Method and system for testing reliability of short-circuit resistance of transformer

Abstract

The application provides a transformer short-circuit resistance reliability test method and system, which collects N groups of short-circuit stress test data of the transformer under multiple working conditions, including short-circuit stress, test results and response monitoring data, builds a Gaussian process regression proxy model, generates M groups of virtual stress response samples, fuses the original data to obtain an augmented data set, extracts equipment development parameters, combines the similarity of transformer parameters to solve the prior hyperparameters, constructs a hierarchical Bayesian model with ordered constraints and multi-level degradation thresholds, establishes a joint probability model based on the augmented data set, fuses the monitoring data and the degradation threshold to obtain the failure and deletion probabilities by using the likelihood function, and integrates the virtual data prediction variance into the marginalization integral process to reflect the prediction uncertainty, samples the model posterior distribution to deduce the failure distribution and the degradation threshold parameter, solves the conditional reliability function, outputs a two-dimensional reliability curve, and completes the short-circuit resistance reliability evaluation.

Description

Technical Field

[0001] This application belongs to the field of testing, and in particular relates to a method and system for testing the reliability of transformer short-circuit withstand capability. Background Technology

[0002] Power transformers are core equipment in power grid operation, and their stability and reliability directly affect the safe operation of the power system. Among the various types of faults that transformers may suffer, winding deformation, structural loosening, or component damage caused by external short circuits are among the most dangerous and common types of faults in engineering. Sudden short-circuit tests are an important means of verifying the short-circuit withstand capability of transformers, and can realistically reflect the mechanical strength and structural stability of the equipment under short-circuit impact. However, short-circuit tests on large-capacity transformers are costly, complex to organize and implement, and the test process is somewhat destructive, resulting in a very limited number of short-circuit stress test samples available in actual engineering. This "small sample" characteristic makes traditional reliability assessment methods that rely on large sample statistical laws difficult to apply directly, easily leading to model overfitting, unstable parameter estimation, or insufficient confidence in the assessment results. Therefore, how to quantitatively assess the reliability of transformers under different short-circuit stress impacts under limited test sample conditions has become an urgent problem to be solved in the field of power equipment reliability assessment.

[0003] For reliability assessment under small sample conditions, existing methods, while capable of parameter estimation using some statistical models, often struggle to fully utilize test process monitoring data and prior information from historical equipment when dealing with complex physical degradation processes such as transformer short-circuit withstand capability. Due to the lack of effective fusion of physical state response characteristics during short-circuit impacts, existing methods fail to reveal the deep degradation patterns from slight deformation to structural instability when sample sizes are insufficient. Relying solely on a small number of discrete test results for modeling makes it difficult to construct a joint distribution model capable of characterizing failure probability changes under complex operating conditions, and also lacks a systematic processing mechanism for multi-level degradation thresholds and censored test data. These issues of insufficient data utilization and limited modeling dimensions not only restrict the quantitative understanding of transformer short-circuit withstand capability degradation mechanisms but also make it difficult to continuously demonstrate the conditional reliability trends of equipment under different stress conditions. Therefore, it is necessary to propose a novel testing method that can overcome the limitations of the number of real test samples, integrate historical prior knowledge and process monitoring status information, and achieve high-precision conditional reliability assessment of transformer short-circuit withstand capability. Summary of the Invention

[0004] To improve the accuracy of transformer short-circuit withstand reliability assessment in small sample scenarios, this invention provides a method and system for testing the reliability of transformer short-circuit withstand capability.

[0005] In a first aspect, the present invention provides a method for testing the short-circuit withstand capability reliability of a transformer, comprising the following steps:

[0006] N sets of short-circuit stress test data of the target transformer under different test conditions are obtained. The test data includes short-circuit stress, test results and process monitoring data, wherein the process monitoring data represents the transformer response under short-circuit impact. Based on the N sets of short-circuit stress test data, a Gaussian process regression surrogate model is constructed to map the short-circuit stress to the physical state response. The model is used to generate M sets of virtual stress and corresponding predicted responses and prediction variances. The generated virtual data is combined with the original test data to form an augmented dataset. The development parameters of the transformer are extracted, and prior hyperparameters are calculated by weighting the similarity between the development parameters of the target transformer and historical transformers of the same family. A hierarchical Bayesian model is constructed based on the prior hyperparameters. In the hierarchical Bayesian model, the multi-level degradation threshold is modeled as a random variable subject to ordered constraints. A joint probability model is constructed based on the augmented dataset. The likelihood function in the joint probability model includes the failure probability and censoring probability constructed by combining process monitoring data with multi-level degradation thresholds. The likelihood contribution of the virtual data is distributed and integrated according to the prediction variance to adjust for uncertainty marginalization. Based on the joint probability model, the posterior distribution of failure distribution parameters and multi-level degradation thresholds is inferred. The conditional reliability function of the target transformer is calculated by sampling the posterior distribution, and a two-dimensional conditional reliability curve is generated.

[0007] Furthermore, the step of constructing a Gaussian process regression surrogate model based on the N sets of short-circuit stress test data to map short-circuit stress to physical state response includes: The winding displacement deformation and insulation medium response characteristics in the process monitoring data are extracted as physical state response vectors, and the scalar norm of the physical state response vectors is calculated to reduce the dimension of the response vectors and fuse them into one-dimensional continuous physical response scalar values. A training sample set was constructed using the experimental short-circuit stress and the scalar value of the one-dimensional continuous physical response. A Gaussian process regression network was constructed using a combined covariance function containing a squared exponential kernel and a Matrn kernel. By maximizing the marginal likelihood function to optimize the hyperparameters of the combined covariance function, a nonlinear mapping relationship is established between the input stress and the scalar value of the one-dimensional continuous physical response.

[0008] Furthermore, the extraction of transformer development parameters and the weighted calculation of prior hyperparameters based on the similarity between the development parameters of the target transformer and historical transformers of the same family include: Extract the transformer's rated capacity, per-unit short-circuit impedance, and winding geometry as a vector of development parameters; Calculate the Mahalanobis distance between the target transformer and the development parameter vectors of each transformer in the historical family database; Based on the Mahalanobis distance, the similarity weight coefficients of the failure distribution parameters of each historical transformer are calculated using the Gaussian radial basis kernel function. The mean hyperparameter of the failure distribution prior model of the target transformer is obtained by weighting and summing the mean parameters of the failure distribution model of the historical transformer using the weight coefficients, and the variance hyperparameter is calculated based on the inverse proportional mapping of the sum of the similarity weight coefficients before normalization.

