CN115017654B - Method for measuring and minimizing urban rail network vulnerabilities - Google Patents

Method for measuring and minimizing urban rail network vulnerabilities Download PDF

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CN115017654B
CN115017654B CN202210016545.6A CN202210016545A CN115017654B CN 115017654 B CN115017654 B CN 115017654B CN 202210016545 A CN202210016545 A CN 202210016545A CN 115017654 B CN115017654 B CN 115017654B
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刘杰
何明卫
占曙光
帅春燕
刘阳
石庄斌
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Kunming University of Science and Technology
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Abstract

The invention relates to the technical field of urban rail transit, in particular to a method for measuring and minimizing urban rail transit network vulnerability, which comprises the following steps: (1) Constructing an urban rail network and defining a travel path acceptable for passengers; (2) evaluating urban rail network service performance; (3) assessing vulnerability; (4) And adding turn-back tracks at stations through an optimal sequence to minimize the vulnerability of the urban rail network. The invention can better measure and minimize the vulnerability of the urban rail network, and can be simply realized through software programming.

Description

Method for measuring and minimizing urban rail network vulnerabilities
Technical Field
The invention relates to the technical field of urban rail transit, in particular to a method for measuring and minimizing the vulnerability of an urban rail transit network (urban rail network for short).
Background
Urban rail transit is one of the most important public transport modes in large cities rapidly due to the advantages of high speed, large capacity, low carbon, energy conservation and the like. Urban rail transit plays an important role in public transport, and means that network interruption caused by operation accidents, natural disasters or equipment faults can cause significant social and economic impacts. Therefore, measuring and reducing the negative influence of network interruption on the service performance of the system are very important for ensuring the normal operation of urban rail transit and improving the service quality of passengers.
The existing research is based on network connectivity and the topological structure of the network to measure the vulnerability of the urban rail network, and the important role of the foldback function provided by the foldback track under the condition of network interruption is seriously ignored. Existing studies evaluate the changes in network structure and network characteristics in the event of an outage. However, the basic function of urban rail networks is to transport passengers, and rail operators are also interested in improving the quality of service and passenger satisfaction of urban rail networks. Therefore, it is necessary to study the vulnerability of the urban rail network from the viewpoint of passenger travel service, considering the role of the reentry rail in the case of network interruption.
Disclosure of Invention
The invention provides a method for measuring and minimizing the vulnerability of an urban rail network, which can evaluate the degree of network service performance deterioration caused by the concentrated interruption of a turn-back interval so as to evaluate the vulnerability of the urban rail network.
According to the invention, the method for measuring and minimizing the vulnerability of the urban rail network comprises the following steps:
(1) Constructing an urban rail network and defining a travel path acceptable for passengers;
(2) Evaluating urban rail network service performance;
(3) Evaluating the vulnerability;
(4) And adding turn-back tracks at stations through an optimal sequence to minimize the vulnerability of the urban rail network.
Preferably, in step (1), constructing an urban rail network and defining a travel path acceptable to passengers includes:
when constructing an urban rail network, the urban rail network is represented as a directed weighted graph G = (N) Net ,E Net )，N Net And E Net Station sets and interval sets of the urban rail network are respectively; the set of non-turnaround stations which are not provided with turnaround tracks and meet the condition of additionally arranging turnaround tracks is N,
Figure BDA0003461190360000021
the set of b non-turn-back stations with turn-back tracks is delta b ,
Figure BDA0003461190360000022
b is an integer between 0 and | N |; b =0 indicates that no turn-back track is added at any station; | N | is the number of stations in the station set N; when no turn-back track is added at any station and station set delta b After the turn-back track is added, the set of turn-back intervals between two adjacent turn-back stations in the network is respectively
Figure BDA0003461190360000023
And
Figure BDA0003461190360000024
passengers can select an acceptable path with the weighted travel time lower than the acceptable time between the station OD pairs for traveling; meanwhile, the acceptable travel time of the passengers with different travel destinations is also different, and the acceptable travel paths of the passengers with different travel destinations between the OD pairs are determined by the formulas (1) and (2):
Figure BDA0003461190360000025
Figure BDA0003461190360000026
wherein
Figure BDA0003461190360000027
Is the path m from station o to d;
Figure BDA0003461190360000028
the method is an acceptable path set when a passenger with the travel purpose of a travels from an o station to a d station;
Figure BDA0003461190360000029
the weighted travel time including transfer penalty time on a path m from the o station to the d station is calculated by adopting a formula (2);
Figure BDA00034611903600000210
representing the acceptable travel time when the passenger with the travel purpose a travels from the o station to the d station; lambda [ alpha ] a Is the tolerance coefficient of the passenger with the travel purpose a;
Figure BDA00034611903600000211
is the shortest weighted travel time from station o to station d;
Figure BDA00034611903600000212
and
Figure BDA00034611903600000213
waiting time, transfer walking time, in-car time, and transfer times on the route m, respectively; beta is a transfer penalty coefficient used for converting the transfer times into transfer penalty time.
Preferably, in step (2), evaluating service performance of the urban rail network includes:
(2.1) calculating the total travel times of passengers, average travel time of each passenger and average transfer times of each passenger when the network normally operates;
total number of trips T of passengers on network trips Average trip time P for each passenger time And average number of transfers P per passenger trans Calculating by adopting formulas (3) to (5), and respectively measuring the service performance of the urban rail network in the aspects of passenger trip times, trip time and trip convenience when the network normally operates:
Figure BDA0003461190360000031
Figure BDA0003461190360000032
Figure BDA0003461190360000033
wherein
Figure BDA0003461190360000034
The number of passengers with the purpose of a going out from the o station to the d station when the network normally operates;
Figure BDA0003461190360000035
and
Figure BDA0003461190360000036
respectively representing the weighted travel time and transfer times when a passenger n with the travel purpose a travels from the o station to the d station when the network normally operates;
(2.2) averaging the generalized travel cost of each passenger when the network is in normal operation;
the method comprises the following steps of comprehensively measuring the service performance of urban rail transit by the generalized travel cost of each average passenger, and calculating the currency cost and the travel fare converted by weighting the travel time:
Figure BDA0003461190360000037
wherein P is cost The generalized travel cost of each passenger is averaged when the network normally operates; α is a time cost parameter for converting travel time into monetary cost;
Figure BDA0003461190360000038
is the fare from the o station to the d station on the subway.
