CN112784204A - Train schedule and passenger flow control robust optimization method facing uncertain demand - Google Patents

Train schedule and passenger flow control robust optimization method facing uncertain demand Download PDF

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CN112784204A
CN112784204A CN202011577170.8A CN202011577170A CN112784204A CN 112784204 A CN112784204 A CN 112784204A CN 202011577170 A CN202011577170 A CN 202011577170A CN 112784204 A CN112784204 A CN 112784204A
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train
station
passenger flow
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flow control
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杨立兴
卢亚菡
戚建国
阴佳腾
李树凯
高自友
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Beijing Jiaotong University
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Abstract

The invention belongs to the technical field of urban rail transit operation management, and relates to a train schedule and passenger flow control robust optimization method for uncertain demands, which comprises the following steps: describing passenger flow characteristics by adopting scene-based random dynamic data, and dividing passenger flow requirements into deterministic requirements and extra requirements; determining the dynamic robustness protection level of each station according to the extra requirements; introducing decision variables related to time, trains and stations based on the passenger flow data representation; constructing a scene coupling constraint, a robust passenger flow control variable and an association constraint among train operation decision variables according to the decision variables; constructing a robust optimization model by taking the extra train capacity required by adding a robust protection level into a minimum as a target according to the constraint; and solving the robust optimization model to obtain a robust control strategy suitable for controlling the passenger flow of the large-city rail transit, and providing theoretical support for safe and efficient operation of the actual rail transit.

Description

Train schedule and passenger flow control robust optimization method facing uncertain demand
Technical Field
The invention belongs to the technical field of urban rail transit operation management, relates to a train timetable and passenger flow control robust optimization method for uncertain demands, and particularly relates to an urban rail transit train timetable and passenger flow control robust collaborative optimization method for dynamic uncertain passenger flow demands.
Background
Urban rail transit has the characteristics of large capacity, high speed, accurate time, little pollution, good safety and the like, and becomes the main artery of an urban traffic network. In recent years, urban rail transit in China develops rapidly, and the total operating line length is estimated to exceed 6000 kilometers in 2021. However, with the increasing length of urban rail service lines, the demand of passengers is continuously and rapidly increasing, and great pressure is brought to the efficient operation of rail transit. By taking Beijing as an example, the daily passenger flow is more than 1000 thousands of people to become a normal state by 2018, wherein more than 40 percent of passengers are commuting passenger flow on working days, and the stations are crowded at early and late peak periods, the overload of trains is frequent, and the accident risk rate and the vehicle delay rate are greatly increased.
With the continuous expansion of road networks, the passenger flow is continuously increased, and the train schedule is only optimized, which is not enough to meet the large-scale travel demand. In order to effectively relieve the excessive congestion phenomenon of the platform in early and late peak periods and improve the trip quality of passengers while reducing the operation risk, the rail transit operation departments in the super-large cities such as Beijing and Shanghai adopt current limiting measures at partial large passenger flow stations. Data shows that nearly hundreds of conventional current-limiting stations exist in Beijing at present, but the current-limiting mode is relatively simple, and generally from the qualitative point of view, according to the experience of a manager, fences of an out-of-station team are set or partial charging gates are closed, so that the dynamic behavior of passenger flow is difficult to deal with. In addition, due to the diversity of passenger travel, the passenger flow demands of different stations on the same subway line are different even in the same operation period. Currently, most of the related research aiming at passenger flow control is based on a determined passenger flow demand environment, and the uncertainty of the passenger flow is not fully considered. In addition, the existing current-limiting strategy does not realize full-line linkage, all uses a station as an operation unit, and is not coupled with train operation.
In view of the above, the invention especially researches an effective optimization method, starts with the two aspects of urban rail transit train schedule optimization and passenger flow control by scientific quantification means and adopting an advanced light robust optimization theory method, and develops cooperative optimization research to maximally match transportation supply and traffic requirements. Specifically, by means of a time discretization method, time-related passenger travel OD (origin-Destination, i.e., a passenger travel starting point and an end point) demand (i.e., passenger flow demand) data is used for describing the dynamics of the passenger flow demand, and different scenes (e.g., each sampling date is used as an independent scene) are used for describing the uncertainty of the passenger flow demand. Based on the data, a robust passenger flow control strategy facing the passenger travel OD requirement is provided, and the number of passengers on the platform and the train is adjusted by controlling the inbound decision variables, so that the excessive congestion of the platform and the continuous overload of the train are avoided. Meanwhile, considering the coupling constraint between scenes, the association constraint between a robust passenger flow control variable and a train operation decision variable, the constraint of the maximum carrying capacity of a train, the arrival time of the train, the departure time and the like, constructing a strict train schedule and a passenger flow control robust optimization model (a robust optimization model for short), designing an effective solving algorithm, and providing a long-term, stable and effective train schedule and subway passenger flow control robust collaborative optimization strategy for operation practice.
Disclosure of Invention
The invention aims to provide a robust passenger flow control and train operation diagram cooperative optimization method for reducing operation accident risks in different scenes by adjusting the platform passenger gathering amount within a safety range by considering uncertainty of passenger traveling demands under the condition of ensuring safe operation of a train, so as to solve the technical problem that the existing passenger flow control method in the background technology is difficult to be oriented to engineering practice.
In order to achieve the purpose, the planning time interval is dispersed into a plurality of equal time intervals, and the following technical scheme is adopted:
an uncertain demand-oriented train schedule and passenger flow control robust optimization method comprises the following steps:
s1: recording a set of all stations in a subway line as S {1,2, …, k, …, S }, wherein k ═ 1 denotes an origin station, k ═ S denotes a destination station, and k ∈ S; note that the set of all available trains on the line is L ═ {1,2, …, i, …, n }; recording a set of traffic scenes as Ω ═ 1, 2. Let the set of discrete time periods T ═ T0,t0+Δ,…,t0+ n Δ }, wherein Δ is a time discrete step size, which represents the length of each small time interval after the dispersion, and n is a natural number; dividing the passenger flow demand in each scene into a deterministic passenger flow demand (deterministic demand for short) and an extra passenger flow demand (extra demand for short), and respectively recording the deterministic passenger flow demand and the extra passenger flow demand as deterministic passenger flow demand (extra demand for short)
Figure BDA0002863690800000031
And
Figure BDA0002863690800000032
wherein T represents a small time interval after dispersion, and T belongs to T; the method comprises the steps that random dynamic data based on scenes are adopted to describe passenger flow characteristics, and passenger flow requirements are divided into deterministic requirements and extra requirements;
s2: determining the expected dynamic robustness protection level between the starting point and the end point OD point pairs of each passenger trip according to the extra passenger flow requirements related to the scene, and recording the protection level as
Figure BDA0002863690800000033
In general, the
Figure BDA0002863690800000034
The value of (A) is more than 80 percent
Figure BDA0002863690800000035
The value of (d);
s3: introducing decision variables related to time, trains and stations into a train schedule and a passenger flow control robust optimization model, wherein the passenger flow control decision variables specifically comprise: robust passenger flow control decision variables
Figure BDA0002863690800000036
And
Figure BDA0002863690800000037
train capacity decision variable gamma additionally required for achieving robust protection level (also called robust protection level)i(ii) a The train operation decision variables include: decision variable d for 0-1 train running statei,k(t) and train departure interval decision variable hi；
Wherein T represents the small time interval after dispersion, T belongs to T, i represents the train, i belongs to L,
Figure BDA0002863690800000038
represents: at station k, the passenger flow to station v among the passengers leaving by train i;
Figure BDA0002863690800000039
represents: at a station k, the total passenger flow amount leaving by taking the train i; h isiRepresents: departure interval between train i and train i + 1; di,k(t) is: a variable of 0 to 1 reflecting the running state of the train, di,k(t) — 1 indicates that the train i does not arrive at time t and does not pass through the station k, di,k(t) ═ 0 indicates that train i has arrived or has passed station k at time t;
s4: analyzing the relation between a train schedule and a passenger flow control strategy, and constructing coupling constraint (scene coupling constraint for short) between scenes, association constraint between a robust passenger flow control decision variable (also called robust passenger flow control variable) and a train operation decision variable, linear constraint such as maximum carrying capacity of a train, arrival time of the train and departure time of the train and the like;
s5: combining the steps S1-S4, and constructing the uncertainty-oriented object with the goal of minimizing the additional train capacity required for adding the robustness protection levelThe method comprises the following steps of determining a required train schedule and a passenger flow control robust optimization model, wherein decision variables of the robust optimization model comprise: decision variable h for train departure intervali0-1 decision variable d of train running statei,k(t) (also known as train operating state decision variables) and robust passenger flow control decision variables
Figure BDA00028636908000000310
S6: and solving the robust optimization model to obtain a global optimal solution which is used as an optimal train schedule and a robust passenger flow control strategy.
