CN108830401B - Dynamic congestion charging optimal rate calculation method based on cellular transmission model - Google Patents

Dynamic congestion charging optimal rate calculation method based on cellular transmission model Download PDF

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CN108830401B
CN108830401B CN201810418436.0A CN201810418436A CN108830401B CN 108830401 B CN108830401 B CN 108830401B CN 201810418436 A CN201810418436 A CN 201810418436A CN 108830401 B CN108830401 B CN 108830401B
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刘志远
王路濛
程启秀
俞俊
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Southeast University
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Abstract

The invention discloses a dynamic congestion charging optimal rate calculation method based on a cellular transmission model, compared with the existing algorithm, the dynamic congestion charging optimal rate calculation method based on the cellular transmission model provided by the invention is fairer and more effective, and the obtained optimal rate enables the time consumption of the whole road network to be minimum; compared with the original cellular transmission model, the cellular transmission model based on the path has the advantages of no need of calculating the waiting time of the cellular and the like, so that the calculation is simpler, more convenient and more efficient.

Description

Dynamic congestion charging optimal rate calculation method based on cellular transmission model
Technical Field
The invention relates to the technical field of urban traffic management and control, in particular to a dynamic congestion charging optimal rate calculation method based on a cellular transmission model.
Background
Along with the rapid advance of urbanization and the increasing improvement of the living standard of residents, the motor vehicle ownership of each large city in China is continuously increased, the problem of urban traffic jam is increased, and the problems of environmental pollution, energy waste and the like are also brought. Urban road congestion charging is a means of traffic demand management, and is characterized in that a car user changes a travel route, adjusts travel time or changes a travel mode by utilizing a price lever, so that the purposes of regulating and controlling traffic demand total and optimizing a traffic travel structure are achieved.
Road congestion charging is one of the economic means of traffic demand management, and is increasingly paid more attention by traffic management departments. Since the policy of road congestion charging was implemented by the government of singapore in 1975, a plurality of countries and cities (such as norway, london, stockholm, milan and the like) have implemented the policy of road congestion charging successively, and remarkable effect is achieved in the aspect of relieving urban traffic congestion. Because the warning line charging mode has the advantages of easy implementation, effectiveness and the like, the warning line charging mode is almost adopted in the regions: in the city part area, a toll area is determined by a warning line, and certain road congestion fees are collected for motor vehicles entering the toll area. However, all cities implementing the road congestion charging policy adopt a single pricing method, the method ignores the travel distance of the vehicle in the guard line and the road congestion condition, and the same fee is collected for users with different travel distances in the guard line under different congestion conditions, which brings the problem of unfair charging. In addition, some users may intentionally increase the use of road segments in the warning line to improve the economic utility of the congestion fees paid by the users, which not only cannot alleviate traffic congestion, but also increases congestion in the warning line. The congestion charging measures are generally targeted, charging is carried out aiming at areas which are easy to cause traffic congestion, such as urban central business areas, express ways, outer loops, urban business areas and the like, the cities are generally provided with higher private motor vehicle traveling proportion and perfect loops, the traffic congestion areas are easy to divide, and unsaturated road networks or developed public transportation which exist outside the congestion charging areas can bear huge traffic volume transferred after congestion charging is carried out.
Disclosure of Invention
The invention aims to provide a dynamic congestion charging optimal rate calculation method based on a cellular transmission model, which can greatly save the total travel time of the whole road network system.
In order to solve the technical problem, the invention provides a dynamic congestion charging optimal rate calculation method based on a cellular transmission model, which comprises the following steps:
(1) inputting relevant data of an urban traffic network, and establishing a traffic network topological graph;
(2) based on the requirement of warning line charging, determining each entrance of a charging area as a charging starting point and each exit as a charging end point, obtaining the length of each path in the charging area, and establishing a general expression of a combined charging function based on distance and congestion, which changes along with time;
(3) a cellular transmission model based on a path is adopted to simulate the propagation process of traffic flow, and further the average route in-transit travel time is calculated;
(4) establishing a double-layer optimization model, and determining a combined congestion rate which meets a dynamic user balance principle and changes along with time and is based on distance and congestion by optimizing the whole system;
(5) and (4) applying a hybrid adaptive gradient projection algorithm and an artificial bee colony algorithm to solve the double-layer optimization model proposed in the step (4) and output the optimal rate.
The dynamic charging provided by the invention is embodied in two aspects, one is that the path selection decision of the vehicle follows the dynamic user balance principle, and the component of the dynamic traffic flow is represented by a path-based cellular transmission model; the second is a combined charging function which is established in the step (2) and changes along with time and is based on distance and congestion.
