Reliability optimization method and system for pressurized water reactor nuclear fuel assemblies based on transfer learning

Abstract

The present invention discloses a method and system for optimizing the reliability of a pressurized water reactor nuclear fuel assembly based on transfer learning, comprising: defining a mathematical model for reliability optimization, determining an objective function and a probability constraint function, and upper and lower limits of the distribution of source domain parameters; randomly sampling input data of multiple source domain training samples, calculating true responses, and fusing the input data and corresponding responses to form source domain training samples; constructing a source domain feedforward neural network model by learning the source domain training samples; randomly sampling input data of multiple migration samples from the source domain, calling the source domain feedforward neural network model to predict responses; fusing the input data of the migration samples and the corresponding responses to form a migration sample; calculating the probability density ratio of the source domain to the target domain, and reconstructing the target domain FNN using weighted migration samples; estimating the failure probability gradient using a first-order fractional function method; transmitting the reliability analysis results and gradient information to an optimizer, updating the iteration point and the target domain, and judging whether convergence has occurred; and obtaining the optimal design if convergence has occurred.

Description

Technical Field

[0001] The present invention relates to the technical field of reliability design, and in particular to a method and system for optimizing the reliability of a pressurized water reactor nuclear fuel assembly based on transfer learning. Background Art

[0002] Nuclear fuel assemblies are the fuel input for pressurized water reactors (PWRs). The nuclear fuel within these assemblies generates heat and electricity through a controlled, self-sustaining chain nuclear fission reaction. As the most important PWR internal components and a core component of the nuclear industry, nuclear fuel assemblies are the source of nuclear energy and are known as the "core" of the nuclear industry. During actual nuclear reactor operation, numerous uncertainties exist in the physical properties of the heat transfer medium and the operating conditions within the reactor. The coupling and propagation of these uncertainties can lead to significant fluctuations in the thermal-hydraulic properties of the fuel assemblies within the reactor, potentially causing safety incidents. Reliability-based design optimization (RBDO) is being applied to the optimal design of nuclear fuel assemblies because it can account for the impact of these uncertainties on design performance fluctuations, thereby generating more reliable design solutions.

[0003] At present, the common theoretical methods for solving reliability design optimization (RBDO) are mainly divided into two categories: probabilistic analytical methods and surrogate model methods. The probabilistic analytical methods can be divided into double-loop methods, single-loop methods and decoupling methods according to different optimization strategies. Among them, the main idea of ​​the decoupling method is to split the reliability analysis and optimization process, and let the reliability analysis and optimization design be carried out alternately, thereby transforming the double-loop RBDO problem into a series of deterministic optimization problems, so that the decoupling method has high accuracy and solution stability while improving the solution efficiency. However, when using the RBDO method based on it to carry out the thermal hydraulic reliability optimization design of nuclear fuel assemblies, there are at least the following difficulties: (1) Using computational fluid dynamics to simulate the thermal hydraulics of nuclear fuel assemblies usually requires a lot of time and computing power costs; (2) The reliability optimization design process requires reliability analysis of the nuclear fuel assembly function function at each iteration step, resulting in the RBDO method having very low computational efficiency in solving the thermal hydraulic reliability design of nuclear fuel assemblies, which is difficult to meet the needs of actual engineering.

[0004] Therefore, a method and system for reliability optimization of pressurized water reactor nuclear fuel assemblies based on transfer learning are provided to achieve high-performance reliability optimization design of thermal hydraulics of nuclear fuel assemblies. Summary of the Invention

[0005] In order to solve the above defects in the prior art, the present invention provides a pressurized water reactor nuclear fuel assembly reliability optimization method and system based on transfer learning.

[0006] The present invention provides a method for optimizing the reliability of a pressurized water reactor nuclear fuel assembly based on transfer learning, comprising the following steps:

[0007] S1. Based on the requirements of thermal-hydraulic reliability optimization design for nuclear fuel assemblies, define a mathematical model for reliability optimization, including determining the design variables, random parameters, and probability distribution of nuclear fuel assemblies, and determine the objective function and probability constraint function;

[0008] S2. Define the input data, corresponding response, and probability distribution information of the initial iteration point as the source domain, and define the input data, corresponding response, and probability distribution information of the subsequent iteration points as the target domain; set the number of iterations, the initial design point, and the positive tolerance;

[0009] S3. Based on the actual operating conditions of nuclear fuel assemblies and the requirements of thermal-hydraulic reliability optimization design, the distribution of the input parameters of the source domain training samples is set to follow a uniform distribution, and its upper and lower limits are determined. S4. The relationship between the input parameters and the output responses in the source domain is established. Based on the upper and lower limits of the input parameters determined in S3, the input data of multiple source domain training samples are randomly sampled through a uniform distribution, and the true responses are calculated using CFD methods. The input data of the source domain training samples and the corresponding responses are then fused to form the source domain training samples.

[0010] S5. Build and train a source domain feedforward neural network model by learning source domain training samples. When the mean square error of the neural network model is less than 0.001, the model training is completed and put into use, and the next step is entered.

