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

Abstract

The invention discloses a neutron transport constitutive relation and low-dimensional control equation modeling method based on sparse regression, and the method comprises the steps: enabling a state parameter of each state and a physical parameter calculated through a neutron transport program to form a neutron current matrix under a plurality of state parameters, and filling a sparse regression physical parameter matrix with the built fitting physical parameter, calculating an inverse matrix by using a least square method to obtain a coefficient matrix, and assembling to obtain a neutron transport constitutive relationship under the plurality of state parameters; according to the method, the characteristic that the calculation result of the neutron transport equation is accurate is utilized, the reason that the calculation result of the neutron diffusion equation is large in deviation in a small-scale range is grasped, the neutron transport constitutive relation is obtained in combination with sparse regression fitting, then the low-dimensional macroscopic neutron transport control equation is obtained, the theoretical derivation difficulty is small, the research progress is easier, and the method has good application prospects. And the effect of improving the calculation precision is obvious, the applicability is better, the difference from the situation under the actual reactor core working condition is small, and certain practicability is achieved.

Description

Technical Field

[0001] The present invention relates to the field of calculation methods for nuclear reactor cores, and in particular to a neutron transport constitutive relation and low-dimensional control equation modeling method based on sparse regression. Background Art

[0002] At present, the calculation methods of nuclear reactor core are mainly divided into two types: 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 relationship between parameters such as neutron flow and neutron flux. The constitutive relationship of the neutron diffusion equation, namely Fick's law, has limitations, which makes the calculation results of the neutron diffusion equation have larger errors than those of the neutron transport equation under small-scale problems. Therefore, how to obtain the constitutive relationship between parameters such as neutron flow and neutron flux in the neutron transport equation at a macroscopic scale and establish the corresponding low-dimensional control equation will have great practical significance and application value.

[0003] Most of the existing modeling methods for neutron transport constitutive relations and low-dimensional control equations start from the perspective of basic theory, and attempt to derive the constitutive relations of parameters such as neutron flow and neutron flux in theory, so as to establish low-dimensional control equations. For example, a more accurate constitutive relation between neutron flow and neutron flux is proposed based on the integral form of the neutron transport equation, but its limitations and disadvantages are as follows:

[0004] 1. It is difficult to theoretically derive the constitutive relationship between parameters such as neutron current and neutron flux, and research progress is difficult. In addition, the effect of applying the derived constitutive relationship to the low-dimensional control equation to improve the calculation accuracy is not significant.

[0005] 2. The derived constitutive relations have poor applicability. Many constitutive relations in research are valid only under various assumptions or ideal conditions of core operating conditions, which are far from the actual core operating conditions and are not practical.

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

[0007] In order to solve the above technical problems, the present invention provides a modeling method of neutron transport constitutive relations and low-dimensional control equations based on sparse regression. The theoretical derivation is less difficult, the research and application are easier, the effect of improving the calculation accuracy is more obvious, the applicability is better, and it has certain practicality.

[0008] The technical solution of the present invention is as follows: a neutron transport constitutive relation and a low-dimensional control equation modeling method based on sparse regression, which consists of the following steps:

[0009] Step S110: Under given reactor state parameters, the neutron transport program is used to calculate the physical quantity parameters under different state parameters, and the state parameters and physical quantity parameters of each state are combined into a neutron flux matrix J under multiple state parameters. matrix ;

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

[0011] Step S130: The neutron flux matrix J under multiple state parameters is matrix Set as the target matrix, assuming J matrix =x i_matrix* a i_matrix , use the sparse regression fitting method to calculate the inverse matrix and get the coefficient matrix a i_matrix , and the neutron flux J of the neutron transport constitutive relation under multiple state parameters is obtained by assembling pred ;

[0012] Step S140: neutron flux J pred The expression is substituted into the neutron diffusion equation, and the neutron flux J in the neutron diffusion equation is replaced to obtain the low-dimensional macroscopic neutron transport control equation.

