A power transformer fault diagnosis method and system, storage medium

By combining data analysis with an improved bacterial foraging algorithm to optimize the support vector machine, the problem of insufficient accuracy in power transformer fault diagnosis was solved, achieving efficient and accurate fault diagnosis.

CN116257796BActive Publication Date: 2026-06-02STATE GRID JIBEI ELECTRIC POWER COMPANY LIMITED CHENGDE POWER SUPPLY +3

Patent Information

Authority / Receiving Office
CN · China
Patent Type
Patents(China)
Current Assignee / Owner
STATE GRID JIBEI ELECTRIC POWER COMPANY LIMITED CHENGDE POWER SUPPLY
Filing Date
2022-12-29
Publication Date
2026-06-02

AI Technical Summary

Technical Problem

Existing power transformer fault diagnosis technologies suffer from insufficient accuracy. In particular, traditional diagnostic techniques are simple and easy to implement but lack accuracy, while intelligent diagnostic techniques also struggle to improve diagnostic accuracy when they lack targeted modeling.

Method used

By combining data analysis and intelligent algorithms, a support vector machine (SVM) is constructed using data obtained from oil chromatography analysis. An improved bacterial foraging algorithm is then used to find the globally optimal parameters of the SVM and optimize the SVM model.

Benefits of technology

This improves the accuracy and efficiency of power transformer fault diagnosis, ensures the efficiency and accuracy of the support vector machine model, avoids local optima traps, and achieves the exploration of globally optimal parameters.

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Abstract

The present application relates to a kind of power transformer fault diagnosis method and system, storage medium.It includes the following steps: the content of each gas in the oil of power transformer obtained by oil chromatographic analysis, obtains original data;Original data is preprocessed, constructs feature vector, and is divided into training sample set and test sample set;Support vector machine is constructed based on training sample set;Support vector machine is optimized by improving bacterial foraging algorithm, and support vector machine fault diagnosis model is obtained;Test sample set T2 is input to support vector machine fault diagnosis model and carries out fault diagnosis.The present application combines data analysis diagnosis technology and intelligent algorithm, constructs support vector machine by obtaining data through oil chromatographic analysis, finds the global optimal parameter of support vector machine by improving bacterial foraging algorithm, improves the accuracy of diagnosis.
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Description

Technical Field

[0001] This invention relates to the field of power system technology, specifically to a method and system for diagnosing power transformer faults, and a storage medium. Background Technology

[0002] With the development of science and the advancement of technology, ensuring the safe and stable operation of power equipment is becoming increasingly important, as the operating status of power equipment directly affects the state of the power system. Power transformers are core equipment for energy transmission and conversion in the power grid, with wide applications and significant roles; their operational reliability is closely related to the safe and economical operation of the power system.

[0003] Power transformers are core equipment for energy transmission and conversion in power grids, widely used and playing a crucial role. Their operational reliability is closely related to the safe and economical operation of the power system. However, with the continuous increase in system short-circuit capacity and short-circuit current, the damage caused by short-circuit impacts to transformers is becoming increasingly severe, exacerbating transformer damage. In addition, factors such as insulation dampness and aging during transformer operation, as well as various adverse operating conditions like overvoltage and overload, combine to reduce transformer lifespan, increasing the probability of transformer failures and impacting grid operation, sometimes even causing unplanned unit outages. Therefore, strengthening the operation and maintenance management of power transformers, conducting accurate fault diagnosis and condition analysis, and promptly eliminating potential transformer faults are crucial for the safe and stable operation of the power grid.

[0004] Currently, power transformer fault diagnosis can be divided into traditional fault diagnosis technology, which utilizes electrical tests and oil gas chromatography data analysis, and intelligent diagnosis technology based on various intelligent algorithms. Traditional diagnostic technology involves judging the nature and type of fault based on analysis results according to test procedures. While simple in principle and easy to implement, its accuracy is insufficient. Intelligent diagnostic technology utilizes computer programming or modeling of algorithms, and then analyzes and judges faults based on this model. However, without targeted modeling and training, diagnostic accuracy can also be insufficient. Summary of the Invention

[0005] This invention provides a method and system for diagnosing power transformer faults. It combines data analysis and diagnostic technology with intelligent algorithms. By acquiring data through oil chromatography analysis to construct a support vector machine, and then using an improved bacterial foraging algorithm to find the globally optimal parameters of the support vector machine, the accuracy is improved.

