A decomposition and optimization method for flexible train turnaround planning based on spatiotemporal network

Abstract

This invention discloses a method for decomposing and optimizing flexible train turnaround plans based on spatiotemporal networks. Based on the dynamic and uneven passenger flow of urban rail transit, and combined with the emerging flexible train formation transportation organization mode, it establishes an optimization model for flexible train turnaround plans based on spatiotemporal networks from a system optimization perspective. This model collaboratively optimizes two core rail transit plans: the train timetable and the train set routing. Based on the constructed collaborative optimization model, a corresponding pricing sub-model is designed to generate new train unit alternative routes. According to the principles of the train generation algorithm and the corresponding restricted master model and pricing sub-model, a branch pricing algorithm is designed. This invention can further explore the transportation capacity of rail transit systems through flexible train formation operation modes, overcome the problem of poor adaptability of existing fixed train formation schemes to dynamic and uneven passenger flow, alleviate passenger congestion, and improve the service level of urban rail transit systems.

Description

Technical Field

[0001] This invention belongs to the field of urban rail transit operation and management technology, and in particular relates to a method for decomposing and optimizing flexible train turnaround plans based on spatiotemporal networks. Background Technology

[0002] Urban rail transit boasts advantages such as high safety, low accident rate, speed and convenience, low environmental pollution, low energy consumption, and a positive user experience, making it a backbone mode of transportation for medium- and long-distance commutes within cities. However, an urban rail transit system is a typical complex mega-system, involving the coordination and control of factors such as train turnaround and passenger flow, exhibiting high complexity and organization. To reduce the complexity and uncertainty of operation plan development, urban rail transit system operation plans typically employ a hierarchical and phased development method, such as... Figure 1 As shown. At the strategic level, train operation plans are determined based on dynamic and time-varying passenger flow; at the tactical level, train timetables are prepared and train rollout and crew plans are completed; finally, at the operational level, the prepared plans are adjusted in real time based on actual operating conditions.

[0003] The compilation of urban rail transit train timetables is a core issue at the tactical level, determining the departure and arrival times of each train based on the operation plan. Then, a rolling stock turnover plan is developed based on the current rolling stock inventory, assigning appropriate rolling stock to each train's operating line, thereby obtaining the turnover path of each rolling stock within the planned time domain. The flexible train formation turnover plan mentioned in this invention can simultaneously achieve coordinated optimization of the train timetable and the rolling stock turnover plan.

[0004] Traditional train operation models typically only allow a single type of train (6-car or 8-car formation) to serve a particular line. However, when high passenger volumes occur on certain sections of a subway line, rail transit operators must shorten train intervals, meaning more train units are deployed. In such cases, a single fixed-formation train results in significant wasted capacity on low-passenger-flow sections. Flexible formation, as a new urban rail transit operation model, allows for flexible adjustment of train capacity through "forming" or "deforming" trains. This model can utilize different formations to further improve the flexibility and efficiency of operational plans. The train unit formation and deformation process is as follows: Figure 2 As shown.

[0005] Analysis of existing urban rail transit operation organization processes and related literature reveals that current methods pay limited attention to the impact of segmental passenger flow on train formation schemes and rolling stock turnover, making it difficult to match the uneven distribution of passengers in both space and time. Furthermore, existing methods for optimizing train formation rarely consider the collaborative optimization of train timetables, limiting further improvements in system transport capacity. In terms of solution methods, existing approaches typically construct mixed-integer programming models, relying on commercial solution software, resulting in poor controllability and difficulty in incorporating real-world operational scenarios.

[0006] Based on the above analysis of issues related to flexible train turnaround planning, this invention proposes a novel decomposition and optimization method for flexible train turnaround planning based on spatiotemporal networks. This method reconstructs the coordinated optimization problem of train timetable and rolling stock turnaround based on vehicle flow spatiotemporal networks and passenger flow spatiotemporal networks, and designs a solution algorithm based on branch pricing. This method can flexibly plan train formations, operating lines, and rolling stock turnaround plans according to passenger flow distribution in different sections, improving the poor coordination of existing fixed train formations with dynamically unbalanced passenger flow, further exploring the capacity of urban rail transit, and providing decision support for scientific operation and organization. Summary of the Invention

[0007] The purpose of this invention is to provide a new method for decomposing and optimizing the turnaround plan of flexible train formations based on spatiotemporal networks, in order to improve the poor adaptability of existing fixed train formations to dynamic and unbalanced passenger flow, improve the turnaround efficiency of fixed equipment in rail transit, and avoid the waste of transportation capacity.

[0008] To achieve the above objectives, this invention proposes a decomposition and optimization method for flexible train turnaround plans based on spatiotemporal networks. Specifically, from a system optimization perspective, it establishes a train operation network, a passenger flow allocation network, and the associated constraints between them, ultimately constructing an integrated mathematical optimization model. This method improves the poor adaptability of fixed-formation trains to dynamically unbalanced passenger flows.

[0009] A method for decomposing and optimizing the turnaround plan of flexible train formations based on spatiotemporal networks, specifically including the following steps:

[0010] Step 1: Describe the processes of train units entering and leaving the depot, marshalling, demarcating, stopping at stations, and running between sections based on a discrete spatiotemporal network flow model; describe the processes of passengers waiting at the platform, boarding and alighting, and taking the train based on a continuous multi-commodity flow model.

