Multi-element segmented degradation modeling and life prediction method for permanent magnet of permanent magnet synchronous motor

Abstract

The invention relates to the technical field of life prediction of a permanent magnet in a permanent magnet synchronous motor, and discloses a multi-element segmented degradation modeling and life prediction method for the permanent magnet of the permanent magnet synchronous motor, which comprises the following steps of: firstly, selecting a back electromotive force fundamental frequency amplitude drop rate and a characteristic harmonic content rise rate as degradation characteristic quantities; respectively representing overall and local demagnetization behaviors of the permanent magnet; the method comprises the following steps: acquiring data through an accelerated degradation test, and establishing a segmented Wiener process model to describe stage characteristics of degradation; utilizing a t-Copula function to construct a joint distribution model of the multivariate degradation characteristic quantity; extrapolating model parameters under normal stress by combining an Arrhenius model and a segmentation point-temperature model; and life prediction is carried out by adopting Monte Carlo simulation, and the model precision is verified through an area ratio index and a cross validation method. According to the method, the degradation process of the permanent magnet can be comprehensively and accurately represented, high-precision life prediction is realized, and the method is suitable for reliability evaluation and preventive maintenance of a motor system.

Description

Technical Field

[0001] This invention relates to the field of life prediction technology for permanent magnets in permanent magnet synchronous motors, and in particular to a multi-segment degradation modeling and life prediction method for permanent magnets in permanent magnet synchronous motors. Background Technology

[0002] Permanent magnet synchronous motors (PMSMs) are widely used in electric vehicles, rail transportation, aerospace, industrial drive systems, and home appliances due to their high power density, high efficiency, and low losses. As the core component of the motor, permanent magnets are subjected to a combination of stresses from high temperatures, current, vibration, and magnetic field fluctuations during long-term operation, leading to gradual degradation of their magnetic properties. This degradation results in problems such as decreased motor output torque, reduced efficiency, and increased vibration and noise, and in severe cases, even motor failure. For example, in electric vehicles, permanent magnet degradation directly affects the output performance and driving range of the drive motor; in the aerospace field, motor reliability is directly related to flight safety. Therefore, accurate monitoring of the degradation status and lifespan prediction of permanent magnets is of great significance for improving motor system reliability, developing preventative maintenance strategies, and reducing operating costs.

[0003] To better monitor and predict permanent magnets, those skilled in the art have proposed the following technical solutions:

[0004] 1. A method for predicting the magnetic properties of permanent magnets based on magnetoviscous experiments and recovery curves (other publication number: CN114925535B). This method has the advantage of utilizing the intrinsic properties of materials for modeling, resulting in good universality. However, it primarily addresses magnetic changes at the material level, neglecting the multi-physics coupling effects (such as electromagnetic, thermal, and vibration) experienced by the permanent magnet during actual motor operation, and also failing to handle local demagnetization and non-uniform degradation problems. Furthermore, this method does not introduce multi-dimensional degradation characteristics, making it difficult to comprehensively characterize the complex degradation behavior of motor permanent magnets under actual operating conditions.

[0005] 2. A method for modeling the uniform degradation of permanent magnets in a polarized magnetic system relay, disclosed in CN115630519B, predicts lifetime by constructing a demagnetization rate model for permanent magnet materials and a Kriging substitution model. The limitation of this method is that it targets permanent magnets in relays, whose operating environment and degradation mechanisms differ significantly from those in motors. Motor permanent magnets face higher temperature gradients and electromagnetic interference during operation, and their degradation process often exhibits segmented characteristics. This patent does not address segmented modeling or consider the correlation between multiple degradation characteristics, resulting in limited applicability for lifetime prediction of motor permanent magnets.

[0006] 3. A method for evaluating the lifespan of permanent magnets based on accelerated life testing, with publication number CN117434484A, evaluates the lifespan by selecting functional performance indicators and establishing a life prediction model. While this method has strong engineering applicability, its models are mostly driven by a single stress or a single characteristic quantity, failing to consider the segmented characteristics of the degradation process and the coupling relationship between multiple characteristic quantities. In the application of permanent magnets in motors, degradation often exhibits a segmented trend of "rapid initial degradation followed by slow degradation," with both overall and local demagnetization coexisting. This patented method struggles to accurately describe such complex degradation behavior, limiting its prediction accuracy.

[0007] Therefore, those skilled in the art urgently need a modeling and life prediction method that can comprehensively and accurately describe the degradation process of permanent magnets in motors and has both physical mechanisms and data-driven characteristics. Summary of the Invention

[0008] The purpose of this invention is to solve the following problems:

[0009] 1. Failed to comprehensively consider the multi-dimensional degradation characteristics of permanent magnets during motor operation (such as overall demagnetization and local demagnetization);

[0010] 2. Lack of modeling ability for segmented characteristics during degradation;

[0011] 3. The correlation between multiple degenerate features was not effectively handled;

[0012] 4. The model relies on historical data or a single physical parameter, and the extrapolation to constant stress conditions results in a large error.

[0013] Furthermore, a multi-segment degradation modeling and life prediction method for permanent magnet synchronous motor permanent magnets was designed.

[0014] The technical solution of the present invention to achieve the above objectives is a method for multi-segment degradation modeling and life prediction of permanent magnets in a permanent magnet synchronous motor, which includes the following steps:

[0015] Step one involves determining the degradation characteristic quantities. The rate of decrease in the fundamental frequency amplitude of the back electromotive force and the rate of increase in the characteristic harmonic content are selected as the degradation characteristic quantities of the permanent magnet. The purpose is to determine observational indicators that can comprehensively and accurately characterize the degradation state of the permanent magnet. This step is the foundation and starting point of the entire method, solving the problem of "what to use to measure degradation." Selecting two characteristic quantities that respectively reflect "overall demagnetization" and "local demagnetization" lays the cornerstone for subsequent multivariate modeling. If only one characteristic quantity is used, the subsequent joint modeling (step four) becomes meaningless.