Preferably, in step (3), the vulnerability is assessed by:
(3.1) the reduction rate of the total trip times of passengers when the turn-back interval set is interrupted;
the vulnerability of the urban rail network is measured from the perspective of passenger travel feasibility by the reduction rate of the total travel times of the passengers; the interruption of the turn-back interval set can interrupt an acceptable path between station OD pairs; if there is no acceptable path of communication between the OD pairs, the passenger turns to other mode of transportation; therefore, when the set of the turnaround intervals s,
Figure BDA0003461190360000039
total number of trips of passengers going out on urban rail network during interruption
Figure BDA00034611903600000310
Calculated from equation (7):
Figure BDA00034611903600000311
wherein
Figure BDA00034611903600000312
Is a binary decision variable, when the set of turn-back intervals is interrupted, if the passenger with the travel purpose of a does not have a connected acceptable path, the decision variable is used for judging whether the passenger has the travel destination of a
Figure BDA00034611903600000313
If not, then the mobile terminal can be switched to the normal mode,
Figure BDA00034611903600000314
reduction rate of total travel times of passengers when the set of turn-back intervals is interrupted
Figure BDA00034611903600000315
Calculated using equation (8):
Figure BDA0003461190360000041
(3.2) averaging the travel time increasing rate of each passenger when the foldback interval set is interrupted;
the average trip time increasing rate of each passenger measures the vulnerability of the urban rail network from the aspect of the trip time of the passenger; the total travel time of the passengers comprises the travel time of the passengers taking urban rail transit and the travel time of the passengers transferring to other transportation modes due to the interruption of the acceptable path between the station OD pairs; assuming that passengers transferred to other modes adopt taxis for going out, wherein the travel distance of the taxis is equal to the shortest distance between the station OD pairs; thus, the travel time transferred to a taxi passenger is equal to the ratio of the taxi travel distance to the taxi speed; when the set of turn-back intervals is interrupted, the travel time of each passenger on the urban rail network is averaged
Figure BDA0003461190360000042
And average rate of increase of travel time per passenger
Figure BDA0003461190360000043
Calculated using equations (9) and (10), respectively:
Figure BDA00034611903600000414
Figure BDA0003461190360000044
wherein
Figure BDA0003461190360000045
Is the travel time of the passenger n with the travel purpose a from the o station to the d station when the set of turn-back sections is interrupted,
Figure BDA0003461190360000046
the distance from the o station to the d station when a taxi is taken is equal to the shortest distance from the o station to the d station on an urban rail network; v. of taxi Is the speed of the taxi cab or taxi cab,
Figure BDA0003461190360000047
the taxi transfer system is used for judging whether the passenger transfers to the taxi or not; if it is used
Figure BDA0003461190360000048
After the turn-back interval is intensively broken, no acceptable communication path exists between the station o and the station d, and the passenger transfers to the taxi; otherwise, the passenger still adopts a circuitous route to go out on the urban rail network;
(3.3) the increasing rate of the average transfer times of each passenger when the turn-back interval set is interrupted;
from the perspective of convenience of passenger travel, the vulnerability of the urban rail network is measured by adopting the increasing rate of the average transfer times of each passenger; when the set of turn-back intervals is interrupted, the transfer times of each passenger on the urban rail network are averaged
Figure BDA0003461190360000049
And an increase rate of the average number of transfers per passenger
Figure BDA00034611903600000410
Calculated using equations (11) and (12), respectively:
Figure BDA00034611903600000411
Figure BDA00034611903600000412
wherein
Figure BDA00034611903600000413
When the return interval set is interrupted, the number of times of transfer from the o station to the d station of a passenger n with the purpose of a going out on the urban rail network is counted;
(3.4) average increasing rate of generalized travel cost of each passenger when the reentry interval set is interrupted;
when the set of turn-back intervals is interrupted, some passengers can transfer to taxies or travel on an urban rail network by adopting a roundabout route, and at the moment, the generalized travel cost of each passenger on the urban rail network is averaged
Figure BDA0003461190360000051
And average rate of increase of generalized travel cost per passenger
Figure BDA0003461190360000052
Respectively adopting formulas (13) and (14) to calculate; comprehensively measuring the vulnerability of the urban rail network when the reentry interval set is interrupted by averaging the increment rate of the generalized travel cost of each passenger;
Figure BDA0003461190360000053
Figure BDA0003461190360000054
wherein
Figure BDA0003461190360000055
The taxi fee from the station o to the station d is estimated according to the taxi journey distance from the station o to the station d;
(3.5) comprehensively evaluating the vulnerability of the urban rail network by considering the weight of the reentry interval set;
calculating the increase rate of the average generalized travel cost of each passenger considering the weight of each turn-back interval set through a formula (15) so as to comprehensively measure the vulnerability of the urban rail network; the weight of the turn-back interval set is equal to the ratio of the total length of the intervals contained in the turn-back interval set to the total length of all the intervals in the network;
Figure BDA0003461190360000056
wherein
Figure BDA0003461190360000057
Is the station set delta b The station in the middle adds the vulnerability of the urban rail network after turning back the track, | s Is the total length of the concentration section of the turn-back section,
Figure BDA0003461190360000058
preferably, in the step (4), turning back tracks are additionally arranged at a station through an optimal sequence so as to minimize the vulnerability of the urban rail network; the realization idea is that firstly, station sets with turn-back tracks are optimized, and then the sequence of adding the turn-back tracks at the stations is optimized; when optimizing the station set additionally provided with the turn-back tracks, the station set delta is obtained b The vulnerability of the urban rail network is minimized by adding the foldback track, and the model is as follows:
Figure BDA0003461190360000059
the limiting conditions are as follows:
Figure BDA00034611903600000510
Figure BDA00034611903600000511
Figure BDA00034611903600000512
∑ j∈Nxj ＝b (20)
Figure BDA00034611903600000618
wherein the objective function (16) represents the set delta maximized at the station b Adding a reduction value of the vulnerability of the urban rail network after the return track is added;
Figure BDA0003461190360000061
and
Figure BDA0003461190360000062
respectively show the set delta when no turn-back track is added and at the station b Adding the vulnerability of an urban rail network after turning back a rail; in constraint (17)
Figure BDA0003461190360000063
Showing that no turn-back track is added at any station; constraints (18) specify x j Is a binary decision variable; if j stations are selected to add the turn-back tracks, x j =1; otherwise, x j =0; constraint (19) specifies if x j If not, then the station j belongs to the station set delta b (ii) a The number of stations for additionally arranging the turning-back tracks is b by constraint (20); constraint (21) ensures that the total cost of adding the turn-back tracks at the b stations is lower than a specified budget; c and C are respectively the average cost and the total budget for additionally arranging the turn-back track at a station;
station set delta for optimizing additional turn-back tracks b After at delta b Station in the middle adds the optimal sequence of the turn-back orbitFor sequencers
Figure BDA0003461190360000064
Show, optimize using the following model
Figure BDA0003461190360000065
Figure BDA0003461190360000066
The limiting conditions are as follows:
Figure BDA0003461190360000067
Figure BDA0003461190360000068
Figure BDA0003461190360000069
Figure BDA00034611903600000610
wherein the objective function (22) represents a sequence of stations maximized at an optimum
Figure BDA00034611903600000611
I =1,2,3 \ 8230b stations k, k ∈ δ b A reduced value of network vulnerability after the foldback track is added; in the case of the formula (23),
Figure BDA00034611903600000612
and
Figure BDA00034611903600000613
respectively equal to at the optimal station sequence
Figure BDA00034611903600000614
The first i-1 stations and the first i stations add the vulnerability of the urban rail network after the return track; in the constraint (24), x i,k Is a binary decision variable; if a station k is selected, k ∈ δ b As an optimal station sequence
Figure BDA00034611903600000615
Station i in (1), then x i,k =1; otherwise, x i,k =0; constraint (25) limits selection of only delta b As a station in
Figure BDA00034611903600000616
The ith station in; constraint (26) specification
Figure BDA00034611903600000617
Equal to the vulnerability of the urban rail network when no turn-back rail is additionally arranged at any station.