[0013] The neutron transport constitutive relation and low-dimensional control equation modeling method based on sparse regression, wherein the step S110 is composed of the following steps:

[0014] Step S111: According to the given typical operating state of the reactor, select the range of key state parameters, and select N for each state parameter within the upper and lower limits of the reasonable operating range. i parameters, i represents each state parameter;

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

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

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

[0018] The neutron transport constitutive relation and low-dimensional control equation modeling method based on sparse regression, wherein the step S120 is composed of the following steps:

[0019] Step S121: Select several parameters from the state parameters in step S110 and the calculated physical quantity parameters based on experience to form the fitting physical quantity parameters x i , requiring the fitting of physical quantity parameters x i The dimension of is consistent with that of the neutron flux J;

[0020] Step S122: construct a N-dimensional grid ×N i OK, N xi The sparse regression matrix x of columns i_matrix , and the fitting physical quantity parameter x i Fill in the sparse regression physical quantity matrix x i_matrix middle.

[0021] The neutron transport constitutive relation and low-dimensional control equation modeling method based on sparse regression, wherein the step S130 is composed of the following steps:

[0022] Step S131: The neutron flux matrix J under multiple state parameters is matrix Set as the target matrix, assuming J matrix =x i_matrix* a i_matrix ,Right now:

[0023]

[0024] The inverse matrix is ​​calculated by the least squares method, and N is obtained. xi The coefficient matrix a with 1 row and 1 column i_matrix ;

[0025] Step S132: For the x and y directions, each fitting physical quantity parameter x i Multiply by its corresponding coefficient a i And sum them up to get the neutron flux J of the neutron transport constitutive relation for multiple state parameters pred :

[0026] J pred =a 1 × 1 +a 2 × 2 +…+a Nxi × Nxi .

[0027] The neutron transport constitutive relation and low-dimensional control equation modeling method based on sparse regression, wherein in step S131, the coefficient matrix a is obtained by calculating the inverse matrix by replacing the least squares method with a genetic algorithm. i_matrix .

[0028] The neutron transport constitutive relation and low-dimensional control equation modeling method based on sparse regression, wherein in step S131, the coefficient matrix a is obtained by calculating the inverse matrix by replacing the least squares method with a neural network algorithm. i_matrix .

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

[0030]

[0031] The neutron transport constitutive relation and low-dimensional control equation modeling method based on sparse regression provided by the present invention utilize the accurate calculation results of the neutron transport equation, grasp the main reason for the large deviation of the calculation results of the neutron diffusion equation in a small scale range, namely, the key problem that the neutron diffusion constitutive relation is not applicable in a small scale range, and obtain the neutron transport constitutive relation by combining sparse regression fitting. Then, by embedding the new constitutive relation into the traditional neutron diffusion equation, a low-dimensional macroscopic neutron transport control equation is obtained. The theoretical derivation difficulty is relatively small, the research and application are easier, the effect of improving the calculation accuracy is more obvious, the applicability is better, and it has certain practicality, providing a theoretical basis and calculation basis for core parameter detection. BRIEF DESCRIPTION OF THE DRAWINGS

[0032] The drawings described herein are only for explanation purposes and are not intended to limit the scope of the present invention in any way; the shapes and proportional dimensions of the components in the drawings are only schematic and are used to help understand the present invention, but are not intended to specifically limit the shapes and proportional dimensions of the components of the present invention; those skilled in the art can select various possible shapes and proportional dimensions to implement the present invention according to the specific circumstances under the guidance of the present invention.

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

[0034] The specific implementation modes and embodiments of the present invention will be described in detail below in conjunction with the accompanying drawings. The described specific embodiments are only used to explain the present invention and are not used to limit the specific implementation modes of the present invention.

