Self-adaptive cross-domain battery pack health state probability prediction method

Abstract

The invention relates to a self-adaptive cross-domain battery pack health state probability prediction method, and belongs to the technical field of battery health management. The method comprises the following steps: S1, data preprocessing and trajectory normalization; s2, carrying out adaptive step length difference modeling based on local curvature; s3, multi-modal noise modeling; and S4, cross-domain migration probability prediction based on GMM-UT. Quantizing the nonlinear characteristics of the degradation track through local curvature, and dynamically adjusting the difference step length so as to accurately extract the degradation characteristics; a Gaussian mixture model is adopted to fit multi-modal noise, and accurate quantification of uncertainty is achieved; multi-modal noise statistical characteristics and unscented transformation are fused, cross-domain migration and uncertainty propagation are optimized, and battery pack degradation track multi-modal probability distribution is output. According to the method, the defects that an existing method is poor in fixed step length adaptability, inaccurate in noise modeling and insufficient in cross-domain prediction robustness are overcome, the health prediction precision and reliability of the battery pack are improved, and the method is suitable for health management of the battery pack under complex working conditions.

Description

Technical Field

[0001] This invention belongs to the field of battery health management technology, specifically relating to an adaptive cross-domain probability prediction method for battery pack health status. Background Technology

[0002] With the rapid development of the new energy industry, lithium-ion batteries, with their advantages of high energy density, long cycle life, and environmental friendliness, have been widely used in electric vehicles, energy storage systems, and portable electronic devices. In practical applications, batteries typically operate in battery packs, and the health status of the battery pack (such as remaining capacity and degree of aging) directly determines the reliability, safety, and lifespan of the entire system. Therefore, achieving accurate prediction of battery pack health status is of significant practical importance for early warning of faults, optimizing operation and maintenance strategies, and reducing usage risks.

[0003] Among existing technical solutions, there is a migration-driven prediction framework based on the deep fusion of Adaptive Differential Model Decomposition (ADMD) and Unscented Transform (UT). First, in the source domain modeling stage, the aging trajectories of multiple battery cells are normalized using Euclidean distance to extract a baseline trajectory. Measurement noise and stable degradation trends are separated through initial and secondary decomposition, thus constructing a high-precision single-cell differential dynamic model. Second, in the cross-domain migration stage, to overcome the domain differences between battery cells and the battery pack, an online adaptive scaling factor is introduced. An optimization framework based on the Lagrange multiplier method and Newton's method is constructed to minimize the Euclidean distance between the battery pack trajectory and the single-cell baseline trajectory. Finally, in the target domain prediction stage, the optimized scaling factor is incorporated into the state transition equation, and the uncertainty in nonlinear prediction is handled by combining Unscented Transform (UT). By propagating the state mean and covariance, the probability distribution of the battery pack degradation trajectory, including confidence intervals, is finally output. Subsequently, relevant experiments were conducted to verify the accuracy and robustness of the method using multiple battery cells and battery packs with different configurations through multiple charge-discharge cycles.

[0004] The fixed-step differential mechanism in the ADMD method has significant limitations, restricting its practical application. The fixed-step design struggles to adapt to the inherent nonlinear time-varying characteristics of battery degradation. Battery degradation typically manifests as slow early decay followed by a rapid acceleration later. A fixed step size tends to over-amplify measurement noise in the stable early stages, while failing to effectively respond to the rapidly evolving kinetic characteristics in the later stages of rapid capacity reduction, resulting in the model's inability to accurately represent the actual nonlinear degradation trend. Furthermore, the noise modeling method in the degradation trajectory decomposition process also has flaws. This process inevitably introduces two types of uncertainty: measurement noise originating from fluctuations in the original data and differential noise originating from degradation kinetics. ADMD ignores the significant multimodal characteristics of these two types of noise; their probability density function (PDF) exhibits a multi-peak shape in reality, yet it is still simplified to a single Gaussian distribution for modeling. Fitting multimodal noise to a single-peak distribution severely reduces the accuracy of noise modeling, causing the uncertainty interval to fail to effectively cover the actual degradation fluctuations, thus compromising the reliability of the battery migration prediction system. Summary of the Invention

[0005] The purpose of this invention is to provide an adaptive cross-domain battery pack health state probability prediction method with a simple structure and reasonable design in order to solve the above problems.

