Flexible marshalling city rail train timetable and car bottom connection coordination optimization method

Abstract

This application discloses a method for the coordinated optimization of urban rail transit train timetables and trainset connections under flexible formation, relating to the field of urban rail transit. The method includes: acquiring urban rail transit line structure and passenger demand data; establishing a mixed-integer programming model for simultaneously optimizing train timetables, train formation schemes, and trainset connection plans; the mixed-integer programming model aims to minimize the average passenger waiting time and the total travel distance of the trainsets, and uses the storage capacity constraint of turnaround stations as a constraint condition for the trainset connection plan; based on the urban rail transit line structure and passenger demand data, a multi-strategy adaptive coordinated optimization algorithm is used to solve the mixed-integer programming model, outputting train timetables, train formation schemes, and trainset connection schemes. This application reduces operating costs while meeting passenger demand.

Description

Technical Field

[0001] This application relates to the field of urban rail transit, and in particular to a method for the coordinated optimization of urban rail train timetables and trainset connections under flexible formation. Background Technology

[0002] Urban Rail Transit (URT) systems have become a primary mode of transportation for commuters and play a vital role in alleviating traffic congestion and reducing carbon emissions. However, passenger demand for urban rail transit exhibits significant imbalances both temporally and spatially. For example, during weekday morning and evening rush hours, passengers primarily travel between suburbs and the city center, leading to a significant surge in passenger traffic during these periods; conversely, passenger demand is relatively low during off-peak hours. Simultaneously, operational strategies must meet passenger demand while minimizing operating costs to maximize efficiency. Therefore, a balance must be struck between passenger satisfaction and operating costs in urban rail transit service design. To address these challenges, relevant applications and practices have proposed three main strategies from an operational perspective: (1) Flexible Time Heading (FTH) The FTH (Frequency-to-Hand) strategy allows for increased train service frequency during peak hours and decreased frequency during off-peak hours. This strategy not only ensures passenger satisfaction across different time periods but also reduces train deployment during off-peak hours, thereby lowering overall operating costs. However, this strategy cannot effectively address the issue of spatially uneven passenger flow.

[0003] (2) Flexible Train Routing (FTR) The FTR strategy refers to the operation of express and local trains on different sections. Specifically, some trains may use shorter routes. For example, the combination of full routes and short turnaround routes is the most common implementation. In this case, full route trains run from the start to the end of the line, while short turnaround trains depart from and turn back at intermediate stations with turnaround capabilities to enhance capacity supply in key sections.

[0004] (3) Flexible Train Formation (FTF) The Fast Formation (FT) strategy allows train services to consist of multiple flexibly combinable car units, such as a single car unit containing three carriages. The Train Formation Problem (TFP) in urban rail transit primarily determines the number of car units used in a train and their connection relationships between different train services. Therefore, this strategy requires decoupling / coupling operations at specific stations, such as turnaround stations or depots, and allows for dynamic adjustments to train formations during operation to achieve a higher demand-capacity match.

[0005] The first two types of strategies have been widely used in large-scale URT systems in multiple cities. The third type of strategy has been proposed or piloted on some lines. All three types of strategies aim to improve service quality and enhance adaptability to passenger flow demand, but their introduction has also significantly increased the complexity of operation planning, including the Train Timetabling Problem (TTP), Train Formation Problem (TFP), and Train Unit Circulation Problem (TUCP).

[0006] Most related technologies address issues such as train timetable compilation, train formation planning and rolling stock connection planning, train stop planning, and train route design. However, these applications typically overlook the storage capacity of turnaround stations' sidings. In reality, turnaround stations have a limited number of storage tracks, and their available capacity dynamically changes with train operation; train coupling and decoupling operations must also be carried out under conditions where available storage tracks are available. Therefore, when adopting flexible route strategies and flexible formation strategies, the capacity of turnaround stations should be taken into account.

[0007] Furthermore, URT operators typically address uneven passenger flow by increasing departure frequencies or using shorter routes, but rarely coordinate this with flexible train formation strategies. This leads to overcapacity and high operating costs during off-peak hours. If strategies such as Flexible Time Heading (FTH), Flexible Train Routing (FTR), and Flexible Train Formation (FTF) are applied in tandem to more accurately match passenger demand, operating costs can be reduced to a lower level while maintaining service quality. Summary of the Invention

[0008] The purpose of this application is to provide a method for coordinating and optimizing urban rail transit timetables and trainset connections under flexible train formation, which can reduce operating costs while meeting passenger demand. To achieve the above objective, this application provides the following solution: Firstly, this application provides a method for coordinated optimization of urban rail transit timetables and trainset connections under flexible train formation, including: S1. Obtain data on urban rail transit line structure and passenger flow demand; S2. Establish a mixed integer programming model for synchronously optimizing train timetables, train formation schemes, and train set connection plans. The mixed integer programming model aims to minimize the average passenger waiting time and the total travel distance of the train set, and uses the storage capacity constraint of the turnaround station as a constraint condition for the train set connection plan. The storage capacity constraint of the turnaround station is used to track and limit the real-time occupancy of storage lines in each depot, ensuring that the de-staging operation is performed only when there are available storage lines. S3. Based on the urban rail transit line structure and passenger flow demand data, a multi-strategy adaptive collaborative optimization algorithm is used to solve the mixed integer programming model, and the train timetable, train formation scheme and train set connection scheme are output.

[0009] According to the specific embodiments provided in this application, this application has the following technical effects: This application provides a method for the coordinated optimization of urban rail train timetables and trainset connections under flexible train formation. By constructing a mixed-integer programming model with the objective of minimizing average passenger waiting time and total trainset travel distance, the method coordinates and optimizes train timetables, train formation schemes, and trainset connection plans. This effectively reduces the total trainset travel distance while meeting dynamic passenger flow demands, thereby improving operational economy and ensuring passenger travel experience. By introducing constraints on turnaround station storage capacity, the method tracks the occupancy status of storage lines in each depot in real time and ensures that de-formation operations are only performed when available lines are available. This makes the output results more consistent with actual physical constraints, improving the feasibility and refinement of the scheduling scheme. Furthermore, a multi-strategy adaptive coordinated optimization algorithm is used to solve the coordinated optimization problem, accelerating the solution speed of the mixed-integer programming model and improving operational efficiency. Attached Figure Description

[0010] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0011] Figure 1 This application presents a schematic diagram of bidirectional train operation under flexible routes and train formations, (a) is a schematic diagram of an urban rail transit line, (b) is a schematic diagram of various flexible train routes on a bidirectional urban rail line, and (c) is a schematic diagram of train formations. Figure 2The diagram shows a train cycle that takes into account siding storage, (a) is a dismantling operation, and (b) is a marshalling operation. Figure 3 This is a flowchart illustrating the collaborative optimization method for urban rail train timetable and rolling stock connection under flexible train formation according to this application. Figure 4 This is a schematic diagram of the multi-strategy adaptive collaborative optimization algorithm framework; Figure 5 A diagram illustrating the strategy for adjusting service frequency; Figure 6 This is a diagram illustrating a flexible routing strategy. Figure 7 A diagram illustrating the strategy for adjusting departure intervals; Figure 8 A simplified schematic diagram of light rail line 5 in a certain area; Figure 9 The diagram shows the distribution of OD demand in three time periods: (a) is a diagram of the morning peak period, (b) is a diagram of the evening peak period, and (c) is a diagram of the off-peak period. Figure 10 The following are schematic diagrams of the layout of the constructed numerical simulation examples: (a) is a schematic diagram of the layout of the 5-station urban rail transit line simulation example, and (b) is a schematic diagram of the layout of the corresponding two-way train service network. Figure 11 The diagrams show a comparison of the computational efficiency of the MSAO, LS and ALNS algorithms. (a) is a diagram of the convergence curve for scenario 1, (b) is a diagram of the convergence curve for scenario 2, and (c) is a diagram of the convergence curve for scenario 3. Figure 12 Here are some diagrams illustrating the impact of search strategy 3: departure frequency on convergence speed. (a) is a diagram of the convergence curve for scenario 1, (b) is a diagram of the convergence curve for scenario 2, and (c) is a diagram of the convergence curve for scenario 3. Figure 13 The train operation diagrams for three weight combinations under scenario 1 are shown in (a) and (b) shows the train operation diagrams for scenario 1. (b) is the train operation diagram under scenario 1. (c) is the train operation diagram under scenario 1. ); Figure 14 The train schedules are shown in scenarios 2 and 3, (a) is the train schedule for scenario 2. (b) are the train operation diagrams under scenario 3. ). Detailed Implementation

[0012] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0013] To make the objectives, features and advantages of this application more apparent and understandable, the application will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0014] This application studies TTP, TFP, and TUCP, and comprehensively considers FTH, FTR, FTF, and DCTS. TTP, TFP, and TUCP are core optimization problems in urban rail transit operation. TTP primarily determines the specific arrival and departure times of each train at each station; TFP aims to reduce operating costs during off-peak hours by using fewer train units, or to better meet passenger demand during peak hours by using more train units; and TUCP aims to improve train connection efficiency under various operating strategies, particularly FTH, FTR, and FTF strategies. Furthermore, the main contributions of this application are as follows: (1) A unified programming model integrating optimization of TTP, TFP and TUCP is proposed, which breaks through the limitation of traditional applications that first give TTP and TFP and then solve TUCP.

[0015] (2) The number of available train units at turnaround stations is constrained in TFP and TUCP, which can characterize the dynamic capability (DCTS) of turnaround stations, making up for the shortcomings of existing applications that generally ignore the capability of turnaround stations, and making the model closer to the real operating environment.

[0016] (3) By synergistically applying multiple operational strategies such as FTH, FTF, and FTR, the supply of transport capacity and passenger flow demand can be matched more accurately. Compared with applications that only use FTH or FTR, this application can reduce operating costs to a lower level while ensuring service quality.

[0017] (4) A Multi-Strategy Adaptive Collaborative Optimization (MSAO) algorithm is designed, which includes five search strategies, three repair strategies, and one acceleration strategy, namely tabu search, which can efficiently solve real cases of different scales. Compared with benchmark algorithms such as LS and ALNS, the MSAO algorithm proposed in this application can improve the quality of the solution by an average of about 4.1%.

[0018] Furthermore, in order to realize the technology of this application, it is necessary to clarify the problems that exist: (1) the difficulty in coordinating train timetables, train formations and rolling stock connections with flexible routes and flexible formation strategies. (2) the use of turnaround station capacity.

