Train operation air resistance low-order equivalent method based on sensitivity analysis method

Abstract

The invention provides a train operation air resistance low-order equivalent method based on a sensitivity analysis method, and belongs to the technical field of train aerodynamic modeling and optimizing.The method comprises the steps that all parameters are selected, an influence parameter matrix is formed, and range setting is conducted; based on Monte Carlo sampling and matrix column vectors, interchanging to obtain a parameter sample matrix; calculating to obtain an output value Y matrix; performing global sensitivity analysis by using a sobol index method to obtain a parameter importance sequence; carrying out single parameter analysis; and reconstructing a tunnel piston wind model by using a single parameter analysis result to obtain a simplified tunnel piston wind model, substituting the simplified tunnel piston wind model into a train operation air resistance formula, and simplifying to obtain a simplified tunnel air resistance model. According to the method, how to realize degradation equivalence of an air resistance model in a complex scene is researched from the air resistance problem in train operation, so that engineering application requirements are met, and efficient calculation and optimization of train operation air resistance are realized.

Description

Technical Field

[0001] The invention belongs to the technical field of train aerodynamic modeling and optimization, and in particular relates to a low-order equivalent method for train running air resistance based on a sensitivity analysis method. Background Art

[0002] The existing technology proposes a non-steady flow theoretical calculation method to solve the problem of air resistance when a train passes through a tunnel. It systematically analyzes the influence of factors such as tunnel length, blocking ratio, train speed and train length on the additional air resistance, and clarifies the changing law of piston wind and additional air resistance. The research results provide theoretical support for the low-order equivalence of the refined model of air resistance under complex working conditions. However, it has the following disadvantages:

[0003] 1) High-order theoretical models have many parameters and strong nonlinear coupling, which limits their prediction accuracy and engineering applications in complex scenarios.

[0004] 2) Existing research focuses on few technologies for transforming high-order models into low-order equivalent models. The computational complexity is too high to meet the needs of real-time optimization.

[0005] 3) Not suitable for practical engineering applications: Many models fail to accurately identify key parameters, and the randomness and interaction of parameters may affect the prediction accuracy and model adaptability.

[0006] Therefore, one of the keys to improving train operation efficiency is to optimize aerodynamic performance to reduce energy consumption and improve operation stability. Summary of the invention

[0007] In view of the above-mentioned deficiencies in the prior art, the present invention provides a low-order equivalent method of train running air resistance based on sensitivity analysis method, which solves the problems of complexity of high-order models, insufficient real-time optimization capability and poor engineering adaptability in current train aerodynamic modeling.

[0008] In order to achieve the above objectives, the technical solution adopted by the present invention is: a low-order equivalent method of train running air resistance based on sensitivity analysis method, comprising the following steps:

[0009] S1. Select all parameters according to the tunnel piston wind model to form an influencing parameter matrix, and set the range of all parameters in the influencing parameter matrix;

[0010] S2, based on Monte Carlo sampling and matrix column vectors, the parameter sample matrix is ​​obtained by exchanging;

[0011] S3, substituting the parameter sample matrix into the tunnel piston wind model to obtain the output value Y matrix;

[0012] S4. Select the parameter matrix near the tunnel length threshold in the output value Y matrix, use the Sobol index method to perform global sensitivity analysis, and obtain the parameter importance ranking by calculating the first-order effect coefficient and the total effect coefficient corresponding to each parameter;

[0013] S5. Select the parameters with the highest importance, perform single parameter analysis, and obtain the single parameter analysis results;

[0014] S6. Reconstruct the tunnel piston wind model using the single parameter analysis results to obtain a simplified tunnel piston wind model;

[0015] S7. Substitute the simplified tunnel piston wind model into the train running air resistance formula and simplify it to obtain a low-order equivalent model of train running air resistance in the tunnel environment.

[0016] The beneficial effects of the present invention are as follows: the present invention uses the Sobol index method global sensitivity analysis combined with the single parameter analysis method to accurately screen the key parameters in the train tunnel air resistance model, eliminate minor factors, and greatly simplify the complex high-order model. In view of the problems of strong nonlinear coupling of parameters and low computational efficiency of traditional air resistance models under complex working conditions, a low-order equivalent method for dynamically adjusting the parameter range is proposed, which realizes the adaptability and robustness of the model under various operating conditions. By combining the influence law of piston wind with the matrix design of simplified models, the computational efficiency and engineering operability of the model are significantly improved, which makes up for the shortcomings of traditional methods in real-time optimization and complex scene prediction. Overall, the present invention has significant technical advantages in model simplification, computational efficiency improvement and practical engineering applicability.

[0017] Furthermore, the expression of the parameter sample matrix is ​​as follows:

[0018]

[0019] Where i=1,2,...,D, i represents the index of the parameter, D represents the parameter dimension, AB i represents the matrix obtained by replacing the i-th column in matrix A with the i-th column in matrix B, θ n,12 represents the element in the nth row and 12th column of matrix A, θ n ' ,i Represents the element in the nth row and the ith column of the matrix B, BA i represents the matrix obtained by replacing the i-th column in matrix B with the i-th column in matrix A, θ n,i represents the element in the nth row and ith column of matrix A, θ n ' ,12 Represents the element in the nth row and 12th column of matrix B.

[0020] The beneficial effect of the above further scheme is: by constructing AB i and BA i The matrix can effectively simulate single parameter changes to evaluate the independent impact of each parameter on the system output. This column replacement operation facilitates parameter sensitivity analysis.

[0021] Furthermore, the expression of the output value Y matrix is ​​as follows:

[0022]

[0023] Among them, Y X Represents the output value matrix calculated according to the Sobol index method, with a dimension of n'×1, where n' represents the number of sampling points, and y n' Indicates the output value corresponding to the n'th sampling point.

[0024] The beneficial effect of the above further solution is: by constructing the output value matrix Y X , can systematically express the model output results of different sampling points, providing a clear and structured result form for sensitivity analysis. The matrix form of expression is convenient for subsequent statistical analysis and effect evaluation, and helps to quickly calculate the influence of parameters on output, thereby improving analysis efficiency and accuracy.

