Transportation-burnup coupling calculation method based on dynamic mode decomposition
By introducing boundary control model and DMD prediction in the transportation-fuel consumption coupling calculation method, the problem of large nuclide error when the fuel consumption depth increases in the existing method is solved, and efficient and accurate calculation results are achieved.
Patent Information
- Application Number
- CN202411326692.9
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2024-09-23
- Publication Date
- 2025-02-07
AI Technical Summary
The existing DMD-based transportation-fuel consumption coupling calculation method causes large relative errors to some nuclides as the fuel consumption depth deepens.
A transportation-fuel consumption coupling calculation method based on dynamic modal decomposition is proposed. By introducing a boundary control model, high-fidelity transportation-fuel consumption coupling calculation and DMD prediction are used to reduce the transportation calculation frequency and improve the calculation efficiency.
On the premise of ensuring calculation accuracy, the transportation calculation frequency is significantly reduced, the calculation efficiency is improved, and the relative error of each nuclide density is within an acceptable range.
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Figure CN119400262A_ABST
Abstract
Description
Technical Field
[0001] The present invention belongs to the field of reactor physics calculation, and in particular relates to a transport-burnup coupling calculation method based on dynamic modal decomposition. Background Art
[0002] During reactor operation, fuel composition changes due to neutron irradiation and the radioactive decay of the material itself. Burnup calculations aim to capture the timescale evolution of the fuel composition. Currently, most reactor burnup calculations are based on traditional transport-burnup coupled methods. This method relies on repeatedly invoking transport and burnup calculations to complete the coupled calculations over the entire burnup lifetime. Transport calculations, whether using Monte Carlo or deterministic methods, require multiple solutions to the neutron transport equation. With the development of high-fidelity models, computational costs have increased significantly, limiting their application in practical engineering calculations. To improve the efficiency of reactor transport-burnup coupled calculations, research is needed on novel coupled calculation methods based on traditional methods. These include novel transport-burnup coupled methods based on dynamic mode decomposition (DMD). However, existing DMD-based transport-burnup coupled methods can introduce significant relative errors for some nuclides as the burnup depth increases. Therefore, it is necessary to develop a transport-fuel coupling calculation method based on dynamic modal decomposition, introduce a boundary control model, and effectively reduce the frequency of transport calculations while ensuring calculation accuracy, thereby significantly improving calculation efficiency. Summary of the Invention
[0003] The technical problem addressed by this invention is that existing DMD-based transport-burnup coupling methods can result in significant relative errors for some nuclides as the burnup depth increases. Therefore, this invention proposes a transport-burnup coupling calculation method based on dynamic modal decomposition (DMD), which accounts for the changes in the burnup matrix between adjacent burnup steps. This method significantly reduces computational time and costs throughout the coupled calculation process while maintaining accuracy.
[0004] In order to achieve the above technical features, the object of the present invention is achieved as follows: a transport-fuel consumption coupling calculation method based on dynamic modal decomposition, the method comprising the following steps: S1: Use high-fidelity transport-burnup coupled calculation to obtain the nuclide density and burnup matrix at each burnup step; S2: Determine whether the current fuel consumption step has reached the maximum fuel consumption step: If the current high-fidelity calculation fuel consumption step has not reached the maximum fuel consumption step, then enter S3; If the current high-fidelity calculation burnup step reaches the maximum burnup step, the high-fidelity and predicted calculation data sets are output; S3: Determine whether the high-fidelity calculation boundary has been reached: If the high-fidelity calculation boundary is not reached, enter S1; If the high-fidelity calculation boundary is reached, enter S4; S4: Generate snapshot matrices for the high-fidelity nuclide density and burnup matrix data sets of the time series; S5: DMD prediction is performed on the snapshot matrices of nuclide density and burnup matrix respectively; S6: Determine the nuclide density prediction boundary based on the predicted nuclide density and burnup matrix: S7: Determine whether the current fuel consumption step has reached the maximum fuel consumption step: If the current high-fidelity calculation fuel consumption step has not reached the maximum fuel consumption step, then enter S1; If the current high-fidelity calculation burnup step reaches the maximum burnup step, the high-fidelity and predicted calculation data sets are output.