Preferably, in step (2), the general expression of the joint charging function based on distance and congestion is established as follows: let road network G be (N, a) with N nodes and a links, W denotes the set of origin-destination pairs OD, PWRepresents the set of paths between the ODs, assuming that the minimum and maximum lengths within the toll fence are l0And lKThe distance of the journey may be divided into K equal intervals, the distance length function of each interval may be represented by two end points, and the vertex of each charging interval is defined as l (l)0,l1,…,lk,…,lK)TThe corresponding charges are respectively phi ═ phi0,φ1,…,φk,…,φK)T，
Figure BDA0001650002730000026
Representing the length of the kth charging interval, the charging interval distance-based charging function phi (l) can be expressed as:
Figure BDA0001650002730000021
representing the road congestion condition by using the delay time of each road section, and then using the OD point to carry out charging function based on the congestion level on the path p between the W
Figure BDA0001650002730000022
Is expressed as:
Figure BDA0001650002730000023
wherein
Figure BDA0001650002730000025
Indicating the congestion-based charging rate on the path p between the OD point pair w,
Figure BDA0001650002730000024
indicating the delay time, t0The free road travel time of the road section is represented, beta is a congestion rate, and a general expression of a combined charging function tau (l, beta) based on distance and congestion can be represented as follows:
Figure BDA0001650002730000031
equation (3) represents a joint charging function based on distance and congestion, wherein,
Figure BDA0001650002730000032
represents the combined charging rate on the path p between the OD point pair w based on the distance and the congestion, theta1And theta2Weights representing distance charge and congestion charge, respectively;
the method takes half an hour as a time unit, and calculates the congestion rate again according to the traffic demand every half an hour; assume that the time charged per day is from 7 in the morning: 30 to 18 pm: 30, the present invention was studied for morning commute traffic only; thus, the time charged is from morning 7:30 to 9: 30. because the charging is changed once every half hour, the two hours are divided into four subintervals, and then a charging expression of the four subintervals is generated;
for the d sub-interval, a combined dynamic congestion charging function tau based on distance and congestiondThe general expression of (l, β) can be expressed as：
Figure BDA0001650002730000033
Figure BDA0001650002730000034
Representing the joint charging rate based on distance and congestion on the path p between the OD point pair w in the d interval;
the exact time the vehicle arrives in the toll fence area can be determined from the travel time from the starting point to the inside of the toll fence area, so that the toll time interval d can be correctly calculated:
Figure BDA0001650002730000035
wherein the content of the first and second substances,
Figure BDA0001650002730000036
and t' and l respectively represent the travel time from the starting point to the charging warning line area and the time interval length in minutes in the cellular transmission model.
Preferably, in the step (3), a cellular transmission model based on a path is adopted to simulate a traffic flow propagation process, and further the average route journey time is calculated, and the specific method is as follows: the road segment is divided into basic units, the duration is divided into time intervals, and the basic form of the cellular transmission model can be expressed as follows:
Figure BDA0001650002730000037
Figure BDA0001650002730000038
Figure BDA0001650002730000039
representing the amount of traffic of cell i at the t time step,
Figure BDA00016500027300000310
represents the amount of traffic, Q, moving from the upstream cell i to the downstream cell i +1 at the t-th time stepiIndicates the maximum number of vehicles flowing from cell i, Ni+1Representing the congestion propagation coefficient, gamma representing the ratio of the backward inflow speed to the forward speed, the cell length usually being the distance that the vehicle travels at the speed of the free stream in one time step;
three conditions of unit cell connection, road section convergence and divergence need to be considered when a cellular transmission model is constructed, and the constructed cellular transmission model comprises five cells and three links; the five kinds of cells are respectively: the upper and lower streams of the common cells are respectively connected with a cell; the upper stream of the divergent cells is only connected with one cell, and the lower stream of the divergent cells is connected with two cells; converging cells, wherein the upstream is connected with two cells, and the downstream is connected with only one cell; a source cell having only downstream cells and no upstream cells; a terminal cell, having only an upstream cell and no downstream cell; the three links are respectively a common link, a divergent link and a convergent link;
setting values of the initialized cells:
Figure BDA0001650002730000041
Figure BDA0001650002730000042
five types of cells can be represented as:
a source cell:
Figure BDA0001650002730000043
a normal cell:
Figure BDA0001650002730000044
divergent cells and convergent cells:
Figure BDA0001650002730000045
a final cell:
Figure BDA0001650002730000046
wherein the content of the first and second substances,
Figure BDA0001650002730000047
representing the occupancy of the cell i on the path p at the beginning of the time interval t,
Figure BDA0001650002730000048
represents the amount of traffic, h, flowing from cell i to cell j on path p at the beginning of time interval tp,tIndicating the rate of departure of the vehicle on the path p during the time interval t. For parameter
Figure BDA0001650002730000049
When cell i (or j, k) is on path p,
Figure BDA00016500027300000410
otherwise
Figure BDA00016500027300000411
CD、CM、CR、CS、CORespectively representing the set of divergent cells, convergent cells, source cells, final cells and common cells; gamma-shapediIs a downstream set of cells from cell i,
Figure BDA0001650002730000051
an upstream set of cells that is cell i; t represents a maximum time range;
three types of links can be represented as:
and (3) common linking:
Figure BDA0001650002730000052
divergent chaining:
Figure BDA0001650002730000053
convergence linking:
Figure BDA0001650002730000054
wherein E isD、EM、EORespectively representing a set of divergent links, convergent links and normal links,
Figure BDA0001650002730000055
the aggregate occupancy of time interval t initially that will flow from divergent cell i into cell j,
Figure BDA0001650002730000056