Based on the above technical solution, the total amount of the passenger flow demand in each scene described in step S1 is the sum of the deterministic passenger flow demand and the extra passenger flow demand
Figure BDA0002863690800000041
On the basis of the above technical solution, the specific steps of S2 are: selecting more than 80% of scenes for each discrete small time interval t and each station k
Figure BDA0002863690800000042
For example, when 5 scenes are considered, to determine the robustness protection level at time t, a positive integer larger than the number of passengers additionally arriving in 4 scenes needs to be selected as the expected robustness protection level at the time
Figure BDA0002863690800000043
The level of robustness protection
Figure BDA0002863690800000044
Representing additional passenger flow demands that are expected to be met in addition to meeting deterministic passenger flow demands; the level of robustness protection
Figure BDA0002863690800000045
The scenes are coupled.
Based on the technical scheme, the specific method of S4The method comprises the following steps: leading in a decision variable d of 0-1 of the train running statei,k(t)，di,k(t) '1' indicates that the train i does not arrive at time t and does not pass through the station k, di,kWhen t is 0, the train i arrives or passes through the station k at the time t, so the train operation state 0-1 decision variables form a matrix and are non-decreasing matrices, and the following constraint is satisfied, as shown in formula (1):
di,k(t)≥di,k(t+1) (1)
wherein the content of the first and second substances,
Figure BDA0002863690800000046
k∈S,t∈T,t+1∈T，di,k(t +1) represents a decision variable of 0-1 of the train running state at the moment of t + 1;
on the basis of the technical scheme, the train departure interval decision variable hiAnd a decision variable d of 0-1 train running statei,k(t) is constrained by a linear relationship as shown in equation (2):
Figure BDA0002863690800000047
wherein the content of the first and second substances,
Figure BDA0002863690800000048
on the basis of the technical scheme, the departure time of the train at the starting station is uniquely determined by the departure time and the departure interval of the train 1, and the departure time of the train 1 at the starting station is given in advance, so that the departure time and the train departure interval of the train at the starting station make a decision variable hiThere is a linear constraint between them, as shown in equation (3):
Figure BDA0002863690800000051
wherein the content of the first and second substances,
Figure BDA0002863690800000052
wherein the content of the first and second substances,
Figure BDA0002863690800000053
representing the time of the origination of the train i,
Figure BDA0002863690800000054
represents the time of the start of train i + 1; i.e. the time at which the train i departs from the origin.
In actual operation, the simplest and most widely applied parking strategy is that all trains are parked at all stations to provide service for passengers, and the parking time is the same, and furthermore, due to the rapid development of the technology of the automatic train operation system, the operation time of the same type of train in the same section is usually very small in deviation, so that the operation time of each train in each section is assumed to be a constant related to the distance of the section to be traveled. In summary, the operation time of the train i in each section and the stop time of each station are fixed, so that the operation state of the train i on the whole line can be tracked according to the following constraints, wherein the constraints are shown in formulas (4) and (5):
Figure BDA0002863690800000055
Figure BDA0002863690800000056
wherein the content of the first and second substances,
Figure BDA0002863690800000057
k-1∈S，k∈S，
Figure BDA0002863690800000058
representing the time at which train i arrives at station k,
Figure BDA0002863690800000059
represents the departure time of the train i from the station k and the arrival time of the train i at the station k
Figure BDA00028636908000000510
For the moment it leaves station k-1
Figure BDA00028636908000000511
Plus the running time from station k-1 to station k
Figure BDA00028636908000000512
Moment when train i leaves station k
Figure BDA00028636908000000513
For the moment it arrives at station k
Figure BDA00028636908000000514
Plus the stop time at station k
Figure BDA00028636908000000515
Since the departure time of the train 1 at the station 1 is predetermined, i.e. the departure time is advanced
Figure BDA00028636908000000516
As is known, the running time and the stop time are known, so that the running state of any one train on the whole line can be tracked according to the above constraints.
Further, to ensure all deterministic traffic demands
Figure BDA00028636908000000517
Is satisfied, i.e. all arriving passengers eventually leave by car, said robust passenger flow control decision variable
Figure BDA00028636908000000518
The following constraints are satisfied, as shown in equation (6):
Figure BDA00028636908000000519
wherein the content of the first and second substances,
Figure BDA00028636908000000520
further, to refine robust protection, the passenger flow requirement and the expected achieved robust protection level are divided among each OD point pair, and the following linear constraints are established, as shown in formulas (7), (8) and (9):
Figure BDA0002863690800000061
Figure BDA0002863690800000062
Figure BDA0002863690800000063
wherein the content of the first and second substances,
Figure BDA0002863690800000064
k∈S，v∈S，t∈T，ω∈Ω，
Figure BDA0002863690800000065
the passenger flow demand arriving at the station k at the time t and destined to the station v is multiplied by the time-varying OD proportion delta(k,v)(t) is obtained by(k,v)And (t) is a proportion matrix from any station k to stations of the stations behind the station at the time t, and particularly, the sum of the proportions from any station k to the stations of the stations behind the station k is 1.
