CN116796915B - Proxy model method for intelligent railway route selection - Google Patents
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Abstract
The invention discloses a railway three-dimensional route selection optimization method based on a Kriging proxy model. By improving Latin hypercube sampling method, introducing maximum and minimum distance constraint, improving uniformity of sampling sample points, providing mixed worst point adding criterion, accelerating Kriging proxy model construction speed, introducing intelligent optimization algorithm to solve Kriging proxy model parameters, constructing Kriging proxy model suitable for railway three-dimensional line selection, and applying the model to line design.
Description
Technical Field
The invention relates to the field of railway three-dimensional line selection optimization, and also belongs to the field of computers and statistics, in particular to a Kriging proxy model sampling method and a point adding criterion are improved, and the Kriging proxy model sampling method and the point adding criterion are applied to a railway three-dimensional line selection optimization method.
Background
The railway route selection is a complex decision problem that a plurality of factors need to be evaluated to bring forward the specific shape of a route, the route selection design is one of very important links in the early stage of railway project construction, and the route scheme is reasonably and efficiently brought forward to influence the process and economic benefit of the railway project. Therefore, the research of the intelligent railway line selection algorithm for improving the design efficiency of railway line selection has important significance. The existing railway route selection method is mostly combined with algorithms such as a dynamic programming method, a distance changing method and a genetic algorithm to search for a route theoretical optimal scheme which is infinitely close to a real optimal scheme under multiple complex constraint conditions, but the problems of complex route constraint and heavy calculation load are faced, for example, documents [1]Wei Li,Hao Pu,Paul Schonfeld,et al.Methodology for optimizing constrained3-dimensional railway alignments in mountainous terrain [ J ]. Transportation Research Part CEmerging Technologies,2016,68:549-565. The existing railway route selection technology depends on digital ground models such as GIS and the like, and a large number of geographic entities need to be evaluated, for example, in the literature [2]Ghoreishi B,Shafahi Y,Hashemian S E.A Model for Optimizing Railway Alignment Considering Bridge Costs,Tunnel Costs,and Transition Curves[J ]. Urban Rail Transit,2019,5 (4): 207-224. However, no matter what intelligent algorithm and calculation model are adopted for intelligent route selection, the problems of large calculation load and long time consumption during target evaluation are always faced, and the calculation time of the line optimization is decomposed in the existing literature [3]KANG M W,SCHONFELD P,YANG N.Prescreening and repairing in a genetic algorithm for highway alignment optimization[J ]. Computer-Aided Civil and Infrastructure Engineering,2009,24 (2): 109-119 ], so that the conclusion is that most of the calculation time of the model is spent on a detailed evaluation program, and the total evaluation time accounts for 99.15% of the running time of the program. Therefore, a low-cost proxy model is used for replacing a real objective function with expensive calculation cost, so that the calculation load of objective evaluation is reduced, the calculation efficiency is improved, and the method has important value and research significance for railway three-dimensional route selection.
Disclosure of Invention
The invention constructs a railway three-dimensional route selection optimization method based on a Kriging proxy model. The method aims to overcome the defect of low calculation efficiency caused by the existing method due to the fact that the number of evaluation targets is large, the calculation load is large, the data volume of the triangle network interpolation search is complex and the like. When the existing intelligent algorithm solves the problem of three-dimensional route selection optimization of the railway, most of time is spent on detailed route evaluation, and the generation time of the route is less. The agent model can replace a real numerical analysis model which is complex in calculation and time-consuming in the process of optimizing design, and plays roles of greatly improving the optimizing efficiency and reducing the calculating complexity. The main idea of the Kriging proxy model is to take the dynamic construction of known sample data as a precondition, fully consider the spatial association characteristics between the data, construct the approximate function relation of the object problem, and simulate the position information of any point. The invention improves from two aspects of sampling method and dotting criterion based on the existing classical Kriging proxy model. Then, the Kriging agent model is combined with an intelligent optimization algorithm, wherein the intelligent optimization algorithm plays a role in solving super parameters in the Kriging agent model, and common intelligent optimization algorithms include a particle swarm algorithm, a genetic algorithm, an ant colony algorithm and the like.