Preferably, in the step (4), a multi-population genetic algorithm is applied to determine a station set for adding a retracing track so as to minimize the vulnerability of the urban rail network, and the steps are as follows:
(4.1) determining the number of stations additionally provided with the turn-back tracks and initializing a population;
calculating and determining the number b of stations with the additional turn-back tracks by using an equation (21) according to the budget C and the average cost C of the additional turn-back tracks at one station; generating M sub-populations, each sub-population having h individual chromosomes; encoding the chromosome of an individual by adopting binary coding 1 or 0, wherein the encoding length is | N |; each code may be represented as x j J =1,2, \8230, | N |, equal to "1" or "0", respectively, represents whether a retrace track is added at station j; the total number of codes for "1" on each chromosome equals b;
(4.2) calculating the fitness of the individuals and identifying the elite individuals in the sub-population;
adding a reentry track for a station according to the chromosome code of each chromosome, and calculating the vulnerability of the network after adding the reentry track; urban rail network vulnerability V after adding turn-back rail for station according to p-th chromosome in s-th sub-population s,p The fitness of the chromosome is calculated by equation (27):
Figure BDA0003461190360000071
wherein, f s,p Is the fitness of the p-th chromosome in the s-th subgroup; h is the number of chromosomes in each subpopulation;
the higher the fitness of the chromosome, the higher the quality of the solution corresponding to the chromosome, and therefore, the individual with the highest fitness in the subgroups is selected as an elite individual;
(4.3) crossover, variation and selection of chromosomes in the sub-populations;
the probabilities of chromosome crossing and mutation are randomly generated between 0.7 and 0.9 and between 0.001 and 0.05 respectively, and the chromosome crossing adopts single-point crossing; during the chromosome mutation process, the code "1" on the chromosome becomes "0" and "0" becomes "1";
selecting chromosomes in the sub-population using a roulette algorithm; selecting the p-th chromosome in the s-th sub-population as the next generation chromosome if the fitness of the p-th chromosome satisfies formula (28);
Figure BDA0003461190360000072
in which ξ p ∈[0,1]Is a uniformly distributed random number;
(4.4) calculating the individual fitness and inserting the elite individual into the new sub-population;
calculating the total number of codes of '1' on each chromosome, wherein the codes of '1' indicate that a reentry track is additionally arranged on a corresponding station; if the total number exceeds b, the number of stations for additionally arranging the reentry tracks exceeds an upper limit, and at the moment, the fitness of the chromosome is set to be a smaller value; otherwise, calculating the vulnerability of the urban rail network after the reentry rail is additionally arranged for the station according to the chromosome;
inserting the elite individuals determined in step (4.2) into a new sub-population in order to avoid the disappearance of elite individuals during chromosome selection, crossing and mutation;
(4.5) sub-population migration;
respectively identifying chromosomes with highest fitness and chromosomes with lowest fitness in each sub-population; randomly selecting the best and worst chromosomes when the fitness of the plurality of chromosomes is equal to the highest or lowest fitness in the sub-population; replacing the worst chromosome in one sub-population with the optimal chromosome in the other sub-population by using a migration operator;
(4.6) obtaining the optimal individuals and generating a sub-population to replace the worst sub-population;
the optimal individual is the chromosome with the highest fitness among all the sub-populations; generating a sub-population to replace the worst sub-population every 5 generations of inheritance; determining the worst sub-population according to the harmonic average value of the individual fitness in the sub-population;
(4.7) stopping genetic judgment;
if the optimal individual keeps 10 generations unchanged, the genetic iteration is ended; otherwise, the step (4.2) is returned.
Preferably, in the step (4), the optimized set delta of the additionally-arranged retraced railway station is obtained b Then, the station set delta needs to be determined b Optimal station sequence of medium b stations additionally provided with turn-back tracks
Figure BDA0003461190360000081
The method comprises the following implementation steps:
step 1, calculating the vulnerability of the urban rail network when no turn-back track is additionally arranged at any station
Figure BDA0003461190360000082
Let i =1, let the optimal station sequence of the added turn-back track
Figure BDA0003461190360000083
Is empty;
step 2, determining the optimal station sequence
Figure BDA0003461190360000084
The ith station in (2);
step 2.1, selecting a station k, wherein k belongs to delta b As
Figure BDA0003461190360000085
The ith station in (1);
step 2.2, calculate at
Figure BDA0003461190360000086
After the first i station and the first i-1 station in the system are additionally provided with the turn-back tracks, the vulnerability of the network is ensured
Figure BDA0003461190360000087
And
Figure BDA0003461190360000088
step 2.3, calculate Δ V using equation (23) i,k ；
Step 2.4, return to Step 2.1 until all Δ V's have been calculated i,k ,
Figure BDA0003461190360000089
Selecting Δ V i,k Station corresponding to the maximum value as
Figure BDA00034611903600000810
The ith station in (1);
step3, let i = i +1; if i is<b, returning to Step 2; otherwise, adding the optimal station sequence of the turn-back tracks
Figure BDA00034611903600000811
Has already been determined.
The method adopts the degree of network service performance deterioration caused by the centralized interruption of the turn-back intervals to evaluate the vulnerability of the urban rail network. The service performance of the urban rail network is measured from different angles by adopting (1) the total travel times of passengers, (2) the average travel time of each passenger and (3) the average transfer times of each passenger. And calculating the average generalized travel cost of each passenger according to the three indexes, and using the generalized travel cost to comprehensively evaluate the service performance of the urban rail network. The interruption of the set of turn-back intervals may cause the interruption of travel routes of some passengers, and the passengers may turn to other transportation ways to travel, thereby increasing the total travel time and the transfer times of the passengers. Therefore, when the foldback interval set is interrupted, the vulnerability of the urban rail network is measured from three aspects of the travel feasibility, the travel time and the travel convenience of passengers respectively by adopting (1) the reduction rate of the total travel times of the passengers, (2) the increase rate of the average travel time of each passenger and (3) the increase rate of the average transfer times of each passenger. And (4) integrating the three angles, and calculating the increase rate of the generalized travel cost of each average passenger to measure the vulnerability of the urban rail network when the return interval set is interrupted. In order to consider the condition of interruption of different turn-back interval sets, the weight of each turn-back interval set is determined, and the vulnerability of the urban rail network is weighted and evaluated according to the vulnerability of the network when the different turn-back interval sets are interrupted. The number of sections between two adjacent turning-back stations is reduced by adding the turning-back tracks to the stations, so that the negative influence caused by concentrated disconnection of the turning-back sections can be effectively reduced, and therefore, a model is established to determine the optimal sequence of adding the turning-back tracks to the stations, and the vulnerability of the urban rail network is minimized. The method can effectively and reasonably evaluate the vulnerability of the urban rail network, and can guide an operator to orderly additionally provide the turning back tracks at a proper station, thereby minimizing the vulnerability of the network.
Drawings
Fig. 1 is a flowchart of a method for measuring and minimizing urban rail network vulnerability in embodiment 1;
fig. 2 is a schematic diagram of vulnerability of an urban rail network when reentry tracks are additionally arranged on stations in different sequences in embodiment 1;
FIG. 3 is a schematic diagram of one chromosome in example 1;
FIG. 4 is a schematic diagram of chromosome crossing in example 1;
FIG. 5 is a schematic diagram showing chromosomal variations in example 1;
FIG. 6 is a flowchart of the multi population genetic algorithm in example 1.
Detailed Description
For a further understanding of the present invention, reference is made to the following detailed description taken in conjunction with the accompanying drawings and examples. It is to be understood that the examples are illustrative of the invention and not limiting.
Example 1
As shown in fig. 1, the present embodiment provides a method for measuring and minimizing urban rail network vulnerabilities, comprising the steps of:
(1) Constructing an urban rail network and defining a travel path acceptable for passengers;
urban rail network definition and station sequence additionally provided with turn-back tracks
When constructing an urban rail network, the urban rail network is represented as a directed weighted graph G = (N) Net ,E Net )，N Net And E Net Station sets and interval sets of the urban rail network are respectively; the set of non-retracing stations which are not provided with a retracing track and meet the condition of additionally arranging the retracing track is N,
Figure BDA0003461190360000101
the set of b non-turn-back stations with turn-back tracks is delta b ,
Figure BDA0003461190360000102
b is an integer between 0 and | N |; b =0 indicates that no turn-back track is added at any station; | N | is the number of stations in the station set N; before and after the return track is added to any station, the return interval sets between two adjacent return stations in the network are respectively
Figure BDA0003461190360000103
And
Figure BDA0003461190360000104
acceptable path of passenger on urban rail network
Passengers can select an acceptable path with the weighted travel time lower than the acceptable time between the station OD pairs for traveling; meanwhile, the acceptable travel time of the passengers with different travel destinations is also different, and the acceptable travel paths of the passengers with different travel destinations between the OD pairs are determined by the formulas (1) and (2):
Figure BDA0003461190360000105
Figure BDA0003461190360000106
wherein
Figure BDA0003461190360000107
Is the route m from station o to d;
Figure BDA0003461190360000108
the method is an acceptable path set when a passenger with the travel purpose of a travels from an o station to a d station;
Figure BDA0003461190360000109
the weighted travel time including transfer penalty time on a path m from the o station to the d station is calculated by adopting a formula (2);
Figure BDA00034611903600001010
represents an acceptable travel time; lambda [ alpha ] a Is the tolerance coefficient of a passenger for travel purpose;
Figure BDA00034611903600001011
is the shortest weighted travel time from station o to station d;
Figure BDA00034611903600001012
and
Figure BDA00034611903600001013
waiting time, transfer walking time, in-car time, and transfer times on the route m, respectively; beta is a transfer penalty coefficient used for converting the transfer times into transfer penalty time.
