∂\mathrm{\σ}}{\∂x}+\frac{1}{\mathrm{\τ}}[\mathrm{\σ}-{\mathrm{\σ}}^2\·V_e({\mathrm{\ρ}}_m,\mathrm{\η})]\end{array}\right.$

Wherein, η >0 is state variable, and σ >0 is state variable, and x is and the position of road starting point, and unit is m, and t is the time, and unit is s, V _e(ρ _m, η) be equivalent speed function, c ₀for spreading of disturbance, unit is m/s, and τ is slack time, and unit is s, c ₀, τ is constant, ρ _msaturation flow density during for traffic congestion, unit is veh/m;

3, solve the equilibrium solution of introducing the Jiang-Wu-Zhu macroscopic traffic flow after substitution of variable:

When system reaches stable (balance) state, should have:

$\left\{\begin{array}{c}\frac{\∂\mathrm{\η}}{\∂t}=0\\ \frac{\∂\mathrm{\σ}}{\∂t}=0\end{array}\right.$

Suppose to enter with the vehicle flowrate rolling away from and equate by ring road, i.e. π [r (x, t), s (x, t)]=0, now, model equilibrium point all meets following balance equation:

$\left\{\begin{array}{cc}(\mathrm{\η}-{\mathrm{\ρ}}_m{\mathrm{\η}}^2)\frac{\∂\mathrm{\σ}}{\∂x}+\mathrm{\σ}\frac{\∂\mathrm{\η}}{\∂x}=0& \left(a\right)\\ (c_0-\frac{1}{\mathrm{\σ}})\frac{\∂\mathrm{\σ}}{\∂x}+\frac{1}{\mathrm{\τ}}[\mathrm{\σ}-{\mathrm{\σ}}^2\·V_e({\mathrm{\ρ}}_m,\mathrm{\η})]=0& \left(b\right)\end{array}\right.$

To solving after (a) formula conversion in above-mentioned equation, have

And (b) formula of bringing into can be obtained the equilibrium solution of the rear model of conversion:

Wherein, η >0 is state variable, and σ >0 is state variable, and x is and the position of road starting point, and unit is m, and t is the time, and unit is s, V _e(ρ _m, η) be equivalent speed function, c ₀for spreading of disturbance, unit is m/s, c _mspreading of disturbance during for traffic jam, unit is m/s, ρ _msaturation flow density during for traffic congestion, unit is veh/m, V _ffor free travel speed, unit is m/s, and τ is slack time, and unit is s, c ₀, τ is constant, k ₀for or not 0 constant;

Conventionally, equivalent speed function V _e(ρ) can choose the following expression-form being proposed by people such as Del Castillo and (see document Del Castillo J M, Bennitez F G.On functional form of the speed-density relationship-I: General theory; II: Empirical investigation.Transpn Res B, 1995,29:373～406):

$V_e\left(\mathrm{\ρ}\right)=V_f\{1-\exp [1-\exp \left(\frac{c_m}{V_f}(\frac{{\mathrm{\ρ}}_m}{\mathrm{\ρ}}-1)\right)\left]\right\}$

Wherein, V _ffor free stream velocity, unit is m/s, c _mspreading of disturbance during for traffic jam, unit is m/s, ρ _mtraffic flow density while occurring blocking for traffic, unit is veh/m, and η >0 is state variable, and it is V after substitution of variable _e(ρ _m, being expressed as η):

$V_e({\mathrm{\ρ}}_m,\mathrm{\η})=V_f\{1-\exp [1-\exp \left(\frac{c_m}{V_f({\mathrm{\ρ}}_m\mathrm{\η}-1)}\right)\left]\right\}$

Wherein, V _ffor free travel speed, unit is m/s, c _mspreading of disturbance during for traffic jam, unit is m/s, ρ _msaturated traffic density during for traffic congestion, unit is veh/m, η >0 is state variable;

4, the equilibrium point from solving, searching system take-off point, carries out specificity analysis to the system take-off point meeting within the scope of actual physics meaning:

Meeting actual physics meaning scope is set as follows: because China's highway general speed is limited in 120km/h, be 33.33m/s, consider that the situation of hypervelocity often appears in vehicle in the performance of vehicle and actual traffic, here by speed restriction at 40m/s, during traffic congestion, speed is 0, therefore have σ ∈ [0.025 ,+∞); Consider that the size of automobile, generally at 4～5m, arranges ρ _m=0.2veh/m, therefore have

be η ∈ [5 ,+∞);

