A wind turbine gearbox fault feature extraction method based on Gaussian mixture modeling

Abstract

This invention relates to the field of wind power equipment fault diagnosis technology, and provides a method for extracting fault features from wind turbine gearboxes based on Gaussian mixture modeling. The method includes: modeling a mathematical model of the wind turbine gearbox observation signal by superimposing impact fault feature vectors and multi-source noise vectors, wherein the fault feature vector is the product of a redundant dictionary D and a sparse coefficient vector; modeling the multi-source noise vector as a Gaussian mixture distribution; constructing an objective function within a Bayesian framework to solve for the sparse coefficient vector in the mathematical model using maximum a posteriori probability estimation; simplifying the objective function; and using the EM algorithm and ADMM algorithm in a joint alternating iterative solution to obtain the optimized sparse coefficient vector; and reconstructing the impact fault feature vector in the wind turbine gearbox observation signal. This method improves the accuracy and robustness of extracting fault features from the observation signal of offshore wind turbine gearboxes.

Description

Technical Field

[0001] This invention relates to the field of wind power equipment fault diagnosis technology, specifically to a method for extracting fault features of wind turbine gearboxes based on Gaussian mixture modeling. Background Technology

[0002] Offshore wind power, as an important component of clean and renewable energy, plays a crucial role in the transformation of the energy structure. As the core component of the main drive train of offshore wind turbines, the gearbox operates for a long time in harsh marine environments such as typhoons, salt spray, and waves, and faces frequently changing operating conditions. This results in the presence of complex multi-source noise in its monitoring signals. These noise sources include random noise from the marine environment, inherent system noise, and component coupling noise, and the noise distribution is unknown and difficult to characterize.

[0003] In existing technologies, the paper "WPConvNet: An Interpretable Wavelet Packet Kernel-Constrained Convolutional Network for Noise-Robust Fault Diagnosis" (Li, T.Li, C. Sun, X. Chen and R. Yan. IEEE Transactions on Neural Networks and Learning Systemsvol) proposes WPConvNet, which combines multi-scale basis functions of wavelet packets with convolutional kernel constraints to achieve noise-robust deep feature learning. Although the multi-scale structure has a certain degree of robustness to noise, the noise resistance performance of the model under different noise distributions may still be uncertain if there is a lack of explicit characterization of the statistical properties of noise. For example, Chinese invention patent application CN115774195A, entitled "A Method for Diagnosing Composite Faults of Hybrid Eccentricity and Demagnetization in Permanent Magnet Motors," proposes a fault diagnosis method for permanent magnet synchronous motors that combines sparse representation and machine learning. However, it uses an orthogonal matching pursuit algorithm to solve for sparse coefficients. This greedy algorithm is prone to getting trapped in local optima and is difficult to obtain a globally optimal sparse solution, resulting in limited accuracy in fault feature representation. Furthermore, it does not consider the complex distribution characteristics of noise in the signal, and simple preprocessing alone cannot effectively suppress noise interference. Chinese invention patent application CN121300329A, entitled "A Fault Detection Method Based on Sparse Representation," proposes a fault detection method based on sparse representation, but it uses... Solving for sparse coefficients using norms is an NP-hard optimization problem that requires empirical numerical methods. Constraining sparsity can easily lead to insufficient accuracy or excessive computational complexity in sparse solutions, affecting real-time detection. The fault dictionary in this patent is constructed solely based on normal operating condition data, without incorporating fault feature information. Therefore, the fault dictionary has limited ability to represent fault signals and struggles to accurately capture the differences between fault and normal states. The shortcomings of existing technologies can be summarized as follows: noise modeling lacks specificity, either failing to explicitly characterize noise statistical properties or relying solely on simple preprocessing to address noise, neither of which can adapt to complex noise distributions, resulting in uncertain noise reduction effects; sparse coefficient solving has defects; and the fault dictionary construction is incomplete, with limited ability to represent fault signals and difficulty in accurately distinguishing between fault and normal states.

[0004] Therefore, there is an urgent need for a fault feature extraction method that can accurately model multi-source unknown noise to meet the requirements of reliable fault diagnosis of offshore wind turbine gearboxes. Summary of the Invention

[0005] The technical problem to be solved by this invention is how to improve the accuracy and robustness of extracting fault features from observation signals of offshore wind turbine gearboxes.

[0006] The present invention solves the above-mentioned technical problems through the following technical means:

[0007] This invention provides a method for extracting fault features of wind turbine gearboxes based on Gaussian mixture modeling, comprising the following steps: S1. By superimposing the impact-type fault feature vector and the multi-source noise vector. Modeling wind turbine gearbox observation signals The mathematical model, in which the fault feature vector is a redundant dictionary D and a sparse coefficient vector. The product of the redundant dictionary is a wavelet inverse transform matrix constructed using the db4 wavelet function; S2. The multi-source noise vector in the mathematical model... Modeled as a Gaussian mixture distribution; S3. Within the Bayesian framework, construct a method to solve the sparse coefficient vector in the mathematical model using maximum a posteriori probability estimation. The objective function; S4. By introducing latent variables and lower bound function The objective function obtained by simplifying step S3; S5. The simplified objective function is solved by combining the EM algorithm and the ADMM algorithm in an alternating iterative manner until the optimal solution is approximated, resulting in the optimized sparse coefficient vector. ; S6. Using the optimized sparse coefficient vector A linear combination of the redundant dictionary D is used to reconstruct the wind turbine gearbox observation signal. The feature vector of impact-type faults in the model.