(2) Evaluating urban rail network service performance;
(2.1) calculating the total travel times of passengers, average travel time of each passenger and average transfer times of each passenger when the network normally operates;
total number of trips T of passengers on network trips Average trip time P for each passenger time And average number of transfers P per passenger trans Calculating by adopting formulas (3) to (5), respectively measuring the service performance of the urban rail network in the aspects of passenger trip times, trip time and trip convenience when the network normally operates:
Figure BDA0003461190360000111
Figure BDA0003461190360000112
Figure BDA0003461190360000113
wherein
Figure BDA0003461190360000114
The number of passengers with the purpose of a going out from the o station to the d station when the network normally operates;
Figure BDA0003461190360000115
and
Figure BDA0003461190360000116
respectively, the weighted travel time and the transfer times of the passenger n with the travel purpose a when the network is in normal operation and the passenger n travels from the o station to the d station.
(2.2) averaging the generalized travel cost of each passenger when the network is in normal operation;
the method comprises the following steps of comprehensively measuring the service performance of urban rail transit by using the generalized travel cost of each average passenger, and calculating the total travel times of the passengers according to the currency cost converted by travel time, the travel fare and the travel fare:
Figure BDA0003461190360000117
wherein P is cost The generalized travel cost of each passenger is averaged when the network normally operates; α is a time cost parameter for converting travel time into monetary cost;
Figure BDA0003461190360000118
is the fare from station o to station d on subway.
(3) Evaluating the vulnerability;
(3.1) the reduction rate of the total trip times of passengers when the turn-back interval set is interrupted;
according to the reduction rate of the total travel times of passengers, the vulnerability of the urban rail network is measured from the perspective of the travel feasibility of the passengers; the interruption of the turn-back interval set interrupts an acceptable path between station OD pairs; if there is no acceptable path of communication between the OD pairs, the passenger switches to other modes of transportation; therefore, when the set of the turnaround intervals s,
Figure BDA0003461190360000119
total number of trips of passengers going out on urban rail network during interruption
Figure BDA00034611903600001110
Calculated from equation (7):
Figure BDA0003461190360000121
wherein
Figure BDA0003461190360000122
Is a binary decision variable, when the turn-back interval set is interrupted, if the passenger with the travel purpose of a does not have a connected acceptable path, the binary decision variable is used for judging whether the passenger has the travel destination of a
Figure BDA0003461190360000123
If not, then,
Figure BDA0003461190360000124
when the set of turn-back intervals is interrupted, the passengers always go outRate of reduction of line count
Figure BDA0003461190360000125
Calculated using equation (8):
Figure BDA0003461190360000126
(3.2) averaging the travel time increasing rate of each passenger when the turning-back interval set is interrupted;
the average trip time increasing rate of each passenger measures the vulnerability of the urban rail network in the aspect of the trip time of the passenger; the total travel time of the passengers comprises the travel time of the passengers taking the subway and the travel time of the passengers transferring to other modes for traveling due to interruption of the acceptable path between the station OD pairs; supposing that passengers transferred to other modes adopt taxies for going out, wherein the travel distance of the taxies is equal to the shortest distance between the station OD pairs; thus, the travel time transferred to a taxi passenger is equal to the ratio of the taxi travel distance to the taxi speed; when the set of turn-back intervals is interrupted, the average travel time of each passenger on the urban rail network
Figure BDA0003461190360000127
And average rate of increase of travel time per passenger
Figure BDA0003461190360000128
Calculated using equations (9) and (10), respectively:
Figure BDA0003461190360000129
Figure BDA00034611903600001210
wherein
Figure BDA00034611903600001211
When the return interval set is interrupted, the passenger n with the trip purpose a arrives from the o stationd the travel time of the station is shown,
Figure BDA00034611903600001212
the distance from the o station to the d station when a taxi is taken is equal to the shortest distance from the o station to the d station on an urban rail network; v. of taxi Is the speed of the taxi cab or taxi cab,
Figure BDA00034611903600001213
the taxi transfer system is used for judging whether the passenger transfers to the taxi or not; if it is not
Figure BDA00034611903600001214
After the turn-back interval is intensively broken, no acceptable path connection exists between the station o and the station d, and the passenger transfers to the taxi; otherwise, the passenger still adopts a circuitous route to go out on the urban rail network;
(3.3) the increasing rate of the average transfer times of each passenger when the turn-back interval set is interrupted;
from the perspective of convenience in passenger travel, the vulnerability of the urban rail network is measured by adopting the increasing rate of the average transfer times of each passenger; passengers transferred to the taxi are increased for one transfer; when the set of turn-back intervals is interrupted, the transfer times of each passenger on the urban rail network are averaged
Figure BDA00034611903600001215
And an increase rate of the average number of transfers per passenger
Figure BDA00034611903600001216
Calculated using equations (11) and (12), respectively:
Figure BDA0003461190360000131
Figure BDA0003461190360000132
wherein
Figure BDA0003461190360000133
When the return interval set is interrupted, the number of times of transfer from the o station to the d station of a passenger n with the purpose of a going out on the urban rail network is counted;
(3.4) average increasing rate of generalized travel cost of each passenger when the reentry interval set is interrupted;
when the set of the turning-back intervals is interrupted, some passengers can be transferred to taxies or travel on an urban rail network by adopting a roundabout route, and at the moment, the generalized travel cost of each passenger on the average on the urban rail network
Figure BDA0003461190360000134
And average rate of increase in generalized travel cost per passenger
Figure BDA0003461190360000135
Respectively adopting formulas (13) and (14) to calculate; comprehensively measuring the vulnerability of the urban rail network when the reentry interval set is interrupted by averaging the increment rate of the generalized travel cost of each passenger;
Figure BDA0003461190360000136
Figure BDA0003461190360000137
wherein
Figure BDA0003461190360000138
The taxi fee from the station o to the station d is estimated according to the taxi journey distance from the station o to the station d;
(3.5) comprehensively evaluating the vulnerability of the urban rail network by considering the weight of the turn-back interval;
calculating the increase rate of the average generalized travel cost of each passenger considering the weight of each turn-back interval set through a formula (15) so as to comprehensively measure the vulnerability of the urban rail network; the weight of the turn-back interval set is equal to the ratio of the total length of the intervals contained in the turn-back interval set to the total length of all the intervals in the network;
Figure BDA0003461190360000139
wherein
Figure BDA00034611903600001310
The turn-back track is added to a station set delta b Vulnerability of the post urban rail network,/ s Is the total length of the concentration section of the turnaround section,
Figure BDA00034611903600001311
(4) And adding turn-back tracks at stations through an optimal sequence to minimize the vulnerability of the urban rail network.