In the same way, the extra passenger flow demand arriving at station k at time t and destined for station v can be obtained
Figure BDA0002863690800000066
As shown in formula (8); desired level of robust protection between OD point pairs
Figure BDA0002863690800000067
As shown in formula (9);
building robust guestsFlow control decision variables
Figure BDA0002863690800000068
And a decision variable d of 0-1 train running statei,k(t) is bound by the relationship as shown in equation (10):
Figure BDA0002863690800000069
wherein the content of the first and second substances,
Figure BDA00028636908000000610
k < v, k ∈ S, v ∈ S, i ∈ L/{1} denotes: i belong to all the trains except the 1 st train, such as the 2 nd, 3 rd and 4 th trains and the like,
Figure BDA00028636908000000611
the destination is the passenger flow demand of the station v among the passengers waiting for the train i at the station k; when the train 1 arrives at the station k, the number of passengers waiting for going to the station v at the station k station hall is the number of all arriving passengers before because no passenger gets on or off the station; when a subsequent train i arrives, the station hall of the station k waits for the train i, wherein the number of passengers arriving at the station k and having the destination of the station v is the number of passengers and the passenger flow demand of the train leaving the station before all the passengers have taken the train i
Figure BDA00028636908000000612
A difference of (d);
waiting for the passenger variable of the train i in the deterministic passenger flow demand of each station under the robust passenger flow control strategy
Figure BDA00028636908000000613
The following constraints are satisfied, as shown in equations (11) and (12):
Figure BDA00028636908000000614
Figure BDA00028636908000000615
wherein k is<v，k∈S，v∈S，
Figure BDA00028636908000000616
The number of passengers who arrive at station k is necessarily greater than the number of passengers who have been served, and therefore, the number of passengers waiting for a train at station k is the number of passengers who have served the train at station v
Figure BDA0002863690800000071
Must be equal to or greater than 0. In addition, the total amount of passenger flow waiting for the train at station k
Figure BDA0002863690800000072
For the demand of passenger flow to the following station
Figure BDA0002863690800000073
And (4) summing.
On the basis of the technical scheme, aiming at the deterministic passenger flow demand, the passenger flow control linear constraint is shown as formulas (13), (14) and (15):
Figure BDA0002863690800000074
Figure BDA0002863690800000075
Figure BDA0002863690800000076
wherein the content of the first and second substances,
Figure BDA0002863690800000077
k<v, k, v belongs to S, the number of people leaving the train i at the station k is determined by the passenger flow control strategy by taking the station v as the destination
Figure BDA0002863690800000078
The number of persons waiting for the train at station k is not greater than the number of persons at station v as destination
Figure BDA0002863690800000079
And controlling the number of boarding people
Figure BDA00028636908000000710
Is inevitably equal to or greater than 0; total amount of passenger flow leaving by train i at station k
Figure BDA00028636908000000711
For the number of persons getting on the bus with the next station as the destination
Figure BDA00028636908000000712
And (4) summing.
On the basis of the technical scheme, under a passenger flow control strategy, aiming at deterministic passenger flow demands, the train passenger flow dynamically loads linear constraints as shown in formulas (16) to (20):
Figure BDA00028636908000000713
wherein the content of the first and second substances,
Figure BDA00028636908000000714
k ∈ S, k ∈ S/{1, S } denotes: k belong to all stations except the originating station and the terminating station, such as 2 nd station, 3 rd station and 4 th station, etc.,
Figure BDA00028636908000000715
wherein the content of the first and second substances,
Figure BDA00028636908000000716
k∈S，
Figure BDA00028636908000000717
wherein，
Figure BDA00028636908000000718
k ∈ S, k ∈ S/{1} denotes: k belong to all stations except the originating station, such as 2 nd, 3 rd and 4 th stations and so on,
Figure BDA00028636908000000719
wherein the content of the first and second substances,
Figure BDA00028636908000000720
k∈S，
Figure BDA0002863690800000081
wherein the content of the first and second substances,
Figure BDA0002863690800000082
k ∈ S, k ∈ S/{1} denotes: k belong to all stations except the originating station, such as 2 nd, 3 rd and 4 th stations and so on,
Figure BDA0002863690800000083
the passenger carrying capacity when the train i leaves the station k,
Figure BDA0002863690800000084
the passenger carrying capacity when the train i leaves the station k-1,
Figure BDA0002863690800000085
the getting-on passenger flow of the train i at the station k,
Figure BDA0002863690800000086
the passenger flow of getting-off when the train i is at the station k, CmaxThe station m is any station between the starting station and the station k for the maximum carrying capacity of the train,
Figure BDA0002863690800000087
indicating from the initial vehicleThe passenger flow to the station k among passengers getting on the train at any station in the station-to-station k-1,
Figure BDA0002863690800000088
the remaining capacity of the train, i.e. how much passenger volume the train can still carry.
Specifically, at station 1, the passenger capacity of train i when leaving
Figure BDA0002863690800000089
For controlling the number of persons getting on the bus
Figure BDA00028636908000000810
In the last station, namely the station s, passengers can get off the train at the terminal station, and passengers are not allowed to get on the train at the terminal station, namely, no passenger gets on the train at the terminal station, and after all passengers get off the train, the train is in an empty state and is driven to the train section. Passenger carrying capacity when train i leaves
Figure BDA00028636908000000811
Is 0; passenger carrying capacity when train i leaves at other intermediate station k
Figure BDA00028636908000000812
For the passenger carrying capacity of the train leaving the previous station k-1
Figure BDA00028636908000000813
Minus the number of persons getting off at station k
Figure BDA00028636908000000814
Plus the number of people getting on the bus at the station
Figure BDA00028636908000000815
To ensure train operation safety, train i is not allowed to overload. Number of persons getting off at station k
Figure BDA00028636908000000816
The number of passengers getting on the train is controlled from the station 1 to the station k-1, and the passenger number takes the station k as the destination.In addition, under the passenger flow control strategy, the number of people who leave the train i at the station k
Figure BDA00028636908000000817
Is not larger than the train residual capacity of the train i after the train i finishes the getting-off behavior at the station k
Figure BDA00028636908000000818
Otherwise, the train i will be overloaded, which brings great potential safety hazard. At the station 1, when the train i arrives, no passenger needs to get off the train, and the remaining capacity of the train is at the moment
Figure BDA00028636908000000819
I.e. the maximum carrying capacity C of the trainmax(ii) a When the train i arrives at each subsequent station k, the passengers on the train need to finish the getting-off action firstly, and after all the passengers destined for the station k get-off, the residual capacity of the train is at the moment
Figure BDA00028636908000000820
For maximum carrying capacity C of trainmaxMinus the passenger capacity of the train leaving the previous station k-1
Figure BDA00028636908000000821
Plus additional empty seats added by passengers alighting
Figure BDA00028636908000000822
On the basis of the technical scheme, the robust passenger flow control constraint is constructed by considering the extra passenger flow requirements of each scene, as shown in formulas (21) to (24):
Figure BDA0002863690800000091
wherein the content of the first and second substances,
Figure BDA0002863690800000092
k<v, k ∈ S, v ∈ S, i ∈ L/{1} denotes: i belonging to all but the 1 st trainNext, such as column 2, column 3, and column 4,
Figure BDA0002863690800000093
wherein the content of the first and second substances,
Figure BDA0002863690800000094
k∈S，
Figure BDA0002863690800000095
wherein the content of the first and second substances,
Figure BDA0002863690800000096
Figure BDA0002863690800000097
wherein the content of the first and second substances,
Figure BDA0002863690800000098
k∈S，
Figure BDA0002863690800000099
in order to wait for passengers of the train i at the station k, the destination is the number of stations v,
Figure BDA00028636908000000910
for summation of the traffic volume waiting for train i at station k, comprising both deterministic traffic demand and extra traffic demand, γiRepresents the additional train capacity required for train i to achieve the desired level of robustness protection; considering the extra passenger flow demand of each scene, the total passenger flow volume when the train 1 leaves each station k is the passenger flow volume considering only the deterministic passenger flow demand
Figure BDA00028636908000000911
And the number of additional required arrivals of train 1 at each station k; each subsequent trainThe total passenger flow volume when leaving each station k is the passenger flow volume considering only the deterministic passenger flow demand
Figure BDA00028636908000000912
And the number of arrival persons required for additional passenger flow in the time interval between arrival of train i and train i-1 at each station k.