[0011] Step 2: Construct train operation organization constraints and passenger flow allocation constraints based on train unit paths.

[0012] The constraints on train operation organization include: train route selection, travel coverage, and train safety interval constraints; the constraints on passenger behavior include: passenger flow balance constraints on trains and platforms, total passenger disembarkation balance constraints, and train capacity constraints.

[0013] Step 3: Construct a collaborative optimization model with the objective function of minimizing the total waiting cost of passengers at the station and the total transportation cost of the train.

[0014] The total transportation cost of the train includes the fixed operating cost of the train unit and the electricity cost of train operation.

[0015] Step 4: Construct a pricing sub-problem for the flexible train turnaround plan collaborative optimization model based on spatiotemporal network obtained in Step 3, and design a branch pricing algorithm to solve it until a feasible and better collaborative optimization scheme is obtained.

[0016] Based on the above scheme, step 1 specifically includes:

[0017] For the train operation organization process, firstly, based on the planned time domain and accuracy requirements, the time range is discretized into a set of timestamps, forming a time domain set. Then, based on the discrete time domain set, all physical nodes are spatiotemporally extended. Specifically, according to the characteristics of the flexible train formation operation organization mode, this includes spatiotemporal nodes such as train arrival at the station, departure from the station, the starting point of the depot, and the ending point of the depot. Finally, based on the set of spatiotemporal nodes, a corresponding set of spatiotemporal arcs is constructed. Specifically, this includes train running arcs, waiting arcs, depot entry / exit arcs, station stopping arcs, and connecting virtual arcs. The relationship between the relevant spatiotemporal nodes and spatiotemporal arcs in the train operation organization process is as follows: Figure 3 As shown.

[0018] For passenger behavior on the platform and on the train, the platform is first spatiotemporally expanded at the same time granularity. Then, passenger arrival arcs, boarding arcs, and alighting arcs are constructed to describe passenger behavior between the platform and the train; for passengers who cannot board on time, waiting arcs are constructed to describe their waiting behavior on the platform. The relationship between relevant spatiotemporal nodes and spatiotemporal arcs in the passenger behavior process on the platform and on the train is as follows: Figure 4 As shown.

[0019] Based on the above scheme, the train operation organization constraints mentioned in step 2 specifically include:

[0020] (1) The train route selection constraint: For each train unit, only one route can be selected from its corresponding train unit route candidate set, as shown in formula (1):

[0021]

[0022] Where: index k represents the k-th train unit belonging to the train unit set K, and index p represents the train route candidate set P corresponding to train unit k. k The p-th train route; decision variable y p This indicates whether train path p has been selected. If path p is selected, then y... p =1, otherwise y p =0.

[0023] (2) The travel coverage constraint: For each train operation arc in the constructed spatiotemporal network, according to the requirements of flexible train operation conditions, the number of covered train units shall not exceed the coverage limit, as shown in formula (2):

[0024]

[0025] Wherein, index k represents the k-th train unit belonging to the train unit set K, and index p represents the train route candidate set P corresponding to train unit k. k The p-th train path; index a represents the set of train running arcs. The a-th train running arc; parameter γ a p This represents whether the train path p passes through the spacetime arc a. If the train path p passes through the spacetime arc a, then γ a p =1, otherwise γ a p =0; parameter N a max The upper limit of the number of times a train unit covers the train operation arc a; decision variable y p This indicates whether train path p has been selected. If path p is selected, then y... p =1, otherwise y p =0.

[0026] (3) The train safety interval constraint is used to ensure that there is no spatiotemporal conflict between each train unit, so as to ensure the safe operation of the train, as shown in formula (3).

[0027]

[0028] Wherein, index k represents the k-th train unit belonging to the train unit set K, and index p represents the train route candidate set P corresponding to train unit k. k The p-th train route, where index k' represents the k'-th train unit belonging to the set K (excluding train unit k), and index p' represents the train route candidate set P corresponding to train unit k'. k’ The p'th train path; index a' represents the a-th spatiotemporal arc belonging to the set of incompatible arcs of train operation arc a; parameter γp a’ This indicates whether the train path p passes through the spacetime arc a'. If the train path p passes through the spacetime arc a', then γ p a’ =1, otherwise γ p a’ =0; Decision variable y p This indicates whether train path p has been selected. If path p is selected, then y... p =1, otherwise y p =0.

[0029] To represent the incompatibility between different train running arcs, the set Ω a The set of all incompatible arcs representing train running arc a is determined by formula (4). For a train running arc a = (i,j,t,t') in the spatiotemporal network, where indices i and j are the i-th and j-th station nodes in the station node set I, respectively; t and t' are the timestamps of the train departing from station i and arriving at station j, and T is the set of all timestamps. For any other spatiotemporal arc (i,j,τ,τ') in the spatiotemporal network that passes through station i and station j at times τ and τ', if it coincides with the departure time window [t,t+h], D [i,j)] or reach the side time window [t',t'+h] A If there is an intersection between [i,j], then it is defined as an incompatible arc and placed into the set Ω. a .

