Reactor neutron noise source prediction method based on least square method

Abstract

A reactor neutron noise source prediction method based on a least square method comprises the following steps: according to reactor geometry and material parameters input by a user, solving a frequency domain neutron noise Green function equation by using a numerical discretization method to obtain a reactor neutron noise transfer matrix, and calculating a neutron noise source according to the dimension of the neutron noise source; the method comprises the following steps: performing iterative updating to obtain the number and position of non-zero elements, constructing a selection matrix according to the number and position of the current non-zero elements, calculating to obtain a matrix Q and a vector x according to the selection matrix and a neutron noise transfer matrix in a reactor, constructing a least square problem, and obtaining an optimal residual error through comparison according to a series of input residual errors after solving. And the neutron noise source prediction of the reactor is realized by using the frequency domain neutron noise Green function equation and through two-layer cycle search of optimal results of the number and the position of the noise source under the least square meaning.

Description

Technical Field

[0001] The present invention relates to a technology in the field of reactor control, in particular to a method for predicting reactor neutron noise sources based on the least square method. Background Art

[0002] Reactor neutron noise refers to the random fluctuation of neutron flux density around the steady-state value during the actual steady-state operation of a nuclear reactor. The neutron noise phenomenon causes the reactor's operating state to deviate from normal, which has an adverse impact on the economy and safety of reactor operation. 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] In view of the fact that the existing technology has high computational complexity and is difficult to apply to the prediction of multiple noise sources, and has insufficient computational accuracy when dealing with cores with complex geometric configurations and fine energy spectrum structures, the present invention proposes a reactor neutron noise source prediction method based on the least squares method. The method uses the frequency domain neutron noise Green's function equation and searches for the optimal results of the number and position of noise sources in the least squares sense through two layers of loops to achieve the prediction of reactor neutron noise sources.

[0004] The present invention is achieved through the following technical solutions:

[0005] The present invention relates to a method for predicting reactor neutron noise sources based on the least square method, comprising:

[0006] Step 1: According to the geometric and material parameters of the reactor, the reactor neutron noise transfer matrix is ​​constructed by numerically discretizing 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 According to prior knowledge from engineering practice, the neutron noise source is a sparse vector, in which the number of non-zero elements is significantly smaller than the number of zero elements.

[0008] Step 2: Initialize the number and position of non-zero elements in the neutron noise source according to the observed value of the reactor neutron noise spectrum; construct a selection matrix P according to the current position of the non-zero elements, and use the selection matrix to calculate the matrix Q and the vector x;

[0009] The neutron noise spectrum observation value is an m-dimensional vector δφ in the complex domain, satisfying According to prior knowledge from engineering practice, the neutron noise spectrum observation value is a full vector in which all elements are non-zero.

[0010] Step 3: Solve the noise source under the condition of the current given number and position of non-zero elements by the least square method, and iteratively update the optimal noise source under the current number of non-zero elements, that is, the solution with the minimum residual among all possible non-zero element positions and its residual;

[0011] Step 4: By comparing the relative change degree of the residuals and the ratio of the residual standard deviation to the mean, the global optimal solution of the neutron noise source is calculated.

[0012] The invention relates to a reactor neutron noise source prediction system for realizing the above 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 method solution module and a residual comparison module, wherein: the frequency domain neutron noise Green's function module solves the frequency domain neutron noise Green's function equation according to the reactor geometry and material parameters input by the user by using a numerical discrete method to obtain a reactor neutron noise transfer matrix; the non-zero element iteration module iteratively updates the number and position of non-zero elements according to the dimension of the neutron noise source; the selection matrix construction module constructs a selection matrix according to the number and position of the current non-zero elements; the least squares construction module calculates the matrix Q and the vector x according to the selection matrix and the reactor neutron noise transfer matrix, and constructs a least squares problem; the least squares method solution module calculates the solution of the least squares problem according to the input least squares problem; the residual comparison module obtains the optimal residual by comparison according to a series of input residuals. Technical Effects

[0013] The present invention, based on the transfer matrix of reactor neutron noise obtained by using the Green's function method, regards the noise source prediction problem as an underdetermined linear regression problem. By enumerating subsets of the number and position of noise sources, the problem is decomposed into a series of overdetermined linear regression sub-problems, which can be solved by the least square method. Finally, the optimal estimate of the number and position of noise sources is determined by the residual of the least squares, thereby solving the reactor neutron noise source prediction problem. Compared with the prior art, the present invention significantly improves the computational efficiency of noise source prediction and reduces the computational complexity, and can handle the neutron noise source prediction problem when high-risk components are unknown. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] Figure 1 It is a flow chart of the present invention;

