Method for predicting neutron noise sources in a reactor based on least squares

Abstract

A reactor neutron noise source prediction method based on least squares method, according to the user input reactor geometry and material parameters, the numerical discrete method is used to solve the frequency domain neutron noise Green function equation, after the reactor neutron noise transfer matrix is obtained, according to the dimension of the neutron noise source, the number and position of non-zero elements are iteratively updated, then according to the number and position of the current non-zero elements, the selection matrix is constructed and the matrix Q and vector x are calculated according to the selection matrix and the reactor neutron noise transfer matrix, and the least squares problem is constructed, and the optimal residual is obtained by comparing the input series of residuals. By using the frequency domain neutron noise Green function equation and through two layers of loop search, the optimal result of the number and position of the noise source in the least squares sense is obtained, and the prediction of the reactor neutron noise source is realized.

Description

Technical Field

[0001] This invention relates to a technology in the field of reactor control, specifically a reactor neutron noise source prediction method based on the least squares method. Background Technology

[0002] Neutron noise in a nuclear reactor refers to the random fluctuations in neutron flux density around its steady-state value during the actual steady-state operation of a nuclear reactor. Neutron noise causes the reactor's operating state to deviate from normal, adversely affecting the reactor's economic efficiency and safety. The impact of neutron noise on the reactor's operating state can be simulated by solving the frequency domain neutron noise equation. Summary of the Invention

[0003] This invention addresses the problems of high computational complexity in existing technologies, which make them difficult to apply to multiple noise source prediction problems, and insufficient computational accuracy when dealing with reactor cores with complex geometries and fine energy spectrum structures. It proposes a reactor neutron noise source prediction method based on the least squares method. This method utilizes the Green's function equation for neutron noise in the frequency domain and searches for the optimal results of the number and location of noise sources in the least squares sense through two-level loops to achieve the prediction of reactor neutron noise sources.

[0004] This invention is achieved through the following technical solution:

[0005] This invention relates to a reactor neutron noise source prediction method based on the least squares method, comprising:

[0006] Step 1: Based on the reactor's geometric and material parameters, construct the reactor neutron noise transfer matrix by numerically discretizing and solving the frequency domain neutron noise Green's function equation.

[0007] The neutron noise source is an n-dimensional vector S in the complex field, satisfying... Based on prior knowledge of engineering practice, a neutron noise source is a sparse vector in which the number of non-zero elements is significantly less than the number of zero elements.

[0008] Step 2: Based on the observed values ​​of the reactor neutron noise spectrum, initialize the number and position of non-zero elements in the neutron noise source; construct the selection matrix P based on the current positions of the non-zero elements, and use the selection matrix to calculate matrix Q and vector x;

[0009] The observed neutron noise spectrum is an m-dimensional vector δφ in the complex field, satisfying... Based on prior knowledge of engineering practice, the observed neutron noise spectrum is a full vector in which all elements are non-zero.

[0010] Step 3: Solve for the noise source under the given number and position of non-zero elements using the least squares method. Iterate and update the optimal noise source under the current number of non-zero elements, that is, the solution with the smallest residual among all possible non-zero element positions and its residual.

[0011] Step 4: Calculate the global optimal solution for the neutron noise source by comparing the relative changes in the residuals and the ratio of the residual standard deviation to the mean.

[0012] This invention relates to a reactor neutron noise source prediction system that implements the above-described method, comprising: a frequency domain neutron noise Green's function module, a non-zero element iteration module, a selection matrix construction module, a least squares problem construction module, a least squares solution module, and a residual comparison module. Specifically: the frequency domain neutron noise Green's function module solves the frequency domain neutron noise Green's function equation using a numerical discretization method based on user-input reactor geometry and material parameters to obtain the reactor neutron noise transfer matrix; the non-zero element iteration module iteratively updates the number and position of non-zero elements based on the dimension of the neutron noise source; the selection matrix construction module constructs a selection matrix based on the current number and position of non-zero elements; the least squares construction module calculates matrix Q and vector x based on the selection matrix and the reactor neutron noise transfer matrix, and constructs a least squares problem; the least squares solution module calculates the solution to the least squares problem based on the input least squares problem; and the residual comparison module obtains the optimal residual by comparing a series of input residuals.