[0016] Step two involves acquiring degradation data. Accelerated degradation experiments are designed and implemented to obtain degradation data of the permanent magnet under different temperature stresses. The purpose is to acquire degradation data containing different stress levels, necessary for building and calibrating the model, within a reasonable timeframe. This step serves as the "data fuel" and experimental foundation for the model; the feature quantities selected in Step one need to be experimentally transformed into specific, time-varying data sequences in this step. Without this step, all subsequent modeling efforts would be futile. Furthermore, the different temperature stresses set in this step provide the basis for parameter extrapolation in Step five.

[0017] Step three involves single-feature degradation modeling. Two degenerate features are modeled separately, resulting in two corresponding degradation feature models. The aim is to model each degradation feature individually, accurately describing its phased degradation patterns and randomness over time. This step precisely characterizes a single degradation path. It first "refines" the degradation behavior of each feature, identifying and modeling its segmented characteristics. The output of this step (the segmented model parameters for each feature) serves as the input and foundation for the joint modeling in step four. If the models of individual features are inaccurate, the accuracy of the joint model is meaningless.

[0018] Step four involves constructing a multivariate segmented degradation model, analyzing the correlation and dependency between the two degradation feature models, and using the t-Copula function to construct a joint distribution model of the multivariate degradation features. The aim is to characterize the correlation and dependency between the two degradation features, merging two independent univariate models into a unified multivariate degradation model. This step is the core of achieving "multivariate" modeling. It solves the problem of the two models operating independently in Step three by probabilistically linking them through the Copula function, thus more realistically reflecting the physical reality of simultaneous and mutually influential overall and local degradation of the permanent magnet. This step, together with Step three, constitutes the core modeling system of this method.

[0019] Step five involves deriving the model parameters. Based on the Arrhenius model and the piecewise point-temperature model, the parameters of the degradation model under constant stress are extrapolated (i.e., derived). The purpose is to transform the model established under accelerated stress into a model under actual use (constant stress) conditions. This step serves as a bridge connecting accelerated experiments with practical applications. Steps two through four are all performed under accelerated conditions, but the ultimate goal is to predict the lifespan under actual operating conditions. This step utilizes the accelerated model (such as the Arrhenius model) and the established piecewise point-temperature relationship to "translate" the model parameters to the constant stress level, giving the model practical predictive value.

[0020] Step six, lifetime prediction, is based on the established multivariate segmented degradation model. Monte Carlo simulation is used to generate degradation trajectories, and lifetime prediction is performed in conjunction with failure thresholds. The aim is to predict the product's lifetime distribution through extensive stochastic simulations based on the established degradation model extrapolated to constant stress. This step is the specific application and output of the model. It utilizes the complete model built in the first five steps to simulate thousands of possible degradation processes through computer simulation, thereby statistically determining key indicators such as the lifetime probability density function and reliability function, directly providing the answer to "what is the lifetime?" for engineering design.

[0021] Step seven, model validation, verifies the accuracy of the lifetime prediction results using the area ratio index and cross-validation. Its purpose is to evaluate the accuracy and reliability of the model and prediction results built using this method. This step is a quality assurance step to ensure the effectiveness and credibility of the method. It retrospectively tests the prediction results from step six using statistical methods such as cross-validation and the area ratio index. If the validation passes, it proves that the entire technical chain from step one to step six is ​​scientific and reliable; if it fails, it may be necessary to retrospectively adjust the preceding model or parameters.

[0022] It should be noted that steps one and two define the problem and provide the raw materials, while steps three and four are the core technologies, which process the raw materials and build a high-fidelity mathematical model. Step five calibrates the model to make it suitable for the target environment. Step six uses the calibrated model to produce the final product - the life prediction result. Step seven is to conduct quality inspection on the final product.

[0023] The formula for calculating the rate of decrease in the fundamental frequency amplitude of the back electromotive force is as follows:

[0024]

[0025] In the formula, E1 is the fundamental frequency amplitude of the back electromotive force under normal conditions, E 1，t It represents the fundamental frequency amplitude of the back electromotive force at time t, degenerated to that value.

[0026] The formula for calculating the rate of increase of the characteristic harmonic content is as follows:

[0027]

[0028] In the formula, H0 is the initial characteristic harmonic content, H t This represents the characteristic harmonic content at time t, which degenerates to that at time t.

[0029] The mathematical expression for the segmented Wiener process model is:

[0030]

[0031] In the formula, τ is the segmentation point, μ1 and μ2 are both drift coefficients, σ1 and σ2 are both diffusion coefficients, and B(t) is the standard Brownian motion.

[0032] The mathematical expression for the t-Copula function is:

[0033]

[0034] In the formula, u1 and u2 are both marginal distribution functions, ρ is the correlation coefficient, and v is the degree of freedom.

[0035] The mathematical expression for the segmented point-temperature model is:

[0036] τ(T)=a·exp(b·T)

[0037] In the formula, a and b are model parameters, obtained by least squares fitting.

[0038] The mathematical expression for the degradation rate of the joint distribution model under constant stress, as used by the Arrhenius model, is as follows:

[0039]

[0040] In the formula, y(T) is the degradation rate, A is a constant, and E is a constant. a The activation energy is k, and the Boltzmann constant is k.