[0009] Furthermore, the likelihood function in the joint probability model includes failure probability and censoring probability constructed from process monitoring data combined with multi-level degradation thresholds, including: Multiple degradation state threshold parameters are set that are subject to monotonically increasing constraints. The difference between adjacent threshold parameters is modeled as an independent random variable that follows a positive domain truncated normal distribution to ensure order constraint. When calculating the likelihood function, for data points derived from experimental data, if they are not invalid, the maximum one-dimensional continuous physical response scalar value reached by the process monitoring data of the invalid data points is used to determine the censoring state that has not exceeded the highest degradation threshold, and the contribution of the invalid data points to the likelihood function is constructed based on the probability that the one-dimensional continuous physical response scalar value has not exceeded the highest degradation threshold. If the data point has failed, its contribution to the likelihood function is calculated as the failure probability that the scalar value of the one-dimensional continuous physical response exceeds the highest degradation threshold.

[0010] Furthermore, the likelihood contribution of the virtual data is integrally distributed based on the prediction variance to perform uncertainty marginalization adjustment, including: For each set of virtual data in the augmented dataset, a Gaussian prediction probability density function of the physical state response is constructed using the predicted response of the data as the mean and the predicted variance as the variance. The probability density function of the physical response under a given short-circuit stress condition, defined by the failure distribution parameters, is used as the model likelihood function of the unlabeled virtual response. The model likelihood function is then multiplied by the Gaussian prediction probability density function, and marginal integration is performed over the entire domain of the physical response to obtain the expected likelihood function of the virtual data. The expected likelihood function, after being discounted by virtual sample weights, is used as the likelihood contribution of the virtual data and added to the joint probability model. This allows the prediction uncertainty of the virtual data to be reflected in the joint posterior inference process through marginalization integration, thus avoiding the direct inclusion of the virtual prediction response as a definite observation in the joint probability model.

[0011] Further, the step of inferring the posterior distribution of failure distribution parameters and multi-level degradation thresholds based on the joint probability model, and generating a two-dimensional conditional reliability curve by sampling the posterior distribution to calculate the conditional reliability function of the target transformer, includes: Joint posterior samples of failure distribution parameters and multi-level degradation thresholds are extracted from the joint probability model; Based on the extracted posterior sample set, the probability distribution of the physical state response not exceeding the failure threshold under any given short-circuit stress condition within the applicable stress range of the model is calculated. The expectation of the probability distribution in the posterior sample space is calculated, and a two-dimensional conditional reliability curve of the target transformer within the applicable stress range of the model is generated by fitting.

[0012] Secondly, the present invention also provides a transformer short-circuit withstand capability reliability testing system, comprising the following modules: A construction module is used to acquire N sets of short-circuit stress test data of the target transformer under different test conditions. The test data includes short-circuit stress, test results, and process monitoring data, wherein the process monitoring data represents the transformer's response to short-circuit impact. Based on the N sets of short-circuit stress test data, a Gaussian process regression surrogate model is constructed to map the short-circuit stress to the physical state response. The model is used to generate M sets of virtual stress and corresponding predicted responses and predicted variances. The generated virtual data is combined with the original test data to form an augmented dataset. The adjustment module is used to extract the development parameters of the transformer and calculate the prior hyperparameters by weighting the similarity between the development parameters of the target transformer and historical transformers of the same family. Based on the prior hyperparameters, a hierarchical Bayesian model is constructed. In the hierarchical Bayesian model, the multi-level degradation thresholds are modeled as random variables subject to ordered constraints. Based on the augmented dataset, a joint probability model is constructed. The likelihood function in the joint probability model includes the failure probability and censoring probability constructed by combining process monitoring data with multi-level degradation thresholds. The likelihood contribution of the virtual data is distributed and integrated according to the prediction variance to adjust for uncertainty marginalization. The generation module is used to infer the posterior distribution of failure distribution parameters and multi-level degradation thresholds based on the joint probability model, and to generate a two-dimensional conditional reliability curve by sampling the posterior distribution to calculate the conditional reliability function of the target transformer.

[0013] Preferably, the step of constructing a Gaussian process regression surrogate model based on the N sets of short-circuit stress test data to map the short-circuit stress to the physical state response includes: The winding displacement deformation and insulation medium response characteristics in the process monitoring data are extracted as physical state response vectors, and the scalar norm of the physical state response vectors is calculated to reduce the dimension of the response vectors and fuse them into one-dimensional continuous physical response scalar values. A training sample set was constructed using the experimental short-circuit stress and the scalar value of the one-dimensional continuous physical response. A Gaussian process regression network was constructed using a combined covariance function containing a squared exponential kernel and a Matrn kernel. By maximizing the marginal likelihood function to optimize the hyperparameters of the combined covariance function, a nonlinear mapping relationship is established between the input stress and the scalar value of the one-dimensional continuous physical response.

[0014] Preferably, the step of extracting the development parameters of the transformer and calculating the prior hyperparameters based on the similarity between the development parameters of the target transformer and historical transformers of the same family includes: Extract the transformer's rated capacity, per-unit short-circuit impedance, and winding geometry as a vector of development parameters; Calculate the Mahalanobis distance between the target transformer and the development parameter vectors of each transformer in the historical family database; Based on the Mahalanobis distance, the similarity weight coefficients of the failure distribution parameters of each historical transformer are calculated using the Gaussian radial basis kernel function. The mean hyperparameter of the failure distribution prior model of the target transformer is obtained by weighting and summing the mean parameters of the failure distribution model of the historical transformer using the weight coefficients, and the variance hyperparameter is calculated based on the inverse proportional mapping of the sum of the similarity weight coefficients before normalization.

[0015] Preferably, the likelihood function in the joint probability model includes failure probability and censoring probability constructed from process monitoring data combined with multi-level degradation thresholds, including: Multiple degradation state threshold parameters are set that are subject to monotonically increasing constraints. The difference between adjacent threshold parameters is modeled as an independent random variable that follows a positive domain truncated normal distribution to ensure order constraint. When calculating the likelihood function, for data points derived from experimental data, if they are not invalid, the maximum one-dimensional continuous physical response scalar value reached by the process monitoring data of the invalid data points is used to determine the censoring state that has not exceeded the highest degradation threshold, and the contribution of the invalid data points to the likelihood function is constructed based on the probability that the one-dimensional continuous physical response scalar value has not exceeded the highest degradation threshold. If the data point has failed, its contribution to the likelihood function is calculated as the failure probability that the scalar value of the one-dimensional continuous physical response exceeds the highest degradation threshold.

