Neutron transport constitutive relation and low-dimensional control equation modeling method based on sparse regression

Abstract

The application discloses a neutron transport constitutive relation and low-dimensional control equation modeling method based on sparse regression, which comprises the following steps: a neutron flow matrix under multiple state parameters is formed by state parameters of each state and physical quantity parameters calculated through a neutron transport program; fitting physical quantity parameters are filled into a sparse regression physical quantity matrix; a coefficient matrix is obtained by using a least square method to calculate an inverse matrix; and a neutron transport constitutive relation under multiple state parameters is obtained by assembling. The application utilizes the accurate characteristics of the calculation result of the neutron transport equation, grasps the cause of the larger deviation of the calculation result of the neutron diffusion equation in a small scale range, combines sparse regression fitting to obtain the neutron transport constitutive relation, and further obtains a low-dimensional macroscopic neutron transport control equation. The theoretical derivation is less difficult, the research progress is easier, the effect of improving the calculation precision is more obvious, the applicability is better, the gap with the situation under the actual reactor core working condition is not large, and the application has certain practicality.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of nuclear reactor core calculation method, in particular to a modeling method of neutron transport constitutive relation and low-dimensional control equation based on sparse regression. BACKGROUND

[0002] The current nuclear reactor core calculation method is mainly divided into neutron transport equation and neutron diffusion equation; among them, the neutron transport equation has wide applicability and accurate calculation results, but there is no detailed constitutive relation between the parameters such as neutron flow and neutron flux; and the constitutive relation of neutron diffusion equation, that is, Fick's law, also has limitations, so that the calculation results of neutron diffusion equation are larger than the calculation results of neutron transport equation in small scale problems; therefore, how to obtain the constitutive relation between the parameters such as neutron flow and neutron flux in neutron transport equation in macroscopic scale, and establish the corresponding low-dimensional control equation, will have great practical significance and application value.

[0003] And the existing modeling method of neutron transport constitutive relation and low-dimensional control equation mostly starts from the basic theory, and tries to derive the constitutive relation between the parameters such as neutron flow and neutron flux in theory, so as to establish the low-dimensional control equation, for example: the more accurate constitutive relation between neutron flow and neutron flux is proposed according to the integral form of neutron transport equation, but its limitations and shortcomings are as follows:

[0004] 1. It is difficult to theoretically derive the constitutive relation between the parameters such as neutron flow and neutron flux, the research progress is difficult, and the effect of applying the constitutive relation obtained by derivation to the low-dimensional control equation to improve the calculation accuracy is not significant.

[0005] 2. The applicability of the constitutive relation derived is poor, and many research constitutive relations are established under various assumptions or ideal conditions, which is far from the actual core working condition, and does not have practicality.

[0006] Therefore, the prior art still needs to be improved and developed. SUMMARY

[0007] To solve the above technical problems, the present application provides a modeling method of neutron transport constitutive relation and low-dimensional control equation based on sparse regression, which has smaller theoretical derivation difficulty, easier research and application, more obvious effect of improving calculation accuracy, better applicability and certain practicality.

[0008] The technical scheme of the present application is as follows: the modeling method of neutron transport constitutive relation and low-dimensional control equation based on sparse regression is composed of the following steps:

[0009] Step S110: Given the reactor state parameters, calculate the physical quantity parameters under different state parameters using the neutron transport program, and combine the state parameters and physical quantity parameters of each state to form a neutron flux matrix J under multiple state parameters. matrix ;

[0010] Step S120: Combining neutron transport theory, construct the fitting physical quantity parameters, and simultaneously build the sparse regression physical quantity matrix x of the neutron transport constitutive relation. i_matrix And fit the physical quantity parameter x i Fill into the sparse regression physical quantity matrix x i_matrix middle;

[0011] Step S130: Calculate the neutron flow matrix J under multiple state parameters. matrix Let J be the target matrix. matrix =x i_matrix* a i_matrix The inverse matrix is ​​calculated using the sparse regression fitting method, resulting in the coefficient matrix a. i_matrix And by assembling, a neutron flux J is obtained for the constitutive relations of neutron transport under multiple state parameters. pred ;

[0012] Step S140: Neutron stream J pred Substituting the expression into the neutron diffusion equation and replacing the neutron flux J in the neutron diffusion equation, we obtain the low-dimensional macroscopic neutron transport control equation.