representing the average value of occupancy rates of the cells i on the path p at the beginning of the time interval t, wherein mu is a number which is infinitesimal and larger than zero;
in the above-described cell transmission model, the initial cell occupancy and the outgoing traffic volume are assumed to be zero in equations (8) to (9); equations (10) - (13) represent a path-based cell update procedure; equations (14) - (16) represent path-based traffic flow propagation constraints; wherein the conversion rate of the path-based cellular transport model is not exogenous, but is uniquely determined by the supply and demand of upstream and downstream cells;
in the invention, a path selection decision of a vehicle is set to follow a dynamic user balance principle, a cellular transmission model based on a path is applied to a dynamic user balance problem, and the uniquely determined average on-road travel time of a vehicle which starts at the same time is determined by calculating the actual path travel time output from the cellular transmission model;
the equalization condition for an ideal dynamic user equalization state may be expressed as: the total generalized cost generated by passengers who simultaneously depart between each OD pair is equal and minimum;
the mathematical expression of the dynamic user balance theory is as follows:
Figure BDA0001650002730000061
Figure BDA0001650002730000062
wherein the content of the first and second substances,
Figure BDA0001650002730000063
Tde T denotes a set of charging time intervals over time,
Figure BDA0001650002730000064
represents the minimum path travel cost of OD to w in the time interval t, alpha is the time value,
Figure BDA0001650002730000065
represents the generalized travel cost away from the OD for the path p on w during the time interval t. Path flow
Figure BDA0001650002730000066
Is about the congestion rate
Figure BDA0001650002730000067
Thus, the objective of the dynamic customer balancing problem translates into finding a feasible flow that satisfies both equations (17) and (18), as well as the demand flow conservation principle (20) and the non-negative constraint (21)
Figure BDA0001650002730000068
Figure BDA0001650002730000069
f≥0,u≥0 (21)
Wherein u represents
Figure BDA00016500027300000610
The vector of (a), i.e.,
Figure BDA00016500027300000611
the dynamic user equalization problem described by equations (17) to (21) above is equivalent to the problem of the finite variable inequality in equation (22):
Figure BDA00016500027300000612
in the formula (22), the upper corner mark represents the optimal solution, and Ψ represents
Figure BDA00016500027300000613
The column vector of (a) is,
Figure BDA00016500027300000614
Ω represents a set of feasible solutions that satisfy the demand flow conservation principle (20) and the non-negative constraints (21);
the output of the cellular transmission model is the cellular occupancy of each time interval, and the accumulated leaving traffic volume of the source cell r on the path p at the beginning of the time interval t
Figure BDA0001650002730000071
Equal to the cumulative amount of traffic on path p leaving from cell r at the beginning of time interval t-1
Figure BDA0001650002730000072
And the traffic volume of the cell r on the path p flowing out in the time interval t
Figure BDA0001650002730000073
The sum of (a); the final cell s on the path p is in the initial time intervalCumulative amount of traffic arriving at the start of ω
Figure BDA0001650002730000074
Equal to the cumulative arrival traffic volume of the final cell s on the path p at the beginning of the time interval omega-1
Figure BDA0001650002730000075
And the accumulated inflow traffic volume of the terminal cell s on the path p in the time interval omega-1
Figure BDA0001650002730000076
The sum of (c) can be expressed as a mathematical formula:
Figure BDA0001650002730000077
Figure BDA0001650002730000078
wherein the content of the first and second substances,
Figure BDA0001650002730000079
the accumulated traffic volume leaving from the cell r on the path p at the beginning of the time interval t; in fact, in the case of time discretization, the slave cells on the path p within the time interval trCumulative amount of traffic leaving
Figure BDA00016500027300000710
Not all of the endpoints s are necessarily reached simultaneously; the invention adopts the average on-road travel time to replace the actual on-road travel time of the vehicles which leave the path at the same time, and introduces two arrival time coefficients omega1And ω2，ω1Is to satisfy
Figure BDA00016500027300000711
Minimum time coefficient value of condition, ω2Is to satisfy
Figure BDA00016500027300000712
Condition of the mostSmall time coefficient values. The average travel time of the traffic flow leaving the path p connecting the OD pair w within the time interval t
Figure BDA00016500027300000713
The expression of (a) is:
Figure BDA00016500027300000714
preferably, in the step (4), a double-layer optimization model is established, and a combined congestion rate which meets the dynamic user balance principle, changes along with time and is based on distance and congestion is determined by optimizing the whole system; the upper layer is a road network system optimization model, and the purpose is to minimize the total travel time of a road network; the lower layer is a dynamic user equilibrium state and can be described as a multivariate inequality; the specific expression is as follows:
and (3) upper layer:
Figure BDA0001650002730000081
the lower layer:
Figure BDA0001650002730000082
preferably, in the step (5), the method for solving the double-layer optimization model in the step (4) adopts a self-adaptive gradient projection algorithm to solve the problem of the lower-layer multivariate inequality; an artificial bee colony algorithm is adopted to solve an upper-layer optimization model; the method for solving the problem of the lower multivariate inequality by adopting the adaptive gradient projection algorithm specifically comprises the following steps:
(51) is provided with
Figure BDA00016500027300000814
u∈[0.5,1]；ε>0，ρmax>0，ρ0>0，f0∈Ω；γ0＝ρ0，k＝0；
(52) Find satisfaction
Figure BDA0001650002730000083
Minimum non-negative integer of condition lkUpdating non-shortest path traffic that satisfies the constraint (29):
Figure BDA0001650002730000084
Figure BDA0001650002730000085
wherein
Figure BDA0001650002730000086
Is the shortest path between OD and w in the kth iteration;
Figure BDA0001650002730000087
Fkrepresenting a vector
Figure BDA0001650002730000088
Figure BDA0001650002730000089
Representing a vector
Figure BDA00016500027300000810
Updating the shortest path traffic:
Figure BDA00016500027300000811
(53) if inequality (30) is satisfied, then
Figure BDA00016500027300000812
Else gammak+1＝ρk+1；
Figure BDA00016500027300000813
(54) If the predetermined convergence criterion is satisfied, stopping the algorithm to obtain the final result fk+1(ii) a Otherwise, let k be k +1,and returning to the step (52).