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

[0036] Step S110: Under given reactor state parameters, use the neutron transport program to calculate physical quantity parameters under different state parameters, including but not limited to neutron flux J, neutron flux density Φ, and neutron flux density gradient Φ in the x direction. x and source intensity gradient q x , y-direction neutron flux density gradient Φ y and source intensity gradient q y The state parameters and physical quantity parameters of each state are combined into a neutron flow matrix J under multiple state parameters matrix ;

[0037] Step S120: According to the state parameters in step S110 and the calculated physical quantity parameters, combined with the neutron transport theory and the dimension of the neutron flux J, a suitable fitting physical quantity parameter x is constructed. i , requiring the fitting physical quantity parameter x i The dimension of is equal to the dimension of neutron flux J; at the same time, the sparse regression physical quantity matrix x of the neutron transport constitutive relation is constructed. i_matrix , and the fitting physical quantity parameter x i Fill in the sparse regression physical quantity matrix x i_matrix middle;

[0038] Step S130: The neutron flux matrix J under multiple state parameters is matrix Set as the target matrix, assuming J matrix =x i_matrix* a i_matrix , use the sparse regression fitting method to calculate the inverse matrix and get the coefficient matrix a i_matrix , a i_matrix Each element coefficient a in i Corresponding to a fitting physical quantity parameter x i , the neutron transport constitutive relation for multiple state parameters is obtained by assembling, namely, the neutron flux J pred Expression of

[0039] Step S140: The neutron transport constitutive relation under multiple state parameters, i.e., the neutron flux J predSubstituting the expression into the existing neutron diffusion equation and replacing the neutron flux J in the traditional neutron diffusion equation, we can obtain the low-dimensional macroscopic neutron transport control equation.

[0040] Specifically, step S110 includes the following steps:

[0041] Step S111: According to a given typical operation state of the reactor, select a range of key state parameters, including but not limited to the total cross-section Σ t , scattering cross section Σ s and the distribution of source strength q, and select N for each state parameter within a reasonable operating range i parameter value, which must include the upper and lower limits of the reasonable operating range, i represents the i-th state parameter;

[0042] Step S112: Calculate the physical parameters under different state parameters using the neutron transport program, including but not limited to neutron flux J, neutron flux density Φ, neutron flux density gradient Φ x and source intensity gradient q x ; and assemble each state parameter and physical quantity parameter into N grid ×N i A vector of rows and columns, N grid Indicates the number of grids.

[0043] Specifically, step S120 includes the following steps:

[0044] Step S121: For the x direction, select several parameter vectors from the state parameters in step S110 and the calculated physical quantity parameters based on experience to form N xi A fitted physical quantity parameter vector x i , each x i The size is N grid ×N i A vector of rows and 1 column, for example: Φ, Φ x / Σ t For the y direction, a similar method is used to construct the fitting physical quantity parameter vector x in the y direction. i , such as: Φ, Φ y / Σ t etc.; Fitting physical quantity parameter x i The selection of should be closely related to the fitting effect, and the fitting physical quantity parameter x i The dimension of is consistent with that of the neutron flux J;

[0045] Step S122: construct a N-dimensional grid ×N i OK, N xi The sparse regression matrix x of columns i_matrix, and the fitting physical quantity parameter x i Fill in the sparse regression physical quantity matrix x i_matrix middle.

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

[0047] Step S131: The neutron flux matrix J under multiple state parameters is matrix Set as the target matrix, assuming J matrix =x i_matrix* a i_matrix ,Right now:

[0048]

[0049] Coefficient matrix a i_matrix N xi A matrix of 1 row and 1 column, whose value can be calculated by the least squares method;

[0050] Step S132: Based on each fitting physical quantity parameter x i There are corresponding coefficients a i , thus each fitting physical quantity parameter x i Multiply by its corresponding coefficient a i By summing them up, we can get the neutron transport constitutive relation for multiple state parameters, namely, the neutron flux J pred The detailed expression of:

[0051] J pred =a 1 × 1 +a 2 × 2 +…+a Nxi × Nxi . Specific embodiment:

[0053] [Step S111] Taking uniform material as an example, the calculation model is selected as a 1cm×1cm square single grid element, the boundary condition is a reflection boundary, and the number of grids is N x =N y =201, the total cross section of the key state parameters given by the reactor Σ t =1, scattering cross section Σ S =1, the source strength q is set as follows:

[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] According to the operating experience of the upper and lower limits of the reasonable operating range of the core, (Σ t1 =0.9,Σ s1 =0.45)、(Σ t2 =0.95,Σ s2 =0.475)、(Σ t3 =1.05,Σ s3 =0.525)、(Σ t4 =1.1,Σ s4 =0.55) There are four groups of cross sections, that is, select N Σ =4 parameters;