[0013] Technical effect

[0014] This invention, based on the Green's function method to obtain the transfer matrix of reactor neutron noise, transforms the noise source prediction problem into an underdetermined linear regression problem. By enumerating subsets of the number and location of noise sources, the problem is decomposed into a series of overdetermined linear regression subproblems. These subproblems can be solved using the least squares method. Finally, the optimal estimate of the number and location of noise sources is determined by the least squares residuals, thus solving the reactor neutron noise source prediction problem. Compared with existing technologies, this invention significantly improves the computational efficiency and reduces the computational complexity of noise source prediction, while also being able to handle neutron noise source prediction problems when high-risk components are unknown. Attached Figure Description

[0015] Figure 1 This is a flowchart of the present invention;

[0016] Figure 2 This is a schematic diagram of the geometric structure and material arrangement of an embodiment;

[0017] Figure 3 This is a schematic diagram of the distribution of sub-noise spectrum observations in the embodiment;

[0018] Figure 4This is a schematic diagram of the predicted residual distribution of the sub-noise source in the example. Detailed Implementation

[0019] like Figure 1 As shown in this embodiment, a reactor neutron noise source prediction method based on the least squares method is involved, including:

[0020] Step 1: Based on the reactor's geometry and material parameters, construct the reactor neutron noise transfer matrix by numerically discretizing and solving the frequency domain neutron noise Green's function equation. This specifically includes:

[0021] 1.1 The Green's function equation for sub-noise in the frequency domain is constructed as follows:

[0022] Where: Ω and Ω' are angular variables, Ω s For the angle variable of the noise source term, Let r be a differential operator, and r be a spatial variable. s Let ω be the spatial variable of the noise source term, and ω be the frequency variable. s Let g be the frequency variable of the noise source term, and g' be the energy group number. s The energy group number for the noise source term. Let be the distribution of the Green's function of the neutron noise in the g-th group with respect to the spatial, angular, and frequency variables. Similarly, Σ s,g′→g,0 (r,Ω'→Ω) represents the distribution of the steady-state neutron scattering cross section from group g' to group g and from angle Ω' to angle Ω with respect to spatial and angular variables, ν is the average number of neutrons produced per fission, and Σ f,g′,0 (r) represents the distribution of the steady-state neutron fission cross section of the g' group with respect to spatial variables. Let be the distribution of the total dynamic neutron cross section of the g-th group with respect to spatial and frequency variables. Let be the distribution of the dynamic neutron energy spectrum of the g-th group with respect to spatial and frequency variables. For the position located at the gth s Group, spatial location r s Angle is Ω s The frequency is ω s The distribution of a unit point source with respect to energy group, spatial variables, angular variables, and frequency variables, k eff,0 It is a steady-state effective growth factor.

[0023] 1.2 For energy groups g and g s Spatial variables r and r s , angular variables Ω and Ω s Frequency variables ω and ω s After numerical discretization, the neutron noise Green's function Converting to discrete form, the m×n dimensional neutron noise transfer matrix A is obtained by solving the Green's function equation for neutron noise in the frequency domain, satisfying... Based on prior knowledge of engineering practice, m is usually significantly smaller than n.

[0024] Step 2: Based on the observed neutron noise spectrum values ​​from the reactor, initialize the number and position of non-zero elements in the neutron noise source vector S; construct the selection matrix P based on the current positions of the non-zero elements, specifically including:

[0025] 2.1 When the number and position of non-zero elements in the sub-noise source are known, construct a selection matrix P that satisfies... Where: k is the number of non-zero elements in the neutron noise source. In the selection matrix, the element in the i-th row and j-th column is P. ij Then P ij satisfy: Then the observed neutron noise spectrum value δφ=AP T PS and has the same solution as the equation δφ=AS, where: A is the neutron noise transfer matrix; S is the neutron noise source, P T To select the transpose of matrix P.