[0016] Preferably, the likelihood contribution of the virtual data is integrally distributed based on the prediction variance to perform uncertainty marginalization adjustment, including: For each set of virtual data in the augmented dataset, a Gaussian prediction probability density function of the physical state response is constructed using the predicted response of the data as the mean and the predicted variance as the variance. The probability density function of the physical response under a given short-circuit stress condition, defined by the failure distribution parameters, is used as the model likelihood function of the unlabeled virtual response. The model likelihood function is then multiplied by the Gaussian prediction probability density function, and marginal integration is performed over the entire domain of the physical response to obtain the expected likelihood function of the virtual data. The expected likelihood function, after being discounted by virtual sample weights, is used as the likelihood contribution of the virtual data and added to the joint probability model. This allows the prediction uncertainty of the virtual data to be reflected in the joint posterior inference process through marginalization integration, thus avoiding the direct inclusion of the virtual prediction response as a definite observation in the joint probability model.

[0017] Preferably, the step of inferring the posterior distribution of failure distribution parameters and multi-level degradation thresholds based on the joint probability model, and generating a two-dimensional conditional reliability curve by sampling the posterior distribution to calculate the conditional reliability function of the target transformer, includes: Joint posterior samples of failure distribution parameters and multi-level degradation thresholds are extracted from the joint probability model; Based on the extracted posterior sample set, the probability distribution of the physical state response not exceeding the failure threshold under any given short-circuit stress condition within the applicable stress range of the model is calculated. The expectation of the probability distribution in the posterior sample space is calculated, and a two-dimensional conditional reliability curve of the target transformer within the applicable stress range of the model is generated by fitting.

[0018] This invention effectively expands the original test data by acquiring a limited number of short-circuit stress test data and constructing a Gaussian process regression surrogate model to generate virtual samples containing predicted responses and predicted variances, thus enhancing the data support foundation for reliability assessment. By leveraging the development experience of historical transformers from the same family and introducing prior information based on the similarity of development parameters, a hierarchical Bayesian model containing ordered multi-level degradation thresholds is constructed. This allows the model to better conform to the gradual degradation pattern of equipment conditions, improving the utilization efficiency of historical prior information. In the joint probability model, the likelihood contribution of the predicted variance to the virtual data is used for uncertainty marginalization, avoiding the direct inference of virtual prediction results as definite observations, thereby reducing parameter estimation bias caused by the uncertainty of virtual samples. The final generated two-dimensional conditional reliability curve can more intuitively and continuously characterize the short-circuit withstand capability and reliability level of the target transformer under different short-circuit stress conditions. Attached Figure Description

[0019] Figure 1 A flowchart of a method for testing the short-circuit withstand capability reliability of a transformer; Figure 2 A schematic diagram of the probability density of physical state response prediction; Figure 3 This is a schematic diagram of the threshold probability density of the degenerate state. Detailed Implementation

[0020] In this application, the terms "exemplarily" or "for example" are used to indicate that something is being described as an example, illustration, or illustration. Any embodiment or design described as "exemplarily" or "for example" in this application should not be construed as being more preferred or advantageous than other embodiments or designs. Specifically, the use of terms such as "exemplarily" or "for example" is intended to present the relevant concepts in a specific manner.

[0021] Reference Figure 1 This invention proposes a method for testing the short-circuit withstand capability reliability of a transformer, comprising the following steps: S1, acquire N sets of short-circuit stress test data of the target transformer under different test conditions. The test data includes short-circuit stress, test results and process monitoring data, wherein the process monitoring data represents the transformer's response under short-circuit impact.

[0022] By deploying voltage transformers, current transformers, piezoelectric vibration sensors, winding displacement monitoring sensors, and insulation dielectric response testing units on the target transformer, the three-phase voltage and current waveforms, vibration acceleration signals on the tank surface, axial or radial displacement responses of the windings, and insulation dielectric response data obtained from repeated measurements before, during, or after the short-circuit impact are collected. The winding displacement monitoring sensors can be fiber optic displacement sensors, inductive displacement sensors, or non-contact displacement measurement devices matched to the test structure. The insulation dielectric response testing unit can be a frequency domain dielectric spectrum testing device or a dielectric loss testing device, used to obtain complex capacitance, dielectric loss factor, or equivalent insulation response characteristics. The collected signals are converted into digital signals by a synchronous data acquisition card and stored in a comma-separated value file format. The file content is read, and short-circuit stress characteristics containing the effective value and duration of the short-circuit current are extracted. Whether structural deformation or electrical breakdown occurs is marked as a binary classification test result. The power spectral density of the vibration acceleration signal was calculated using vibration signal processing methods, and combined with the winding displacement response and insulation medium response characteristics as process monitoring data representing the response, resulting in N sets of complete short-circuit stress test data.

[0023] To avoid inconsistencies in the definition of the abscissa of the two-dimensional conditional reliability curve due to differences in short-circuit duration, one implementation uses the short-circuit duration as a control variable within the standard short-circuit test duration or a preset duration range. Another implementation normalizes the effective value of the short-circuit current and the duration separately, then merges them into an equivalent short-circuit stress scalar. For example, this can be achieved by multiplying the short-circuit current multiple by the normalized duration, or by using an index related to short-circuit thermal shock. This equivalent short-circuit stress scalar is then used for the abscissa of the subsequent Gaussian process regression surrogate model and the two-dimensional conditional reliability curve. When the short-circuit duration is fixed, the effective value of the short-circuit current is used as the equivalent short-circuit stress scalar.

[0024] S2. Based on the N sets of short-circuit stress test data, construct a Gaussian process regression surrogate model to map the short-circuit stress to the physical state response. Use the model to generate M sets of virtual stresses and corresponding predicted responses and predicted variances. Combine the generated virtual data with the original test data to form an augmented dataset.