In the aspect of sampling methods, a maximum and minimum distance algorithm is introduced into a Latin hypercube sampling method, a Latin hypercube sampling method with distance constraint is provided, a Coordinates exchange algorithm is adopted, uniformity indexes of three sampling methods in a table 1 are calculated, ten sampling experiments are carried out on the three sampling methods, and a method of averaging through multiple tests is adopted, so that the sampling result of the Latin hypercube sampling method with distance constraint is verified to be more uniform, and the Latin hypercube sampling method has better uniformity indexes, as shown in the table 1.
TABLE 1 uniformity index
The Latin hypercube sampling method with the distance constraint specifically comprises the following steps:
step 1: let t=0, t the initial number of times of Latin hypercube sampling method, t the maximum number of times of sampling max ；
Step 2, a Latin hypercube sampling method is adopted to obtain a sample X, t=t+1;
step 3, calculating the distance D between every two sample points, wherein the maximum value is D max Minimum value is D min ；
Step 4, if D min ＞λ＞D max And outputting the sampling result, ending the sampling, and otherwise, continuing the next step. Lambda is a parameter of the extremum distance constraint sampling method, and in order to obtain a proper lambda value, a formula for defining lambda is as follows:
wherein n is the dimension of the design space, m is the number of sample points, eta is the scaling factor of the value according to the actual use condition, and 0 < eta < 1. Obtaining lambda by adopting a square equipartition method, dividing the design space into 3X 3 grid patterns when 9 evenly distributed sample points are obtained in the two-dimensional design space, taking the center of each grid as a sample point, and then the ratio of the maximum distance to the minimum distance between the sample points is as follows:
when 27 sample points which are uniformly distributed are acquired in a three-dimensional design space, the design space is equally divided into a lattice diagram of 3 multiplied by 3, taking the center of each lattice as a sample point, the ratio of the maximum distance and the minimum distance between the sample points is as follows:
the formula is generalized to an n-dimensional design space, and when m sample points are obtained, the lambda calculation formula is as follows:
the lambda calculated by the formula is ideal and uniform, and the samples obtained by the Latin hypercube sampling method have certain randomness, so that a scaling coefficient eta is defined, and the lambda calculation formula is scaled.
And 5, outputting the most uniform primary sampling result if t is greater than the maximum sampling times, otherwise, returning to the step 2.
In the aspect of the point adding criterion, a mixed worst point adding criterion is provided to improve the prediction capability of the Kriging proxy model in the whole design space and accelerate the convergence capability of the Kriging proxy model.
TABLE 2 Performance evaluation index
The mixed worst point adding criterion specifically comprises the following steps:
step 1, setting a numerical value N;
step 2, defining a worst point interpolation point adding criterion as follows:
the worst point interpolation point adding criterion is a point adding criterion for adding a sample point (worst point) corresponding to the maximum value of the relative error into a training sample set based on the constructed agent model and tested by using a test set sample to calculate the relative error of a sample prediction response value and a real response value.
The calculation formula of the relative error of the sample predicted value and the true value is as follows:
the WI-point criterion uses a maximum value max (delta (x)) of the relative error smaller than a minimum value epsilon as an optimized termination condition, namely:
max(δ(x))＜ε
wherein epsilon is determined according to the specific application (e.g., epsilon=0.01 is set in the invention);
and 3, in the early stage of updating the proxy model, if the sample capacity of the training set is smaller than N, the point adding is carried out simultaneously by adopting an optimal point interpolation (OI) point adding criterion and an worst point interpolation (WI) point adding criterion, the optimal point and the worst point are added into the training sample set, and the point adding is stopped after the optimal termination conditions of the two point adding criteria are respectively reached, otherwise, the step 4 is carried out.
And 4, in the later stage of updating the proxy model, the sample capacity of the training set is larger than N, the point adding is carried out by adopting a point adding criterion of maximizing the Expected Improvement (EI), the first N sample points with the maximum expected value are added into the training sample set each time, and the point adding is stopped after the optimal termination condition is reached.