[0030]

[0031] Where the parameter h D (i,j) and h A (i,j) represent the minimum safe train intervals on the departure and arrival sides of the section between station i and station j, respectively. In urban rail transit systems, the minimum interval between two trains is typically set to 2 minutes. For example... Figure 5 As shown, the set of incompatible arcs for a train's spatial arc (i,j,2,3) can be represented as Ω(i,j,2,3)={(i,j,2,4),(i,j,3,4),(i,j,3,5)}. If a train's spatial arc (i,j,2,3) is occupied by a certain train unit, no other spatial arc in the set Ω(i,j,2,3) can be occupied by other trains.

[0032] Based on the above scheme, the passenger flow balance constraint between trains and platforms mentioned in step 2 specifically includes:

[0033] (1) The passenger flow balance constraint on the train, based on the spatiotemporal network, can be used to characterize the passenger activity behavior on the train. For any spatiotemporal network node, the difference between the total flow of the train entering the arc and the total flow of the train waiting to leave the arc is the total disembarkation flow of the train at that network node, as shown in formula (5).

[0034]

[0035] Where, index 'a' represents the a-th spatiotemporal arc belonging to the set of trains waiting to exit the arc or trains entering the arc; index 'i' represents the i-th station node belonging to the set of station nodes I; and index 't' is the t-th moment in the time domain set T. Let (i,t) be the set of trains waiting to exit the arc in the spatiotemporal network. Let f be the set of train entry arcs for the spatiotemporal network node (i,t); Decision variable f a The number of passengers on the train's running or waiting arc a; variable α it down The total number of passengers alighting at node i and time t. Similarly, for any spatiotemporal network node, the difference between the total flow of trains running out of the arc and the total flow of trains waiting to enter the arc is the total flow of trains boarding at that network node, as shown in formula (6).

[0036]

[0037] Where, index 'a' represents the a-th spatiotemporal arc belonging to the set of trains waiting to exit the arc or trains entering the arc; index 'i' represents the i-th station node belonging to the set of station nodes I; and index 't' is the t-th moment in the time domain set T. Let the set of train running arcs be the set of the spatiotemporal network node (i,t). Let f be the set of trains waiting to enter the arc at node (i,t) in the spatiotemporal network; decision variable f a The number of passengers on the train's running or waiting arc a; variable α it up This represents the total number of passengers boarding at station node i and time t.

[0038] (2) The passenger flow balance constraint at the station, the change in passenger flow at station node i at timestamp t can be achieved by accumulating passenger flow CP at the platform. it In terms of flow balance, the cumulative passenger flow at the platform at spatiotemporal node (i,t) should be equal to the cumulative passenger flow at spatiotemporal node (i,t-1), minus the number of passengers boarding α. it up And add the number of people entering the station P it in As shown in formula (7).

[0039]

[0040] Where index i represents the i-th node in the station node set I, and index t represents the t-th time in the time domain set T; decision variable CP i,t CP represents the cumulative number of passengers at station node i at time t. i,t-1 The variable α represents the cumulative number of passengers at station node i at time t-1. it up Represents the total number of passengers boarding at station node i and time t; parameter P it in This represents the total number of passengers entering the station at station node i and time t.

[0041] (3) The total passenger disembarkation balance constraint states that the sum of all passengers disembarking at station node i within the planning time domain should equal the number of passengers exiting the station within the planning time domain, P. i out As shown in formula (8).

[0042]

[0043] Where index i represents the i-th node in the station node set I, and index t represents the t-th time in the time domain set T; variable α it down Represents the total number of passengers alighting at station node i and time t; parameter P i out The number of departing passengers within the overall planning time domain of station node i.

[0044] (4) The train capacity constraint is used to ensure that the number of passengers in each train unit does not exceed its capacity limit, as shown in formula (9):

[0045]

[0046] Wherein, index k represents the k-th train unit belonging to the train unit set K, and index p represents the train route candidate set P corresponding to train unit k. k The p-th train path, index a represents the a-th spatiotemporal arc in the set of train running and waiting spatiotemporal arcs, set and These represent the sets of spatiotemporal arcs for train movement and train waiting, respectively; parameter γ p a This represents whether the train path p passes through the spacetime arc a. If the train path p passes through the spacetime arc a, then γ p a =1, otherwise γ p a =0, parameter Cap represents the maximum passenger capacity of each train unit; decision variable fa Passenger flow representing spatiotemporal arc a; decision variable y p This indicates whether train path p has been selected. If path p is selected, then y... p =1, otherwise y p =0.

[0047] Based on the above scheme, step 3, which uses minimizing the total waiting cost of passengers at the station and the total transportation cost of the train as the objective function, specifically includes:

[0048] The objective function is to minimize the total waiting time cost of passengers at the station and the total transportation cost of the train, i.e., to minimize the total waiting time cost of passengers at the station and the total transportation cost of the train. Since the train cost and passenger cost have different dimensions, weighting coefficients α and β are needed to normalize the two costs, as shown in formula (11).

[0049]

[0050] The first term represents the total cost of waiting for passengers at all station nodes at all times, with parameter α being the weighting coefficient for this cost. The second term represents the total fixed cost of all used train units, with parameter β being the weighting coefficient for this cost. Index i represents the i-th node in the set of station nodes I, index t represents the t-th time in the time domain set T, index k represents the k-th train unit belonging to the set of train units K, and index p represents the train path candidate set P corresponding to train unit k. k The p-th train route; variable F is the objective function variable for total cost, and the optimization objective min is to minimize the total cost; decision variable CP it The decision variable y represents the cumulative number of passengers at station node i at time t. p This indicates whether train path p has been selected. If path p is selected, then y... p =1, otherwise y p =0; parameter C p This represents the fixed cost of the alternative path p for the train unit.