[0026] 2.2 Further, the equation δφ=AP T PS can be rewritten as δφ = Qx, where: matrix Q = AP T ,satisfy Given the number and position of non-zero elements, the neutron noise source vector x is x = PS, satisfying... Based on prior knowledge from engineering practice, the number of non-zero elements k in a neutron noise source is generally significantly smaller than the number of neutron noise spectrum observations m.

[0027] Step 3: Solve for the neutron noise source vector x and its residual e under the given conditions of the number and position of non-zero elements using the least squares method. Iterate and update the optimal solution for the noise source under the current number of non-zero elements, i.e., the solution with the smallest residual among all possible non-zero element positions, and its residual. Specifically, this includes:

[0028] 3.1 According to the least squares method, given the number and position of non-zero elements in the neutron noise source, the solution to the equation δφ=Qx is: x=(Q T Q) -1 Q T δφ, the residual is e=||δφ-Qx||, where: e is the residual.

[0029] 3.2 Introducing a two-loop search for the optimal solution for the number and position of non-zero elements: The outer loop iterates through all possible numbers of non-zero elements, while the inner loop, given the number of non-zero elements, iterates through all possible positions of non-zero elements. The optimal solution for the neutron noise source given the current number and position of non-zero elements is obtained using the least squares method. After the two loops are completed, the number of all possible non-zero elements, k, is obtained. j That is, the number of the j-th possible non-zero elements, corresponding to a series of neutron noise sources S. j That is, the number of neutron noise sources corresponding to the j-th possible non-zero element quantity and the corresponding series of residuals e j That is, the residual corresponding to the number of possible non-zero elements of the j-th element.

[0030] Step 4: Based on the relative changes in the residuals and the ratio of the residual standard deviation to the mean, calculate the global optimal solution for the neutron noise source. Specifically, for the (j+1)th residual e j+1 With the j-th residual e j The degree of relative change between The neutron noise source with the largest relative change in residual is S. t , where: t = argmin j (d j If the ratio of the standard deviation σ of the residuals to the mean μ of the residuals is less than the threshold ∈, then S1 is the optimal solution for the neutron noise source; otherwise, S t It is the optimal solution for the neutron noise source.

[0031] Through specific practical experiments, Figure 2 Taking the MOX benchmark as an example, the setup consists of 9 MOX fuel assemblies, 40 UO2 fuel assemblies, and 32 reflector assemblies, each with a side length of 21.42 cm, and all four boundaries are under vacuum boundary conditions. The neutron dynamics constant is approximated using a single-set delayed neutron, with a delayed neutron fraction β = 0.0049 s. -1 The decay constant of the slow-emitting neutron precursor nuclear is λ = 0.0797 s. -1 The velocity of the fast group neutrons is v1 = 1.82304 × 10⁻⁶. 7 cm / s, hot group neutron velocity v2=4.13067×10 5 cm / s; The homogenization group constants of the UO2 module are: fast group diffusion coefficient D1 = 1.2 cm, hot group diffusion coefficient D2 = 0.4 cm, and fast group absorption cross section Σ a，1 =0.010cm -1 Heat group absorption cross section Σ a，2 =0.100cm -1 , fast group move out of section Σ r，1 =0.020cm -1 Fast group neutron generation cross section vΣ f，1=0.0050cm -1 Hot group neutron generation cross section vΣ f，2 =0.125cm -1 The homogenization group constants of the MOX module are: fast group diffusion coefficient D1 = 1.2 cm, hot group diffusion coefficient D2 = 0.4 cm, and fast group absorption cross section Σ. a，1 =0.015cm -1 Heat group absorption cross section Σ a，2 =0.300cm -1 , fast group move out of section Σ r，1 =0.015cm -1 Fast group neutron generation cross section νΣ f，1 =0.0075cm -1 Hot group neutron generation cross section vΣ f，2 =0.450cm -1 The homogenization group constants of the reflective layer assembly are: fast group diffusion coefficient D1 = 1.2 cm, thermal group diffusion coefficient D2 = 0.2 cm, and fast group absorption cross section Σ. a，1 =0.001cm -1 Heat group absorption cross section Σ a，2 =0.04cm -1 , fast group move out of section Σ r，1 =0.050cm -1 Fast group neutron generation cross section νΣ f，1 =0cm -1 The hot group neutron production cross section νΣ f，2 =0cm -1 .enter Figure 3 The observed neutron noise spectrum values ​​are used to predict the neutron noise source vector and its residual using this method, as shown below. Figure 4 As shown, the results indicate that the residuals of this technical solution in this embodiment do not exceed 0.02%.