Due to the limitation of resources and funds and the interference of the construction process of additionally arranging the retracing tracks on the operation of the urban rail network, the retracing tracks are required to be sequentially and not simultaneously additionally arranged. The sequence of adding the turn-back tracks to the stations can obviously influence the efficiency of reducing the vulnerability of the urban rail network. Fig. 2 shows the decreasing value of the vulnerability of the urban rail network after the station a, the station B and the station C add the foldback tracks in different orders. When the turn-back tracks are additionally arranged on all three stations, the reduction values of the urban rail network vulnerability are the same, but the addition of the turn-back tracks according to the sequence of the station C, the station B and the station A is the best sequence, because the rate of the reduction of the urban rail network vulnerability is the fastest when the turn-back tracks are additionally arranged according to the sequence.
Adding a turn-back track at a station through an optimal sequence so as to minimize the vulnerability of an urban rail network; the realization idea is that firstly, station sets with turn-back tracks are optimized, and then the sequence of adding the turn-back tracks at the stations is optimized; when optimizing the station set additionally provided with the turn-back tracks, the station set delta is obtained b The vulnerability of the urban rail network is minimized by adding the foldback track, and the model is as follows:
Figure BDA0003461190360000141
the limiting conditions are as follows:
Figure BDA0003461190360000142
Figure BDA0003461190360000143
Figure BDA0003461190360000144
∑ j∈N x j ＝b (20)
Figure BDA00034611903600001410
wherein the objective function (16) represents the set delta maximized at the station b Adding a reduction value of the vulnerability of the urban rail network after the return track is added;
Figure BDA0003461190360000145
and
Figure BDA0003461190360000146
respectively showing the set delta when no turn-back track is added and at the station b Adding the vulnerability of an urban rail network after turning back a rail; in constraint (17)
Figure BDA0003461190360000147
Showing that no turn-back track is added at any station; constraints (18) specify x j Is a binary decision variable; if j stations are selected to add the turn-back tracks, x j =1; otherwise, x j =0; constraint (19) specifies if x j If not, then the station j belongs to the station set delta b (ii) a The number of stations for additionally arranging the turning-back tracks is b through constraint (20); constraint (21) ensures that the total cost of adding the turn-back tracks at the b stations is lower than a specified budget; c and C are respectively the average cost and the total budget for additionally arranging the turn-back track at a station;
determining optimizationStation set delta of back turn-back track b After at delta b Station in the middle for adding optimal sequence of turn-back tracks
Figure BDA0003461190360000148
Expressed, the optimization is as follows:
Figure BDA0003461190360000149
the limiting conditions are as follows:
Figure BDA0003461190360000151
Figure BDA0003461190360000152
Figure BDA0003461190360000153
Figure BDA0003461190360000154
wherein the objective function (22) is maximized at station k, k ∈ δ b The reduction value of the network vulnerability after the addition of the turn-back track is selected as the optimal station sequence
Figure BDA0003461190360000155
I =1,2,3 \ 8230b stations; in the formula (23), the first and second groups,
Figure BDA0003461190360000156
and
Figure BDA0003461190360000157
respectively equal to at the optimal station sequence
Figure BDA0003461190360000158
The first i-1 stations and the first i stations add the vulnerability of the urban rail network after the return track; in the constraint (24), x i,k Is a binary decision variable; if a station k is selected, k ∈ δ b As an optimal station sequence
Figure BDA0003461190360000159
Station i in (1), then xi ,k =1; otherwise, x i,k =0; constraint (25) limits selection of only delta b As a station in
Figure BDA00034611903600001510
The ith station in (1); constraint (26) specification
Figure BDA00034611903600001511
Equal to the vulnerability of the urban rail network when no turn-back rail is additionally arranged at any station.
Evaluation process
Firstly, the service performance and the vulnerability of the urban rail network are evaluated by the steps, and then a multi-population genetic algorithm is applied to determine a station set additionally provided with a retracing track so as to minimize the vulnerability of the urban rail network. Finally, a method is proposed to determine the optimal sequence of adding turn-back tracks at each station in the optimized station set.
Service performance evaluation during normal operation of urban rail network
The total travel times of passengers are determined by processing (AFC) data collected by automatic ticket gates during normal operation of the urban rail network. In order to calculate the average travel time of each passenger, the average transfer times of each passenger and the average generalized travel cost of each passenger during normal operation, an acceptable travel path between OD pairs is determined by using a K short circuit algorithm, and the probability of the path being selected by the passenger is calculated by using a Logit model. And calculating the average travel time of each passenger, the average transfer times of each passenger and the average generalized travel expense of each passenger when the network is in normal operation according to the travel time, the transfer times and the travel cost on the acceptable travel path and the probability of selecting the acceptable travel path by the passenger.
Vulnerability assessment
Station set delta b The vulnerability assessment steps of the urban rail network after the station is additionally provided with the turn-back track are as follows:
(a) Determining the station set delta b Turn-back interval set in network after adding turn-back track at each station
Figure BDA00034611903600001512
(b) A simulated turn-back interval set s,
Figure BDA0003461190360000161
and (4) interrupting. When the simulation is interrupted, the interval in the reentry interval set is unavailable.
(c) And calculating the vulnerability of the urban rail network.
(c-1) determining whether there is a communication path between the station OD pairs and the number of passengers transferred to other transportation means.
When the return interval set s is interrupted, if a passenger n with a trip purpose a does not have an acceptable trip path from the o station to the d station, the passenger transfers to a taxi; otherwise, the passenger still adopts a circuitous path to go out on the urban rail network, and the probability that the passenger selects different acceptable paths when going out from the o station to the d station is calculated by adopting the Logit model.
And (c-2) calculating a reduction rate of the total trip times of the passengers, an increase rate of the average trip time of each passenger, an increase rate of the average transfer times of each passenger and an increase rate of the average generalized trip cost of each passenger, which are caused by the interruption of the turn-back section set s.
When the turn-back section set s is interrupted, the total trip times of the passengers are calculated by adopting a formula (7), and then the reduction rate of the total trip times of the passengers is calculated by adopting a formula (8). And calculating the travel time, transfer times and travel cost of the passengers traveling on the urban rail network according to the probability of selecting the acceptable travel route and the travel time, transfer times and travel cost corresponding to the route. And calculating the travel time and the travel cost transferred to the taxi passengers according to the running distance of the taxis between the station OD pairs, the taxi speed and the taxi fare. Then, the increase rate of the average trip time per passenger, the increase rate of the average transfer times per passenger, and the increase rate of the average generalized trip fare per passenger when the set of turn-around sections is interrupted are calculated using equations (9) to (14).
And (c-3) calculating the vulnerability of the urban rail network when each foldback interval set is interrupted according to the steps from (a) to (c-2), and then comprehensively evaluating the vulnerability of the urban rail network considering the weight of the foldback interval by using a formula (15).
Determining optimized station set with foldback tracks
The method for minimizing the vulnerability of the urban rail network by adding the turn-back tracks in the station set is a nonlinear integer programming model and is difficult to solve by an accurate algorithm. The multi-population genetic algorithm is used here to solve the model because it has a stronger global and local search capability than standard genetic algorithms. As shown in fig. 6, the multi-population genetic algorithm is applied to determine the station set with the turn-back tracks added so as to minimize the vulnerability of the urban rail network, and the specific steps are as follows:
(4.1) determining the number of stations for additionally arranging turn-back tracks and initializing a population;
determining the number b of stations with the foldback tracks added by using an equation (21) according to the budget C and the average cost C of adding the foldback tracks at one station; generating M sub-populations, each sub-population having h individual chromosomes; the chromosomes of the individuals are encoded using binary encoding ("1" or "0"), as shown in fig. 3, with a length of | N |. Each code may be represented as x j J =1,2, \8230, | N |, equal to "1" or "0", respectively, indicating whether a retrace track is added for station j; the total number of codes for "1" on each chromosome equals b;
(4.2) calculating the fitness of the individuals and identifying the elite individuals in the sub-population;
adding a turn-back track at a station according to the chromosome code of each chromosome, and calculating the vulnerability of the network after adding the turn-back track; according to the urban rail network vulnerability V of the p-th chromosome in the s-th sub population after the reentry rail is added at the station s,p The fitness of the chromosome is equal to:
Figure BDA0003461190360000171
wherein, f s,p Is the fitness of the pth chromosome in the s sub-population, and h is the number of chromosomes in each sub-population;
the higher the fitness of the chromosome, the higher the quality of the solution corresponding to the chromosome, and therefore, the individual with the highest fitness in the subgroups is selected as an elite individual;
(4.3) crossover, variation and selection of chromosomes in the sub-populations;
the probabilities of chromosome crossing and mutation are randomly generated between 0.7 and 0.9 and 0.001 and 0.05, respectively, which improves the local and global search capabilities of the algorithm. Chromosome crossing adopts single point crossing, as shown in FIG. 4; during chromosomal variation, the codes "1" and "0" on the chromosome become "0" and "1", respectively; an example of chromosomal variation is shown in FIG. 5.