[0011] S6. Randomly sample input data of multiple transfer samples from the source domain according to a uniform distribution, and use the source domain feedforward neural network model to predict their responses; fuse the input data of the transfer samples and the corresponding responses to form a transfer sample;

[0012] S7. Calculate the probability density ratio between the source domain and the target domain, use the probability density ratio as the weight of the source domain migration sample, and use the weighted migration sample to reconstruct the target domain FNN;

[0013] S8. Perform reliability analysis using the MCS method; based on the marginal distribution of the target domain, generate input data for multiple target domain MCS samples, use the target domain FNN to predict the corresponding response, and then obtain the probability distribution function of the predicted response for reliability analysis;

[0014] S9. Use the first-order fractional function method to directly estimate the failure probability gradient based on the failure conditions of the MCS samples;

[0015] S10. The reliability analysis results obtained in S8 and the gradient information obtained in S9 are passed to the optimizer, the iteration point and the target domain are updated, and convergence is determined. If converged, the optimal design is obtained and S11 is executed. If not, the design point is iteratively updated and steps S7-10 are repeated.

[0016] The convergence condition is that the difference between the current objective function value and the previous objective function value, as well as the difference between the current iteration point and the previous iteration point, is less than 0.001.

[0017] S11. Output the optimal design and end.

[0018] In S1, the mathematical model of reliability optimization design is:

[0019] (1),

[0020] in, and Represent the objective function and Functions, is the counting symbol in the formula, indicating the number of functional functions or constraints. Indicates the number of constraints, that is, the number of functional functions; express dimensional design variable vector, Indicates the mean A vector of random variables, 、 denote the upper and lower bounds of the design variable vector respectively; represents probability calculation, For the The failure probability of a constraint, Indicates the The target failure probability of the constraint; Defined as a failure event, the probability constraint is defined as The probability of failure is less than or equal to the target failure probability , the task of reliability optimization design is to minimize the objective function while satisfying the probability constraints; where t represents the target value.

[0021] In S2, set the initial number of iterations , the initial design point is expressed as , the allowable error is expressed as , and the source domain is represented as S, the input data is In S3, according to the actual operating conditions of nuclear fuel assemblies and the requirements of thermal hydraulic reliability optimization design, the upper and lower limits of the distribution of source domain parameters are determined, that is, the input data of the source domain training samples Obey the uniform distribution, expressed as:

[0022] (2).

[0023] In S4, random sampling is performed according to uniform distribution. Source domain training sample input data , the corresponding response is obtained by computational fluid dynamics simulation: ,Right now:

[0024] (3);

[0025] in, is a counting symbol, indicating the number of training samples;

[0026] Further expressed as:

[0027] (4);

[0028] in, for dimensional vector, Represents the number of functional functions; the input data and the corresponding response are fused to form the source domain training sample, then The training sample of a function is expressed as:

[0029] (5).

[0030] In S5, the constructed source domain FNN is expressed as follows:

[0031] (6);

[0032] in Indicates the The source domain FNN of the functional function, and represents the weight vector and bias vector of the network, represents the hidden layer activation function;

[0033] Select the Tanh function as the activation function of the hidden layer:

[0034] (7);

[0035] Given a training sample , the output of the training sample Output predicted by the feedforward neural network The mean square error (MSE) between is used as the loss function:

[0036] (8);

[0037] The source domain training samples contain the source domain input data and corresponding responses , the obtained FNN captures the mapping relationship between the source domain input data and the response, as the source domain input data and response The neural network model between is expressed as:

[0038] (9);

[0039] in, is an estimate of the actual response.

[0040] In S6, random samples are taken from the source domain according to a uniform distribution. Migration sample input data , call the source domain FNN to calculate the corresponding response estimate ,Right now:

[0041] (10);

[0042] in, is a counting symbol, indicating the number of the migration sample.

[0043] The input data and the corresponding response are fused to form a migration sample. Function The migration samples are expressed as:

[0044] (11);

[0045] In S7, the probability density ratio is calculated as follows:

[0046] (12),

[0047] in and Represent the source domain and target domain random variables respectively The joint probability density distribution function of ; the probability density ratio is used as the weight of the source domain migration sample, and the weighted migration sample is used to reconstruct the target domain FNN. The new loss function expression is:

[0048] (13);

[0049] The Levenberg-Marquardt backpropagation algorithm is used to minimize the loss function to determine the weights and biases of the target domain FNN. The FNN reconstructed based on sample transfer learning can be used as the target domain input data. and response The neural network model between is expressed as:

[0050] (14).

[0051] In S8, according to the marginal distribution of the target domain, the input data of N target domain MCS samples are generated, which are recorded as , using the target domain FNN to estimate Response:

[0052] (15);

[0053] The probability distribution function of the estimated response is obtained for reliability analysis, where the MCS samples are divided into two categories: failure and safety according to the estimated response. When , it indicates failure; when When , it indicates a safe situation; the failure indication function is as follows:

[0054] (16);

[0055] Then, based on the number of failed samples, the failure probability can be approximately estimated:

[0056] (17).

[0057] In S9, the first-order fractional function method is used to estimate the gradient failure probability according to the failure condition of the MCS sample. design variables The partial derivative of is:

[0058] (18);

[0059] in, Indicates that at the iteration point The target domain random variable The joint probability density distribution function of .

