A low-order equivalent method for train running air resistance based on sensitivity analysis

Abstract

The present invention provides a low-order equivalent method for train running air resistance based on sensitivity analysis method, which belongs to the field of train aerodynamic modeling and optimization technology. The method includes selecting all parameters to form an influencing parameter matrix and setting the range; based on Monte Carlo sampling and matrix column vectors, exchanging to obtain a parameter sample matrix; calculating the output value Y matrix; using the Sobol index method to perform global sensitivity analysis to obtain parameter importance ranking; and performing single parameter analysis; using the single parameter analysis results to reconstruct the tunnel piston wind model to obtain a simplified tunnel piston wind model, and substituting it into the train running air resistance formula to simplify and obtain a simplified tunnel air resistance model. Starting from the air resistance problem in train operation, the present invention studies how to achieve the downgraded equivalence of the air resistance model in complex scenarios to meet the needs of engineering applications and to achieve efficient calculation and optimization of train running air resistance.

Description

Technical Field

[0001] The present 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] Existing technology addresses the issue of air resistance when trains pass through tunnels by proposing a non-steady flow theory calculation method. This method systematically analyzes the impact of factors such as tunnel length, blockage ratio, train speed, and train length on additional air resistance, and clarifies the changing patterns of piston wind and additional air resistance. This research provides theoretical support for the low-order equivalent of refined models of air resistance under complex operating conditions. However, this method has the following shortcomings:

[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 lacks technologies for converting high-order models into low-order equivalent models, and the computational complexity is too high to meet real-time optimization requirements.

[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 prediction accuracy and model adaptability.

[0006] Therefore, one of the keys to improving train operation efficiency lies in optimizing aerodynamic performance to reduce energy consumption and improve operational stability. Summary of the Invention

[0007] In response to the above-mentioned deficiencies in the prior art, the present invention provides a low-order equivalent method for train running air resistance based on sensitivity analysis, which solves the problems of high-order model complexity, insufficient real-time optimization capabilities 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 ranges for 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. Substitute 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, perform a global sensitivity analysis using the Sobol index method, 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 ranked high in 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 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 for global sensitivity analysis combined with a 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 response to the problems of strong nonlinear coupling of parameters and low computational efficiency in 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 a simplified model, the computational efficiency and engineering operability of the model are significantly improved, making 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 ith column of 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 at row n and column 12 in matrix B.

[0020] The beneficial effect of the above further solution 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 , which can systematically express model output results at different sampling points, providing a clear and structured form of results for sensitivity analysis. Matrix representation facilitates subsequent statistical analysis and effect assessment, helping to quickly calculate the impact of parameters on outputs, 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 trial i The first-order effect index, S Ti represents the total effect coefficient, represents the parameter θ calculated from the jth trial i The total effect index;

[0031] S404. Obtain parameter importance ranking based on the global sensitivity analysis results.

[0032] The beneficial effect of this further approach is that by performing multiple experiments using the Sobol index method to calculate the first-order effect index and the total effect index of the parameters, and then averaging these, 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 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 the parameter θ i 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, represents the parameter θ ～i Under the influence of the condition, the parameter θ i The expected variance of θ reflects the parameter θ i The global contribution of θ ～i represents the 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, and n represents the total number of trials. Represents the output value matrix corresponding to the j-th trial sampling in matrix B, Represents the corresponding output value matrix obtained after the jth trial 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 this further approach is that, through expressions for first-order and total effect coefficients, we can systematically quantify the direct and overall impact of parameters on the output. Furthermore, by using the decomposition form of Var(Y) and matrix operations, we effectively achieve a global assessment of parameter sensitivity. This approach 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 ranked high in importance based on the results of the global sensitivity analysis.

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

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

[0044] S504: Obtain the optimal fitting method between each parameter and the output data according to the fitting result, and obtain the single parameter analysis result 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 resistance coefficient of the tunnel wall, λ′ represents the friction resistance 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, and A v Indicates the average cross-sectional area of ​​the train, A t Indicates the average cross-sectional area of ​​the tunnel.

[0049] The beneficial effect of this further approach is that, through the results of single-parameter analysis, the fitting relationship between various parameters (such as train speed and tunnel length) and the output results, as well as their degree of influence, is clearly defined. This representation facilitates quantification of the independent contribution of each parameter, providing a concrete basis for model simplification. It also optimizes the description of the parameter's impact on tunnel air resistance, making subsequent model adjustments more intuitive and precise.