[0041] The failure threshold is set as follows:

[0042] The failure threshold L1 = 22% for the rate of decrease in the back EMF fundamental frequency amplitude;

[0043] The failure threshold L2 for the rate of increase of characteristic harmonic content is 233%.

[0044] The formula for calculating the area ratio consistency index is as follows:

[0045]

[0046] In the formula, s(R) a (t)) and s(R) b (t) represents the area enclosed by the two reliability curves and the coordinate axes, respectively.

[0047] The cross-validation method described above verifies the consistency of the prediction results using a three-fold cross-validation approach.

[0048] Compared with the prior art, the present invention has the following beneficial effects:

[0049] 1. This invention can comprehensively characterize degradation behavior by capturing overall and local demagnetization phenomena through multiple degradation feature quantities, thereby achieving a comprehensive description of the degradation process of permanent magnets.

[0050] 2. This invention employs a high-precision segmented modeling method, which accurately depicts the stage changes in the degradation process through a segmented Wiener process, thereby improving the model's ability to fit complex degradation behaviors.

[0051] 3. This invention has strong correlation processing capabilities, effectively capturing the nonlinear dependencies between multiple features by utilizing the t-Copula function, thereby improving the accuracy of joint modeling;

[0052] 4. This invention has good extrapolation performance. It achieves accurate extrapolation of model parameters from accelerated stress to constant stress conditions through a segmented point-temperature model and an Arrhenius acceleration model, making it suitable for practical engineering applications.

[0053] 5. This invention designs a reliable verification mechanism, introducing an area ratio consistency index and a cross-validation method to ensure the reliability and stability of the model prediction results;

[0054] 6. This invention has strong engineering applicability. It is based on actual motor operating data, does not require complex material parameters, and is easy to promote and use in industrial sites. Attached Figure Description

[0055] Figure 1 This is a flowchart of a method for multi-segment degradation modeling and life prediction of permanent magnets in a permanent magnet synchronous motor, as described in this invention.

[0056] Figure 2 This is a graph showing the trend of demagnetization rate of the permanent magnet described in this invention;

[0057] Figure 3 This is a graph of the γ index at different temperatures as described in this invention;

[0058] Figure 4 These are physical and structural diagrams of the accelerated degradation test platform described in this invention;

[0059] Figure 5 This is a graph showing the variation of degradation characteristic quantities at different temperatures as described in this invention;

[0060] Figure 6 This is the back electromotive force and demagnetization rate fitting curve of the present invention;

[0061] Figure 7 This is a degradation trajectory diagram of the degradation characteristic quantity under accelerated stress described in this invention;

[0062] Figure 8 This is a graph showing the lifetime probability density and reliability of the permanent magnet under accelerated stress as described in this invention.

[0063] Figure 9 This is a graph showing the lifetime probability density and reliability curves of the permanent magnet under constant stress as described in this invention.

[0064] Figure 10 This is a diagram showing the distribution of sample points under different stresses as described in this invention;

[0065] Figure 11 This is an analysis diagram of the harmonic components of the back electromotive force under different states as described in this invention;

[0066] Figure 12 This is a table showing the parameter estimation results of the Wiener process model described in this invention;

[0067] Figure 13 This is a table showing the consistency test results of the failure mechanism described in this invention;

[0068] Figure 14 This is a table of model parameter estimation results for the degradation feature quantity one described in this invention;

[0069] Figure 15 This is a table showing the model parameter estimation results for the second degradation feature described in this invention;

[0070] Figure 16 This is a table of estimated values ​​for the model parameters described in this invention;

[0071] Figure 17 This is a table showing the model parameter estimation results for the degradation characteristic quantity 1 under constant stress as described in this invention;

[0072] Figure 18 This is a table showing the model parameter estimation results for the second degradation characteristic quantity under constant stress as described in this invention. Detailed Implementation

[0073] The present invention will now be described in detail with reference to the accompanying drawings;

[0074] In the technical solution of this application, the experimental design process for accelerated degradation of the permanent magnet of the motor is as follows:

[0075] First, select the degradation characteristic quantities;

[0076] Permanent magnet synchronous motors are susceptible to environmental factors such as high temperatures during operation, which can lead to a decrease in the overall magnetic properties of the permanent magnets. In addition, uneven temperature distribution inside the motor may also cause local demagnetization.

[0077] Uniform demagnetization characteristics can be characterized by introducing a demagnetization coefficient. The demagnetization coefficient is defined as follows:

[0078]

[0079] Where, k d k is the demagnetization coefficient, representing the degree of demagnetization. d The smaller the value, the more severe the demagnetization; the range is 0. <k d ≤1; For permanent magnet flux linkage under uniform demagnetization fault; ψ f This refers to the permanent magnet flux linkage under normal conditions.

[0080] The expressions for the motor parameters under uniform demagnetization fault conditions are as follows:

[0081]

[0082] Among them, i d i q ψ represents the current along the d-axis and q-axis, respectively; ω is the electric angular velocity of the motor; ψ d ψ q T represents the stator flux linkages along the d-axis and q-axis, respectively; e It is electromagnetic torque; p n It is the number of pole pairs of the motor; L d L q These represent the inductance along the d-axis and q-axis, respectively.

[0083] As can be seen from equation (2), uniform demagnetization directly reduces the amplitude of the back electromotive force by decreasing the amplitude of the flux linkage, thereby affecting the voltage, current, and electromagnetic torque of the motor. Since the spatial distribution of the flux linkage remains unchanged, uniform demagnetization has little impact on the harmonic components of the back electromotive force.