[0006] The solution of the present invention to the above-mentioned technical problems is as follows: A method for diagnosing faults in power transformers, comprising the following steps:

[0007] The content of various gases in the oil of the power transformer was obtained by chromatographic analysis of the oil, and the raw data was obtained.

[0008] The original data is preprocessed, and feature vectors are constructed based on the preprocessed data. The data is then divided into a training sample set T1 and a test sample set T2.

[0009] Construct a support vector machine based on the training sample set T1;

[0010] By improving the bacterial foraging algorithm, the support vector machine is optimized to obtain a support vector machine fault diagnosis model;

[0011] The test sample set T2 is input into the support vector machine fault diagnosis model for fault diagnosis.

[0012] Preferably, the raw data includes the volumes of hydrogen, methane, ethylene, ethane, and acetylene.

[0013] Condition diagnosis of power transformers requires the support of various state variables and the collection of various gas data to improve the accuracy of condition diagnosis.

[0014] Preferably, the preprocessing of the raw data includes normalizing the raw data, and the specific calculation formula is as follows:

[0015]

[0016] Where, o i1 ,...,o i5 These are hydrogen, methane, ethylene, ethane, and other gases in the chromatographic data, respectively.

[0017] The original volume of acetylene gas; o i1 ′ represents the percentage of hydrogen volume relative to the total gas volume;

[0018] o i2 ′,...,o i5 ′ represents the percentage of the volume of methane, ethylene, ethane, and acetylene relative to the volume of the hydrocarbon gas, respectively.

[0019] Preferably, the step of constructing a support vector machine based on the training sample set T1 includes:

[0020] Based on the training sample set T1={(x e ,y e e = 1, 2, ..., l, x e ∈R n ,y e ∈{+1,-1}}, R n Given an n-dimensional set of real numbers; by defining the kernel function k(x) p ,x qThe nonlinearly separable input vector is mapped to a high-dimensional space, and the optimal hyperplane is found in this new space to correctly distinguish the samples. The corresponding optimal classification function f(x) is:

[0021]

[0022] Where, α e * The data points corresponding to the non-zero values ​​are the support vectors, b is the slack variable, and C is the penalty coefficient.

[0023] Where the RBF kernel function k(x) p ,x q )for:

[0024] k(x p ,x q )=exp(-γ||x p -x q ||) 2

[0025] Where γ is the kernel parameter, x p For the p-th training sample, x q Let q be the q-th training sample.

[0026] Preferably, the improved bacterial foraging algorithm is obtained by optimizing the step size c(i) of the forward swimming in the bacterial foraging algorithm using a differential evolution algorithm.

[0027] Preferably, the step size c(i) for forward swimming in the bacterial foraging algorithm is optimized using the differential evolution algorithm, and the specific formula is as follows:

[0028]

[0029] Where c(i) is the step size of the forward swim. new c(i) represents the step length for the next forward swim. max V represents the maximum stride length for swimming forward; V represents bacterial susceptibility.

[0030]

[0031] Where j represents the chemotactic operation step of an individual in the population, j max For the largest step of chemotaxis in an individual population; j min `rand` represents the minimum number of steps required for chemotaxis in an individual population; `rand` is a random number.

[0032] Bacterial Foraging Optimization (BFA) is a novel biomimetic intelligent optimization algorithm with advantages such as parallel search, easy escape from local minima, and fast convergence speed. Its core idea is to explore optimal parameters by simulating the foraging behavior of *E. coli*. The foraging behavior of each individual is simulated, defining j chemotaxis operations, k replication operations, and l migration operations. The parallel search and global optimization capabilities of BFA enable its application in various fields. The core of the BFA algorithm is the chemotaxis operation, and the step size c(i) is one of the key parameters of the BFA chemotaxis operator, controlling the convergence and diversity of the population. If the value of c(i) is too small, the bacteria will be confined to a small portion of the optimization interval. If the value of c(i) is too large, the bacteria will rapidly move to the target region, even exceeding the optimal region, causing the bacteria to get trapped in local optima. Differential evolution and adaptive step size adjustment strategies are introduced, and the chemotaxis operation results in the bacterial foraging algorithm are optimized using the DE algorithm. It corrected the dimensional degradation phenomenon in the evolutionary process, increased the diversity of the population, and preserved the best individuals in the population.

