CN110390421A - Congestion subway line passenger flow control method for coordinating based on time-space network - Google Patents
Congestion subway line passenger flow control method for coordinating based on time-space network Download PDFInfo
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Abstract
The invention belongs to urban rail transit technology fields, it is related to a kind of congestion subway line passenger flow control method for coordinating based on time-space network, method comprehensively considers train space-time trajectory, each OD passenger's dynamic need, train load, passenger's Path selection, waiting time under set operation figure etc., under the premise of not influencing existing operation figure, optimize passenger's trip, guarantee the safe and efficient operation of rail traffic, reduces passenger waiting time, and equilibrium paths traffic resource.Train operation time-space network is constructed under the premise of operation figure determines especially by time discretization method, passenger flow control problem is converted into passenger's trip moment select permeability, establishes the mathematic optimal model for accurately portraying problem.It is further the trip shortest path Path Selection based on individual passenger by model conversion by Lagrange relaxation method, and designs the algorithm of Solve problems, the rail line under the set operation figure of Efficient Solution cooperates with current limliting strategy.
Description
Technical Field
The invention belongs to the technical field of urban rail transit, relates to a method for coordinating and controlling passenger flow of congested subway lines based on a space-time network, and particularly relates to a method for optimally controlling the passenger flow of the cooperation of the subway lines under a given operating diagram.
Background
At present, the urban rail transit system plays an increasingly important role in relieving traffic jam of a large city by the characteristics of punctuality, convenience, large capacity, low emission and the like. However, as the operation scale of subway systems expands, the demand of passengers on a subway as a trip mode in a large city also rapidly increases. Taking the beijing subway system as an example, the number of passengers served by the beijing subway shows a trend of increasing sharply in recent years. According to statistics, the passenger flow total of Beijing subway in 2007 is only 5 hundred million people, and in 2016, the passenger flow total in year is over 35 hundred million people, which is nearly 7 times of the passenger flow total in 2007.
With the rapid increase of the demand of passengers, the congestion problem of the subway system becomes more and more serious. Especially in a modern large city where public transportation service plays a crucial role in resident travel, subway travel becomes the first choice for more and more passengers to commute in the morning and at night due to punctuality and rapidity, and further overcrowding of subway lines at the peak of the large city in the morning and at the night is caused. Relevant survey data show that at least 9 lines are in a congestion state in the morning and evening peak hours of Beijing, and taking eight-way lines of Beijing subway as an example, the full load rate of part of the lines exceeds 100%, so that the problem of overstock of passengers is serious. The passenger transships, can reduce the quality of service of subway operation on the one hand, reduces the comfort level of track traffic trip, especially under the condition of big passenger flow, and some passengers need wait for the longer time at the station and just can take the train, and the passenger trip is experienced and is declined, influences the appeal of subway trip. On the other hand, a large amount of platform passengers are overstocked, so that train operation delay is easily caused, passenger pushing phenomena are frequent, and the occurrence frequency of operation accidents and the safety risk of rail transit operation are increased. In actual operation, in order to solve the problem, current limiting measures are taken at partial sites, which is an effective means often adopted by railway operation departments; the method comprises the following steps: by closing part of station entrances, setting an out-station fence and closing part of automatic toll gate (AFC) to change the space-time distribution of passenger flow, the congestion condition of the passenger flow at busy stations is relieved. For example, in beijing subway line 13, six stations take current limiting measures in the morning and evening rush hour, and the congested passenger flow is transferred to the outside of the stations, so that the subway operation risk is reduced.
However, since there is a complex coupling relationship between the passenger flow and the train flow, in practice, the passenger flow control measures taken by the operation department are usually performed independently after being judged according to the expertise and experience, and the interaction relationship between the passenger flow and the train flow between different stations is not fully considered. At present, most of the existing current-limiting strategies are based on stations and platforms, and the individual trip behaviors of passengers on the whole subway line are less considered, so that the adopted control strategy is usually only a better control strategy of a single station, and the system optimal strategy of the whole subway operation system is difficult to achieve. And under the existing information technology, it becomes possible to conveniently acquire the travel behaviors of the passengers. In view of this, the present invention particularly provides an effective optimization method, which can cooperatively optimize the passenger flow control strategy of the whole subway line based on the travel selection of passengers.
