Fan Health Assessment Method Based on Koopman Space Transformation and Mahalanobis Distance

Abstract

The fan health assessment method based on Koopman space transformation and Mahalanobis distance disclosed by the present invention first collects the low-frequency signals and high-frequency vibration signals of the system. After preprocessing the data, it uses extended dynamic mode decomposition to construct a second-order polynomial basis function, and solves the Koopman operator to achieve the global linearization of the nonlinear system. Key modes with low attenuation rates and matching the characteristic frequencies of gear wear and bearing faults are selected through eigenvalue decomposition. The real-time data is projected into the modal space, and the standardized Mahalanobis distance from its normal state distribution is calculated. A health index model is constructed in combination with historical data to achieve health quantification and hierarchical warning. The present invention combines Koopman theory with statistical distance, solves the high-dimensional modeling problem of nonlinear fan systems, realizes the accurate mapping of fault severity and life loss, is applicable to the health management of key components such as gearboxes and bearings, and is extended to rotating machinery fields such as water pumps and compressors.

Description

Technical Field

[0001] The present invention belongs to the technical field of condition monitoring and fault diagnosis of wind turbines, and specifically relates to a method for evaluating the health of wind turbines based on Koopman space transformation and Mahalanobis distance. Background Technique

[0002] Currently, the mainstream methods for evaluating the health of wind turbines include vibration analysis, SCADA data modeling, data-driven AI, etc. However, the above methods still face the following key problems:

[0003] Low efficiency of multi-source data fusion: Existing condition monitoring methods for wind turbines rely mostly on simple weighting or manual feature extraction for the fusion of multi-source heterogeneous data such as vibration, temperature, and power, and fail to effectively mine the dynamic coupling relationships between data, resulting in low information utilization rate.

[0004] Lack of physical interpretability: The diagnostic models used in traditional fault detection methods are generally black-box models based on deep learning (such as LSTM, CNN). Although LSTM and CNN can fit complex fault patterns, they lack explicit expression of fault mechanisms and are difficult to support causal reasoning for operation and maintenance decisions. Summary of the Invention

[0005] The purpose of the present invention is to provide a method for evaluating the health of wind turbines based on Koopman space transformation and Mahalanobis distance, which solves the problems of low efficiency of multi-source data fusion and lack of physical interpretability in the life prediction method of nonlinear wind turbine systems.

[0006] The technical solution adopted by the present invention is as follows: A method for evaluating the health of wind turbines based on Koopman space transformation and Mahalanobis distance, and the specific operation steps are as follows:

[0007] Step 1: Collect multi-source operation data of the wind turbine, including low-frequency signals based on the Supervisory Control and Data Acquisition (SCADA) system and high-frequency vibration signals based on the Condition Monitoring System (CMS).

[0008] Step 2: Perform missing value imputation, noise reduction processing, and time alignment on the multi-source operation data to generate a standardized data set.

[0009] Step 3: Linearly represent the standardized data set in Step 2 through Koopman space transformation; use the EDMD expansion module to decompose and approximately solve the K operator.

[0010] Step 4: Extract characteristic modes representing system dynamics in the Koopman space, and calculate the Mahalanobis distance between the fault state data and the standardized data set.

[0011] Step 5: Based on the correlation between the Mahalanobis distance and historical health data, establish a wind turbine health evaluation model to quantify the health status of the equipment.

[0012] The features of the present invention also lie in that

[0013] Step 2 is specifically as follows: perform wavelet threshold denoising on the high-frequency vibration signal; use linear interpolation method to interpolate the missing values of the low-frequency signal; perform time alignment on the high-frequency vibration data by using the moving window average method to make it consistent with the low-frequency signal frequency.

[0014] First, decompose the high-frequency vibration signal into low-frequency approximation coefficients and high-frequency detail coefficients :

[0015] Among them, the low-frequency approximation coefficients represent the operation trend of the device, and the high-frequency detail coefficients represent noise or fault pulses;

[0016] Then, perform soft threshold processing on :

[0017]

[0018] Among them, the threshold is the noise standard deviation ; Q is the signal length; is the detail coefficient of the highest layer in the wavelet transform decomposition;

[0019] Perform low-pass filtering and downsampling on to obtain the low-frequency approximation ;

[0020] Then, reconstruct the high-frequency vibration signal:

[0021] .