Selecting chromosomes in the sub-population using a roulette algorithm; selecting the p-th chromosome in the s-th sub-population as the next generation chromosome if the fitness of the p-th chromosome satisfies formula (28);
Figure BDA0003461190360000172
in which ξ p ∈[0,1]Is a uniformly distributed random number;
(4.4) calculating the individual fitness and inserting the elite individuals into the new sub-population;
calculating the total number of codes of '1' on each chromosome, wherein the codes of '1' indicate that a reentry track is additionally arranged on a corresponding station; if the total number exceeds b, the number of stations for which the reentry tracks are added exceeds the upper limit, and the fitness of the chromosome is set to a smaller value (e.g., 0.3); otherwise, calculating the vulnerability of the urban rail network after the reentry track is additionally arranged on the station according to the chromosome;
inserting the elite individuals determined in step (4.2) into a new sub-population in order to avoid the disappearance of the elite individuals during the process of chromosome selection, crossing and mutation;
(4.5) sub-population migration;
respectively identifying chromosomes with highest fitness and chromosomes with lowest fitness in each sub-population; randomly selecting the best and worst chromosomes when the fitness of the plurality of chromosomes is equal to the highest or lowest fitness in the sub-population; replacing the worst chromosome in one sub-population with the optimal chromosome in another sub-population by using a migration operator;
(4.6) obtaining the optimal individuals and generating a sub-population to replace the worst sub-population;
the optimal individual is the chromosome with the highest fitness among all the sub-populations; generating a sub-population to replace the worst sub-population every 5 generations of inheritance; determining the worst sub-population according to the harmonic average value of the individual fitness in the sub-population;
(4.7) stopping genetic judgment;
if the optimal individual keeps 10 generations unchanged, the genetic iteration is ended; otherwise, the step (4.2) is returned.
Determining an optimized station sequence
Obtaining the optimized station set delta needing to add the turn-back track b Later, the station set δ needs to be determined b The optimal station sequence of the turn-back tracks is added to the middle b stations, and the implementation steps are as follows:
step 1, calculating the vulnerability of the urban rail network when no turn-back track is additionally arranged at any station
Figure BDA0003461190360000181
Let i =1, let the optimal station sequence of the added turn-back track
Figure BDA0003461190360000182
Is empty;
step 2, determining the optimal station sequence
Figure BDA0003461190360000183
The ith station in (1);
step 2.1, selecting a station k, wherein k belongs to delta b As
Figure BDA0003461190360000184
The ith station in (1);
step 2.2, calculate at
Figure BDA0003461190360000185
After the first i station and the first i-1 stations are additionally provided with the retracing tracks, the vulnerability of the network
Figure BDA0003461190360000186
And
Figure BDA0003461190360000187
step 2.3, calculate Δ V using equation (23) i,k ；
Step 2.4, return to Step 2.1 until all Δ V's have been calculated i,k ,
Figure BDA0003461190360000188
Selecting Δ V i,k Station corresponding to the maximum value of (a) as
Figure BDA0003461190360000191
The ith station in (1);
step3, let i = i +1; if i<b, returning to Step 2; otherwise, adding the optimal station sequence of the turn-back tracks
Figure BDA0003461190360000192
Has already been determined.
Measuring and reducing the vulnerability of urban rail networks are very important for weakening the influence of network interruption on network performance and guaranteeing the transport service quality of the network. The embodiment measures the vulnerability of the urban rail network, and evaluates the total trip times of passengers, the average transfer times of each passenger, the average trip time of each passenger and the change of the generalized trip cost of each passenger when the return interval set is interrupted.
The method is applied to a metro network of a metro. The results show that when the set of foldback intervals is interrupted, the vulnerability of the network in terms of average transfer times per passenger and total travel times of passengers is low, but the vulnerability in terms of average travel time per passenger and generalized travel cost is high. This means that the total number of passengers' trips and the number of passenger transfers do not vary significantly during the turn-back section convergence period, while the variation in the travel time and travel cost of the passengers is large. Through analysis results, a key turn-back interval set which seriously influences network service performance is determined, and guidance can be provided for key protection of intervals. And analyzing the condition of the reduction of the vulnerability of the urban rail network after the turn-back tracks are additionally arranged on the non-turn-back stations according to the optimal station sequence under the certain budget constraint and without the budget constraint. The analysis result shows that when the number of stations with the foldback tracks is small, the vulnerability of the network is rapidly reduced, and when the number of stations with the foldback tracks reaches a certain number, the vulnerability of the network begins to be reduced slowly. Therefore, a small number of stations can be selected to add the retracing tracks for the stations in the optimal sequence, so that the cost for adding the retracing tracks can be effectively reduced, and the vulnerability of the network can be rapidly reduced. Case analysis also finds that the vulnerability of the urban rail network can be effectively reduced by additionally arranging the retracing track at the transfer station. Therefore, the method provided by the invention has a strong application value in the two aspects of vulnerability assessment of the urban rail network and reduction of the vulnerability of the urban rail network by additionally arranging the foldback track.
The above embodiments are only preferred embodiments of the present invention, and the protection scope of the present invention is not limited thereto; therefore, if the person skilled in the art receives the teaching, without departing from the spirit of the invention, the person skilled in the art shall not inventively design the similar structural modes and embodiments to the technical solution, but shall fall within the scope of the invention.