[0005] Preferably, in S3, the high-fidelity boundary comprises the following steps: S301: Create a snapshot matrix for the currently calculated nuclide density data set : ;(1) Where: for The column vector composed of different nuclides collected at the time; S302: Divide into submatrices and as follows: ;(2) ;(3) S303: Construct the relationship between formula (2) and formula (3) as follows: ;(4) S304: Matrix B The calculation process can be regarded as a Frobenius norm minimization problem: ;(5) ;(6) S305: Use singular value decomposition method to Make a low-order approximation: ;(7) In formula (7), is a left singular matrix, is the transpose of the right singular matrix, is a diagonal matrix with singular values on the main diagonal; S306: Constructing the modal matrix : ;(8) Where: is the truncated left singular matrix, is the transpose of the truncated left singular matrix, is the truncated right singular matrix, is the inverse of the truncated singular value matrix; S307: Calculation The eigenvalues and eigenvectors of : ;(9) In formula (9) is a diagonal matrix with eigenvalues on the diagonal elements. W is the eigenvector matrix; S308: Calculate DMD mode: ;(10) ;(11) S309: Order , Reconstruct the system's DMD agent model: ;(12) S310: Order , Equation (12) can be simplified to obtain the evolution law of nuclide density over time: ; (13) Where: is the time evolution matrix of nuclide density; S311: Use the Frobenius norm to describe the changes in the nuclide evolution matrix: ;(14) ;(15) Where: is a matrix The result of subtraction after normalization is obtained by the difference between the two sets of norms Determine the trend of nuclide density changes to determine the high-fidelity calculation boundary.
[0006] Preferably, in S5, the DMD prediction includes the following steps: S501: Collect the burnup matrix and nuclide density of each step of the high-fidelity calculation and construct a snapshot matrix: ; (16) ; (17) Where: For the front m A snapshot of the burnup matrix for the step, For the front m A snapshot of the nuclide density at the next step; S502: The burnup matrix and nuclide density DMD proxy model can be reconstructed through equations (1) to (13): ; (18) ; (19) S502: The burnup matrix and nuclide density prediction results can be obtained through the DMD proxy model: ; (20) ;(twenty one) Where: for m +1 to k The prediction results of the fuel consumption matrix of the step, For the m +1 to k The predicted results of the nuclide density in the next step.
[0007] Preferably, in S6, the nuclide density prediction boundary is determined by predicting the obtained nuclide density and the burnup matrix: S601: Combine the predicted burnup matrix and reconstruct a new nuclide density matrix: ;(twenty two) ;(twenty three) Where: The first k The nuclide density prediction results of the first step are: For the m +1 to k The nuclide density prediction result is obtained by the burnup matrix; S602: Determine the prediction boundary: ;(twenty four) Where: It is the difference between the nuclide density DMD prediction result and the nuclide density obtained from the DMD prediction burnup matrix.
[0008] S603: When reaching the boundary, the nuclide density of the previous step of the boundary is Transfer to high-fidelity transport calculations.
[0009] Beneficial effects of the present invention: 1. The present invention proposes a transport-burnup coupled calculation method based on dynamic modal decomposition. This method first performs a high-fidelity transport-burnup coupled calculation, selects an appropriate burnup snapshot for prediction using the F-norm, and determines the burnup step boundary for DMD prediction based on the relationship between the burnup matrix and the nuclear density evolution model. Once the prediction boundary is reached, a high-fidelity calculation with an adaptive burnup step number is performed again, and this process is repeated until the end of the lifespan.