Based on the current application situation, the invention considers the arriving dynamics of passengers, converts the passenger flow control problem into the passenger travel time selection problem by constructing a train operation spatio-temporal network on the premise of determining the operation diagram, constructs a strict mathematical optimization model and designs an effective solving algorithm. The method comprises the following steps: by means of time discretization, a time-space network of passengers and trains is constructed, train running tracks are described by adopting time-space paths, and arrival rules of the passengers are quantized by adopting time-related matrixes. Under the technical background, the passenger flow control problem is converted into a passenger path selection problem, the passenger requirements of all stations on a line are comprehensively considered, a space-time path shortest-path model is constructed, and a passenger flow cooperative optimization control method for seeking the system optimization is designed.
Disclosure of Invention
The invention aims to: the subway line cooperative passenger flow control optimization method under the established operation diagram is provided, the purpose of relieving platform congestion, reducing the waiting time of passengers, realizing the balanced configuration of rail traffic resources and improving the service quality of rail traffic is realized under the condition that the rail traffic operation diagram is not influenced. The invention takes into account, among other things, the dynamics of the passenger, including: passenger arrival time, train loading capacity, and passenger travel OD (origin-destination).
In order to achieve the purpose, the time is discretized, a train operation space-time network is constructed on the basis of the existing track traffic diagram, the train operation track is converted into a space-time track, a passenger trip selection optimization model is established, the track traffic resource allocation is optimized, and the cooperative current limiting of the whole line station and the system level optimization are achieved. In addition, the model is decomposed by utilizing the Lagrange relaxation idea, the original problem is converted into a series of mutually independent shortest path problems related to passengers, and the model is effectively solved by respectively adopting optimization software and a heuristic algorithm based on Lagrange relaxation, and the specific technical scheme is as follows.
A method for coordinating and controlling passenger flow of a congested subway line based on a space-time network comprises the following steps:
s1, establishing a passenger trip selection shortest-path optimization model based on a spatio-temporal network, and specifically comprising the following steps:
s11, constructing a spatio-temporal network
Discretizing the planned time interval of the train, converting the planned time interval into a series of time intervals with unit length delta, and adopting T ═ T { (T)0,t1,t2,…,tMDenotes the discretized time node, where t0For the start of the planned time period, tMIs the end time;
with time nodes as abscissa, station s of rail transit line1,s2,…,snAs a vertical coordinate, a coordinate plane is formed;
mapping the train running track to the coordinate plane to construct a space-time network;
s12, defining decision variables
Defining decision variables
Wherein the subscripts represent the spatio-temporal path (i, j, t, t ') indicating that passenger p leaves the i station at time t and arrives at the j station at time t';
and makes the following hypothetical decisions:
if passenger p selects spatiotemporal path (i, j, t, t'), thenIf not, then,
wherein P belongs to P, and P is a passenger set in a planning time interval;
s13, determining constraint conditions
The constraint conditions include: based on passenger flow balance constraints and loading capacity constraints;
the flow balance constraint is expressed as formula (2) and is marked as constraint (2),
wherein, OpIs the origin of the passenger p,the time when passenger p arrives at the origin station, A is the set of space-time arcs;as virtual passenger arrival nodes, tMFor passenger arrivalThe time of (d);
the loading capacity constraint is expressed as a formula (3) and is marked as a constraint (3),
wherein C is train loading capacity;
s14, constructing an objective function, and establishing a passenger trip selection shortest path optimization model based on a space-time network
The method specifically comprises the following steps:
s141, calculating waiting time of passenger p in spatio-temporal networkAs shown in the formula (5),
wherein,a decision variable waiting at the origin for passenger p;
s142, the objective function is shown as formula (8),
wherein, wpComprises the following steps: the weight for any passenger p, as shown in equation (7),
wherein σ is a constant;
s143, the model based on the space-time network is shown as a formula (9),
;
s2, solving passenger trip selection shortest path optimization model based on space-time network
The method specifically comprises the following steps:
s21, initializing Lagrange multiplier muijtt′Making the iteration number n equal to 1;
s22, solving the optimal solution of the sub-problem model by using a label method, taking the optimal solution as a lower bound of the model solution based on the spatio-temporal network, and updating the lower bound LB (n);
s23, calculating an objective function (8) by adopting a step-by-step adjustment method, and updating an upper bound UB (n);
s24, updating Lagrange multiplier mu by using a sub-gradient methodijtt′;
S25, when either of the following two conditions is satisfied (i.e. condition (r) and-
Or ②), the iteration is stopped,
the conditions are as follows: the relative error of the upper and lower bounds is less than a given allowable error value ξ;
condition two: the iteration number n reaches the set maximum iteration number;
otherwise, the process returns to step S22 to continue the subsequent steps.