Similarly, in order to ensure the operation safety, the train i is required not to be overloaded, and in order to meet all the passenger flow demands, some carrying capacity gamma is additionally added to each train iiThe decision variable gammaiThe numerical value of (A) can provide theoretical basis for the flexible marshalling of subway trains.
On the basis of the technical scheme, a parameter alpha is introduced to limit the number of waiting passenger flows in the robust optimization scheme to a certain extent, as shown in formula (25):
Figure BDA00028636908000000913
wherein the content of the first and second substances,
Figure BDA00028636908000000914
when only the deterministic passenger flow demand is considered, the number of waiting passengers under the obtained optimal passenger flow control strategy is solved;
the robust optimization model is composed of equations (26), (1) - (12), (14) - (16), and (18) - (25), as follows:
Figure BDA0002863690800000101
di,k(t)≥di,k(t+1) (1)
Figure BDA0002863690800000102
Figure BDA0002863690800000103
Figure BDA0002863690800000104
Figure BDA0002863690800000105
Figure BDA0002863690800000106
Figure BDA0002863690800000107
Figure BDA0002863690800000108
Figure BDA0002863690800000109
Figure BDA00028636908000001010
Figure BDA00028636908000001011
Figure BDA00028636908000001012
Figure BDA00028636908000001013
Figure BDA00028636908000001014
Figure BDA00028636908000001015
Figure BDA00028636908000001016
Figure BDA00028636908000001017
Figure BDA00028636908000001018
Figure BDA00028636908000001019
Figure BDA0002863690800000111
Figure BDA0002863690800000112
Figure BDA0002863690800000113
Figure BDA0002863690800000114
the gamma isiThe solution is obtained by equation (23) and equation (24).
The invention has the following beneficial technical effects:
the invention provides a train schedule and passenger flow control robust collaborative optimization method which adjusts the gathering quantity of passengers at a platform within a safety range by considering uncertainty of passenger travel requirements under the condition of ensuring safe operation of a train so as to reduce operation accident risks under different scenes, solves the technical problem that the existing passenger flow control method in the prior art is difficult to be oriented to engineering practice, obtains a robust control strategy suitable for large-city rail transit passenger flow control, and provides theoretical support for safe and efficient operation of actual rail transit.
Drawings
The invention has the following drawings:
FIG. 1 is a schematic flow chart of an urban rail transit train timetable and passenger flow control robust collaborative optimization method for dynamic uncertain passenger flow demand according to an embodiment of the present invention;
FIG. 2 is a schematic view showing the dynamic arrival of passengers at each station of the eight-way traffic early peak of Beijing subway;
fig. 3 is a schematic diagram illustrating an arrangement state of the entering equipment of the get-off station and a dynamic boarding of passengers under the robust passenger flow control strategy according to the embodiment of the present invention;
FIG. 4 is a schematic diagram of a deterministic passenger flow demand curve according to an embodiment of the present invention;
FIG. 5 is a schematic diagram of an extra passenger flow demand curve according to an embodiment of the present invention;
FIG. 6 is a schematic diagram of a train operation state 0-1 decision variable matrix form.
Detailed Description
In order to more clearly illustrate the invention, the invention is further described below with reference to preferred embodiments and the accompanying drawings. 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.
As shown in fig. 1, the method for robust collaborative optimization of urban rail train timetable and passenger flow control for dynamic uncertain passenger flow demand according to the embodiment of the present invention specifically includes the following steps:
firstly, establishing a robust optimization model
1. Key parameter
The key technology of the invention lies inThe scene, the train schedule and the robust passenger flow control are coupled, and further, a train schedule and robust passenger flow control collaborative optimization method facing the OD requirements of dynamic uncertain passenger travel is provided. To achieve this object, the invention first treats a continuous control period as a series of discrete time intervals, each time interval having a length Δ, by means of a time discretization. In reality, the traveling demands of passengers are dynamically changed along with time, and in order to describe the dynamic demand distribution of the passengers, a curve of the arrival of passengers per minute at each station of a certain subway line peak is given as shown in fig. 2. As can be seen from the figure, the passenger flow is very dynamic. In one embodiment of the invention, the invention uses time-dependent parameters in order to scientifically characterize the dynamics of the passenger flow
Figure BDA0002863690800000121
Is shown at (t)0,t0+Δ]The time interval passenger travel demand, i.e. the deterministic demand, specifically,
Figure BDA0002863690800000122
is shown at (t)0,t0+Δ]Arriving at station k in time and destined for the number of passengers at station v. By means of the method, all passenger travel OD demands in a planning period can be represented. Further, the invention introduces scene-related parameters
Figure BDA0002863690800000123
The uncertainty of the passenger flow demand, i.e. the extra demand, is characterized. Generally, for safety, the operation time of the subway train between each section and the stop time of each station are fixed, and are respectively marked as
Figure BDA0002863690800000124
And
Figure BDA0002863690800000125
wherein i represents a train, i belongs to L, and the set of all stations in the subway line is S ═ {1,2, …, k, …, S }, where k ═ 1 represents the originationAnd the station, k ∈ S, represents the terminal.
2. Decision variables
In a specific embodiment of the invention, the decision variables comprise robust passenger flow control decision variables
Figure BDA0002863690800000126
And
Figure BDA0002863690800000127
and train operation decision variables. Wherein the robust passenger flow control decision variables comprise:
Figure BDA0002863690800000128
at station k, the amount of traffic to station v among passengers leaving by train i
Figure BDA0002863690800000129
Total amount of passenger flow leaving by train i at station k
Train operation decision variables include:
decision variable h for train departure intervali: departure interval between train i and train i + 1;
decision variable d for 0-1 train running statei,k(t): a variable of 0 to 1 reflecting the running state of the train, di,k(t) — 1 indicates that the train i does not arrive at time t and does not pass through the station k, di,kAnd (t) ═ 0 represents that the train i arrives or passes through the station k at the time t, and the train operation state 0-1 decision variables form a matrix which is a non-decreasing matrix and is in the form shown in fig. 6.