[0019] (1) The challenges in coordinating train timetables, train formations, and rolling stock connections with flexible routes and flexible formation strategies include: The comprehensive optimization of train timetables, train route plans, train formations, and rolling stock connection plans includes multiple aspects of train operation, arrival, departure, turnaround, formation, and dismantling, as follows: Figure 1 As shown. Figure 1 (a) illustrates a bidirectional urban rail line, comprising |N| platforms, |S| sections, a depot, two directions of train operation (UP and DR), a joint section, a turnaround point, a short-turn route, a full-length route, and a short-turn route. In this urban rail line, trains traveling in opposite directions use different platforms to operate independently, using a set N. Σ Mark the turnaround stations for both directions separately. Stations can be divided into two categories: one category handles only train arrival and departure operations, such as... , , Another type is equipped with sidings for train turnaround, marshalling, and demarcation, such as , , , Typically, two train yards are set up at the starting and ending stations of the entire line, directly connected to the station platforms. The technical operations carried out in the train yards include vehicle maintenance, storage and safekeeping, and marshalling / demarshalling operations.

[0020] Flexible routes refer to trains operating on either full or short routes, as follows: Figure 1 As shown in (a), sections where multiple roads intersect are called joint road sections. Figure 1 (b) describes the full route and possible short routes. A full route includes the train's departure from the depot to the station after formation, passenger boarding and alighting on the platform, passenger disembarkation at the terminal station, and train decoupling or turnback to perform new services. Conversely, the terminal station for a short route must be a turnback station, such as n3, n7, n12, and n16. To describe the problem more clearly, each turnback station, such as the downlink stations n3 and n7, is divided into a platform used only for receiving and dispatching trains, and a virtual depot for train turnback, train storage, and train decoupling. The platforms and virtual depots are connected by virtual track lines. The platform receiving the train is marked as... , ..., parking lots and virtual parking lots are marked as , Regarding the selection of appropriate routes for all train services within the timetable to reduce operating costs while ensuring that passenger demand is fully met, a comprehensive optimization combining train timetables and train routes is required.

[0021] Flexible formation strategies aim to balance unevenly distributed passenger demand with operator costs. Figure 1 (c) shows an example of train formation and disassembly. Flexible formation trains consist of single or double train units and can be converted between each other through formation and disassembly operations in the depot or virtual depot.

[0022] (2) Dynamic capability strategy of turnaround station To effectively reduce operating costs and alleviate conflicts between turnaround and storage lines, train cycle cycles resulting from flexible routes and formation strategies must be considered. Once all passengers have disembarked at the terminal station platform, the train may be disassembled, reassembled, or turned back, depending on the established train formation plan. Typically, a turnaround station can accommodate multiple train sets, allowing for consecutive turnarounds on a first-come, first-served basis. Disassembly and reassembly operations occur only in the (virtual) yard and follow a first-come, first-served principle.

[0023] Figure 2 (a) shows a double-unit train undergoing decoupling in a virtual depot; the decoupling train will run to the storage line and wait for a single-unit train to couple, or the train will turn back and move to the siding. Similarly, Figure 2 (b) shows that when single-unit trains need to be assembled at a station, they run from the siding to the platform. It should be noted that the disassembly and assembly time, including the technical work time related to the train body and parking, is greater than the turnaround time. Therefore, the availability of the siding limits the assembly / disassembly operations in the depot. This application considers the depot siding capacity based on discretized train arrival and departure time windows to propose a more realistic and practical method for train timetables, train assembly, and rolling stock connection.

[0024] In response to the above problems, such as Figure 3 As shown, this application provides a method for coordinated optimization of urban rail train timetables and trainset connections under flexible formation, which includes the following S1 to S3. Wherein: S1. Obtain data on urban rail transit line structure and passenger flow demand.

[0025] S2. Establish a mixed integer programming model for synchronously optimizing train timetables, train formation schemes, and train set connection plans. The mixed integer programming model aims to minimize the average passenger waiting time and the total travel distance of the train set, and uses the storage capacity constraint of the turnaround station as a constraint condition for the train set connection plan. The storage capacity constraint of the turnaround station is used to track and limit the real-time occupancy of storage lines in each depot, ensuring that the de-staging operation is performed only when there are available storage lines.

[0026] S3. Based on the urban rail transit line structure and passenger flow demand data, a multi-strategy adaptive collaborative optimization algorithm is used to solve the mixed integer programming model, and the train timetable, train route plan, train formation scheme and car body connection scheme are output.

[0027] By implementing steps S1 to S3 above, and setting the optimization objective and constraints of the mixed-integer programming model, this application minimizes operating costs while meeting passenger demand. Furthermore, by using a multi-strategy adaptive collaborative optimization algorithm to solve the mixed-integer programming model, the solution speed of this application is accelerated, and its operational efficiency is improved.

[0028] The core of the mixed-integer programming model in S2 begins with four fundamental assumptions that simplify the real-world problem. These four assumptions form the basis for constructing the mathematical model. By reasonably simplifying the complexity of the real world, the mixed-integer programming model becomes solvable and computable while maintaining its core optimization functionality. The four fundamental assumptions are as follows: Assumption 1: The vehicles (train sets) are homogeneous, which means that each train set has the same characteristics in terms of operating speed and passenger capacity.

[0029] Assumption 2: The time costs for different trains to dismantle, assemble, and turn around are fixed.

[0030] Assumption 3: Passengers from all different OD pairs are fully integrated on the platform and follow the first-come, first-served principle.

[0031] Assumption 4: Train service is directional and terminates when a train turns around or enters a depot or virtual depot.

[0032] These assumptions are important and common in existing applications. Assumptions 1 and 2 can reduce the number of parameters in the mixed-integer programming model. In Assumption 3, the number of passengers getting on and off can be calculated more proportionally to the total number of passengers, avoiding consideration of the origin-destination (OD) for each passenger and reducing the complexity of passenger calculations. Assumption 4 is consistent with reality, where train service terminates upon reaching the destination of its route.

[0033] Based on the four fundamental assumptions above, an objective function is constructed, namely the optimization objective of the mixed-integer programming model: Urban rail transit managers typically increase train operations during peak hours to meet growing passenger demand, leading to increased operating costs. This application aims to minimize operating costs while meeting passenger demand through FTR and FTF strategies. Therefore, two sub-objectives are considered: passenger travel costs and operator operating costs.

[0034] Based on two sub-objectives, the objective function comprises the standardization of two weighted objectives: the first sub-objective function sets the minimum average passenger waiting time, considering various scenarios for different train formations across each timetable; the second sub-objective function aims to minimize the total running distance of all operating train units. Accordingly, and This represents the weights of the two sub-objectives. The expression for the optimization objective of the mixed-integer programming model is: (1); Where Z is the overall objective function. For the first sub-objective function, for The theoretical minimum value, for The theoretical maximum value, For the second sub-objective function, for The theoretical minimum value, for The theoretical maximum value, The weights of the first sub-objective function are... The weights are for the second sub-objective function.

[0035] For the first sub-objective function, for both uplink and downlink directions, the set and It contains train indexes for each direction (except for the originating train). Total passenger waiting time includes the travel costs of all passengers, including those who successfully boarded and those waiting for the next train. Formula (2) represents the average passenger waiting time under all possible circumstances, including passenger demand waiting time, in a passenger flow of... The first train to depart Previously, and when passenger flow was The last train to arrive in both directions Then, the expression for the first sub-objective function is: (2); in, This represents the average waiting time for passengers under all circumstances; This represents a set of multiple cases; k represents the index of the case. This represents the total number of passengers in case k. This represents the number of passengers who have arrived before the first train departs, in case k, from platform n' to platform n. This indicates the departure time of the first train from platform n′; Indicates the departure time window; This represents the number of passengers who have not boarded the last train after it arrives, in case k, among the passenger demand from platform n′ to platform n. Indicates the arrival time window; This indicates the departure time of the last train from platform n′; This represents the set of trains operating in one direction; i represents the index of the train column operating in one direction. Indicates the assembly of trains heading upwards; Indicates the group of trains heading south; Indicates the platform index. This represents the set of platforms in the direction of train i's travel; This represents the passenger demand for train i-1 at platform n under condition k; This represents the number of passengers boarding train i-1 at platform n in case k. This indicates the departure time of train i at platform n; This indicates the departure time of train i-1 at platform n; Indicate the number of passengers boarding train i at platform n under case k; This indicates the arrival time of train i-1 at platform n.

[0036] For the second sub-objective function, operating costs are represented by the travel distance of the train sets. Furthermore, the empty mileage incurred by trains leaving the depot is considered as additional penalty costs. Formula (3) represents the total travel distance of the train sets, consisting of two components: the travel distance of trains with different configurations and the associated penalty costs. Terminology Let represent the additional empty mileage when train i leaves depot m. The expression for the second sub-objective function is: (3); in, Represents the total travel distance of the train set; I represents the set of all trains; i represents the index of the train running in one direction; R represents the set of all routes; The route r represents the travel distance; f represents the train formation type; and F represents the set of train formation types. This indicates the train set in the F formation mode, with 1 for single-car train mode and 2 for double-car train and fixed mode. This represents a 0-1 variable. If train i has a formation mode of f, uses route r, and the train set originates from trains in depot m... , =1, otherwise 0; I' represents the set of trains traveling in the opposite direction. Ψ represents the index of trains traveling in the opposite direction; Ψ represents the set of depots, including virtual depots, and m is the depot index; This represents a 0-1 variable, where the train set in train i originates from a train in depot m. , =1, otherwise 0; This indicates the additional travel distance the train travels when it leaves the depot (m).

[0037] After outlining the sub-objective functions that measure service quality and operational efficiency, this application establishes a series of constraints considering train scheduling and dynamic passenger flow redistribution. These constraints include: (1) timetable constraints on flexible routes, (2) train operation safety time constraints, (3) train carriage connection plans on flexible routes, (4) turnaround station storage capacity constraints, and (5) passenger demand constraints. The implementation of each constraint is as follows: (1) Constraints of timetable on flexible routes: It is used to calculate and verify the arrival and departure times of each station based on the train operation routes and stopping schemes, combined with the section running time, the minimum stop time of the station, the maximum stop time of the station and the turnaround operation time, so as to generate an operation plan with time sequence feasibility.

[0038] Since this application considers short-haul routes, it is necessary to first determine the originating and intermediate stations for each train. An auxiliary binary variable is introduced. Let represent whether train i departs from the depot m shared with the turnaround station. This variable is determined by the path decision variable. Export.