[0051] Based on the above scheme, step 4, which describes the construction of a pricing sub-problem for a flexible train turnaround plan collaborative optimization model based on spatiotemporal networks, specifically includes:

[0052] The collaborative optimization model for flexible train turnaround planning based on spatiotemporal networks is abbreviated as M1, and model M1 is the restricted master model (RMP). A pricing sub-model SP is constructed based on the dual variable values ​​of the restricted master model RMP. Specifically, the pricing sub-problem is decomposable for train units; the pricing sub-problem for the k-th train unit belonging to the set K is denoted as SP. k Among them, the pricing sub-model SP kThe objective function is shown in formula (12).

[0053]

[0054] Where, indices i and j represent the i-th or j-th station node belonging to station node set I, indices t and t' are the t-th and t'-th times in time domain set T, index (i,j,t,t') represents the spatiotemporal arc from time t of station node i to time t' of station node j, index k represents the k-th train unit belonging to train unit set K, and set A A Let A be the set of all train spatiotemporal arcs, including train running arcs, waiting arcs, train depot entry / exit arcs, station stopping arcs, connecting virtual arcs, etc. T Let A be the set of all train running arcs. W Let x be the set of all train waiting arcs; decision variable x k i,j,t,t’ This indicates whether train k occupies the spatiotemporal arc from station node i at time t to station node j at time t'. If train k occupies the spatiotemporal arc (i,j,t,t'), then x k i,j,t,t’ =1, otherwise x k i,j,t,t’ =0; variable Z k Let c be the objective function variable for the pricing subproblem of the k-th train unit; c is the parameter. i,j,t,t’ The cost is the spacetime arc (i,j,t,t'), where Cap represents the maximum passenger capacity of each train unit, and π represents the cost. 1 k ,π 2 i,j,t,t’ ,π 3 k,i,j,t,t’ ,π 4 i,j,t,t’ These are the values ​​of the dual variables of constraints (1), (2), (3), and (9), respectively.

[0055] Regarding the pricing sub-model SP k The constraints are as follows: for each train unit k, it is necessary to ensure that the feasible path starts from the spatiotemporal origin (i). D (k), t0(k)) to its spacetime endpoint (i D (k), t end (k) is connected end to end, as shown in formula (13).

[0056]

[0057] Where, indices i and j represent the i-th or j-th station node belonging to the station node set I, indices t and t' are the t-th and t'-th times in the time domain set T, index (j,t') represents the spatiotemporal node at station node j and time t', index (i,t) represents the spatiotemporal node at station node i and time t, and index k represents the k-th train unit belonging to the train unit set K. In particular, index (i D (k),t0(k)) represents the starting station node i of the k-th train unit. D (k) and the spatiotemporal node of the starting time t0(k), index (i D (k),t end (k) represents the depot node i of the k-th train unit. D (k) and end time t end (k) is a spatiotemporal node.

[0058] Based on the above scheme, the branch pricing algorithm described in step 4, such as Figure 6 As shown, it specifically includes:

[0059] Step 1: Based on the initial passenger demand information, using the dynamic programming algorithm, for each train unit, solve a set of train spatiotemporal paths using the dynamic programming algorithm, which serve as the initial train alternative path combinations.

[0060] Step 2: Based on the initial set of train alternative paths, use the simplex algorithm for linear programming to solve the relaxed model (denoted as RMP-LP) of the restricted master model RMP (i.e., model M1), obtaining the decision variables for the selection weight of each alternative path and the dual variables of each constraint. If all selection weight decision variables are integers, then a feasible solution to the original problem is obtained, and the global upper bound is updated.

[0061] Step 3: Based on the dual variables in Step 7.2, construct the pricing sub-model SP. For each train unit, solve the pricing sub-model SP using a dynamic programming algorithm to obtain a new train alternative route, and add the new alternative route to the alternative route set. After solving the subproblems for all train units, calculate the verification value (i.e., the objective function value of the pricing sub-model) for each sub-model. If all verification values ​​are greater than zero, the current solution is proven to be the optimal solution. Otherwise, repeat Step 7.2 iteratively based on the newly obtained alternative route set until all verification values ​​for the subproblems are positive.

[0062] Step 4: The branch pricing algorithm includes a branching strategy and a pricing strategy.

[0063] The branching strategy includes: if solving the relaxed model RMP-LP of the restricted master model RMP does not yield train path selection weight decision variables that are all integers, then the decision variable with the weight closest to 0.5 is selected, fixed to 0 or 1, and two decision branches are created. Each decision branch inherits the original restricted master model and the corresponding fixed variable values.

[0064] The pricing strategy includes: solving the pricing sub-model SP and a bounding process. For the bounding process, if the relaxed model RMP-LP that currently restricts the main model RMP is infeasible, or the objective function value of the current relaxed model is greater than the global upper bound, or a feasible integer optimal solution is obtained, then the node is not proceeded to the subsequent node branch.