[0008] Furthermore, the mathematical model described in step S1 is as follows:

[0009] in, Where N is the number of observation points, and N is the number of observation points. The redundant dictionary D serves as the basis for providing sparse representations of impact-type fault features. Each column vector in the redundant dictionary D corresponds to an impact feature pattern. It is a sparse coefficient vector; It is a multi-source noise vector that follows a Gaussian mixture distribution.

[0010] Furthermore, the construction of the redundant dictionary in step S1 specifically includes: The standard db4 wavelet function is scaled and time-shifted to generate several wavelet basis functions with different scales and positions. The scale parameter determines the frequency range of the wavelet to adapt to the characteristics of impact faults at different frequencies, and the shift parameter determines the time position of the wavelet to cover the full time domain of the signal. These wavelet basis functions are used as column vectors, forming a structure with dimension . The redundant dictionary D is determined by adjusting the wavelet decomposition level, scale range, and translation step size, where M is the number of atoms in the redundant dictionary.

[0011] Furthermore, the mixed Gaussian distribution described in step S2 is as follows:

[0012] in, This represents the number of Gaussian sub-distributions in the mixture Gaussian distribution, and its value is determined based on the complexity of the multi-source noise. For the first The mixing ratio parameters of the Gaussian sub-distributions satisfy... and ; For the first The mean of each Gaussian sub-distribution, taking into account the symmetry and zero-mean property of the noise, ; For the first The variance of the distribution of the nth Gaussian sub-distribution characterizes the variance of the nth Gaussian sub-distribution. The intensity of noise type; Let be the probability density function of the standard Gaussian distribution, expressed as follows:

[0013] in This represents an exponential function.

[0014] Further, step S3 includes the following steps: S31, Given a sparse coefficient vector In this case, construct the observation signal Likelihood probability of occurrence As shown in the following formula:

[0015] in, For observation signal The One element, The first sparse representation of the reconstructed signal One element, For the first The noise residual of each element; S32, Given a sparse coefficient vector In this case, construct the observation signal Prior probability of occurrence As shown in the following formula:

[0016] in, is the scaling parameter of the Laplace distribution, used to adjust the strength of the prior probability; for Norms, by summing the absolute values ​​of the coefficients, achieve sparsity constraints; S33. Based on steps S31 and S32, observe the signal. The posterior probability is given by the following formula:

[0017] Therefore, the objective of maximum a posteriori probability estimation is to find the posterior probability that... Largest sparse coefficient vector As shown in the following formula:

[0018] Taking the natural logarithm of both sides of equation (7), we transform the product form into a summation form, as shown below:

[0019] S34. Substitute equations (4) and (5) into equation (8), and ignore the... Ignoring the irrelevant constant terms, we obtain the optimization objective function, as follows:

[0020] The first term is the data fidelity term, used to ensure the consistency between the reconstructed signal and the observed signal, and to achieve accurate fitting of multi-source noise through the logarithmic form of the Gaussian mixture distribution; the second term is the sparsity regularization term, used to promote the sparse coefficient vector. The sparsity ensures that only valid information corresponding to impact-type fault characteristics is retained.

[0021] Further, step S4 includes the following steps: S41. Introducing Latent Variables Its physical meaning is when This indicates that the observed signal is the first... Noise residual of each element From the A Gaussian sub-distribution; This indicates that it does not come from the first... Gaussian subdistribution; latent variables Satisfy constraints ; Based on latent variables Likelihood probability It can be rewritten as:

[0022] S42. Based on Jensen's inequality, transform the logarithmic form of the likelihood probability to introduce the probability distribution of the latent variable. Construct a lower bound function for the optimization objective function. As shown in the following formula:

[0023] in, The set of parameters to be estimated. Let be the set of variances of each Gaussian sub-distribution; substituting equations (10) and (5) into equation (11), expanding and simplifying, we obtain the lower bound function. The specific form is as follows:

[0024] in, For redundant dictionaries The row vectors Latent variables The expectation, denoted as , indicating the first The noise residual belongs to the first The probability of a Gaussian sub-distribution; further simplifying equation (12), ignoring the probability of a Gaussian sub-distribution; Irrelevant constant terms yield a simplified lower bound function. As shown in the following formula:

[0025] in, To and Irrelevant constant terms can be ignored during the optimization process; therefore, the sparse coefficient vector... The optimization objective function can be transformed into:

[0026] S43. Define the reweighting matrix. It is a diagonal matrix, and its diagonal elements are:

[0027] Equation (14) can then be transformed into:

[0028] In the formula, This is a regularization parameter used to balance the weights of data fidelity terms and sparse regularization terms; For weighted Norm, through reweighted matrix Different weights are assigned to residuals of different noise intensities to suppress strong noise components.

[0029] Further, step S5 includes the following steps: S51. Use the EM algorithm to estimate the parameters of the mixture Gaussian distribution; S52. The ADMM algorithm is used to solve the objective function through variable separation and alternating updates. S53. Alternately execute the EM algorithm and ADMM algorithm until the convergence condition is met, and output the optimized sparse coefficient vector. .