[0039] (4); in, This represents a 0-1 variable; if train i originates from depot m, Otherwise, it is 0. This indicates that the entire route starts from parking lot m, and r is the index. , Represents the set of all routes; Indicates train formation type Represents a set of grouping types; This represents a 0-1 variable. If the formation mode of train i is f, the route r is used; otherwise, it is 0. It is a set of depots, including virtual depots, with m as the index. Formula (4) indicates whether train i originates from depot m.

[0040] This application specifically addresses all trains stopping at stations. In this operating mode, the station stopping pattern is entirely determined by the originating depot and train formation plan for each train. A binary parameter is defined. Its scale is This indicates whether platform n belongs to route r. Whether train i stops at platform n can be expressed by formula (5). A key operational constraint is that, ultimately, trains on all routes in each direction must ensure coverage of all passenger services during the application period.

[0041] (5); in, This represents a 0-1 variable; if train i stops at platform n, Otherwise, it is 0; This represents a 0-1 variable, where the route r includes station n. Otherwise, it is 0; Let n represent the set of platforms in the direction of train i's travel, and n be the index; I is the set of all trains, including those going up and down. Formula (5) means that the stopping status of train i at platform n depends on whether the selected route r of the train includes that platform.

[0042] Meanwhile, the route and formation pattern of each train should be unique in the timetable, as shown in the following formula (6): (6); Trains depart from the depot or virtual depot. On the platform, trains stop according to the first-come, first-served (FCFS) principle for each direction. This includes sections between adjacent lines and sections connecting the depot to the relevant platforms. For any section ,if Connecting the depot to the platform means the travel time is the same as the travel time from the depot to the platform. Conversely, if... Connecting the platform to the parking lot will then use the travel time... (Return travel time) is represented by formula (7), which constrains the arrival time of each train at the platform. It is calculated by adding the departure time from the previous platform to the travel time of the interval. This indicates the section connecting the platforms.

[0043] (7); in, Let n be an integer variable representing the arrival time of train i at platform n. ; This represents an integer variable, indicating the departure time of train i at platform n. ; Indicates the section the train passes through. The runtime; This represents the segment in the direction of train i's travel, and s is the index.

[0044] In this application, the platform of the turnaround station and the virtual depot are connected by a virtual section. Formulas (8)-(11) restrict the departure and arrival times of the turnaround station and require that the minimum stopping time and running time be met. Indicates connecting stations and the segment of n, This represents the virtual segment connecting the turnaround station and its associated virtual yard. Specifically, if platform n is the starting point of train i in the short route, its arrival time at platform n follows formulas (8) and (9). Similarly, if platform n is the destination of train i in the short route, its arrival time at platform n+1 depends on the departure time from platform n, as shown in formulas (10) and (11).

[0045] (8); (9); (10); (11); in, Indicates the section the train passes through. The runtime; A very large number; train Whether via connecting station and the segment of n; train Whether it passes through the virtual section of the turnaround station and its associated virtual depot; This represents an integer variable, indicating the departure time of train i at platform n. ; Represents an integer variable, indicating the arrival time of train i at platform n+1; This represents an integer variable, indicating the arrival time of train i in depot m. ; Indicates the section the train passes through. The runtime; This represents the set of originating stations for short-haul routes, where n is the index. ; This represents the set of terminal stations for short-route services, where n is the index. .

[0046] Finally, each train must meet the stop time constraint when stopping at the platform, as shown in formula (12). The last scheduled train must run the entire route to ensure service to all passengers during train operation.

[0047] (12); in, This represents the minimum allowed dwell time at platform n; This indicates the maximum allowed dwell time at platform n.

[0048] (2) Train operation safety time constraints: These are used to ensure that the time interval between any two trains departing consecutively in the same depot, and between any two trains arriving and departing consecutively on the same platform, is not less than the minimum departure interval specified by the depot and the minimum arrival and departure interval specified by the station, respectively.

[0049] Specifically, to ensure operational safety, the interval between two consecutive trains must meet the minimum departure and arrival interval requirements. For any two consecutive trains i-1 and i departing from the same depot m, the departure interval must meet the minimum threshold. See equation (13) below.

[0050] (13); in, This represents an integer variable, indicating the departure time of train i in depot m. ; Represents an integer variable, indicating the departure time of train i-1 in depot m; This represents a 0-1 variable. If train i-1 originates from depot m, Otherwise, it is 0; Indicates the minimum allowable departure interval of the depot; This represents a set of parking lots, including virtual parking lots, where m is the index.

[0051] Accordingly, two trains stopping at the same platform should meet the minimum headway constraint, as shown in equation (14).

[0052] (14); in, This represents a 0-1 variable. If train i-1 stops at platform n, Otherwise, it is 0; This indicates the minimum allowed arrival and departure time at the station.

[0053] (3) Constraints of the car body connection plan on flexible routes: used to determine the connection relationship between car body units and different train services, the connection relationship including de-marshalling, marshalling and turnaround operations, and to ensure the minimum time required for each operation.

[0054] For a given urban rail line, the number of available train sets is limited by scale, as shown in formula (15): (15); in, This indicates the train set in the F formation mode, with 1 for single-car train mode and 2 for double-car train and fixed mode. This represents the number of available train formation types f; This represents a set of train formation types, including fixed single-vehicle train formations. Fixed double train formation Flexible bicycle grouping and flexible double train formation f is the index. and .

[0055] Considering the maximum number of train sets at turnaround stations, we describe the train set cycle process. This process involves three key technical operations: decoupling (separating train sets), marshalling (merging train sets), and turnaround (reversing direction). Each service initiates its next operation at the terminal or turnaround station.

[0056] use This indicates that a train inherits a subset of a trainset, where the train... It inherited the train group from train i. Three variables. , and These are used to represent the disassembly, marshalling, and turnaround operations of trains i' and i in yard m, respectively. If there is a cyclic relationship between trains i' and i in yard m, then... The value is 1 if it is not 0 otherwise, and the calculation method is formula (16): (16); in, This represents a 0-1 variable. If the train set in train i comes from train i' in depot m, Otherwise, it is 0; This represents a 0-1 variable; if train i' disintegrates at yard m, Otherwise, it is 0; This represents a 0-1 variable. If train i' is grouped in yard m, Otherwise, it is 0; This represents a 0-1 variable. If train i' turns back at depot m, Otherwise, it is 0.

[0057] It should be noted that if train i has no preceding train relationship, it is considered to originate directly from its designated depot: ; Formula (17) ensures that the starting point of each train is unique.

[0058] (17); Formulas (18)-(20) ensure train The time interval between the arrival and departure of train i is sufficient for dismantling, marshalling, and turnaround operations to be carried out respectively.

[0059] (18); (19); (20); in, This represents an integer variable, indicating the departure time of train i in depot m. ; Represents an integer variable, trains in depot m. The arrival time; This indicates the time required for dismantling, grouping, and rerouting operations.

[0060] All possible train set cycle operations and variables are shown in Table 1.

[0061] Table 1. Variable values ​​for different tasks

[0062] Disassembly and marshalling operations are carried out within turnaround stations, and due to the availability of sidings, these operations allow for simultaneous arrival and departure. When a preceding train needs to be disassembled or marshalled at a turnaround station, a following train can only arrive after this operation is completed. In addition to maintaining safe intervals between trains, time constraints for disassembly and marshalling operations must also be considered, as described in formulas (21) and (22).

[0063] (twenty one); (twenty two); (4) Car storage capacity constraint at turnaround stations: used to track and limit the real-time occupancy of storage lines in each yard, ensuring that the unloading operation is performed only when there are available storage lines.

[0064] The vehicle storage capacity constraint of the turnaround station is implemented in the following way: The number of storage lines already occupied in the depot when the current train arrives at the depot is dynamically calculated using integer variables.

[0065] The integer variable is constrained to not exceed the maximum capacity of the parking lot.

[0066] The constraint on dismantling operations for flexibly assembled double-car trains is only permitted if there are available storage tracks in the depot when the train arrives.

[0067] The constraint on the formation of flexible single-car train sets is only permitted if there are already car body units occupying the storage lines in the depot when the train departs.

[0068] Specifically, to ensure that parking lot capacity constraints are met, integer variables are used. This indicates the number of storage lines occupied in depot m (including the virtual depot of the turnaround station) when train i arrives. Formula (23) considers train i that is temporarily stored in the depot before its arrival but has not yet been used for train formation unit calculation. This equation restricts the formation of trains to a condition where at least one train unit is available on the available lines. Furthermore, formula (24) ensures that the maximum storage capacity of the depot is not exceeded.

[0069] (twenty three); (twenty four); in, Indicates train Is it in the parking lot? After being dismantled, it becomes a new train. , Indicates train Is it in the parking lot? After being assembled, they become a new train. , This indicates the maximum capacity of parking lot m.

[0070] For each depot (including virtual depots), the decompilation operation must simultaneously meet two conditions: (a) Dismantling: can only be performed on flexible double-car trains, as in formulas (25) and (26); requires at least the siding capability of a single-car train, as in formula (27): (25); (26); (27); in, This indicates that trains are gathering in the opposite direction. For indexing; Represents a 0-1 variable, if the train The train formation mode is f, the route is r, and the train sets come from trains in depot m. , =1, otherwise 0; This indicates flexible double-car formation.

[0071] (b) Formation: This can only be done on flexible formation single-car trains, as shown in formulas (28) and (29); at least one train unit is required on the side rail, as shown in formula (30): (28); (29); (30); This application achieves efficient and refined utilization of the limited car storage capacity of turnaround stations by dynamically calculating and constraining the number of occupied storage lines, avoiding resource waste or bottlenecks caused by traditional static allocation. By binding the triggering conditions for dismantling and marshalling operations to the real-time car storage status, it fundamentally ensures the physical feasibility of various car-mounted operations under flexible marshalling strategies, improving the reliability of the plan. The constraints deeply couple the yard storage capacity, train marshalling plan, and operational routes, enabling synergistic optimization of car-mounted turnaround schemes, flexible route plans, and infrastructure capacity, enhancing the feasibility and robustness of the overall operational plan.

[0072] (5) Passenger demand constraint: Based on time-varying passenger demand, it dynamically calculates the number of waiting passengers at each station when each train arrives, combines the real-time remaining capacity of the train with the platform carrying capacity, determines the actual number of passengers getting on and off the train, and updates the remaining capacity of the train at subsequent stations accordingly, so as to achieve the matching of capacity supply and passenger demand.

[0073] Specifically, to calculate the actual number of passengers boarding each train, it is first necessary to determine the number of passengers on each train at each platform. The passenger demand of train i at platform n can be calculated from the total number of passengers from platform n to subsequent stations, as shown in formula (31). The actual number of passengers boarding is the minimum between the remaining capacity of the train and the total number of passengers waiting on the platform, as shown in formula (32).