[0006] The present invention achieves the above objectives through the following technical solutions:

[0007] An adaptive cross-domain battery pack health state probability prediction method includes the following steps:

[0008] Includes the following steps:

[0009] S1. Data preprocessing and trajectory normalization: Remove outliers from the battery cell degradation trajectory, calculate the Euclidean distance between each battery cell degradation trajectory and all other cell trajectories, select the trajectory with the smallest sum of distances to other trajectories as the reference trajectory, assign scaling factors to the remaining cell trajectories and minimize their Euclidean distance with the reference trajectory through scaling operations, and complete the normalization process.

[0010] S2. Adaptive step-size differential modeling based on local curvature: The normalized trajectory is initially decomposed to separate the stable degenerate trajectory from the measurement noise; the local curvature of the stable degenerate trajectory is calculated using the three-point curvature estimation method; the low curvature stage and the high curvature stage are selected based on the local curvature; the differential step size is dynamically adjusted through an adaptive step-size adjustment mechanism; the differential decomposition is performed using the adjusted adaptive step size to extract the nonlinear features of the degenerate trajectory and separate the differential noise.

[0011] S3. Multimodal noise modeling: Gaussian mixture model (GMM) is used to model the measurement noise and differential noise obtained in step S2. The model parameters are initialized by K-Means clustering algorithm, and the model parameters are iteratively optimized by combining log-likelihood function (LLF) and expectation-maximization (EM) algorithm to achieve accurate fitting of multimodal noise distribution.

[0012] S4. Cross-domain migration probability prediction based on GMM-UT: A scaling factor is introduced to adjust the battery pack degradation trajectory. The scaling factor is optimized by the Lagrange multiplier method and Newton's method to minimize the domain difference between the battery pack trajectory and the source domain reference trajectory. For each Gaussian noise component in GMM, a Sigma point set is generated by combining the current state estimate and the noise covariance of the corresponding component. Each Sigma point set is propagated through the state transition function and the mean and covariance of the predicted state under the corresponding mode are calculated. The mixed weights of GMM are used to perform weighted fusion of the statistics of each mode to output the multimodal probability distribution of the battery pack degradation trajectory.

[0013] As a further optimization of the present invention, in step S1, the criterion for judging abnormal values ​​is: if the absolute value of the capacity difference between adjacent cycles exceeds 10% of the maximum capacity of the battery, it is judged as an abnormal point and removed.

[0014] As a further optimization of the present invention, in step S2, the initial decomposition process extracts the dynamic characteristics of measurement noise through smoothing splines to separate the stable degradation trajectory from the measurement noise. In step S2, the local curvature of the stable degradation trajectory is calculated using the three-point curvature estimation method, and the nonlinear intensity of each data point is quantified by the coordinate information of three consecutive data points. In step S2, the differential decomposition uses an adaptive step size to calculate the differential result of the stable degradation trajectory. After smoothing the differential result, the differential noise is separated. The low curvature threshold is the 20th percentile of the curvature distribution, and the maximum step size is the 80th percentile of the segment length sequence of the flat stage. The adjustment coefficient is obtained by normalizing the standard deviation of the local curvature. A constant to prevent division by zero is introduced during the normalization process. In step S2, the differential decomposition uses an adaptive step size to calculate the differential result of the stable degradation trajectory. After smoothing the differential result, the differential noise is separated.

[0015] As a further optimization of the present invention, the initial decomposition process of the normalized trajectory and the nonlinear dynamics of the degenerate trajectory can be expressed as follows:

[0016]

[0017] in, This represents the initial degradation trajectory during a charge-discharge cycle. This indicates a stable degradation trajectory. This represents the noise uncertainty in the measurement process, i.e. and The measurement noise extracted between them This indicates the uncertainty of state transitions. Indicates description from arrive The nonlinear state transition function.