[0025] Furthermore, step S4 includes the following steps:

[0026] S401, selecting a parameter matrix in the output value Y matrix that is near the tunnel length threshold;

[0027] S402. Perform global sensitivity analysis using the Sobol index method to calculate the first-order effect coefficient and total effect coefficient corresponding to each parameter;

[0028] S403. Based on the first-order effect coefficient and the total effect coefficient corresponding to each parameter, the global sensitivity analysis result is obtained by taking the average multiple times using the following formula:

[0029]

[0030] Among them, S i represents the first-order effect coefficient, M represents the number of experiments, j represents the index of the experiment, represents the parameter θ calculated from the jth test i The first-order effect index, S Ti represents the total effect coefficient, represents the parameter θ calculated from the jth test i The total effect index;

[0031] S404. Obtain parameter importance ranking according to the global sensitivity analysis results.

[0032] The beneficial effect of the above further scheme is: by performing multiple experiments using the Sobol index method to calculate the first-order effect index and the total effect index of the parameter, and averaging them, it is possible to comprehensively evaluate the direct and interactive impact of each parameter on the system output, thereby clarifying the importance ranking of the parameters. This method efficiently and systematically quantifies the contribution of the parameters, provides a scientific basis for model simplification and optimization, and improves the accuracy and reliability of the analysis.

[0033] Furthermore, the expressions of the first-order effect coefficient and the total effect coefficient are as follows:

[0034]

[0035] Var(Y)=Var(Y A +Y B )

[0036]

[0037]

[0038] in, represents the fixed parameter θ i When the parameter θ ～i The variance of the output value Y caused by the influence of θ～i (Y|θ i )) represents the fixed parameter θ i When the parameter θ ～i The conditional expectation of i The direct contribution to the output value Y, Var(Y) represents the total variance of the output value Y matrix, Y represents the output value matrix, Denotes the parameter θ ～i Under the influence of the condition, the parameter θ i The expected variance of , which reflects the parameter θ i The global contribution of ～i Indicates the difference parameter θ i All elements except Y A Represents the output value matrix corresponding to matrix A, Y B represents the output value matrix corresponding to matrix B, Var() represents the variance operator, n represents the total number of trials, represents the output value matrix corresponding to the j-th test sample in matrix B, It means that the corresponding output value matrix is ​​obtained after the jth test sampling by replacing the i-th column of matrix A with the i-th column of matrix B. Represents the output result of the reduced-order model corresponding to the j-th trial sampling in matrix A.

[0039] The beneficial effect of the above further scheme is that the direct and overall impact of the parameters on the output can be systematically quantified through the expressions of the first-order effect coefficient and the total effect coefficient. At the same time, based on the decomposition form and matrix operation of Var(Y), the global evaluation of parameter sensitivity is effectively achieved. This method accurately identifies key parameters and their interactions, significantly improving the scientific nature of model simplification and the efficiency of sensitivity analysis.

[0040] Furthermore, step S5 includes the following steps:

[0041] S501. Filter parameters with higher importance according to the global sensitivity analysis results;

[0042] S502, sampling Y matrix data output by the to-be-reduced model when the parameters screened out in step S501 vary individually; wherein Y represents the output value matrix calculated according to the Sobol index method;

[0043] S503, based on the sampling results, determine whether the parameter-data fitting method is to select a fixed strategy or a closest strategy. If the fixed strategy is selected, the parameter-data is fitted using a fixed fitting formula. If the closest strategy is selected, the parameter-data is fitted using the fitting formula in turn.

[0044] S504: According to the fitting result, the optimal fitting method between each parameter and the output data is obtained, and the single parameter analysis result is obtained by using the optimal fitting method.

[0045] The beneficial effect of the above further scheme is that by sampling and fitting the parameters ranked high in importance and selecting the optimal fitting method, the relationship between the parameters and the output data can be accurately established, ensuring that the main influence of the key parameters is retained during the model simplification process, while improving the accuracy and efficiency of the fitting calculation.

[0046] Furthermore, the expression of the single parameter analysis result is as follows:

[0047]

[0048] Among them, K U , K L , K l , K λ′ , K λ and K β They all represent the impact of each parameter changing individually on the output of the reduced-order model. U , C L , C l , C λ′ , C λ and C βThey represent the fitting relationship coefficients between the output of the model to be reduced when each parameter changes independently, λ represents the friction coefficient of the tunnel wall, λ′ represents the friction coefficient of the train wall, U represents the train speed, l represents the train length, L represents the tunnel length, C, δ β are constants, β is the ratio of the average cross-sectional area of ​​the train to the average cross-sectional area of ​​the tunnel, A v Represents the average cross-sectional area of ​​the train, A t Indicates the average cross-sectional area of ​​the tunnel.

[0049] The beneficial effect of the above further scheme is that through the single parameter analysis results, the fitting relationship between each parameter (such as train speed, tunnel length, etc.) and the output result and its influence are clarified. This expression form is convenient for quantifying the independent contribution of each parameter, providing a specific basis for model simplification, and optimizing the description of the influence of parameters on tunnel air resistance, making subsequent model adjustments more intuitive and accurate.

[0050] Furthermore, the simplified tunnel piston wind model is expressed as follows:

[0051] u=C u K u Uln(C x K x x+1)

[0052] C u =C U C L C l C λ′ C λ C β

[0053]

[0054] Where u represents the piston wind speed, U represents the train speed, and C x is a constant, whose value is C0C L C l , K x is a constant whose value is x represents the distance between the train and the tunnel exit, C0 represents a constant, K U , K L , K l , K λ′ , K λ and K β They represent the impact of each parameter changing individually on the output of the reduced-order model, C U , C L , C l , C λ′ , C λ and C βBoth represent the fitting relationship coefficient between each parameter and the output of the model to be reduced when it changes individually, δ β represents a constant, λ represents the friction coefficient of the tunnel wall, λ' represents the friction coefficient of the train wall, U represents the train speed, l represents the train length, L represents the tunnel length, C u is a constant whose value is the product of the fitting relationship coefficients, K u is a constant whose value is d represents the hydraulic diameter of the tunnel, and d' represents the hydraulic diameter of the train.