[0074] (31); (32); in, Indicates the distance from platform n to ( The set K represents the OD pairs; the set K represents the passenger flow under different conditions (i.e., morning peak, evening peak, and off-peak hours); the set This represents the passenger arrival rate within the time window under condition k. Let k be an integer variable representing the passenger demand for train i at platform n in case k. ; This refers to the collection of platforms on both urban rail lines. This represents the set of trains running in one direction, where i is the index; Let k be an integer variable representing the number of passengers boarding train i at platform n in case k. ; This represents the maximum passenger capacity of platform n; Let k represent an integer variable, which is the passenger flow demand from station n to station n' when train i is at station n in case k. Let k represent the number of passengers carried by train i when it stops at platform n, given the following cases. ; Let k represent the number of passengers carried by train i when it stops at platform n, given the following cases. ; This indicates the maximum capacity of train formation type f.

[0075] Train i begins service at platform n on route r, with the platform serving as its originating station, and its capacity equal to the train's maximum capacity. Otherwise, the train's effective capacity should be determined by considering the cumulative passenger flow at all preceding platforms, as shown in formula (33):

[0076] (33); in, Let k represent an integer variable, and let k be the case where train i is at platform k. The number of passengers boarding; Let k represent an integer variable, indicating the passenger disembarking from train i at platform n in case k.

[0077] In Assumption 3, the number of passengers boarding each train is calculated proportionally at each station based on the total number of passengers under that station class, following a predetermined method. Therefore, from The proportion of actual passengers boarding at point n out of all passengers boarding is equal to the proportion of passengers boarding at point n. Passenger demand at platform n The proportion of total demand is given by formula (34).

[0078] (34); in, Let k represent an integer variable, and let k be the case where train i is at platform k. OD pairs ( The number of passengers boarding the train (n). Let k represent an integer variable, and let k be the case where train i is at platform k. The demand of passengers.

[0079] Formula (35) represents the proportion of passengers boarding or alighting from train i to the total number of passengers boarding at the originating station, where The cumulative number of passengers boarding from all preceding stations was quantified.

[0080] (35); in, Let k represent an integer variable, indicating the case where passengers disembark from train i at platform n. .

[0081] The accumulated passengers on the platform at the time of train departure include the remaining passengers from the previous train and new passengers arriving during the time interval. Arriving passengers, such as (36).

[0082]

[0083] (36); in, Let k represent an integer variable, and let k be the case where train i is at platform k. From arrive The demand for passenger flow; This represents an integer variable, where car i-1 is at platform k in case k. From arrive The demand for passenger flow; This represents an integer variable, where car i-1 is at platform k in case k. OD pairs ( , The number of passengers boarding the train; Indicates the OD pair under case k ( , The passenger arrival rate can be represented as a periodic function; This represents an integer variable, indicating that train i is at platform i. Departure time; This represents an integer variable, where train i-1 is on platform i. The departure time.

[0084] This application constructs a mixed-integer programming model that integrates four constraints: timetable feasibility, operational safety intervals, rolling stock turnover, and dynamic passenger flow matching. This model achieves integrated and coordinated optimization of flexible routes, train formation, rolling stock utilization, and timetables. While strictly ensuring operational safety and operating time, this mixed-integer programming model significantly improves the utilization efficiency of key resources such as line capacity and depots, enabling the operation plan to accurately respond to time-varying passenger flow demands. This, in turn, ensures the feasibility, economy, and service level of a high-density, flexible formation operation mode.

[0085] Because the mixed-integer programming model contains nonlinear constraints, such as equations (32) and (34), it is impossible to directly solve the mixed-integer programming model to obtain an exact solution due to these nonlinear constraints. Specifically, equation (32) involves a comparison of minimum values, requiring the introduction of auxiliary variables to determine the values ​​of the decision variables. Equation (34) involves the proportion of passengers with corresponding ODs in the total number of passengers, which is a function of time and requires discretization.

[0086] Linearization of Equation (32): Equation (37) is linearized by introducing a set of auxiliary binary variables. To indicate whether This can be linearized into formula (38). We construct linear constraint formulas (38)-(42) to represent integer variables using the Big M method. .in, This represents an auxiliary 0-1 variable, representing passenger demand. Exceeding remaining passenger capacity Otherwise, it is 0.

[0087] (37); (38); (39); (40); (41); (42); Linearization of formula (34) by Discretize the probability of boarding For each passenger boarding ratio Actual passenger demand The range of values ​​is Introducing an auxiliary binary variable Let represent the proportion of passengers on platform n who board train i under condition k. and corresponding passenger flow variables and , indicating that the ratio is OD is The actual number of passengers boarding and the demand from origin and destination (OD) are considered.

[0088] Formula (34) is linearized using formulas (43)-(47) based on the given occupancy rate. Formula (43) constrains train i stopping at platform n to have only one occupancy rate. Formulas (44)-(46) calculate the occupancy rate when the occupancy rate is... When, OD is Passenger flow variables are The number of passengers boarding the bus. Formula (47) recalculates the number of passengers boarding the bus. .

[0089] (43); (44); (45); ( 46); (47); in, This indicates the percentage of passengers boarding at the origin-destination (OD) pair. ; This indicates the percentage of passengers boarding the vehicle. ; This represents an auxiliary 0-1 variable, indicating the percentage of passengers disembarking at platform n for train i under case k. , Otherwise, it is 0; Let represent an auxiliary integer variable, and let OD pair (n, n') have a passenger disembarkation rate equal to , ; Indicates the possible loading ratio granularity, .

[0090] This application also provides a complexity analysis of the mixed integer programming model, as shown in Table 2. This table summarizes the relationship between decision variables, constraints, and problem dimensions. The solution space complexity of this model increases exponentially with the increase of train service frequency.

[0091] Table 2. Number of variables and constraints in the mixed-integer programming model.

[0092] in,( / () is the number of trains going up / down. For unidirectional platform counting, ( ) is the case count, ( ) is the route type count, ( ) is the grouping type count, ( ) is the count of dismantling / grouping nodes, ( ) is a discrete passenger ratio count, ( ) is an interval number, ( / () represents the total number of stops on the short-haul route.

[0093] Based on the complexity analysis of mixed-integer programming models and the precision of the required computation process, this application proposes a multi-strategy adaptive collaborative optimization algorithm class to obtain practically feasible solutions to mixed-integer programming models. The solutions in this application represent train operation strategies. Specifically: S3. Based on urban rail transit line structure and passenger demand data, the multi-strategy adaptive collaborative optimization algorithm is used to solve the mixed-integer programming model, outputting train timetables, train formation schemes, and train set connection schemes. This specifically includes the following S31-S33, wherein: S31. Generate an initial solution based on the urban rail transit line structure and passenger flow demand data, and set the initial solution as the first train operation strategy and the global optimal train operation strategy.

[0094] S32. Enter the iteration loop and repeat the following steps until the termination condition is met: S32-1. Obtain the current selection weight corresponding to each search strategy, and determine the train target search strategy based on the current selection weight corresponding to each search strategy using the roulette wheel selection algorithm. The search strategies include: Search Strategy 1 (SS-1): Adjust service frequency; Search Strategy 2 (SS-2): Flexible route strategy; Search Strategy 3 (SS-3): Adjust departure interval time; Search Strategy 4 (SS-4): Flexible train formation adjustment strategy; Search Strategy 5 (SS-5): Adjust train carriage connection relationship.

[0095] S32-2. The first train operation strategy is adjusted using the train target search strategy to obtain the second train operation strategy.

[0096] S32-3. Determine whether the second train operation strategy is feasible. If it is not feasible, call the repair strategy corresponding to the train target search strategy to repair it and obtain the third train operation strategy. The repair strategy includes: (1) first repair strategy (RS-1): adjust the train connection relationship, (2) second repair strategy (RS-2): adjust the timetable structure, (3) third repair strategy (RS-3): update the station available capacity.

[0097] S32-4. Calculate the objective function values ​​of the third train operation strategy and the first train operation strategy, and based on the objective function values, determine whether to update the first train operation strategy with the third train operation strategy according to the simulated annealing criterion.

[0098] Specifically, if the objective function value of the third train operation strategy is less than the objective function value of the first train operation strategy, then the first train operation strategy is updated using the third train operation strategy; if the objective function value of the third train operation strategy is not less than the objective function value of the first train operation strategy, then the simulated annealing acceptance probability condition is satisfied. Accept the third train operation strategy; The probability is a randomly generated uniform distribution. For temperature parameters, This is the proportionality coefficient. Let the objective function value be the value of the third train's operating strategy. Let be the objective function value of the first train's operating strategy.

[0099] S32-5. If updating, perform the following steps: Determine whether the third train operation strategy is superior to the global optimal train operation strategy. If so, update the current global optimal train operation strategy based on the third train operation strategy.

[0100] If no update is made and this low-quality solution is rejected, the corresponding train object involved in this iteration will be given a tabu state. This state will prevent the same operator from reselecting the train object in subsequent n iterations, where n represents the dynamically adjusted tabu period.

[0101] S32-6. Update the selection weight of the train target search strategy in the current iteration according to the current globally optimal train operation strategy.

[0102] S33. After the iteration terminates, the optimized train timetable, train formation scheme and rolling stock connection scheme are obtained based on the output global optimal train operation strategy.

[0103] This application introduces a tabu search strategy to avoid getting stuck in a loop of suboptimal train selection during the optimization process. The mechanism operates as follows: when an operator encounters and rejects a low-quality train operation strategy, the corresponding train object involved in that iteration is given a tabu state. This state will prevent the same operator from reselecting the same train object in subsequent n iterations, where n represents a dynamically adjusted tabu period. The specific implementation process is as follows: In S31, the taboo list is set to empty. In S32-1, after determining the train target search strategy using the roulette wheel selection algorithm based on the current selection weight corresponding to each search strategy, the process includes determining whether the adjustment operation to be performed by the train target search strategy for the first train operation strategy is prohibited based on the current taboo list; if prohibited, the adjustment operation object is adjusted until a feasible adjustment that is not prohibited is selected.

[0104] In S32-4, after determining whether to update the first train operation strategy with the third train operation strategy based on the simulated annealing criterion, the method further includes: if the first train operation strategy is not updated with the third train operation strategy, the operation objects involved in the third train operation strategy are added to the taboo list; if the first train operation strategy is updated with the third train operation strategy, the globally optimal train operation strategy is updated when the third train operation strategy is superior to the globally optimal train operation strategy.