On the basis of the above technical solution, the obtaining of the subproblem model in step S22 specifically includes the following steps:
s221, introducing a Lagrange multiplier muijtt′And relaxing the loading capacity constraint into an objective function (8) to obtain a relaxation problem objective function shown as a formula (10),
s222, introducing intermediate variablesAs shown in the formula (11),
by intermediate variablesCombining equations (10) and (11), the objective function of the relaxation problem is further simplified to equation (12),
the model of the relaxation problem is then expressed as: considering equation (13) under the flow balance constraint described by equation (2),
s223, according to the mutual independence of the passenger path selection behaviors, further decomposing the formula (13) into a subproblem model selected by the p trips of each passenger, as shown in the formula (14),
on the basis of the above technical solution, the specific steps of step S23 are as follows:
counting passengers per trainWherein k is train number; according to the method of vehicle-by-vehicle and station-by-station, inquiring passenger selection behaviors which do not meet the loading capacity constraint, sequencing all passengers who select to take the train and get off after the station i according to the arrival time of the passengers at the starting station for the train k exceeding the loading capacity constraint when the train k exceeds the station i, and according to the first-in first-out principle, sequencing the exceeding partAdjusting to k +1 times of train; recalculating the train loading condition every time of adjustment until all the train loads meet the loading capacity constraint, calculating an objective function (8),as the lower bound of the model solution based on the spatio-temporal network, and updates the upper bound ub (n).
On the basis of the above technical solution, the step S24 specifically includes the following steps:
s241, setting a gradient as shown in a formula (15),
s242, Lagrange multiplier muijtt′As shown in equation (16), update is performed:
wherein λ isnIs the iteration step size.
On the basis of the technical scheme, the iteration step length lambda isnGiven the standard step size, as shown in equation (17),
wherein, thetanThe value range is 0 < thetan≤2。
The invention has the following beneficial technical effects:
the method comprehensively considers factors such as train space-time tracks, dynamic requirements of all OD passengers, train loading capacity, passenger path selection, waiting time and the like under the established operation diagram, optimizes the travel behaviors of the passengers on the premise of not influencing the existing operation diagram, ensures safe and efficient operation of rail transit, reduces the waiting time of the passengers and balances the utilization of rail transit resources. Specifically, a train operation time-space network is constructed through a time discretization method on the premise that an operation diagram is determined, the passenger flow control problem is converted into a passenger travel time selection problem, and a mathematical optimization model accurately describing the problems is established. Further, the model is converted into a travel shortest path selection problem based on individual passengers by means of a Lagrange relaxation method, and an efficient algorithm for solving the problem is designed. The technology can efficiently solve the rail transit line cooperative current limiting strategy under a given operation diagram.
Drawings
The invention has the following drawings:
FIG. 1 is a schematic diagram of rail transit operation;
FIG. 2 is a schematic diagram of a train operation spatiotemporal network structure based on time dispersion;
FIG. 3 is a schematic diagram of passenger arrival distribution at an early peak section of a Beijing subway eight-way line;
FIG. 4 is a schematic diagram of passenger travel path selection in the spatio-temporal network;
FIG. 5 is a schematic diagram of a train operation diagram and train loading conditions solved by CPLEX in example verification;
FIG. 6 is a statistical plot of individual passenger waiting times solved using CPLEX in example validation;
FIG. 7 is a schematic illustration of an actual passenger inbound distribution and a passenger inbound distribution under control for example verification;
FIG. 8 is a graph of iterative convergence of a design algorithm in an example verification.