[0055] The beneficial effect of the above further scheme is: through the simplified tunnel piston wind model expression, the relationship between the piston wind speed and parameters such as train speed and tunnel length is clarified, which simplifies the calculation process and makes it more suitable for the rapid calculation requirements in actual engineering.

[0056] Furthermore, the expression of the low-order equivalent model of train running air resistance is as follows:

[0057]

[0058] f1(U,x)=ln 2 (C x K x x+1)U 2

[0059] f2(U,x)=U 2

[0060] f3(U,x)=ln(C x K x x+1)U 2

[0061] Among them, F represents the air resistance of train running in the matrix form under the tunnel environment, f1, f2, f3 represent the core functions that describe the influence of train speed and displacement in the tunnel on the air resistance of train running in the tunnel, φ1, φ2, φ3, φ4, φ5, φ6, φ7, φ8, φ9 represent the weights of different influencing factors in the formula, U represents the train speed, and x represents the distance between the train and the tunnel exit.

[0062] The beneficial effect of the above further scheme is that by expressing the train running air resistance model in matrix form, the influence of train speed and position on air resistance is clarified, and combined with the key weight parameter φ i and function f i The description of the influence of (U, x) on each part simplifies the computational complexity of the model, making it more suitable for practical engineering applications and fast solution requirements. BRIEF DESCRIPTION OF THE DRAWINGS

[0063] Figure 1The figure is a flow chart of the method of the present invention. DETAILED DESCRIPTION

[0064] The specific implementation modes of the present invention are described below so that those skilled in the art can understand the present invention. However, it should be clear that the present invention is not limited to the scope of the specific implementation modes. For those of ordinary skill in the art, as long as various changes are within the spirit and scope of the present invention as defined and determined by the attached claims, these changes are obvious, and all inventions and creations utilizing the concept of the present invention are protected.

[0065] Before describing the technical solution of the present invention in detail, its basic theory and core model are summarized, including the principle of the refined model of air resistance in a train tunnel and the Sobol index method.

[0066] 1. Refined model of air resistance in train tunnels

[0067] (1) Tunnel piston wind model:

[0068]

[0069] Among them, u represents the piston wind speed, a, b, c, q are all calculation coefficients and have no actual physical meaning.

[0070]

[0071] q=4ac-b 2

[0072] Among them, λ represents the friction coefficient of the tunnel wall, U represents the train speed (m / s), L represents the tunnel length (m), l represents the train length (m), and A t Indicates the average cross-sectional area of ​​the tunnel (m 2 ), A ν Indicates the average cross-sectional area of ​​the train (A ν ), C h Indicates the train head pressure difference coefficient, C t represents the pressure difference coefficient at the rear of the train, λ' represents the friction coefficient of the train wall, K e represents the pressure loss coefficient at the tunnel entrance, C1 and C2 represent constants, d represents the hydraulic diameter of the tunnel, and d' represents the hydraulic diameter of the train.

[0073] (2) Air resistance model when all trains enter the tunnel:

[0074] F=D+ΔP h A v +ΔP t A t (2)

[0075] Among them, F represents the air resistance of the train running in the tunnel environment, D represents the friction resistance of the vehicle body, and ΔP h A v Indicates the pressure difference resistance at the front of the vehicle, ΔP t A t Indicates the pressure difference resistance at the rear of the vehicle.

[0076]

[0077] Where w represents the superposition speed of the train speed U and the annular space airflow speed w', w' represents the annular space airflow speed, ΔP h Indicates the pressure difference at the front of the vehicle, ΔP t represents the pressure difference at the rear of the train, β represents the ratio of the average cross-sectional area of ​​the train to the average cross-sectional area of ​​the tunnel, C t represents the pressure difference coefficient at the rear of the train, ρ represents the air density, S t represents the average wetted perimeter of the tunnel, S v Represents the average wet cycle of the train.

[0078] Air resistance in train tunnels study assumptions:

[0079] 1) Define the annular gap between the train and the tunnel wall as the annular space;

[0080] 2) Ignore the influence of volume density changes in the tunnel, do not consider the compressibility of the gas in the dangerous passage, the physical properties of the gas such as density are the same, and the temperature of the gas at all places in the tunnel is the same, and ignore the turbulent flow area;

[0081] 3) The static pressure on the cross section of the annular space is a constant.

[0082] 2. Sobol Index Method

[0083] The core of the Sobol index method is to use the method of decomposing variance to decompose the system into single parameters and functions of the combination of parameters, and to analyze the importance of parameters and the degree of mutual influence between parameters by calculating the influence of the variance of single parameters and parameter combinations on the total variance. The specific idea is as follows.

[0084] Assume the model Y = f(X) = f(x1,x2,...,x n ), where x i ~U(0,1), we can get:

[0085]

[0086] Among them, Y and f(X) represent the mapping relationship between the input and output of the model, x nrepresents the nth parameter, i represents the index of the parameter, which is used to represent a specific single input variable, j represents another index of the input variable, which is used to represent the interaction between variables, n represents the total number of input variables, that is, the number of all parameters in the system that may affect the output, and f ij Represents the input variable x i and x j The interaction effect function describes x i and x j The common impact on the system output, x i represents the i-th input variable of the system, f i Represents the input variable x i The individual effect function of x i The impact of individual on system output, x j represents the jth input variable of the system, f 1,2,...,n represents all variables x1,x2,...,x n The high-order interaction effect function between them describes the influence of all variables on the system output. f0 represents a constant, and the integral of each added variable is 0, that is:

[0087]

[0088] Where 1≤i1<... s ≤n,1≤k≤n, Represents input variables The joint effect function of represents the i-th s input variables, Represents the integral variable of small increments.