[0033] Preferably, the step of optimizing the support vector machine using the improved bacterial foraging algorithm to obtain the support vector machine fault diagnosis model includes:

[0034] Configure parameters to improve the bacterial foraging algorithm;

[0035] The penalty coefficient C and kernel parameter γ of the support vector machine are optimized by improving the bacterial foraging algorithm;

[0036] Substituting the optimal parameters {C, γ} into the support vector machine yields the support vector machine fault diagnosis model.

[0037] Preferably, the improved bacterial foraging algorithm includes the following steps for optimizing the C parameter of the support vector machine and the kernel parameter γ:

[0038] The parameters for setting the improved bacterial foraging algorithm are as follows:

[0039]

[0040] Where, θ i Let represent the position of individual i in the population, j represent the directional operation, k represent the replication operation, l represent the migration operation, and c(i) represent the step size for moving forward. Δ (i) represents a unit vector in a random direction; Δ i (i) represents the i-th power unit vector in a random direction; θ i (j,k,l) ​​represents the directional, replicative, and migration operations at the location of individual i in the population; θ i(j+1,k,l) ​​represents the next directional operation, replication operation, or migration operation at position i in the population.

[0041] Using its current position {C, γ} as the algorithm parameters, and employing k-fold cross-validation of the training sample set, the accuracy C is determined. r The fitness function F(i,j,k,l) ​​is expressed as follows:

[0042]

[0043] Where k is the number of folds in the cross-validation; l t Let be the number of samples in the t-th validation set; This represents the number of samples correctly classified in the validation set.

[0044] Let k-fold cross-validation accuracy The maximum value is the objective function of the optimization problem, and the range of values ​​for the parameters {C, γ} is a constraint condition, i.e.:

[0045]

[0046] In the formula, Obj is the minimum value obtained by solving the quadratic programming problem; st is the constraint on the function; C min C is the minimum penalty coefficient; max γ is the maximum penalty coefficient; min The minimum kernel parameter; γ max The maximum kernel parameter;

[0047] Find the objective function, perform chemoattraction, replication, and migration operations, and obtain the {C, γ} that maximizes fitness; these are the optimal parameters.

[0048] A power transformer fault diagnosis system, comprising:

[0049] The data acquisition module is used to obtain the content of various gases in the oil of power transformers through chromatographic analysis of oil, and to obtain raw data;

[0050] The data processing module is used to preprocess the raw data, construct feature vectors based on the preprocessed data, and divide them into training sample set T1 and test sample set T2.

[0051] The Support Vector Machine (SVM) building module is used to build a support vector machine based on the training sample set T1.

[0052] The Support Vector Machine (SVM) optimization module is used to optimize the SVM by improving the bacterial foraging algorithm, thereby obtaining an SVM fault diagnosis model.

[0053] The fault diagnosis module is used to input the test sample set T2 into the support vector machine fault diagnosis model for fault diagnosis.

[0054] The present invention provides a computer storage medium, wherein the computer-readable storage medium stores a computer program, and the computer program, when executed by a processor, implements the steps of the power transformer fault diagnosis method described above.

[0055] The beneficial effects of this invention are as follows:

[0056] 1. This invention collects multiple gas data in power transformer oil through chromatographic analysis, enhances the collection of state variables, and constructs a support vector machine based on the gas data to improve the accuracy of state diagnosis.

[0057] 2. This invention optimizes the parameters of the support vector machine by improving the bacterial algorithm, obtaining the optimal penalty coefficient C and kernel parameter γ, thus ensuring the efficiency and accuracy of the support vector machine model and improving the accuracy of power transformer fault diagnosis.

[0058] 3. This invention optimizes the bacterial foraging algorithm through differential evolution, which increases the diversity of the bacterial population, preserves the best individuals in the population, avoids bacteria getting trapped in local optima, and achieves the exploration of globally optimal parameters.

[0059] In summary, the power transformer fault diagnosis method proposed in this invention combines the advantages of traditional chromatographic data analysis with intelligent algorithms, providing a new approach for power transformer fault diagnosis.