[0105] This application's multi-strategy adaptive collaborative optimization algorithm integrates multi-dimensional search strategies with targeted repair strategies to achieve integrated collaborative optimization and feasibility assurance of train timetables, train formation schemes, and rolling stock connection plans. This design significantly improves the efficiency of the optimization process and the overall quality of train operation strategies, ensuring that the final solution simultaneously meets the requirements of operational efficiency, safety constraints, and resource limitations, thereby increasing the efficiency of developing highly complex and feasible flexible operation plans.

[0106] Regarding the specific steps of the aforementioned multi-strategy adaptive collaborative optimization algorithm, this application details the multi-strategy adaptive collaborative optimization algorithm from three aspects: (i) multi-strategy adaptive collaborative optimization algorithm, (ii) search strategy design, and (iii) repair strategy design.

[0107] (I) Multi-strategy adaptive cooperative optimization algorithm (1) Framework of multi-strategy adaptive cooperative optimization algorithm The multi-strategy adaptive collaborative optimization algorithm generates an initial solution based on known parameters, including OD demand and urban rail transit line structure. It then enters an iterative phase, implementing strategic adjustments through search strategies, including: adjusting service frequency, switching to shorter routes, changing departure intervals, adopting flexible train formation modes, and altering trainset connections. These operations may generate new infeasible solutions, requiring conflict detection and remediation. Corresponding remediation strategies include: adjusting trainset connections, ensuring safe train intervals, and ensuring station safety capacity.

[0108] The solution is updated according to the simulated annealing acceptance criterion. The "adaptability" is reflected in the dynamic adjustment of strategy weights through roulette wheel selection and the establishment of a tabu list to record poor solutions. This continuous adaptive mechanism enables the algorithm to accelerate convergence and identify the optimal search direction. For a detailed algorithm flowchart, please refer to [link to algorithm details]. Figure 4 An initial solution is generated based on the urban rail transit line structure and passenger flow demand data, and this initial solution is set as the current solution and the global optimal solution. An iterative loop is then entered, and the following steps are repeated until the termination condition is met: Based on the current selection weights of multiple search strategies, a roulette wheel selection algorithm is adopted, that is, the roulette wheel rule randomly selects a search strategy as the train target search strategy; the current solution is adjusted using the train target search strategy to generate a new solution; Determine if the new solution is feasible. If the new solution is not feasible, invoke the repair strategy corresponding to the train target search strategy to repair it and obtain a feasible new solution. Calculate the objective function values ​​of the feasible new solution and the current solution, and based on the objective function values, decide whether to update the current solution with the feasible new solution according to the simulated annealing criterion. If the updated current solution is better than the global optimal solution, update the global optimal solution. Update the selection weight of the selected strategy in the current iteration according to the updated global optimal solution. After the iteration terminates, the optimized train timetable, train formation scheme and rolling stock connection scheme are obtained based on the output global optimal solution.

[0109] Regarding the steps of the multi-strategy adaptive collaborative optimization algorithm, in step S31, the initial solution generation scheme, through the simplification rule of full intersection and fixed two-unit grouping, provides a high-quality and highly feasible iterative starting point for subsequent complex collaborative optimization algorithms. The initial solution generation includes the following steps: Train operation lines are generated according to the minimum departure interval; Each train operates on a fixed full-route schedule; The train carriage succession plan is determined by using a first-in, first-out (FIFO) sorting method. All operational services are uniformly allocated to a dual-unit flexible train formation mode.

[0110] The initial solution in this application has a simple structure and clear rules, which can be generated very quickly with almost no computation time. For the MSAO algorithm, iterative optimization starting from the initial solution can enter the promising search region faster than starting from a random or empty state, effectively accelerating the convergence of the entire optimization process.

[0111] In S32, to improve computational efficiency during the iteration process, two complementary optimization mechanisms are introduced: roulette wheel selection algorithm and tabu search algorithm.

[0112] For the roulette wheel selection algorithm, each strategy is assigned a weight, which determines the probability of each search strategy being selected. This method prioritizes objects with better adjustment effects while maintaining a certain degree of randomness, thereby improving the iterative efficiency of the multi-strategy adaptive collaborative optimization algorithm. Each object in the roulette wheel is assigned an evaluation function; the higher the function value, the greater the probability of the object being selected for adjustment.

[0113] The specific evaluation criteria vary depending on the strategy type: for the search strategy, the probability of a train being selected is positively correlated with the degree to which its solution deviates from the initial conditions, and trains with larger deviations are prioritized for adjustment; for the repair strategy, the probability of selecting a train is positively correlated with the number of stranded passengers, aiming to prioritize the restoration of train services during periods of severe passenger congestion.

[0114] After each iteration, the policy weights are adaptively updated based on the improvement in the quality of the solution: if the objective function value generated by the policy... Better than the first The current objective function value in the next iteration If the strategy performs well, its weight will increase; conversely, the weight of a poorly performing strategy will decrease. We use... This indicates that the strategy is in Weights in the next iteration To adjust the step size, the formula for calculating the weight update in each iteration is as follows: ; Furthermore, S2 utilizes simulated annealing and employs the Metropolis acceptance criterion to handle the feasible new solutions. This selection mechanism operates under two different conditions: (a) if the objective function value of the third train operation strategy is better than that of the first train operation strategy, then the third train operation strategy is accepted as the globally optimal train operation strategy; (b) if the objective function value of the third train operation strategy is worse than that of the first train operation strategy, then the third train operation strategy is a poor train operation strategy. Poor train operation strategies maintain a certain non-zero acceptance probability to ensure that the algorithm can escape local optima. This probabilistic acceptance of poor train operation strategies gradually decays during the cooling process, thus systematically reducing the exploration intensity when the train operation strategies tend to converge.

[0115] (II) Search Strategy Design (1) Search Strategy 1 (SS-1): Adjust service frequency To more effectively explore the optimal train operation strategy, search strategy 1 (SS-1) includes two possible operations: increasing the total number of trains in service ( or reduce the total number of train services. ),like Figure 5 As shown. The search process for SS-1 begins with adjusting the number of trains, followed by using the formula Recalculate the uniform arrival and departure intervals, where This indicates the total operating period duration. Then, a new timetable is generated based on the updated intervals.

[0116] It is worth noting that SS-1 only modifies the number of train services without changing the model's objective function. Therefore, the selection process for this strategy avoids a roulette wheel approach, thus ensuring a deterministic exploration of service frequency adjustments.

[0117] (2) Search Strategy 2 (SS-2): Flexible Route Strategy Search strategy 2 (SS-2) is designed for train routes and includes two possible operations: (a) randomly select a long-distance train service, and (b) restore a randomly selected short-distance service to full-route operation, such as... Figure 6 As shown.

[0118] We use This indicates the passenger losses resulting from the change, used to assess the impact on service fairness; This represents the number of delayed passengers. The corresponding roulette wheel evaluation function is shown below: ; (3) Search Strategy 3 (SS-3): Adjust departure interval time Similarly, search strategy 3 (SS-3) modifies the train departure schedule in two ways: (a) randomly selects an operating track and advances the departure time of the previous train on it. (b) Delay the departure time of another randomly selected service. ,like Figure 7 As shown.

[0119] The evaluation function is constructed using the total number of waiting passengers when the train with the highest passenger capacity arrives at the station, where Indicates train Arrival Platform The number of waiting passengers. Specifically, the former operation targets the trains with the highest load, reducing both immediate platform congestion and cumulative passenger waiting time by advancing their departure times.

[0120] ; (4) Search Strategy 4 (SS-4): Flexible Grouping Adjustment Strategy Search strategy 4 (SS-4) also adjusts the train formation strategy through two operations: (a) converting a randomly selected fixed formation unit ("6-car formation") into a flexible formation ("3+3 formation"), and (b) restoring a randomly selected flexible formation unit ("3+3 formation") to a fixed formation ("6-car formation").

[0121] To maintain fleet capacity constraints, the first operation prioritizes trains with lower capacity utilization. Conversely, the second operation selects routes based on empty mileage penalties—after calculating the specific penalty value for each route, all flexible formation trains on the route with the highest cumulative penalty value are converted to fixed formations. The formula for the evaluation function is shown below, where... Indicates route The total number of passengers boarding the bus, and Indicates the execution of the route The total empty mileage of the trains.

[0122] ; (5) Search Strategy 5 (SS-5): Adjust the connection between the bottom of the car Search strategy 5 (SS-5) modifies trainset connections through two complementary adjustments: (1) converting a randomly selected "3-car trainset" unit to a "3+3 trainset," and (2) restoring a randomly selected "3+3 trainset" unit to a "3-car trainset." When performing the first adjustment, priority is given to trains associated with stranded passengers, and services where passenger boarding failures have occurred are prioritized as target routes. The evaluation functions for performing both operations are as follows: ; (III) Repair Strategy Design (1) First repair strategy (RS-1): Adjust the connection relationship of the undercarriage The first repair strategy is used to eliminate connection conflicts by dispatching or returning empty trains from the depot, corresponding to the repair of the infeasibility of train connection caused by the implementation of the adjusted service frequency or the flexible route search strategy.

[0123] Specifically, the repair process begins immediately after an infeasible connection is generated in SS-1 or SS-2. This process first identifies the first terminating train at each turnaround station. Subsequently, repair strategy 1 (RS-1) will... The nearest available departure train in terms of time Matching is performed sequentially to establish a feasible connection. For any remaining unconnected trains, this strategy will take either of the following actions: (1) arrange for the terminal unit to run empty to the nearest depot, or (2) dispatch an empty unit from the nearest depot to serve the departing train.

[0124] (2) Second repair strategy (RS-2): Adjust the timetable structure The second repair strategy is used to restore the minimum safe interval constraint by adjusting train schedules, thereby fixing the timetable infeasibility problem caused by executing the adjusted departure interval search strategy.

[0125] Specifically, when search strategy 3 (SS-3) violates the departure interval or turnaround time constraints, maintenance strategy 2 (RS-2) will systematically shift the operation of the affected trains in the timetable until all operational constraints are met.

[0126] (3) Third Repair Strategy (RS-3): Update Station Available Capacity The third repair strategy is used to adjust the real-time occupancy status of the depot storage lines and schedule the transfer of car units when the capacity is over-limited or insufficient, thereby repairing the problem of infeasibility of turnaround station storage capacity constraints caused by the execution of the flexible grouping adjustment strategy or the car unit connection search strategy.