In model, other parameter initial value designs are as follows: V _f=33m/s, τ=7s, k ₀=1.5, ρ _m=0.2veh/m, c ₀=c _mwhether=6m/s, just can directly utilize the Systems Theory analytical approach of branch-and-bound to there will be and block up or other abnormal occurrence from global stability angle judgement traffic;

Owing to being for the stability problem of discussing system under long-time, when processing PDE model, regarding parameter or constant as with respect to the partial derivative of position x after speed, density equivalence transformation, discuss a lot of compared with the traffic flow non-linear phenomena in shorted segment, these a lot of are coupled together compared with shorted segment is exactly the overall section that we will study, it is processed thinking and is similar to Integral Thought, think under synchronization the small or steady change of the traffic flow in short space, certainly, this short space is relative; Because system exists numerous equilibrium point, take one of them equilibrium point is example as system initial value, utilize MatCont to paint individual software, by choosing different parameters as the continuous variable parameter of system, and then obtain system branch: the parameter that τ in model is considered as to continuous variable, other parameters are considered as invariant parameter, given system balancing point U ₀=(5.6,0.037), Selecting All Parameters ρ _m=0.2veh/m, V _f=33m/s, c ₀=c _m=6m/s, π [r (x, t), s (x, t)]=-0.000711, the branching diagram of drawing system, obtain occurring Hopf branch at place, τ=13.930272, its Lyapunov coefficient is-2.540208e-002 to be less than 0, Ji Gai Hopf branch is postcritical, be that periodic solution is stable, in τ=7 and place, τ=7.442513 there is LP(Limit Point) branch, at place, τ=7.139606, there is neutral type saddle point (neutral saddle), at this some place, there is the real character value of a pair of equal and opposite in direction, opposite direction, choose τ, c ₀for parameter, the stable Hopf branch producing when above-mentioned τ is changed carries out two-parameter variation continuation, has obtained GH (Generalized Hopf (Bautin)) branch and BT (Bogdanov-Takens) branch; Choose other parameters and can also obtain more take-off point as continuous variable parameter, due to traffic non-linear phenomena corresponding to different take-off points, the characteristic of take-off point can be from the reason of explaining that in essence corresponding traffic non-linear phenomena produces, the existence of as precritical in system Hopf branch is steady state solution and reason unstable solution and that deposit of system;

5, for unsettled system take-off point design control program, make the unstable take-off point of system become Stable Branch point, or unstable take-off point disappeared become the stabilized zone of system, and then make traffic flow in steady operational status:

Supposing the system traffic flow stable operation is at equilibrium point (η ₀, σ ₀) locate, at equilibrium point place, to model, this carries out linearization process, has:

$\left[\begin{array}{c}\frac{\∂\mathrm{\η}}{\mathrm{\σt}}\\ \frac{\∂\mathrm{\σ}}{\∂t}\end{array}\right]={\left[\begin{array}{cc}{f}_{11}(\mathrm{\η},\mathrm{\σ},{\mathrm{\ρ}}_m)& f_{12}(\mathrm{\η},\mathrm{\σ},{\mathrm{\ρ}}_m)\\ f_{21}(\mathrm{\η},\mathrm{\σ},{\mathrm{\ρ}}_m)& f_{22}(\mathrm{\η},\mathrm{\σ},{\mathrm{\ρ}}_m)\end{array}\right]}_{\begin{array}{c}\mathrm{\η}={\mathrm{\η}}_0\\ \mathrm{\σ}={\mathrm{\σ}}_0\end{array}}\left[\begin{array}{c}\mathrm{\η}\\ \mathrm{\σ}\end{array}\right]+\left[\begin{array}{c}{\mathrm{\Δ}}_{\mathrm{\η}}(\mathrm{\η},\mathrm{\σ},{\mathrm{\ρ}}_m,\frac{\∂\mathrm{\η}}{\∂x},\frac{\∂\mathrm{\σ}}{\∂x},\frac{{\∂}^2\mathrm{\η}}{\∂x^2},\frac{{\∂}^2\mathrm{\σ}}{\∂x^2})\\ {\mathrm{\Δ}}_{\mathrm{\σ}}(\mathrm{\η},\mathrm{\σ},{\mathrm{\ρ}}_m,\frac{\∂\mathrm{\σ}}{\∂x},\frac{\∂\mathrm{\η}}{\∂x},\frac{{\∂}^2\mathrm{\η}}{\∂x^2},\frac{{\∂}^2\mathrm{\σ}}{\∂x^2})\end{array}\right]$