[0032] Compared with existing technologies, this method is based on transport theory and does not rely on diffusion theory, making it more applicable to reactor analysis problems with strong absorption and strong anisotropy. A neutron noise source prediction method based on the reactor neutron noise transfer matrix is ​​established using the Green's function method, which eliminates the need for complete neutron noise theory calculations in the neutron noise source prediction stage. It can obtain the global optimal solution of the neutron noise source in the least squares sense while significantly reducing computational complexity, and has higher and more stable computational accuracy.

[0033] The above-described specific implementations can be partially adjusted by those skilled in the art in different ways without departing from the principles and purpose of the present invention. The scope of protection of the present invention is defined by the claims and is not limited to the above-described specific implementations. All implementation schemes within the scope of the claims are bound by the present invention.

Claims

1. A reactor neutron noise source prediction method based on the least squares method, characterized in that, include: Step 1: Based on the reactor's geometric and material parameters, construct the reactor neutron noise transfer matrix by numerically discretizing and solving the frequency domain neutron noise Green's function equation. Step 2: Based on the observed values ​​of the reactor neutron noise spectrum, initialize the number and position of non-zero elements in the neutron noise source; construct the selection matrix P based on the current positions of the non-zero elements, and use the selection matrix to calculate matrix Q and the neutron noise source vector x; Step 3: Solve for the noise source under the given number and position of non-zero elements using the least squares method. Iterate and update the optimal noise source under the current number of non-zero elements, that is, the solution with the smallest residual among all possible non-zero element positions and its residual. Step 4: Calculate the global optimal solution for the neutron noise source by comparing the relative changes in the residuals and the ratio of the residual standard deviation to the mean. The neutron noise source is an n-dimensional vector S in the complex field, satisfying... The observed neutron noise spectrum is an m-dimensional vector δφ in the complex field, satisfying...

2. The reactor neutron noise source prediction method based on least squares method according to claim 1, characterized in that, Step 1 specifically includes: 1.1 The Green's function equation for sub-noise in the frequency domain is constructed as follows: Where: Ω and Ω' are angular variables, Ω s For the angle variable of the noise source term, Let r be a differential operator, and r be a spatial variable. s Let ω be the spatial variable of the noise source term, and ω be the frequency variable. s Let g be the frequency variable of the noise source term, and g' be the energy group number. s The energy group number for the noise source term. Let be the distribution of the Green's function of the neutron noise in the g-th group with respect to the spatial, angular, and frequency variables. Similarly, Σ s,g′→g,0 (r,Ω'→Ω) represents the distribution of the steady-state neutron scattering cross section from group g' to group g and from angle Ω' to angle Ω with respect to spatial and angular variables, ν is the average number of neutrons produced per fission, and Σ f,g′,0 (r) represents the distribution of the steady-state neutron fission cross section of the g' group with respect to spatial variables. Let be the distribution of the total dynamic neutron cross section of the g-th group with respect to spatial and frequency variables. Let be the distribution of the dynamic neutron energy spectrum of the g-th group with respect to spatial and frequency variables. For the position located at the gth s Group, spatial location r s Angle is Ω s The frequency is ω s The distribution of a unit point source with respect to energy group, spatial variables, angular variables, and frequency variables, k eff,0 It is a stable and effective growth factor; 1.2 For energy groups g and g s Spatial variables r and r s , angular variables Ω and Ω s Frequency variables ω and ω s After numerical discretization, the neutron noise Green's function Converting to discrete form, the m×n dimensional neutron noise transfer matrix A is obtained by solving the Green's function equation for neutron noise in the frequency domain, satisfying... Based on prior knowledge of engineering practice, m is less than n.