[0060] In S6, in the source domain and target domain When the feature space and conditional probability distribution are the same, the optimal neural network model is reconstructed for the target domain based on sample transfer learning when there is no response information in the target domain; specifically, assuming that the target domain is reconstructed with The neural network model with parameters is based on the empirical risk minimization theory. It is obtained by solving the following optimization problem:

[0061] (19);

[0062] in represents the empirical risk, yes The loss function of

[0063] By using Bayes’ theorem and the definition of expectation, the probability density ratio calculation formula (12) can be rewritten as:

[0064] (20);

[0065] That is, using the responsive samples in the source domain, we can learn by minimizing the weighted expected risk. ;

[0066] Since the conditional probability distributions of the target domain and the source domain are already the same, that is, ; The joint probability distribution can be decomposed into ,get , formula (20) is further rewritten as:

[0067] (twenty one);

[0068] That is, the sample The weights of the target domain and the source domain in the data point The ratio of the marginal distribution probability density at ;

[0069] Given a source domain, there are corresponding migration samples , by defining the probability density ratio , formula (21) is rewritten as:

[0070] (twenty two).

[0071] At the same time, the present invention also provides a pressurized water reactor nuclear fuel assembly reliability optimization system based on transfer learning, which uses the pressurized water reactor nuclear fuel assembly reliability optimization method based on transfer learning as described above to perform reliability optimization.

[0072] The technical solution of the present invention has the following advantages:

[0073] The transfer learning-based pressurized water reactor nuclear fuel assembly reliability optimization method provided by the present invention effectively alleviates the repeated reliability analysis in the RBDO optimization iteration process through the transfer learning method; when reconstructing the target domain model, based on the fact that the marginal probability distributions of the source domain and the target domain are known during the RBDO solution process, the probability density ratio of the migration samples can be accurately calculated. Through the accurate probability density ratio, an optimal model can be constructed for the target domain, fundamentally ensuring the accuracy of the reliability analysis, thereby achieving high-precision solution to the RBDO problem; the present invention extracts source domain samples with a uniform distribution and constructs an FNN source domain model. In addition to the source domain samples, no additional function evaluation is required during the entire optimization process, which effectively reduces the number of experiments and simulations and achieves efficient solution to complex RBDO problems. BRIEF DESCRIPTION OF THE DRAWINGS

[0074] In order to more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the specific embodiments or the description of the prior art. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0075] Figure 1 This is a flow chart of the method for reliability optimization of pressurized water reactor nuclear fuel assemblies based on transfer learning according to the present invention;

[0076] Figure 2 This is a schematic structural diagram of a nuclear fuel assembly according to Example 3 of the present invention;

[0077] Figure 3 This is a schematic diagram of a nuclear fuel assembly simulation model according to Example 3 of the present invention;

[0078] Figure 4 This is a schematic diagram of the heat flux density area division described in Example 3 of the present invention;

[0079] Figure 5 This is a graph showing the convergence process of the objective function described in Example 3 of the present invention;

[0080] Figure 6 This is a constraint failure probability change curve diagram described in Example 3 of the present invention. DETAILED DESCRIPTION

[0081] The technical solution of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the embodiments described are only some embodiments of the present invention, not all embodiments. All other embodiments obtained by ordinary technicians in this field based on the embodiments of the present invention without making any creative efforts shall fall within the scope of protection of the present invention.

[0082] In addition, the technical features involved in the different embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0083] Example 1: Figure 1 As shown, this embodiment provides a method for optimizing the reliability of a pressurized water reactor nuclear fuel assembly based on transfer learning, comprising the following steps:

[0084] S1. Based on the requirements of thermal-hydraulic reliability optimization design for nuclear fuel assemblies, define a mathematical model for reliability optimization, including determining the design variables, random parameters, and probability distribution of nuclear fuel assemblies, and determine the objective function and probability constraint function;

[0085] Specifically, in S1, the mathematical model of reliability optimization design is:

[0086] (1),

[0087] in, and Represent the objective function and Functions, is the counting symbol in the formula, indicating the number of functional functions or constraints. Indicates the number of constraints, that is, the number of functional functions; express dimensional design variable vector, Indicates the mean A vector of random variables, 、 denote the upper and lower bounds of the design variable vector respectively; represents probability calculation, For the The failure probability of a constraint, Indicates the The target failure probability of the constraint; Defined as a failure event, the probability constraint is defined as The probability of failure is less than or equal to the target failure probability , the task of reliability optimization design is to minimize the objective function while satisfying the probability constraints; where t represents the target value.

[0088] S2. Set the initial number of iterations , the initial design point is expressed as , the allowable error is expressed as , and the source domain is represented as S, the input data is ;

[0089] S3. Based on the actual operating conditions of nuclear fuel assemblies and the requirements of thermal-hydraulic reliability optimization design, the distribution of the input parameters of the source domain training samples is set to follow a uniform distribution, and its upper and lower limits are determined;

[0090] Specifically, in S3, according to the actual operating conditions of the nuclear fuel assembly and the requirements of thermal hydraulic reliability optimization design, the upper and lower limits of the distribution of the source domain parameters, that is, the input data of the source domain training samples, are determined. Obey the uniform distribution, expressed as:

[0091] (2).