[0084] Local demagnetization differs from uniform demagnetization. Local demagnetization causes distortion in the magnetic flux distribution of permanent magnets, thereby introducing higher harmonic components into the magnetic field. The characteristic harmonics caused by local demagnetization can be expressed as:

[0085]

[0086] Where f1 is the fundamental frequency component; p n is the number of pole pairs of the motor; k is a positive integer.

[0087] As can be seen from equation (3), the harmonic components caused by local demagnetization include multiple frequency components.

[0088] Based on the above theoretical foundation, this application introduces two degradation characteristic quantities. For example... Figure 11 As shown, the fundamental frequency amplitude of the back electromotive force (EMF) is strongly correlated with the magnetic strength of the permanent magnet, and its decrease can characterize the overall decline in the magnetic properties of the permanent magnet. Therefore, this application defines the rate of decrease in the fundamental frequency amplitude of the back EMF as a degradation characteristic quantity, and its expression is as follows:

[0089]

[0090] Where E1 is the fundamental frequency amplitude of the back electromotive force under normal operating conditions of the permanent magnet synchronous motor; E 1,t The fundamental frequency amplitude of the back electromotive force at time t when the permanent magnet degenerates;

[0091] When the magnetic properties of a permanent magnet are not uniformly distributed, the back electromotive force (EMF) will generate a series of characteristic harmonics as shown in equation (3). The content of characteristic harmonics in the back EMF reflects the degree of non-uniformity of the permanent magnet's magnetic properties. The expression for the content of characteristic harmonics in the back EMF is:

[0092]

[0093] To reflect the changing trend of the non-uniformity of the magnetic properties of permanent magnets with degradation time, this application defines the rate of increase of the characteristic harmonic content of the back electromotive force as the degradation characteristic quantity two, and its expression is:

[0094]

[0095] Secondly, select the accelerating stress;

[0096] During long-term operation of a PMSM, the internal permanent magnets will inevitably be subjected to the combined effects of various external stresses, including temperature, current, mechanical vibration, external magnetic fields, and chemical corrosion. These factors can all cause degradation of the magnetic properties of the permanent magnets. Among these, temperature is widely considered to be the main stress that triggers demagnetization of permanent magnets. Therefore, this application selects temperature as the accelerating stress for the accelerated degradation test of permanent magnets.

[0097] To ensure the reasonableness of the model parameters extrapolated from the accelerated degradation test results to normal stress conditions, the failure mechanism should remain consistent across all accelerated stress levels. Considering the high time and cost investment required for whole-machine degradation testing, this section selects NdFeB for magnetic property degradation testing. By modeling its degradation characteristic quantities and verifying the consistency of the failure mechanism, the effective applicable range of accelerated stress is determined.

[0098] According to the characteristics of permanent magnets, the degradation rate of their surface magnetic flux density (hereinafter referred to as demagnetization rate) is positively correlated with the degradation rate of remanence. Therefore, by measuring the trend of the demagnetization rate of a permanent magnet, the degree of demagnetization of the permanent magnet can be directly represented. The calculation formula is as follows:

[0099]

[0100] Among them, h irr B is the demagnetization rate. T0 B is the magnetic flux density value at temperature T0 before the experiment; T The magnetic flux density value is the value that recovers to the temperature T0 after an accelerated degradation test at temperature T.

[0101] In the magnetic property degradation test, the highest operating temperature of NdFeB, 180℃, was selected as the highest temperature stress T1, and the other temperature stresses T2, T3, and T4 were set to 170℃, 160℃, and 150℃, respectively.

[0102] The magnetic flux density of the NdFeB surface was measured using a gaussmeter and substituted into equation (7) to obtain the trend of demagnetization rate of the permanent magnet at four temperatures, as shown below. Figure 2 As shown in the figure, curves in groups A, B, C, and D correspond to the trends in demagnetization rate of permanent magnets at 180℃, 170℃, 160℃, and 150℃, respectively. Numbers 1-3 represent the measurement results of three different samples at each temperature. After the experiment, the average demagnetization rates of the permanent magnets at 180℃, 170℃, 160℃, and 150℃ were 25.6%, 17.4%, 11.7%, and 8.5%, respectively.

[0103] Depend on Figure 2 It is evident that under different temperature stresses, the demagnetization rate of permanent magnets increases significantly with increasing temperature. Furthermore, within 15 cycles, the demagnetization rate of the permanent magnets gradually increases with degradation time, exhibiting a faster demagnetization rate initially, which then slows down and approaches stability after reaching a certain point. Therefore, this characteristic of change should be considered when establishing a permanent magnet degradation model, and segmented modeling should be employed.

[0104] Segmentation point identification is a crucial step in segmented modeling. Its main purpose is to determine the segmentation locations during the NdFeB degradation process, enabling accurate modeling of degradation behavior at different stages. The experimental results show that a relatively obvious demagnetization velocity inflection point exists in the NdFeB variation trend under different temperature stresses. Based on this characteristic, the quantitative indices for the two-stage segmentation point and degradation velocity change are defined as τ and γ. By calculating the time t corresponding to the maximum value of γ, the segmentation point τ can be determined. The calculation steps are as follows:

[0105] Step 1: Calculate the demagnetization rate v(t). The demagnetization data x(t) is a degradation characteristic quantity that changes with time t. The demagnetization rate v(t) is the derivative of x(t) with respect to t. The calculation method is as follows:

[0106]

[0107] Step 2: Calculate the γ index. To quantify the change in demagnetization rate and eliminate the influence of absolute values, the γ index is defined as follows:

[0108]

[0109] Step 3: Find the maximum value of the γ indicator. The breakpoint τ corresponds to the maximum value of the γ indicator; finding the position of the maximum value of the γ indicator is the breakpoint.