Preferably, in the step (5), the optimization model for solving the upper layer by using the artificial bee colony algorithm specifically comprises:
(a) setting the population size NcNumber of leading bees NeNumber of follower bees NoThreshold limit; setting an iteration initial value I to be 1 and a maximum iteration number Imax；
(b) Randomly generating an initial solution, calculating the fitness value of each solution corresponding to each leading bee, and initializing a threshold value to be zero;
(c) and carrying out local search according to the existing solution and estimating the fitness value of the solution. If the solution is better, replacing the existing solution with the newly generated local solution, and resetting the threshold value to be zero; otherwise, the existing solution is unchanged but the threshold is set to 1;
(d) and selecting leading bees by adopting a roulette method for each following bee, namely generating a uniformly distributed random number r at [0,1], and if the probability is greater than r, generating a new solution around the existing solution by the following bee and calculating the fitness value of the new solution. If this solution is more optimal, the existing solution is replaced by the neighboring solution, otherwise the existing solution is not changed and the threshold is increased to 1;
(e) selecting a solution with the highest fitness value according to the existing solution, if a solution which cannot continuously improve the fitness value within a threshold value but is not the optimal solution with the highest fitness value exists, converting the corresponding leading bee role into a scout bee, searching again in an adjacent space to generate a new random solution, and resetting the threshold value to be 0;
(f) if I is equal to I +1, judging whether the algorithm meets the termination condition, and if I is equal to I +1, judging whether the algorithm meets the termination condition<ImaxGo to step (c); otherwise, the algorithm is terminated to obtain the optimal solution.
The invention has the beneficial effects that: different congestion charging functions can enable a road network system to generate different balance flow and total social benefits, and if the path selection behavior of a user obeys the generalized random user balancing principle, the optimal rate calculation problem aims to find a proper rate so that the total social benefits reach the maximum value; compared with the existing algorithm, the dynamic congestion charging optimal rate calculation method based on the cellular transmission model is fairer and more effective, and the obtained optimal rate enables the time consumption of the whole road network to be minimum; compared with the original cellular transmission model, the cellular transmission model based on the path has the advantages of no need of calculating the waiting time of the cellular and the like, so that the calculation is simpler, more convenient and more efficient.
Drawings
FIG. 1 is a schematic flow chart of the method of the present invention.
Fig. 2 is a schematic diagram of the Nguyen-Dupuis network with police line charging area according to the present invention.
FIG. 3 is a schematic diagram of the cell type of the Nguyen-Dupuis network of the present invention.
Detailed Description
As shown in fig. 1, a dynamic congestion charging optimal rate calculation method based on a cellular transmission model includes the following steps:
(1) inputting relevant data of an urban traffic network, and establishing a traffic network topological graph;
(2) based on the requirement of warning line charging, determining each entrance of a charging area as a charging starting point and each exit as a charging end point, obtaining the length of each path in the charging area, and establishing a general expression of a combined charging function based on distance and congestion, which changes along with time;
(3) a cellular transmission model based on a path is adopted to simulate the propagation process of traffic flow, and further the average route in-transit travel time is calculated;
(4) establishing a double-layer optimization model, and determining a combined congestion rate which meets a dynamic user balance principle and changes along with time and is based on distance and congestion by optimizing the whole system;
(5) and (4) applying a hybrid adaptive gradient projection algorithm and an artificial bee colony algorithm to solve the double-layer optimization model proposed in the step (4) and output the optimal rate.
The method comprises the following steps: and inputting relevant data (points, lines and partitions) of the target urban traffic network to obtain a traffic network topological graph.
Fig. 2 shows a network structure of an embodiment, which uses an Nguyen-Dupuis network in this example, and includes 13 nodes, 19 segments, 25 paths, and 4 pairs of OD points, and a charging area of a fence line is shown in a dashed line. The traffic demand and route information of the present network are shown in table 1.