[0022] Use the linear interpolation method to interpolate the missing values of the low-frequency signal. The low-frequency signal includes power P ( t ), rotational speed R ( t ), and temperature T ( t );

[0023] Calculate the power missing value , within the time interval P ( t ), then:

[0024]

[0025] Among them, and respectively represent the power data collected at the and moments; the missing value processing methods for the rotational speed R ( t ) and the temperature T ( t ) are the same as those for the power P ( t );

[0026] The time alignment of the high-frequency vibration data is performed using the moving window average method as follows:

[0027] Define the sampling window width of the high-frequency vibration data , and take the average of all high-frequency vibration signal data within the window. The formula is as follows:

[0028]

[0029] where is the vibration signal of the target frequency, is the number of window data points, is the vibration signal at the moment, is the sampling interval,

[0030] Step 3 is specifically as follows:

[0031] Construct a second-order polynomial basis function;

[0032] where the second-order polynomial basis function includes the following terms: Use the system data processed in Step 2 as the state observation data, including the current state and the next moment state where is the time step; ∈Rn is the state vector, and Rn represents the spatial domain;

[0033]

[0034] Construct a second-order polynomial basis function vector, and the second-order polynomial basis function vector includes square terms and cross terms:

[0035]

[0036] where : temperature, P : power, : rotational speed, : vibration effective value, : bearing outer ring fault frequency energy, : Wind speed, : Represents frequency ~ Band energy, : Extracted by wavelet packet decomposition Band energy; Represents the decomposition level, Represents the decomposition node.

[0037] Let the total number of basis functions be , and the order 2 needs to be selected according to the system complexity, n is the system characteristic number, and this basis function is designed with a total of 8 characteristics;

[0038]

[0039] Map each state point x i to the basis function space to form a basis function matrix:

[0040]

[0041]

[0042] Solve the approximate Koopman operator matrix by the least squares method , m is the number of basis functions;

[0043]

[0044]

[0045] Among them represents the Moore - Penrose pseudoinverse, and the calculation method is:

[0046]

[0047] Then perform eigenvalue decomposition on the Koopman operator matrix K :

[0048]

[0049] Among them, Λ is the eigenvalue diagonal matrix, Φ is the eigenvector matrix, and the eigenvectors correspond to the spatial distribution of the modes;

[0050] Then use the Koopman operator to predict the system dynamics:

[0051]

[0052] Compare the prediction results with the real data, calculate the mean square error MSE or visualize the trajectory to verify the accuracy of the Koopman operator matrix.

[0053] Step 4 is specifically as follows:

[0054] Step 4.1: Perform eigenvalue decomposition on the Koopman operator matrix K :

[0055] KΦ = ΦΛ

[0056] where Λ is the diagonal matrix of eigenvalues, Λ = diag(λ1, λ2, …, λ m ); the amplitudes and phases of the eigenvalues λ1, λ2, …, λ m reflect the decay rate and frequency of the mode, Φ is the matrix of eigenvectors, , represents the mode;

[0057] Step 4.2: Screen the Koopman modes related to physical faults:

[0058] Step 4.2.1: Decay rate screening: Select the modes with the real part of the eigenvalue close to zero, Re(λ i ) ≥ , = 0.1; these modes correspond to the long-term dynamics or steady-state oscillations of the system;

[0059] Step 4.2.2: Energy ranking: Calculate the modal energy corresponding to the modes screened in Step 4.2.1 , and retain the top k modes with the highest energy; k takes values from 10 to 50;

[0060] Step 4.2.3: Combine the fault characteristic frequencies and further screen the modes corresponding to the fault frequency bands on the basis of Step 4.2.2; the fault characteristics include gear wear, broken teeth, tooth surface peeling, gear eccentricity, inner and outer ring faults of bearings, and shaft cracks;