[0013] The method for modeling neutron transport constitutive relations and low-dimensional governing equations based on sparse regression, wherein step S110 consists of the following steps:

[0014] Step S111: Based on the given typical operating conditions of the reactor, select the range of key state parameters, and for each state parameter, select N within the upper and lower limits of the reasonable operating range. i There are several parameters, where i represents each state parameter;

[0015] Step S112: Calculate the physical quantity parameters under different state parameters using the neutron transport procedure, and assemble each state parameter and physical quantity parameter into N. grid ×N i A matrix of N rows and 1 column. grid Indicates the number of grid cells.

[0016] The aforementioned method for modeling neutron transport constitutive relations and low-dimensional governing equations based on sparse regression, wherein the key state parameters in step S111 include at least the total cross section Σ. t , scattering cross section Σ s The distribution of source strength q.

[0017] The aforementioned method for modeling neutron transport constitutive relations and low-dimensional governing equations based on sparse regression, wherein the physical quantity parameters in step S112 include at least the neutron flux J, neutron flux density Φ, and neutron flux density gradient Φ in the x-direction. x and source gradient q x , neutron flux density gradient Φ in the y direction y and source gradient q y ;

[0018] The method for modeling neutron transport constitutive relations and low-dimensional governing equations based on sparse regression, wherein step S120 consists of the following steps:

[0019] Step S121: Based on experience, select several parameters from the state parameters in step S110 and the calculated physical quantity parameters to form the fitted physical quantity parameters x. i The physical quantity parameter x is required to be fitted. i The dimensions of the neutron flux J are consistent with those of the neutron flux J.

[0020] Step S122: Construct a structure of size N for the x and y directions respectively. grid ×N i Okay, N xi sparse regression physical quantity matrix x of the column i_matrix And fit the physical quantity parameter x i Fill the sparse regression physical quantity matrix x i_matrix middle.

[0021] The method for modeling neutron transport constitutive relations and low-dimensional governing equations based on sparse regression, wherein step S130 consists of the following steps:

[0022] Step S131: Convert the neutron flow matrix J under multiple state parameters. matrix Let J be the target matrix. matrix =x i_matrix* a i_matrix ,Right now:

[0023]

[0024] The inverse matrix N is calculated using the least squares method. xi The coefficient matrix a has 1 row and 1 column. i_matrix ;

[0025] Step S132: For the x and y directions respectively, calculate the parameters of each fitted physical quantity x. i Multiply by its corresponding coefficient a i Summing these values ​​yields the neutron flux J, which represents the neutron transport constitutive relation for multiple state parameters. pred :

[0026] J pred = a1 x x1 + a2 x x2 + … + a Nxi x x Nxi .

[0027] The neutron transport constitutive relation and low-dimensional control equation modeling method based on sparse regression, wherein, in the step S131, the coefficient matrix a i_matrix .

[0028] The neutron transport constitutive relation and low-dimensional control equation modeling method based on sparse regression, wherein, in the step S131, the coefficient matrix a i_matrix .

[0029] The neutron transport constitutive relation and low-dimensional control equation modeling method based on sparse regression, wherein, in the step S140, the neutron diffusion equation is:

[0030]

[0031] The neutron transport constitutive relation and low-dimensional control equation modeling method based on sparse regression provided by the application has the characteristics of accurate calculation results of the neutron transport equation, grasps the key problem that the neutron diffusion constitutive relation is not applicable in a small scale range, which is the main reason for the large deviation of the calculation results of the neutron diffusion equation in a small scale range, combines the neutron transport constitutive relation obtained through sparse regression, and then embeds the new constitutive relation into the traditional neutron diffusion equation to obtain a low-dimensional macroscopic neutron transport control equation, which has smaller theoretical derivation difficulty, easier research and application, more obvious calculation precision improvement effect, better applicability, and certain practicality, and provides a theoretical basis and calculation basis for the reactor core parameter detection. BRIEF DESCRIPTION OF DRAWINGS

[0032] The drawings described herein are only for the purpose of explanation, and are not intended to limit the scope of the present application in any way; the shapes and scales of the components in the drawings are only illustrative, and are used to help understand the application, and are not specific limitations on the shapes and scales of the components; those skilled in the art can select various possible shapes and scales according to specific conditions to implement the application under the guidance of the application.