Wherein, $\left\{\begin{array}{c}{f}_{11}(\mathrm{\η},\mathrm{\σ},{\mathrm{\ρ}}_m)=2\mathrm{\η}\·\mathrm{\π}[r(x,t),s(x,t)]\\ f_{12}(\mathrm{\η},\mathrm{\σ},{\mathrm{\ρ}}_m)=0\\ f_{21}(\mathrm{\η},\mathrm{\σ},{\mathrm{\ρ}}_m)=\frac{1}{\mathrm{\τ}}\left\{\right[1-2\mathrm{\σ}\·V_e({\mathrm{\ρ}}_m,\mathrm{\η})]\\ f_{22}(\mathrm{\η},\mathrm{\σ},{\mathrm{\ρ}}_m)=-\frac{1}{\mathrm{\τ}}[{\mathrm{\σ}}^2\·\frac{\∂V_e({\mathrm{\ρ}}_m,\mathrm{\η})}{\∂\mathrm{\η}}]\end{array}\right.,$

with

all about η, σ, ρ _m,

high-order infinitely small, note matrix ${\left[\begin{array}{cc}{f}_{11}(\mathrm{\η},\mathrm{\σ},{\mathrm{\ρ}}_m)& f_{12}(\mathrm{\η},\mathrm{\σ},{\mathrm{\ρ}}_m)\\ f_{21}(\mathrm{\η},\mathrm{\σ},{\mathrm{\ρ}}_m)& f_{22}(\mathrm{\η},\mathrm{\σ},{\mathrm{\ρ}}_m)\end{array}\right]}_{\begin{array}{c}\mathrm{\η}={\mathrm{\η}}_0\\ \mathrm{\σ}={\mathrm{\σ}}_0\end{array}}$ For A, when the eigenwert of A matrix is all containing negative real part, while dropping on S Left half-plane, model is at equilibrium point (η ₀, σ ₀) compared with stable in small neighbourhood;

From the expression formula of matrix A, can find out, the rate of change of the density that the size of the eigenwert of A and equivalent speed and the vehicle flowrate that is entered or rolled away from by ring road cause is relevant, by regulating its size can make traffic flow stable operation, therefore, by controlling the opening/closing time of Entrance ramp adjuster bar, regulate r (x, t) size, thereby change π [r (x, t), s (x, t)] size, regulates the size of equivalent speed by variable information display board is set, can make eigenwert that system meets matrix A all containing the condition of negative real part;

In actual control operation process, in requiring the section of controlling, according to actual traffic state, by control Entrance ramp adjuster bar opening/closing time, in the upper section in the larger section of flow or more front section, set the size of variable information display board speed, the ring road of usining enters blocked road flow as mode input, variable information display board is forced the display device of output regulated quantity as pressure speed and ring road, when regulating equivalent speed size, vehicles passing in and out number to ring road is controlled, thus realize traffic flow steadily, effectively operation.

Claims (1)

1. for a macroscopic traffic flow branch-and-bound control and analysis for blocked road, its feature comprises the following steps:

(1) macroscopic traffic flow of setting up according to hydrokinematics model can be unified to be described below:

Wherein, the average density that ρ is vehicle, unit is veh/m, the average velocity that v is vehicle, unit is m/s, and t is the time, and unit is s, and x is and the distance of road starting point, and unit is m, and ρ, v are the function about x, t, V _e(ρ) be equivalent speed function, f be about

v _e(ρ) function, the rate of change of the density that the vehicle flowrate that π [r (x, t), s (x, t)] is entered or rolled away from by ring road for the t moment, x section causes, unit is veh/ms ^-1, r (x, t)=r _o(x, t)+r _qthe vehicle flowrate that (x, t) entered by ring road for the t moment, x section, s (x, t)=s _o(x, t)+s _qthe vehicle flowrate that (x, t) rolled away from by ring road for the t moment, x section, r _o(x, t), s _o(x, t) normal vehicle flowrate for sailing into, roll away from by ring road, r _qthe flow reducing amount that (x, t) causes for the ring road highway of controlling that No entry, s _qthe flow recruitment that (x, t) forces outgoing vehicles to cause for information display board, flux unit is veh/s;

(2) introducing state variable is:

Wherein, ρ _mtraffic flow density while occurring blocking for traffic, unit is veh/m, ρ (x, t) be the average density of vehicle, unit is veh/m, v (x, t) be the average velocity of vehicle, unit is m/s, as long as state variable η (x, t) >0 or σ (x, t) value of >0 increases, just illustrate that density increases or speed reduces, whole traffic system trends towards unstable, as state variable η (x, when t) >0 is tending towards infinite, that represent traffic density is tending towards saturated traffic density, produce traffic congestion, as state variable σ (x, when t) >0 is tending towards infinite, represent that vehicle average velocity goes to zero, produce traffic congestion,