The railway three-dimensional route selection optimization method based on the Kriging proxy model specifically comprises the following steps:
step 1, determining an objective function F of a proxy model, and calculating a real response value of an initial training sample set, wherein the calculation formula of F is as follows:
F＝C c +C F +C L +C R +C B +C T
wherein C is C Is the digging cost, C F Is the filling cost, C L Is the track cost, C R Is the land solicitation cost, C B Is the bridge cost, C T Is the tunnel cost;
step 2, sampling to obtain an initial training sample set by using a Latin hypercube sampling method with distance constraint;
step 3, constructing or updating a Kriging agent model based on the training sample set and the real response value thereof;
step 4, sampling to obtain a test sample set by using a Latin hypercube sampling method with distance constraint;
step 5, obtaining a real response value of a test sample set through a functional or simulation experiment, evaluating the constructed proxy model, checking whether the optimization convergence is achieved, if so, performing the next step 6, if not, performing dotting on the training sample set by adopting a mixed worst point dotting criterion, returning to the step 3, and updating the Kriging proxy model;
step 6, initializing intelligent optimization algorithm variables by using the optimal points of the training sample set as initial solutions of the adopted intelligent optimization algorithm;
step 7, predicting a response value of the sample point by using a Kriging proxy model;
step 8, updating an intelligent optimization algorithm;
step 9, updating the response value of the sample point predicted by the Kriging proxy model, calculating the real response value of the sample point with the minimum predicted response value by using an objective function F, adding the variable into a training sample set if the relative error of the variable is larger than epsilon, updating the Kriging proxy model, otherwise, returning to the step 8;
and step 10, outputting a current optimal solution and a corresponding three-dimensional line shape if the maximum iteration number of the intelligent optimization algorithm is reached, otherwise, returning to the step 8 to continue iteration.
The calculation formula of the cost variable in the objective function F is as follows:
F＝C C +C F +C L +C R +C B +C T
(1) Excavation squareCost C C
Wherein Lc represents the number of excavated sections of the earthwork, d represents the calculated distance (unit: m) of the earthwork, c c Representing the unit price per cubic meter (unit: yuan/m) of earth and stone engineering excavation 3 )，A lc Indicating the lc-th cross section;
(2) Cost of filling C F
Lf represents the number of filling sections of the earthwork, d represents the calculated distance (unit: m) of the earthwork, and c f Representing unit price per cubic meter (unit: yuan/m) of earth and stone engineering fill 3 )，A lf Indicating the lf-th cross section;
(3) Track cost C L
C L ＝c L ·LC
Wherein c L Representing the unit price per linear meter of the track (unit: yuan/m), LC representing the total mileage length of the railway line (unit: m);
(4) Cost C for land investigation R
Wherein c R Representing the unit price per square meter (unit: yuan/m) of land solicitation 2 )，L Rl The width (unit: m) of the land solicitation representing the first cross section is calculated from the following formula:
L Rl ＝B+m·ΔH l
(5) Bridge cost C B
Wherein Q represents the number of bridges, c B Representing the unit price (unit: yuan/m) of the bridge per linear meter, L q Represents the length (unit: m) of the q-th bridge;
(6) Tunnel cost C T
Wherein T represents the number of tunnels, c T Represents the unit price per linear meter (unit: yuan/m) of the tunnel, L t The length of the q-th tunnel (unit: m) is represented.
Based on the three-dimensional railway line selection model and decision variables used by the three-dimensional railway line selection model, the constraint of the railway line is determined as follows:
(1) The number of the plane intersection points and the number of the longitudinal section slope change points are not smaller than the minimum value (N min ,M min ) Cannot be greater than the maximum value (N max ,M max )：
N min ≤N≤N max
M min ≤M≤M max
(2) The radius of the plane circular curve cannot be smaller than the radius R of the minimum circular curve min Cannot be greater than the maximum circle curve radius R max ：
R min ≤R≤R max
(3) In the complete model, the length l of the relaxation curve 0n Cannot be smaller than the allowable minimum relaxation curve length l 0min ：
l 0n ≥l 0min
(4) Length L of straight line between two adjacent curves jn Cannot be smaller than the minimum clamp straight line length L jmin ：
L jn ≥L jmin
Wherein alpha is n The steering angle indicating the nth intersection point is calculated by the following equation:
wherein d n Representing the intersection HI of planes n Intersection with plane HI n+1 The distance between them is calculated by the following formula:
(5) Grade i of longitudinal section m The absolute value of (a) cannot be greater than the limit gradient i max ：
(6) Gradient difference delta i of adjacent slope sections m Cannot be greater than the maximum longitudinal slope difference delta i max ：
Δi m ＝|i m -i m-1 |≤Δi max
(7) Length L of slope section pm Cannot be smaller than the minimum slope section length L pmin ：
L pm ＝S m+1 -S m ≥L pmin
Wherein n=1, 2, …, N; m=1, 2, …, M.