3. The reactor neutron noise source prediction method based on least squares method according to claim 1, characterized in that, Step 2 specifically includes: 2.1 When the number and position of non-zero elements in the sub-noise source are known, construct a selection matrix P that satisfies... Where: k is the number of non-zero elements in the neutron noise source, and in the selection matrix, the element in the i-th row and j-th column is P. ij Then P ij satisfy: Then the observed neutron noise spectrum value δφ=AP T PS has the same solution as the equation δφ=AS, where: A is the neutron noise transfer matrix; S is the neutron noise source; and PT is the transpose of the selection matrix P. 2.2 Further, the equation δφ=AP T PS can be rewritten as δφ = Qx, where: matrix Q = AP T ,satisfy Given the number and position of non-zero elements, the neutron noise source vector x is x = PS, satisfying... Based on prior knowledge from engineering practice, the number of non-zero elements k in a neutron noise source is less than the number of neutron noise spectrum observations m.

4. The reactor neutron noise source prediction method based on least squares method according to claim 1, characterized in that, Step 3 specifically includes: 3.1 According to the least squares method, given the number and position of non-zero elements in the neutron noise source, the solution to the equation δφ=Qx is: x=(Q T Q) -1 Q T δφ, the residual is e=||δφ-Qx||, where: e is the residual, Q is the selection matrix calculation matrix, and δφ is the neutron noise spectrum observation value, which is an m-dimensional vector in the complex field; 3.2 Introducing a two-loop search for the optimal solution for the number and position of non-zero elements: The outer loop iterates through all possible numbers of non-zero elements, while the inner loop, given the number of non-zero elements, iterates through all possible positions of non-zero elements. The optimal solution for the neutron noise source given the current number and position of non-zero elements is obtained using the least squares method. After the two loops are completed, the number of all possible non-zero elements, k, is obtained. j That is, the number of the j-th possible non-zero elements, corresponding to a series of neutron noise sources S. j That is, the number of neutron noise sources corresponding to the j-th possible non-zero element quantity and the corresponding series of residuals e j That is, the residual corresponding to the number of possible non-zero elements of the j-th element.

5. The reactor neutron noise source prediction method based on least squares method according to claim 1, characterized in that, Step 4, specifically, involves: for the (j+1)th residual e j+1 With the j-th residual e j The degree of relative change between The neutron noise source with the largest relative change in residual is S. t , where: k j For the number of all possible non-zero elements, t = argmin j (d j If the ratio of the standard deviation σ of the residuals to the mean μ of the residuals is less than the threshold ∈, then S1 is the optimal solution for the neutron noise source; otherwise, S t It is the optimal solution for the neutron noise source.

6. A reactor neutron noise source prediction system for implementing the method of any one of claims 1-5, characterized in that, include: The system comprises the following modules: a frequency-domain neutron noise Green's function module, a non-zero element iteration module, a selection matrix construction module, a least squares problem construction module, a least squares solution module, and a residual comparison module. Specifically: the frequency-domain neutron noise Green's function module solves the frequency-domain neutron noise Green's function equation using a numerical discretization method based on the user-input reactor geometry and material parameters, obtaining the reactor neutron noise transfer matrix; the non-zero element iteration module iteratively updates the number and position of non-zero elements based on the dimension of the neutron noise source; the selection matrix construction module constructs a selection matrix based on the current number and position of non-zero elements; the least squares construction module calculates matrix Q and vector x based on the selection matrix and the reactor neutron noise transfer matrix, and constructs a least squares problem; the least squares solution module calculates the solution to the least squares problem based on the input least squares problem; and the residual comparison module obtains the optimal residual by comparing a series of input residuals.

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