Relation between η (x, t), σ (x, t) density and speed is brought in the unified macroscopic traffic flow of describing, and macroscopic traffic flow new after substitution of variable is described as:

Wherein, η >0 is state variable, and σ >0 is state variable, and x is and the position of road starting point, and unit is m, and t is the time, and unit is s, V _e(ρ _m, η) be equivalent speed function, ρ _msaturated traffic density during for traffic congestion, unit is veh/m, when state variable η is tending towards infinite, that represent traffic density is tending towards saturated traffic density, produces traffic congestion, when state variable σ is tending towards infinite, represent that vehicle average velocity goes to zero, produce traffic congestion;

(3) according to the new macroscopic traffic flow after substitution of variable, the equilibrium solution of solving system:

When system reaches stable (balance) state, should have:

Suppose to enter with the vehicle flowrate rolling away from and equate by ring road, i.e. π [r (x, t), s (x, t)]=0, now, the equilibrium point of the model after substitution of variable meets following balance equation:

To solving after (a) formula conversion in above-mentioned equation, have

And (b) formula of bringing into can obtain conversion after the equilibrium solution of model be:

Conventionally, equivalent speed function V _e(ρ) can choose the following expression-form being proposed by people such as Del Castillo and (see document Del Castillo J M, Bennitez F G. On functional form of the speed-density relationship-I: General theory; II: Empirical investigation.Transpn Res B, 1995,29:373～406):

Wherein, V _ffor free stream velocity, unit is m/s, c _mspreading of disturbance during for traffic jam, unit is m/s, ρ _mtraffic flow density while occurring blocking for traffic, unit is veh/m, and η >0 is state variable, and it is V after substitution of variable _e(ρ _m, being expressed as η):

Each parameter meaning is the same;

(4) take equilibrium point as initial point, utilize MatCont software searching system take-off point, the system take-off point meeting within the scope of actual physics meaning is carried out to specificity analysis, be i.e. traffic non-linear phenomena corresponding to different take-off points:

The scope that meets actual physics meaning refers to that the selection of model parameters under different road conditions meets the condition of actual traffic operation, the corresponding actual density of scope of state variable η, σ, the boundary value of speed;

Owing to being for the stability problem of discussing system under long-time, when processing PDE model, regarding parameter or constant as with respect to the partial derivative of position x after speed, density equivalence transformation, discuss a lot of compared with the traffic flow non-linear phenomena in shorted segment, these a lot of are coupled together compared with shorted segment is exactly the overall section that we will study, it is processed thinking and is similar to Integral Thought, think under synchronization the small or steady change of the traffic flow in short space, certainly, this short space is relative; Selecting system equilibrium point, as system initial value, utilizes MatCont to paint individual software, by choosing different parameters as the continuous variable parameter of system, and then obtains system branch, as Hopf branch, An Jie branch, BT branch, GH branch etc.;

(5) for unsettled system take-off point design control program, make the unstable take-off point of system become Stable Branch point, or make unstable take-off point disappearance become the stabilized zone of system, i.e. the steady operational status of traffic flow:

Supposing the system traffic flow stable operation is at equilibrium point (η ₀, σ ₀) locate, at equilibrium point place, to model, this carries out linearization process, has:

Wherein,

f ₂₁(η, σ, ρ _m), f ₂₂(η, σ, ρ _m) the expression of value second equation in model decide, and relevant to the size of equivalent speed,

with

all about η, σ, ρ _m,

high-order infinitely small, note matrix

for A, when the eigenwert of A matrix is all containing negative real part, while dropping on S Left half-plane, model is at equilibrium point (η ₀, σ ₀) compared with stable in small neighbourhood;

From the expression formula of matrix A, can find out, the rate of change of the density that the size of the eigenwert of A and equivalent speed and the vehicle flowrate that is entered or rolled away from by ring road cause is relevant, by regulating its size can make traffic flow stable operation, therefore, by controlling the opening/closing time of ramp metering bar, regulate r (x, t) size, thereby change π [r (x, t), s (x, t)] size, regulates the size of equivalent speed by variable information display board is set, can make eigenwert that system meets matrix A all containing the condition of negative real part;

In actual control operation process, in requiring the section of controlling, according to actual traffic state, by control Entrance ramp adjuster bar opening/closing time, in the upper section in the larger section of flow or more front section, set the size of variable information display board speed, the ring road of usining enters blocked road flow as mode input, variable information display board is forced the display device of output regulated quantity as pressure speed and ring road, when regulating equivalent speed size, vehicles passing in and out number to ring road is controlled, thus realize traffic flow steadily, effectively operation.