TABLE 1 traffic demand and Path information Table for networks
Figure BDA0001650002730000101
Figure BDA0001650002730000111
Step two: based on the requirement of warning line charging, determining each entrance of a charging area as a charging starting point and each exit as a charging end point, obtaining the length of each path in the charging area, and establishing a general expression of a combined charging function based on distance and congestion.
The minimum and maximum lengths of the warning line charging area in this example are 3.2 km and 5.6 km, respectively, and the range is 2.4 km. Therefore, we assume that the piecewise linear charging function has 4 vertices and 3 linear charging intervals, each interval being 0.8km in length.
In the example, the congestion rate is calculated again according to the traffic demand every half hour by taking half an hour as a time unit. The invention is only studied for morning commuter traffic. Thus, the time charged is from morning 7:30 to 9 in the morning: 30. since the charging rate varies every half hour, the two hours are divided into four sub-intervals, and a charging expression of the four sub-intervals is generated.
Let θ1＝0.6，θ2＝0.4，φmin＝1.0，φmaxFor the d-th sub-interval, the general expression of the combined dynamic congestion charging function based on distance and congestion can be expressed as:
Figure BDA0001650002730000112
wherein the content of the first and second substances,
Figure BDA0001650002730000121
Figure BDA0001650002730000122
step three: and simulating the propagation process of traffic flow by adopting a cellular transmission model based on the path, and further calculating the average route journey time. This example includes 63 cells, as shown in FIG. 3.
In the example, the free flow speed is 48km/h, the congestion density is 125vehicles/km, the cell length is 0.8km, the backward impact speed is 18km/h, the length of each time interval is 1 minute, and the traffic flow carrying capacity on the double lanes is 1800vehicles/h per lane.
The road segment is divided into basic units, the duration is divided into time intervals, and the basic form of the cellular transmission model can be expressed as follows:
Figure BDA0001650002730000123
Figure BDA0001650002730000124
according to the output of the cellular transmission model, the average travel time of the traffic flow leaving from the path p connecting the OD pair w in the time interval t is calculated
Figure BDA0001650002730000125
Figure BDA0001650002730000126
Step four: a dynamic congestion rate calculation method is provided, a double-layer optimization model is established, and a combined congestion rate which meets the dynamic user balance principle and changes with time and is based on distance and congestion is determined by optimizing the whole system.
The double-layer optimization model is as follows:
and (3) upper layer:
Figure BDA0001650002730000127
the lower layer:
Figure BDA0001650002730000131
step five: and a mixed self-adaptive gradient projection algorithm and an artificial bee colony algorithm are adopted to solve the double-layer optimization model in the fourth step.
A. Solving the lower-layer multivariate inequality by adopting a self-adaptive gradient projection algorithm
Step 0: is provided with
Figure BDA00016500027300001313
u＝0.6，ρ0＝1.0，ε＝0.001，f0E is omega; let gamma0＝ρ0，k＝0。
Step 1: find satisfaction
Figure BDA0001650002730000132
Minimum non-negative integer of condition lkUpdating the non-shortest path flow that satisfies the constraint (40):
Figure BDA0001650002730000133
Figure BDA0001650002730000134
wherein
Figure BDA0001650002730000135
Is the shortest path between OD and w in the kth iteration;
Figure BDA0001650002730000136
Fkrepresenting a vector
Figure BDA0001650002730000137
Figure BDA0001650002730000138
Representing a vector
Figure BDA0001650002730000139
Then the shortest path flow is updated:
Figure BDA00016500027300001310
step 2: if the condition of inequality (29) is satisfied, then select
Figure BDA00016500027300001311
Else gammak+1＝ρk+1。
Figure BDA00016500027300001312
And step 3: stopping the algorithm to obtain f if a predetermined convergence criterion is metk+1As a final result; otherwise, setting k to k +1, and returning to the step 1.
B. And the artificial bee colony algorithm is adopted to solve the problem of system optimization of the upper layer.
Step 1, set population size NcNumber of leading bees N40eSet the threshold limit to 2, and the maximum number of iterations I at 20max＝500。
Step 2: an initial solution (e.g., honey source) is randomly generated, a fitness value of each solution corresponding to each leading bee is calculated, and an initialization threshold value is zero.
And step 3: and carrying out local search according to the existing solution, estimating the fitness value of the solution, if the solution is better, replacing the existing solution with the newly generated local solution, and resetting the threshold value to be zero, otherwise, the existing solution is not changed but the threshold value is set to be 1.
And 4, step 4: selecting leading bees by roulette method for each following bee, namely in [0,1]]Generating a uniformly distributed random numberrIf the probability is greater thanrThe follower bee generates a new solution around the existing solution and calculates its fitness value. If this solution is better, the existing solution is replaced with the neighboring solution, otherwise the existing solution is not changed and the threshold is increased to 1.