[0061] Step 4.2 is specifically as follows:

[0062] ① Screen the Koopman eigenvalue λ i = a i + j b i ,

[0063] The real part K of the matrix a i : Determines the stability or decay rate of the mode. If a iis negative, and this mode will decay over time; if a i is positive, the mode will grow;

[0064] The matrix K learns the evolution relationship of the basis functions over time. For the FFT frequency band energy term , its evolution is modeled as a linear combination:

[0065]

[0066] The FFT frequency band energy term affects the imaginary part of the eigenvalues of the Koopman matrix K through the periodicity of its time evolution. Even if the energy term itself is real, its dynamic changes (such as fluctuations modulated by the rotational speed) will be K captured as the oscillation frequency of the complex eigenvalues. This method adds a rotational speed design basis function to ensure that the fault frequency is transmitted to the eigenvalues through the coupling mechanism.

[0067] Therefore, the imaginary part j of the matrix K : the oscillation frequency of the corresponding mode, that is, the vibration frequency of the system in this mode, j represents the imaginary part; the key modes are screened by calculating the eigenvalue frequencies; b i :

[0068] Calculate the modal frequency :

[0069]

[0070] ② Screen the key modes that match the fault characteristic frequencies: The screening condition is: , and a key mode matrix is formed after screening;

[0071] is the fault characteristic frequency threshold, is the occurrence frequency of each type of fault, obtained according to expert experience;

[0072] ③ After screening out the key modes, perform normal state modeling;

[0073] Obtain the state data of normal operating units from the preprocessed standardized dataset and substitute them into the basis functions to obtain

[0074]

[0075]

[0076] where, is the key mode matrix after screening; , is the number of selected modes, m is the total number of basis functions;

[0077] ④ Calculate the mean and covariance of:

[0078]

[0079]

[0080] wherein, represents the i th sample, is the number of samples of, N ≥ 100, indicates each sample in;

[0081] ⑤ According to the method in step ③, project the fault data onto the same modal space, , and finally calculate the Mahalanobis distance in the fault state.

[0082] The calculation of the Mahalanobis distance in step 4 is specifically as follows:

[0083]

[0084] : Pseudo-inverse of the regularized covariance matrix;

[0085] Step 5 is specifically:

[0086] Dynamically associate the Mahalanobis distance with the health index and output a health score of 0% - 100%; the health index is used to trigger hierarchical early warning; the hierarchical early warning includes three states: normal, early warning, and emergency; calculate the Mahalanobis distance between the fault data and the normal state, and set the distance threshold Dth;

[0087] Taking gear wear as an example, according to historical data statistics, the average Mahalanobis distance of the normal state of gear wear is D ≤ 20, the range of D is about 20 - 80 for slight wear, and D > 80 for severe wear; therefore, according to expert experience, Dth = 80 is set, that is, when D ≥ 80, it is considered that the wear has affected the operation;

[0088] Health index

[0089] where α is the time decay coefficient, and the initial value is 0.5%; β: distance decay coefficient, and the initial value is 2%; T over: Cumulative time when the Mahalanobis distance exceeds the distance threshold Dth, Tover = ∑(ti | Di ≥ Dth);D max: The maximum Mahalanobis distance during a fault;

[0090] Through the health index formula, a linear regression model is constructed, historical data of the SCADA and CMS systems are obtained, the time series Mahalanobis distance D, the over-threshold duration Tover, and the maximum Mahalanobis distance Dmax during a fault are calculated, the regression model is fed, and α and β are optimized, and Dth is adjusted.

[0091] When HI < 50%, it indicates an emergency state;

[0092] When 50% ≤ HI ≤ 70%, it indicates a warning state;

[0093] When HI > 70%, it indicates a normal state.