The beneficial effects of the invention are as follows:
compared with the prior art, the method has the advantages that the Kriging agent model is combined with the intelligent optimization algorithm to be applied to railway three-dimensional route selection, and can replace a real numerical analysis model which is complex in calculation and time-consuming in the target evaluation process, so that the optimization efficiency is improved, and the calculation complexity is reduced. Secondly, on the basis of a classical Kriging proxy model, improvement measures are provided from two aspects of a super-parameter solving method and a sampling method, an intelligent optimization algorithm is adopted for super-parameter solving, a maximum and minimum distance algorithm is introduced into a Latin super-cubic sampling method, a Latin super-cubic sampling method with distance constraint is provided, and experimental results show that the sampling result of the method is more uniform and has better uniformity indexes through comparing the grid sampling method and the Latin super-cubic sampling method. In addition, by testing the proposed point adding criterion on low-order, high-order, low-dimensional and high-dimensional test functions and adopting performance evaluation indexes, the Root Mean Square Error (RMSE) and the Relative Root Mean Square Error (RRMSE) are used for respectively carrying out experiments on different point adding criteria, and experimental results show that the proposed WI point adding criterion and the mixed worst point adding criterion have excellent performance on low-order, high-order, low-dimensional and high-dimensional problems.
Drawings
FIG. 1 is a schematic illustration of a Latin hypercube sampling method with distance constraints.
FIG. 2 is a flow chart of the mixed worst point plus point criterion.
Fig. 3 is a flow chart of a railway three-dimensional route selection optimization method.
Fig. 4 is a schematic diagram of an example optimization result of railway three-dimensional line selection optimization.
Detailed Description
The present invention will be further described with reference to the following examples and drawings for the purpose of making the objects, technical solutions and features of the present invention more apparent, and the exemplary embodiments and descriptions of the present invention are provided for explaining the present invention without limiting the present invention thereto.
Examples:
selecting a terrain area, wherein the terrain area mainly comprises hills, designing a line with the length of about 9 km, the maximum height difference is 130 m, the starting point coordinates of the railway line are (10914585.631, 2515177.246, 866.930), the ending point coordinates are (10910516.785, 2508738.804, 870.667), the aviation distance of the starting point is 7616.307 m, the elevation difference is 3.734 m, the average natural gradient of the ground of the line determined by the starting point is 0.49 per mill, and the adopted model parameters and cost indexes are shown in the following table.
Table 3 model parameters and algorithm parameters
TABLE 4 expense index
Step 1, determining an objective function F of a proxy model, and calculating a real response value of an initial training sample set, wherein the calculation formula of F is as follows:
F＝C c +C F +C L +C R +C B +C T
step 2, sampling to obtain an initial training sample set by using a Latin hypercube sampling method with distance constraint;
step 3, constructing or updating a Kriging agent model based on the training sample set and the real response value thereof;
step 4, sampling to obtain a test sample set by using a Latin hypercube sampling method with distance constraint;
step 5, obtaining a real response value of a test sample set through a functional or simulation experiment, evaluating the constructed proxy model, checking whether the optimization convergence is achieved, if so, performing the next step 6, if not, performing dotting on the training sample set by adopting a mixed worst point dotting criterion, returning to the step 3, and updating the Kriging proxy model;
step 6, initializing intelligent optimization algorithm variables by using the optimal points of the training sample set as initial solutions of the adopted intelligent optimization algorithm;
step 7, predicting a response value of the sample point by using a Kriging proxy model;
step 8, updating an intelligent optimization algorithm;
step 9, updating the response value of the sample point predicted by the Kriging proxy model, calculating the real response value of the sample point with the minimum predicted response value by using an objective function F, adding the variable into a training sample set if the relative error of the variable is larger than epsilon, updating the Kriging proxy model, otherwise, returning to the step 8;
and step 10, outputting a current optimal solution and a corresponding three-dimensional line shape if the maximum iteration number of the intelligent optimization algorithm is reached, otherwise, returning to the step 8 to continue iteration.