Detailed Description
In order to more clearly illustrate the present invention, the present invention is further described below with reference to examples and the accompanying drawings.
The following describes the specific implementation of the proposed method in detail according to the two parts of model building and model solving. It is to be understood by persons skilled in the art that the following detailed description is illustrative and not restrictive, and is not to be taken as limiting the scope of the invention.
Establishing a model based on a space-time network
1. Building spatio-temporal networks
The method has the key technology that the rail transit running track is mapped into a space-time network and converted into a space-time track, so that the passenger flow control problem is converted into the passenger departure time selection problem, and a passenger trip selection shortest-path optimization model is established.
To achieve the above object, the present invention first proceeds with the planning period consideredDiscretization processing, which converts the planning period into a series of time intervals with unit time length delta (i.e. discrete time steps). Here, T ═ T is used0,t1,t2,…,tMDenotes the discretized time node, i.e. the set of discrete time intervals, where t0For the start of the planned time period, tMIs the end time.
In practice, the train is usually operated periodically according to a predetermined operation diagram based on the route shown in fig. 1, where s1,s2,…,snRepresenting a rail transit line stop. For convenience of description, the train moving track may be mapped to a space-time network as shown in fig. 2, wherein an abscissa represents a time node, an ordinate represents a rail transit line station, solid arrows connected by dotted arrows respectively represent the moving track of each train, and the dotted arrows represent a stop arc of the train at the station.
Since the traveling demands of passengers are dynamically changed with time in reality, in order to describe the dynamic demand distribution of passengers, an eight-way line early peak part station passenger arrival statistical curve diagram (i.e. an early peak part station passenger arrival distribution diagram) shown in fig. 3 is given. As can be seen from the figure, the early peak passenger arrival is far beyond the peak flat period. To further characterize the dynamics of the passenger flow, the invention uses a time-dependent OD matrix P (t)k) Is shown at (t)k-1,tk]The trip demand of the passengers arriving in the time interval is shown in the formula (1),
wherein p isij(tk) Represents (t)k-1,tk]The number of passengers arriving at station i in time and traveling OD i → j. By means of the method, all passenger travel demands in the planning period can be represented dynamically.
On the basis of the construction of the spatio-temporal network, the subway line passenger flow control process can be represented as a selection process of individual passenger travel paths in the spatio-temporal network as shown in fig. 4. Specifically, by regarding a single passenger with different travel demands as a control unit, on the premise of considering the constraint of train loading capacity, the optimal passenger travel path of the system is found, so as to realize the balanced configuration of system resources. The solid black nodes in the figure are the time-space nodes (i.e., passenger arrival nodes) at which passengers arrive, the dashed arrows indicate that passengers are waiting at the origin (i.e., passenger waiting arc in fig. 4), and the solid arrows indicate that passengers leave the origin (i.e., passenger outgoing arc in fig. 4) in a corresponding train. On the basis of the technology, a shortest path model for passenger travel selection is constructed to seek an optimal full-line passenger flow cooperative control optimization method of the system.
2. Defining decision variables
Generally speaking, the passenger routing variables are binary variables, and the following decision variables are adopted by the invention to describe the routing behavior of the passenger.
Defining decision variables
And makes the following hypothetical decisions:
if passenger p selects spatiotemporal path (i, j, t, t'), thenIf not, then,
the above decision variablesReflecting the travel track of the passenger in different time dimensions, and indicating that the passenger waits at the station i or the station j when the station i is j (for example, the train does not depart or the train on which the passenger stops at the station); when i is not equal to j, the passenger is indicated to be in a running state by taking the train, and the passenger leaves the station i at the time t and arrives at the station j at the time t'; (i, j, t, t') (also called spatio-temporal arc segments) are: and the train running track in the space-time network is used for indicating that the train departs from the station i at the time t and arrives at the station j at the time t'.