[0030] Further, step S51 includes the following steps: S511. Calculate the expected value of latent variables. Assuming the current sparse coefficient vector and mixture Gaussian distribution parameters Given that the latent variables can be calculated using Bayes' theorem... posterior probability expectation That is, the first The noise residual belongs to the first The probability of a Gaussian sub-distribution is given by the following formula:

[0031] Wherein, the denominator is the probability-weighted sum of all Gaussian sub-distributions under the current residual, ensuring This conforms to the basic properties of probability; S512. Update the parameters of the Gaussian mixture distribution based on the expected value of the latent variables obtained in step S511. By maximizing the lower bound function Update the parameters of the Gaussian mixture distribution. and ; By all noise residuals belonging to the first Estimate by the probability mean of a Gaussian sub-distribution As shown in the following formula:

[0032] Estimate by the mean of the weighted squared residuals As shown in the following formula:

[0033] Where the weights are the expected values ​​of the latent variables. .

[0034] Further, step S52 includes the following steps: S521, Introducing Auxiliary Variables Equation (16) is transformed into a constrained optimization problem, as shown below:

[0035] S522. Incorporate the constraints into the optimization objective to construct the augmented Lagrangian function, as shown below:

[0036] in, As a dual variable, it is used to penalize the degree of violation of constraints; This is a penalty parameter used to balance the weights of data fidelity terms and constraint penalty terms; S523, based on the ADMM algorithm, through alternating updates , and The solution is gradually approximated as optimal, specifically as follows: fixed and ,right Solving unconstrained optimization problems, based on the objective function with respect to... It is a convex quadratic function, and the closed-form solution can be obtained by differentiation, as shown in the following equation:

[0037] fixed and ,right Solving unconstrained optimization problems, the essence of which is... The soft threshold of the norm is solved as follows:

[0038] in, For soft thresholding operators, The threshold parameter is calculated using the following formula:

[0039] in, For input vectors The One element, It is a symbolic function; Dual variable update rules based on ADMM algorithm The update is as follows:

[0040] in, Indicates the number of iterations.

[0041] Furthermore, the convergence condition described in step S53 satisfies the following equation:

[0042] The convergence condition is that the difference between the objective functions of two iterations is less than 1. .

[0043] The advantages of this invention are: (1) This invention constructs a sparse representation model based on the db4 wavelet dictionary, introduces a mixture Gaussian distribution to accurately model multi-source mixed noise, breaks through the limitations of the traditional single Gaussian distribution assumption, utilizes its universal approximation property to comprehensively characterize the complex statistical characteristics of noise from different sources and intensities, and at the same time ensures the sparsity of fault features through the Laplace prior distribution, making up for the shortcomings of traditional methods in adapting to complex noise.

[0044] (2) This invention introduces latent variables based on the Bayesian framework to solve the problem of modeling Gaussian mixture distribution. It constructs the next Q function to transform the optimization problem with latent variables into a convex optimization problem that can be solved efficiently. Then, it adopts the EM-ADMM joint optimization algorithm. The EM algorithm adaptively estimates the key parameters such as the mixing ratio and variance of the Gaussian mixture distribution. With the help of the ADMM algorithm, the sparse coefficients are solved efficiently through variable separation and reweighting mechanism. The synergistic optimization of parameter estimation and sparse solution is achieved, which not only ensures the convergence stability of the model, but also improves the solution efficiency.

[0045] (3) This invention enhances the ability to focus on the features of weak impact faults and suppresses strong noise interference by leveraging the time-frequency localization characteristics of the db4 wavelet dictionary and the differentiated weight allocation of the reweighted matrix. Then, by reconstructing the fault feature signal through a linear combination of the dictionary and optimal sparse coefficients, it achieves effective separation of noise and features, outputting intuitive fault features with clear physical meaning. Finally, through a complete "modeling-optimization-reconstruction" process, it achieves accurate extraction of fault features, providing a scientific basis for mechanical equipment fault diagnosis. Attached Figure Description

[0046] Figure 1 This is a flowchart illustrating a method for extracting fault features of wind turbine gearboxes based on Gaussian mixture modeling, according to an embodiment of the present invention. Detailed Implementation

[0047] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below in conjunction with the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0048] Example 1 This embodiment provides a method for extracting fault features of wind turbine gearboxes based on Gaussian mixture modeling. It combines Gaussian mixture distribution modeling with Bayesian sparse representation theory for extracting impact fault features in offshore wind turbine gearboxes. First, a Gaussian mixture distribution is used to accurately fit multi-source mixed noise. Utilizing its universal approximation property, this method overcomes the limitations of the traditional single Gaussian distribution assumption, comprehensively characterizing the complex statistical characteristics of noise from different sources and intensities. Then, relying on a Bayesian framework, likelihood probability and prior probability are organically integrated to construct an optimization objective that combines data fidelity and sparsity constraints. This ensures that fault feature extraction conforms to the signal probability distribution law while also being able to extract features through... Norms enable efficient feature focusing. Furthermore, latent variables and Q-functions are introduced to address the challenge of solving mixed distributions. Combined with the EM-ADMM joint optimization algorithm, adaptive estimation of noise parameters and efficient solution of sparse coefficients are achieved, ensuring both the accuracy of parameter estimation and improving the algorithm's convergence speed. Finally, fault feature reconstruction is performed based on the db4 wavelet dictionary, effectively separating mixed noise from impact features. Through a deep integration of mixed Gaussian noise modeling and Bayesian sparse representation, the model exhibits both strong noise resistance and high accuracy, efficiently extracting weak impact fault features even under multi-source mixed noise interference. The specific implementation process of the method is as follows... Figure 1 As shown, it includes the following steps: S1. By superimposing the impact-type fault feature vector and the multi-source noise vector. Modeling wind turbine gearbox observation signals The mathematical model, in which the fault feature vector is a redundant dictionary D and a sparse coefficient vector. The product of the two terms, wherein the redundant dictionary is a wavelet inverse transform matrix constructed using the db4 wavelet function; the mathematical model is as follows:

[0049] in, Where N is the number of observation points, and N is the number of observation points. The redundant dictionary D serves as the basis for providing sparse representations of impact-type fault features. Each column vector in the redundant dictionary D corresponds to an impact feature pattern. It is a sparse coefficient vector, most of its elements are 0 or close to 0, and only a few non-zero elements correspond to the weights of the impact fault features. Sparsity is the core prerequisite for realizing feature extraction. It is a multi-source noise vector containing mixed noise from multiple sources in the mechanical system, and follows a mixture Gaussian distribution.

[0050] To ensure that impact-type fault features can be efficiently and sparsely represented on the dictionary, the construction of the redundant dictionary specifically includes: The standard db4 wavelet function is scaled and time-shifted to generate several wavelet basis functions with different scales and positions. The scale parameter determines the frequency range of the wavelet to adapt to the characteristics of impact faults at different frequencies, and the shift parameter determines the time position of the wavelet to cover the full time domain of the signal. These wavelet basis functions are used as column vectors, forming a structure with dimension . The redundant dictionary D, where M is the number of atoms in the redundant dictionary, is determined by adjusting the wavelet decomposition level (usually set to ). The layers, scale range (corresponding to the impact characteristic frequency range), and translation step size (to ensure no omission of atomic coverage) are determined to ensure that the dictionary can fully cover impact-type fault characteristic modes of different frequencies and amplitudes.

[0051] Step S1 above lays a solid foundation for fault feature extraction through precise signal decomposition and targeted redundant dictionary design. Signal modeling strictly follows the physical essence of "feature + noise," clearly defining the mathematical definitions and physical meanings of each component to ensure that the model can truly reflect the compositional rules of vibration signals and avoid feature extraction distortion caused by modeling bias. The redundant dictionary design focuses on the characteristics of impact faults, selecting the db4 wavelet function and optimizing the scale and translation parameters to ensure a high degree of matching between dictionary atoms and impact features, thereby improving the efficiency and accuracy of sparse feature representation. Balancing fitting ability and computational efficiency, the dictionary dimensions are reasonably set to ensure full coverage of impact features of different frequencies and amplitudes while reducing computational complexity by leveraging the tight support characteristics of the db4 wavelet, thus adapting to the real-time diagnostic needs of engineering projects. The redundant dictionary is highly adaptable and can be adapted to impact fault feature extraction scenarios of different equipment such as bearings and gearboxes by adjusting parameters such as the number of decomposition layers and scale range, thereby improving the versatility of the method.

[0052] S2. Facing the high complexity of noise sources and unknown distribution in the monitoring signals of offshore wind turbine gearboxes. This embodiment assumes a complex problem involving multiple noise sources. Each element It follows an independent and identically distributed Mixture of Gaussian (MoG) distribution. The core advantage of the MoG distribution lies in its universal approximation property; by weighted combination of multiple Gaussian sub-distributions, it can accurately fit any continuous probability distribution, thus comprehensively characterizing the complex statistical properties of multi-source mixed noise. This relates to the multi-source noise vector in the mathematical model. The model is a Gaussian mixture distribution; specifically, as shown in the following formula:

[0053] in, The number of Gaussian sub-distributions in the mixture Gaussian distribution is determined based on the complexity of the multi-source noise. In this embodiment, it is selected between 3 and 8 using the cross-validation method. For the first The mixing ratio parameters of the Gaussian sub-distributions satisfy... and Its physical meaning is the first The weight of a noise type in the total noise; For the first The mean of each Gaussian sub-distribution, taking into account the symmetry and zero-mean property of the noise, ; For the first The variance of the distribution of the nth Gaussian sub-distribution characterizes the variance of the nth Gaussian sub-distribution. The intensity of different types of noise The values ​​of correspond to noise components of different intensities (such as...). Smaller subdistributions correspond to lower intensity intrinsic noise. Larger sub-distributions correspond to high-intensity impulse interference noise. Let be the probability density function of the standard Gaussian distribution, expressed as follows:

[0054] in This represents an exponential function.

[0055] In a mixture Gaussian distribution, each Gaussian sub-distribution corresponds to a type of noise component, determined by the mixing ratio parameter. and variance parameter The combination of these methods allows for the precise quantification of the contribution and intensity characteristics of different noise sources. For example, the inherent friction noise during normal equipment operation can be characterized by a low-variance Gaussian sub-distribution, while interference noises such as component collisions and external impacts can be characterized by a high-variance Gaussian sub-distribution. The mixing ratio reflects the dynamic change of the proportion of each type of noise in the total noise as the operating conditions change. This modeling approach essentially characterizes the statistical properties of multi-source mixed noise, providing a solid theoretical foundation for subsequent noise suppression.