[0127] Specifically, train formation adjustments caused by SS-4 / SS-5 may require updating station capacity. Maintenance Strategy 3 (RS-3) continuously monitors dynamic station capacity and enforces the following adjustments: If Exceeding station capacity Then set the dynamic capacity to Immediately dispatch an empty train to send the undercarriage unit back to the depot; if the capacity is less than 0, set it to 0, and dispatch an empty train to move the undercarriage unit to the turnaround station before the reconfiguration operation.

[0128] Finally, regarding the iterative termination condition of the multi-strategy adaptive collaborative optimization algorithm: The stopping criteria for inner-layer strategy search include: a single strategy reaching its maximum number of iterations; and the strategy having yielded a certain number of better train operation strategies. The stopping criteria for total iterations include: the total number of iterations reaching its maximum; and no better train operation strategies being updated.

[0129] The multi-strategy adaptive collaborative optimization algorithm provided in this application performs an efficient and collaborative global search in the train operation strategy space by simultaneously adjusting multiple key decision dimensions such as service frequency, route mode, departure interval, train formation scheme, and rolling stock connection. This allows for rapid exploration and approximation of the overall optimal operation scheme, overcoming the tendency of single-strategy or sequential optimization to get trapped in local optima. By correcting strategies such as adjusting connection and ensuring safe intervals and capacity, the algorithm can instantly identify and correct infeasible train operation strategies resulting from strategy adjustments. This ensures that the scheme during the iteration process and the final output satisfy all operational and safety constraints, significantly improving the algorithm's practicality and the reliability of the output results.

[0130] To verify the performance and effectiveness of the proposed model and algorithm, this application constructs a real-world case study based on a local light rail line 5, which includes: (i) case description, (ii) experimental design, (iii) parameter settings, (iv) small-scale case study results, and (v) large-scale case study results.

[0131] (I) Case Description like Figure 8 As shown, a certain light rail line 5 comprises 30 stations (60 platforms, 30 in each direction; 58 sections, 29 in each direction). Train depot ZSL is located at the first station in the downward direction, while another depot SL is adjacent to the upward direction station SL. HYS and YBY stations are turnaround stations, where short-route trains originate or terminate.

[0132] The line includes four train routes: a full route from YGDD to TD, and short routes from YGDD to HYS, YBY to HYS, and YBY to TD. The line is equipped with 60 flexible unit trains and 20 fixed-formation trains. Three types of trains are permitted to operate: single-unit trains, double-unit trains, and fixed-formation trains. All trains operate on a station-stopping mode (i.e., stopping at every station along the route).

[0133] To reveal the differences in passenger flow demand across different time periods, based on historical passenger flow demand data from the automated ticketing system, passenger flow demand during three typical periods—morning peak, evening peak, and off-peak—was selected. Each period lasted two hours, specifically 7:00-9:00, 17:00-19:00, and 12:00-14:00. The total time window (120 minutes) for each period was discretized into 10-minute time intervals, thereby obtaining the average arrival rate of each origin-destination (OD) pair across 12 time segments. Figure 9 As shown, passenger arrival rates under these three scenarios were determined based on passenger flow data from Line 5 of a certain light rail system. The passenger flow exhibits a spatiotemporally uneven distribution, highlighting the importance of employing FTH, FTF, and FTR strategies.

[0134] (II) Experimental Design This application evaluates the effectiveness of the adopted strategy by validating the model and algorithm through both small-scale and large-scale computational examples. First, a simplified small-scale computational example is constructed to test the effectiveness of the model, algorithm, and proposed strategy. This example aims to obtain an exact solution and evaluate the difference between the proposed MSAO algorithm and the exact solution, thereby verifying the efficiency of the algorithm in solving large-scale computational examples. To systematically evaluate the performance of the MSAO algorithm, the global optimal solution obtained by the Gurobi solver and the solution results of another heuristic algorithm—Local Search (LR)—are selected as benchmarks. Based on model complexity analysis, comprehensive tests involving five different scenarios are conducted to systematically evaluate the model's performance under different operational scales, specifically: small-scale example 1 (8 trains / 4 pairs within the planned time period), small-scale example 2 (10 trains / 5 pairs within the planned time period), small-scale example 3 (12 trains / 6 pairs within the planned time period), small-scale example 4 (14 trains / 7 pairs within the planned time period), and small-scale example 5 (16 trains / 8 pairs within the planned time period). By using the scenarios of different scales described above, the computational cost of different solution methods can be quantitatively obtained.

[0135] Subsequently, large-scale computational examples in multiple scenarios were constructed to evaluate the model and algorithm. First, to verify the applicability of the proposed method in real-world cases, a set of experiments was designed to test its performance under different passenger flow demand scenarios, specifically Scenario 1: morning peak hours, Scenario 2: evening peak hours, and Scenario 3: off-peak hours. The passenger flow distribution during these three periods is as follows: Figure 9As shown in the figure. Based on the above scenario, the objective function values ​​of the traditional fixed grouping mode and the proposed flexible grouping mode are compared to reveal the effectiveness of the proposed method in reducing costs. Furthermore, the performance of the proposed MSAO algorithm is compared with other widely used algorithms, namely LS and Adaptive Large Neighborhood Search (ALNS), to demonstrate the advantages of the proposed solution algorithm. The adaptive mechanisms of RW and TS in the MSAO algorithm are shown to maintain the quality of the solution while accelerating convergence, as evidenced by the superior objective function values ​​in all test scenarios.

[0136] (III) Parameter Settings This paper determines the relevant parameter categories based on the following aspects: (a) operational parameters derived from the technical requirements of a certain light rail line 5; (b) model parameters constrained by the range of variable values ​​and physical limitations; and (c) algorithm parameters tuned according to the complexity and convergence characteristics of the computational examples. Specific parameter values ​​are shown in Table 3.

[0137] All numerical experiments were performed in the Visual Studio environment, with the following computing platform configuration: Intel(R) Core(TM) 13th i7-13700KF 8-core processor (3.40 GHz), 32 GB of memory, and Windows 11 operating system. For benchmark comparison, exact solutions were obtained using the Gurobi Optimizer 12.0.1 solver under the same hardware configuration.

[0138] Table 3 Parameter Values

[0139] Based on the timetable and passenger demand published by a certain light rail line 5, a uniform minimum departure interval of 60 seconds was adopted for the entire route in all test cases, thereby maximizing passenger satisfaction while ensuring operational feasibility. In SS-4, the train formation optimization order with cost optimization as the objective was flexible single-unit trains, flexible double-unit trains, and fixed-formation trains, fully verifying the advantages of the FTF strategy. The pre-operation diagrams and cumulative passenger volumes generated by large-scale tests in scenarios 1 to 3 are shown in Table 4. This result will serve as the initial solution for the algorithm proposed in Section 6.5.

[0140] Table 4: Initial Solution and Passenger Backlog Situation of Large-Scale Cases

[0141] Note: 1, 2, and 3 represent passenger flow conditions over a continuous 2-hour period under three different scenarios; OT represents the total number of trains in operation during this period; AWT represents the average waiting time for passengers; TUTD represents the total travel distance of a train unit; Z represents the normalized objective function value; APAP represents the average peak passenger backlog, calculated using the following formula: MPAP represents the peak number of passengers in the backlog; U and D represent the up and down directions of train operation, respectively.

[0142] (iv) Results of small-scale numerical examples To effectively obtain an accurate solution, five stations on the two-way section of Light Rail Line 5 between HML Station and DLS Station in a certain area were selected as the application object, such as... Figure 10 As shown in Table 5, to compare the performance of heuristic algorithms such as MSAO with that of exact solvers, the solution results of each algorithm are presented in five small-scale examples. The calculated lower bounds, solutions, CPU time, and solution gaps are shown in Table 5.

[0143] As shown in Table 5, with the extension of the planning period and the increase in the number of operating train pairs, the CPU time required to calculate AWT using the exact solution method (using Gurobi) increases significantly, and it cannot obtain a zero-gap optimal solution within 14400 seconds (as in scenarios 2 to 5). Specifically, when solving cases with a duration exceeding 1 hour, Gurobi cannot even find a feasible solution (as in scenarios 4 and 5), indicating that this solver cannot be used to accurately solve larger-scale cases. Numerical experimental results show that as the Gurobi solution time increases, the convergence speed of the current optimal solution slows down significantly, while the lower bound continues to tighten for optimality verification. In contrast, the proposed algorithm exhibits a significantly faster convergence speed than Gurobi in terms of CPU time, and obtains a higher-quality near-optimal solution compared to the LS algorithm. In summary, the results show that the proposed MSAO algorithm is superior in solving real-world TTP, TFP, and TUCP problems.

[0144] Table 5: Solution results for small-scale examples

[0145] Note: Trains run in pairs within a time period, counted as the total number of trains running in both directions. Algorithms include the Gurobi solver, LS algorithm, and MSAO algorithm. NFS indicates no feasible solution was found. Gap (%) represents the solution gap, calculated using the Gurobi solver formula: The calculation formulas for LS and MSAO algorithms are as follows: .

[0146] (v) Results of large-scale numerical experiments This section evaluates the model, algorithm, and proposed strategy based on large-scale real-world examples covering morning peak, evening peak, and off-peak scenarios, verifying the robustness of the proposed integrated optimization model and algorithm.

[0147] (1) Performance under different passenger flow scenarios To systematically evaluate the model and algorithm, nine sets of experimental data were generated, and sub-objectives and normalized objectives were constructed, namely, average waiting time (AWT), train unit travel distance (TUTD), and the Pareto front of Z, as shown in Table 6. In the experiment, if TUTD (i.e., ...) is not considered... and By considering the minimum AWT and the maximum TUTD, a near-optimal solution for the sub-objective can be obtained. This solution can be considered as the minimum of AWT and the maximum of TUTD. Conversely, if AWT is not considered (i.e., ... and Then, the minimum value of TUTD and the maximum value of AWT can be obtained in the experiment. These results provide the necessary maximum and minimum values ​​for the sub-objective normalization defined by equation (1).

[0148] Experimental results show that, considering both AWT and TUTD ( In the case of FTF and FTR strategies, the objective function value Z of the running graph was reduced by 64.26% and 57.98% respectively compared to the pre-run graph. Furthermore, the results show that the optimization performance of FTF and FTR strategies is significantly higher during peak periods (Scenarios 1 and 2) than during off-peak periods (Scenario 3). In addition, the proposed algorithm can solve near-optimal solutions for 2-hour scale examples in approximately 6 minutes, indicating its high applicability.

[0149] Table 6 Comparative Analysis under Different Passenger Flow Scenarios

[0150] Note: D and C represent the number of uncompilation and cascading operations, respectively; PreZ represents the normalized objective function value of the generated pre-run graph; Improve represents the percentage reduction in the total objective function value after optimization compared to PreZ, calculated using the following formula: .