[0060] The above description is merely an overview of the technical solution of the present invention. In order to better understand the technical means of the present invention and to implement it according to the contents of the specification, the preferred embodiments of the present invention are described in detail below with reference to the accompanying drawings. Specific embodiments of the present invention are given in detail in the following embodiments and their accompanying drawings. Attached Figure Description

[0061] The accompanying drawings, which are included to provide a further understanding of the invention and form part of this application, illustrate exemplary embodiments of the invention and, together with their description, serve to explain the invention and do not constitute an undue limitation thereof. In the drawings:

[0062] Figure 1 This is a flowchart of a power transformer fault diagnosis method according to Embodiment 1 of the present invention;

[0063] Figure 2 This describes the optimization process of the support vector machine parameters in the improved bacterial foraging algorithm of Embodiment 1 of the present invention;

[0064] Figure 3 This is a flowchart of the bacterial foraging algorithm optimized by differential evolution in Embodiment 1 of the present invention;

[0065] Figure 4 This refers to the diagnostic results after the sample set is input into the support vector machine fault diagnosis model in Embodiment 1 of the present invention. Detailed Implementation

[0066] The principles and features of the present invention are described below with reference to the accompanying drawings. The examples given are only for explaining the present invention and are not intended to limit the scope of the present invention.

[0067] like Figure 1 As shown, the present invention provides a method for diagnosing power transformer faults, including the following steps.

[0068] S1. Obtain the volumes of hydrogen, methane, ethylene, ethane, and acetylene in the oil of the power transformer through chromatographic analysis to obtain the raw data;

[0069] S2. Preprocess the original data, construct feature vectors based on the preprocessed data, and divide them into training sample set T1 and test sample set T2.

[0070] The preprocessing process includes normalizing the raw data, and the specific calculation formula is shown below:

[0071]

[0072] Where, o i1 ,...,o i5 These are hydrogen, methane, ethylene, ethane, and other gases in the chromatographic data, respectively.

[0073] The original volume of acetylene gas; o i1 ′ represents the percentage of hydrogen volume relative to the total gas volume;

[0074] o i2 ′,...,o i5 ′ represents the percentage of the volume of methane, ethylene, ethane, and acetylene relative to the volume of the hydrocarbon gas, respectively.

[0075] S3. Construct a support vector machine based on the training sample set T1, which includes the following steps;

[0076] Based on the training sample set T1={(x e ,y e e = 1, 2, ..., l, x e ∈R n ,y e ∈{+1,-1}}, R n Given an n-dimensional set of real numbers; by defining the kernel function k(x) p ,x qThe nonlinearly separable input vector is mapped to a high-dimensional space, and the optimal hyperplane is found in this new space to correctly distinguish the samples. The corresponding optimal classification function f(x) is:

[0077]

[0078] Where, α e * The data points corresponding to the non-zero values ​​are the support vectors, b is the slack variable, and C is the penalty coefficient.

[0079] Where the RBF kernel function k(x) p ,x q )for:

[0080] k(x p ,x q )=exp(-γ||x p -x q ||) 2

[0081] Where γ is the kernel parameter, x p For the p-th training sample, x q Let q be the q-th training sample.

[0082] S4. Optimize the support vector machine by improving the bacterial foraging algorithm, such as... Figure 2 As shown, the support vector machine fault diagnosis model is obtained, including the following steps;

[0083] The penalty coefficient C and kernel parameter γ of the support vector machine are optimized by improving the bacterial foraging algorithm;

[0084] The parameters for setting the improved bacterial foraging algorithm are as follows:

[0085]

[0086] Where, θ i Let represent the position of individual i in the population, j represent the directional operation, k represent the replication operation, l represent the migration operation, and c(i) represent the step size for moving forward. Δ (i) represents a unit vector in a random direction; Δ i (i) represents the i-th power unit vector in a random direction; θ i (j,k,l) ​​represents the directional, replicative, and migration operations at the location of individual i in the population; θ i (j+1,k,l) ​​represents the next directional operation, replication operation, or migration operation at position i in the population.