[0056] Step S2 above achieves an accurate characterization of multi-source mixed noise through scientific noise distribution assumptions and parameter definitions. The noise distribution closely matches engineering practice, utilizing the universal approximation property of the mixture Gaussian distribution to overcome the limitations of a single distribution assumption. It can comprehensively fit the complex statistical characteristics of multi-source mixed noise, providing theoretical support for strong noise suppression. The parameters have clear physical meanings, and the mixing ratios are... With variance The contribution and intensity of noise sources are quantified separately, ensuring that noise modeling not only has a mathematical basis but also reflects the actual action mechanism of different noise components, facilitating subsequent parameter optimization and noise suppression. The modeling is highly flexible; by adjusting the number K of Gaussian sub-distributions, it can adapt to noise scenarios of varying complexity, ensuring efficient fitting of simple noise while accurately characterizing multi-source strong interference noise, thus improving the method's adaptability to different operating conditions. The calculations are simplified without sacrificing accuracy, assuming the mean of each sub-distribution. While reducing model complexity, signal preprocessing ensures the rationality of assumptions, thus achieving a balance between modeling accuracy and computational efficiency.

[0057] S3. Within the Bayesian framework, construct a method to solve the sparse coefficient vector in the mathematical model using maximum a posteriori probability estimation. The objective function; the specific implementation includes the following steps: S31, Given a sparse coefficient vector In this case, construct the observation signal Likelihood probability of occurrence As shown in the following formula:

[0058] in, For observation signal The One element, The first sparse representation of the reconstructed signal One element, For the first The noise residual of each element; S32, To ensure the sparse coefficient vector The sparsity of the coefficients is assumed to follow a Laplace distribution, which has a heavy-tailed property that effectively promotes coefficient sparsity. Given a sparse coefficient vector... In this case, construct the observation signal Prior probability of occurrence As shown in the following formula:

[0059] in, is the scaling parameter of the Laplace distribution, used to adjust the strength of the prior probability; for Norms, by summing the absolute values ​​of the coefficients, achieve sparsity constraints; S33. Based on steps S31 and S32, observe the signal. The posterior probability is given by the following formula:

[0060] Therefore, the objective of maximum a posteriori probability estimation is to find the posterior probability that... Largest sparse coefficient vector As shown in the following formula:

[0061] Taking the natural logarithm of both sides of equation (7) (the logarithmic function is a monotonically increasing function and does not change the optimization result), the product form is transformed into a summation form, as shown in the following equation:

[0062] S34. Substitute equations (4) and (5) into equation (8), and ignore the... Irrelevant constant terms are used to obtain the optimization objective function, ultimately transforming the maximization problem into a minimization problem, as shown in the following equation:

[0063] The first term is the data fidelity term, used to ensure the consistency between the reconstructed signal and the observed signal, and to achieve accurate fitting of multi-source noise through the logarithmic form of the Gaussian mixture distribution; the second term is the sparsity regularization term, used to promote the sparse coefficient vector. The sparsity ensures that only valid information corresponding to impact-type fault characteristics is retained.

[0064] Step S3 above constructs an optimization objective based on Bayesian theory, achieving an organic unity between data fidelity and sparsity constraints. It has a solid theoretical foundation, combining likelihood probability and prior probability to give sparse representation clear statistical meaning, ensuring that the optimization objective not only pursues numerical optimality but also conforms to the probability distribution of the signal, thus improving the reliability of feature extraction. The objective function is scientifically designed; the data fidelity term accurately fits the noise using a Gaussian mixture distribution logarithmic form, while the sparsity regularization term employs... Norms effectively promote coefficient sparsity, and the weights of both are adjusted through regularization parameters. It is flexible and can be dynamically adapted to noise intensity and feature sparsity; the optimization direction is clear, transforming the maximization of posterior probability into a minimization problem, which is simple in form and easy to solve, providing a clear goal orientation for the design of subsequent joint optimization algorithms; it is robust, embracing the uncertainty of noise distribution through probabilistic modeling, and has a stronger adaptability to fluctuations in noise statistical characteristics compared to traditional deterministic models, thus improving the stability of fault feature extraction.

[0065] S4. Since the objective function contains a logarithmic summation term from a Gaussian mixture distribution, it is extremely difficult to solve directly. Therefore, we introduce latent variables... and lower bound function The objective function obtained in simplified step S3 is described below. The specific implementation includes the following steps: S41. Introducing Latent Variables Its physical meaning is when This indicates that the observed signal is the first... Noise residual of each element From the A Gaussian sub-distribution; This indicates that it does not come from the first... Gaussian subdistribution; latent variables Satisfy constraints (Each noise residual belongs to only one Gaussian sub-distribution); Based on latent variables Likelihood probability It can be rewritten as:

[0066] S42. Based on Jensen's inequality, transform the logarithmic form of the likelihood probability to introduce the probability distribution of the latent variable. Construct a lower bound function for the optimization objective function. As shown in the following formula:

[0067] in, The set of parameters to be estimated. Let be the set of variances of each Gaussian sub-distribution; substituting equations (10) and (5) into equation (11), expanding and simplifying, we obtain the lower bound function. The specific form is as follows:

[0068] in, For redundant dictionaries The row vectors Latent variables The expectation, denoted as , indicating the first The noise residual belongs to the first The probability of a Gaussian sub-distribution; further simplifying equation (12), ignoring the probability of a Gaussian sub-distribution; Irrelevant constant terms yield a simplified lower bound function. As shown in the following formula:

[0069] in, To and Irrelevant constant terms can be ignored during the optimization process; therefore, the sparse coefficient vector... The optimization objective function can be transformed into:

[0070] S43. To make equation (14) consistent with the classical sparse representation, a reweighted matrix is ​​defined. It is a diagonal matrix, and its diagonal elements are:

[0071] Equation (14) can then be transformed into:

[0072] In the formula, This is a regularization parameter used to balance the weights of data fidelity terms and sparse regularization terms; For weighted Norm, through reweighted matrix Different weights are assigned to residuals of different noise intensities to suppress strong noise components.