[0089] From equations (4) and (5), we can infer that all the addition terms in equation (4) are orthogonal, that is: where i1,...,i s ≠j1,...,j t , Represents input variables The joint effect function of Represents input variables The joint effect function of x represents the input variable. Another conclusion is that the decomposition form of f(x) is unique, which can be obtained through the following process. Integrating both sides of formula (5) with respect to x can be obtained:

[0090] f0=∫f(X)dX (6)

[0091] For both sides of equation (4), except for x i Integrating all parameters except , we can get: ​

[0092]

[0093] Among them, x ～i Indicates division by x i The set of all input variables except i (x i ) represents the variable x i The individual effect function of .

[0094] For both sides of equation (4), except for x i ,x j Integrating all parameters except , we can get:

[0095]

[0096] Among them, f i,j (x i ,x j ) represents the input variable x i and x j The joint effect function, x ～{i,j} Indicates division by x i and x j The set of all input variables except j (x j ) represents the input variable x j The individual effect function of .

[0097] The functions of the decomposed parts on the right side of the equal sign in formula (4) can be derived from the higher-order terms of similar formulas. According to the Sobol index method, the total variance V of f(x) is defined as:

[0098] V = ∫f 2 (X)dX-f0 2 (9)

[0099] Among them, f 2 Represents the square value of f(X), where X represents the input variable.

[0100] According to each decomposition term in formula (4), the partial variance of a single parameter and multiple input parameters can be calculated as:

[0101] V i =∫f i 2 (x i )dx i (10)

[0102]

[0103] Among them, V i Represents a single input variable x i The partial variance, Represents input variables The joint partial variance of Represents input variables The square of the joint effect coefficient, x s represents an input variable; Represents a specific set of variables A variable in .

[0104] Where s = 1, ..., n", s represents the number of variables involved in the joint variable effect, and n" represents the total number of input variables. The total variance V represents the impact of all parameters on the system output, and the partial variance V i and They represent the impact of a single parameter on the system output and the impact of the interaction between multiple input parameters on the system. The sensitivity parameter is defined as:

[0105]

[0106] in, Represents input variables The sensitivity index is used to measure the contribution of these variables and their interactions to the system output variance. Represents input variables The joint partial variance of , V represents the total variance of the system output.

[0107] According to the above reasoning, the sum of all sensitivity indicators is 1, that is:

[0108]

[0109] Among them, S(i) represents the i-th input variable x i The first-order sensitivity index of , i represents the index of an input variable, j represents the index of another input variable, and n represents the total number of input variables in the system; S(i,j) represents the variable x i and x j The second-order sensitivity index between them; S(1,2,...,n) represents the high-order interaction sensitivity index of all n input variables, which is used to describe the common contribution of all variables to the system output variance.

[0110] According to formula (6), formula (7) and formula (8), we can get:

[0111] f0=E(Y) (14)

[0112] f i =E(Y|x i )-E(Y) (15)

[0113] f ij =E(Y|xi ,x j )-E(Y)-f i -f j (16)

[0114] Among them, E(Y) represents the expectation of the system output Y, f i Represents the input variable x i The first-order effect function of j Represents the input variable x j The first-order effect function, f ij Represents the input variable x i and x j The second-order joint effect function of .

[0115] Next, the sensitivity indicators of the Sobol index method: main effect index, total effect index and interaction effect index were analyzed.

[0116] According to mathematical statistics, the variance can be decomposed into:

[0117] V(Y)=E(V(Y|x i ))+V(E(Y|x i )) (17)

[0118] Where V(Y) represents the total variance of the system output Y, E(V(Y|x i )) indicates that at a fixed x i Input variable x i Under the condition of , the expected value of the conditional variance of the output Y, V(E(Y|x i )) represents the variance caused by the impact of the input variables alone on the output Y.

[0119] Since E(V(Y|x i )) on the uncertainty of Y. According to formula (21), the definition of sensitivity analysis index can be obtained as follows:

[0120]

[0121] in, It is the main effect indicator, also called the first-order effect indicator, which reflects the influence of a single variable on the variance of Y, and its value range is between [0,1].

[0122] Divide the input variable into X i and X -i , then E(V(Y|x -i )) means except X -i The contribution of other variables to the variance of Y. Therefore, X i The total effect index is:

[0123]

[0124] In the formula, Represents X i The main effect index and X i The effect of interactions with other variables on the variance of Y, V(E(Y|x -i )) means dividing variable x i The contribution of other variables to the variance Y.

[0125]

[0126] Among them, V(E(Y|x i x j )) represents the variable x i and x j The combined effect of two variables on the output Y, V(E(Y|x i )) represents the variable x i The individual contribution to the variance of output Y, Represents variable x i The main effect index (also called the first-order effect index) of Represents the variable x i The total effect index of the two variables is the total effect index of the two variables, which indicates the degree of influence of the main effect index of the two variables and the interaction effect between the two parameters on the variance of Y.

[0127] Based on the above train air resistance refinement model and the basic theory of the Sobol index method, the technical content of the present invention is introduced below.

[0128] like Figure 1 As shown, the present invention provides a low-order equivalent method of train running air resistance based on sensitivity analysis method, and its implementation method is as follows:

[0129] S1. Select all parameters according to the tunnel piston wind model to form an influencing parameter matrix, and set the range of all parameters in the influencing parameter matrix;

[0130] In this embodiment, all parameters are selected according to the tunnel piston wind model to be simplified to form an influencing parameter matrix, where D is the number of parameters: θ = [θ1, θ2, ..., θ D ], taking the piston wind model as an example, D = 12, θ = [U, L, l, A v ,A t ,C h ,C t ,λ′,λ,d′,d], U represents the train speed, l represents the train length, L represents the tunnel length, A v Represents the average cross-sectional area of ​​the train, A trepresents the average cross-sectional area of ​​the tunnel, λ represents the friction coefficient of the tunnel wall, λ′ represents the friction coefficient of the train wall, d represents the hydraulic diameter of the tunnel, d′ represents the hydraulic diameter of the train, and C h Indicates the train head pressure difference coefficient, C t Indicates the pressure difference coefficient at the rear of the train.