[0057] [Step S112] Use the neutron transport program to calculate the physical quantity parameters of each group of cross sections, including the neutron flux J in the x direction. x , neutron flux J in the y direction y , neutron flux density Φ, combined with the finite difference method, calculate the neutron flux density gradient Φ in the x direction x , the neutron flux density gradient in the y direction Φ y , the source intensity density gradient q in the x direction x and the neutron flux density gradient q in the y direction y ; Integrate all state parameters and physical quantity parameters, and assemble each parameter into a vector of 4×201×201 rows and 1 column; Similarly, the total cross section Σ t and scattering cross section Σ S Also assemble like this;

[0058] [Step S121] Based on the above key state parameters and physical quantity parameters, according to the core operation experience, the above key state parameters and physical quantity parameters are combined to obtain the fitting physical quantity x i , and the fitting physical quantity x i The dimension of the neutron flux J should be equal to 1 / (cm 2 ·s), for example: Φ x / Σ t , Φ y / Σ s ,q x / Σ t 2 ,q y / Σ s 2 etc.; the fitting error is also related to the selected fitting physical quantity x i is closely related, so the appropriate fitting physical quantity x should be selected as much as possible on the basis of the same dimension. i ;

[0059] Neutron flux J in the x direction xSelect Φ here x / Σ t ,q x / Σ t 2 There are two fitting physical quantities, namely N xi =2; Similarly, for the neutron flux J in the y direction y The two selected fitting physical quantities are Φ y / Σ t ,q y / Σ t 2 ;

[0060] [Step S122] For the x and y directions, all the fitted physical quantities x i Fill to x i_matrix In the matrix, reassemble and construct a two-dimensional matrix of 4 × 201 × 201 rows and 2 columns;

[0061] [Step S131] x direction, the neutron flow J matrix =x i_matrix* a i_matrix Set as the target matrix, where a i_matrix is a constant vector. Using the least squares method to calculate the inverse matrix, we get a constant vector a with 2 rows and 1 column. i_matrix , a i_matrix =[-0.1005,0.471], constant vector a in the y direction i_matrix and the constant vector a in the x direction i_matrix same;

[0062] [Step S132] The constant vector a i_matrix Each element of and its corresponding fitting physical quantity x i Multiply and sum to get the fitted neutron flux J in the x and y directions respectively. x_pred and J y_pred The detailed expression of the constitutive relation is:

[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 flux J x_pred and J y_pred Substitute the relationship into the neutron diffusion equation In the above equation, the neutron flux J in the traditional neutron diffusion equation is replaced to obtain the low-dimensional macroscopic neutron transport control equation:

[0066]

[0067] This embodiment uses the sparse regression method to obtain the constitutive relationship between neutron flux and neutron flow at the macroscopic scale and the corresponding low-dimensional macroscopic neutron transport control equation. The low-dimensional macroscopic neutron transport control equation has a certain generalization and can provide a theoretical basis and calculation foundation for core parameter detection.

[0068] It should be noted that the present invention is based on the neutron transport constitutive relationship and low-dimensional control equation modeling method of sparse regression. Based on the neutron transport equation, the sparse regression method is used to obtain the constitutive relationship of the neutron transport equation at a macroscopic scale and the corresponding low-dimensional control equation, which can provide a theoretical basis and calculation basis for core parameter detection. It does not use complex steps to achieve simple functions, nor does it use conventional or simple features for combination or stacking. Therefore, it conforms to the common sense of technological improvement and is practical.

[0069] The contents not described in detail in this specification belong to the prior art known to ordinary technicians in this field.

[0070] It should be understood that the above description is only a preferred embodiment of the present invention and is not sufficient to limit the technical solution of the present invention. For ordinary technicians in this field, within the spirit and principle of the present invention, the above description can be added, subtracted, replaced, transformed or improved. For example, the construction of the sample can use more core state parameters to obtain more fitting variables, and the coefficient matrix a i_matrix The solution can also be obtained by matrix inversion, and the construction of the constitutive relationship can also be achieved by other methods such as neural network algorithm, genetic algorithm, etc.; and all these technical solutions after addition, subtraction, replacement, transformation or improvement should fall within the scope of protection of the claims attached to the present invention.