[0073] Step S4 above addresses the challenge of solving mixed distribution problems through latent variables, providing an efficient solution path for the optimization objective. It effectively simplifies complex problems, transforming a non-smooth optimization problem containing logarithmic summation terms of a mixed Gaussian distribution into a convex optimization problem based on the Q-function, reducing the difficulty of direct solution and improving the algorithm's convergence speed. The latent variables have clear physical meanings. Clearly characterizing the attribution of noise residuals makes the modeling process of mixed distributions more intuitive and easier to understand and interpret parameters; the lower bound property of the Q function is reliable, and the Q function constructed based on Jensen's inequality can strictly approximate the lower bound of the original optimization objective, ensuring that the objective function is monotonically increasing during the iteration process and guaranteeing the convergence of the algorithm; key information is preserved, and only constant terms unrelated to the optimization variables are ignored during the simplification of the Q function, without losing core information, ensuring the accuracy of sparse coefficient estimation.

[0074] The objective function of S5 and Equation (16) is a non-smooth convex optimization problem, and involves the joint estimation of Gaussian mixture distribution parameters and sparse coefficients. The simplified objective function is solved by alternating iterations of the EM algorithm and ADMM algorithm until the optimal solution is approximated, and the optimized sparse coefficient vector is obtained. The core idea is to use the EM algorithm to estimate the parameters of the Gaussian mixture distribution. With latent variable expectation The ADMM algorithm is used to solve for the sparse coefficient vector. The two processes are iterated alternately until convergence. The specific implementation includes the following steps: S51. Estimate the parameters of the mixture Gaussian distribution using the EM algorithm; including the following steps: S511. Calculate the expected value of latent variables. Assuming the current sparse coefficient vector and mixture Gaussian distribution parameters Given that the latent variables can be calculated using Bayes' theorem... posterior probability expectation That is, the first The noise residual belongs to the first The probability of a Gaussian sub-distribution is given by the following formula:

[0075] Wherein, the denominator is the probability-weighted sum of all Gaussian sub-distributions under the current residual, ensuring This conforms to the basic properties of probability; S512. Update the parameters of the Gaussian mixture distribution based on the expected value of the latent variables obtained in step S511. By maximizing the lower bound function Update the parameters of the Gaussian mixture distribution. and ; By all noise residuals belonging to the first Estimate by the probability mean of a Gaussian sub-distribution As shown in the following formula:

[0076] Estimate by the mean of the weighted squared residuals As shown in the following formula:

[0077] Where the weights are the expected values ​​of the latent variables. .

[0078] S52. The ADMM algorithm is used to solve the objective function through variable separation and alternating updates; including the following steps: S521, Introducing Auxiliary Variables Equation (16) is transformed into a constrained optimization problem, as shown below:

[0079] S522. Incorporate the constraints into the optimization objective to construct the augmented Lagrangian function, as shown below:

[0080] in, As a dual variable, it is used to penalize the degree of violation of constraints; This is a penalty parameter used to balance the weights of the data fidelity term and the constraint penalty term; in this embodiment, it is determined through simulation experiments. It can balance convergence speed and optimization accuracy.

[0081] S523, based on the ADMM algorithm, through alternating updates , and The solution is gradually approximated as optimal, specifically as follows: fixed and ,right Solving unconstrained optimization problems, based on the objective function with respect to... It is a convex quadratic function, and the closed-form solution can be obtained by differentiation, as shown in the following equation:

[0082] fixed and ,right Solving unconstrained optimization problems, the essence of which is... The soft threshold of the norm is solved as follows:

[0083] in, For soft thresholding operators, The threshold parameter is calculated using the following formula:

[0084] in, For input vectors The One element, It is a symbolic function; Dual variable update rules based on ADMM algorithm The update is as follows:

[0085] in, Indicates the number of iterations.

[0086] S53. Alternately execute the EM algorithm and ADMM algorithm until the convergence condition is met, and output the optimized sparse coefficient vector. The convergence condition satisfies the following equation:

[0087] The convergence condition is that the difference between the objective functions of two iterations is less than 1. .

[0088] Step S5 above achieves efficient estimation of parameters and coefficients through the collaborative use of two algorithms, balancing accuracy and efficiency. The parameter estimation is highly adaptive; the EM algorithm iterates alternately between expectation and maximization steps, automatically estimating the mixing ratio, variance, and latent variable expectation of the Gaussian mixture distribution without manual intervention, adapting to dynamic changes in noise distribution and improving the method's adaptability. The sparse solution is efficient and stable; the ADMM algorithm decomposes the complex optimization problem into three simple subproblems through variable separation, each with a closed-form solution or an efficient solution method, avoiding the convergence and oscillation problems of traditional gradient descent methods, thus improving solution efficiency and stability. The reweighting mechanism is highly targeted, using a reweighting matrix... Differential weights are assigned to residuals of varying noise intensities to achieve precise suppression of strong noise components and improve the separation of fault features from noise; iterative convergence is guaranteed by explicitly setting convergence conditions (the difference in the objective function is less than...). This avoids excessive or insufficient iteration, ensuring that the algorithm achieves a balance between accuracy and computational cost.