[0015] Figure 2 A schematic diagram of the geometric structure and material arrangement of an embodiment;

[0016] Figure 3 Schematic diagram of the distribution of neutron noise spectrum observation values ​​in the embodiment;

[0017] Figure 4Schematic diagram of the residual distribution of neutron noise source prediction in the embodiment. DETAILED DESCRIPTION

[0018] like Figure 1 As shown, this embodiment relates to a method for predicting reactor neutron noise sources based on the least squares method, comprising:

[0019] Step 1: According to the geometric and material parameters of the reactor, the reactor neutron noise transfer matrix is ​​constructed by numerically discretizing the frequency domain neutron noise Green's function equation, which specifically includes:

[0020] 1.1 Constructing the frequency domain neutron noise Green's function equation is as follows:

[0021] Among them: Ω and Ω' are angle variables, Ω s is the angular variable of the noise source term, is the differential operator, r is the spatial variable, r s is the spatial variable of the noise source term, ω is the frequency variable, ω s is the frequency variable of the noise source term, g and g' are the energy group numbers, g s is the energy group number of the noise source term, is the distribution of the g-th group neutron noise Green's function with respect to spatial variables, angle variables and frequency variables, Similarly, Σ s,g′→g,0 (r,Ω'→Ω) is the distribution of the steady-state neutron scattering cross section from the g'th group to the g'th group and from the Ω' angle to the Ω angle with respect to spatial variables and angular variables, ν is the average number of neutrons produced per fission, Σ f,g′,0 (r) is the distribution of the steady-state neutron fission cross section of the g'th group with respect to spatial variables, is the distribution of the total dynamic neutron cross section of the g-th group with respect to spatial variables and frequency variables, is the distribution of the dynamic neutron energy spectrum of the gth group with respect to spatial variables and frequency variables, For the g s Group, spatial position is r s , angle is Ω s , frequency is ω s The distribution of the unit point source with respect to energy group, spatial variable, angle variable and frequency variable, k eff,0 It is a steady-state effective proliferation factor.

[0022] 1.2 For energy groups g and g s , spatial variables r and r s , angle variables Ω and Ω s , frequency variables ω and ω s After numerical discretization, the neutron noise Green's function Converted into discrete form, the m×n dimensional neutron noise transfer matrix A is obtained by solving the frequency domain neutron noise Green's function equation, satisfying According to prior knowledge from engineering practice, m is usually significantly smaller than n.

[0023] Step 2: Initialize the number and position of non-zero elements in the neutron noise source vector S according to the observed value of the reactor neutron noise spectrum; construct a selection matrix P according to the current position of non-zero elements, specifically including:

[0024] 2.1 When the number and position of non-zero elements in the neutron 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 neutron noise spectrum observation 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 is the transpose of the selection matrix P.

[0025] 2.2 Further transform the equation δφ=AP T PS is rewritten as δφ=Qx, where: matrix Q=AP T ,satisfy The neutron noise source vector x under the condition of the current number and position of non-zero elements is x = PS, satisfying According to prior knowledge from engineering practice, the number k of non-zero elements in the neutron noise source is generally significantly smaller than the number m of neutron noise spectrum observations.

[0026] Step 3: Solve the neutron noise source vector x and its residual e under the condition of the current given number and position of non-zero elements by the least square method, and iteratively update the optimal noise source under the current number of non-zero elements, that is, the solution with the minimum residual among all possible non-zero element positions and its residual, specifically including:

[0027] 3.1 According to the least squares method, under the current given conditions of the number and position of non-zero elements in the neutron noise source, the solution of the equation δφ=Qx is: x=(Q T Q) -1 Q T δφ, the residual is e=||δφ-Qx||, where: e is the residual.

[0028] 3.2 Introduce two-layer loops to search for the optimal solution for the number and position of non-zero elements: the outer loop traverses all possible numbers of non-zero elements, and the inner loop traverses all possible positions of non-zero elements under the condition of a given number of non-zero elements. The optimal solution of the neutron noise source under the current number and position of non-zero elements is obtained by the least squares method. After the two-layer loop is completed, all possible numbers of non-zero elements k are obtained. j , that is, the number of possible non-zero elements of the jth, corresponding to a series of neutron noise sources S j , that is, the neutron noise source corresponding to the number of possible non-zero elements of j and the corresponding series of residuals e j , which is the residual corresponding to the jth possible number of non-zero elements.