[0110] τ=argmax[γ(t k (10)

[0111] Following the steps above, Figure 2 The demagnetization data of NdFeB were segmented and identified, and the results are shown in the figure below.

[0112] from Figure 3 As can be seen, the degradation rate is faster and the segmentation point appears earlier under high temperature conditions, while the degradation rate is slower and the segmentation point appears significantly later under low temperature conditions. Based on the analysis results of the above experimental data, the NdFeB degradation process exhibits obvious stage characteristics [15,16]. Through segmentation point identification, it is confirmed that the degradation process of permanent magnets can be divided into two stages. Therefore, a two-stage Wiener process model is used to model it:

[0113]

[0114] Where τ is the time of occurrence of the segment point; μ1 and σ1 represent the drift coefficient and diffusion coefficient of the first stage of degradation process, respectively; X(τ) is the initial value of the second stage of degradation, i.e. the degradation amount at the segment point; μ2 and σ2 represent the drift coefficient and diffusion coefficient of the second stage of degradation process, respectively; B(t) is the standard Brownian motion.

[0115] The parameters of the Wiener process model at various temperature levels were estimated using the maximum likelihood estimation method, and the results are as follows: Figure 12 As shown:

[0116]

[0117] according to Figure 12 The parameter estimates show a significant difference between the λ estimate at 180℃ and the values ​​under the other three temperature stresses. Therefore, it can be determined that the failure mechanism has changed at 180℃. Based on this, the consistency of the failure mechanism needs to be tested using the t-statistic for the test objects under the other three stresses. Considering that the NdFeB degradation model is divided into two stages, the consistency test of the failure mechanism of NdFeB using the above method is also performed in stages, and the results are as follows. Figure 13 As shown in the table, U1 and U2 are the test statistics for μ / σ² and λ, respectively, and the critical values ​​of the rejection regions are calculated from the degrees of freedom and the significance level α.

[0118] Through consistency testing of failure mechanisms, it was found that the failure mechanisms were consistent at 170℃, 160℃, and 150℃. This result provides a scientific basis for the design and implementation of subsequent degradation tests of the permanent magnets in the entire machine.

[0119] Finally, the accelerated degradation test was determined;

[0120] The accelerated stress level used for extrapolating model parameters and lifespan was determined above. Based on this, an accelerated degradation test platform for motor permanent magnets was designed to simulate the motor's operating environment and obtain real data on the permanent magnet degradation process. The structural diagram and schematic diagram of the test platform are shown below. Figure 4 As shown:

[0121] Accelerated degradation tests were conducted. After the tests, the degradation patterns and characteristic change trends of the permanent magnets were analyzed based on the measured back electromotive force waveforms and characteristic quantity data. The obtained test data are as follows: Figure 5 As shown:

[0122] from Figure 5 It can be seen that under temperature conditions of 150℃, 160℃ and 170℃, the rate of decrease of the fundamental frequency amplitude of the back EMF and the rate of increase of the characteristic harmonic content of the back EMF both show a significant increasing trend with the increase of degradation time, but exhibit different characteristics and patterns.

[0123] Characteristic quantity one (the rate of decrease in the fundamental frequency amplitude of the back electromotive force) serves as a key indicator of the overall magnetic performance degradation of permanent magnets, and its changing trend reflects the global decay characteristics of the magnetic induction intensity of the permanent magnet. Experimental results show that characteristic quantity one exhibits a significantly accelerated degradation trend with increasing temperature, with average degradation rates of 8.18%, 10.08%, and 14.88% at 150℃, 160℃, and 170℃, respectively. Furthermore, different samples under various temperature conditions exhibited obvious segmented characteristics, with rapid initial growth followed by a gradual plateauing, indicating that the overall magnetic performance degradation of permanent magnets has a phased characteristic.

[0124] Characteristic quantity two (the rate of increase of the characteristic harmonic content of the back electromotive force) is a characterization index of local demagnetization, and its changing trend reflects the spatial distribution characteristics of magnetic field distortion. Experimental results show that characteristic quantity two exhibits a more continuous growth trend, but the curve fluctuates significantly, reflecting the non-uniformity of local demagnetization. Different temperature conditions have significant effects on the degradation rate and amplitude. The average degradation rates at 150℃, 160℃, and 170℃ are 23.42%, 31.99%, and 56.53%, respectively. Among these, the growth rate and final value of both characteristic quantities are highest at 170℃, indicating that high temperature significantly accelerates the degradation process of the permanent magnet, while degradation is slowest at 150℃. Furthermore, the degradation curves of different samples at the same temperature are generally consistent, but characteristic quantity two shows some fluctuations among different samples, which may be related to differences in local magnetic properties.

[0125] The multi-segment degradation modeling process of permanent magnets is as follows:

[0126] First, single-feature degradation modeling based on the Wiener process;

[0127] The segmented Wiener process introduces segmentation points to model each stage separately, thereby more accurately reflecting the stage characteristics of degradation behavior

[20] . Assuming the degradation process occurs at times t1, t2, ..., t... k If there are k segmentation points, then the segmented Wiener process can be represented as:

[0128]

[0129] Where X(t) i ) is the segmentation point t i The state of degradation at that location.

[0130] For the degradation trajectory X1(t) of the rise rate of the fundamental amplitude of the back EMF, degradation test data show that X1(t) exhibits obvious piecewise degradation characteristics. To accurately describe the piecewise characteristics, a degradation modeling method based on the piecewise Wiener process is adopted. For the degradation trajectory X2(t) of the rise rate of the characteristic harmonic content of the back EMF, statistical analysis of the experimental data of characteristic quantity two in the accelerated degradation test of the motor permanent magnet reveals that characteristic quantity two does not exhibit obvious piecewise characteristics during the degradation process. Therefore, a non-piecewise Wiener process model is used to model it. The models are represented as follows:

[0131]

[0132] X2(t)=μ′t λ′ +σ′B(t λ′ (15)

[0133] The unknown parameters in the model are estimated using the maximum likelihood function, and the corresponding parameter estimates are obtained as follows: Figure 14 As shown.