And in the early stage of updating the agent model by taking the capacity of the training sample set as 100 in the mixed worst point adding criterion, optimizing and updating the Kriging agent model by adopting the WI adding criterion and the OI adding criterion, after the sample capacity of the training set reaches 100, adding the first five maximum expected values into the training sample set each time by adopting the EI adding criterion to update the Kriging agent model, and taking the maximum EI value as a termination condition or the sample capacity of the training set as 200. Because the number of plane intersection points and the number of longitudinal section slope changing points of the line are changed, the maximum value of the number of plane intersection points is 4, and 9 decision variables (the number of plane intersection points and 8 plane guide line variables) are contained; the maximum value of the number of the longitudinal section variable slope points is 4, and 9 decision variables (the number of the longitudinal section variable slope points and 8 longitudinal section guide line variables) are included, so that the total number of the decision variables in the case is 18.
The plane guide line obtained by the Kriging agent method comprises three plane intersection points, the whole line is 8959 m, the elevation of the area where the line passes is lower, and the detour scheme is selected for plane elevation barriers such as hills and the like with higher elevation. The method is based on the secondary optimization of the Kriging proxy method, so that the trend of the plane guide line obtained by the method is consistent with that of the plane guide line of the Kriging proxy method, and the plane guide line also comprises three plane intersection points, and the mileage of the plane guide line is slightly longer than that of the plane guide line obtained by the Kriging proxy method, and the total length of the plane guide line is 9016 meters. The front half section of the plane guide line obtained by the non-agent method is consistent with the plane guide line of the Kriging agent method, but the rear half section of the plane guide line is selected to bypass the plane obstacle from the right side, so that the length of the plane guide line is the shortest, and is 8610 meters.
Table 5 comparison of optimized results
The optimal solution of the Kriging agent method is used as an initial solution of an algorithm, and is secondarily optimized, so that the total cost of the optimal solution of the Kriging agent method is 72319954.4 yuan, the line length of the solution is longer, but the line solution is more attached to the ground line, and the cost of the earth and stone side of the solution is lowest.
Claims (1)
1. A railway three-dimensional route selection optimization method based on a Kriging proxy model is characterized by comprising the following steps:
the Kriging proxy model is a semi-parameterized interpolation model based on a Gaussian random process, and new sample points are guided to be added by generating an estimated value and a mean square error of the estimated value, and then the proxy model is updated;
step 1, determining an objective function F of a proxy model, and calculating a real response value of an initial training sample set, wherein the calculation formula of F is as follows:
F＝C c +C F +C L +C R +C B +C T
wherein C is C Is the digging cost, C F Is the filling cost, C L Is the track cost, C R Is the land solicitation cost, C B Is the bridge cost, C T Is the tunnel cost;
step 2, sampling to obtain an initial training sample set by using a Latin hypercube sampling method with distance constraint;
the Latin hypercube sampling method with the distance constraint comprises the following steps of:
step i, setting the initial sampling times t=0 and the maximum sampling times t of Latin hypercube sampling method max ；
Step II, a Latin hypercube sampling method is adopted to obtain a sample X, t=t+1;
step III, calculating the distance D between every two sample points, wherein the maximum value is D max Minimum value is D min ；
Step IV, if D max ＞λ＞D min Outputting the sampling result, ending the sampling, otherwise continuing the next step;
lambda is a parameter of the extremum distance constraint sampling method, and in order to obtain a proper lambda value, a formula for defining lambda is as follows:
wherein n is the dimension of the design space, m is the number of sample points, eta is the scaling factor of the value according to the actual use condition, and 0 < eta < 1;
step V, if t is greater than the maximum sampling time t max Outputting the most uniform primary sampling result, otherwise returning to the step II;
step 3, constructing or updating a Kriging agent model based on the training sample set and the real response value thereof;
step 4, sampling to obtain a test sample set by using a Latin hypercube sampling method with distance constraint;