3. Determining constraints
(1) Passenger-based flow balance constraints
After the passengers arrive at the starting station, the passengers need to wait at the station or take a train to go to the destination station, so that the optimal space-time path from the starting station to the destination station needs to be found for each passenger, based on the above consideration, a flow balance constraint is constructed, as shown in formula (2), and is marked as constraint (2),
wherein the left term of the equation represents the number of times that the passenger enters or exits the node j, OpIs the origin of the passenger p,the time when a passenger P arrives at an origin station, namely the departure time, P is a passenger set in a planning time interval, and A is a set of space-time arcs; in order to ensure the rationality of the constraint, a purpose station virtual node is specially addedWherein,as virtual passenger arrival nodes, tMVirtual node arrival for passengersTime of (d). Then the space-time arc segmentRepresents passenger at tkConstantly driving away from the actual destination node DpAnd at tMTime of arrival at a virtual nodeNote that from actual node DpTo the virtual nodeHas a travel time of 0, i.e. tk=tM。
(2) Load capacity constraint
The travel of passengers is limited by the constraint of train loading capacity, in order to ensure the reasonability of the model, the constraint of the loading capacity is established, as shown in formula (3), and is marked as constraint (3),
in principle, to ensure travel safety, the number of passengers on the train must not exceed the train loading capacity, reflected on the spatio-temporal network, i.e. the number of passengers with i ≠ j does not exceed the train loading capacity C, selecting the same path (i, j, t, t').
4. Constructing an objective function
As mentioned above, the present invention aims to guarantee: on the premise that the rail transit operation diagram is not influenced, the travel paths of passengers are optimized to reasonably distribute rail transit resources, the waiting time of the passengers is reduced, and the cooperative optimization control of subway line passenger flow is realized. The waiting time of the passenger p is the difference between the departure time and the arrival time of the passenger, as shown in the formula (4),
wherein,in order to be the waiting time for the passenger p,the time at which passenger p leaves the origin station,time of arrival at the origin for passenger p;
on a time discrete basis, the waiting time of the passenger p in the spatio-temporal network, i.e. the sum of all waiting arcs thereof, is multiplied by a discrete time step delta, as shown in equation (5),
wherein,a decision variable for passenger p to wait at the origin. Considering that the present invention aims to minimize the total waiting time of all passengers in the system, further, the waiting time t for all passengerswaitMake a calculation of the waiting time for all individual passengersThe addition of (b) is represented by the formula (6),
considering that if only the shortest total waiting time is pursued, the later passenger can take a car preferentially, and the first passenger waits, in order to ensure the passenger service to follow the principle of first-in first-out, the first passenger needs to be given a certain priority to ensure that the first passenger takes a train preferentially, and for the convenience of calculation, the following weight w is given to any passenger pp:
Wherein, the sigma is a given constant and is given according to specific requirements.
The invention aims to optimize the total waiting time of passengers, considers the first-in first-out principle and combines the formulas (6) and (7) to provide a model objective function shown in the formula (8),
in summary, the model based on the spatio-temporal network is represented as: equation (9) under the flow balance constraint of equation (2) and the load capacity constraint of equation (3),
second, solving model based on space-time network
The model solving based on the spatio-temporal network comprises three parts, namely, firstly, the model is needed to be decomposed, secondly, an iterative algorithm based on Lagrangian relaxation is designed, and finally, example verification is carried out.
1. Model decomposition based on Lagrange relaxation
In order to solve the model based on the space-time network, the loading capacity constraint (3) is analyzed and known to be difficult, the loading capacity constraint needs to be relaxed, and in order to solve the problem, a Lagrange multiplier mu is introducedijtt′,(i,j,t,t′)∈A。
By introducing lagrange multiplier muijtt′Relaxing the loading capacity constraint into a model objective function (8) to obtain a relaxation problem objective function shown as a formula (10),
to simplify the description, intermediate variables are introducedAs shown in the formula (11),
by intermediate variablesCombining equations (10) and (11), the objective function of the relaxation problem is further simplified to equation (12),
the model of the relaxation problem can be expressed as: considering equation (13) under the flow balance constraint described by equation (2),
in the relaxation model (13), the routing behaviors of all passengers are independent from each other, so that the model can be further decomposed into a series of sub-problems (sub (p)) related to individual passengers to form a sub-problem model of the p travel selection of each passenger, as shown in formula (14),
from the sub-problem model equation (14), it can be seen that each sub-problem is a classical shortest-path problem that only considers the flow balance constraint, and it is solved precisely by the label method.