[0025] A Gaussian process regression surrogate model is constructed using a covariance kernel function with a white noise term. It takes N sets of equivalent short-circuit stress scalars as input and the corresponding one-dimensional continuous physical response scalar values ​​as output for model parameter fitting. M sets of uniformly distributed virtual stresses are generated within the training sample coverage or physically feasible interpolation range of the equivalent short-circuit stress scalars. These M sets of virtual stresses are input into the trained Gaussian process regression surrogate model, and the mean and variance of the predicted physical state response corresponding to each set of virtual stresses are calculated. When using a program interface that returns the predicted standard deviation, the predicted standard deviation is obtained first, and then the predicted standard deviation is squared to obtain the predicted variance, thereby ensuring that the variance parameter in the subsequent Gaussian prediction probability density function matches the square dimension of the physical response. These M sets of virtual data containing the predicted response and predicted variance are concatenated row-wise with the original N sets of experimental data to generate an augmented dataset containing N+M samples.

[0026] In one possible embodiment, the step of constructing a Gaussian process regression surrogate model based on the N sets of short-circuit stress test data, mapping short-circuit stress to physical state response, includes: The winding displacement deformation and insulation medium response characteristics in the process monitoring data are extracted as physical state response vectors, and the scalar norm of the physical state response vectors is calculated to reduce the dimension of the response vectors and fuse them into one-dimensional continuous physical response scalar values. A training sample set was constructed using the experimental short-circuit stress and the scalar value of the one-dimensional continuous physical response. A Gaussian process regression network was constructed using a combined covariance function containing a squared exponential kernel and a Matrn kernel. By maximizing the marginal likelihood function to optimize the hyperparameters of the combined covariance function, a nonlinear mapping relationship is established between the input stress and the scalar value of the one-dimensional continuous physical response.

[0027] The process involves extracting winding displacement deformations (such as axial and radial displacements, typically ranging from 0.1 to 5.0 mm) from monitoring data with a sampling frequency of at least 10 kHz, and insulation dielectric response characteristics (such as complex capacitance or dielectric loss factor, typically ranging from 0.1% to 2.0%) from frequency domain dielectric spectroscopy testing within the range of 1 mHz to 1 kHz. These two parameters are then cascaded after range standardization to form a multidimensional physical state response vector. Using a predefined weighted diagonal matrix (e.g., displacement weight 0.7, insulation weight 0.3), and with the baseline physical response vector under healthy conditions or the mean vector of training samples as the reference center, the Mahalanobis distance norm or weighted L2 norm of the current physical state response vector relative to the reference center is calculated. This vector is then dimensionality-reduced and mapped into a single one-dimensional continuous physical response scalar value, typically mapped to the range of 0 to 10. For example, when the maximum measured axial displacement is 1.2 mm and the dielectric loss factor is 0.5%, the scalar response value of 2.35 can be obtained through weighted norm calculation. The equivalent short-circuit stress scalar is set as the input independent variable, and a training sample set is constructed by pairing it with the scalar value. If the short-circuit duration of each group of experiments is fixed under the same standard duration, the equivalent short-circuit stress scalar can be the effective value of the short-circuit current, which is usually 1.0 to 2.5 times the rated current. If there are differences in the short-circuit duration, the effective value of the short-circuit current and the duration are normalized and then synthesized into the equivalent short-circuit stress scalar. In the surrogate model construction stage, the Gaussian process regression network uses a squared exponential kernel to extract the global smoothing trend, with a preferred signal variance of 0.1 to 5.0 and a preferred length scale of 0.5 to 2.0. It is combined with a Matrn kernel to fit local mutation features, with the smoothing parameter usually taken as three-half or five-half, and the scale parameter preferably taken as 0.1 to 1.0. The combined covariance function is then used.

[0028] This Gaussian process regression network comprises an input layer, a covariance calculation kernel layer, and an output inference layer. The input layer receives the equivalent short-circuit stress scalar as model input. The covariance calculation kernel layer uses a squared exponential kernel and a Matrn kernel to form a combined covariance function. The squared exponential kernel characterizes the overall smooth trend of the relationship between stress and physical response, while the Matrn kernel enhances the model's ability to fit local abrupt changes or non-smooth response variations. Based on this combined covariance function, the model constructs the covariance relationships between training samples and between the stress point to be predicted and the training samples. In the output inference layer, the predicted mean and predicted variance of the one-dimensional continuous physical response scalar value corresponding to the stress point to be predicted are calculated according to Bayesian inference rules.

[0029] During model training, the observed noise variance was used as one of the parameters to be optimized. The optimal hyperparameter combination was obtained by jointly optimizing the squared exponential kernel, the Matrn kernel, and relevant hyperparameters in the noise term by maximizing the log-marginal likelihood function. The trained Gaussian process regression network can establish a nonlinear mapping relationship between the equivalent short-circuit stress and the scalar value of the one-dimensional continuous physical response. It also outputs the predicted mean and variance of the physical state response under untested short-circuit stress conditions, thus providing a data foundation for subsequent virtual sample generation and uncertainty marginalization. The probability density of the predicted physical state response is illustrated in the diagram below. Figure 2 As shown.

[0030] S3, extract the development parameters of the transformer and calculate the prior hyperparameters by weighting the similarity between the development parameters of the target transformer and historical transformers of the same family.

[0031] The rated capacity, per-unit short-circuit impedance, and winding geometry of the target transformer and historical transformers are extracted from the transformer technical documents as a development parameter vector. In other embodiments, the rated voltage can also be included as an extended development parameter. The Mahalanobis distance between the target transformer development parameter vector and the development parameter vectors of each historical transformer is calculated, and a normalized similarity weight is calculated based on the Mahalanobis distance using a Gaussian radial basis function kernel. The historical mean values ​​of the scale and shape parameters of the historical transformers under the Weibull distribution assumption are extracted, and the historical mean values ​​are weighted and summed using the obtained similarity weights to obtain the Weibull distribution prior hyperparameters used for subsequent inference, namely the prior shape parameter and the prior scale parameter. The Weibull distribution is a probability distribution commonly used in reliability analysis, which can be used to describe the probability change of equipment failure under stress with time or response level increases.

[0032] In one possible embodiment, the step of extracting the development parameters of the transformer and calculating the prior hyperparameters based on the similarity between the development parameters of the target transformer and historical transformers of the same family includes: Extract the transformer's rated capacity, per-unit short-circuit impedance, and winding geometry as a vector of development parameters; Calculate the Mahalanobis distance between the target transformer and the development parameter vectors of each transformer in the historical family database; Based on the Mahalanobis distance, the similarity weight coefficients of the failure distribution parameters of each historical transformer are calculated using the Gaussian radial basis kernel function. The mean hyperparameter of the failure distribution prior model of the target transformer is obtained by weighting and summing the mean parameters of the failure distribution model of the historical transformer using the weight coefficients, and the variance hyperparameter is calculated based on the inverse proportional mapping of the sum of the similarity weight coefficients before normalization.