[0010] 2. This method significantly reduces the frequency of transport calculations throughout the burnup calculation process, improves computational efficiency, and ensures that the relative error of each nuclide density is within an acceptable range. This method can provide a reference for effectively improving the efficiency of high-fidelity burnup calculations. BRIEF DESCRIPTION OF THE DRAWINGS
[0011] The present invention will be further described below with reference to the accompanying drawings and examples.
[0012] Figure 1 This is a flow chart of a transport-fuel coupling calculation method based on dynamic modal decomposition of the present invention.
[0013] Figure 2 is the relative deviation of 21 actinide nuclides in the transport-burnup coupling calculation method based on dynamic mode decomposition of the present invention.
[0014] Figure 3 The predicted results of the nuclide density from step 71 to step 141 for 21 actinide nuclides in the existing dynamic mode decomposition transport-burnup coupling calculation method. DETAILED DESCRIPTION
[0015] The present invention will be further described in detail below through specific examples. The following examples can enable those skilled in the art to more fully understand the present invention, but are not intended to limit the present invention in any way.
[0016] This example uses the JAEA-published MOX fuel grid cell benchmark to perform DMD predictions for nuclide density. This benchmark uses a constant power of 36.6 W / gHM, a uniform step size of 0.5 GW⋅d / tU, and a final burnup depth of 70 GW⋅d / tU. The burnup database uses the simplified burnup chain provided by MVP, which includes 21 heavy nuclides, 49 fission products, and one pseudonuclide.
[0017] like Figure 1 As shown, the transport-fuel coupling calculation method based on boundary-controlled dynamic mode decomposition provided in this embodiment is specifically performed in the following steps: In the first step, high-fidelity transport-burnup coupled calculations are used to obtain the nuclide density and burnup matrix at each burnup step. The second step is to determine whether the current fuel consumption step has reached the maximum fuel consumption step: If the current high-fidelity calculation fuel consumption step has not reached the maximum fuel consumption step, then enter the third step; If the current high-fidelity calculation burnup step reaches the maximum burnup step, the high-fidelity, predicted calculation data set is output; The third step is to create a snapshot matrix of the currently calculated nuclide density data set : ;(1) Where: for The column vector composed of different nuclides collected at the time; The fourth step is to Divide into submatrices and as follows: ;(2) ;(3) The fifth step is to construct the relationship between formula (2) and formula (3) as follows: ;(4) Step 6: Matrix B The calculation process can be regarded as a Frobenius norm minimization problem: ;(5) ;(6) The seventh step is to use the singular value decomposition method to Make a low-order approximation: ;(7) In formula (7), is a left singular matrix, is the transpose of the right singular matrix, is a diagonal matrix with singular values on the main diagonal; Step 8: Construct the modal matrix : ;(8) Where: is the truncated left singular matrix, is the transpose of the truncated left singular matrix, is the truncated right singular matrix, is the inverse of the truncated singular value matrix; Step 9: Calculation The eigenvalues and eigenvectors of : ;(9) In formula (9) Is a diagonal matrix, and the diagonal elements are eigenvalues, W is the eigenvector matrix; Step 10: Calculate the DMD mode: ;(10) ;(11) Step 11: , The DMD agent model of the system can be reconstructed: ;(12) Step 12: , Equation (12) can be simplified to obtain the evolution law of nuclide density over time: ; (13) In the formula is the time evolution matrix of nuclide density; In the thirteenth step, the Frobenius norm (F norm) is used to describe the changes in the nuclide evolution matrix: ;(14) ;(15) In the formula is a matrix The result of subtraction after normalization is obtained by the difference between the two sets of norms Determine the linearity of the nuclide density change trend to determine the high-fidelity calculation boundary.