[0131] In this embodiment, the range is set as: θ i ∈[θ imin ,θ imax ], where θ i represents the i-th influencing parameter, θ imin Denotes each parameter θ i The minimum value range of θ imax Denotes each parameter θ i The maximum value range of θ represents the vector composed of various parameters. D represents the Dth influencing parameter;

[0132] S2, based on Monte Carlo sampling and matrix column vectors, the parameter sample matrix is ​​obtained by exchanging;

[0133] S3, substituting the parameter sample matrix into the tunnel piston wind model to obtain the output value Y matrix;

[0134] In this embodiment, the expression of the tunnel piston wind model is shown in formula (1).

[0135] S4. Select the parameter matrix near the tunnel length threshold in the output value Y matrix, use the Sobol index method to perform global sensitivity analysis, and obtain the parameter importance ranking by calculating the first-order effect coefficient and the total effect coefficient corresponding to each parameter;

[0136] In this embodiment, the Sobol index method is used to perform global sensitivity analysis according to equations (26) and (27) to obtain the parameter importance ranking. For example, the piston wind model parameter importance ranking is: U, L, λ', l, A t ,d′,A v ,λ,d,Ch,Ct,ρ, the implementation method is as follows:

[0137] S401, selecting a parameter matrix in the output value Y matrix that is near the tunnel length threshold;

[0138] S402. Perform global sensitivity analysis using the Sobol index method to calculate the first-order effect coefficient and total effect coefficient corresponding to each parameter;

[0139] S403. Based on the first-order effect coefficient and the total effect coefficient corresponding to each parameter, the global sensitivity analysis result is obtained by taking the average multiple times using the following formula: and

[0140] Among them, S i represents the first-order effect coefficient, M represents the number of experiments, j' represents the index of the experiment, represents the parameter θ calculated from the j'th test i The first-order effect index, S Ti represents the total effect coefficient, represents the parameter θ calculated from the j'th test i The total effect index.

[0141] S404. According to the results of global sensitivity analysis, the parameter importance ranking is obtained: U, L, λ', l, A t ,d′,A v ,λ,d,Ch,Ct,ρ.

[0142] S5. Select the parameters with the highest importance, perform single parameter analysis, and obtain the single parameter analysis results. The implementation method is as follows:

[0143] S501. Filter parameters with higher importance according to the global sensitivity analysis results;

[0144] S502, sampling Y matrix data output by the to-be-reduced model when the parameters screened out in step S501 vary individually; wherein Y represents the output value matrix calculated according to the Sobol index method;

[0145] S503, based on the sampling results, determine whether the parameter-data fitting method is to select a fixed strategy or a closest strategy. If the fixed strategy is selected, the parameter-data is fitted using a fixed fitting formula. If the closest strategy is selected, the parameter-data is fitted using the fitting formula in turn.

[0146] In this embodiment, a fixed strategy is selected, that is, a commonly used fitting method such as one of equations (28)-(33) is fixedly used to fit the parameter-output data. Where X represents the input parameter, and θ represents the parameter set of the fitting function (such as slope, intercept, etc.): f 定 =f(X,θ), where f 定 It represents the form of the commonly used fitting function used under the fixed strategy, and f() represents the function model used for fitting.

[0147] In this embodiment, the closest strategy is selected, that is, the commonly used fitting methods such as equations (28)-(33) are used in turn to fit the parameter-output data, and the fitting method with the smallest error is selected. Where Y is the actual output value: f best represents the fitting function with the smallest selection error under the closest strategy, f krepresents the kth fitting function in a set of commonly used fitting functions, and f() represents the function model used for fitting.

[0148] S504, according to the fitting results, obtaining the best fitting method between each parameter and the output data, and obtaining the single parameter analysis result by using the best fitting method;

[0149] S6. Reconstruct the tunnel piston wind model using the single parameter analysis results to obtain a simplified tunnel piston wind model;

[0150] S7. Substitute the simplified tunnel piston wind model into the train running air resistance formula and simplify it to obtain a simplified tunnel air resistance model.

[0151] In this embodiment, it should be noted that this method is versatile and can be applied to the low-order equivalent of the air resistance of train operation in other complex environments.

[0152] The present invention is further described below in conjunction with the above process.

[0153] (1) Method for establishing a simplified model of tunnel air resistance

[0154] According to the above tunnel piston wind model, the influencing parameter matrix of piston wind can be defined as:

[0155] θ=[U,L,l,A v ,A t ,C h ,C t ,ρ,λ′,λ,d′,d] (21)

[0156] Among them, θ represents the parameter matrix, which contains the selected influencing parameters.

[0157] According to formula (22), the range of all parameters in the parameter matrix is ​​set:

[0158] θ i ∈[θ imin ,θ imax ](twenty two)

[0159] Among them, θ i represents the i-th influencing parameter; θ imin Denotes each parameter θ i The minimum value range of θ imax Denotes each parameter θ i The maximum value range of . For example, L∈[0,30000] in the parameter matrix, unit (m).

[0160] The parameters are sampled based on the Monte Carlo sampling method, where the parameter dimension D'=12 and the number of sampling points is n, and two sample matrices A and B are obtained:

[0161]

[0162] Put A(:,i)=B(:,i),B(:,i)=A(:,i) to get AB i and BA i , where i = 1, 2, ..., D', A(:, i) represents the i-th column vector of matrix A, B(:, i) represents the i-th column vector of matrix B, i represents the index of the parameter, and D' represents the parameter dimension.

[0163]

[0164] Among them, AB i represents the matrix obtained by replacing the i-th column in matrix A with the i-th column in matrix B, θ n,12 represents the element in the nth row and 12th column of matrix A, θ n ' ,i Represents the element in the nth row and the ith column of the matrix B, BA i represents the matrix obtained by replacing the i-th column in matrix B with the i-th column in matrix A, θ n,i represents the element in the nth row and ith column of matrix A, θ n ' ,12 Represents the element in the nth row and 12th column of matrix B.

[0165] Substitute each set of parameter sample matrices into the piston wind calculation formula (1) to calculate the output value Y X matrix:

[0166]

[0167] Among them, Y X Represents the output value matrix calculated according to the Sobol index method, with a dimension of n'×1, where n' represents the number of sampling points, and y n' Indicates the output value corresponding to the n'th sampling point.