[0092] S4. Construct a relationship between the input parameters and the output responses in the source domain. Based on the upper and lower bounds of the input parameters determined in S3, randomly sample the input data of multiple source domain training samples using a uniform distribution, and calculate the true responses using CFD methods. Then, fuse the input data of the source domain training samples and the corresponding responses to form the source domain training samples.

[0093] Specifically, in S4, random sampling is performed according to uniform distribution. Source domain training sample input data , the left side of the equation represents all input data, which contains many categories, and each individual X on the right side represents each individual category, such as Indicates flow rate, Indicates heat flux density, etc. Each letter X on the right also contains ns of actual sampled numbers;

[0094] The corresponding response obtained by computational fluid dynamics simulation is: ,Right now:

[0095] (3);

[0096] in, is a counting symbol, indicating the number of training samples;

[0097] Further expressed as:

[0098] (4);

[0099] in, for dimensional vector, Represents the number of functional functions; the input data and the corresponding response are fused to form the source domain training sample, then The training sample of a function is expressed as:

[0100] (5);

[0101] S5. Build and train a source domain feedforward neural network model by learning source domain training samples. When the mean square error of the neural network model is less than 0.001, the model training is completed and put into use, and the next step is entered.

[0102] Specifically, in S5, the constructed source domain FNN is expressed as follows:

[0103] (6);

[0104] in Indicates the The source domain FNN of the functional function, and represents the weight vector and bias vector of the network, represents the hidden layer activation function;

[0105] Select the Tanh function as the activation function of the hidden layer:

[0106] (7);

[0107] Given a training sample , the output of the training sample Output predicted by the feedforward neural network The mean square error (MSE) between is used as the loss function:

[0108] (8);

[0109] The Levenberg-Marquardt backpropagation algorithm is used to minimize the loss function to determine the weights and biases of the target domain FNN; the source domain training sample contains the source domain input data and corresponding responses , the obtained FNN captures the mapping relationship between the source domain input data and the response, as the source domain input data and response The neural network model between is expressed as:

[0110] (9);

[0111] in, is an estimate of the actual response.

[0112] S6. Randomly sample input data of multiple transfer samples from the source domain according to a uniform distribution, and use the source domain feedforward neural network model to predict their responses; fuse the input data of the transfer samples and the corresponding responses to form a transfer sample;

[0113] Specifically, in S6, random samples are taken from the source domain according to a uniform distribution. Migration sample input data , call the source domain FNN to calculate the corresponding response estimate ,Right now:

[0114] (10);

[0115] in, is a counting symbol, indicating the number of the migration sample.

[0116] The input data and the corresponding response are fused to form a migration sample. Function The migration samples are expressed as:

[0117] (11).

[0118] S7. Calculate the probability density ratio between the source domain and the target domain, use the probability density ratio as the weight of the source domain migration sample, and use the weighted migration sample to reconstruct the target domain FNN;

[0119] Specifically, in S7, the probability density ratio calculation formula is as follows:

[0120] (12),

[0121] in and Represent the source domain and target domain random variables respectively The probability density distribution function of the source domain is used as the weight of the source domain migration sample. The target domain FNN is reconstructed using the weighted migration sample. The new loss function expression is:

[0122] (13);

[0123] The Levenberg-Marquardt backpropagation algorithm is used to minimize the loss function to determine the weights and biases of the target domain FNN; the FNN reconstructed based on sample transfer learning can be used as the target domain input data and response The neural network model between is expressed as:

[0124] (14).

[0125] S8. Perform reliability analysis using the MCS method; based on the marginal distribution of the target domain, generate input data for multiple target domain MCS samples, use the target domain FNN to predict the corresponding response, and then obtain the probability distribution function of the predicted response for reliability analysis;

[0126] Specifically, in S8, according to the edge distribution of the target domain, the input data of N target domain MCS samples are generated, which are recorded as , using the target domain FNN to estimate Response:

[0127] (15);

[0128] The probability distribution function of the estimated response is obtained for reliability analysis, where the MCS samples are divided into two categories: failure and safety according to the estimated response. When , it indicates failure; when When , it indicates a safe situation; the failure indication function is as follows:

[0129] (16);

[0130] Then, based on the number of failed samples, the failure probability can be approximately estimated:

[0131] (17).

[0132] S9. Using the first-order fractional function method, the failure probability gradient is directly estimated according to the failure situation of the MCS sample. Specifically, in S9, the first-order fractional function method is used to estimate the gradient failure probability according to the failure situation of the MCS sample. design variables The partial derivative of is:

[0133] (18);

[0134] in, Indicates that at the iteration point The target domain random variable The joint probability density distribution function of .

[0135] S10. The reliability analysis results obtained in S8 and the gradient information obtained in S9 are passed to the optimizer, the iteration point and the target domain are updated, and convergence is determined. If converged, the optimal design is obtained and S11 is executed. If not, the design point is iteratively updated and steps S7-10 are repeated.

[0136] The convergence condition is that the difference between the current objective function value and the previous objective function value, as well as the difference between the current iteration point and the previous iteration point, is less than 0.001.