And 5: and selecting the solution with the highest fitness value according to the existing solutions. If one solution is not the optimal solution with the highest fitness value but cannot be continuously improved within the threshold value, the corresponding leading bee role is changed into a scout bee, searching is carried out again in the adjacent space, a new random solution is generated, and the threshold value is reset to be 0. Step 6: making I ═ I + 1; judging whether the algorithm meets the termination condition, if I<ImaxGo to step 3, otherwise terminate the algorithm to get the best solution.
The invention discloses a method for calculating an optimal charge rate based on dynamic congestion of a cellular transmission model, which comprises the following steps:
table 2 benefit evaluation table based on dynamic congestion charging optimum rate calculation method of cellular transmission model amount
Figure BDA0001650002730000141
Table 2 shows the total travel time of the road network system for different β values. The last column in table 2 is the rate of reduction of the total travel time in the system, which is calculated from the difference between each objective function value and 448,666 divided by each objective function value.
As shown in table 2, when β is 06, the two-layer optimization model has the optimal solution, and the corresponding optimal congestion rate is
Figure BDA0001650002730000142
Wherein each behavior is from 07:30 to 09:30, the fee value represented by the vertex of the piecewise linear charging function of each sub-period in the whole modeling range. Wherein the road section traffic flow of the whole network is zero at 07: 30.

Claims (4)

1. The dynamic congestion charging optimal rate calculation method based on the cellular transmission model is characterized by comprising the following steps of:
(1) inputting relevant data of an urban traffic network, and establishing a traffic network topological graph;
(2) based on the requirement of warning line charging, determining each entrance of a charging area as a charging starting point and each exit as a charging end point, obtaining the length of each path in the charging area, and establishing a general expression of a combined charging function based on distance and congestion, which changes along with time; the general expression of establishing the joint charging function which changes along with time and is based on the distance and the congestion is concretely as follows: let road network G be (N, a) with N nodes and a links, W denotes the set of origin-destination pairs OD, PWRepresents the set of paths between the ODs, assuming that the minimum and maximum lengths within the toll fence are l0And lKDividing the distance of the journey into K equal intervals, wherein the distance length function of each interval is represented by two end points, and the vertex of each charging interval is defined as l ═ l (l)0,l1,…,lk,…,lK)TThe corresponding charges are respectively phi ═ phi0,φ1,…,φk,…,φK)T，
Figure FDA0002989781910000011
Representing the length of the kth charging interval, the charging interval distance-based charging function phi (l) is then expressed as:
Figure FDA0002989781910000012
representing the road congestion condition by using the delay time of each road section, and then using the OD point to carry out charging function based on the congestion level on the path p between the W
Figure FDA0002989781910000013
Is expressed as:
Figure FDA0002989781910000014
wherein
Figure FDA0002989781910000015
Indicating the congestion-based charging rate on the path p between the OD point pair w,
Figure FDA0002989781910000016
indicating the delay time, t0The general expression of the joint charging function tau (l, beta) based on distance and congestion is represented as:
Figure FDA0002989781910000017
equation (3) represents a joint charging function based on distance and congestion, wherein,
Figure FDA0002989781910000018
represents the combined charging rate on the path p between the OD point pair w based on the distance and the congestion, theta1And theta2Weights representing distance charge and congestion charge, respectively;
for the d sub-interval, a combined dynamic congestion charging function tau based on distance and congestiondThe general expression of (l, β) is expressed as:
Figure FDA0002989781910000021
Figure FDA0002989781910000022
represents the link based on distance and congestion on the path p between the OD point pair w in the d intervalA flat rate;
the exact time the vehicle arrives in the toll fence area is determined from the travel time from the starting point to the toll fence area, thus correctly calculating the toll time interval d:
Figure FDA0002989781910000023
wherein the content of the first and second substances,
Figure FDA0002989781910000024
represents a minimum integer greater than or equal to the value in parentheses, t' and
Figure FDA0002989781910000025
respectively representing the travel time and the time interval length from a starting point to a charging warning line area in the cellular transmission model, and taking minutes as a unit;
(3) a cellular transmission model based on a path is adopted to simulate the propagation process of traffic flow, and further the average route in-transit travel time is calculated; the method specifically comprises the following steps: the road section is divided into basic units, the duration is divided into time intervals, and the basic form of a cellular transmission model is represented as follows:
Figure FDA0002989781910000026
Figure FDA0002989781910000027
Figure FDA0002989781910000028
representing the amount of traffic of cell i at the t time step,
Figure FDA0002989781910000029
at the t-th time stepThe amount of traffic, Q, that moves from the upstream cell i to the downstream cell i +1iIndicates the maximum number of vehicles flowing from cell i, Ni+1Representing a congestion propagation coefficient, gamma representing the ratio of backward inflow speed to forward speed, and the cell length is the distance length which a vehicle passes through at the speed of free flow in a time step;
three conditions of unit cell connection, road section convergence and divergence need to be considered when a cellular transmission model is constructed, and the constructed cellular transmission model comprises five cells and three links; the five kinds of cells are respectively: the upper and lower streams of the common cells are respectively connected with a cell; the upper stream of the divergent cells is only connected with one cell, and the lower stream of the divergent cells is connected with two cells; converging cells, wherein the upstream is connected with two cells, and the downstream is connected with only one cell; a source cell having only downstream cells and no upstream cells; a terminal cell, having only an upstream cell and no downstream cell; the three links are respectively a common link, a divergent link and a convergent link;