[0033] Figure 1 is a flow chart of the neutron transport constitutive relation and low-dimensional control equation modeling method based on sparse regression of the application. DETAILED DESCRIPTION

[0034] The specific embodiments and examples of the present application will be described in detail below with reference to the accompanying drawings, and the specific embodiments described are only used to explain the present application, and are not used to limit the specific embodiments of the present application.

[0035] As shown in Figure 1 , in order to solve the problem of lack of constitutive relation of neutron transport equation, the present application provides a modeling method for obtaining the constitutive relation of neutron transport in a nuclear reactor based on sparse regression method and establishing low-dimensional control equation, which generally includes the following steps:

[0036] Step S110, under the condition of given reactor state parameters, the physical quantity parameters under different state parameters are calculated by using the neutron transport program, including but not limited to neutron flow J, neutron flux density Φ, x-direction neutron flux density gradient Φ x and source intensity gradient q x , y-direction neutron flux density gradient Φ y and source intensity gradient q y , and the state parameters and physical quantity parameters of each state are combined to form the neutron flow matrix J matrix under multiple state parameters;

[0037] Step S120, according to the state parameters in step S110 and the calculated physical quantity parameters, combining the neutron transport theory and the dimension of neutron flow J, the fitting physical quantity parameter x i is constructed, which requires the dimension of the fitting physical quantity parameter x i equal to the dimension of the neutron flow J; meanwhile, the sparse regression physical quantity matrix x i_matrix of the constitutive relation of neutron transport is constructed, and the fitting physical quantity parameter x i is filled into the sparse regression physical quantity matrix x i_matrix ;

[0038] Step S130, the neutron flow matrix J matrix under multiple state parameters is set as the target matrix, assuming J matrix =x i_matrix* a i_matrix , the inverse matrix is calculated by using the sparse regression fitting method to obtain the coefficient matrix a i_matrix , each element coefficient a i_matrix in a i corresponds to a fitting physical quantity parameter x i , and the expression of the constitutive relation of neutron transport under multiple state parameters, i.e., the neutron flow J pred is obtained by assembling;

[0039] Step S140, the constitutive relation of neutron transport under multiple state parameters, i.e., the neutron flow J predThe expression is substituted into the existing neutron diffusion equation to replace the neutron flow J in the traditional neutron diffusion equation, so as to obtain a low-dimensional macroscopic neutron transport control equation.

[0040] Specifically, the step S110 comprises the following steps.

[0041] Step S111, according to the given typical operating state of the reactor, the range of the key state parameters is selected, including but not limited to the total cross section Σ t , scattering cross section Σ s and source intensity q, for each state parameter, N i parameter values are selected in the reasonable operating range, the parameter values must include the upper and lower limits of the reasonable operating range, i represents the i th state parameter;

[0042] Step S112, the physical quantity parameters under different state parameters are calculated by using the neutron transport program, including but not limited to the neutron flow J, the neutron flux density Φ, the neutron flux density gradient Φ x and the source intensity gradient q x ; and each state parameter and the physical quantity parameter are assembled into an N grid ×N i row, 1 column vector, N grid represents the number of grids.

[0043] Specifically, the step S120 comprises the following steps.

[0044] Step S121, for the x direction, a plurality of parameter vectors are selected from the state parameters in step S110 and the calculated physical quantity parameters according to experience, to form N xi fitting physical quantity parameter vectors x i , each x i is a vector with the size of N grid ×N i row, 1 column, for example: Φ, Φ x / Σ t , etc.; for the y direction, the fitting physical quantity parameter vectors x i in the y direction are constructed in a similar manner, such as: Φ, Φ y / Σ t , etc.; the selection of the fitting physical quantity parameter x i should be closely related to the fitting effect, and the dimension of the fitting physical quantity parameter x i should be consistent with the dimension of the neutron flow J;

[0045] Step S122, for the x and y directions respectively, a sparse regression physical quantity matrix x grid with the size of N i ×N xi row, N i_matrix column is constructed.And fit the physical quantity parameter x i Fill the sparse regression physical quantity matrix x i_matrix middle.