[0137] S11. Output the optimal design and end.

[0138] At the same time, as a transformable implementation, in S6, in the source domain and target domain When the feature space and conditional probability distribution are the same, the optimal neural network model is reconstructed for the target domain based on sample transfer learning when there is no response information in the target domain; specifically, assuming that the target domain is reconstructed with The neural network model with parameters is based on the empirical risk minimization theory. It is obtained by solving the following optimization problem:

[0139] (19);

[0140] in represents the empirical risk, yes The loss function of

[0141] By using Bayes’ theorem and the definition of expectation, the probability density ratio calculation formula (12) can be rewritten as:

[0142] (20);

[0143] That is, using the responsive samples in the source domain, we can learn by minimizing the weighted expected risk. ;

[0144] Since the conditional probability distributions of the target domain and the source domain are already the same, that is, ; The joint probability distribution can be decomposed into ,get , formula (20) is further rewritten as:

[0145] (twenty one);

[0146] That is, the sample The weights of the target domain and the source domain in the data point The ratio of the marginal distribution probability density at ;

[0147] Given a source domain, there are corresponding migration samples , by defining the probability density ratio , formula (21) is rewritten as:

[0148] (twenty two).

[0149] Example 2: Based on Example 1, this example further provides a pressurized water reactor nuclear fuel assembly reliability optimization system based on transfer learning. The system uses the pressurized water reactor nuclear fuel assembly reliability optimization method based on transfer learning as described in Example 1 to perform reliability design on the pressurized water reactor nuclear fuel assembly.

[0150] Example 3: Based on Example 1, this example further provides a specific example application of a pressurized water reactor nuclear fuel assembly reliability optimization method based on transfer learning.

[0151] During nuclear reactor operation, the maximum fuel rod surface temperature is a critical parameter, directly impacting the thermal performance and safety of the reactor. To ensure safe operation, the thermal design guidelines for the reactor core stipulate that the maximum fuel rod surface temperature must be below a certain limit to maintain reactor integrity and prevent the leakage of radioactive materials. The fuel rod surface temperature is dependent on a variety of factors, including the cooling water inlet velocity, the distribution of heat flux density across the fuel rods, and overall heat transfer efficiency. Specifically, a lower maximum fuel rod surface temperature translates to a greater safety margin, a longer cladding life, and the potential for serious accidents caused by cladding melt. However, excessively low maximum fuel rod surface temperatures can reduce the overall heat transfer efficiency of the core, leading to lower economic benefits. Therefore, during nuclear reactor operation, factors such as overall heat transfer efficiency and pressure drop must be comprehensively considered to ensure economic benefits. High heat exchange efficiency can ensure that the heat inside the fuel assembly is fully utilized, which helps to improve the thermal efficiency of the nuclear reactor; reasonable control of the pressure drop of cooling water can ensure the normal operation of the nuclear fuel assembly and cooling system, reduce system pressure fluctuations, and at the same time reduce the system's energy consumption and improve economy.

[0152] like Figure 2 、 3 As shown in Figure 4, this example aims to minimize the maximum temperature of the fuel rod outer surface, takes the total heat transfer efficiency and cooling water pressure drop as constraints, and performs reliability optimization design on a 5×5 grid fuel assembly with stirring fins under a constant total heat flux density.

[0153] S1. Based on the requirements of thermal-hydraulic reliability optimization design of nuclear fuel assemblies, define a mathematical model for reliability optimization, including determining the design variables, random parameters, and probability distribution of nuclear fuel assemblies, and determine the objective function and probability constraint function.

[0154] In this embodiment, the temperature rise of the cooling water represents the heat transfer efficiency of the fuel assembly, so the temperature rise is used as the heat transfer efficiency constraint. At the same time, the cooling water inlet velocity and the heat flux power factor of each fuel rod region are used as random variables, and the mean of the random variables is used as the design variable. For the heat flux distribution of the fuel assembly, the main consideration is the difference in heat flux density of the fuel rods in different radial regions; for example, Figure 4 As shown in the figure, the 5×5 fuel assembly is divided into three areas A, B, and C, each containing 1, 8, and 16 fuel rods. According to the radial power distribution characteristics of the core under actual operating conditions, the heat flux density power factor of different radial areas is determined. The value range and the total power factor, the axial heat flux density distribution is set according to the axial power factor confirmation function:

[0155] ,

[0156] in, is the heat flux power factor, is the axial distance of the fuel rods.

[0157] The random variable and parameter distribution information is shown in Table 1. Since the total heat flux density is constant, the power factor of the heat flux density of the fuel rods in area B can be directly calculated after the power factors of the heat flux density of the fuel rods in areas A and C are designed. Therefore, the power factor of the heat flux density of the fuel rods in area B is used as a random parameter.

[0158] Table 1 Random variables and parameter distribution information

[0159]

[0160] The cooling water inlet flow rate and the power factor of the fuel rod heat flux in different regions of the 5×5 grid fuel assembly with mixing wings are used as design variables. The maximum temperature of the fuel rod surface is used as the design target, and the fuel assembly heat transfer efficiency and cooling water pressure drop are used as constraints. The RBDO mathematical model of the fuel assembly is established as follows:

[0161] (1-1);

[0162] in, Indicates the maximum temperature of the fuel rod surface, Indicates the cooling water temperature rise, represents the cooling water pressure drop; based on the model simulation results and previous experience, the minimum cooling water temperature rise is set to 2°C and the maximum cooling water pressure drop is set to 4400Pa; is the target failure probability.