[0168] The section effect index and the total effect index are calculated according to the Sobol index method:

[0169]

[0170] Where, Var(Y)=Var(Y A +Y B ),

[0171] in, represents the fixed parameter θ i When the parameter θ ～i The variance of the output value Y caused by the influence of θ～i(Y|θ i )) represents the fixed parameter θ i When the parameter θ ～i The conditional expectation of i The direct contribution to the output value Y, Var(Y) represents the total variance of the output value Y, Y represents the output value matrix, Denotes the parameter θ ～i Under the influence of the condition, the parameter θ i The expected variance of , which reflects the parameter θ i The global contribution of ～i Indicates the difference parameter θ i All elements except Y A Represents the output value matrix corresponding to matrix A, Y B represents the output value matrix corresponding to matrix B, Var() represents the variance operator, n represents the total number of trials, represents the output value matrix corresponding to the j-th test sample in matrix B, It means that the corresponding output value matrix is ​​obtained after the jth test sampling by replacing the i-th column of matrix A with the i-th column of matrix B. Represents the output value matrix corresponding to the j-th trial sampling in matrix A.

[0172] The above is the theoretical derivation of the sensitivity analysis of the Sobol index method. Ten parameter matrices located near the tunnel length L = 1000 (m), L = 2000 (m), and L = 3000 (m) in the sample matrix are selected to calculate the first-order effect coefficient S corresponding to each parameter. i and the total effect coefficient S Ti , taking the average value of the data at each position to obtain the final result. In the results of the sensitivity analysis of the piston wind model, the first-order effect coefficient and the total effect coefficient of wind speed U at each position are greater than 0.6, which is the largest among all parameters, indicating that it has the greatest impact on the piston wind speed; while the train head pressure difference coefficient C h and the pressure difference coefficient C at the rear of the train t The first-order effect coefficient and the total effect coefficient at each position are less than 0.001, indicating that it has little effect on the piston wind speed.

[0173] According to the analysis results, the importance of the parameters in the piston wind model is ranked as follows: U, L, λ', l, A t, d′,A v, λ,d,Ch,Ct,ρ.

[0174] Considering the importance of each parameter and whether the value changes significantly due to environmental changes, we can get:

[0175] The main factors are: train speed, tunnel length, train length;

[0176] Important influencing factors include: vehicle body friction coefficient, tunnel wall friction coefficient, average tunnel cross-sectional area, and average train cross-sectional area.

[0177] Global sensitivity analysis can only get the ranking of the importance of different input parameters to the output, but cannot get the precise influence law, and cannot determine the specific influence of the parameter on the output. Therefore, in order to simplify the piston wind formula, a single parameter analysis is further performed based on the global sensitivity results.

[0178] The parameters with the highest importance in the global analysis results are selected for analysis, as their influence is the most significant and important. The process of single parameter analysis is to change the value of a certain parameter while keeping the values ​​of other parameters unchanged, and obtain the relationship between the parameter and the model output. The mathematical relationship is obtained by fitting. For the fitting process, the present invention provides two strategies, one is a fixed strategy, that is, one of the commonly used fitting methods is fixedly used to fit the parameter-output data; the other is a closest strategy, that is, the commonly used fitting methods are used in turn to fit the parameter-output data, and the fitting method with the smallest error is selected. Common fitting methods are as follows:

[0179] Linear Fit:

[0180] y=mx+b (28)

[0181] Quadratic fit:

[0182] y=ax 2 +bx+c (29)

[0183] Cubic fit:

[0184] y=ax 3 +bx 2 +cx+d (30)

[0185] Exponential fit:

[0186] y=ae bx (31)

[0187] Logarithmic fit:

[0188] y=alog(x)+b (32)

[0189] Power function fitting:

[0190] y=ax b (33)

[0191] In the present invention, the closest strategy is selected, and the influence law of train speed is closest to linear through analysis, and the specific form is: K U =C U U, where CU Represents a constant. By the same method, the influence rules of other influencing parameters can be obtained as follows:

[0192]

[0193] Among them, K U , K L , K l , K λ′ , K λ and K β They all represent the impact of each parameter changing individually on the output of the reduced-order model. U , C L , C l , C λ′ , C λ and C β They represent the fitting relationship coefficients between the output of the model to be reduced when each parameter changes independently, λ represents the friction coefficient of the tunnel wall, λ′ represents the friction coefficient of the train wall, U represents the train speed, l represents the train length, L represents the tunnel length, C, δ β are constants, β is the ratio of the average cross-sectional area of ​​the train to the average cross-sectional area of ​​the tunnel, A v Represents the average cross-sectional area of ​​the train, A t Indicates the average cross-sectional area of ​​the tunnel.

[0194] According to the form of piston wind, in order to facilitate the design of subsequent formulas and controllers, the initial form of the simplified piston wind formula is designed to be logarithmic, and the time variable is converted into the position variable of the train entering the tunnel. The simplified piston wind formula is:

[0195] u=C u K u Uln(C x K x x+1) (35)

[0196] Among them, C u =C U C L C l C λ′ C λ C β , C x =C0C L C l ,

[0197] Where u represents the piston wind speed, U represents the train speed, and C x is a constant, whose value is C0C L Cl , K x is a constant whose value is x represents the distance between the train and the tunnel exit, C0 represents a constant, K U , K L , K l , K λ′ , K λ and K β They represent the impact of each parameter changing individually on the output of the reduced-order model, C U , C L , C l , C λ′ , C λ and C β Both represent the fitting relationship coefficient between each parameter and the output of the model to be reduced when it changes individually, δ β represents a constant, λ represents the friction coefficient of the tunnel wall, λ' represents the friction coefficient of the train wall, U represents the train speed, l represents the train length, L represents the tunnel length, C u is a constant whose value is the product of the fitting relationship coefficients, K u is a constant whose value is d represents the hydraulic diameter of the tunnel, and d' represents the hydraulic diameter of the train.