[0029] Step 4: By comparing the relative change of the residuals and the ratio of the residual standard deviation to the mean, the global optimal solution of the neutron noise source is calculated. Specifically, for the j+1th residual e j+1 and the jth residual e j The relative change between Then the neutron noise source with the largest relative variation of residual is S t , where: t = argmin j (d j ); When the ratio of the residual standard deviation σ to the residual mean μ is less than the threshold ∈, S 1 is the optimal solution for the neutron noise source; otherwise S t It is the optimal solution for the neutron noise source.

[0030] After specific practical experiments, Figure 2 As an example of the MOX benchmark shown in the figure, in 9 MOX fuel assemblies, 40 UO 2 The fuel assembly and 32 reflector assemblies, each with a side length of 21.42 cm, are placed under vacuum boundary conditions on all four boundaries. The neutron kinetic constant is approximated by a single group of delayed neutrons, and the delayed neutron fraction β = 0.0049s -1 , the delayed neutron precursor decay constant λ=0.0797s -1 , the fast group neutron velocity v 1 =1.82304×10 7 cm / s, thermal group neutron velocity v 2 =4.13067×10 5 cm / s; UO 2 The homogenization group constant of the component is: the fast group diffusion coefficient D 1 =1.2cm, thermal group diffusion coefficient D 2 =0.4cm, fast group absorption cross section Σ a，1 =0.010cm -1 , thermal group absorption cross section Σ a，2=0.100cm -1 , the fast group moves out of the cross section Σ r，1 =0.020cm -1 , fast group neutron production cross section vΣ f，1 =0.0050cm -1 , thermal group neutron production cross section vΣ f，2 =0.125cm -1 ; The homogenization group constant of the MOX component is: the fast group diffusion coefficient D 1 =1.2cm, thermal group diffusion coefficient D 2 =0.4cm, fast group absorption cross section Σ a，1 =0.015cm -1 , thermal group absorption cross section Σ a，2 =0.300cm -1 , the fast group moves out of the cross section Σ r，1 =0.015cm -1 , fast group neutron production cross section νΣ f，1 =0.0075cm -1 , thermal group neutron production cross section vΣ f，2 =0.450cm -1 ; The homogenization group constant of the reflector component is: fast group diffusion coefficient D 1 =1.2cm, thermal group diffusion coefficient D 2 =0.2cm, fast group absorption cross section Σ a，1 =0.001cm -1 , thermal group absorption cross section Σ a，2 =0.04cm -1 , the fast group moves out of the cross section Σ r，1 =0.050cm -1 , fast group neutron production cross section νΣ f，1 =0cm -1 , thermal neutron production cross section νΣ f，2 =0cm -1 .enter Figure 3 The neutron noise spectrum observation value shown in the figure is predicted by this method to obtain the neutron noise source vector and its residual as shown in the figure Figure 4 As shown, the results show that the residual of the present technical solution in this embodiment does not exceed 0.02%.

[0031] Compared with the existing technology, this method is based on the derivation of transport theory and does not rely on diffusion theory. It has a wider applicability to reactor analysis problems with strong absorption and strong anisotropic media. The Green's function method is used to establish a neutron noise source prediction method based on the reactor neutron noise transfer matrix, which makes it unnecessary to carry out complete neutron noise theoretical calculations in the neutron noise source prediction stage. The global optimal solution of the neutron noise source in the least squares sense can be obtained while significantly reducing the computational complexity, and has higher and more stable computational accuracy.

[0032] The above-mentioned specific implementation can be partially adjusted in different ways by those skilled in the art without departing from the principle and purpose of the present invention. The protection scope of the present invention shall be based on the claims and shall not be limited by the above-mentioned specific implementation. Each implementation scheme within its scope shall be subject to the constraints of the present invention.