[0163] S2. Set the input parameters, corresponding responses, and the probability distribution information of the two that obey the data in Table 1 as the source domain, and set the input data, corresponding responses, and the probability distribution information of the two at each subsequent iteration point as the target domain; set the number of iterations , initial design point , set the allowable error , the allowable error is set to a small positive number.

[0164] S3. According to the actual operating conditions of nuclear fuel assemblies and the requirements of thermal hydraulic reliability optimization design, determine the upper and lower limits of the distribution of source domain parameters, that is, the input data of the source domain training samples Obey the uniform distribution:

[0165] (2-1);

[0166] S4. Maximum fuel rod surface temperature Design variables only function of the cooling water temperature rise and pressure drop is a random variable function, so two different sets of training samples are required and .

[0167] Assume that the two sets of data each contain 40 samples. Among them, the training samples Used to train the maximum temperature of the fuel rod surface The neural network model has input data uniformly distributed Randomly generated, the power factor of the fuel rod heat flux density in area B is based on Calculated; training samples Used to train cooling water temperature rise and pressure drop The neural network model has input data uniformly distributed and After randomly generating and obtaining the input data, ANSYS FLUENT software is used to calculate the corresponding response.

[0168] S5. Construct a source domain feedforward neural network model based on the training samples. The size of the hidden layer is set to 1, and the MSE is used as the loss function. The Levenberg-Marquardt backpropagation algorithm is used for backpropagation to minimize the loss function.

[0169] Since feedforward neural networks are generally unbiased in their estimation of training sample points, training samples cannot directly verify the accuracy of the constructed network. Therefore, 10 more sample points were generated using the same sample generation method described above to verify the accuracy of the constructed feedforward neural network. Tables 2 and 4 provide information on the test sample points, and Tables 3 and 5 show the accuracy test results of the neural network model. The maximum relative errors of the results were 0.08%, 0.53%, and 1.59%, respectively, all within acceptable ranges. Therefore, the constructed feedforward neural network model can be used in subsequent reliability optimization design.

[0170] Table 2 Fuel rod surface maximum temperature test sample points

[0171]

[0172] Table 3 Fuel rod surface maximum temperature test results

[0173]

[0174] Table 4 Cooling water temperature rise and pressure drop test sample points

[0175]

[0176] Table 5 Cooling water temperature rise and pressure drop test results

[0177]

[0178] S6. Randomly sample from the source domain according to uniform distribution Migration sample input data , call the source domain FNN to calculate the corresponding response estimate ,Right now:

[0179] (10-1);

[0180] The input data and the corresponding response are fused to form the migration sample. Function The migration samples are expressed as:

[0181] (11-1);

[0182] S7. Calculate the probability density ratio between the source domain and the target domain and reconstruct the target domain FNN. The probability density ratio calculation formula is as follows:

[0183] (12-1);

[0184] in, and Represent the source domain and target domain random variables respectively The joint probability density distribution function of ; the probability density ratio is used as the weight of the source domain migration sample, and the weighted migration sample is used to reconstruct the target domain FNN. The new loss function expression is:

[0185] (13-1);

[0186] The Levenberg-Marquardt backpropagation algorithm is used to minimize the loss function to determine the weights and biases of the target domain FNN. The FNN reconstructed based on sample transfer learning can be used as the target domain input data. and response The neural network model between is expressed as:

[0187] (14-1);

[0188] S8. After using the migration samples to build an accurate target domain FNN model, the MCS method is used for reliability analysis. According to the marginal distribution of the target domain, the input data of N target domain MCS samples is generated, which is recorded as In this example, we set the sampling rate to 10 6 Then use the target domain FNN to estimate Response:

[0189] (15-1);

[0190] The probability distribution function of the estimated response is obtained for reliability analysis, where the MCS samples are divided into two categories: failure and safety according to the estimated response. When , it indicates failure; when When , it indicates a safe situation; the failure indication function is as follows:

[0191] (16-1);

[0192] Then, based on the number of failed samples, the failure probability is approximately estimated:

[0193] (17-1);

[0194] S9. Using the first-order fractional function method, the gradient is directly estimated according to the failure of the MCS sample; the failure probability is design variables The partial derivative of is:

[0195] (18-1);

[0196] in, Indicates that at the iteration point The target domain random variable The joint probability density distribution function of .

[0197] S10. Pass the reliability analysis results obtained in S8 and the gradient information obtained in S9 to the optimizer, update the iteration point and target domain, and check whether they converge; if converged, the optimal design is obtained , execute step 11; if it does not converge, set the number of iterations , repeat S7-S10.

[0198] S11. Output optimal design ,Finish.

[0199] Using the above steps, from the initial iteration point Initially, this embodiment obtained the optimal design after 28 iterations. , the convergence process of the objective function is as follows Figure 5 As shown; Figure 6 (a) and (b) respectively show the changes in the failure probability of the two probabilistic constraints during the optimization process.