[0151] Based on the superior performance of the MSAO algorithm in balancing sub-objective tradeoffs, benchmark tests were conducted to evaluate the performance of the MSAO algorithm relative to the LS and ALNS algorithms, thereby revealing its performance in peak-hour scenarios. The improvement effect under ( ). For example, Figure 11 As shown, the minimum objective function value proves the effectiveness of the MSAO efficient adaptive search mechanism, which benefits from the RW and TS mechanisms. In scenario 1, it achieves optimization improvements of 10.66% and 6.88% respectively compared to the LS and ALNS algorithms. Figure 11 (a) shows that in scenario 2, LS and ALNS achieved improvements of 2.97% and 3.92% respectively. Figure 11 (b)). Furthermore... Figure 11(c) indicates that during off-peak hours with lower passenger flow, the adaptive mechanism of MSAO can effectively accelerate the initial optimization process. However, compared with the LS and ALNS algorithms, this mechanism has limited advantages in optimal solution search. Among these, the best objective... The optimal objective value is defined by , where Iteration is the number of iterations, and Minimum objective is the minimum objective value.

[0152] Table 7 summarizes key indicators including the number of trains in service, the number of decoupling / coupling operations, the number of trains operating on the entire route / short route, and the average / maximum platform congestion. These results reveal the impact of multi-strategy integration on the optimization objective.

[0153] The results show that as the weight of AWT decreases and the weight of TUTD increases, the operational strategy tends to reduce operating costs by decreasing the number of trains in service to lower the normalized objective function value, while still meeting passenger demand. However, during peak hours, a combination of short-route and flexible train formation strategies is necessary. This highlights the strong adaptability and flexibility of the proposed model and algorithm under different passenger flow scenarios.

[0154] Specifically, the proportion of short-haul trains increased significantly, reaching a peak of 0.61 during morning and evening rush hours. Combined with the declining trend of TUTD (Turn-Off-Demand) values, this highlights the effectiveness of the FTR (Frequency-Trip-Return) strategy in reducing operating costs.

[0155] Table 7. Strategy Adoption and Passenger Backlog in Three Scenarios

[0156] Table 7 (continued): Strategy adoption and passenger backlog in three scenarios

[0157] Note: FR represents the number of trains operating on the entire route; FF represents the number of fixed-formation trains; SF represents the number of flexible single-unit trains (including flexible double-unit trains, with double-unit trains counted as 2 units). and These represent the proportions of flexible formation strategy and short-route strategy respectively, and the calculation formulas are as follows: and The larger the value, the more significant the impact of the adjustment strategy on the solution. "0" and "--" represent that the strategy was not adopted and was adopted, respectively.

[0158] (2) The improvement effect of FTR and FTF strategies We evaluated the impact of FTR and FTF strategies on AWT, TUTD, normalized objective function value Z, train routing schemes, train formation schemes, and passenger backlog.

[0159] Table 8 shows the optimization effect of the FTR strategy in three scenarios. By operating short-route trains, the normalized objective function value Z was significantly reduced (the maximum reduction was 42.62%, 36.8%, and 13.81% in the three scenarios, respectively), mainly due to the reduction in TUTD. Compared with the pre-operational schedule using uniform minimum departure intervals, the near-optimal solution increased the number of collinear sections (see...). Figure 1 (a) The frequency of train service on the platform is reduced, which in turn reduces the frequency of service on other platforms. This leads to a slight worsening of passenger congestion, which in most cases manifests as an increase in AWT.

[0160] Compared to the scheme without the FTR strategy, the APAP and MPAP values ​​show that the passenger backlog did not worsen in most experiments. During the morning peak (Scenario 1), the average increase in APAP was only slight, not exceeding 40 people; while during the evening peak (Scenario 2), the increase in APAP was less than 25 people. Furthermore, during off-peak hours (Scenario 3), although the application of the FTR strategy was less affected by the uniform distribution of passenger flow, the maximum decrease in the objective function value Z still reached 13.81%, effectively mitigating the risk of passenger backlog (the peak backlog was reduced by 58 people), thus verifying the robustness of the proposed algorithm under different scenarios.

[0161] Table 8 shows the optimization results of the FTR strategy in eight scenarios.

[0162] Accordingly, Table 9 shows the optimization effect after adopting the FTF strategy. The FTF strategy allows for train decoupling and coupling operations at turnaround stations and depots, thereby accelerating the turnaround of rolling stock and reducing the penalty cost of trains leaving the depot. The results show that the number of trains in operation decreased by 12, 8, and 16 trains, respectively; compared with the pre-operation schedule, the maximum reduction in the normalized objective function value during the morning peak, evening peak, and off-peak periods reached 40.01%, 37.50%, and 24.49%, respectively. Furthermore, it can be found that in scenario 3 (off-peak period), due to the relatively uniform passenger flow distribution, a high-quality solution can be obtained even without adopting the FTF strategy.

[0163] Furthermore, compared to the initial timetable using minimum headway service, reducing train size leads to a decrease in platform service frequency, resulting in increased passenger congestion and AWT growth. Experimental results show that when TUTD reaches its maximum reduction of 13.75%, 20.00%, and 14.08% in the three scenarios, the corresponding increases in AWT are 10.34%, 11.58%, and 14.74%, respectively. Nevertheless, according to the APAP index, the number of passengers with platform congestion only increases by about 10 people in most cases. Similar to the optimization effect of the FTR strategy, in both directions of Scenario 1 (morning peak) and in the upward direction of Scenarios 2 and 3, MPAP did not show a significant increase under most weight combination experiments, indicating that a more balanced trade-off between AWT and TUTD was achieved while meeting more passenger demand.

[0164] Table 9 shows the optimization effects of the FTF strategy in three scenarios.

[0165] Table 9 (continued) Optimization effects of FTF strategy in three scenarios

[0166] (3) FTH strategy sensitivity This section constructs an experiment by setting the departure interval parameter of the pre-running schedule to 120s to 240s (with a step size of 30s). To evaluate the sensitivity of the algorithm, Figure 12 The convergence curves are shown, and the results indicate that a larger initial headway significantly accelerates the optimization process. Initial headway is the initial headway distance. Notably, in scenarios 2 and 3, when the headway parameter is set to 120s, the number of trains required for the optimal solution exceeds the number of available wagons. Furthermore, the headway setting has a more significant impact on the final near-optimal solution in scenario 3, and the objective function value decreases as this parameter increases, indicating that a high-density train service frequency is unnecessary during off-peak hours.

[0167] (4) Weighted combination sensitivity experiment In constructing the objective function, this paper comprehensively considers the perspectives of both passengers and urban rail transit operators. To explore the trade-off between the two sub-objectives, different cost weight coefficients were set and a series of experiments were conducted to analyze their impact on the overall optimization results. Multiple experiments were performed for each set of weights, and the optimal result was selected.

[0168] As shown in Table 10, the higher the weight assigned to the train unit's travel distance, the greater the improvement in the solution compared to the pre-operation plan. This indicates that the strategy proposed in this application has a significant advantage in reducing the operator's operating costs. After optimization, compared to the pre-operation plan, the weighted total objective function value in scenario 1 shows a maximum reduction of 90.11% and a minimum reduction of 24.85%; in scenario 2, the reductions are 92.64% and 35.77%, respectively; and in scenario 3, the reductions are 74.64% and 50.99%, respectively.

[0169] Table 10: The impact of different weights on the objective function value in three scenarios

[0170] Figure 13 The diagram illustrates train schedules for three different weight combinations under Scenario 1. Notably, compared to the fixed formation mode in the initial schedule, all downline trains have been adjusted to a flexible formation mode. Specifically, with the weighting set as follows... At that time, the 3rd and 7th down-line trains and the 2nd up-line train were adjusted to flexible double-unit formations; in a balanced setting... At that time, the fourth and fifth down-line trains also adopted a flexible double-unit formation; while in a balance setting... At that time, two additional flexible double-unit trains were added, and the departure window was moved forward. These trains are categorized as: Fixed double-unit trains, Flexible double-unit trains, Flexible single-unit trains, Connection, From / to the depot, and Coupling operation.

[0171] The proportion of short-route trains varies across different times during the morning rush hour. Approximately between 7:00 and 9:45 (see...) Figure 13 (a) The number of short-route trains and full-route trains in the downbound direction was roughly equal; however, due to uneven distribution of tidal passenger flow, only two short-route trains operated in the upbound direction. Similarly, in Figure 13 In (b), the number of short-route trains in the upward direction increases significantly. Considering passengers who must take the full-route train to stations on non-co-located sections, an additional waiting time penalty needs to be included in the sub-objective; therefore, the algorithm prioritizes full-route trains, and only operates one short-route train at 7:45 (see...). Figure 13 (b)).

[0172] In terms of capacity utilization, the FS-1's undercarriage turnover strategy can be used... Figure 13 This was verified in (b). The second down train was adjusted to a double-car train in the flexible formation mode. Since the previous train was also planned as a flexible double-car train and had already occupied one of the storage lines at TD station, the train was arranged to return to the depot empty after arrival.

[0173] Furthermore, simultaneously balancing service quality and operating cost targets may lead to significant passenger flow accumulation on platforms. For example... Figure 13 As shown in (b), the algorithm prioritized short-interval double-unit trains in the downbound direction. However, severe passenger congestion still occurred at platforms SQP, XTZ, and ZYGYX. Severe passenger congestion was also observed at platforms DLS, SJB, GYDD, and LJG in the upbound direction.

[0174] Figure 14 The scenarios 2 and 3 are shown respectively with fixed weight combinations ( The optimal operating graph under the condition of ) can be seen. It can be seen that the combination of FTF and FTR strategies can achieve optimal performance during the evening peak hours (see Figure 14 (a) Effectively balances service quality and operating costs while alleviating peak passenger congestion. The FTF strategy is also applicable during off-peak hours (see...). Figure 14 (b) By arranging unit-formation trains to provide higher frequency services.