[0089] S6. Using the optimized sparse coefficient vector A linear combination of the redundant dictionary D is used to reconstruct the wind turbine gearbox observation signal. The impact-type fault feature vector is reconstructed as follows:

[0090] In the formula, The reconstructed impact-type fault feature vector has been effectively separated from multi-source noise, and the periodic impact features are clearly visible. It can be directly used for subsequent diagnostic tasks such as fault type identification and fault location.

[0091] Step S6 above uses the redundant dictionary D and the optimized sparse coefficient vector The linear combination of these features enables accurate reconstruction and output of fault characteristics. The physical meaning of the reconstructed features is clear; based on the "atomic combination" principle of sparse representation, the reconstruction process is reversible and interpretable, clearly tracing the source of fault characteristics and improving the reliability of diagnostic results. The output results are highly practical; the reconstructed feature signals have effectively separated multi-source noise, and periodic impact characteristics are clearly visible, allowing direct use for subsequent diagnostic tasks such as fault type identification and location positioning without additional processing. The results have broad adaptability; the output signal has the same dimension as the original observed signal, facilitating integration with traditional signal analysis methods (such as envelope spectrum analysis), compatibility with existing diagnostic processes, and lowering the barrier to engineering applications. Features are fully preserved; during the sparse coefficient estimation process... Norm constraints only remove invalid coefficients corresponding to noise, without losing key information about fault characteristics, ensuring that the reconstructed features can truly reflect the equipment fault state.

[0092] 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 extracting fault features of wind turbine gearboxes based on Gaussian mixture modeling, characterized in that, Includes the following steps: S1. By superimposing the impact-type fault feature vector and the multi-source noise vector. Modeling wind turbine gearbox observation signals The mathematical model, in which the fault feature vector is a redundant dictionary D and a sparse coefficient vector. The product of the redundant dictionary is a wavelet inverse transform matrix constructed using the db4 wavelet function; S2. The multi-source noise vector in the mathematical model... Modeled as a Gaussian mixture distribution; S3. Within the Bayesian framework, construct a method to solve the sparse coefficient vector in the mathematical model using maximum a posteriori probability estimation. The objective function; S4. By introducing latent variables and lower bound function The objective function obtained by simplifying step S3; S5. The simplified objective function is solved by combining the EM algorithm and the ADMM algorithm in an alternating iterative manner until the optimal solution is approximated, resulting in the optimized sparse coefficient vector. ; S6. Using the optimized sparse coefficient vector A linear combination of the redundant dictionary D is used to reconstruct the wind turbine gearbox observation signal. The feature vector of impact-type faults in the model.

2. The method for extracting fault features of wind turbine gearboxes based on Gaussian mixture modeling according to claim 1, characterized in that, The mathematical model described in step S1 is as follows: in, Where N is the number of observation points, and N is the number of observation points. The redundant dictionary D serves as the basis for providing sparse representations of impact-type fault features. Each column vector in the redundant dictionary D corresponds to an impact feature pattern. It is a sparse coefficient vector; It is a multi-source noise vector that follows a Gaussian mixture distribution.

3. The method for extracting fault features of wind turbine gearboxes based on Gaussian mixture modeling according to claim 1, characterized in that, The construction of the redundant dictionary in step S1 specifically includes: The standard db4 wavelet function is scaled and time-shifted to generate several wavelet basis functions with different scales and positions. The scale parameter determines the frequency range of the wavelet to adapt to the characteristics of impact faults at different frequencies, and the shift parameter determines the time position of the wavelet to cover the full time domain of the signal. These wavelet basis functions are used as column vectors, forming a structure with dimension . The redundant dictionary D is determined by adjusting the wavelet decomposition level, scale range, and translation step size, where M is the number of atoms in the redundant dictionary.

4. The method for extracting fault features of wind turbine gearboxes based on Gaussian mixture modeling according to claim 1, characterized in that, The mixture Gaussian distribution mentioned in step S2 is as follows: in, This represents the number of Gaussian sub-distributions in the mixture Gaussian distribution, and its value is determined based on the complexity of the multi-source noise. For the first The mixing ratio parameters of the Gaussian sub-distributions satisfy... and ; For the first The mean of each Gaussian sub-distribution, taking into account the symmetry and zero-mean property of the noise, ; For the first The variance of the distribution of the nth Gaussian sub-distribution characterizes the variance of the nth Gaussian sub-distribution. The intensity of noise type; Let be the probability density function of the standard Gaussian distribution, expressed as follows: in This represents an exponential function.