[0033] At least 10 transformers of the same family were selected from the target transformer and historical database. Development parameters were extracted to construct a standardized development parameter vector, including rated capacity (typically ranging from 10MVA to 1000MVA), per-unit short-circuit impedance (e.g., 4% to 24%), and winding geometry characteristics covering inner diameter, outer diameter, and axial height (e.g., outer diameter 1200mm, height 1500mm). The inverse of the historical sample covariance matrix was used as the weight matrix in the Mahalanobis distance calculation to determine the Mahalanobis distance between the target transformer and the development parameter vector of each historical transformer. For example, when the target capacity is 50MVA and impedance is 8.5%, while a historical model has 63MVA and impedance is 9.0%, the Mahalanobis distance calculated after eliminating the influence of parameter dimensions is approximately 0.85.

[0034] A Gaussian radial basis function kernel with a bandwidth parameter preferably ranging from 0.5 to 2.0 is selected. If set to 1.0, the Mahalanobis distance is converted into nonlinear similarity weight coefficients; for example, an initial weight of approximately 0.697 corresponds to a distance of 0.85. All coefficients are then normalized. Assuming the failure distribution parameters of historical equipment are known, such as the mean of a Weibull distribution, the mean parameters of the historical failure model are weighted and summed using the normalized weight coefficients. This yields the mean hyperparameter of the prior model for the failure distribution of the target transformer. The variance hyperparameter is calculated by multiplying the basic population variance by an adjustment function, which is inversely proportional to the sum of the similarity weights before normalization. The adjustment coefficient is typically taken as 1.0 to 5.0. This mechanism allows the variance hyperparameter to decay and shrink when multiple highly similar historical samples exist in the database, reducing the uncertainty of the prior distribution and injecting prior information into subsequent inferences.

[0035] S4. Based on the prior hyperparameters, a hierarchical Bayesian model is constructed, in which the multi-level degradation threshold is modeled as a random variable subject to ordered constraints.

[0036] A hierarchical Bayesian model is constructed in a probabilistic programming environment, and the prior hyperparameters obtained above are used to construct the prior distribution of the failure distribution parameters. The failure distribution model represents the conditional distribution of a one-dimensional continuous physical response scalar value under a given equivalent short-circuit stress condition. The scaling parameter of the Weibull distribution is set as a monotonic function of the mean of the Gaussian process regression prediction corresponding to the equivalent short-circuit stress, and the shape parameter is used to characterize the degree of response dispersion, so that the reliability under any short-circuit stress condition can be expressed as the probability that the conditional response distribution has not exceeded the highest-level degradation threshold. Since both the scaling and shape parameters of the Weibull distribution are required to be positive, both are modeled using prior distributions with positive domains, such as log-normal, semi-normal, or gamma distributions, instead of directly using the ordinary normal distribution without positive constraints as the sampling distribution for the scaling parameter. To represent the multi-stage degradation process of the transformer from slight deformation to complete failure, three levels of degradation thresholds are set, and the probability density of the degradation state thresholds is illustrated as follows. Figure 3 As shown. By using ordered transformation or adjacent threshold positive difference modeling, the three multi-level degradation thresholds are applied with a monotonically increasing constraint that the first level is less than the second level and the second level is less than the third level during the Markov chain Monte Carlo sampling process. The above three multi-level degradation thresholds are modeled as random variables subject to ordered constraints.

[0037] S5. Construct a joint probability model based on the augmented dataset. The likelihood function in the joint probability model includes the failure probability and censoring probability constructed by combining process monitoring data with multi-level degradation thresholds. The likelihood contribution of the virtual data is distributed and integrated according to the prediction variance to adjust for uncertainty marginalization.

[0038] Based on the aforementioned hierarchical Bayesian model and augmented dataset, a custom likelihood function for the joint probability model is constructed. For N sets of original experimental data, the likelihood contributions of failed and unfailed samples are constructed according to the relationship between the one-dimensional continuous physical response scalar values ​​corresponding to the experimental results and process monitoring data and the highest-level degradation threshold. Specifically, the likelihood contribution of failed samples is the probability of exceeding the threshold under the current Weibull distribution failure model and degradation threshold conditions, while the likelihood contribution of unfailed samples is the censoring probability of not exceeding the highest-level degradation threshold. If the experimental records contain degradation level labels such as mild or moderate (not fully failed), the interval censoring probability is calculated based on the adjacent degradation threshold intervals in which the sample is located. If no intermediate degradation level labels are provided, the censoring probability of not exceeding the highest-level degradation threshold is used.

[0039] For M sets of virtual data, no labels are assigned to actual failures or non-failures; instead, they are treated as potential physical response samples with predictive uncertainty. For each set of virtual data, a model probability density function for the physical response scalar value under the corresponding stress condition is constructed based on its equivalent short-circuit stress and the failure distribution parameters under the current posterior sampling. Simultaneously, a Gaussian prediction probability density function is constructed using the mean of the predicted response output by the Gaussian process regression as the mean and the variance as the variance. Multiplying the model probability density function and the Gaussian prediction probability density function and integrating them over the domain of physical response yields the unlabeled expected likelihood function of the virtual data. To avoid the virtual samples repeatedly amplifying the original experimental information, a discount weight is applied to the likelihood contribution of the virtual data, ensuring that the total weight of the virtual data does not exceed the total weight of the original experimental data.

[0040] In one possible embodiment, the likelihood function in the joint probability model includes failure probabilities and censoring probabilities constructed from process monitoring data combined with multi-level degradation thresholds, including: Multiple degradation state threshold parameters are set that are subject to monotonically increasing constraints. The difference between adjacent threshold parameters is modeled as an independent random variable that follows a positive domain truncated normal distribution to ensure order constraint. When calculating the likelihood function, for data points derived from experimental data, if they are not invalid, the maximum one-dimensional continuous physical response scalar value reached by the process monitoring data of the invalid data points is used to determine the censoring state that has not exceeded the highest degradation threshold, and the contribution of the invalid data points to the likelihood function is constructed based on the probability that the one-dimensional continuous physical response scalar value has not exceeded the highest degradation threshold. If the data point has failed, its contribution to the likelihood function is calculated as the failure probability that the scalar value of the one-dimensional continuous physical response exceeds the highest degradation threshold.