2. Solving algorithm
The subproblems obtained after the original problem is decomposed are classical shortest-path problems, and can be solved by means of a labeling method. The obtained relaxation problem objective function (12) can be regarded as the lower bound of the original problem solution, and in order to search the feasible solution of the original problem, the invention designs a gradual adjustment algorithm, and the solution obtained after relaxation is adjusted into the feasible solution, and the algorithm is divided into three parts.
1) Finding feasible solutions
Solving the optimal solution of the subproblems by using the shortest-path algorithm, and calculating the number of passengers on each train based on the optimal solutionWherein k is the train number. Querying for passenger selection behavior that does not meet load-carrying capacity constraints on a car-by-car, station-by-station basisFor the train k beyond the loading capacity constraint, at the station i beyond the train k, sorting all passengers who choose to take the train and get off after the station i according to the arrival time of the passengers at the starting station, and according to the first-in first-out principle, sorting the exceeding partAdjust to k +1 trains. And recalculating the train loading condition every time of adjustment until all the train loads meet the loading capacity constraint, obtaining a feasible solution of the problem, calculating an objective function (8), and updating the upper bound.
2) Updating lagrange multiplier muijtt′
Using the sub-gradient method, the Lagrangian multiplier μ is updated as followsijtt′。
First, a gradient is set, as shown in equation (15),
the definition of the method is that,and representing the optimal solution of all the subproblems obtained by the nth iteration.
Lagrange multiplier muijtt′As shown in equation (16), update is performed:
wherein λ isnFor the iteration step, in the present invention, the iteration step λnGiven the standard step size, as shown in equation (17),
wherein, thetanThe value range is 0 < thetanLess than or equal to 2, is used for adjusting the step length and ensures the nonnegativity of the penalty value.
The specific steps for solving the algorithm are as follows,
step 1: (initialization). Initializing lagrange multiplier muijtt′Making the iteration number n equal to 1;
step 2: (calculate sub-problem optimal solution). Solving the optimal solution of the sub-problem model by using a shortest-path algorithm (namely a labeling method), and updating a lower bound LB (n);
step 3: (generating a feasible solution). And adjusting the generated infeasible solution into a feasible solution of the original problem according to the provided step-by-step adjustment method for searching the feasible solution, and updating the upper bound UB (n).
Step 4: (update multiplier). Updating Lagrange multiplier mu using a sub-gradient methodijtt′。
Step 5: (algorithm termination). The algorithm terminates when either of the following two conditions is satisfied.
(1) The relative error of the upper and lower bounds is less than a given allowable error value ξ;
(2) and the iteration reaches the set maximum iteration number.
Otherwise, returning to Step 2 and continuing to calculate n as n + 1.
3. Example verification
In order to more clearly illustrate the invention, the invention is further described below with reference to preferred examples and the accompanying drawings. A small-scale numerical example is designed to verify the inventive content, as shown in table 1, assuming that a one-way rail transit line with 6 stations runs 5 trains, a planning time domain (i.e. planning time interval) is 30min, which can be dispersed into 30 time intervals, a discrete step δ is 1min, and the dynamic arrival of passengers is given as follows:
(1) assuming that [5,15] is a peak time period, the arrival number of passengers per unit time among all trips OD is randomly given in an interval [20,40], and the passenger demand exceeds the loading capacity of a train;
(2) the rest time periods are peak-off time periods, and the arrival number of passengers per unit time among all trips OD is randomly given in the interval [5,10 ].
The number of passengers generated in the above manner may exceed the train capacity during peak hours. To ensure that passengers can ride the train during the scheduled time period, the train is finally operated according to the schedule shown in table 1, given that no passengers arrive at the station after the last train departure from the station, and finally the total number of passengers 6399.