[0087] The optimization interval is set with bacterial population s = 50, directional operation j = 2, replication operation k = 4, migration operation l = 10, and migration probability P. ed =0.25;

[0088] Using its current position {C, γ} as the algorithm parameters, and employing k-fold cross-validation of the training sample set, the accuracy C is determined. r The fitness function F(i,j,k,l) ​​is expressed as follows:

[0089]

[0090] Where k is the number of folds in the cross-validation; l t Let be the number of samples in the t-th validation set; This represents the number of correctly classified samples in the validation set; the optimization interval is set to 2. -5 <C<2 10 ,2 -5 <γ<2 5 The cross-validation fold number k = 10;

[0091] Let k-fold cross-validation accuracy The maximum value is the objective function of the optimization problem, and the range of values ​​for the parameters {C, γ} is a constraint condition, i.e.:

[0092]

[0093] In the formula, Obj is the minimum value obtained by solving the quadratic programming problem; st is the constraint on the function; C min C is the minimum penalty coefficient; max γ is the maximum penalty coefficient; min The minimum kernel parameter; γ max The maximum kernel parameter;

[0094] Find the objective function, perform chemoattraction, replication, and migration operations, and obtain the {C, γ} that maximizes fitness; these are the optimal parameters.

[0095] Substituting the optimal parameters {C, γ} into the support vector machine yields the support vector machine fault diagnosis model.

[0096] The improved bacterial foraging algorithm is derived by optimizing the step size c(i) of the forward swimming in the bacterial foraging algorithm using the differential evolution algorithm, such as... Figure 3 As shown, the specific formula is as follows:

[0097]

[0098] Where c(i) is the step size of the forward swim. new c(i) represents the step length for the next forward swim.max V represents the maximum stride length for swimming forward; V represents bacterial susceptibility.

[0099]

[0100] Where j represents the chemotactic operation step of an individual in the population, j max For the largest step of chemotaxis in an individual population; j min `rand` represents the minimum number of steps required for chemotaxis in an individual population; `rand` is a random number.

[0101] S5. Input the test sample set T2 into the support vector machine fault diagnosis model to perform fault diagnosis. The diagnosis results are as follows: Figure 4 As shown, the diagnostic model based on the improved bacterial foraging algorithm and support vector machine, on the test set T2, shows that the model misclassified only 3 samples, with an accuracy exceeding 90%. The fault diagnosis model based on the improved bacterial foraging algorithm and support vector machine has better classification ability, the highest accuracy, the lowest misclassification rate, good robustness, and stronger global optimization ability.

[0102] Example 2

[0103] A power transformer fault diagnosis system, comprising:

[0104] The data acquisition module is used to obtain the content of various gases in the oil of power transformers through chromatographic analysis of oil, and to obtain raw data;

[0105] The data processing module is used to preprocess the raw data, construct feature vectors based on the preprocessed data, and divide them into training sample set T1 and test sample set T2.

[0106] The Support Vector Machine (SVM) building module is used to build a support vector machine based on the training sample set T1.

[0107] The Support Vector Machine (SVM) optimization module is used to optimize the SVM by improving the bacterial foraging algorithm, thereby obtaining an SVM fault diagnosis model.

[0108] The fault diagnosis module is used to input the test sample set T2 into the support vector machine fault diagnosis model for fault diagnosis.

[0109] Example 3

[0110] A computer storage medium storing a computer program, characterized in that, when executed by a processor, the computer program implements the steps of the power transformer fault diagnosis method as described in Embodiment 1.

[0111] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any way. Those skilled in the art can readily implement the present invention based on the accompanying drawings and the above description. However, any modifications, alterations, or variations made by those skilled in the art without departing from the scope of the present invention, utilizing the disclosed technical content, are equivalent embodiments of the present invention. Furthermore, any modifications, alterations, or variations made to the above embodiments based on the essential technology of the present invention are still within the protection scope of the present invention.