[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, its 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 effects 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 βThey all represent the fitting relationship coefficient between each parameter when it changes individually and the output of the model to be reduced, 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 that 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 fast calculation needs 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] Where F represents the air resistance of train operation in the tunnel environment in matrix form, f1, f2, and f3 represent the core functions that describe the effects of train speed and displacement in the tunnel on the air resistance of train operation in the tunnel, φ1, φ2, φ3, φ4, φ5, φ6, φ7, φ8, and 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 solution 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 1Flow chart of the method of the present invention. DETAILED DESCRIPTION

[0064] The specific embodiments of the present invention are described below to facilitate understanding of the present invention by those skilled in the art. However, it should be clear that the present invention is not limited to the scope of the specific embodiments. For those skilled in the art, as long as various changes are within the spirit and scope of the present invention as defined and determined by the appended claims, these changes are obvious, and all inventions and creations utilizing the concepts 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 principles 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 Tunnel

[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] Where λ is the friction coefficient of the tunnel wall, U is the train speed (m / s), L is the tunnel length (m), l is 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 are 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 train 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 Indicates the average wet perimeter of the train.

[0078] Assumptions for the study of air resistance in train tunnels:

[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 such as gas density are the same, and the temperature of the gas at all points in the tunnel is the same, and ignore the flow zone;

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

[0082] 2. Sobol Index Method

[0083] The core of the Sobol index method is to use variance decomposition to decompose the system into individual parameters and functions of their combinations. By calculating the impact of the variance of individual parameters and their combinations on the total variance, the importance of parameters and the degree of mutual influence between parameters are analyzed. The specific concept is as follows.

[0084] Assume that 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, f ij Represents the input variable x i and x j The interaction effect function of 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 A separate effect function describing x i The impact on system output alone, 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, Indicates the integral variable 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, and this decomposition form 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 influence 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 the 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 methods, 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 x is fixed 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 effect 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 degree of influence of a single variable on the variance of Y, and its value range is between [0,1].

[0122] Divide the input variables 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. i The full effect index is:

[0123]

[0124] Where, 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 contribution of the individual to the variance of the output Y, Represents the 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 ; it represents 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 refined model of train air resistance 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 for 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 ranges for 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 Indicates 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, θ 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. Substitute 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, perform a global sensitivity analysis using the Sobol index method, 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 trial i The first-order effect index, S Ti represents the total effect coefficient, represents the parameter θ calculated from the j'th trial 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 ranked high in importance and perform single parameter analysis to obtain the single parameter analysis results. The implementation method is as follows:

[0143] S501. Filter parameters ranked high in importance based on the results of the global sensitivity analysis.

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

[0145] S503. Based on the sampling results, determine whether to select a fixed strategy or a closest approach strategy for parameter-data fitting. If the fixed strategy is selected, the parameter-data is fitted using a fixed fitting formula. If the closest approach 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 fixed fitting method such as one of the common fitting methods (28)-(33) is 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 common 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 that minimizes the 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: Obtain the optimal fitting method between each parameter and the output data according to the fitting result, and obtain the single parameter analysis result using the optimal 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 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 will be 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 ith column of 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 at row n and column 12 in 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] Calculate the section effect index and the total effect index 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 the parameter θ i Direct contribution to the output value Y, Var(Y) represents the total variance of the output value Y, Y represents the output value matrix, represents the parameter θ ～i Under the influence of the condition, the parameter θ i The expected variance of θ reflects the parameter θ i The global contribution of θ ～i represents the 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, and n represents the total number of trials. Represents the output value matrix corresponding to the j-th trial sampling in matrix B, Represents the corresponding output value matrix obtained after the jth trial 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 sample 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 , take the average value of the data at each location to get 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 location are both 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 train tail pressure difference coefficient C 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 its value changes significantly due to environmental changes, we can get:

[0175] The main factors are: train speed, tunnel length, and 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 rank the importance of different input parameters to the output, but cannot accurately determine the influence pattern or the specific impact of each 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. This mathematical relationship is obtained by fitting. For the fitting process, the present invention provides two strategies, one is a fixed strategy, that is, fixedly using one of the commonly used fitting methods to fit the parameter-output data; the other is a closest strategy, that is, using the commonly used fitting methods in turn to fit the parameter-output data, and selecting the fitting method with the smallest error. 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 fitting:

[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. Through analysis, it is found that the influence of train speed is closest to linearity. The specific form is: K U =C U U, where CU Represents a constant, and the same method can be used to obtain the influence of other influencing parameters:

[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 resistance coefficient of the tunnel wall, λ′ represents the friction resistance 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 represents the ratio of the average cross-sectional area of ​​the train to the average cross-sectional area of ​​the tunnel, A v Indicates the average cross-sectional area of ​​the train, A t Indicates the average cross-sectional area of ​​the tunnel.