[0065] The beneficial effects of this invention are as follows: Based on a spatiotemporal network model, this invention constructs a spatiotemporal network for train operation organization and passenger flow allocation, and proposes a flexible train turnaround plan decomposition and optimization method based on the spatiotemporal network. This improves the poor adaptability of existing fixed-formation train turnaround plans to dynamically unbalanced passenger flow. Furthermore, it is worth noting that the branch pricing method based on the spatiotemporal network model proposed in this invention does not rely on integer programming software, is applicable to parallel computing conditions, and can provide estimates of the upper and lower bounds of the current results. The method proposed in this invention can be extended to more collaborative optimization problems related to urban rail transit train timetables, fully exploring the redundancy capabilities generated by train transportation organization models. This is of great significance for alleviating the current problems of tight rail transit capacity, low transportation efficiency, and low operational service levels. Attached Figure Description

[0066] The present invention includes the following figures:

[0067] Figure 1 This is a diagram illustrating the decision-making process for urban rail transit operation planning.

[0068] Figure 2 It is a diagram of the formation and de-formation process of flexible trains;

[0069] Figure 3 This is a schematic diagram of the spatiotemporal nodes and spatiotemporal arcs related to the train operation organization process;

[0070] Figure 4 This is a schematic diagram of the spatiotemporal nodes and spatiotemporal arcs related to the passenger's behavior on the platform and on the train;

[0071] Figure 5 This is a schematic diagram of train safety interval constraints;

[0072] Figure 6 It is the algorithm framework for solving the problem;

[0073] Figure 7 This is an example of an urban rail transit route map;

[0074] Figure 8 These are the train timetables and rolling stock turnover plans obtained using this method;

[0075] Figure 9 These are the parameters for the train unit operation plan;

[0076] Figure 10 These are the basic parameters of the train unit;

[0077] Figure 11 This is information about each train unit;

[0078] Figure 12 This is a statistical analysis of the solution process for the branch pricing method. Detailed Implementation

[0079] The preferred embodiments are described in detail below with reference to the accompanying drawings. It should be emphasized that the following description is merely exemplary and is not intended to limit the scope of the invention or its application.

[0080] This invention proposes a novel method for decomposing and optimizing flexible train turnaround plans based on spatiotemporal networks by applying flexible train formation modes to the compilation of train timetables and rolling stock turnover plans for urban rail transit lines. The specific implementation method of the invention is described in detail below.

[0081] Step 1: Predetermine the following necessary parameters and data:

[0082] (1) Urban rail transit route map, i.e., the distribution structure of depots;

[0083] (2) Passenger flow at each station at different times;

[0084] (3) Basic train operation parameters, including train interval time, station stop time, and interval running time.

[0085] Step 2: Given the above conditions, write code in C++ or Python to build the model framework proposed in this method and obtain the corresponding train turnaround plan.

[0086] Step 3: The results can be adjusted to a certain extent according to the actual situation.

[0087] The following is a detailed explanation based on Example 1:

[0088] like Figure 7 The diagram shows a section of a double-track urban rail transit line with 3 stations, given the following conditions:

[0089] (1) This section takes an urban rail transit line as an example to verify the performance of the established model and algorithm. Figure 7The layout of this rail transit line includes three physical stations (S1 to S3) and two train depots (D1 and D2). The planning time domain is set from 0 min to 35 min, with a discrete time granularity of 1 min, including a total of 35 time nodes. In particular, stations S1, S2 and S3 can all provide train unit marshalling, demarcation and turnaround operations. Figure 9 The required parameters for the operation plan of the train unit are listed.

[0090] (2) Basic parameters of the train unit, such as Figure 10 As shown, all train units can be pre-planned to depots D1 or D2. Furthermore, regarding passenger flow, stations S2 and S3 receive passengers at a rate of 60 passengers / min in each direction from 1 to 15 minutes, while station S1 receives passengers at 60 passengers / min from 1 to 5 minutes and 11 to 15 minutes, and at 120 passengers / min from 6 to 10 minutes. This flexible train combination pattern effectively adapts to this uneven and oversaturated passenger flow distribution, improving service quality and saving operating costs.

[0091] When the relative gap reaches the threshold of 3%, the algorithm can obtain a feasible optimization solution with a lower bound objective function value of 4606.61, an upper bound value of 4735.31, and a relative gap of 2.71%. Figure 8 The train turnaround plan and passenger flow allocation results of the collaborative optimization scheme are visualized. Different colored running lines represent the spatiotemporal trajectories of different train units, and thick running lines indicate that the train journey is operated by two train units. The collaborative optimization results use a total of 4 train units. In order to meet the needs of the peak passenger flow period from 0min to 10min, train units 1 and 2 operate as double-unit trains in train journey 1, while maintaining a minimum safe interval (2min) between train journeys 1 and 2. Figure 11 The spatiotemporal path information of each train unit is statistically analyzed, including the train unit's entry and exit times from the depot, and its arrival and departure times from each station.

[0092] During peak hours (0 min–10 min), to accommodate the oversaturated passenger demand, and given that the safe interval between trains cannot be further reduced, the collaborative optimization results employ double-train unit formations for route 1, maintaining a short safe interval with route 3, thus preventing secondary waiting for passengers at stations. During periods of stable passenger flow (10 min–35 min), single-train formations are sufficient to meet passenger service needs, therefore, a large number of double-train formations are not required for operation.