Claims

1. A method for diagnosing faults in power transformers, characterized in that, Includes the following steps: The content of various gases in the oil of the power transformer was obtained by chromatographic analysis of the oil, and the raw data was obtained. The original data is preprocessed, and feature vectors are constructed based on the preprocessed data. The data is then divided into a training sample set T1 and a test sample set T2. Construct a support vector machine based on the training sample set T1; The support vector machine (SVM) was optimized by improving the bacterial foraging algorithm to obtain an SVM fault diagnosis model; the test sample set T2 was input into the SVM fault diagnosis model for fault diagnosis. Among them, the improved bacterial foraging algorithm optimizes the step size of forward swimming in the bacterial foraging algorithm through differential evolution algorithm. c ( i It was obtained, specifically as follows: in, c ( i () represents the stride length of the swimmer. c(i) new The stride length for the next forward swimming movement; c(i) max This is the maximum stride length for swimming forward; V Susceptibility to bacteria; in, j For the chemotactic operation steps of individuals in a population, j max This represents the largest step in the chemotaxis operation of an individual in the population. j min The minimum step for chemotaxis in an individual population; rand It is a random number; The raw data includes the volumes of hydrogen, methane, ethylene, ethane, and acetylene. The preprocessing of the raw data includes normalization, and the specific calculation formula is as follows: Where, in the formula These are the original values ​​of the gas volumes in the chromatographic data; This represents the percentage of hydrogen volume relative to the total gas volume. These represent the percentage of the volume of methane, ethylene, ethane, and acetylene relative to the volume of other hydrocarbon gases.

2. The power transformer fault diagnosis method according to claim 1, characterized in that, The step of constructing a support vector machine based on the training sample set T1 includes: Based on training sample set R n Given an n-dimensional set of real numbers; by defining a kernel function The process involves mapping a nonlinearly separable input vector to a high-dimensional space and then searching for the optimal hyperplane within this new space to correctly distinguish the samples. The corresponding optimal classification function is... f(x) for: Where, in the formula The data points corresponding to the non-zero values ​​are the support vectors. b As slack variables, C Penalty coefficient, in RBF Kernel function for: Where, in the formula γ For kernel parameters, x p For the first p training samples, x q For the first q Training samples.

3. The power transformer fault diagnosis method according to claim 2, characterized in that, The improved bacterial foraging algorithm optimizes the support vector machine to obtain the support vector machine fault diagnosis model. The steps include: By improving the penalty coefficient of the support vector machine in the bacterial foraging algorithm. C and kernel parameters γ To perform optimization; The optimal parameters { C , γ Substituting these values ​​into the support vector machine yields the support vector machine fault diagnosis model.

4. The power transformer fault diagnosis method according to claim 3, characterized in that, The improved bacterial foraging algorithm is compared to support vector machines. C and kernel parameters γ The optimization process includes: The parameters for setting the improved bacterial foraging algorithm are as follows: in, θ i For individuals in a population i Location, j This is a directional operation; k For copying operations; l This is a migratory operation; c ( i () represents the stride length of the swimmer. (i) Represents a unit vector in a random direction; θ i (j,k,l) For individuals in a population i Positional directional operations, replication operations, and migration operations; θ i (j+1,k,l) For individuals in a population i The next directional operation, replication operation, or migration operation at the current location; With its current position { C , γ } represents the algorithm parameters, and uses the training sample set. k - Accuracy of cross-validation As a fitness function F ( i , j , k , l Its expression is: in, k The number of folds for cross-validation; For the first t The number of samples in each validation set; This represents the number of samples correctly classified in the validation set. make k - Accuracy of cross-validation The maximum value is the objective function of the optimization problem, and the parameters { C , γ The range of values ​​for} is the constraint condition, that is: (8) Obj is the minimum value obtained by solving the quadratic programming problem; st is the constraint on the function; C min The minimum penalty coefficient; C max This is the maximum penalty coefficient; γ min Minimum kernel parameters; γ max The maximum kernel parameter; Find the objective function, perform chemoattraction, replication, and migration operations, and obtain the result when the fitness is maximized. C , γ The optimal parameters are}.

5. A power transformer fault diagnosis system, characterized in that, The steps of performing the method as described in claim 1 include: The data acquisition module is used to obtain the content of various gases in the oil of power transformers through chromatographic analysis of oil, and to obtain raw data; The data processing module is used to preprocess the raw data, construct feature vectors based on the preprocessed data, and divide them into training sample set T1 and test sample set T2. The Support Vector Machine (SVM) building module is used to build a support vector machine based on the training sample set T1. The Support Vector Machine (SVM) optimization module is used to optimize the SVM by improving the bacterial foraging algorithm, thereby obtaining an SVM fault diagnosis model. The fault diagnosis module is used to input the test sample set T2 into the support vector machine fault diagnosis model for fault diagnosis.

6. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is executed by the processor, it implements the steps of the power transformer fault diagnosis method as described in any one of claims 1-5.