[0194] Based on 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, its 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 effects 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 They all represent the fitting relationship coefficient between each parameter when it changes individually and the output of the model to be reduced, 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, its 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 Indicates the pressure difference at the rear of the vehicle, ΔP h A v Indicates the pressure difference resistance of the vehicle head, 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] Where F represents the air resistance of the train running in the tunnel environment in matrix form (the only unknown variables are the train speed and the train displacement in the tunnel), f1, f2, and f3 are core functions that describe the influence of the train speed and the displacement in the tunnel on the air resistance of the train running in the tunnel, φ1, φ2, φ3, φ4, φ5, φ6, φ7, φ8, and 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] During the simplification process, the simplified piston wind formula was first substituted into the tunnel air resistance formula, eliminating minor parameters with minimal impact on resistance. The model was then expressed in a matrix form that facilitates subsequent analysis and controller design. The resulting model not only retains the key physical properties of tunnel air resistance but also significantly improves computational efficiency and model adaptability, providing a more efficient computational 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 operating conditions, the Sobol index method global sensitivity analysis is used to identify key parameters, 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 to flexibly adjust the parameter range based on different operating conditions to improve 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 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 its core goal, helps to improve energy saving and operation stability of trains, and thus provides 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 by: The following steps are involved: S1. Select all parameters according to the tunnel piston wind model to form an influencing parameter matrix, and set ranges for all parameters in the influencing parameter matrix; S2. Use Monte Carlo to sample the parameters in the influencing parameter matrix and obtain the matrix , and the matrix The column vectors of are exchanged to obtain the parameter sample matrix; S3. Substitute 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, perform a global sensitivity analysis using the Sobol index method, 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 ranked high in 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 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: in, , i Indicates the index of the parameter, represents the parameter dimension, Indicates that the matrix The Column matrix No. The matrix obtained after column replacement is, Representation matrix Middle The element in row 1 and column 12, Representation matrix Middle Row, No. Elements of the column, Indicates that the matrix The Column matrix No. The matrix obtained after column replacement is, Representation matrix Middle Row, No. Elements of the column, Representation matrix Middle The element at row 1 and column 12.

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: in, Represents the output value matrix calculated according to the Sobol index method, with dimensions of , represents the number of sampling points, Indicates the The output value corresponding to each 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: in, represents the first-order effect coefficient, represents the number of experiments, represents the index of the experiment, Indicates the Parameters calculated from the test The first-order effect index, represents the total effect coefficient, Indicates the Parameters calculated from the test The total effect index; S404. Obtain parameter importance ranking based on 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: in, Indicates fixed parameters When the parameter The output value generated by the influence of The variance of Indicates fixed parameters When the parameter The conditional expectation of For output value direct contribution, Indicates the output value The total variance of the matrix, represents the output value matrix, Representation parameters Under the conditions of influence, the parameters The expected variance of The global contribution of Indicates the division parameter All elements except Representation matrix The corresponding output value matrix, Representation matrix The corresponding output value matrix, represents the variance operator, represents the total number of trials, Representation matrix Middle The output value matrix corresponding to the experimental sampling is, Indicates that the matrix No. Columns replaced by matrices No. The matrix obtained by column The corresponding output value matrix obtained after the trial sampling is: Representation matrix Middle The output results of the reduced-order model corresponding to the experimental sampling.

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 ranked high in importance based on the results of the global sensitivity analysis. S502, when the parameters selected in step S501 are changed individually, the output of the reduced-order model Matrix data is sampled; where Represents the output value matrix calculated according to the Sobol index method; S503. Based on the sampling results, determine whether to select a fixed strategy or a closest approach strategy for parameter-data fitting. If the fixed strategy is selected, the parameter-data is fitted using a fixed fitting formula. If the closest approach strategy is selected, the parameter-data is fitted using the fitting formula in turn. S504: Obtain the optimal fitting method between each parameter and the output data according to the fitting result, and obtain the single parameter analysis result using the optimal fitting method.

7. The low-order equivalent method for 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: in, 、 、 、 、 and They all represent the impact of each parameter changing individually on the output of the reduced-order model. 、 、 、 、 and Respectively represent the fitting relationship coefficient between each parameter when it changes independently and the output of the model to be reduced, represents the tunnel wall friction resistance coefficient, represents the train wall friction resistance coefficient, Indicates the train speed, represents the length of the train, Indicates the tunnel length, are constants, It represents the ratio of the average cross-sectional area of ​​the train to the average cross-sectional area of ​​the tunnel. represents the average cross-sectional area of ​​the train, 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 7 is characterized in that: The simplified tunnel piston wind model is expressed as follows: in, Indicates piston wind speed, Represents a constant whose value is , Represents a constant whose value is , Indicates the distance between the train and the tunnel exit. represents a constant, Represents a constant, whose value is the product of the fitting relationship coefficients. Represents a constant whose value is , represents the hydraulic diameter of the tunnel, Indicates the hydraulic diameter of the train.

9. The low-order equivalent method for 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: in, The matrix represents the air resistance of the train running in the tunnel environment. 、 、 represents the core function that describes the influence of train speed and displacement in the tunnel on the air resistance of the train running in the tunnel, 、 、 、 、 、 、 、 、 Indicates the weights of different impact factors in the formula.

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