[0200] The results of the design variable optimization are shown in Table 6. From the results, it can be seen that the maximum temperature of the fuel rod surface dropped from 336.94℃ in the initial design to 334.34℃ in the optimal design. In order to verify the accuracy of the reliability analysis, Table 7 compares the failure probability values ​​of sample transfer learning and computational fluid dynamics simulation at the initial point and the optimal point. The comparison shows that the failure probability values ​​calculated by the two are very close, whether at the initial point or the optimal point; only at the optimal point, the computational fluid dynamics simulation is very close to the failure probability values ​​calculated by the two. The calculated failure probability is 0.0107, which is slightly higher than the 0.0054 of sample transfer learning.

[0201] These results demonstrate that, by optimizing the cooling water inlet velocity and fuel rod heat flux density in each region, the optimal design reduces the maximum fuel rod surface temperature by 2.6°C compared to the initial design. Furthermore, it ensures a cooling water temperature rise of greater than 2°C and a pressure drop of less than 4400 Pa with a probability of over 99%, thereby improving fuel assembly safety while ensuring its economic benefits. Furthermore, this method achieves reliability optimization at a low computational cost. A total of 80 simulations were performed, including 40 neural network models for the maximum fuel rod surface temperature and 40 neural network models for the cooling water temperature rise and pressure drop.

[0202] Table 6 Design variable optimization results

[0203]

[0204] Table 7 Comparison of failure probability results

[0205]

[0206] Obviously, the above embodiments are merely examples for clarity of explanation and are not intended to limit the implementation methods. Those skilled in the art will readily appreciate that other variations or modifications based on the above descriptions are possible. It is not necessary and impossible to enumerate all implementation methods here. Obvious variations or modifications arising therefrom remain within the scope of protection of the present invention.

Claims

1. A reliability optimization method for pressurized water reactor nuclear fuel assemblies based on transfer learning, characterized in that: The following steps are involved: S1. Based on the requirements of thermal-hydraulic reliability optimization design for nuclear fuel assemblies, define a mathematical model for reliability optimization, including determining the design variables, random parameters, and probability distribution of nuclear fuel assemblies, and determine the objective function and probability constraint function; S2. Define the input data, corresponding response, and probability distribution information of the initial iteration point as the source domain, and define the input data, corresponding response, and probability distribution information of the subsequent iteration points as the target domain; set the number of iterations, the initial design point, and the positive tolerance; S3. Based on the actual operating conditions of nuclear fuel assemblies and the requirements of thermal-hydraulic reliability optimization design, the distribution of the input parameters of the source domain training samples is set to follow a uniform distribution, and its upper and lower limits are determined; S4. Construct a relationship between the input parameters and the output responses in the source domain. Based on the upper and lower bounds of the input parameters determined in S3, randomly sample the input data of multiple source domain training samples using a uniform distribution, and calculate the true responses using CFD methods. Then, fuse the input data of the source domain training samples and the corresponding responses to form the source domain training samples. S5. Build and train a source domain feedforward neural network model by learning source domain training samples. When the mean square error of the neural network model is less than 0.001, the model training is completed and put into use, and the next step is entered. S6. Randomly sample input data of multiple transfer samples from the source domain according to a uniform distribution, and use the source domain feedforward neural network model to predict their responses; fuse the input data of the transfer samples and the corresponding responses to form a transfer sample; S7. Calculate the probability density ratio between the source domain and the target domain, use the probability density ratio as the weight of the source domain migration sample, and use the weighted migration sample to reconstruct the target domain FNN; S8. Perform reliability analysis using the MCS method; based on the marginal distribution of the target domain, generate input data for multiple target domain MCS samples, use the target domain FNN to predict the corresponding response, and then obtain the probability distribution function of the predicted response for reliability analysis; S9. Use the first-order fractional function method to directly estimate the failure probability gradient based on the failure conditions of the MCS samples; S10. The reliability analysis results obtained in S8 and the gradient information obtained in S9 are passed to the optimizer, the iteration point and the target domain are updated, and convergence is determined. If converged, the optimal design is obtained and S11 is executed. If not, the design point is iteratively updated and steps S7-10 are repeated. The convergence condition is that the difference between the current objective function value and the previous objective function value, as well as the difference between the current iteration point and the previous iteration point, is less than 0.

001. S11. Output the optimal design and end.

2. The method for optimizing the reliability of pressurized water reactor nuclear fuel assemblies based on transfer learning according to claim 1, characterized in that: In S1, the mathematical model of reliability optimization design is: ， in, and Represent the objective function and Functions, is the counting symbol in the formula, indicating the number of functional functions or constraints. Indicates the number of constraints, that is, the number of functional functions; express dimensional design variable vector, Indicates the mean A vector of random variables, 、 denote the upper and lower bounds of the design variable vector respectively; represents probability calculation, For the The failure probability of a constraint, Indicates the The target failure probability of the constraint; Defined as a failure event, the probability constraint is defined as The probability of failure is less than or equal to the target failure probability , the task of reliability optimization design is to minimize the objective function while satisfying the probability constraints; Indicates the target value.