3. Constraint conditions
In one embodiment of the present invention, the train operation state tracking constraint is represented by equations (1) to (5):
di,k(t)≥di,k(t+1) (1)
wherein the content of the first and second substances,
Figure BDA0002863690800000131
k∈S,t,t+1∈T；
Figure BDA0002863690800000132
wherein the content of the first and second substances,
Figure BDA0002863690800000133
Figure BDA0002863690800000134
wherein the content of the first and second substances,
Figure BDA0002863690800000135
Figure BDA0002863690800000136
wherein the content of the first and second substances,
Figure BDA0002863690800000137
k-1,k∈S
Figure BDA0002863690800000138
wherein the content of the first and second substances,
Figure BDA0002863690800000139
k∈S，di,kwhen t is 1, the train i does not arrive and passes through the station k, di,kWhen t is 0, the train i arrives or passes through the station k, hiIs the departure interval between the train i and the train i +1,
Figure BDA00028636908000001310
the time when the train i arrives at the station k,
Figure BDA00028636908000001311
the time when the train i departs from the station k. Column(s) ofThe time at which vehicle i arrives at station k is equal to the time at which it departs from station k-1
Figure BDA00028636908000001312
Plus the time required to travel from station k-1 to station k
Figure BDA00028636908000001313
Time when train i departs from station k
Figure BDA00028636908000001314
Equal to the moment when it arrives at station k
Figure BDA00028636908000001315
Time of parking at station k
Figure BDA00028636908000001316
And thus, the above constraint is obtained.
In the actual passenger flow control process, all passengers arriving at the station k finally leave the train, so the following constraint is obtained, as shown in equation (6):
Figure BDA00028636908000001317
wherein the content of the first and second substances,
Figure BDA00028636908000001318
when the train 1 arrives at the station k, the number of passengers waiting for going to the station v at the station k station hall is equal to the number of all passengers arriving before, because no passenger gets on or off the station; when a subsequent train i arrives, the number of passengers waiting for the train i in the station k station hall is equal to the number of all passengers arriving at the station k and destined for the station v minus the passenger leaving the train before the train i, so that the following constraint is obtained, as shown in equation (10):
Figure BDA0002863690800000141
wherein the content of the first and second substances,
Figure BDA0002863690800000142
k＜v,k,v∈S，
in the actual passenger flow control process, there is no possibility that more passengers are served than the passenger flow demand, so the following constraint is obtained, as shown in equation (11):
Figure BDA0002863690800000143
wherein the content of the first and second substances,
Figure BDA0002863690800000144
k<v,k,v∈S，
for station k, when a train i arrives, the number of passengers waiting in the station hall is the total passenger flow of each station after the destination is station k, so the following constraint is obtained, as shown in equation (12):
Figure BDA0002863690800000145
wherein the content of the first and second substances,
Figure BDA0002863690800000146
k∈S，
in the actual passenger flow control process, the number of passengers allowed for inbound traffic is unlikely to be greater than waiting passengers, and therefore the following constraint is obtained, as shown in equation (13):
Figure BDA0002863690800000147
wherein the content of the first and second substances,
Figure BDA0002863690800000148
k<v,k,v∈S，
for the station k, when the train i arrives, the number of passengers allowed to enter the station is the total passenger flow volume of each station after the destination is the station k, so the following constraint is obtained, as shown in equation (15):
Figure BDA0002863690800000149
wherein the content of the first and second substances,
Figure BDA00028636908000001410
k∈S，
in the actual passenger flow control process, when the train i arrives at the station k, the train possibly has no residual capacity, and passengers are not allowed to enter the station at the moment; once the train has empty seats, at least 1 passenger is allowed to get in and get out of the train, in other words, the amount of passenger flow allowed to get in must be greater than or equal to 0, so the following constraint is obtained, as shown in equation (14):
Figure BDA00028636908000001411
wherein the content of the first and second substances,
Figure BDA00028636908000001412
k∈S
since passengers have strong uncertainty, in an embodiment of the present invention, a robustness protection level to be implemented between each OD point pair is determined according to extra arriving passenger flow related to a scene, and it is ensured that a train that arrives most recently can leave after the passengers arrive at a station, further, when robust passenger flow control is performed, a sum of passenger flows of waiting trains i at the station k needs to be calculated, including two parts of deterministic passenger flow demand and extra passenger flow demand, so as to obtain the following constraint, as shown in equation (21):
Figure BDA0002863690800000151
wherein the content of the first and second substances,
Figure BDA0002863690800000152
k<v,k,v∈S，
Figure BDA0002863690800000153
among passengers waiting for train i at station k, the destination is the number of stations v. Considering the extra passenger flow requirement of each scene, when the train 1 arrives at the station k, the number of waiting persons is the number of waiting persons only considering the deterministic passenger flow requirement
Figure BDA0002863690800000154
And the number of additional passenger flow demands arriving at train 1 arriving at station k; when each subsequent train arrives at the station k, the waiting number is the waiting number only considering the deterministic requirement
Figure BDA0002863690800000155
And the number of arrival persons required for additional passenger flow in the time interval between arrival of train i and train i-1 at each station k.
For station k, when the train i arrives, the number of passengers waiting in the station hall is the total passenger flow of each station after the destination is station k, so the following constraint is obtained as shown in equation (22):
Figure BDA0002863690800000156
wherein the content of the first and second substances,
Figure BDA0002863690800000157
k∈S，
to ensure the quality of a robust solution, it is necessary to control the percentage increase of the number of waiting passengers with respect to the waiting traffic volume obtained when only deterministic traffic demands are considered, thus obtaining the constraint, as shown in equation (25), in which,
Figure BDA0002863690800000158
the number of waiting passengers under the obtained optimal passenger flow control strategy is solved when only the deterministic passenger flow demand is considered.