[0093] The solution process and related computational performance of the branch-and-price algorithm are as follows: Figure 12As shown, each level of the branch-and-bound search tree is considered an iteration. Statistical results show that, based on the method proposed in this patent, the first feasible solution is obtained in the 5th iteration; simultaneously, an optimized solution with a relative gap reduced to 3% is obtained. During parallel computation, each iteration takes approximately 1–3 seconds, and the total running time of the optimization algorithm is 14.81 seconds.

[0094] In summary, this invention proposes a novel method for optimizing train timetables and route selection by employing the modeling concept of multi-granularity spatiotemporal networks. This method effectively balances the problem size and solution efficiency of traffic spatiotemporal networks, effectively improving upon the shortcomings of traditional methods.

[0095] It should be noted that the above description is merely one specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any person skilled in the art can easily modify and transform the proposed model within the technical scope of the present invention, and such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the various embodiments of the present invention. The scope of protection of the present invention is defined only by the appended claims.

[0096] The contents not described in detail in this specification are existing technologies known to those skilled in the art.

Claims

1. A method for decomposing and optimizing the turnaround plan of flexible train formations based on spatiotemporal networks, characterized in that, Specifically, the steps include the following: Step 1: Describe the train unit's entry and exit from the depot, marshalling, demarcation, stopping at stations, and operation between sections based on the discrete spatiotemporal network flow model; describe the passenger waiting at the platform, boarding and alighting, and boarding the train based on the continuous multi-commodity flow model; Step 2: Construct train operation organization constraints and passenger flow allocation constraints based on train unit paths; The constraints on train operation organization include: train route selection, travel coverage, and train safety interval constraints. Passenger behavior constraints include: passenger flow balance constraints on trains and platforms, total passenger disembarkation balance constraints, and train capacity constraints. Step 3: Construct a collaborative optimization model with the objective functions of minimizing the total waiting cost of passengers at the station and the total transportation cost of the train; The total transportation cost of the train includes the fixed operating cost of the train unit and the electricity cost of train operation; Step 4: Construct a pricing sub-problem for the flexible train turnaround plan collaborative optimization model based on spatiotemporal network obtained in Step 3, and design a branch pricing algorithm to solve it until a feasible and better collaborative optimization scheme is obtained.

2. The method for decomposing and optimizing the turnaround plan of flexible train formation based on spatiotemporal networks as described in claim 1, characterized in that, Step 1 specifically includes: For the train operation organization process, according to the planning time domain and accuracy requirements, the time range is discretized into a set of timestamps to form a time domain set; Based on the discrete time domain set, all physical nodes are spatiotemporally extended; specifically, based on the characteristics of the flexible train formation operation organization mode, the spatiotemporal nodes of train arrival at the station, departure from the station, the starting point of the depot, and the ending point of the depot are included. Based on the set of spatiotemporal nodes, a corresponding set of spatiotemporal arcs is constructed; specifically, these include train running arcs, waiting arcs, train depot entry / exit arcs, and turnaround arcs. For passenger behavior on the platform and on the train, the platform is spatiotemporally expanded at the same time granularity. Construct passenger station waiting nodes, virtual destinations, waiting arcs, arrival arcs, boarding arcs, and alighting arcs to describe passenger behavior on the platform and between trains; For passengers who cannot board the train on time, construct a waiting arc to describe their waiting behavior on the platform.

3. The method for decomposing and optimizing the turnaround plan of flexible train formation based on spatiotemporal networks as described in claim 1, characterized in that, The train operation organization constraints mentioned in step 2 specifically include: The train route selection constraint is as follows: For each train unit, only one route can be selected from its corresponding train unit route candidate set, as shown in the formula: ; Where: index k This represents a set of train units. K The k Train unit, index p This represents a train unit. k Corresponding train route alternatives The p Train routes; decision variables y p Represents the train route p Whether it is selected, if the path p If selected y p =1, otherwise y p =0; The travel coverage constraint is as follows: For each train operation arc in the constructed spatiotemporal network, according to the requirements of flexible train formation operation, the number of covered train units shall not exceed the coverage limit, as shown in the formula: ; Among them, index k This represents a set of train units. K The k Train unit, index p This represents a train unit. k Corresponding train route alternatives The p Train routes; index This represents the set of train running arcs. The Train running arc; parameters Represents the train route p Has it passed through a spacetime arc? If the train route p Passing through the spacetime arc but ,on the contrary ;parameter Represents the train's running arc The maximum number of times a train unit can be covered; decision variables Represents the train route p Whether it is selected, if the path p If selected ,on the contrary ; The train safety interval constraint is used to ensure that no spatiotemporal conflicts occur between train units, thereby ensuring the safe operation of the train, as shown in the formula: ; Among them, index k This represents a set of train units. K The k Train unit, index p This represents a train unit. k Corresponding train route alternatives P k The p Train routes, index k’ This represents a set of train units. K， But excluding train units k The k’ Train unit, index p’ This represents a train unit. k’ Corresponding train route alternatives The p’ Train routes; index This represents the train's running arc. The a-th spacetime arc in the incompatible arc set; parameters Represents the train route p Has it passed through a spacetime arc? If the train route p Passing through the spacetime arc but ,on the contrary Decision variables Represents the train route p Whether it is selected, if the path p If selected ,on the contrary ; To represent the incompatibility between different train running arcs, a set Represents the train's running arc The set of all incompatible arcs, which is determined by formula (4); for a train running arc in a spatiotemporal network , where index i and j Each represents a set of station nodes. I The i The and the first j Each station node; t and t’ For the train from the station i Departure and arrival stations j timestamp, T For the set of all timestamps; for any other spatiotemporal arc in the spatiotemporal network ( i , j , τ , τ' ) respectively in time τ , τ' Passing the station i and the station j If it coincides with the departure time window Or reach the side time window If there is an intersection, then the arcs are defined as incompatible and added to the set. ； ; Where parameters and Stations i To the station j The minimum safe interval between trains on the departure and arrival sides of a section; in urban rail transit systems, the minimum interval between two trains is usually set to 2 minutes; during train operation ( i , j The set of incompatible arcs of ( , 2, 3) can be represented as If the train is running in an empty arc ( i , j , 2, 3) Occupied by a certain train unit, set No spacetime arc in the system may be occupied by other trains.