[0198] In the above process, the present invention uses the Sobol index method sensitivity analysis and single parameter sensitivity analysis to conduct an in-depth study of the relevant parameters of the piston wind model. Through a rigorous and systematic analysis process, the core parameters that have a key impact on the model results are accurately identified, and those parameters with low contribution to the model and weak influence are successfully eliminated, thereby constructing a simplified piston wind model with a significantly reduced number of parameters. This simplified model effectively improves the computational efficiency and operability of the model while retaining key physical properties and predictive capabilities. More importantly, it provides a more efficient and accurate tool for further in-depth exploration of tunnel aerodynamic phenomena. In the subsequent content of the present invention, this simplified piston wind model will be fully utilized to establish a simplified model of tunnel air resistance.

[0199] Substituting the simplified piston wind formula u into formula (2) yields:

[0200]

[0201] in,

[0202] Where D is the body friction resistance, ρ is the air density, l is the train length, λ′ is the train wall friction resistance coefficient, S vrepresents the average wetted perimeter of the train, β represents the ratio of the average cross-sectional area of ​​the train to the average cross-sectional area of ​​the tunnel, u represents the piston wind speed, U represents the train speed, φ1, φ2, φ3 represent the weight coefficients in solving the equation for the body friction force, C x is a constant, whose value is C0C L C l , K x is a constant whose value is x represents the distance between the train and the tunnel exit.

[0203] Similarly, we can get:

[0204]

[0205] in,

[0206]

[0207] in,

[0208] Where ΔP t Represents the pressure difference at the rear of the vehicle, ΔP h A v Indicates the pressure difference resistance of the front of the vehicle, S v represents the average wetted perimeter of the train, ρ represents the air density, β represents the ratio of the average cross-sectional area of ​​the train to the average cross-sectional area of ​​the tunnel, φ4, φ5, φ6 represent the weight coefficients in solving the pressure difference equation at the rear of the train, φ7, φ8, φ9 represent the weight coefficients in solving the pressure difference equation at the rear of the train, C h Indicates the train head pressure difference coefficient, C t Indicates the pressure difference coefficient at the rear of the train.

[0209] For the subsequent controller design, the tunnel resistance formula is rewritten into matrix form:

[0210]

[0211] Where f1(U,x)=ln 2 (C x K x x+1)U 2 , f2(U,x)=U 2 ,f3(U,x)=ln(C x K x x+1)U 2 ;

[0212] Among them, F represents the air resistance of train running in the tunnel environment in matrix form (the only unknown variables are the train speed and the displacement of the train in the tunnel), f1, f2, f3 represent the core functions that describe the influence of train speed and displacement in the tunnel on the air resistance of train running in the tunnel, φ1, φ2, φ3, φ4, φ5, φ6, φ7, φ8, φ9 represent the weights of different influencing factors in the formula, U represents the train speed, and x represents the distance between the train and the tunnel exit.

[0213] In the simplification process, the simplified piston wind formula is first substituted into the tunnel air resistance formula, and the minor parameters with little effect on the resistance are eliminated, and then the model is expressed in a matrix form that is convenient for subsequent analysis and controller design. The final model not only retains the key physical characteristics of tunnel air resistance, but also effectively improves the calculation efficiency and model adaptability, providing a more efficient calculation tool for train aerodynamic optimization and practical engineering applications.

[0214] In summary, in order to solve the above problems, the present invention proposes a simplified air resistance modeling method based on the combination of global sensitivity analysis of the Sobol index method and single parameter sensitivity analysis. Through in-depth analysis of the aerodynamic characteristics under tunnel operation conditions, the key parameters are identified by global sensitivity analysis of the Sobol index method, and combined with single parameter sensitivity analysis, the influence of each parameter on the model is accurately quantified, and minor influencing factors are eliminated, thereby greatly reducing the complexity of the high-order air resistance model. In order to further improve the engineering applicability of the model, the present invention introduces a dynamic adjustment mechanism, flexibly adjusts the parameter range based on different operating conditions, and improves the adaptability and robustness of the model. In practical applications, the adjusted low-order air resistance model can not only accurately reflect complex aerodynamic characteristics, but also significantly improve the calculation efficiency. It is suitable for complex conditions of tunnel operation and realizes efficient calculation and optimization of air resistance. The present invention takes the optimization of air resistance in tunnel operation as the core goal, helps train energy saving and improves operation stability, thereby providing an efficient and accurate technical tool for the optimization of train aerodynamics under complex operating environments.

Claims

1. A low-order equivalent method for train running air resistance based on sensitivity analysis method, characterized in that: The following steps are involved: S1. Select all parameters according to the tunnel piston wind model to form an influencing parameter matrix, and set the range of all parameters in the influencing parameter matrix; S2, based on Monte Carlo sampling and matrix column vectors, the parameter sample matrix is ​​obtained by exchanging; S3, substituting the parameter sample matrix into the tunnel piston wind model to obtain the output value Y matrix; S4. Select the parameter matrix near the tunnel length threshold in the output value Y matrix, use the Sobol index method to perform global sensitivity analysis, and obtain the parameter importance ranking by calculating the first-order effect coefficient and the total effect coefficient corresponding to each parameter; S5. Select the parameters with the highest importance, perform single parameter analysis, and obtain the single parameter analysis results; S6. Reconstruct the tunnel piston wind model using the single parameter analysis results to obtain a simplified tunnel piston wind model; S7. Substitute the simplified tunnel piston wind model into the train running air resistance formula and simplify it to obtain a low-order equivalent model of train running air resistance in the tunnel environment.

2. The low-order equivalent method of train running air resistance based on sensitivity analysis method according to claim 1 is characterized in that: The expression of the parameter sample matrix is ​​as follows: Where i = 1, 2, ..., D', i represents the index of the parameter, D' represents the parameter dimension, AB i represents the matrix obtained by replacing the i-th column in matrix A with the i-th column in matrix B, θ n,12 represents the element in the nth row and 12th column of matrix A, θ n ' ,i Represents the element in the nth row and the ith column of the matrix B, BA i represents the matrix obtained by replacing the i-th column in matrix B with the i-th column in matrix A, θ n,i represents the element in the nth row and ith column of matrix A, θ n ' ,12 Represents the element in the nth row and 12th column of matrix B.