Claims

1. Neutron transport constitutive relation and low-dimensional control equation modeling method based on sparse regression, characterized in that: It consists of the following steps: Step S110: Under given reactor state parameters, the neutron transport program is used to calculate the physical quantity parameters under different state parameters, and the state parameters and physical quantity parameters of each state are combined into a neutron flux matrix J under multiple state parameters. matrix ; Step S120: Combine the neutron transport theory to build the fitting physical quantity parameters and construct the sparse regression physical quantity matrix x of the neutron transport constitutive relation. i_matrix , and the fitting physical quantity parameter x i Fill in the sparse regression matrix x i_matrix middle; Step S130: The neutron flux matrix J under multiple state parameters is matrix Set as the target matrix, assuming J matrix =x i_matrix* a i_matrix , use the sparse regression fitting method to calculate the inverse matrix and get the coefficient matrix a i_matrix , and the neutron flux J of the neutron transport constitutive relation under multiple state parameters is obtained by assembling pred ; Step S140: neutron flux J pred The expression is substituted into the neutron diffusion equation, and the neutron flux J in the neutron diffusion equation is replaced to obtain the low-dimensional macroscopic neutron transport control equation.

2. The neutron transport constitutive relation and low-dimensional control equation modeling method based on sparse regression according to claim 1, characterized in that: The step S110 consists of the following steps: Step S111: According to the given typical operating state of the reactor, select the range of key state parameters, and select N for each state parameter within the upper and lower limits of the reasonable operating range. i parameters, i represents each state parameter; Step S112: Calculate the physical quantity parameters under different state parameters using the neutron transport program, and assemble each state parameter and physical quantity parameter into N grid ×N i A matrix of rows and columns, N grid Indicates the number of grids.

3. The neutron transport constitutive relation and low-dimensional control equation modeling method 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 and the distribution of source strength q.

4. The neutron transport constitutive relation and low-dimensional control equation modeling method based on sparse regression according to claim 2, characterized in that: The physical quantity parameters in step S112 at least include neutron flux J, neutron flux density Φ, and neutron flux density gradient Φ in the x direction. x and source intensity gradient q x , y-direction neutron flux density gradient Φ y and source intensity gradient q y。 5. The neutron transport constitutive relation and low-dimensional control equation modeling method based on sparse regression according to claim 1, characterized in that: The step S120 consists of the following steps: Step S121: Select several parameters from the state parameters in step S110 and the calculated physical quantity parameters based on experience to form the fitting physical quantity parameters x i , requiring the fitting of physical quantity parameters x i The dimension of is consistent with that of the neutron flux J; Step S122: construct a N-dimensional grid ×N i OK, N xi The sparse regression matrix x of columns i_matrix , and the fitting physical quantity parameter x i Fill in the sparse regression physical quantity matrix x i_matrix middle.

6. The neutron transport constitutive relation and low-dimensional control equation modeling method based on sparse regression according to claim 1, characterized in that: The step S130 consists of the following steps: Step S131: The neutron flux matrix J under multiple state parameters is matrix Set as the target matrix, assuming J matrix =x i_matrix* a i_matrix ,Right now: The inverse matrix is ​​calculated by the least squares method, and N is obtained. xi The coefficient matrix a with 1 row and 1 column i_matrix ; Step S132: For the x and y directions, each fitting physical quantity parameter x i Multiply by its corresponding coefficient a i And sum them up to get the neutron flux J of the neutron transport constitutive relation for multiple state parameters pred : I pred =a1×x1+a2×x2+…+a Nxi ×x Nxi .

7. The neutron transport constitutive relation and low-dimensional control equation modeling method based on sparse regression according to claim 6, characterized in that: In step S131, the coefficient matrix a is obtained by using a genetic algorithm to replace the least squares method to calculate the inverse matrix. i_matrix .

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

9. The neutron transport constitutive relation and low-dimensional control equation modeling method based on sparse regression according to claim 1, characterized in that: The neutron diffusion equation in step S140 is:

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