[0018] Step 14: Determine whether the high-fidelity calculation boundary has been reached: If the high-fidelity calculation boundary has not been reached, proceed to the first step; If the high-fidelity calculation boundary is reached, proceed to step 15; Step 15: Create snapshot matrices for the high-fidelity nuclide density and burnup matrix data sets of the time series. In the sixteenth step, the snapshot matrices of the nuclide density and burnup matrix are selected and DMD prediction is performed separately. The burnup matrix and nuclide density of each step of the high-fidelity calculation are collected and the snapshot matrix is constructed: ; (16) ; (17) In the formula For the front m A snapshot of the burnup matrix for the step, For the front m A snapshot of the nuclide density at the next step; In the seventeenth step, the burnup matrix and nuclide density DMD proxy model can be reconstructed through equations (1) to (13): ; (18) ; (19) In the eighteenth step, the burnup matrix and nuclide density prediction results can be obtained through the DMD proxy model: ; (20) ;(twenty one) In the formula for m +1 to k The prediction results of the fuel consumption matrix of the step, for m +1 to k The predicted results of the nuclide density in the next step.
[0019] In the nineteenth step, the predicted nuclide density and burnup matrix are used to determine the nuclide density prediction boundary. Combined with the predicted burnup matrix, a new nuclide density matrix is reconstructed: ;(twenty two) ;(twenty three) In the formula The first k The nuclide density prediction results of the first step are: for m +1 to k The nuclide density prediction result is obtained by the burnup matrix; Step 20: Determine the prediction boundary: ;(twenty four) In the formula It is the difference between the nuclide density DMD prediction result and the nuclide density obtained from the DMD prediction burnup matrix.
[0020] Step 21: When reaching the boundary, the nuclide density of the previous step of the boundary is Transfer to high-fidelity transport calculations.
[0021] Step 22: Determine whether the current fuel consumption step has reached the maximum fuel consumption step: If the current high-fidelity calculation fuel consumption step has not reached the maximum fuel consumption step, then enter the first step; If the current high-fidelity calculation burnup step reaches the maximum burnup step, the high-fidelity, predicted calculation data set is output and the current calculation ends.
[0022] According to the above process, the data set used in the present invention is derived from numerical simulation.
[0023] Figure 2are the relative deviations of the 21 actinide nuclides in the transport-burnup coupling calculation method based on dynamic mode decomposition of the present invention. During the entire burnup process, the transport-burnup coupling calculation method based on dynamic mode decomposition can keep the relative deviations of the 21 actinide nuclides within an acceptable range.
[0024] Figure 3 The predicted results of the nuclide density from step 71 to step 141 for 21 actinide nuclides in the existing dynamic mode decomposition transport-burnup coupling calculation method. Figure 2 and Figure 3 Compared with the existing dynamic modal decomposition transport-burnup coupling calculation method, as the burnup depth increases, some nuclides will cause large relative deviations. The transport-burnup coupling calculation method based on boundary-controlled dynamic modal decomposition can better ensure the calculation accuracy and ensure that the relative deviation of nuclides remains within 1%.
[0025] Although the preferred embodiments of the present invention have been described above in conjunction with the accompanying drawings, the present invention is not limited to the above-mentioned specific embodiments. The above-mentioned specific embodiments are merely illustrative and not restrictive. Under the guidance of the present invention, ordinary technicians in this field can also make many forms of specific changes without departing from the scope of protection of the invention and the claims. These all fall within the scope of protection of the present invention.
Claims
1. A transport-fuel coupling calculation method based on dynamic mode decomposition, characterized in that: The method comprises the following steps: S1: Use high-fidelity transport-burnup coupling calculation to obtain the nuclide density and burnup matrix at each burnup step; S2: Determine whether the current fuel consumption step has reached the maximum fuel consumption step: If the current high-fidelity calculation fuel consumption step does not reach the maximum fuel consumption step, enter S3; If the current high-fidelity calculation burnup step reaches the maximum burnup step, the high-fidelity and predicted calculation data sets are output; S3: Determine whether the high-fidelity computing boundary has been reached: If the high-fidelity computing boundary is not reached currently, enter S1; If the high-fidelity computing boundary is reached, enter S4; S4: Generate snapshot matrices for the high-fidelity nuclide density and burnup matrix data sets of the time series; S5: DMD prediction is performed for the snapshot matrices of nuclide density and burnup matrix respectively; S6: Determine the nuclide density prediction boundary by predicting the obtained nuclide density and burnup matrix: S7: Determine whether the current fuel consumption step has reached the maximum fuel consumption step: If the current high-fidelity calculation fuel consumption step does not reach the maximum fuel consumption step, enter S1; If the current high-fidelity calculation burnup step reaches the maximum burnup step, the high-fidelity and predicted calculation data sets are output.