Figure BDA0002863690800000159
In a specific embodiment of the present invention, in the dynamic loading process of train passenger flow, if only the deterministic passenger flow requirement is considered, the number of passengers on the train i leaving the station k is the number of passengers on the train i leaving the previous station k-1 minus the number of passengers off the station, plus the number of passengers on the train i leaving the station, specifically, when leaving the first station, the number of passengers on the train is the number of passengers on the train, and when leaving the last station, the number of passengers on the train is 0, so the following constraint is obtained, as shown in equation (16):
Figure BDA0002863690800000161
wherein the content of the first and second substances,
Figure BDA0002863690800000162
k∈S，
in the actual passenger flow control process, since the train capacity has a certain limit, and the train cannot be overloaded to ensure the operation safety, the following constraint is obtained, as shown in formula (17):
Figure BDA0002863690800000163
wherein the content of the first and second substances,
Figure BDA0002863690800000164
k∈S，
when the train i arrives at the station k, the departure passenger flow is the passenger with the destination of the station k in the arrival passenger flow of all the stations in front, so the following constraint is obtained as shown in the formula (18):
Figure BDA0002863690800000165
wherein the content of the first and second substances,
Figure BDA0002863690800000166
k∈S，
in the actual passenger flow control process, due to the limitation of train capacity, at a station k, the passenger flow allowed to get on a train i cannot exceed the remaining capacity of the train, otherwise, the number of passengers on the train exceeds the train capacity, and potential safety hazards are brought, so that the following constraints are obtained, as shown in formula (19):
Figure BDA0002863690800000167
wherein the content of the first and second substances,
Figure BDA0002863690800000168
k∈S，
when a train i arrives at a station k, after passengers get off, the remaining capacity is the total capacity minus the number of passengers on board when arriving at the station, plus the number of passengers getting off at the station, and particularly, at an origin station, the train arrives empty, and the remaining capacity is the total capacity, so that the following constraint is obtained, as shown in formula (20):
Figure BDA0002863690800000169
wherein the content of the first and second substances,
Figure BDA00028636908000001610
k∈S，
in one embodiment of the invention, the level of robustness protection that is desired to be achieved between each OD point pair is determined from the scene-dependent extra arriving passenger flow and ensures that after it arrives at the station it must leave with the most recently arriving train, then during the overcrowding period, considering only the deterministic passenger flow requirements, the train may be fully loaded with no remaining capacity to carry extra passengers unless overloaded. In order to ensure the operation safety, the train i is required not to be overloaded, and in order to meet all passenger flow demands, some carrying capacity gamma is additionally added to each train iiThe decision variable gammaiThe value of (a) represents the additional train capacity required for train i to achieve the desired level of robustness protection, resulting in the following constraints, as shown in equations (23) and (24):
Figure BDA0002863690800000171
wherein the content of the first and second substances,
Figure BDA0002863690800000172
Figure BDA0002863690800000173
wherein the content of the first and second substances,
Figure BDA0002863690800000174
k∈S，
wherein, considering the extra passenger flow demand of each scene, the total passenger flow when the train 1 leaves each station k is the passenger flow volume considering only the determinacy demand
Figure BDA0002863690800000175
And the number of additional required arrivals of train 1 at each station k; the total passenger flow volume when each subsequent train leaves each station k is the passenger flow volume only considering the deterministic passenger flow demand
Figure BDA0002863690800000176
And the number of arrival persons required for additional passenger flow in the time interval between arrival of train i and train i-1 at each station k.
4. Robust optimization model
In summary, with the aim of minimizing the extra train capacity required by adding the robustness protection level, an urban rail transit train schedule and a passenger flow control robust optimization model (abbreviated as a robust optimization model) for uncertain passenger flow demands are constructed, and the specific form of the robust optimization model is composed of formulas (26), (1) - (12), (14) - (16) and (18) - (25), and is as follows:
Figure BDA0002863690800000177
di,k(t)≥di,k(t+1) (1)
Figure BDA0002863690800000178
Figure BDA0002863690800000179
Figure BDA00028636908000001710
Figure BDA00028636908000001711
Figure BDA00028636908000001712
Figure BDA00028636908000001713
Figure BDA0002863690800000181
Figure BDA0002863690800000182
Figure BDA0002863690800000183
Figure BDA0002863690800000184
Figure BDA0002863690800000185
Figure BDA0002863690800000186
Figure BDA0002863690800000187
Figure BDA0002863690800000188
Figure BDA0002863690800000189
Figure BDA00028636908000001810
Figure BDA00028636908000001811
Figure BDA00028636908000001812
Figure BDA00028636908000001813
Figure BDA00028636908000001814
Figure BDA00028636908000001815
Figure BDA00028636908000001816
solving robust optimization model
And solving the constructed robust optimization model to obtain an optimal robust passenger flow control scheme and a matched train schedule. The specific solving method is as follows: according to the train schedule and passenger flow control robust optimization model for the dynamic uncertain passenger flow demand, both a passenger flow control decision variable and a train operation decision variable are integer variables from the viewpoint of decision variables, and the complexity of the robust optimization model mainly depends on the number of stations, the number of service trains and the number of discrete time points. In addition, all the constructed constraint conditions are linear constraints, so that the solution can be carried out by using common commercial optimization software (such as CPLEX and the like), and a better passenger flow control strategy and a matched train schedule of the system can be obtained.
Third, example verification
The urban rail transit train timetable and passenger flow control robust collaborative optimization method for the dynamic uncertain passenger flow disclosed by the invention is further explained by combining specific examples and drawings. FIG. 3 is a schematic diagram showing arrangement states of the station getting-off and station getting-on devices and dynamic passengers getting-on under a robust passenger flow control strategy;
the content of the invention is verified by taking a certain one-way urban rail transit line with four stations as an example, the names of the stations are A, B, C and D, 9 trains are driven within 30min of planning time, the carrying capacity of the trains is 900 persons/train, the time discrete step delta is 1min, 5 scenes are considered, the deterministic passenger flow demand of each station under each scene is shown in FIG. 4, the extra passenger demand (i.e. extra passenger flow demand) is taken as an example of the station A, as shown in FIG. 5, and the robustness protection level expected to be realized at each time of each station is determined according to the extra passenger demand. The total passenger demand at each station in each scenario is shown in table 1. Assume a peak time period of [7,17] during which the passenger demand exceeds the train capacity, and the remaining time period is a flat peak time period. And no passenger arrives at the station after the last bus leaves the station. The running time of the train in each interval is assumed to be 3min, the stop time of each station is assumed to be 1min, the minimum train inter-vehicle interval of two continuous trains is 2min, the maximum train inter-vehicle interval is 4min, and the departure time of the train 1 is the 2 nd moment.
TABLE 1 passenger flow demand schematic table for each scene
Scene 1 Scene 2 Scene 3 Scene 4 Scene 5
Station A 4145 4167 4150 4167 4133
Station B 2006 2010 2004 2008 2027
Station C 3083 3081 3078 3074 3086
According to the given passenger arrival data, codes are written in MATLAB to construct a frame of the method, and CPLEX optimization software is further called to solve the problems, so that a better robust passenger flow control strategy and a matched train schedule can be obtained. First, only the deterministic traffic demand is used for solving, and after about 1 second of calculation, 9009 waiting people are obtained. If only passenger flow control is carried out without collaborative optimization of the train schedule, the waiting number is 10854 persons; if the operation is carried out according to the optimized schedule without adopting a passenger flow control strategy and the number of waiting persons is 11375, the passenger flow control and train schedule collaborative optimization strategy can be adopted, so that the retention condition of passengers can be greatly relieved. Further, the deterministic passenger flow and the extra passenger flow demand are used for solving, different values of alpha are set, and the obtained results are shown in table 2. To better verify the effectiveness of the present invention, each single scenario will be verified separately considering only the deterministic traffic demand, i.e. a solution derived without considering the uncertainty of the passengers and a robust solution, the results of which are shown in table 3. The comparison shows that the scheme obtained based on the light robust method provided by the invention can meet the better results of all scenes and has strong robustness.
TABLE 2 robust collaborative optimization results schematic table
α Objective function Number of waiting persons Calculating time
0.10 97 9099 1s
0.13 14 10181 1s
0.15 0 10234 1s
0.20 0 10234 1s
TABLE 3 robustness comparison results schematic Table
Figure BDA0002863690800000201
In conclusion, the urban rail transit train timetable and passenger flow control robust optimization model for the dynamic uncertain passenger flow demand is constructed, and a better robust passenger flow control scheme and a matched timetable can be quickly solved through optimization software. Under the robust passenger flow control strategy, each scene can achieve system optimization, and the method can provide long-term, stable and effective subway robust passenger flow control measures for operation practice.