[0041] Q multi-level degradation state threshold parameters are set to characterize the state evolution of the transformer from slight deformation and moderate damage to short-circuit instability. Typically, Q is set to 3. To ensure that the multi-level thresholds satisfy the monotonically increasing constraint at the probability distribution level, the baseline primary threshold is set to follow a positive interval restricted normal distribution, and the difference between adjacent thresholds is modeled as mutually independent positive-domain truncated normal random variables. The mean of these difference random variables ranges from 1.0 to 5.0, and the standard deviation ranges from 0.1 to 1.0. For example, when the baseline threshold is 2.5 and the differences between two adjacent thresholds are 1.5 and 2.0, the subsequent degradation thresholds are 4.0 and 6.0 respectively, thus ensuring that the degradation thresholds increase in an orderly manner at each level.

[0042] In the likelihood function calculation of the joint probability model, each observed sample in the experimental dataset is evaluated. For samples identified as not failed after the experiment, if they only have the "not failed" label, the probability that their physical response does not exceed the highest-level degradation threshold is used as the censoring likelihood contribution; if they also have mild or moderate degradation level labels, the interval censoring probability is calculated based on their adjacent degradation threshold intervals. For samples that have failed after a short-circuit impact, the cumulative distributed failure probability that their physical response exceeds the highest-level degradation threshold is calculated, and this probability is used as the failure likelihood contribution. In this way, multi-level degradation thresholds and censoring data processing are uniformly incorporated into the likelihood function, improving the model's ability to identify boundary state samples and the stability of posterior inference.

[0043] In one possible embodiment, the likelihood contribution of the virtual data is integrally distributed based on the prediction variance to perform uncertainty marginalization adjustment, including: For each set of virtual data in the augmented dataset, a Gaussian prediction probability density function of the physical state response is constructed using the predicted response of the data as the mean and the predicted variance as the variance. The probability density function of the physical response under a given short-circuit stress condition, defined by the failure distribution parameters, is used as the model likelihood function of the unlabeled virtual response. The model likelihood function is then multiplied by the Gaussian prediction probability density function, and marginal integration is performed over the entire domain of the physical response to obtain the expected likelihood function of the virtual data. The expected likelihood function, after being discounted by virtual sample weights, is used as the likelihood contribution of the virtual data and added to the joint probability model. This allows the prediction uncertainty of the virtual data to be reflected in the joint posterior inference process through marginalization integration, thus avoiding the direct inclusion of the virtual prediction response as a definite observation in the joint probability model.

[0044] For the augmented virtual dataset output by the Gaussian process surrogate model, the nondeterministic prediction results corresponding to each virtual stress point are extracted one by one. For a single virtual data point, its predicted response is not treated as a fixed observation value. Instead, a Gaussian prediction probability density function of the physical state response at that point is constructed using the mean and variance of the predicted response output by the Gaussian process regression as parameters. If the model interface outputs the prediction standard deviation, it is first squared to convert it into the prediction variance. This preserves the prediction uncertainty of the surrogate model in different stress regions.

[0045] During model fusion, the conditional response probability density function determined by the current failure distribution parameters is multiplied by the Gaussian prediction probability density function. Marginalized integration is then performed within the effective domain of the physical response using methods such as Gauss-Hermitian quadrature or adaptive numerical integration to obtain the expected likelihood function of the virtual sample. When the one-dimensional continuous physical response scalar value is constrained to non-negative values, the Gaussian prediction probability density function is truncated to the positive domain and normalized before integration. Through this processing, the prediction variance can be used as a characterization of the uncertainty of the virtual sample in posterior inference, preventing the virtual response from being directly included in the joint probability model as a definite observation. This reduces the estimation bias introduced by the virtual data and improves the stability of the parameter inference results.

[0046] S6. Based on the joint probability model, infer the posterior distribution of failure distribution parameters and multi-level degradation thresholds. By sampling the posterior distribution, calculate the conditional reliability function of the target transformer and generate a two-dimensional conditional reliability curve.

[0047] Markov chain Monte Carlo sampling is performed in the joint probabilistic model. A no-turn-around sampler is used as the implementation method for Hamiltonian Monte Carlo sampling. Multiple sampling chains are set up, and after discarding preheated samples, a posterior distribution sample set of failure distribution scale parameters, shape parameters, and multi-level degradation thresholds is obtained. Based on the Weibull distribution parameters in the posterior samples, the conditional scale parameters or the mean of the Gaussian process predicted response corresponding to a given equivalent short-circuit stress, and the highest-level degradation threshold, the probability that the one-dimensional continuous physical response scalar value under the short-circuit stress condition does not exceed the highest-level failure threshold is calculated, and this probability is used as the conditional reliability value of the target transformer.

[0048] Using short-circuit stress values ​​as the abscissa and conditional reliability values ​​as the ordinate, the calculation results for each stress node are connected or fitted to generate a two-dimensional conditional reliability curve characterizing the short-circuit withstand capability as a function of stress. The short-circuit stress values ​​are limited to the training sample coverage range or the physically feasible interpolation range of the Gaussian process regression surrogate model. The physically feasible interpolation range is the stress interval between the minimum and maximum stress values ​​of the training samples, where the Gaussian process prediction variance does not exceed a preset upper limit of variance. For example, when the effective value range of the short-circuit current covered by the training samples is 1.0 to 2.5 times the rated current, the calculation interval of the two-dimensional conditional reliability curve is correspondingly limited to 1.0 to 2.5 times the rated current, or limited to the equivalent short-circuit stress scalar interval corresponding to this range.

[0049] In one possible embodiment, the step of inferring the posterior distribution of failure distribution parameters and multi-level degradation thresholds based on the joint probability model, and generating a two-dimensional conditional reliability curve by sampling the posterior distribution to calculate the conditional reliability function of the target transformer, includes: Joint posterior samples of failure distribution parameters and multi-level degradation thresholds are extracted from the joint probability model; Based on the extracted posterior sample set, the probability distribution of the physical state response not exceeding the failure threshold under any given short-circuit stress condition within the applicable stress range of the model is calculated. The expectation of the probability distribution in the posterior sample space is calculated, and a two-dimensional conditional reliability curve of the target transformer within the applicable stress range of the model is generated by fitting.