step 5, obtaining a real response value of a test sample set through a functional or simulation experiment, evaluating the constructed proxy model, checking whether the optimization convergence is achieved, if so, performing the next step 6, if not, performing dotting on the training sample set by adopting a mixed worst point dotting criterion, returning to the step 3, and updating the Kriging proxy model;
wherein, the mixed worst point adding rule comprises the following steps:
step I, setting the sample capacity N of a training set;
step II, defining the worst point interpolation point adding criterion as follows:
the worst point interpolation point adding criterion is a point adding criterion for adding a sample point corresponding to the maximum value of the relative error into a training sample set based on the constructed agent model and tested by using a test set sample to calculate the relative error of a sample prediction response value and a real response value;
sample prediction valueThe calculation formula of the relative error delta (x) with the true value y (x) is as follows:
the WI-point criterion uses a maximum value max (delta (x)) of the relative error smaller than a minimum value epsilon as an optimized termination condition, namely:
max(δ(x))＜ε
wherein epsilon is determined according to specific application conditions;
step III, in the early stage of updating the proxy model, if the sample capacity of the training set is smaller than N, the optimal point interpolation point adding criterion and the worst point interpolation point adding criterion are adopted to simultaneously carry out point adding, the optimal point and the worst point are added into the training sample set, and the point adding is stopped after the optimal termination conditions of the two point adding criteria are respectively reached, otherwise, the step IV is carried out;
step IV, in the later stage of updating the proxy model, the sample capacity of the training set is larger than N, the point adding is carried out by adopting a maximized expected improved point adding criterion, the first N sample points with the maximum expected value are added into the training sample set each time, and the point adding is stopped after the optimized termination condition is reached;
step 6, initializing intelligent optimization algorithm variables by using the optimal points of the training sample set as initial solutions of the adopted intelligent optimization algorithm;
step 7, predicting a response value of the sample point by using a Kriging proxy model;
step 8, updating an intelligent optimization algorithm;
based on the following railway linear constraint, updating an intelligent optimization algorithm;
(1) The number N of the plane intersection points and the number M of the longitudinal section variable slope points cannot be smaller than the minimum value N min ,M min Cannot be greater than the maximum value N max ,M max ：
N min ≤N≤N max
M min ≤M≤M max
(2) The radius R of the plane circular curve cannot be smaller than the radius R of the minimum circular curve min Cannot be greater than the maximum circle curve radius R max ：
R min ≤R≤R max
(3) In the complete model, the length l of the relaxation curve 0n Cannot be smaller than the allowable minimum relaxation curve length l 0min ：
l 0n ≥l 0min
(4) Length L of straight line between two adjacent curves jn Cannot be smaller than the minimum clamp straight line length L jmin ：
L jn ≥L jmin
Wherein alpha is n The steering angle indicating the nth intersection point is calculated by the following equation:
wherein d n Representing the intersection HI of planes n Intersection with plane HI n+1 The distance between them is calculated by the following formula:
(5) Grade i of longitudinal section m The absolute value of (a) cannot be greater than the limit gradient i max ：
|i m |≤i max
(6) Gradient difference delta i of adjacent slope sections m Cannot be greater than the maximum longitudinal slope difference delta i max ：
Δi m ＝|i m -i m-1 |≤Δi max
(7) Length L of slope section pm Cannot be smaller than the minimum slope section length L pmin ：
L pm ≥L pmin
Wherein n=1, 2, …, N; m=1, 2, …, M;
step 9, updating the response value of the sample point predicted by the Kriging proxy model, calculating the real response value of the sample point with the minimum predicted response value by using an objective function F, adding the variable into a training sample set if the relative error of the variable is larger than epsilon, updating the Kriging proxy model, otherwise, returning to the step 8;
and step 10, outputting a current optimal solution and a corresponding three-dimensional line shape if the maximum iteration number of the intelligent optimization algorithm is reached, otherwise, returning to the step 8 to continue iteration.
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