[0134] Secondly, modeling of the joint distribution model of multiple features based on the copula function;

[0135] Copula functions take various forms, among which the t-Copula function has significant advantages in describing nonlinear correlations between variables and can more accurately describe the dependencies between degenerate features. Therefore, this application chooses t-Copula as the function form for joint modeling. Its univariate and bivariate probability density functions t... v and t v,ρ It can be represented as:

[0136]

[0137] Where Γ(·) is the gamma function. v represents the degrees of freedom, indicating the nonlinear correlation between variables. P is the correlation coefficient matrix, characterizing the correlation between variables, and its expression is:

[0138]

[0139] Where ρ is the correlation coefficient, the larger the value, the stronger the correlation between the variables.

[0140] The distribution function and density function of t-Copula can be expressed as:

[0141]

[0142] Among them, T v Let u be the distribution function of a univariate t-distribution. i =F i (Δxi).

[0143] The t-Copula distribution function is derived from the t-distribution and is used to construct the joint distribution. Its density function is relatively complex. To utilize the t-Copula to jointly represent the two degenerate features, the original data of the marginal distributions must first be mapped to the Copula space, i.e.:

[0144] (u i,1 ,u i,2 )=[F1(ΔX i,1 ),F2(ΔX i,2 (19)

[0145] Furthermore, to determine the parameters of the Copula model, the maximum likelihood estimation method is used to estimate the parameters v and ρ in the Copula model. According to equation (26), its log-likelihood function can be expressed as:

[0146]

[0147] This application uses the segmentation point of degradation feature quantity one as the boundary to divide degradation feature quantity two into two stages as well. Joint modeling is performed on the first and second stages respectively to estimate the unknown parameters. The estimation results are: correlation coefficient of the first stage. Degrees of freedom The second correlation coefficient Degrees of freedom

[0148] Correlation analysis showed that the correlation coefficients of the two degradation features were both greater than 0.7, indicating a strong positive correlation between them. This means that the t-Copula function can be used for piecewise modeling.

[0149] Based on the above analysis results, the Copula-Wiener multi-element piecewise degradation model of the permanent magnet of the motor is established as follows:

[0150] F(Δx1,Δx2)=C(F1(Δx1),F2(Δx2);θ1,θ2) (21)

[0151] Where C(·) is the selected t-Copula function, F1(Δx1) and F2(Δx2) are the marginal distribution functions of each performance parameter, and θ1 and θ2 are the relevant parameters of the first and second stages of the t-Copula function, respectively.

[0152] The process for predicting the lifetime of permanent magnets is as follows:

[0153] First, the failure threshold is selected;

[0154] Lifetime prediction based on the Wiener process requires defining the product's failure threshold. In engineering practice, lifetime T is usually defined as the moment when the degradation characteristic first reaches the failure threshold D, or simply the first arrival time, as shown in the following formula:

[0155] T=inf{t|X(t)≥D,t>0} (22)

[0156] According to the relevant standards of QC / T893-2011 "Fault Classification and Judgment of Drive Motor Systems for Electric Vehicles" and GB / T29307-2012 "Reliability Test Methods for Drive Motor Systems for Electric Vehicles", when the demagnetization rate of the permanent magnet reaches 25%, the motor performance degrades to the point where it cannot operate normally. Therefore, the fundamental frequency amplitude of the back EMF corresponding to a demagnetization rate of 25% can be regarded as the failure threshold of the first degradation characteristic quantity. To obtain the fundamental frequency amplitude of the back EMF corresponding to a permanent magnet demagnetization rate of 25%, the permanent magnet of the motor was uniformly demagnetized. When the demagnetization rate of the permanent magnet reached 25%, the back EMF of the motor at rated speed was measured. Simultaneously, to further verify the rationality of the failure threshold, the permanent magnet of the motor was uniformly demagnetized to different degrees, and the fundamental frequency amplitude of the demagnetization rate and the back EMF was measured. The fundamental frequency amplitude of the back EMF corresponding to a demagnetization rate of 25% was calculated by curve fitting. The specific results are as follows: Figure 6 As shown. Based on the above experiments and Figure 6 Analysis shows that when the demagnetization rate of the permanent magnet reaches 25%, the fundamental frequency amplitude of the back electromotive force decreases by 22%. Therefore, this application sets the failure threshold of degradation characteristic quantity one to L1 = 22%.

[0157] According to GB / T25123.4-2015 "Rotating Electrical Machines for Electric Traction Rail Locomotives and Road Vehicles - Part 4", when the back electromotive force harmonic content reaches 5% at rated speed, the motor performance will significantly degrade, affecting its normal operation. All motors used were from the same batch, with an initial characteristic harmonic content of approximately 1.5%. For the second characteristic of performance degradation, the failure threshold was calculated as (5% - 1.5%) / 1.5% = 233%. Therefore, the failure threshold was set at L2 = 233% to accurately reflect the degradative effect of harmonic content changes on motor performance.

[0158] Secondly, implement the prediction of the permanent magnet life of the motor under accelerated stress;

[0159] This application employs Monte Carlo simulation to predict the lifespan of permanent magnets in an electric motor. This method can intuitively display the evolution trend of characteristic quantities over time, providing an efficient and visualized solution for lifespan prediction. Degradation trajectories under different accelerating stress conditions are constructed, such as... Figure 7 As shown.