[0094] The beneficial effects of the present invention are as follows: The present invention introduces the Koopman operator into the multi-source data fusion analysis of wind turbines, breaking through the bottleneck of non-linear system modeling; realizing the accurate mapping of fault severity and life loss, being applicable to the health management of key components such as gearboxes and bearings, and extending to the field of rotating machinery such as pumps and compressors. Specific advantages also include: Engineering practicability: Quantifying the impact of faults through the Mahalanobis distance, directly supporting operation and maintenance decisions (such as spare part scheduling, maintenance priorities); Compatibility: It can be extended to the life prediction scenarios of other rotating machinery (pumps, compressors). Brief Description of the Drawings

[0095] Figure 1 is the flow chart of the wind turbine health evaluation method based on Koopman space transformation and Mahalanobis distance of the present invention;

[0096] Figure 2 is the schematic diagram of Koopman space transformation and feature extraction of the present invention. Detailed Description of the Embodiment

[0097] The present invention will be described in detail below in conjunction with the drawings and specific embodiments.

[0098] Embodiment 1

[0099] The present invention proposes a wind turbine health evaluation method based on Koopman space transformation and Mahalanobis distance, as Figure 1-2 shown, mainly including the following steps:

[0100] Step 1: Collect multi-source operation data of the wind turbine, including low-frequency signals based on the Supervisory Control and Data Acquisition (SCADA) system and high-frequency vibration signals based on the Content Management System (CMS);

[0101] Step 2: Perform noise reduction processing, missing value imputation, and time alignment on the multi-source operation data, and generate a standardized data set after standardization;

[0102] Step 3: Linearly represent the canonical data set in Step 2 through Koopman space transformation; use the EDMD expansion module to decompose and approximately solve the K operator;

[0103] Step 4: Extract the characteristic modes representing the system dynamics in the Koopman space, and calculate the Mahalanobis distance between the fault state data and the canonical data set;

[0104] Step 5: Establish a fan health evaluation model based on the correlation between the Mahalanobis distance and the historical health data to quantify the health state of the equipment.

[0105] Example 2

[0106] Based on Example 1, Step 2 is specifically as follows: Perform wavelet threshold denoising on the high-frequency vibration signal; use the linear interpolation method to interpolate the missing values of the low-frequency signal; align the high-frequency vibration data in time by the sliding window mean method to make it consistent with the low-frequency signal frequency.

[0107] First, decompose the high-frequency vibration signal into low-frequency approximation coefficients and high-frequency detail coefficients :

[0108] Among them, the low-frequency approximation coefficient represents the equipment operation trend, and the high-frequency detail coefficient represents noise or fault pulses;

[0109] Then, perform soft threshold processing on :

[0110]

[0111] Among them, the threshold is the noise standard deviation ; Q is the signal length; is the detail coefficient of the highest layer in the wavelet transform decomposition;

[0112] Perform low-pass filtering and downsampling on to obtain the low-frequency approximation ;

[0113] Then, reconstruct the high-frequency vibration signal:

[0114] .

[0115] Use the linear interpolation method to interpolate the missing values of the low-frequency signal. The low-frequency signal includes power P ( t ) and rotational speed R (t ) Temperature T ( t )

[0116] Calculate the power missing value within the time interval , ) If there is a power missing value within the time interval P ( t ) then:

[0117]

[0118] Wherein and respectively represent the power data at the and collection times; the missing value processing methods for the rotational speed R ( t ) and the temperature T ( t ) are the same as those for the power P ( t )

[0119] For high-frequency vibration data, use the sliding window mean method for time alignment, specifically as follows:

[0120] Define the sampling window width of the high-frequency vibration data , and take the mean of all high-frequency vibration signal data within the window. The formula is as follows:

[0121]

[0122] Wherein is the vibration signal at the target frequency, is the number of window data points, is the vibration signal at the sampling time, is the sampling interval,

[0123] Example 3

[0124] Based on Example 2, step 3 is specifically as follows:

[0125] Construct a second-order polynomial basis function;

[0126] Wherein, the second-order polynomial basis function includes the following terms: Use the system data processed in step 2 as the state observation data, including the current state and the next-state Wherein is the time step; ∈Rn is the state vector;

[0127]

[0128] Construct a second-order polynomial basis function vector (including square terms and cross terms)

[0129]