[0018] As a further optimization of the present invention, the adaptive step size adjustment mechanism realizes dynamic adjustment of the step size based on local curvature, low curvature threshold, maximum step size, minimum step size, adjustment coefficient and local curvature standard deviation.

[0019] As a further optimization of the present invention, the adaptive step size adjustment mechanism is based on local curvature calculation as follows:

[0020]

[0021] in, , , It represents the battery capacity corresponding to three adjacent data points. This refers to the calculated local curvature. Furthermore, the curvature of all individual battery cells at the corresponding data points across all charge-discharge cycles can be expressed as:

[0022]

[0023] in, and These represent the m-th charge-discharge cycle and the corresponding capacity curvature of the n-th battery cell, respectively. It refers to a specific charge-discharge cycle of a single battery cell. The loop refers to The local curvature refers to This represents the m-th charge-discharge cycle of the n-th battery cell. The curvature representing the capacity of the nth battery cell in the mth charge-discharge cycle. It refers to "capability," and in expressions, it is used solely as an identifier for "capacity."

[0024] Based on the calculated local curvature, the differential step size is adaptively adjusted. Low curvature and high curvature stages are selected according to the curvature values. The low curvature and high curvature stages correspond to the gentle and steep segments of the degenerate trajectory, respectively. Then, the step size is dynamically adjusted for these two stages. The adaptive step size adjustment is as follows:

[0025]

[0026] in, This represents the local curvature at a point on the trajectory. Indicates the low curvature threshold. and These are the maximum and minimum step sizes, respectively. It is an adjustment coefficient that controls the sensitivity of the step size to changes in curvature. Characterizes the degree of deviation between current curvatures. It is a Gaussian decay function. The dynamic decay portion corresponding to the maximum step size;

[0027] To ensure that the adaptive step size accurately reflects the dynamic degradation characteristics of the battery, the determination method of each variable should be consistent with the physical degradation mechanism and the statistical characteristics of the experimental data;

[0028] First, the curvature is calculated based on the smoothed degenerate trajectory, and the curvature histogram is analyzed to determine the representative curvature threshold. The lower limit of curvature is defined as the 20th percentile of the curvature distribution.

[0029]

[0030] in This refers to the local curvature of the capacity. This refers to the minimum threshold of local curvature. This refers to taking the 20th percentile of the local curvature, which effectively covers most of the data points related to the flattening degradation stage. Based on this threshold, the set of sampling points for the flattening stage is identified as...

[0031]

[0032] Data points with local curvature less than a curvature threshold are grouped into a set and labeled as follows: That is, after extracting the flat phase, the set of sampling points in the flat phase is used to construct a sequence. To quantify the continuous segments satisfied by the sampling points, the length of each segment is calculated as follows.

[0033]

[0034] in Representing the The length of a continuous flat segment (the number of sampling points it contains). Represents the starting position index of the current "flat segment". This represents the "offset" or "steps" extending backward from the starting point. The "length" of a continuous flat segment;

[0035]

[0036] Indicator function Sampling points used to identify and count continuous flat phases;

[0037] This is a standard logical judgment function. If the condition is true, the result is 1; if the condition is false, the result is 0. It is used to identify and count sampling points in continuous flat segments. For a given sampling point n, if it belongs to the k-th continuous flat segment, the function value is 1; otherwise, it is 0. It is a tool for counting in the formula. To obtain a representative reference value for the flat segment step size, the maximum recommended step size is determined as the segment length sequence. The 80th percentile,

[0038]

[0039] Represents the maximum step size. For the segment length sequence calculated above, This represents taking the 80th percentile of the sequence. ;

[0040]

[0041] in, It is the standard deviation of the local curvature. It is a constant to prevent division by zero (usually taken as 1e-6). This represents the baseline adjustment coefficient determined through offline training;

[0042] Difference model construction

[0043] An adaptive step size is used in the difference decomposition to extract the nonlinear features of the degenerate trajectory.

[0044]

[0045] in Represents the difference result, Indicates adaptive step size, and These represent the nth charge-discharge cycle and the corresponding capacity, respectively.

[0046] By smoothing the difference results, the dynamics of the difference noise can be extracted;

[0047]

[0048] in, Indicates the result of smoothing. This represents the dynamics of differential noise.