[0160] Combination Figure 7 From 1000 degradation trajectories and failure thresholds of the two key characteristics, 1000 predicted lifetime values ​​for the motor permanent magnet can be calculated. This allows for the derivation of the lifetime probability density curve and reliability curve of the motor permanent magnet under accelerated stress, as shown below. Figure 8 (a) and Figure 8 As shown in (b).

[0161] Depend on Figure 8 Analysis shows that the mean time to failure (MTTF) of the motor at 170℃, 160℃, and 150℃ are 391h, 1083h, and 2114h, respectively. Further analysis reveals that the lifetime distribution of the permanent magnet under accelerated stress approximates a normal distribution. With increasing temperature, the lifetime probability density curve shifts generally to the left and gradually concentrates near the MTTF. Figure 8 (b) It can be seen that the reliability curves of the permanent magnets of the motor exhibit similar shapes at different temperatures, all showing a slow decline in the initial and final stages, while the decline is more rapid in the middle stage. As the temperature increases, the reliability curves of the permanent magnets of the motor shift to the left as a whole, and the curves gradually become steeper, with the lifespan distribution becoming more concentrated.

[0162] Finally, the life prediction of the permanent magnet of the motor under constant stress was carried out.

[0163] Before extrapolating the lifespan of the permanent magnet in the motor under constant stress, this application fully considers the segmented characteristics exhibited by characteristic quantity one at different temperatures. A segmentation point-temperature model based on accelerated degradation test data is constructed to predict the occurrence time of segmentation points at different temperatures. To establish the segmentation point-temperature model, curve fitting is performed on the occurrence times of segmentation points at temperatures of 170℃, 160℃, and 150℃. Verification based on the AIC index confirms that the relationship between the occurrence time of segmentation points and temperature is exponential:

[0164] τ(T)=aexp(bT) (23)

[0165] Where a and b are correlation coefficients; T is temperature stress.

[0166] The parameters of the piecewise point-temperature model were estimated using the least squares method, and the model parameters were obtained as a = 2.5 × 10⁻⁶. 4 b = -4.743 × 10 -2Based on the fitting results, effective prediction of the segmentation point's variation with temperature can be achieved. Furthermore, by combining this with an acceleration model, the parameters of the degradation model under constant stress can be accurately extrapolated. This application selects temperature as the accelerating stress; therefore, the Arrhenius model can be used as the acceleration model.

[0167]

[0168] Among them, y(T) k ) represents absolute temperature T k The degradation rate is denoted by ; a is a constant related to product performance and testing methods; b is the Boltzmann constant, b = 8.6171 × 10⁻⁶. -5 eV / K; E a The activation energy is related to the characteristics of the product.

[0169] Taking a logarithmic transformation of equation (24) yields:

[0170]

[0171] Based on the two characteristic quantities in this application, the estimated values ​​of the undetermined coefficients and λ are calculated as follows: Figure 16 As shown.

[0172] In practical applications, the internal temperature of the motor will change with variations in operating conditions and other factors. Under normal operating conditions, the temperature of the permanent magnet inside the motor can reach 80℃ to 100℃. Therefore, this application sets the constant stress at 80℃, 90℃, and 100℃.

[0173] Using the estimated values ​​of c1 to c3 and the piecewise temperature model, the parameters μ and σ under constant stress, as well as the estimated values ​​of the piecewise points, can be obtained. The results are as follows: Figure 17 and Figure 18 As shown.

[0174] Based on the above estimation results, 1000 degradation trajectories with two characteristic quantities were randomly generated using the Monte Carlo simulation method. Combining the degradation trajectories with the failure threshold, the lifetime probability density curve and reliability curve of the motor permanent magnet under constant stress conditions were derived. Figure 9 As shown.

[0175] Depend on Figure 9 Analysis shows that the MTTF (Mean Time To Failure) of the motor at 100℃, 90℃, and 80℃ are 55981h, 115882h, and 197482h, respectively. Furthermore, the lifetime distribution of the permanent magnet under normal stress and accelerated stress exhibits consistent characteristics, both approximately following a normal distribution. The lifetime probability density curve shifts to the left as temperature increases, gradually converging towards the vicinity of the MTTF. The reliability curves under different temperature stresses also show similar trends, exhibiting a slow decline in the early and later stages, and a more rapid decline in the middle stage.

[0176] The process for verifying the accuracy of the lifetime prediction results is as follows:

[0177] The goal of lifetime prediction is to estimate the operating time of a product at a given failure threshold, while the reliability function R(t) describes the probability that the product will not fail before time t.

[0178] R(t)=P(T>t)=1-F(t) (26)

[0179] Where F(t) is the cumulative distribution function of lifetime.

[0180] As can be seen from equation (26), when the reliability function R(t) is accurate, the corresponding F(t) also has high accuracy. Therefore, the verification of the accuracy of the life prediction can be transformed into the verification of the accuracy of the reliability curve of the motor permanent magnet.

[0181] Due to the high reliability and long lifespan of permanent magnets, it is difficult to obtain sufficient degradation data within a limited test period under constant stress levels. To address this issue, this application employs a three-fold cross-validation method to verify the consistency of reliability assessment results using limited experimental data. However, while this method can quantitatively assess the error in the reliability assessment results, it is difficult to directly determine whether two reliability curves are consistent based on this result. Therefore, this application further introduces the area ratio method to more intuitively assess the consistency of reliability curves, and defines the corresponding area ratio calculation formula:

[0182]

[0183] Among them, s(R) a (t)) and s(R) b (t) represents the area enclosed by the corresponding reliability curve and the coordinate axis respectively; ε is the area ratio threshold. If w≤ε, it means that the reliability assessment results are consistent.