[0130] where : temperature P : power : rotational speed : effective vibration value : energy of the outer race fault frequency of the bearing : wind speed : representative frequency ~ band energy : extracted by wavelet packet decomposition band energy; represents the decomposition level represents the decomposition node;

[0131] The total number of basis functions is , , select the order to be 2 according to the system complexity n is the system characteristic number n = 8;

[0132]

[0133] Map each state point x i to the basis function space to form a basis function matrix:

[0134]

[0135]

[0136] Solve the approximate Koopman operator matrix by the least squares method , m is the number of basis functions;

[0137]

[0138]

[0139] where represents the Moore-Penrose pseudoinverse, and the calculation method is:

[0140]

[0141] Then perform eigenvalue decomposition on the Koopman operator matrix K :

[0142]

[0143] Among them, Λ is the eigenvalue diagonal matrix, Φ is the eigenvector matrix, and the eigenvectors correspond to the spatial distribution of the modes;

[0144] Then, the Koopman operator is used to predict the system dynamics:

[0145]

[0146] Compare the prediction results with the real data, calculate the mean square error MSE or visualize the trajectory to verify the accuracy of the Koopman operator matrix.

[0147] Example 4

[0148] Based on Example 3, Step 4 is specifically as follows:

[0149] Step 4.1: Perform eigenvalue decomposition on the Koopman operator matrix K :

[0150] KΦ = ΦΛ

[0151] Among them, Λ is the eigenvalue diagonal matrix, Λ = diag(λ1, λ2, …, λ m ); the amplitude and phase of the eigenvalues reflect the decay rate and frequency of the modes, and Φ is the eigenvector matrix, ;

[0152] Step 4.2: Screen the Koopman modes related to physical faults:

[0153] Step 4.2.1: Decay rate screening: Select the modes whose real part of the eigenvalue is close to zero Re(λ i ) ≥ , = 0.1; these modes correspond to the long-term dynamics or steady-state oscillations of the system;

[0154] Step 4.2.2: Energy ranking: Calculate the modal energy corresponding to the modes screened in Step 4.2.1, and retain the top k modes with the highest energy; k takes values from 10 to 50;

[0155] Step 4.2.3: Combine the fault characteristic frequencies and further screen the modes corresponding to the fault frequency bands on the basis of Step 4.2.2; the fault characteristics include gear wear, broken teeth, tooth surface peeling, gear eccentricity, inner and outer ring faults of bearings, and shaft cracks;

[0156] Among them, the gear meshing frequency f GM=

[0157] N m : Number of gear teeth, fr : Shaft rotational speed (Hz)

[0158] Bearing fault frequency (such as outer race fault frequency )

[0159] Nb : Number of rolling elements; d : Diameter of rolling element; D 0: Pitch diameter of bearing; : Contact angle

[0160] Step 4.2 is as follows:

[0161] ① Screen the Koopman eigenvalues λ of modal frequencies i = a i + j b i ,

[0162] Real part a i : Determines the stability or decay rate of the mode. If a i is negative, the mode decays over time; if a i is positive, the mode grows;

[0163] Imaginary part b i : b i The oscillation frequency corresponding to the mode, i.e., the vibration frequency of the system in this mode. j represents the imaginary part;

[0164] Calculate the modal frequency :

[0165]

[0166] ② Screen the key modes that match the fault characteristic frequencies: The screening condition is: , and a key mode matrix is formed after screening;

[0167] is the fault characteristic frequency threshold, is the occurrence frequency of each type of fault, obtained according to expert experience;

[0168] ③ After screening out the key modes, perform normal state modeling, that is, project the normal state data onto the key mode space:

[0169] First, the data of 100 groups of 8 features collected X _normal is extended to the basis function,

[0170]

[0171] wherein, is the key modal matrix after screening; , is the number of screened modes, m is the total number of basis functions;

[0172] ④ Calculate the mean and covariance of:

[0173]

[0174]

[0175] wherein, represents the i th sample, is the number of samples, represents each sample in.

[0176] ⑤ According to the method in step ③, project the fault data onto the same modal space, , and finally calculate the Mahalanobis distance of the fault state.