[0050] For the joint probabilistic model incorporating prior hyperparameters and the expected likelihood function, a Hamiltonian Monte Carlo sampling technique with a no-back-up mechanism is employed to explore the posterior space. During the sampling process, leapfrog integration steps, initial step size, and an adaptive step size mechanism are implemented, and preheating iterations are used to reduce the statistical bias introduced by the initial chain value. After preheating, at least 5000 stable and convergent posterior parameter samples are extracted to form a joint posterior sample pool containing failure distribution shape parameters, scale parameters, and multi-level degradation thresholds.

[0051] In the reliability calculation phase, multiple calculation nodes are divided along the short-circuit stress range of the target transformer. This range is typically 1.0 to 2.5 times the rated current, or the equivalent short-circuit stress scalar range covered by the training samples. Each set of posterior parameters is substituted into the reliability evaluation model to calculate the probability that the next-dimensional continuous physical response scalar value at each stress node does not exceed the highest-level failure threshold, resulting in the corresponding conditional reliability probability distribution cluster. Subsequently, the expectation of the probability distribution at each stress node is calculated along the posterior sample dimension, and confidence interval boundaries, such as the 5% and 95% quantiles, are extracted.

[0052] Using external short-circuit stress as the horizontal axis and expected conditional reliability as the vertical axis, a two-dimensional conditional reliability curve is generated by fitting or interpolating each stress node, covering the applicable stress range of the model and identifying uncertainty boundaries. If it is necessary to evaluate short-circuit stress conditions beyond the coverage of the training samples, corresponding real test samples should be added or a validated extrapolation correction model should be introduced; when no extrapolation correction model is set, results beyond the coverage of the training samples are not considered as valid outputs of the two-dimensional conditional reliability curve.

[0053] This invention also proposes a transformer short-circuit withstand capability reliability testing system, comprising the following modules: A construction module is used to acquire N sets of short-circuit stress test data of the target transformer under different test conditions. The test data includes short-circuit stress, test results, and process monitoring data, wherein the process monitoring data represents the transformer's response to short-circuit impact. Based on the N sets of short-circuit stress test data, a Gaussian process regression surrogate model is constructed to map the short-circuit stress to the physical state response. The model is used to generate M sets of virtual stress and corresponding predicted responses and predicted variances. The generated virtual data is combined with the original test data to form an augmented dataset. The adjustment module is used to extract the development parameters of the transformer and calculate the prior hyperparameters by weighting the similarity between the development parameters of the target transformer and historical transformers of the same family. Based on the prior hyperparameters, a hierarchical Bayesian model is constructed. In the hierarchical Bayesian model, the multi-level degradation thresholds are modeled as random variables subject to ordered constraints. Based on the augmented dataset, a joint probability model is constructed. The likelihood function in the joint probability model includes the failure probability and censoring probability constructed by combining process monitoring data with multi-level degradation thresholds. The likelihood contribution of the virtual data is distributed and integrated according to the prediction variance to adjust for uncertainty marginalization. The generation module is used to infer the posterior distribution of failure distribution parameters and multi-level degradation thresholds based on the joint probability model, and to generate a two-dimensional conditional reliability curve by sampling the posterior distribution to calculate the conditional reliability function of the target transformer.

[0054] The above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit it. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for testing the reliability of a transformer's short-circuit withstand capability, characterized in that, Includes the following steps: N sets of short-circuit stress test data of the target transformer under different test conditions are obtained. The test data includes short-circuit stress, test results and process monitoring data, wherein the process monitoring data represents the transformer response under short-circuit impact. Based on the N sets of short-circuit stress test data, a Gaussian process regression surrogate model is constructed to map the short-circuit stress to the physical state response. The model is used to generate M sets of virtual stress and corresponding predicted responses and prediction variances. The generated virtual data is combined with the original test data to form an augmented dataset. The development parameters of the transformer are extracted, and prior hyperparameters are calculated by weighting the similarity between the development parameters of the target transformer and historical transformers of the same family. A hierarchical Bayesian model is constructed based on the prior hyperparameters. In the hierarchical Bayesian model, the multi-level degradation threshold is modeled as a random variable subject to ordered constraints. A joint probability model is constructed based on the augmented dataset. The likelihood function in the joint probability model includes the failure probability and censoring probability constructed by combining process monitoring data with multi-level degradation thresholds. The likelihood contribution of the virtual data is distributed and integrated according to the prediction variance to adjust for uncertainty marginalization. Based on the joint probability model, the posterior distribution of failure distribution parameters and multi-level degradation thresholds is inferred. The conditional reliability function of the target transformer is calculated by sampling the posterior distribution, and a two-dimensional conditional reliability curve is generated.

2. The method according to claim 1, characterized in that, The construction of a Gaussian process regression surrogate model based on the N sets of short-circuit stress test data, mapping short-circuit stress to physical state response, includes: The winding displacement deformation and insulation medium response characteristics in the process monitoring data are extracted as physical state response vectors, and the scalar norm of the physical state response vectors is calculated to reduce the dimension of the response vectors and fuse them into one-dimensional continuous physical response scalar values. A training sample set was constructed using the experimental short-circuit stress and the scalar value of the one-dimensional continuous physical response. A Gaussian process regression network was constructed using a combined covariance function containing a squared exponential kernel and a Matrn kernel. By maximizing the marginal likelihood function to optimize the hyperparameters of the combined covariance function, a nonlinear mapping relationship is established between the input stress and the scalar value of the one-dimensional continuous physical response.

3. The method according to claim 2, characterized in that, The extraction of transformer development parameters and the weighted calculation of prior hyperparameters based on the similarity between the development parameters of the target transformer and historical transformers of the same family include: Extract the transformer's rated capacity, per-unit short-circuit impedance, and winding geometry as a vector of development parameters; Calculate the Mahalanobis distance between the target transformer and the development parameter vectors of each transformer in the historical family database; Based on the Mahalanobis distance, the similarity weight coefficients of the failure distribution parameters of each historical transformer are calculated using the Gaussian radial basis kernel function. The mean hyperparameter of the failure distribution prior model of the target transformer is obtained by weighting and summing the mean parameters of the failure distribution model of the historical transformer using the weight coefficients, and the variance hyperparameter is calculated based on the inverse proportional mapping of the sum of the similarity weight coefficients before normalization.