Claims

1. A method for predicting reactor neutron noise sources based on the least squares method, characterized in that: include: Step 1: According to the geometric and material parameters of the reactor, the reactor neutron noise transfer matrix is ​​constructed by numerically discretizing the frequency domain neutron noise Green's function equation; Step 2: Initialize the number and position of non-zero elements in the neutron noise source according to the observed value of the reactor neutron noise spectrum; construct a selection matrix P according to the current position of the non-zero elements, and use the selection matrix to calculate the matrix Q and the neutron noise source vector x; Step 3: Solve the noise source under the condition of the current given number and position of non-zero elements by the least square method, and iteratively update the optimal noise source under the current number of non-zero elements, that is, the solution with the minimum residual among all possible non-zero element positions and its residual; Step 4: Calculate the global optimal solution of the neutron noise source by comparing the relative change degree of the residual 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 neutron noise spectrum observation value is an m-dimensional vector δφ in the complex domain, satisfying 2. The method for predicting reactor neutron noise sources based on the least squares method according to claim 1, characterized in that: The step 1 specifically includes: 1.1 Constructing the frequency domain neutron noise Green's function equation is as follows: Among them: Ω and Ω' are angle variables, Ω s is the angular variable of the noise source term, ▽ is the differential operator, r is the spatial variable, r s is the spatial variable of the noise source term, ω is the frequency variable, ω s is the frequency variable of the noise source term, g and g' are the energy group numbers, g s is the energy group number of the noise source term, is the distribution of the g-th group neutron noise Green's function with respect to spatial variables, angle variables and frequency variables, Similarly, Σ s,g′→g,0 (r,Ω'→Ω) is the distribution of the steady-state neutron scattering cross section from the g'th group to the g'th group and from the Ω' angle to the Ω angle with respect to spatial variables and angular variables, ν is the average number of neutrons produced per fission, Σ f,g′,0 (r) is the distribution of the steady-state neutron fission cross section of the g'th group with respect to spatial variables, is the distribution of the total dynamic neutron cross section of the g-th group with respect to spatial variables and frequency variables, is the distribution of the dynamic neutron energy spectrum of the gth group with respect to spatial variables and frequency variables, For the g s Group, spatial position is r s , angle is Ω s , frequency is ω s The distribution of the unit point source with respect to energy group, spatial variable, angle variable and frequency variable, k eff,0 It is a steady-state effective proliferation factor; 1.2 For energy groups g and g s , spatial variables r and r s , angle variables Ω and Ω s , frequency variables ω and ω s After numerical discretization, the neutron noise Green's function Converted into discrete form, the m×n dimensional neutron noise transfer matrix A is obtained by solving the frequency domain neutron noise Green's function equation, satisfying According to prior knowledge from engineering practice, m is usually significantly smaller than n.

3. The method for predicting reactor neutron noise sources based on the least squares method according to claim 1, characterized in that: The step 2 specifically includes: 2.1 When the number and position of non-zero elements in the neutron 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 neutron noise spectrum observation 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 is the transpose of the selection matrix P; 2.2 Further transform the equation δφ=AP T PS is rewritten as δφ=Qx, where: matrix Q=AP T ,satisfy The neutron noise source vector x under the condition of the current number and position of non-zero elements is x = PS, satisfying According to prior knowledge from engineering practice, the number k of non-zero elements in the neutron noise source is generally significantly smaller than the number m of neutron noise spectrum observations.

4. The method for predicting reactor neutron noise sources based on the least squares method according to claim 1, characterized in that: The step 3 specifically includes: 3.1 According to the least squares method, under the current given conditions of the number and position of non-zero elements in the neutron noise source, the solution of the equation δφ=Qx is: x=(Q T Q) -1 Q T δφ, the residual is e=||δφ-Qx||, where: e is the residual; 3.2 Introduce two-layer loops to search for the optimal solution for the number and position of non-zero elements: the outer loop traverses all possible numbers of non-zero elements, and the inner loop traverses all possible positions of non-zero elements under the condition of a given number of non-zero elements. The optimal solution of the neutron noise source under the current number and position of non-zero elements is obtained by the least squares method. After the two-layer loop is completed, all possible numbers of non-zero elements k are obtained. j , that is, the number of possible non-zero elements of the jth, corresponding to a series of neutron noise sources S j , that is, the neutron noise source corresponding to the number of possible non-zero elements of j and the corresponding series of residuals e j , which is the residual corresponding to the jth possible number of non-zero elements.

5. The method for predicting reactor neutron noise sources based on the least squares method according to claim 1, characterized in that: The step 4 is specifically as follows: for the j+1th residual e j+1 and the jth residual e j The relative change between Then the neutron noise source with the largest relative variation of residual is S t , where: t = argmin j (d j ); when the ratio of the residual standard deviation σ to the residual mean μ 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 implementing the method described in any one of claims 1 to 5, characterized in that: include: The module comprises 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, wherein: the frequency domain neutron noise Green's function module solves the frequency domain neutron noise Green's function equation according to the reactor geometry and material parameters input by the user by using a numerical discrete method to obtain the reactor neutron noise transfer matrix; the non-zero element iteration module iteratively updates the number and position of non-zero elements according to the dimension of the neutron noise source; the selection matrix construction module constructs a selection matrix according to the number and position of the current non-zero elements; the least squares construction module calculates the matrix Q and the vector x according to the selection matrix and the reactor neutron noise transfer matrix, and constructs a least squares problem; the least squares solution module calculates the solution of the least squares problem according to the input least squares problem; the residual comparison module obtains the optimal residual by comparison according to a series of input residuals.

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