[0177] The projections in step ③ and step ⑤ can be understood as projecting the system into an infinite-dimensional space. Taking a three-dimensional space as an example, a certain fault mode is a corresponding line in the three-dimensional space, the direction of the line is fixed, and a certain point on the line is the whole line is , projected in a certain direction. This direction represents a certain fault type, the system is linear, and now calculate its Mahalanobis distance according to the projection of the linear system.

[0178] The calculation of the Mahalanobis distance in step 4 is specifically as follows:

[0179]

[0180] : The pseudo-inverse of the regularized covariance matrix;

[0181] Example 5

[0182] On the basis of Example 4, step 5 is specifically:

[0183] Dynamically associate the Mahalanobis distance with the health index and output a health score ranging from 0% to 100%; the health index is used to trigger hierarchical warnings; the hierarchical warnings include three states: normal, warning, and emergency; calculate the Mahalanobis distance between the fault data and the normal state, and set the distance threshold D th =80

[0184] Establish a health index model

[0185] where α is the time decay coefficient with an initial value of 0.5%; β: the distance decay coefficient with an initial value of 2%; T over: the cumulative time when the Mahalanobis distance exceeds the distance threshold Dth, Tover = ∑(t i | D i ≥Dth); D max: the maximum Mahalanobis distance during the fault;

[0186] Through the health index formula, construct a linear regression model, obtain the historical data of the SCADA and CMS systems, calculate the time series Mahalanobis distance D, the over-threshold duration Tover, and the maximum Mahalanobis distance Dmax during the fault, feed the regression model, and optimize α and β.

[0187] When HI < 50%, it indicates an emergency state;

[0188] When 50% ≤ HI ≤ 70%, it indicates a warning state;

[0189] When HI > 70%, it indicates a normal state.

[0190] Example 6

[0191] Step 1: Collect the 30-day operation data of a certain wind turbine, including:

[0192] SCADA low-frequency signal (1Hz): power P(t), rotational speed R(t), gearbox temperature T(t).

[0193] CMS high-frequency vibration signal (10kHz): axial vibration acceleration V(t), extract the bearing outer ring fault frequency f through FFT bpfo =85Hz, gear meshing frequency f mesh =320Hz.

[0194] Step 2: Data preprocessing

[0195] Time alignment:

[0196] For the CMS data (10kHz → 1Hz), use sliding window mean aggregation, Δt = 0.5s, and the number of window data points Nwindow = 10,000.

[0197] The missing values in SCADA data (such as the missing power value at a certain moment t = 12:05) are completed by linear interpolation.

[0198] Wavelet threshold denoising:

[0199] Decompose the vibration signal x(t) to 5 layers, estimate the noise standard deviation σ = 0.12, and the threshold λ = 0.25.

[0200] The signal-to-noise ratio after denoising is increased to SNR = 28.6 dB.

[0201] Step 3: Koopman space transformation

[0202] Construct a second-order polynomial basis function, including , such as the original variables and cross terms, and the total dimension m = 45.

[0203] Solve the Koopman operator matrix K ∈R45×45, and verify that the prediction error MSE = 0.023.

[0204] Step 4: Key mode extraction and Mahalanobis distance calculation

[0205] Eigenvalue screening:

[0206] Decompose K to obtain the eigenvalues λi, and the screening condition: Re(λi) ≥ -0.1, retain the first k = 20 high-energy modes. Match the gear wear characteristic frequency = 320 Hz, and screen out the mode = 318 Hz ( = 2 Hz).

[0207] Normal state modeling:

[0208] Select 100 groups of normal data and project them into the key mode space, and calculate the mean μ = [1.2, -0.4,..., 0.7] T and the covariance matrix Σ.

[0209] Fault data projection:

[0210] Project the fault data X fault after mapping by the basis function to obtain = [2.8, -1.1,..., 3.5] T .

[0211] Calculate the Mahalanobis distance: .