4. The method according to claim 1, characterized in that, The likelihood function in the joint probability model includes failure probability and censoring probability constructed from process monitoring data combined with multi-level degradation thresholds, including: Multiple degradation state threshold parameters are set that are subject to monotonically increasing constraints. The difference between adjacent threshold parameters is modeled as an independent random variable that follows a positive domain truncated normal distribution to ensure order constraint. When calculating the likelihood function, for data points derived from experimental data, if they are not invalid, the maximum one-dimensional continuous physical response scalar value reached by the process monitoring data of the invalid data points is used to determine the censoring state that has not exceeded the highest degradation threshold, and the contribution of the invalid data points to the likelihood function is constructed based on the probability that the one-dimensional continuous physical response scalar value has not exceeded the highest degradation threshold. If the data point has failed, its contribution to the likelihood function is calculated as the failure probability that the scalar value of the one-dimensional continuous physical response exceeds the highest degradation threshold.

5. The method according to claim 1 or 4, characterized in that, The likelihood contribution of the virtual data is integrated based on the predicted variance to adjust for uncertainty marginalization, including: For each set of virtual data in the augmented dataset, a Gaussian prediction probability density function of the physical state response is constructed using the predicted response of the data as the mean and the predicted variance as the variance. The probability density function of the physical response under a given short-circuit stress condition, defined by the failure distribution parameters, is used as the model likelihood function of the unlabeled virtual response. The model likelihood function is then multiplied by the Gaussian prediction probability density function, and marginal integration is performed over the entire domain of the physical response to obtain the expected likelihood function of the virtual data. The expected likelihood function, after being discounted by virtual sample weights, is used as the likelihood contribution of the virtual data and added to the joint probability model. This allows the prediction uncertainty of the virtual data to be reflected in the joint posterior inference process through marginalization integration, thus avoiding the direct inclusion of the virtual prediction response as a definite observation in the joint probability model.

6. The method according to claim 1, characterized in that, Based on the joint probability model, the posterior distribution of failure distribution parameters and multi-level degradation thresholds is inferred. The conditional reliability function of the target transformer is calculated by sampling the posterior distribution, generating a two-dimensional conditional reliability curve, including: Joint posterior samples of failure distribution parameters and multi-level degradation thresholds are extracted from the joint probability model; Based on the extracted posterior sample set, the probability distribution of the physical state response not exceeding the failure threshold under any given short-circuit stress condition within the applicable stress range of the model is calculated. The expectation of the probability distribution in the posterior sample space is calculated, and a two-dimensional conditional reliability curve of the target transformer within the applicable stress range of the model is generated by fitting.

7. A transformer short-circuit withstand capability reliability testing system, characterized in that, Includes the following modules: A construction module is used to acquire N sets of short-circuit stress test data of the target transformer under different test conditions. The test data includes short-circuit stress, test results, and process monitoring data, wherein the process monitoring data represents the transformer's response to short-circuit impact. Based on the N sets of short-circuit stress test data, a Gaussian process regression surrogate model is constructed to map the short-circuit stress to the physical state response. The model is used to generate M sets of virtual stress and corresponding predicted responses and predicted variances. The generated virtual data is combined with the original test data to form an augmented dataset. The adjustment module is used to extract the development parameters of the transformer and calculate the prior hyperparameters by weighting the similarity between the development parameters of the target transformer and historical transformers of the same family. Based on the prior hyperparameters, a hierarchical Bayesian model is constructed. In the hierarchical Bayesian model, the multi-level degradation thresholds are modeled as random variables subject to ordered constraints. Based on the augmented dataset, a joint probability model is constructed. The likelihood function in the joint probability model includes the failure probability and censoring probability constructed by combining process monitoring data with multi-level degradation thresholds. The likelihood contribution of the virtual data is distributed and integrated according to the prediction variance to adjust for uncertainty marginalization. The generation module is used to infer the posterior distribution of failure distribution parameters and multi-level degradation thresholds based on the joint probability model, and to generate a two-dimensional conditional reliability curve by sampling the posterior distribution to calculate the conditional reliability function of the target transformer.

8. The system according to claim 7, characterized in that, The construction of a Gaussian process regression surrogate model based on the N sets of short-circuit stress test data, mapping short-circuit stress to physical state response, includes: The winding displacement deformation and insulation medium response characteristics in the process monitoring data are extracted as physical state response vectors, and the scalar norm of the physical state response vectors is calculated to reduce the dimension of the response vectors and fuse them into one-dimensional continuous physical response scalar values. A training sample set was constructed using the experimental short-circuit stress and the scalar value of the one-dimensional continuous physical response. A Gaussian process regression network was constructed using a combined covariance function containing a squared exponential kernel and a Matrn kernel. By maximizing the marginal likelihood function to optimize the hyperparameters of the combined covariance function, a nonlinear mapping relationship is established between the input stress and the scalar value of the one-dimensional continuous physical response.

9. The system according to claim 7, characterized in that, The extraction of transformer development parameters and the weighted calculation of prior hyperparameters based on the similarity between the development parameters of the target transformer and historical transformers of the same family include: Extract the transformer's rated capacity, per-unit short-circuit impedance, and winding geometry as a vector of development parameters; Calculate the Mahalanobis distance between the target transformer and the development parameter vectors of each transformer in the historical family database; Based on the Mahalanobis distance, the similarity weight coefficients of the failure distribution parameters of each historical transformer are calculated using the Gaussian radial basis kernel function. The mean hyperparameter of the failure distribution prior model of the target transformer is obtained by weighting and summing the mean parameters of the failure distribution model of the historical transformer using the weight coefficients, and the variance hyperparameter is calculated based on the inverse proportional mapping of the sum of the similarity weight coefficients before normalization.

10. The system according to claim 7, characterized in that, The likelihood function in the joint probability model includes failure probability and censoring probability constructed from process monitoring data combined with multi-level degradation thresholds, including: Multiple degradation state threshold parameters are set that are subject to monotonically increasing constraints. The difference between adjacent threshold parameters is modeled as an independent random variable that follows a positive domain truncated normal distribution to ensure order constraint. When calculating the likelihood function, for data points derived from experimental data, if they are not invalid, the maximum one-dimensional continuous physical response scalar value reached by the process monitoring data of the invalid data points is used to determine the censoring state that has not exceeded the highest degradation threshold, and the contribution of the invalid data points to the likelihood function is constructed based on the probability that the one-dimensional continuous physical response scalar value has not exceeded the highest degradation threshold. If the data point has failed, its contribution to the likelihood function is calculated as the failure probability that the scalar value of the one-dimensional continuous physical response exceeds the highest degradation threshold.

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