Claims (5)

1. A method for measuring and minimizing urban rail network vulnerabilities, characterized by: the method comprises the following steps:
(1) Constructing an urban rail network and defining a travel path acceptable for passengers;
(2) Evaluating urban rail network service performance;
(3) Evaluating the vulnerability;
in step (3), the vulnerability is evaluated, which comprises:
(3.1) the reduction rate of the total trip times of passengers when the turn-back interval set is interrupted;
the vulnerability of the urban rail network is measured from the perspective of passenger travel feasibility by the reduction rate of the total travel times of the passengers; the interruption of the turn-back interval set can interrupt an acceptable path between station OD pairs; if there is no acceptable path of communication between the OD pairs, the passenger switches to other modes of transportation; therefore, when the set of foldback intervals s,
Figure FDA0003954056800000011
total number of trips of passengers going out on urban rail network during interruption
Figure FDA0003954056800000012
Calculated from equation (7):
Figure FDA0003954056800000013
wherein
Figure FDA0003954056800000014
Is a binary decision variable, when the set of turn-back intervals is interrupted, if the passenger with the travel purpose of a does not have a connected acceptable path, the decision variable is used for judging whether the passenger has the travel destination of a
Figure FDA0003954056800000015
If not, then,
Figure FDA0003954056800000016
rate of decrease in total passenger travel times when the set of turn-back intervals is interrupted
Figure FDA0003954056800000017
Calculated using equation (8):
Figure FDA0003954056800000018
(3.2) averaging the travel time increasing rate of each passenger when the foldback interval set is interrupted;
the average trip time increasing rate of each passenger measures the vulnerability of the urban rail network from the perspective of the trip time of the passenger; the total travel time of the passengers comprises the travel time of the passengers taking urban rail transit and the travel time of the passengers transferring to other transportation modes due to the interruption of the acceptable path between the station OD pairs; assuming that passengers transferred to other modes adopt taxis for going out, wherein the travel distance of the taxis is equal to the shortest distance between the station OD pairs; thus, the travel time transferred to a taxi passenger is equal to the ratio of the taxi travel distance to the taxi speed; when the set of turn-back intervals is interrupted, the average travel time of each passenger on the urban rail network
Figure FDA0003954056800000019
And average rate of increase of travel time per passenger
Figure FDA00039540568000000110
Calculated using equations (9) and (10), respectively:
Figure FDA00039540568000000111
Figure FDA0003954056800000021
wherein
Figure FDA0003954056800000022
Is the travel time of the passenger n with the travel purpose a from the o station to the d station when the set of the returning interval is interrupted,
Figure FDA0003954056800000023
is the distance from the o station to the d station for taking a taxi,equal to the shortest distance from the o station to the d station on the urban rail network; v. of taxi Is the speed of the taxi cab or taxi cab,
Figure FDA0003954056800000024
the taxi transfer system is used for judging whether the passenger transfers to the taxi or not; if it is not
Figure FDA0003954056800000025
After the turn-back interval is intensively broken, no acceptable communication path exists between the station o and the station d, and the passenger transfers to the taxi; otherwise, the passenger still adopts a circuitous route to travel on the urban rail network;
(3.3) the increasing rate of the average transfer times of each passenger when the turn-back interval set is interrupted;
from the perspective of convenience of passenger travel, the vulnerability of the urban rail network is measured by adopting the increasing rate of the average transfer times of each passenger; when the set of turn-back intervals is interrupted, the transfer times of each passenger on the urban rail network are averaged
Figure FDA0003954056800000026
And an increase rate of the average number of transfers per passenger
Figure FDA0003954056800000027
Calculated using equations (11) and (12), respectively:
Figure FDA0003954056800000028
Figure FDA0003954056800000029
wherein
Figure FDA00039540568000000210
When the return interval set is interrupted, the number of times of transfer from the o station to the d station of a passenger n with the purpose of a going out on the urban rail network is counted;
(3.4) average increasing rate of generalized travel cost of each passenger when the reentry interval set is interrupted;
when the set of the turning-back intervals is interrupted, some passengers can be transferred to taxies or travel on an urban rail network by adopting a roundabout route, and at the moment, the generalized travel cost of each passenger on the average on the urban rail network
Figure FDA00039540568000000211
And average rate of increase of generalized travel cost per passenger
Figure FDA00039540568000000212
Respectively adopting formulas (13) and (14) to calculate; comprehensively evaluating the vulnerability of the urban rail network when the turning-back interval set is interrupted by averaging the increment rate of the generalized travel cost of each passenger;
Figure FDA00039540568000000213
Figure FDA00039540568000000214
wherein
Figure FDA00039540568000000215
Estimating the taxi fee from the station o to the station d according to the taxi travel distance from the station o to the station d;
(3.5) comprehensively evaluating the vulnerability of the urban rail network by considering the weight of the reentry interval set;
calculating the increase rate of the generalized travel cost of each passenger considering the weight of each turn-back interval set through a formula (15) so as to comprehensively measure the vulnerability of the urban rail network; the weight of the turn-back interval set is equal to the ratio of the total length of the intervals contained in the turn-back interval set to the total length of all the intervals in the network;
Figure FDA0003954056800000031
wherein
Figure FDA0003954056800000032
Is the station set delta b Vulnerability of urban rail network after adding turn-back rail at station s Is the total length of the concentration section of the turn-back section,
Figure FDA0003954056800000033
(4) Additionally arranging a retracing track at a station through an optimal sequence to minimize the vulnerability of an urban rail network;
in the step (4), turning back tracks are additionally arranged on a station through an optimal sequence so as to minimize the vulnerability of an urban rail network; the realization thought is that firstly, station sets of the turn-back tracks are optimized, and then the sequence of adding the turn-back tracks at the stations is optimized; when optimizing the station set additionally provided with the turn-back tracks, the station set delta is obtained b The vulnerability of the urban rail network is minimized by adding the foldback track, and the model is as follows:
Figure FDA0003954056800000034
the limiting conditions are as follows:
Figure FDA0003954056800000035
Figure FDA0003954056800000036
Figure FDA0003954056800000037
∑ j∈N x j ＝b (20)
Figure FDA0003954056800000038
wherein the objective function (16) represents the set delta maximized at the station b The reduction value of the vulnerability of the urban rail network after the return track is added;
Figure FDA0003954056800000039
and
Figure FDA00039540568000000310
respectively show the set delta when no turn-back track is added and at the station b The vulnerability of an urban rail network after the return of the rail is added; in constraint (17)
Figure FDA00039540568000000311
Showing that no turn-back track is added at any station; constraints (18) specify x j Is a binary decision variable; if j stations are selected to add the turn-back tracks, x j =1; otherwise, x j =0; constraint (19) specifies if x j If not, then the station j belongs to the station set delta b (ii) a The number of stations for additionally arranging the turning-back tracks is b through constraint (20); constraint (21) ensures that the total cost of adding the turn-back tracks at the b stations is lower than a specified budget; c and C are respectively the average cost and the total budget for additionally arranging the turn-back track at a station;
station set delta for optimizing additionally-arranged turn-back track b After at delta b Station in the middle for adding optimal sequence of turn-back tracks
Figure FDA0003954056800000041
Expressing, optimizing with the following model
Figure FDA0003954056800000042
Figure FDA0003954056800000043
The limiting conditions are as follows:
Figure FDA0003954056800000044
Figure FDA0003954056800000045
Figure FDA0003954056800000046
Figure FDA0003954056800000047
wherein the objective function (22) represents a sequence of stations maximized at an optimum
Figure FDA0003954056800000048
I =1,2,3 \ 8230b stations k, k ∈ δ b Adding a reduction value of network vulnerability after the foldback track is added; in the formula (23), the first and second groups,
Figure FDA0003954056800000049
and
Figure FDA00039540568000000410
respectively equal to in the optimal station sequence
Figure FDA00039540568000000411
The first i-1 stations and the first i stations add the vulnerability of the urban rail network after the return track; in the constraint (24), x i,k Is a binary decision variable; if a station k is selected, k ∈ δ b As an optimal station sequence
Figure FDA00039540568000000412
Station i in (2), then x i,k =1; otherwise, x i,k =0; constraint (25) limits selection of only delta b As a station in
Figure FDA00039540568000000413
The ith station in; constraint (26) specification
Figure FDA00039540568000000414
Equal to the vulnerability of the urban rail network when no turn-back rail is added at any station.