[0212] Step 5: Health assessment and hierarchical warning

[0213] Set the threshold Dth = 80, the cumulative time over the threshold T over = 6h, the maximum distance D max = 92.3;

[0214] Calculation of the health index:

[0215]

[0216] Output result: The current HI = 72.4% > 70%, triggering the warning state, indicating slight wear of the gear.

Claims

1. A fan health assessment method based on Koopman space transformation and Mahalanobis distance, characterized in that The specific steps are as follows: Step 1: Collect multi-source operation data of the fan, including low-frequency signals based on the Supervisory Control and Data Acquisition (SCADA) system and high-frequency vibration signals based on the Content Management System (CMS). Step 2: Perform noise reduction, missing value imputation, and time alignment on the multi-source operation data to generate a standardized dataset, specifically as follows: Perform wavelet threshold denoising on the high-frequency vibration signals; use linear interpolation to impute missing values in the low-frequency signals; perform time alignment on the high-frequency vibration data using the sliding window mean method. In Step 2, for the high-frequency noise and fault impacts in the high-frequency vibration signals, wavelet threshold denoising is adopted, specifically as follows: First, decompose the high-frequency vibration signal into low-frequency approximation coefficients and high-frequency detail coefficients : Among them, the low-frequency approximation coefficient represents the operation trend of the device, and the high-frequency detail coefficient represents noise or fault pulses; Then perform soft threshold processing on : Among them, the threshold noise standard deviation ; Q is the signal length; is the detail coefficient of the highest layer in the wavelet transform decomposition; Pair Low-pass filtering and downsampling are used to obtain a low-frequency approximation ; Then reconstruct the high-frequency vibration signals: ； Step 3: Linearly represent the standardized dataset in Step 2 through Koopman space transformation; use the Extended Dynamic Mode Decomposition (EDMD) module to decompose and approximately solve the Koopman operator K. Step 4: Extract the characteristic modes representing the system dynamics in the Koopman space and calculate the Mahalanobis distance between the fault state data and the standardized dataset. Step 5: Based on the correlation between the Mahalanobis distance and the historical healthy data, establish a fan health evaluation model to quantify the health state of the equipment.

2. The fan health evaluation method based on Koopman space transformation and Mahalanobis distance according to claim 1, characterized in that ①Interpolate the missing values of the low-frequency signal using linear interpolation method, where the low-frequency signal includes power P ( t ), rotational speed R ( t ), temperature T ( t ); Calculated time interval , of the power loss value P ( t ), then: Among them, and respectively represent the power data at the and acquisition moments; the missing value processing methods for the rotational speed R ( t ) and the temperature T ( t ) are the same as those for the power P ( t ); ② Perform time alignment on the high-frequency vibration data using the sliding window mean method, specifically as follows: Define the sampling window width of high-frequency vibration data , and take the average value of all high-frequency vibration signal data within the window. The formula is as follows: Among them, is the vibration signal at the target frequency, is the number of window data points, is the vibration signal at time is the sampling interval, is the sampling time; if the sampling frequency is 1 Hz, set Δt = 0.5 s.

3. The fan health assessment method based on Koopman space transformation and Mahalanobis distance according to claim 2, wherein Step 3 is specifically as follows: Take the system data processed in step 2 as state observation data, where the state observation data includes the current state and the state at the next moment where is the time step; ∈Rn is the state vector; Construct a second-order polynomial basis function: Construct a second-order polynomial basis function vector: Among them, : Temperature, P : Power, : Rotational speed, : Effective vibration value, : Energy of the fault frequency of the outer bearing ring, : Wind speed, : Representative frequency ~ Band energy, : Extracted by wavelet packet decomposition Band energy, Indicates the decomposition level, Indicates the decomposition node; Let the total number of basis functions be , select the order to be 2 according to the system complexity, n is the system characteristic number; Map each state point x i to the basis function space to form a basis function matrix: Solving the approximate Koopman operator matrix by the least squares method : wherein denotes the Moore-Penrose pseudoinverse, and is calculated as follows: Then perform eigenvalue decomposition on the Koopman operator matrix K : where Λ is the diagonal matrix of eigenvalues and Φ is the matrix of eigenvectors; Then use the Koopman operator to predict the system dynamics: Compare the prediction results with the real data to verify the accuracy of the Koopman operator matrix.