2. According to claim 1, a transport-fuel coupling calculation method based on dynamic mode decomposition is characterized in that: In S3, the high-fidelity boundary comprises the following steps: S301: Create a snapshot matrix of the currently calculated nuclide density data set : ； (1) Where: for The column vector composed of different nuclides collected at the time; S302: Divide into sub-matrices and as follows: ；(2) ；(3) S303: The relationship between formula (2) and formula (3) is constructed as follows: ； (4) S304: Matrix B The calculation process can be regarded as the minimization problem of the Frobenius norm: ；(5) ； (6) S305: Use singular value decomposition method to Make a low-order approximation: ； (7) In formula (7), is a left singular matrix, is the transpose of the right singular matrix, is a diagonal matrix with singular values on the main diagonal; S306: Constructing the modal matrix : ； (8) Where: is the truncated left singular matrix, is the transpose of the truncated left singular matrix, is the truncated right singular matrix, is the inverse of the truncated singular value matrix; S307: Calculation The eigenvalues and eigenvectors of are: ； (9) In formula (9) is a diagonal matrix with eigenvalues on the diagonal elements. W is the eigenvector matrix; S308: Calculate DMD mode: ；(10) ； (11) S309: Order , Reconstruct the DMD agent model of the system: ； (12) S310: Order , formula (12) can be simplified to obtain the evolution law of nuclide density over time: ； (13) Where: is the time evolution law matrix of nuclide density; S311: Use the Frobenius norm to describe the changes in the nuclide evolution matrix: ；(14) ； (15) Where: For the matrix The result of subtraction after normalization is obtained by the difference between the two sets of norms Determine the trend of nuclide density changes to determine the high-fidelity calculation boundary.
3. The method for transport-fuel coupling calculation based on dynamic mode decomposition according to claim 1 is characterized in that: In S5, the DMD prediction includes the following steps: S501: Collect the burnup matrix and nuclide density of each step of high-fidelity calculation and construct a snapshot matrix: ；(16) ； (17) Where: For the front m A snapshot of the burnup matrix for the step, For the front m A snapshot of the nuclide density at the next step; S502: The burnup matrix and nuclide density DMD proxy model can be reconstructed through equations (1) to (13): ；(18) ；(19) S502: The burnup matrix and nuclide density prediction results can be obtained through the DMD proxy model: ；(20) ； (21) Where: for m +1 to k The prediction result of the fuel consumption matrix of the step is For the m +1 to k The predicted results of the nuclide density in the next step.
4. The method for transport-fuel coupling calculation based on dynamic mode decomposition according to claim 1 is characterized in that: In S6, the nuclide density prediction boundary is determined by predicting the obtained nuclide density and the burnup matrix: S601: Combine the predicted burnup matrix to reconstruct a new nuclide density matrix: ；(22) ； (23) Where: is obtained by predicting the burnup matrix k The nuclide density prediction results of the first step are: For the m +1 to k The nuclide density prediction result is obtained by the burnup matrix; S602: Determine the prediction boundary: ； (24) Where: It is the difference between the nuclide density DMD prediction result and the nuclide density obtained from the DMD prediction burnup matrix. S603: When reaching the boundary, the nuclide density of the previous step of the boundary is Transfer to high-fidelity transport calculations.