3. The low-order equivalent method of train running air resistance based on sensitivity analysis method according to claim 1 is characterized in that: The expression of the output value Y matrix is ​​as follows: Among them, Y X Represents the output value matrix calculated according to the Sobol index method, with a dimension of n'×1, where n' represents the number of sampling points, and y n' Indicates the output value corresponding to the n'th sampling point.

4. The low-order equivalent method of train running air resistance based on sensitivity analysis method according to claim 1 is characterized in that: The step S4 comprises the following steps: S401, selecting a parameter matrix in the output value Y matrix that is near the tunnel length threshold; S402. Perform global sensitivity analysis using the Sobol index method to calculate the first-order effect coefficient and total effect coefficient corresponding to each parameter; S403. Based on the first-order effect coefficient and the total effect coefficient corresponding to each parameter, the global sensitivity analysis result is obtained by taking the average multiple times using the following formula: Among them, S i represents the first-order effect coefficient, M represents the number of experiments, j' represents the index of the experiment, represents the parameter θ calculated from the j'th test i The first-order effect index, S Ti represents the total effect coefficient, represents the parameter θ calculated from the j'th test i The total effect index; S404. Obtain parameter importance ranking according to the global sensitivity analysis results.

5. The low-order equivalent method of train running air resistance based on sensitivity analysis method according to claim 4 is characterized in that: The expressions of the first-order effect coefficient and the total effect coefficient are as follows: Var(Y)=Var(Y A +Y B ) in, represents the fixed parameter θ i When the parameter θ ～i The variance of the output value Y caused by the influence of θ～i (Y|θ i )) represents the fixed parameter θ i When the parameter θ ～i The conditional expectation of i The direct contribution to the output value Y, Var(Y) represents the total variance of the output value Y matrix, Y represents the output value matrix, Denotes the parameter θ ～i Under the influence of the condition, the parameter θ i The expected variance of , which reflects the parameter θ i The global contribution of ～i represents the difference parameter θ i All elements except Y A Represents the output value matrix corresponding to matrix A, Y B represents the output value matrix corresponding to matrix B, Var() represents the variance operator, n represents the total number of trials, represents the output value matrix corresponding to the j-th test sample in matrix B, It means that the corresponding output value matrix is ​​obtained after the jth test sampling by replacing the i-th column of matrix A with the i-th column of matrix B. Represents the output result of the reduced-order model corresponding to the j-th trial sampling in matrix A.

6. The low-order equivalent method of train running air resistance based on sensitivity analysis method according to claim 1 is characterized in that: The step S5 comprises the following steps: S501. Filter parameters with higher importance according to the global sensitivity analysis results; S502, sampling Y matrix data output by the to-be-reduced model when the parameters screened out in step S501 vary individually; wherein Y represents the output value matrix calculated according to the Sobol index method; S503, based on the sampling results, determine whether the parameter-data fitting method is to select a fixed strategy or a closest strategy. If the fixed strategy is selected, the parameter-data is fitted using a fixed fitting formula. If the closest strategy is selected, the parameter-data is fitted using the fitting formula in turn. S504: According to the fitting result, the optimal fitting method between each parameter and the output data is obtained, and the single parameter analysis result is obtained by using the optimal fitting method.

7. The low-order equivalent method of train running air resistance based on sensitivity analysis method according to claim 6 is characterized in that: The expression of the single parameter analysis result is as follows: Among them, K U , K L , K l , K λ′ , K λ and K β They all represent the impact of each parameter changing individually on the output of the reduced-order model. U , C L , C l , C λ′ , C λ and C β They represent the fitting relationship coefficients between the output of the model to be reduced when each parameter changes independently, λ represents the friction coefficient of the tunnel wall, λ′ represents the friction coefficient of the train wall, U represents the train speed, l represents the train length, L represents the tunnel length, C, δ β are constants, β is the ratio of the average cross-sectional area of ​​the train to the average cross-sectional area of ​​the tunnel, A v Represents the average cross-sectional area of ​​the train, A t Indicates the average cross-sectional area of ​​the tunnel.

8. The low-order equivalent method of train running air resistance based on sensitivity analysis method according to claim 1 is characterized in that: The simplified tunnel piston wind model is expressed as follows: u=C u K u Uln(C x K x x+1) C u =C U C L C l C λ′ C λ C β Where u represents the piston wind speed, U represents the train speed, and C x is a constant, whose value is C0C L C l , K x is a constant whose value is x represents the distance between the train and the tunnel exit, C0 represents a constant, K U , K L , K l , K λ′ , K λ and K β They represent the impact of each parameter changing individually on the output of the reduced-order model, C U , C L , C l , C λ′ , C λ and C β Both represent the fitting relationship coefficient between each parameter and the output of the model to be reduced when it changes individually, δ β represents a constant, λ represents the friction coefficient of the tunnel wall, λ' represents the friction coefficient of the train wall, U represents the train speed, l represents the train length, L represents the tunnel length, C u is a constant whose value is the product of the fitting relationship coefficients, K u is a constant whose value is d represents the hydraulic diameter of the tunnel, and d' represents the hydraulic diameter of the train.

9. The low-order equivalent method of train running air resistance based on sensitivity analysis method according to claim 8 is characterized in that: The expression of the low-order equivalent model of the train running air resistance is as follows: f1(U,x)=ln 2 (C x K x x+1)U 2 f2(U,x)=U 2 f3(U,x)=ln(C x K x x+1)U 2 Among them, F represents the air resistance of train running in the matrix form under the tunnel environment, f1, f2, f3 represent the core functions that describe the influence of train speed and displacement in the tunnel on the air resistance of train running in the tunnel, φ1, φ2, φ3, φ4, φ5, φ6, φ7, φ8, φ9 represent the weights of different influencing factors in the formula, U represents the train speed, and x represents the distance between the train and the tunnel exit.

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