The above embodiments describe the technical solutions of the present invention in detail. It will be clear that the invention is not limited to the described embodiments. Based on the embodiments of the present invention, those skilled in the art can make various changes, but any changes equivalent or similar to the present invention are within the protection scope of the present invention.
Those not described in detail in this specification are within the knowledge of those skilled in the art.

Claims (10)

1. A train schedule and passenger flow control robust optimization method facing to uncertain demands is characterized by comprising the following steps:
s1: recording a set of all stations in a subway line as S {1,2, …, k, …, S }, wherein k ═ 1 denotes an origin station, k ═ S denotes a destination station, and k ∈ S; note that the set of all trains on the link is L ═ {1,2, …, i, …, n }; recording a set of traffic scenes as Ω ═ 1, 2. Let the set of discrete time periods T ═ T0,t0+Δ,…,t0+ n Δ }, where Δ is a time discrete step representing the length of each time interval after the dispersion, and n is a natural number; dividing the passenger flow demand in each scene into a deterministic passenger flow demand and an extra passenger flow demand, and respectively recording the deterministic passenger flow demand and the extra passenger flow demand as the deterministic passenger flow demand and the extra passenger flow demand
Figure FDA0002863690790000011
And
Figure FDA0002863690790000012
wherein T represents a time interval after dispersion, and T belongs to T;
s2: determining the expected dynamic robustness protection level between the starting point and the end point OD point pairs of each passenger trip according to the extra passenger flow demand of the scene, and recording the protection level as
Figure FDA0002863690790000013
The above-mentioned
Figure FDA0002863690790000014
Is greater than
Figure FDA0002863690790000015
The value of (d);
s3: introducing time, train and station decision variables into a train schedule and a passenger flow control robust optimization model, wherein the passenger flow control decision variables comprise: robust passenger flow control decision variables
Figure FDA0002863690790000016
And
Figure FDA0002863690790000017
train capacity decision variable gamma additionally required for achieving robust protection leveli(ii) a The train operation decision variables include: decision variable d for 0-1 train running statei,k(t) and train departure interval decision variable hi；
Wherein i represents a train, i ∈ L,
Figure FDA0002863690790000018
represents: at station k, the passenger flow to station v among the passengers leaving by train i;
Figure FDA0002863690790000019
represents: at a station k, the total passenger flow amount leaving by taking the train i; h isiRepresents: departure interval between train i and train i + 1; di,k(t) is: a variable of 0 to 1 reflecting the running state of the train, di,k(t) — 1 indicates that the train i does not arrive at time t and does not pass through the station k, di,k(t) ═ 0 indicates that train i has arrived or has passed station k at time t;
s4: analyzing the relation between a train schedule and a passenger flow control strategy, constructing coupling constraint between scenes, association constraint between a robust passenger flow control decision variable and a train operation decision variable, and linear constraint on the maximum carrying capacity, the arrival time and the departure time of a train;
s5: combining steps S1-S4 to minimize the additional train capacity required to add a level of robustness protectionAiming at the objective, a train schedule and passenger flow control robust optimization model facing uncertain demands is constructed, and decision variables of the robust optimization model comprise: decision variable h for train departure intervali0-1 decision variable d of train running statei,k(t) and robust traffic control decision variables
Figure FDA0002863690790000021
S6: and solving the robust optimization model to obtain a global optimal solution which is used as an optimal train schedule and a robust passenger flow control strategy.
2. The uncertain demand oriented train schedule and passenger flow control robust optimization method of claim 1, wherein: the total amount of the passenger flow demand in each scene described in step S1 is the sum of the deterministic passenger flow demand and the extra passenger flow demand
Figure FDA0002863690790000022
3. The uncertain demand oriented train schedule and passenger flow control robust optimization method of claim 2, wherein: the specific steps of S2 are: selecting more than 80% of scenes for each discrete time interval t and each station k
Figure FDA0002863690790000023
As the desired level of robustness protection at that moment
Figure FDA0002863690790000024
The level of robustness protection
Figure FDA0002863690790000025
Representing additional passenger flow demands that are expected to be met in addition to meeting deterministic passenger flow demands; the level of robustness protection
Figure FDA0002863690790000026
The scenes are coupled.
4. The uncertain demand oriented train schedule and passenger flow control robust optimization method of claim 3, wherein: the specific steps of S4 are: leading in a decision variable d of 0-1 of the train running statei,k(t)，di,k(t) '1' indicates that the train i does not arrive at time t and does not pass through the station k, di,kWhen t is 0, the train i arrives or passes through the station k at the time t, so the train operation state 0-1 decision variables form a matrix and are non-decreasing matrices, and the following constraint is satisfied, as shown in formula (1):
di,k(t)≥di,k(t+1) (1)
wherein the content of the first and second substances,
Figure FDA0002863690790000027
k∈S,t∈T,t+1∈T，di,k(t +1) represents a decision variable of 0-1 of the train operation state at the time of t + 1.
5. The uncertain demand oriented train schedule and passenger flow control robust optimization method of claim 4, wherein: the decision variable h for the departure interval of the trainiAnd a decision variable d of 0-1 train running statei,k(t) is constrained by a linear relationship as shown in equation (2):
Figure FDA0002863690790000028
wherein the content of the first and second substances,
Figure FDA0002863690790000029
6. the uncertain demand oriented train schedule and passenger flow control robust optimization method of claim 5, wherein: the departure time of the train at the departure station is determined by the departure time and departure interval of the train 1Upon determination, since the departure time of the train 1 at the departure station is given in advance, the departure time of the train at the departure station and the train departure interval decision variable h are determinediThere is a linear constraint between them, as shown in equation (3):
Figure FDA0002863690790000031
wherein the content of the first and second substances,
Figure FDA0002863690790000032
wherein the content of the first and second substances,
Figure FDA0002863690790000033
representing the time of the origination of the train i,
Figure FDA0002863690790000034
represents the time of the start of train i + 1; the running state of the train i on the whole line is tracked according to the following constraints, which are shown as formulas (4) and (5):
Figure FDA0002863690790000035
Figure FDA0002863690790000036
wherein the content of the first and second substances,
Figure FDA0002863690790000037
k-1∈S，k∈S，
Figure FDA0002863690790000038
representing the time at which train i arrives at station k,
Figure FDA0002863690790000039
representsThe time when the train i departs from the station k and the time when the train i arrives at the station k
Figure FDA00028636907900000310
For the moment it leaves station k-1
Figure FDA00028636907900000311
Plus the running time from station k-1 to station k
Figure FDA00028636907900000312
Moment when train i leaves station k
Figure FDA00028636907900000313
For the moment it arrives at station k
Figure FDA00028636907900000314
Plus the stop time at station k
Figure FDA00028636907900000315
The robust passenger flow control decision variable
Figure FDA00028636907900000316
The following constraints are satisfied, as shown in equation (6):
Figure FDA00028636907900000317
wherein the content of the first and second substances,