[0046] Specifically, step S130 includes the following steps:

[0047] Step S131: Convert the neutron flow matrix J under multiple state parameters. matrix Let J be the target matrix. matrix =x i_matrix* a i_matrix ,Right now:

[0048]

[0049] coefficient matrix a i_matrix For N xi A matrix with 1 row and 1 column whose values ​​can be calculated using the least squares method;

[0050] Step S132: Based on each fitted physical quantity parameter x i Each has a corresponding coefficient a i Therefore, each fitted physical quantity parameter x i Multiply by its corresponding coefficient a i By combining these parameters, we can obtain the constitutive relations for neutron transport with respect to multiple state parameters, i.e., the neutron flux J. pred Detailed expression:

[0051] J pred = a1×x1+a2×x2+…+a Nxi ×x Nxi . Specific implementation examples:

[0053]

Step S111

[0054] q=|sin(xπ / L)×cos(yπ / L) / 4π|+0.0001x+0.0001y+0.0001;

[0055] Where (x,y) represents the position of a point in the cell, and x∈[0,1],y∈[0,1], L represents the cell size and L=1;

[0056] Based on operational experience regarding the upper and lower limits of the core's reasonable operating range, (Σ) is selected.t1 = 0.9,∑ s1 = 0.45), (∑ t2 = 0.95,∑ s2 = 0.475), (∑ t3 = 1.05,∑ s3 = 0.525), (∑ t4 = 1.1,∑ s4 = 0.55) four groups of cross sections, i.e. selecting N Σ = 4 parameters;

[0057]

Step S112

[0058]

Step S121

[0059] For the neutron flux J x in x direction, here we select Φ x / Σ t , q x / Σt 2 There are two fitted physical quantities, namely N xi =2; similarly, for the neutron flow J in the y direction y The two physical quantities selected for fitting are Φ y / Σ t q y / Σ t 2 ;

[0060]

Step S122

[0061]

Step S131

[0062]

Step S132

[0063] J x_pred = -0.1005×Φ x / Σ t +0.0471×q x / Σ t 2 ;

[0064] J y_pred = -0.1005×Φ y / Σ t +0.0471×q y / Σ t 2 ;

[0065] [Step S140] Based on the given reactor state parameters, the neutron flow J x_predand J y_pred The relationship is brought into the neutron diffusion equation , the neutron flow J in the traditional neutron diffusion equation is replaced, and a low-dimensional macroscopic neutron transport control equation is obtained:

[0066]

[0067] In this embodiment, the constitutive relationship between the neutron flux and the neutron flow at the macroscopic scale and the corresponding low-dimensional macroscopic neutron transport control equation can be obtained by the sparse regression method, and the low-dimensional macroscopic neutron transport control equation has a certain generalization, which can provide a theoretical basis and a calculation basis for the core parameter detection.

[0068] It should be noted that the neutron transport constitutive relationship and low-dimensional control equation modeling method based on sparse regression is based on the neutron transport equation, uses the sparse regression method to obtain the constitutive relationship of the neutron transport equation at the macroscopic scale and the corresponding low-dimensional control equation, can provide a theoretical basis and a calculation basis for the core parameter detection, does not realize simple functions with complex steps, and does not use conventional or simple features for combination or stacking, so it meets the common sense of technical improvement and has practicality.

[0069] The contents not described in detail in the specification all belong to the prior art known to those skilled in the art.

[0070] It should be understood that the above only describes the preferred embodiments of the present application and is not limited to the technical solutions of the present application. Those skilled in the art can add, replace, transform or improve the above description according to the spirit and principles of the present application, for example, more core state parameters can be used to construct the sample to obtain more fitting variables, the solution of the coefficient matrix a i_matrix may also use matrix inversion, and the construction of the constitutive relationship can also use other methods such as neural network algorithm, genetic algorithm, etc.; and all these added, replaced, transformed or improved technical solutions should belong to the protection scope of the claims of the present application.