[0184] Figure 10 The paper presents reliability assessment results under three different stress levels: 170℃, 160℃, and 150℃. At 170℃, the extrapolated curve lies to the left of the standard curve, indicating a rapid decrease in extrapolated reliability and a certain underestimation of lifespan. At 160℃ and 150℃, the extrapolated curves lie to the right of the standard curve, indicating a slower decrease in extrapolated reliability and a certain overestimation of lifespan. Although the extrapolated reliability curves do not completely coincide with the standard reliability curves, the overall curves show a high degree of agreement, indicating that the model still has high accuracy. The consistency of the reliability assessment method based on the multivariate piecewise degradation model combining the Copula function and the Wiener process model under different stress levels further demonstrates the effectiveness and reliability of this method in predicting the lifespan of permanent magnets in motors.

[0185] This application constructs a degradation characterization method using the fundamental wave and characteristic harmonics of back electromotive force as characteristic quantities, and proposes a degradation modeling and life prediction method suitable for permanent magnets in motors. The degradation characteristic quantities and degradation model of permanent magnets in motors are established, realizing the effective characterization of the degradation state of permanent magnets using motor electrical signals. The design and implementation of accelerated degradation tests for permanent magnets in motors were completed, and effective degradation data were obtained. A multi-element segmented degradation model modeling method suitable for permanent magnets in motors is proposed, achieving an accurate description of the degradation process of permanent magnets in motors and predicting the life of motors under different stress conditions. Finally, the accuracy of the life prediction of permanent magnets in motors is evaluated using cross-validation and the area ratio consistency index. The evaluation results show that the established model has high accuracy and effectiveness in describing the degradation process of permanent magnets and predicting their life, providing certain theoretical support for the reliability assessment and life prediction of permanent magnets in motors at high temperatures, and has engineering applicability.

[0186] The above technical solutions only embody the preferred technical solutions of the present invention. Any modifications that may be made by those skilled in the art to certain parts thereof embody the principles of the present invention and fall within the protection scope of the present invention.

Claims

1. A method for multi-segment degradation modeling and lifetime prediction of permanent magnets in a permanent magnet synchronous motor, characterized in that, The method includes the following steps: Step 1: Determine the degradation characteristic quantities, and select the rate of decrease in the fundamental frequency amplitude of the back electromotive force and the rate of increase in the characteristic harmonic content as the degradation characteristic quantities of the permanent magnet; Step 2: Obtain degradation data, design and implement accelerated degradation tests to obtain permanent magnet degradation data under different temperature stresses; Step 3: Single feature degradation modeling. Model the two degradation features separately and obtain two corresponding degradation feature models. Step 4: Construct a multivariate segmented degradation model, analyze the correlation and dependency between the two degradation feature models, and use the t-Copula function to construct a joint distribution model of the multivariate degradation features; Step 5: Derive the model parameters. Based on the Arrhenius model and the piecewise point-temperature model, derive the parameters of the multivariate piecewise degradation model under constant stress. Step 6, lifetime prediction: Based on the established multivariate segmented degradation model, Monte Carlo simulation is used to generate degradation trajectories, and lifetime prediction is performed in combination with failure thresholds. Step 7: Model validation. The accuracy of the lifetime prediction results is verified using the area ratio index and cross-validation method.

2. The method according to claim 1, characterized in that, The formula for calculating the rate of decrease in the fundamental frequency amplitude of the back electromotive force is as follows: In the formula, E1 is the fundamental frequency amplitude of the back electromotive force under normal conditions, E 1，t It represents the fundamental frequency amplitude of the back electromotive force at time t, degenerated to that value.

3. The method according to claim 1, characterized in that, The formula for calculating the rate of increase of the characteristic harmonic content is as follows: In the formula, H0 is the initial characteristic harmonic content, H t This represents the characteristic harmonic content at time t, which degenerates to that at time t.

4. The method according to claim 1, characterized in that, The mathematical expression for the segmented Wiener process model is: In the formula, τ is the segmentation point, μ1 and μ2 are both drift coefficients, σ1 and σ2 are both diffusion coefficients, and B(t) is the standard Brownian motion.

5. The method according to claim 1, characterized in that, The mathematical expression for the t-Copula function is: C(u1,u2;ρ,v)=T v，ρ (T v -1 (u1),T v -1 (u2)) In the formula, u1 and u2 are both marginal distribution functions, ρ is the correlation coefficient, and v is the degree of freedom.

6. The method according to claim 1, characterized in that, The mathematical expression for the segmented point-temperature model is: τ(T) = a·exp(b·T) In the formula, a and b are model parameters, obtained by least squares fitting.

7. The method according to claim 1, characterized in that, The mathematical expression for the degradation rate of the joint distribution model under constant stress, as used by the Arrhenius model, is as follows: In the formula, y(T) is the degradation rate, A is a constant, and E is a constant. a The activation energy is k, and the Boltzmann constant is k.

8. The method according to claim 1, characterized in that, The failure threshold is set as follows: The failure threshold L1 = 22% for the rate of decrease in the back EMF fundamental frequency amplitude; The failure threshold L2 for the rate of increase of characteristic harmonic content is 233%.

9. The method according to claim 1, characterized in that, The formula for calculating the area ratio consistency index is as follows: In the formula, s(R) a (t)) and s(R) b (t) represents the area enclosed by the two reliability curves and the coordinate axes, respectively.

10. The method according to claim 1, characterized in that, The cross-validation method described above verifies the consistency of the prediction results using a three-fold cross-validation approach.

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