TABLE 1 train timetable
Train numbering | Station 1 | Station 2 | Station 3 | Station 4 | Station 5 | Station 6 |
Train 1 | (-,2) | (4,5) | (7,8) | (11,12) | (14,15) | (17,-) |
Train 2 | (-,5) | (7,8) | (10,11) | (14,15) | (17,18) | (20,-) |
Train 3 | (-,8) | (10,11) | (13,14) | (17,18) | (20,21) | (23,-) |
Train 4 | (-,12) | (14,15) | (17,18) | (21,22) | (24,25) | (27,-) |
Train 5 | (-,15) | (17,18) | (20,21) | (24,25) | (27,28) | (30,-) |
According to the given train timetable and passenger arrival data, firstly, a CPLEX solver is used for solving a model (9) to obtain a passenger travel selection scheme and a train running loading condition under a cooperative passenger flow control strategy, and the total waiting time of the passengers is calculated to be 15102 min. In order to better show the obtained travel plan and the retention situation of passengers, a train operation diagram, a passenger getting on/off situation of each station and a train loading situation of each station are given as shown in fig. 5. According to the path selection of the passengers, the specific waiting time of each passenger can be obtained, and as shown in fig. 6, the statistical data shows that the waiting time of most passengers at the starting station is short, and the passengers can travel by the train quickly. To better verify the effectiveness of the present invention, CPLEX is used to simulate the travel of passengers without control measures (riding rules on duty) (shown by black solid blocks in fig. 7, corresponding passenger inbound distribution, i.e. legend is no control), and compare with the above passenger inbound scheme (shown by black open blocks in fig. 7, corresponding passenger inbound distribution, i.e. legend is under control) obtained by solving the model (9) under the cooperative passenger flow control strategy. In the two situations, as shown in fig. 7, the obtained passenger inbound distribution can find that the inbound time of the passenger is greatly different from the inbound distribution without control.
In order to further verify the effectiveness of the designed calculation method, the algorithm designed by the application is used for solving the problem, the algorithm can be converged to a better solution within a short time (7.3s), the difference between the total waiting time of the passenger, which is 15180min, and the optimal solution obtained by the CPLEX solver is only 78min, and the convergence condition of the algorithm is shown in FIG. 8. Further, the passenger riding conditions and the passenger waiting time levels obtained by the two methods are compared, as shown in tables 2 and 3. It can be seen from the two tables that although there is some difference between the calculation method and the CPLEX solution result (as shown in bold font part in table 2), the flow control measures are similar from the global perspective, and the solution obtained by the calculation method is close to the optimal solution in terms of the total waiting time, and can be quickly solved in a short time, thereby verifying the effectiveness of the calculation method provided by the invention.
TABLE 2 passenger loading scheme comparison table obtained by two methods
TABLE 3 passenger waiting time statistical table obtained by two solving methods
It should be understood that the above-mentioned embodiments of the present invention are only examples for clearly illustrating the present invention, and are not intended to limit the embodiments of the present invention, and it will be obvious to those skilled in the art that other variations or modifications may be made on the basis of the above description, and all embodiments may not be exhaustive, and all obvious variations or modifications may be made within the scope of the present invention.
Those not described in detail in this specification are within the knowledge of those skilled in the art.