setting values of the initialized cells:
Figure FDA00029897819100000210
Figure FDA00029897819100000211
the five cells are represented as:
a source cell:
Figure FDA0002989781910000031
a normal cell:
Figure FDA0002989781910000032
divergent cells and convergent cells:
Figure FDA0002989781910000033
a final cell:
Figure FDA0002989781910000034
wherein the content of the first and second substances,
Figure FDA0002989781910000035
representing the occupancy of the cell i on the path p at the beginning of the time interval t,
Figure FDA0002989781910000036
represents the amount of traffic, h, flowing from cell i to cell j on path p at the beginning of time interval tp,tRepresenting the rate of departure of the vehicle on the path p within the time interval t; for parameter
Figure FDA0002989781910000037
When cell i (or j, k) is on path p,
Figure FDA0002989781910000038
otherwise
Figure FDA0002989781910000039
CD、CM、CR、CS、CORespectively representing the set of divergent cells, convergent cells, source cells, final cells and common cells; gamma-shapediIs a downstream set of cells from cell i,
Figure FDA00029897819100000310
an upstream set of cells that is cell i; t represents a maximum time range;
three links are represented as:
and (3) common linking:
Figure FDA00029897819100000311
divergent chaining:
Figure FDA0002989781910000041
convergence linking:
Figure FDA0002989781910000042
wherein E isD、EM、EORespectively representing a set of divergent links, convergent links and normal links,
Figure FDA0002989781910000043
the aggregate occupancy of time interval t initially that will flow from divergent cell i into cell j,
Figure FDA0002989781910000044
representing the average value of occupancy rates of the cells i on the path p at the beginning of the time interval t, wherein mu is a number which is infinitesimal and larger than zero;
in the above-described cell transmission model, the initial cell occupancy and the outgoing traffic volume are assumed to be zero in equations (8) to (9); equations (10) - (13) represent a path-based cell update procedure; equations (14) - (16) represent path-based traffic flow propagation constraints; wherein the conversion rate of the path-based cellular transport model is not exogenous, but is uniquely determined by the supply and demand of upstream and downstream cells;
setting a path selection decision of a vehicle to follow a dynamic user balance principle, applying a cellular transmission model based on a path to a dynamic user balance problem, and determining the uniquely determined average on-road travel time of a vehicle which starts at the same time by calculating the actual path travel time output from the cellular transmission model;
the equalization conditions for an ideal dynamic user equalization state are expressed as: the total generalized cost generated by passengers who simultaneously depart between each OD pair is equal and minimum;
the mathematical expression of the dynamic user balance theory is as follows:
Figure FDA0002989781910000045
Figure FDA0002989781910000046
wherein the content of the first and second substances,
Figure FDA0002989781910000047
Tde T denotes a set of charging time intervals over time,
Figure FDA0002989781910000051
represents the minimum path travel cost of OD to w in the time interval t, alpha is the time value,
Figure FDA0002989781910000052
representing the generalized travel cost of leaving from the OD to the path p on w within the time interval t; path flow
Figure FDA0002989781910000053
Is about the congestion rate
Figure FDA0002989781910000054
Thus, the objective of the dynamic customer balancing problem translates into finding a feasible flow that satisfies both equations (17) and (18), as well as the demand flow conservation principle (20) and the non-negative constraint (21)
Figure FDA0002989781910000055
Figure FDA0002989781910000056
f≥0,u≥0 (21)
Wherein u represents
Figure FDA0002989781910000057
The vector of (a) is determined,
Figure FDA0002989781910000058
the dynamic user equalization problem described by equations (17) to (21) above is equivalent to the problem of the finite variable inequality in equation (22):
Figure FDA0002989781910000059
in the formula (22), the upper corner mark represents the optimal solution, and Ψ represents
Figure FDA00029897819100000510
The column vector of (a) is,
Figure FDA00029897819100000511
Ω represents a set of feasible solutions that satisfy the demand flow conservation principle (20) and the non-negative constraints (21);
the output of the cellular transmission model is the cellular occupancy of each time interval, and the accumulated leaving traffic volume of the source cell r on the path p at the beginning of the time interval t
Figure FDA00029897819100000512
Equal to the cumulative amount of traffic on path p leaving from cell r at the beginning of time interval t-1
Figure FDA00029897819100000513
And the cell r on the path p is in the time zoneTraffic volume flowing out of room t
Figure FDA00029897819100000514
The sum of (a); cumulative arrival traffic volume of final cell s on path p at initial time interval omega
Figure FDA00029897819100000515
Equal to the cumulative arrival traffic volume of the final cell s on the path p at the beginning of the time interval omega-1
Figure FDA00029897819100000516
And the accumulated inflow traffic volume of the terminal cell s on the path p in the time interval omega-1
Figure FDA00029897819100000517
Is expressed as a mathematical formula:
Figure FDA00029897819100000518
Figure FDA00029897819100000519
wherein the content of the first and second substances,
Figure FDA0002989781910000061
the accumulated traffic volume leaving from the cell r on the path p at the beginning of the time interval t; in fact, in the case of time discretization, the slave cells on the path p within the time interval trCumulative amount of traffic leaving
Figure FDA0002989781910000062
Not all of the endpoints s are necessarily reached simultaneously; two arrival time coefficients omega are introduced by using the average journey time instead of the actual journey time of the vehicles leaving the path simultaneously1And ω2，ω1Is to satisfy
Figure FDA0002989781910000063
Minimum time coefficient value of condition, ω2Is to satisfy
Figure FDA0002989781910000064
A minimum time coefficient value for the condition; the average travel time of the traffic flow leaving the path p connecting the OD pair w within the time interval t
Figure FDA0002989781910000065
The expression of (a) is:
Figure FDA0002989781910000066
(4) establishing a double-layer optimization model, and determining a combined congestion rate which meets a dynamic user balance principle and changes along with time and is based on distance and congestion by optimizing the whole system;
(5) and (4) applying a hybrid adaptive gradient projection algorithm and an artificial bee colony algorithm to solve the double-layer optimization model proposed in the step (4) and output the optimal rate.