3. The method for optimizing the reliability of pressurized water reactor nuclear fuel assemblies based on transfer learning according to claim 1, characterized in that: In S2, set the initial number of iterations , the initial design point is expressed as The allowable error is expressed as , and the source domain is represented as S, the input data is In S3, according to the actual operating conditions of nuclear fuel assemblies and the requirements of thermal hydraulic reliability optimization design, the upper and lower limits of the distribution of source domain parameters are determined, that is, the input data of the source domain training samples Obey the uniform distribution, expressed as: .

4. The method for optimizing the reliability of pressurized water reactor nuclear fuel assemblies based on transfer learning according to claim 1, characterized in that: In S4, random sampling is performed according to uniform distribution. Source domain training sample input data , the corresponding response is obtained by computational fluid dynamics simulation: ,Right now: (3); in, is a counting symbol, indicating the number of training samples; Further expressed as: ， in, for dimensional vector, Represents the number of functional functions; the input data and the corresponding response are fused to form the source domain training sample, then The training sample of a function is expressed as: ； In S5, the constructed source domain FNN is expressed as follows: （6）； in Indicates the The source domain FNN of the functional function, and represents the weight vector and bias vector of the network, represents the hidden layer activation function; Select the Tanh function as the activation function of the hidden layer: （7）； Given a training sample , the output of the training sample Output predicted by the feedforward neural network The mean square error (MSE) between is used as the loss function: （8）； The Levenberg-Marquardt backpropagation algorithm is used to minimize the loss function to determine the weights and biases of the target domain FNN; the source domain training sample contains the source domain input data and corresponding responses , the obtained FNN captures the mapping relationship between the source domain input data and the response, as the source domain input data and response The neural network model between is expressed as: （9）； in, is an estimate of the actual response.

5. The method for optimizing the reliability of pressurized water reactor nuclear fuel assemblies based on transfer learning according to claim 1, characterized in that: In S6, random samples are taken from the source domain according to a uniform distribution. Migration sample input data , call the source domain FNN to calculate the corresponding response estimate ,Right now: （10）； in, is a counting symbol, indicating the number of the migration sample, The input data and the corresponding response are fused to form a migration sample. Function The migration samples are expressed as: （11）。 6. The method for optimizing the reliability of pressurized water reactor nuclear fuel assemblies based on transfer learning according to claim 1, characterized in that: In S7, the probability density ratio is calculated as follows: （12）， in and Represent the source domain and target domain random variables respectively The joint probability density distribution function of ; the probability density ratio is used as the weight of the source domain migration sample, and the weighted migration sample is used to reconstruct the target domain FNN. The new loss function expression is: （13）； The Levenberg-Marquardt backpropagation algorithm is used to minimize the loss function to determine the weights and biases of the target domain FNN; The FNN reconstructed based on sample transfer learning can be used as the target domain input data and response The neural network model between is expressed as: （14）。 7. The method for optimizing the reliability of pressurized water reactor nuclear fuel assemblies based on transfer learning according to claim 1, characterized in that: In S8, according to the marginal distribution of the target domain, the input data of N target domain MCS samples are generated, which are recorded as , using the target domain FNN to estimate Response: （15）； The probability distribution function of the estimated response is obtained for reliability analysis, where the MCS samples are divided into two categories: failure and safety according to the estimated response. When , it indicates failure; when When , it indicates a safe situation; the failure indication function is as follows: （16）； Then, based on the number of failed samples, the failure probability can be approximately estimated: （17）。 8. The method for optimizing the reliability of pressurized water reactor nuclear fuel assemblies based on transfer learning according to claim 1, characterized in that: In S9, the first-order fractional function method is used to estimate the gradient failure probability according to the failure condition of the MCS sample. design variables The partial derivative of is: （18）； in, Indicates that at the iteration point The target domain random variable The joint probability density distribution function of .

9. The method for optimizing the reliability of pressurized water reactor nuclear fuel assemblies based on transfer learning according to claim 1, characterized in that: In S6, in the source domain and target domain When the feature space and conditional probability distribution are the same, the optimal neural network model is reconstructed for the target domain based on sample transfer learning when there is no response information in the target domain; specifically, assuming that the target domain is reconstructed with The neural network model with parameters is based on the empirical risk minimization theory. It is obtained by solving the following optimization problem: （19）； in represents the empirical risk, yes The loss function of By using Bayes’ theorem and the definition of expectation, the probability density ratio calculation formula (12) can be rewritten as: （20）； That is, using the responsive samples in the source domain, we can learn by minimizing the weighted expected risk. ; Since the conditional probability distributions of the target domain and the source domain are already the same, that is, ; The joint probability distribution can be decomposed into ,get , formula (20) is further rewritten as: （21）； That is, the sample The weights of the target domain and the source domain in the data point The ratio of the marginal distribution probability density at ; Given a source domain, there are corresponding migration samples , by defining the probability density ratio , formula (21) is rewritten as: （22）。 10. A pressurized water reactor nuclear fuel assembly reliability optimization system based on transfer learning, characterized in that: Reliability optimization is performed using the pressurized water reactor nuclear fuel assembly reliability optimization method based on transfer learning as described in any one of claims 1 to 9.

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