4. The fan health assessment method based on Koopman space transformation and Mahalanobis distance according to claim 3, wherein Step 4 is specifically as follows: Step 4.1: Perform eigenvalue decomposition on the Koopman operator matrix K : K Φ = ΦΛ where, Λ is the eigenvalue diagonal matrix, Λ = diag(λ1, λ2, …, λ m ); the amplitudes and phases of the eigenvalues λ1, λ2, …, λ m reflect the decay rate and frequency of the mode, and the eigenvector matrix ; represents the mode; Step 4.2: Screen the Koopman modes related to physical faults: Step 4.2.1: Attenuation rate screening: Select the modes with the real part of the eigenvalue Re(λ i ) ≥ , where = 0.1; Step 4.2.2: Energy sorting: Calculate the modal energy corresponding to the modes screened in Step 4.2.1 , and retain the top k modes with the highest energy; k The value ranges from 10 to 50; Step 4.2.3: Combine the fault characteristic frequencies and further screen the modes corresponding to the fault frequency bands on the basis of Step 4.2.2; the fault characteristics include gear wear, broken teeth, tooth surface peeling, gear eccentricity, inner and outer ring faults of bearings, and shaft cracks.

5. The fan health assessment method based on Koopman space transformation and Mahalanobis distance according to claim 4, wherein Step 4.2 is specifically as follows: ① Screen the Koopman eigenvalue λ of the modal frequency i = a i +j b i , Matrix K Real part of a i : Determine the stability or decay rate of the mode. If a i is negative, the mode will decay over time; If a i is positive, the mode will grow; Matrix K The imaginary part j of b i : b i The oscillation frequency of the corresponding mode, that is, the vibration frequency of the system in this mode, j represents the imaginary part; the key modes are screened by calculating the eigenvalue frequencies Calculating modal frequencies : ② Screen the key modes that match the fault characteristic frequencies: The screening conditions are: , and after screening, a key mode matrix is formed; is the fault feature frequency threshold, is the occurrence frequency of each type of fault; ③ After screening out the key modes, perform normal state modeling: Obtain from the preprocessed specification dataset N the status data of one normally operating unit and at the same time step into the basis function to obtain ; Among them, is the key modal matrix after screening; , is the number of modes selected, m is the total number of basis functions; ④Calculate for mean and covariance: Among them, represents the i th sample, is the number of samples, indicating each sample in ⑤Project the fault data obtained in step ③ into the same modal space according to the following steps and , and finally calculate the Mahalanobis distance of the fault state.

6. The fan health assessment method based on Koopman space transformation and Mahalanobis distance according to claim 5, wherein The calculation of the Mahalanobis distance in Step 4 is specifically as follows: : Pseudo-inverse of the regularized covariance matrix.

7. The fan health assessment method based on Koopman space transformation and Mahalanobis distance according to claim 6, characterized in that, Step 5 is specifically: Dynamically associate the Mahalanobis distance with the health index formula and output a health score of 0% - 100%; the health index is used to trigger hierarchical warnings and prioritize maintenance; the hierarchical warnings include three states: normal, warning, and emergency. Calculate the Mahalanobis distance between the fault data and the normal state, and set the distance threshold Dth; Health Index HI = 100% - α T over - β D max where α is the time decay coefficient with an initial value of 0.5%; β is the distance decay coefficient with an initial value of 2%; T over is the cumulative time of the Mahalanobis distance exceeding Dth; D max is the maximum Mahalanobis distance during the fault; Through the health index formula, construct a linear regression model, obtain the historical data of the SCADA and CMS systems, calculate the time series Mahalanobis distance D, the duration T over the threshold, and the maximum Mahalanobis distance Dmax during the fault period, input them into the linear regression model, and optimize α and β to adjust Dth; When HI < 50%, it indicates an emergency state; When 50% ≤ HI ≤ 70%, it indicates a warning state; When HI > 70%, it indicates a normal state.

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