2. The method for calculating the optimal rate for dynamic congestion charging based on the cellular transmission model as claimed in claim 1, wherein in the step (4), a double-layer optimization model is established, and a combined congestion rate which meets the dynamic user balance principle, changes with time and is based on distance and congestion is determined by optimizing the whole system; the upper layer is a road network system optimization model, and the purpose is to minimize the total travel time of a road network; the lower layer is a dynamic user equilibrium state and is described as a multivariate inequality; the specific expression is as follows:
and (3) upper layer:
Figure FDA0002989781910000067
the lower layer:
Figure FDA0002989781910000068
3. the method for calculating the optimal charge rate for dynamic congestion charging based on the cellular transmission model as claimed in claim 1, wherein in the step (5), the method for solving the double-layer optimization model in the step (4) adopts an adaptive gradient projection algorithm to solve the problem of the lower-layer multivariate inequality; an artificial bee colony algorithm is adopted to solve an upper-layer optimization model; the method for solving the problem of the lower multivariate inequality by adopting the adaptive gradient projection algorithm specifically comprises the following steps:
(51) is provided with
Figure FDA00029897819100000712
u∈[0.5,1]；ε＞0，ρmax＞0，ρ0＞0，f0∈Ω；γ0＝ρ0，k＝0；
(52) Find satisfaction
Figure FDA0002989781910000071
Minimum non-negative integer of condition lkUpdating non-shortest path traffic that satisfies the constraint (29):
Figure FDA0002989781910000072
Figure FDA0002989781910000073
wherein
Figure FDA0002989781910000074
Is the shortest path between OD and w in the kth iteration;
Figure FDA0002989781910000075
Fkrepresenting a vector
Figure FDA0002989781910000076
Figure FDA0002989781910000077
Representing a vector
Figure FDA0002989781910000078
Updating the shortest path traffic:
Figure FDA0002989781910000079
(53) if inequality (30) is satisfied, then
Figure FDA00029897819100000710
Else gammak+1＝ρk+1；
Figure FDA00029897819100000711
(54) If the predetermined convergence criterion is satisfied, stopping the algorithm to obtain the final result fk+1(ii) a Otherwise, let k equal to k +1, return to step (52).
4. The method for calculating the optimal charge rate for dynamic congestion charging based on the cellular transmission model as claimed in claim 3, wherein in the step (5), the optimization model for solving the upper layer by using the artificial bee colony algorithm specifically comprises:
(a) setting the population size NcNumber of leading bees NeNumber of follower bees NoThreshold limit; setting an iteration initial value I to be 1 and a maximum iteration number Imax；
(b) Randomly generating an initial solution, calculating the fitness value of each solution corresponding to each leading bee, and initializing a threshold value to be zero;
(c) carrying out local search according to the existing solution and estimating the fitness value of the solution; if the solution is better, replacing the existing solution with the newly generated local solution, and resetting the threshold value to be zero; otherwise, the existing solution is unchanged but the threshold is set to 1;
(d) selecting leading bees by each following bee by adopting a roulette method, namely generating a uniformly distributed random number r at [0,1], and if the probability is greater than r, generating a new solution around the existing solution by the following bee and calculating the fitness value of the new solution; if this solution is more optimal, the existing solution is replaced by the neighboring solution, otherwise the existing solution is not changed and the threshold is increased to 1;
(e) selecting a solution with the highest fitness value according to the existing solution, if a solution which cannot continuously improve the fitness value within a threshold value but is not the optimal solution with the highest fitness value exists, converting the corresponding leading bee role into a scout bee, searching again in an adjacent space to generate a new random solution, and resetting the threshold value to be 0;
(f) if I is equal to I +1, judging whether the algorithm meets the termination condition, and if I is less than ImaxGo to step (c); otherwise, the algorithm is terminated to obtain the optimal solution.
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