4. The method for decomposing and optimizing the turnaround plan of flexible train formation based on spatiotemporal networks as described in claim 1, characterized in that, The passenger flow balance constraints on trains and platforms mentioned in step 2 specifically include: The passenger flow balance constraint on the train, based on a spatiotemporal network, is used to characterize passenger activity on the train. For any spatiotemporal network node, the difference between the total flow of trains entering the arc and waiting to exit the arc is the total disembarkation flow of the train at that network node, as shown in the formula: ; Among them, index The first group represents the set of trains waiting to exit the arc or trains entering the arc. Spacetime arc, index i This represents the set of station nodes. I The i Station nodes; index t For time domain set T The first in t A moment; a set For spatiotemporal network nodes ( i , t The train is waiting to depart from the arc and assemble. For spatiotemporal network nodes ( i , t The train's entry arc set; decision variables Represents train operation or waiting arc Number of passengers on board; variable Station Node i and time t The total number of passengers alighting; for any spatiotemporal network node, the difference between the total flow of trains leaving the arc and the total flow of trains waiting to enter the arc is the total flow of trains boarding at that network node, as shown in the formula: ; Among them, index The first group represents the set of trains waiting to exit the arc or trains entering the arc. Spacetime arc, index i This represents the set of station nodes. I The i Station nodes; index t For time domain set T The first in t A moment; a set For spatiotemporal network nodes ( i,t The train runs out of the arc assembly, assembly For spatiotemporal network nodes ( i, t (The train waiting to enter the arc set; decision variables) Represents train operation or waiting arc Number of passengers on board; variable Represents the station node i and time t The total number of passengers boarding; The passenger flow balance constraint at the station, passenger flow at station nodes i timestamp t Changes can be observed through accumulated passenger flow at the platform. The flow balance is used to represent the platform at the spatiotemporal node ( i, t The cumulative passenger flow should be equal to the spatiotemporal node. The cumulative passenger flow, minus the number of passengers boarding. And add the number of people entering the station. As shown in the formula: ； Among them, index i Represents the set of station nodes I The first in i Each node, index t For time domain set T The first in t At any given moment; decision variables Representing station nodes i exist t The cumulative number of passengers at any given time. Representing station nodes i exist t The cumulative number of passengers at time -1, variable Represents the station node i and time t Total number of passengers boarding; parameters Represents the station node i and time t The total number of passengers entering the station; The total passenger disembarkation balance constraint applies to station nodes within the planning time domain. i The sum of all disembarking passengers should equal the number of exiting passengers within the station's planned timeframe. As shown in the formula: ； Among them, index i Represents the set of station nodes I The first in i Each node, index t For time domain set T The first in t At any given moment; variables Represents the station node i and time t Total number of passengers alighting; parameters At the station node i The number of passengers exiting the station within the overall planned timeframe; The train capacity constraint is used to ensure that the number of passengers in each train unit does not exceed its maximum capacity, as shown in the formula: ； Among them, index k This represents a set of train units. K The k Train unit, index p This represents a train unit. k Corresponding train route alternatives The p Train routes, index The first in the set of spacetime arcs representing train movement and waiting A spacetime arc, a collection and These represent the sets of spatiotemporal arcs for train movement and train waiting, respectively; parameters Represents the train route p Has it passed through a spacetime arc? If the train route p Passing through the spacetime arc but ,on the contrary The parameter Cap represents the maximum passenger capacity of each train unit; decision variables Represents spacetime arc Passenger flow; decision variables Represents the train route p Whether it is selected, if the path p If selected ,on the contrary .