Claims (5)
1. A method for coordinating and controlling passenger flow of a congested subway line based on a space-time network is characterized by comprising the following steps:
s1, establishing a passenger trip selection shortest-path optimization model based on a spatio-temporal network, and specifically comprising the following steps:
s11, constructing a spatio-temporal network
Discretizing the planned time interval of the train, converting the planned time interval into a series of time intervals with unit length delta, and adopting T ═ T { (T)0,t1,t2,…,tMDenotes the discretized time node, where t0For the start of the planned time period, tMIs the end time;
with time node T as abscissa, station s of rail transit line1,s2,…,snAs a vertical coordinate, a coordinate plane is formed;
mapping the train running track to the coordinate plane to construct a space-time network;
s12, defining decision variables
Defining decision variables
Wherein the subscripts represent the spatio-temporal path (i, j, t, t ') indicating that passenger p leaves the i station at time t and arrives at the j station at time t';
and makes the following hypothetical decisions:
if passenger p selects spatiotemporal path (i, j, t, t'), thenIf not, then,
wherein P belongs to P, and P is a passenger set in a planning time interval;
s13, determining constraint conditions
The constraint conditions include: based on passenger flow balance constraints and loading capacity constraints;
the flow balance constraint is expressed as formula (2) and is marked as constraint (2),
wherein, OpIs the origin of the passenger p,the time when passenger p arrives at the origin station, A is the set of space-time arcs;as virtual passenger arrival nodes, tMFor passenger arrivalThe time of (d);
the loading capacity constraint is expressed as a formula (3) and is marked as a constraint (3),
wherein C is train loading capacity;
s14, constructing an objective function, and establishing a passenger trip selection shortest path optimization model based on a space-time network
The method specifically comprises the following steps:
s141, calculating waiting time of passenger p in spatio-temporal networkAs shown in the formula (5),
wherein,a decision variable waiting at the origin for passenger p;
s142, the objective function is shown as formula (8),
wherein, wpComprises the following steps: the weight for any passenger p, as shown in equation (7),
wherein σ is a constant;
s143, the model based on the space-time network is shown as a formula (9),
s2, solving passenger trip selection shortest path optimization model based on space-time network
The method specifically comprises the following steps:
s21, initializing Lagrange multiplier muijtt′Making the iteration number n equal to 1;
s22, solving the optimal solution of the sub-problem model by using a label method, taking the optimal solution as a lower bound of the model solution based on the spatio-temporal network, and updating the lower bound LB (n);
s23, calculating an objective function (8) by adopting a step-by-step adjustment method, and updating an upper bound UB (n);
s24, updating Lagrange multiplier mu by using a sub-gradient methodijtt′;
S25, when either of the following two conditions is satisfied, stopping the iteration,
the conditions are as follows: the relative error of the upper and lower bounds is less than a given allowable error value ξ;
condition two: the iteration number n reaches the set maximum iteration number;
otherwise, the process returns to step S22 to continue the subsequent steps.
2. The method for the coordinated control of the passenger flow of the congested subway line based on the spatio-temporal network as claimed in claim 1, wherein: the obtaining of the subproblem model in step S22 specifically includes the following steps:
s221, introducing a Lagrange multiplier muijtt′And relaxing the loading capacity constraint into an objective function (8) to obtain a relaxation problem objective function shown as a formula (10),
s222, introducing intermediate variablesAs shown in the formula (11),
by intermediate variablesCombining equations (10) and (11), the objective function of the relaxation problem is further simplified to equation (12),
the model of the relaxation problem is then expressed as: considering equation (13) under the flow balance constraint described by equation (2),
s223, according to the mutual independence of the passenger path selection behaviors, further decomposing the formula (13) into a subproblem model selected by the p trips of each passenger, as shown in the formula (14),
3. the method for coordinating and controlling the passenger flow of the congested subway line based on the spatio-temporal network as claimed in claim 2, wherein: the specific steps of step S23 are as follows:
counting passengers per trainWherein k is train number; according to the method of vehicle-by-vehicle and station-by-station, inquiring passenger selection behaviors which do not meet the loading capacity constraint, sequencing all passengers who select to take the train and get off after the station i according to the arrival time of the passengers at the starting station for the train k exceeding the loading capacity constraint when the train k exceeds the station i, and according to the first-in first-out principle, sequencing the exceeding partAdjusting to k +1 times of train; and recalculating the train loading condition every time of adjustment until all the train loadings meet the loading capacity constraint, calculating an objective function (8) as a lower bound of the model solution based on the spatio-temporal network, and updating an upper bound UB (n).
4. The method for coordinating and controlling the passenger flow of the congested subway line based on the spatio-temporal network as claimed in claim 3, wherein: the step S24 specifically includes the following steps:
s241, setting a gradient as shown in a formula (15),
s242, Lagrange multiplicationMu seedijtt′As shown in equation (16), update is performed:
wherein λ isnIs the iteration step size.
5. The method for the coordinated control of the passenger flow of the congested subway line based on the spatio-temporal network as claimed in claim 4, wherein: the iteration step λnGiven the standard step size, as shown in equation (17),
wherein, thetanThe value range is 0 < thetan≤2。
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