Claims

1. A method for modeling neutron transport constitutive relations and low-dimensional governing equations based on sparse regression, characterized in that, It consists of the following steps: Step S110: Given the reactor state parameters, calculate the physical quantity parameters under different state parameters using the neutron transport program, and combine the state parameters and physical quantity parameters of each state to form a neutron flux matrix under multiple state parameters. J matrix ; Step S120: Combining neutron transport theory, construct the fitting physical quantity parameters, and simultaneously build the sparse regression physical quantity matrix of the neutron transport constitutive relation. x i_matrix And fit the physical quantity parameters x i Filling into the sparse regression physical quantity matrix x i_matrix middle; Step S130: Calculate the neutron flow matrix under multiple state parameters. J matrix Let it be the target matrix, assuming J matrix = x i_matrix * a i_matrix The inverse matrix is ​​calculated using the sparse regression fitting method, thus obtaining the coefficient matrix. a i_matrix And by assembling, a neutron flux for neutron transport constitutive relations under multiple state parameters is obtained. J pred ; Step S140: Neutron stream J pred Substitute the expression into the neutron diffusion equation, replacing the neutron flux in the neutron diffusion equation. J We obtained the low-dimensional macroscopic neutron transport control equations.

2. The method for modeling neutron transport constitutive relations and low-dimensional governing equations based on sparse regression according to claim 1, characterized in that, Step S110 consists of the following steps: Step S111: Based on the given typical operating conditions of the reactor, select the range of key state parameters, and for each state parameter, select the upper and lower limits within the reasonable operating range. N i One parameter, i Represents each state parameter; Step S112: Calculate the physical quantity parameters under different state parameters using the neutron transport procedure, and assemble each state parameter and physical quantity parameter into... N grid × N i A matrix with 1 row and 1 column. N grid Indicates the number of grid cells.

3. The method for modeling neutron transport constitutive relations and low-dimensional governing equations based on sparse regression according to claim 2, characterized in that: The key state parameters in step S111 include at least the total cross-section. Σ t scattering cross section Σ s He Yuanqiang q The distribution of .

4. The method for modeling neutron transport constitutive relations and low-dimensional governing equations based on sparse regression according to claim 2, characterized in that: The physical quantity parameters in step S112 include at least the neutron flux. J neutron flux density Φ , x directional neutron flux density gradient Φ x Source gradient q x , y directional neutron flux density gradient Φ y Source gradient q y 。 5. The method for modeling neutron transport constitutive relations and low-dimensional governing equations based on sparse regression according to claim 1, characterized in that, Step S120 consists of the following steps: Step S121: Based on experience, select several parameters from the state parameters in step S110 and the calculated physical quantity parameters to form the fitted physical quantity parameters. x i The physical quantity parameters are required to be fitted. x i Dimensions and neutron flux J The dimensions are consistent; Step S122, respectively targeting x , y Direction, construct a size of N grid × N i OK, N xi sparse regression physical quantity matrix of columns x i_matrix And fit the physical quantity parameters x i Fill into the sparse regression physical quantity matrix x i_matrix middle.

6. The method for modeling neutron transport constitutive relations and low-dimensional governing equations based on sparse regression according to claim 1, characterized in that, Step S130 consists of the following steps: Step S131: Convert the neutron flow matrix under multiple state parameters. J matrix Let it be the target matrix, assuming J matrix = x i_matrix * a i_matrix ,Right now: The inverse matrix is ​​calculated using the least squares method, resulting in... N xi A coefficient matrix with 1 row and 1 column a i_matrix ; Step S132, respectively targeting x , y Direction, for each fitted physical quantity parameter x i Multiply by its corresponding coefficient a i Summing these parameters yields the neutron flux for the neutron transport constitutive relations with respect to multiple state parameters. J pred : J pred = a 1 ×x 1 + a 2 ×x 2 +… + a Nxi ×x Nxi 。 7. The method for modeling neutron transport constitutive relations and low-dimensional governing equations based on sparse regression according to claim 6, characterized in that: In step S131, the coefficient matrix is ​​obtained by calculating the inverse matrix using a genetic algorithm instead of the least squares method. a i_matrix .

8. The method for modeling neutron transport constitutive relations and low-dimensional governing equations based on sparse regression according to claim 6, characterized in that: In step S131, the coefficient matrix is ​​obtained by calculating the inverse matrix using a neural network algorithm instead of the least squares method. a i_matrix .

9. The method for modeling neutron transport constitutive relations and low-dimensional governing equations based on sparse regression according to claim 1, characterized in that, The neutron diffusion equation in step S140 is: ▽ ·J+Σ a Φ = q ； in, J Represents neutron flow, Φ Represents neutron flux density. q Representative Yuanqiang.

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