5. The method for extracting fault features of wind turbine gearboxes based on Gaussian mixture modeling according to claim 1, characterized in that, Step S3 includes the following steps: S31, Given a sparse coefficient vector In this case, construct the observation signal Likelihood probability of occurrence As shown in the following formula: in, For observation signal The One element, The first sparse representation of the reconstructed signal One element, For the first The noise residual of each element; S32, Given a sparse coefficient vector In this case, construct the observation signal Prior probability of occurrence As shown in the following formula: in, is the scaling parameter of the Laplace distribution, used to adjust the strength of the prior probability; for Norms, by summing the absolute values ​​of the coefficients, achieve sparsity constraints; S33. Based on steps S31 and S32, observe the signal. The posterior probability is given by the following formula: Therefore, the objective of maximum a posteriori probability estimation is to find the posterior probability that... Largest sparse coefficient vector As shown in the following formula: Taking the natural logarithm of both sides of equation (7), we transform the product form into a summation form, as shown below: S34. Substitute equations (4) and (5) into equation (8), and ignore the... Ignoring the irrelevant constant terms, we obtain the optimization objective function, as follows: The first term is the data fidelity term, used to ensure the consistency between the reconstructed signal and the observed signal, and to achieve accurate fitting of multi-source noise through the logarithmic form of the Gaussian mixture distribution; the second term is the sparsity regularization term, used to promote the sparse coefficient vector. The sparsity ensures that only valid information corresponding to impact-type fault characteristics is retained.

6. The method for extracting fault features of wind turbine gearboxes based on Gaussian mixture modeling according to claim 5, characterized in that, Step S4 includes the following steps: S41. Introducing Latent Variables Its physical meaning is when This indicates that the observed signal is the first... Noise residual of each element From the A Gaussian sub-distribution; This indicates that it does not come from the first... Gaussian subdistribution; latent variables Satisfy constraints ; Based on latent variables Likelihood probability It can be rewritten as: S42. Based on Jensen's inequality, transform the logarithmic form of the likelihood probability to introduce the probability distribution of the latent variable. Construct a lower bound function for the optimization objective function. As shown in the following formula: in, For the set of parameters to be estimated, Let be the set of variances of each Gaussian sub-distribution; substituting equations (10) and (5) into equation (11), expanding and simplifying, we obtain the lower bound function. The specific form is as follows: in, For redundant dictionaries The row vectors Latent variables The expectation, denoted as , indicating the first The noise residual belongs to the first The probability of a Gaussian sub-distribution; further simplifying equation (12), ignoring the probability of a Gaussian sub-distribution; Irrelevant constant terms yield a simplified lower bound function. As shown in the following formula: in, To and Irrelevant constant terms can be ignored during the optimization process; therefore, the sparse coefficient vector... The optimization objective function can be transformed into: S43. Define the reweighting matrix. It is a diagonal matrix, and its diagonal elements are: Equation (14) can then be transformed into: In the formula, This is a regularization parameter used to balance the weights of data fidelity terms and sparse regularization terms; For weighted Norm, through reweighted matrix Different weights are assigned to residuals of different noise intensities to suppress strong noise components.

7. The method for extracting fault features of wind turbine gearboxes based on Gaussian mixture modeling according to claim 6, characterized in that, Step S5 includes the following steps: S51. Use the EM algorithm to estimate the parameters of the mixture Gaussian distribution; S52. The ADMM algorithm is used to solve the objective function through variable separation and alternating updates. S53. Alternately execute the EM algorithm and ADMM algorithm until the convergence condition is met, and output the optimized sparse coefficient vector. .

8. The method for extracting fault features of wind turbine gearboxes based on Gaussian mixture modeling according to claim 7, characterized in that, Step S51 includes the following steps: S511. Calculate the expected value of latent variables. Assuming the current sparse coefficient vector and mixture Gaussian distribution parameters Given that the latent variables can be calculated using Bayes' theorem... posterior probability expectation That is, the first The noise residual belongs to the first The probability of a Gaussian sub-distribution is given by the following formula: Wherein, the denominator is the probability-weighted sum of all Gaussian sub-distributions under the current residual, ensuring This conforms to the basic properties of probability; S512. Update the parameters of the mixture Gaussian distribution based on the expected value of the latent variables obtained in step S511. By maximizing the lower bound function Update the parameters of the Gaussian mixture distribution. and ; By all noise residuals belonging to the first Estimate by the probability mean of a Gaussian sub-distribution As shown in the following formula: Estimate by the mean of the weighted squared residuals As shown in the following formula: Where the weights are the expected values ​​of the latent variables. .

9. The method for extracting fault features of wind turbine gearboxes based on Gaussian mixture modeling according to claim 8, characterized in that, Step S52 includes the following steps: S521, Introducing Auxiliary Variables Equation (16) is transformed into a constrained optimization problem, as shown below: S522. Incorporate the constraints into the optimization objective to construct the augmented Lagrangian function, as shown below: in, As a dual variable, it is used to penalize the degree of violation of constraints; This is a penalty parameter used to balance the weights of data fidelity terms and constraint penalty terms; S523, based on the ADMM algorithm, through alternating updates , and The solution is gradually approximated as optimal, specifically as follows: fixed and ,right Solving unconstrained optimization problems, based on the objective function with respect to... It is a convex quadratic function, and the closed-form solution can be obtained by differentiation, as shown in the following equation: fixed and ,right Solving unconstrained optimization problems, the essence of which is... The soft threshold of the norm is solved as follows: in, For soft thresholding operators, The threshold parameter is calculated using the following formula: in, For input vectors The One element, It is a symbolic function; Dual variable update rules based on ADMM algorithm The update is as follows: in, Indicates the number of iterations.

10. The method for extracting fault features of wind turbine gearboxes based on Gaussian mixture modeling according to claim 9, characterized in that, The convergence condition described in step S53 satisfies the following equation: The convergence condition is that the difference between the objective functions of two iterations is less than 1. .

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