2. The method for measuring and minimizing urban rail network vulnerabilities according to claim 1, characterized in that: in the step (1), constructing an urban rail network and defining a travel path acceptable for passengers, comprising the following steps:
when constructing an urban rail network, the urban rail network is represented as a directed weighted graph G = (N) Net ,E Net )，N Net And E Net Station sets and interval sets of the urban rail network are respectively; the set of non-retracing stations which are not provided with a retracing track and meet the condition of additionally arranging the retracing track is N,
Figure FDA00039540568000000415
b non-turn-back station sets with turn-back tracks are delta b ,
Figure FDA00039540568000000416
b is an integer between 0 and | N |; b =0 indicates that no turn-back track is added at any station; | N | is the number of stations in the station set N; when no turn-back track is added at any station and station set delta b After the turn-back track is added, the set of turn-back intervals between two adjacent turn-back stations in the network is respectively
Figure FDA00039540568000000417
And
Figure FDA00039540568000000418
passengers can select an acceptable path with the weighted travel time lower than the acceptable time between the station OD pairs for traveling; meanwhile, the acceptable travel time of the passengers with different travel destinations is also different, and the acceptable travel routes of the passengers with different travel destinations between the OD pairs are determined by formulas (1) and (2):
Figure FDA0003954056800000051
Figure FDA0003954056800000052
wherein
Figure FDA0003954056800000053
Is the path m from station o to d;
Figure FDA0003954056800000054
the method is an acceptable path set when a passenger with the travel purpose of a travels from an o station to a d station;
Figure FDA0003954056800000055
the weighted trip time containing transfer penalty time on a path m from the o station to the d station is calculated by adopting a formula (2);
Figure FDA0003954056800000056
representing the acceptable travel time when the passenger with the travel purpose of a travels from the o station to the d station; lambda a Is the tolerance coefficient of the passenger with the travel purpose a;
Figure FDA0003954056800000057
is the shortest weighted travel time from station o to station d;
Figure FDA0003954056800000058
and
Figure FDA0003954056800000059
waiting time, transfer walking time, in-car time, and transfer times on the route m, respectively; beta is a transfer penalty coefficient used for converting the transfer times into transfer penalty time.
3. The method for measuring and minimizing urban rail network vulnerabilities according to claim 2, characterized in that: in the step (2), evaluating the urban rail network service performance, comprising:
(2.1) calculating the total travel times of passengers, average travel time of each passenger and average transfer times of each passenger when the network normally operates;
total number of trips T of passengers on network trips Average trip time P of each passenger time And average number of transfers P per passenger trans Calculating by adopting formulas (3) to (5), respectively measuring the service performance of the urban rail network in the aspects of passenger trip times, trip time and trip convenience when the network normally operates:
Figure FDA00039540568000000510
Figure FDA00039540568000000511
Figure FDA00039540568000000512
wherein
Figure FDA00039540568000000513
The number of passengers with the purpose of a going from the o station to the d station when the network normally operates;
Figure FDA00039540568000000514
and
Figure FDA00039540568000000515
respectively weighting travel time and transfer times when a passenger n with a travel purpose a travels from an o station to a d station when a network normally operates;
(2.2) averaging the generalized travel cost of each passenger when the network is in normal operation;
the method comprises the following steps of comprehensively measuring the service performance of urban rail transit by the generalized travel cost of each average passenger, and calculating the currency cost and the travel fare converted by weighting the travel time:
Figure FDA0003954056800000061
wherein P is cost The generalized travel cost of each passenger is averaged when the network normally operates; α is a time cost parameter for converting travel time into monetary cost;
Figure FDA0003954056800000062
is the fare from station o to station d on subway.
4. The method for measuring and minimizing urban rail network vulnerabilities according to claim 3, characterized in that: in the step (4), a multi-population genetic algorithm is applied to determine a station set additionally provided with a retracing track so as to minimize the vulnerability of the urban rail network, and the specific steps are as follows:
(4.1) determining the number of stations for additionally arranging turn-back tracks and initializing a population;
calculating and determining the number b of stations with the additional turn-back tracks by using an equation (21) according to the budget C and the average cost C of the additional turn-back tracks at one station; generating M sub-populations, each sub-population having h individual chromosomes; encoding the chromosome of an individual by adopting binary coding 1 or 0, wherein the encoding length is | N |; each code may be represented as x j J =1,2, \ 8230 |, | N |, equal to "1" or "0' respectively representing whether a turn-back track is additionally arranged at the station j; the total number of codes for "1" on each chromosome equals b;
(4.2) calculating the fitness of the individuals and identifying the elite individuals in the sub-population;
adding a turn-back track for a station according to the chromosome code of each chromosome, and calculating the vulnerability of the network after adding the turn-back track; urban rail network vulnerability V after adding turn-back rail at station according to pth chromosome in mth sub population s,p The fitness of the chromosome is calculated by equation (27):
Figure FDA0003954056800000063
wherein f is s,p Is the fitness of the p-th chromosome in the s-th subgroup; h is the number of chromosomes in each subpopulation;
the higher the fitness of the chromosome, the higher the quality of the solution corresponding to the chromosome, and therefore, the individuals with the highest fitness in the subgroup are selected as elite individuals;
(4.3) crossover, variation and selection of chromosomes in the sub-populations;
the probabilities of chromosome crossing and mutation are randomly generated between 0.7 and 0.9 and between 0.001 and 0.05 respectively, and the chromosome crossing adopts single-point crossing; during the chromosome mutation process, the code "1" on the chromosome is changed into "0", and "0" is changed into "1";
selecting chromosomes in the sub-population using a roulette algorithm; selecting the p-th chromosome in the s-th sub-population as the next generation chromosome if the fitness of the p-th chromosome satisfies formula (28);
Figure FDA0003954056800000071
in which ξ p ∈[0,1]Is a uniformly distributed random number;
(4.4) calculating the individual fitness and inserting the elite individual into the new sub-population;
calculating the total number of codes of '1' on each chromosome, wherein the codes of '1' indicate that a reentry track is additionally arranged on a corresponding station; if the total number exceeds b, the number of stations for additionally arranging the reentry tracks exceeds an upper limit, and at the moment, the fitness of the chromosome is set to be a smaller value; otherwise, calculating the vulnerability of the urban rail network after the reentry rail is additionally arranged for the station according to the chromosome;
inserting the elite individuals determined in step (4.2) into a new sub-population in order to avoid the disappearance of elite individuals during chromosome selection, crossing and mutation;
(4.5) sub-population migration;
respectively identifying chromosomes with highest fitness and chromosomes with lowest fitness in each sub-population; randomly selecting the best and worst chromosomes when the fitness of the plurality of chromosomes is equal to the highest or lowest fitness in the sub-population; replacing the worst chromosome in one sub-population with the optimal chromosome in another sub-population by using a migration operator;
(4.6) obtaining the optimal individuals and generating a sub-population to replace the worst sub-population;
the optimal individual is the chromosome with the highest fitness among all the sub-populations; generating a sub-population to replace the worst sub-population for every 5 generations of inheritance; determining the worst sub-population according to the harmonic average value of the individual fitness in the sub-population;
(4.7) stopping genetic judgment;
if the optimal individual keeps 10 generations unchanged, the genetic iteration is ended; otherwise, returning to the step (4.2).
5. The method for measuring and minimizing urban rail network vulnerabilities according to claim 4, characterized in that: in the step (4), the optimized station set delta for additionally arranging the turn-back track is obtained b Later, the station set δ needs to be determined b Optimal station sequence of medium b stations additionally provided with turn-back tracks
Figure FDA0003954056800000072
The method comprises the following implementation steps:
step 1, calculate and doVulnerability of urban rail network when adding turn-back rail at any station
Figure FDA0003954056800000073
Let i =1, let the optimal station sequence of the added turn-back track
Figure FDA0003954056800000074
Is empty;
step 2, determining the optimal station sequence
Figure FDA0003954056800000075
The ith station in (1);
step 2.1, selecting a station k, wherein k belongs to delta b As
Figure FDA0003954056800000081
The ith station in (1);
step 2.2, calculate at
Figure FDA0003954056800000082
After the first i station and the first i-1 stations are additionally provided with the retracing tracks, the vulnerability of the network
Figure FDA0003954056800000083
And
Figure FDA0003954056800000084
step 2.3, calculate Δ V using equation (23) i,k ；
Step 2.4, return to Step 2.1 until all are calculated
Figure FDA0003954056800000085
Selecting Δ V i,k Station corresponding to maximum value as
Figure FDA0003954056800000086
The ith station in (1);
step3, let i = i +1; if i is<b, returning to Step 2; otherwise, adding the optimal station sequence of the turn-back track
Figure FDA0003954056800000087
Has already been determined.
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