Figure FDA00028636907900000318
dividing the passenger flow demand and the expected robustness protection level between each OD point pair, and establishing the following linear constraints as shown in formulas (7), (8) and (9):
Figure FDA00028636907900000319
Figure FDA00028636907900000320
Figure FDA00028636907900000321
wherein the content of the first and second substances,
Figure FDA00028636907900000322
k∈S，v∈S，t∈T，ω∈Ω，
Figure FDA00028636907900000323
the passenger flow demand arriving at the station k at the time t and destined to the station v is multiplied by the time-varying OD proportion delta(k,v)(t) is obtained by(k,v)(t) is a proportion matrix from any station k to stations of each station behind the station at the moment t, and the sum of the proportions from any station k to stations of each station behind the station k is 1;
additional passenger flow demand arriving at station k at time t and destined for station v
Figure FDA0002863690790000041
As shown in formula (8); desired level of robust protection between OD point pairs
Figure FDA0002863690790000042
As shown in formula (9);
constructing robust passenger flow control decision variables
Figure FDA0002863690790000043
And a decision variable d of 0-1 train running statei,k(t) is bound by the relationship as shown in equation (10):
Figure FDA0002863690790000044
wherein the content of the first and second substances,
Figure FDA0002863690790000045
k < v, k ∈ S, v ∈ S, i ∈ L/{1} denotes: i belong to all the trains except the 1 st train,
Figure FDA0002863690790000046
the destination is the passenger flow demand of the station v among the passengers waiting for the train i at the station k; when the train 1 arrives at the station k, the number of passengers waiting for going to the station v at the station k station hall is the number of all arriving passengers before because no passenger gets on or off the station; when a subsequent train i arrives, the station hall of the station k waits for the train i, wherein the number of passengers arriving at the station k and having the destination of the station v is the number of passengers and the passenger flow demand of the train leaving the station before all the passengers have taken the train i
Figure FDA0002863690790000047
A difference of (d);
waiting for the passenger variable of the train i in the deterministic passenger flow demand of each station under the robust passenger flow control strategy
Figure FDA0002863690790000048
The following constraints are satisfied, as shown in equations (11) and (12):
Figure FDA0002863690790000049
Figure FDA00028636907900000410
wherein k is<v，k∈S，v∈S，
Figure FDA00028636907900000411
7. The uncertain demand oriented train schedule and passenger flow control robust optimization method of claim 6, wherein: for deterministic traffic demand, the traffic control linear constraints are as shown in equations (13), (14) and (15):
Figure FDA00028636907900000412
Figure FDA0002863690790000051
Figure FDA0002863690790000052
wherein the content of the first and second substances,
Figure FDA0002863690790000053
k<v, k, v belongs to S, the number of people leaving the train i at the station k is determined by the passenger flow control strategy by taking the station v as the destination
Figure FDA0002863690790000054
The number of persons waiting for the train at station k is not greater than the number of persons at station v as destination
Figure FDA0002863690790000055
And controlling the number of boarding people
Figure FDA0002863690790000056
Is inevitably equal to or greater than 0; total amount of passenger flow leaving by train i at station k
Figure FDA0002863690790000057
For the number of persons getting on the bus with the next station as the destination
Figure FDA0002863690790000058
And (4) summing.
8. The uncertain demand oriented train schedule and passenger flow control robust optimization method of claim 7, wherein: under the passenger flow control strategy, aiming at the deterministic passenger flow demand, the train passenger flow dynamically loads linear constraints, as shown in formulas (16) to (20):
Figure FDA0002863690790000059
wherein the content of the first and second substances,
Figure FDA00028636907900000510
k ∈ S, k ∈ S/{1, S } denotes: k belongs to all stations except the originating station and the terminating station,
Figure FDA00028636907900000511
wherein the content of the first and second substances,
Figure FDA00028636907900000512
k∈S，
Figure FDA00028636907900000513
wherein the content of the first and second substances,
Figure FDA00028636907900000514
k ∈ S, k ∈ S/{1} denotes: k belong to all stations except the originating station,
Figure FDA00028636907900000515
wherein the content of the first and second substances,
Figure FDA00028636907900000516
k∈S，
Figure FDA00028636907900000517
wherein the content of the first and second substances,
Figure FDA00028636907900000518
k ∈ S, k ∈ S/{1} denotes: k belong to all stations except the originating station,
Figure FDA00028636907900000519
the passenger carrying capacity when the train i leaves the station k,
Figure FDA00028636907900000520
the passenger carrying capacity when the train i leaves the station k-1,
Figure FDA0002863690790000061
the getting-on passenger flow of the train i at the station k,
Figure FDA0002863690790000062
the passenger flow of getting-off when the train i is at the station k, CmaxThe station m is any station between the starting station and the station k for the maximum carrying capacity of the train,
Figure FDA0002863690790000063
indicating the passenger flow to the station k among the passengers getting on the train from the initial station to any one of the stations k-1,
Figure FDA0002863690790000064
the remaining capacity of the train.
9. The uncertain demand oriented train schedule and passenger flow control robust optimization method of claim 8, wherein: and (3) constructing robust passenger flow control constraints by considering the additional passenger flow requirements of each scene, as shown in formulas (21) to (24):
Figure FDA0002863690790000065
wherein the content of the first and second substances,
Figure FDA0002863690790000066
k<v, k ∈ S, v ∈ S, i ∈ L/{1} denotes: i belong to all the trains except the 1 st train,
Figure FDA0002863690790000067
wherein the content of the first and second substances,
Figure FDA0002863690790000068
k∈S，
Figure FDA0002863690790000069
wherein the content of the first and second substances,
Figure FDA00028636907900000610
Figure FDA00028636907900000611
wherein the content of the first and second substances,
Figure FDA00028636907900000612
k∈S，
Figure FDA00028636907900000613
in order to wait for passengers of the train i at the station k, the destination is the number of stations v,
Figure FDA00028636907900000614
for summation of the traffic volume waiting for train i at station k, comprising both deterministic traffic demand and extra traffic demand, γiRepresenting the additional train capacity required for train i to achieve the desired level of robustness protection.
10. The uncertain demand oriented train schedule and passenger flow control robust optimization method of claim 9, wherein: the number of waiting streams is limited by introducing a parameter alpha, as shown in formula (25):
Figure FDA00028636907900000615
wherein the content of the first and second substances,
Figure FDA00028636907900000616
when only the deterministic passenger flow demand is considered, the number of waiting passengers under the obtained optimal passenger flow control strategy is solved;
the robust optimization model is composed of equations (26), (1) - (12), (14) - (16), and (18) - (25), as follows:
Figure FDA0002863690790000071
di,k(t)≥di,k(t+1) (1)
Figure FDA0002863690790000072
Figure FDA0002863690790000073
Figure FDA0002863690790000074
Figure FDA0002863690790000075
Figure FDA0002863690790000076
Figure FDA0002863690790000077
Figure FDA0002863690790000078
Figure FDA0002863690790000079
Figure FDA00028636907900000710
Figure FDA00028636907900000711
Figure FDA00028636907900000712
Figure FDA00028636907900000713
Figure FDA00028636907900000714
Figure FDA00028636907900000715
Figure FDA00028636907900000716
Figure FDA00028636907900000717
Figure FDA00028636907900000718
Figure FDA0002863690790000081
Figure FDA0002863690790000082
Figure FDA0002863690790000083
Figure FDA0002863690790000084
Figure FDA0002863690790000085
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