[0175] This paper proposes a hybrid integer programming (MIP) model that integrates train timetable planning (TTP), train formation planning (TFP), and train unit cycle planning (TUCP). The objective is to minimize the weighted sum of average passenger waiting time (service quality) and train unit mileage (operating cost), while also considering operational modes such as flexible train formation (FTF) and short turn strategy (FTR). To efficiently solve this model, a multi-strategy adaptive optimization (MSAO) algorithm integrating a roulette wheel selection mechanism and a tabu search acceleration strategy is designed based on an adaptive large neighborhood search algorithm. Using a light rail line 5 in a certain area as an example, the proposed model and algorithm are validated through small-scale and large-scale real-world cases. Small-scale case experiments show that the MSAO algorithm can obtain near-optimal solutions, and its convergence speed is significantly faster than commercial solvers. Large-scale real-world case validation based on three scenarios shows that compared with mainstream heuristic algorithms such as local search (optimization rate of 10.66% in the morning rush hour scenario) and adaptive large neighborhood search, the MSAO algorithm has superior global optimization performance (optimization rate of 6.88% compared to adaptive large neighborhood search in the morning rush hour scenario), especially achieving a balanced optimization of average passenger waiting time and train unit travel mileage during peak hours. Compared with traditional fixed-formation, full-mileage timetables, the near-optimal solutions using the Short Turning (FTR) strategy and Flexible Train Formation (FTF) can reduce operating costs (train unit travel mileage) by approximately 11.03% (scenario 3) and 20% (scenario 2), respectively.

[0176] Finally, the method proposed in this application achieves integrated optimization of urban rail transit timetable compilation, train route planning, and train unit circulation, and incorporates strategies such as Flexible Train Formation (FTF) and Short Turning (FTR). Empirical analysis based on a local light rail line 5 shows that this method has significant practical value for operation management, especially suitable for urban rail transit systems with significant tidal passenger flow characteristics (such as commuter lines connecting residential and commercial areas). It is recommended to prioritize the flexible train formation mode, which can effectively balance passenger demand satisfaction and operational efficiency. Through this model, urban rail transit operators can more accurately match transport capacity with passenger demand, further reducing operating costs while ensuring service quality. For example, operators can adopt the following strategies: (a) during off-peak hours when passenger demand is low, use unit train formation and full-mileage operation mode; (b) during peak hours, implement flexible train formation strategies to address the temporal imbalance of passenger flow; (c) when some stations experience high passenger flow concentration, switch trains from full-mileage operation mode to short turning strategy to alleviate spatial imbalance of passenger flow.

[0177] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0178] This document uses specific examples to illustrate the principles and implementation methods of this application. The descriptions of the above embodiments are only for the purpose of helping to understand the methods and core ideas of this application. Furthermore, those skilled in the art will recognize that, based on the ideas of this application, there will be changes in the specific implementation methods and application scope. Therefore, the content of this specification should not be construed as a limitation of this application.

Claims

1. A method for coordinated optimization of urban rail train timetable and trainset connection under flexible train formation, characterized in that, The method includes: S1. Obtain data on urban rail transit line structure and passenger flow demand; S2. Establish a mixed integer programming model for synchronously optimizing train timetables, train formation schemes, and train set connection plans. The mixed integer programming model aims to minimize the average passenger waiting time and the total travel distance of the train set, and uses the storage capacity constraint of the turnaround station as a constraint condition for the train set connection plan. The storage capacity constraint of the turnaround station is used to track and limit the real-time occupancy of storage lines in each depot, ensuring that the de-staging operation is performed only when there are available storage lines. S3. Based on the urban rail transit line structure and passenger flow demand data, a multi-strategy adaptive collaborative optimization algorithm is used to solve the mixed integer programming model, and the train timetable, train formation scheme and train set connection scheme are output.

2. The method for coordinated optimization of urban rail train timetable and rolling stock connection under flexible formation as described in claim 1, characterized in that, The expression for the optimization objective of the mixed integer programming model is: ； Where Z is the overall objective function. For the first sub-objective function, for The theoretical minimum value, for The theoretical maximum value, For the second sub-objective function, for The theoretical minimum value, for The theoretical maximum value, The weights of the first sub-objective function are... The weights are for the second sub-objective function.

3. The method for coordinated optimization of urban rail train timetable and rolling stock connection under flexible formation as described in claim 2, characterized in that, The expression for the first sub-objective function is: ； in, This represents the average waiting time for passengers under all circumstances; This represents a set of multiple cases; k represents the index of the case. This represents the total number of passengers in case k. This represents the number of passengers who have arrived before the first train departs, in case k, from platform n' to platform n. This indicates the departure time of the first train from platform n′; Indicates the departure time window; This represents the number of passengers who have not boarded the last train after it arrives, in case k, among the passenger demand from platform n′ to platform n. Indicates the arrival time window; This indicates the departure time of the last train from platform n′; This represents the set of trains operating in one direction; i represents the index of the train column operating in one direction. Indicates the assembly of trains heading upwards; Indicates the group of trains heading south; Indicates the platform index. This represents the set of platforms in the direction of train i's travel; This represents the passenger demand for train i-1 at platform n under condition k; This represents the number of passengers boarding train i-1 at platform n in case k. This indicates the departure time of train i at platform n; This indicates the departure time of train i-1 at platform n; Indicate the number of passengers boarding train i at platform n under case k; This indicates the arrival time of train i-1 at platform n.

4. The method for coordinated optimization of urban rail train timetable and rolling stock connection under flexible formation as described in claim 2, characterized in that, The expression for the second sub-objective function is: ； in, Represents the total travel distance of the train set; I represents the set of all trains; i represents the index of the train running in one direction; R represents the set of all routes; The route r represents the travel distance; f represents the train formation type; and F represents the set of train formation types. This indicates the train set in the F formation mode, with 1 for single-car train mode and 2 for double-car train and fixed mode. This represents a 0-1 variable. If train i has a formation mode of f, uses route r, and the train set originates from trains in depot m... , =1, otherwise 0; I' represents the set of trains traveling in the opposite direction. Ψ represents the index of trains traveling in the opposite direction; Ψ represents the set of depots, including virtual depots, and m is the depot index; This represents a 0-1 variable, where the train set in train i originates from a train in depot m. , =1, otherwise 0; This indicates the additional travel distance the train travels when it leaves the depot (m).

5. The method for coordinated optimization of urban rail train timetable and rolling stock connection under flexible formation as described in claim 1, characterized in that, The vehicle storage capacity constraint of the turnaround station is implemented in the following way: The number of storage lines already occupied in the depot when the current train arrives at the depot is dynamically calculated using integer variables. The integer variable is constrained to not exceed the maximum capacity of the parking lot; The constraint on dismantling flexible double-car trains is only permitted if there is available storage track in the depot when the train arrives. The constraint on the formation of flexible single-car train sets is only permitted if there are already car body units occupying the storage lines in the depot when the train departs.

6. The method for coordinated optimization of urban rail train timetable and trainset connection under flexible formation as described in claim 1, characterized in that, The mixed-integer programming model also includes the following constraints: Constraints of the timetable on flexible routes: It is used to calculate and verify the arrival and departure times of each station based on the train operation routes and stopping plans, combined with the section running time, the minimum stop time at the station, the maximum stop time at the station, and the turnaround operation time, so as to generate an operation plan with time sequence feasibility; Train operation safety time constraints: These are used to ensure that the time interval between any two trains departing consecutively in the same depot, and between any two trains arriving and departing consecutively on the same platform, is not less than the minimum departure interval specified by the depot and the minimum arrival and departure interval specified by the station, respectively. The constraints of the rolling stock connection plan on flexible routes are used to determine the connection relationship between rolling stock units and different train services. The connection relationship includes de-marshalling, marshalling and turnaround operations, and ensures the minimum time required for each operation. Passenger demand constraint: Based on time-varying passenger demand, it dynamically calculates the number of waiting passengers at each station when each train arrives, combines the train's real-time remaining capacity with the platform's carrying capacity, determines the actual number of passengers getting on and off the train, and updates the train's remaining capacity at subsequent stations accordingly, so as to achieve a match between capacity supply and passenger demand.

7. The method for coordinated optimization of urban rail train timetable and rolling stock connection under flexible formation according to any one of claims 1 to 6, characterized in that, Based on urban rail transit line structure and passenger flow demand data, a multi-strategy adaptive collaborative optimization algorithm is used to solve the mixed integer programming model, outputting train timetables, train formation schemes, and rolling stock connection schemes, specifically including: S31. Generate an initial solution based on the urban rail transit line structure and passenger flow demand data, and set the initial solution as the first train operation strategy and the global optimal train operation strategy. S32. Enter the iteration loop and repeat the following steps until the termination condition is met: Obtain the current selection weight corresponding to each search strategy, and determine the train target search strategy based on the current selection weight corresponding to each search strategy using the roulette wheel selection algorithm. The first train operation strategy is adjusted using the train target search strategy to obtain the second train operation strategy; Determine whether the second train operation strategy is feasible. If it is not feasible, call the repair strategy corresponding to the train target search strategy to repair it and obtain the third train operation strategy. Calculate the objective function values ​​of the third train operation strategy and the first train operation strategy, and based on the objective function values, determine whether to update the first train operation strategy with the third train operation strategy according to the simulated annealing criterion. If an update is needed, perform the following steps: Determine whether the third train operation strategy is better than the global optimal train operation strategy. If so, update the current global optimal train operation strategy based on the third train operation strategy. Update the selection weights of the train target search strategy in the current iteration based on the current globally optimal train operation strategy; S33. After the iteration terminates, the optimized train timetable, train formation scheme and rolling stock connection scheme are obtained based on the output global optimal train operation strategy.

8. The method for coordinated optimization of urban rail train timetable and trainset connection under flexible formation according to claim 7, characterized in that, The initial solution generation includes the following steps: Train operation lines are generated according to the minimum departure interval; Each train operates on a fixed full-route schedule; The train carriage succession plan is determined by using a first-in, first-out (FIFO) sorting method. All operational services are uniformly allocated to a dual-unit flexible train formation mode.

9. The method for coordinated optimization of urban rail train timetable and rolling stock connection under flexible formation as described in claim 7, characterized in that, The search strategies include: adjusting service frequency, flexible route strategies, adjusting departure intervals, flexible train formation adjustment strategies, and adjusting train carriage connection relationships.

10. The method for coordinated optimization of urban rail train timetable and trainset connection under flexible formation according to claim 9, characterized in that, The repair strategy includes: The first repair strategy is used to eliminate connection conflicts by dispatching or returning empty trains from the depot, and correspondingly repairs the problem of infeasible train connection caused by the implementation of the adjusted service frequency or the flexible route search strategy. The second repair strategy is used to restore the minimum safe interval constraint by adjusting the train schedule, which correspondingly repairs the timetable infeasibility problem caused by executing the adjusted departure interval search strategy. The third repair strategy is used to adjust the real-time occupancy status of the depot storage lines and schedule the transfer of car units when the capacity is over-limited or insufficient, thereby repairing the problem of infeasibility of turnaround station storage capacity constraints caused by the execution of the flexible grouping adjustment strategy or the car unit connection search strategy.

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