5. The method for decomposing and optimizing the turnaround plan of flexible train formation based on spatiotemporal networks as described in claim 3, characterized in that, Step 3 describes using the minimum total passenger waiting time cost at the station and the total train transportation cost as the objective function, i.e., minimizing the total passenger waiting time cost at the station and the total train transportation cost. Since train costs and passenger costs have different dimensions, weighting coefficients are required. α and β The two costs are normalized as shown in the formula: ； The first term represents the total cost of waiting for passengers at all station nodes at all times. α The weighting coefficient for this cost item, the second item is the sum of the fixed costs of all train units used, parameters β The weighting factor for this cost; index i Represents the set of station nodes I The first in i Each node, index t For time domain set T The first in t At that moment, index k This represents a set of train units. K The k Train unit, index p This represents a train unit. k Corresponding train route alternatives The p Train routes; variables F Let total cost be the objective function variable, and the optimization objective min is to minimize total cost; decision variables... Representing station nodes i exist t Cumulative number of passengers at any given time, decision variables y p Represents the train route p Whether it is selected, if the path p If selected ,on the contrary ;parameter C p Representative train unit alternative routes p Fixed costs.

6. The method for decomposing and optimizing the turnaround plan of flexible train formation based on spatiotemporal networks as described in claim 4, characterized in that, Step 4, which describes the pricing sub-problem of constructing a collaborative optimization model for flexible train turnaround planning based on spatiotemporal networks, specifically includes: The collaborative optimization model for flexible train turnaround planning based on spatiotemporal networks is abbreviated as M1, and model M1 is the restricted master model (RMP). A pricing sub-model SP is constructed based on the dual variable values ​​of the restricted master model RMP. The pricing sub-problem is decomposable for train units, and for each set of train units… K The k The pricing subproblem of individual train units is denoted as SP. k， Among them, the pricing sub-model SP k The objective function is shown in the formula: ； Among them, index i and j This represents the set of station nodes. I The i The or the first j Each station node, index t and t’ For time domain set T The first in t The and the first t’ Time, Index ( i, j, t, t' ) represents from the station node i of t Departure time, arrival station node j of t’ Spacetime arc of a moment, index k This represents a set of train units. K The k Each train unit, assembled This set includes all train spatiotemporal arcs, including train running arcs, waiting arcs, train depot entry / exit arcs, station stopping arcs, and connecting virtual arcs. For the set of all train running arcs, the set For the set of all train waiting arcs; decision variables Indicates train k Does it occupy the station node? i of t Departure time, arrival station node j of t’ The spacetime arc of a moment, like a train k Occupied spacetime arc ( i , j , t , t’ ), then x k i,j,t,t’ =1, otherwise x k i,j,t,t’ =0; variable Z k For the first k Objective function variables and parameters for the pricing subproblem of individual train units; c i, j, t, t’ For spacetime arc ( i, j, t, t' The cost of ) is represented by the parameter Cap, which indicates the maximum passenger capacity of each train unit. The values ​​of the dual variables of constraints (1), (2), (3), and (9) are respectively. Regarding the pricing sub-model SP k The constraints, for each train unit k To ensure a feasible path, it is necessary to start from the spatiotemporal origin. To its spacetime endpoint Connecting the beginning and end, as shown in the formula: ； Among them, index i and j This represents the set of station nodes. I The i The or the first j Each station node, index t and t’ For time domain set T The first in t The and the first t’ Time, Index ( j, t' ) represents the station node j and time t’ spatiotemporal nodes, index ( i, t ) represents the station node i and time t spatiotemporal nodes, index k This represents a set of train units. K The k Each train unit, specifically, index Representing the k The starting depot node of each train unit and starting time spatiotemporal nodes, index Representing the k Depot nodes for individual train units and end time The spatiotemporal nodes.

7. The method for decomposing and optimizing the turnaround plan of flexible train formation based on spatiotemporal networks as described in claim 3, characterized in that, The branch pricing algorithm described in step 4 specifically includes: Step 7.1: Based on the initial passenger demand information, according to the dynamic programming algorithm, for each train unit, solve a set of train spatiotemporal paths using the dynamic programming algorithm, as the initial train alternative path combination; Step 7.2: Based on the initial set of train alternative paths, use the simplex algorithm for linear programming to solve the relaxed model of the restricted master model RMP, denoted as RMP-LP, to obtain the decision variables of the selection weight of each alternative path and the dual variables of each constraint; if all selection weight decision variables are integers, then the feasible solution of the original problem is obtained, and the global upper bound is updated. Step 7.3: Based on the dual variables in Step 7.2, construct the pricing sub-model SP; for each train unit... k The pricing sub-model SP is solved using a dynamic programming algorithm. k This process yields a new alternative train route, which is then added to the alternative route set. After solving all the subproblems of the train units, the verification values ​​of each sub-model are calculated. If all verification values ​​are greater than zero, the current solution is proven to be the optimal solution. Otherwise, step 7.2 is repeated based on the newly obtained alternative route set to continue iterating until all the verification values ​​of the subproblems are positive. Step 7.4: The branch pricing algorithm includes a branching strategy and a pricing strategy; The branching strategy includes: if solving the relaxed model RMP-LP of the restricted master model RMP does not yield train path selection weight decision variables that are all integers, then select the decision variable with the weight closest to 0.5, fix it to 0 or 1, and create two decision branches, where each decision branch inherits the original restricted master model and the corresponding fixed variable values. The pricing strategy includes: solving the pricing sub-model SP and the bounding process. For the bounding process, if the current relaxed model RMP-LP that restricts the main model RMP is infeasible, or the objective function value of the current relaxed model is greater than the global upper bound, or a feasible integer optimal solution is obtained, then the node is not performed and subsequent node branches are performed.

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