[0049] As a further optimization of the present invention, in step S3, the Gaussian mixture model fits the noise distribution by a linear weighted combination of multiple Gaussian sub-components, each Gaussian sub-component containing weights, mean vectors, and covariance matrix parameters.

[0050] As a further optimization of the present invention, the PDF definition of the GMM used to fit measurement noise and differential noise is as follows:

[0051]

[0052] in, The observed samples represent noise, and M is the number of Gaussian sub-components. Representative parameter set, including , , , Let represent the weight of the m-th Gaussian component, and satisfy .

[0053]

[0054] The weights of each Gaussian component should be between 0 and 1, and sum to 1.

[0055] The likelihood function is defined as the product of the joint probability densities of all samples, as shown below:

[0056]

[0057] Likelihood function It indicates that in the parameter set Under these conditions, the joint probability density of the entire noise sample set was observed; The multiplication symbol indicates that the probability density values ​​of all samples are multiplied together. Represents the total number of samples. For sample index, For the first One observation sample, This represents the probability density of a single sample, i.e., a single noise point. In parameters The probability density of the following occurrences;

[0058] The LLF is formed using the natural logarithm of the likelihood function, and it is expressed as follows:

[0059]

[0060] The parameter optimization problem of GMM can be transformed into finding the parameters that maximize LLF.

[0061]

[0062] After establishing LLF as the objective function for parameter optimization, the EM algorithm is used for iterative solution to maximize the function. First, the K-Means clustering algorithm is used for parameter initialization to obtain the initial values ​​of the model. Then, in step E, each sample is calculated. Posterior probability of belonging to the m-th Gaussian component The formula is as follows:

[0063]

[0064] The log-likelihood function transforms the product in formula (15) into a summation. This represents the summation of the log probabilities of all N samples, where M is the number of Gaussian components. This represents the total probability density of each noise sample under the mixed distribution. The weight representing the m-th Gaussian component. Represents the Gaussian probability density function. This represents the mean and covariance of the m-th component;

[0065] Subsequently, in the M-step, the weights are updated using the posterior probabilities from the E-step. Mean vector Covariance Matrix To maximize LLF;

[0066]

[0067] This represents the updated weight, which is the new proportion of the m-th Gaussian component in the mixture model. The probability of all data points belonging to the m-th component is added together and divided by the total number of samples to calculate the average contribution rate of that component in the overall data.

[0068]

[0069] The vector represents the updated mean vector, indicating the center position of the m-th Gaussian distribution. For the i-th observation sample;

[0070] Represents the weighting coefficient, if The sample size is large, indicating that the sample size is large. This has a significant impact on the position of the mean of this component; if Very small, indicating that the sample size is small. It has almost no effect on this component. This represents the sum of the posterior probabilities of that component for all samples;

[0071]

[0072] This represents the updated covariance matrix. As a bias vector, sample points Compared with the newly calculated mean The distance between them The square of the deviation is used to calculate the variance. These are the weighting coefficients, which are part of a weighted variance calculation process. They measure the weighting of data points belonging to the m-th component around the new mean. The degree of looseness of the distribution;

[0073] The E-step and M-step are executed alternately, and the LLF value is evaluated after each iteration. When the difference between the LLF values ​​of two consecutive iterations is lower than a preset threshold, the algorithm is considered to have converged; the parameter set obtained at this time is the optimal solution.

[0074] As a further optimization of the present invention, in step S4, the domain difference is defined as the average of the minimum Euclidean distances between each data point on the target domain trajectory and all points on the source domain reference trajectory. The optimization process of the scaling factor is a minimization problem with the constraint of S>0 inequality, which is transformed into an unconstrained extremum problem by using the Lagrange multiplier method, and S is iteratively updated using Newton's method.

[0075] The beneficial effects of this invention are as follows:

[0076] 1. This invention overcomes the inherent limitations of traditional fixed-step differential models, solving the technical bottleneck of easily amplifying measurement noise in the early stages of battery degradation and struggling to accurately capture rapid evolution dynamics in the later stages. By introducing an adaptive adjustment mechanism based on local curvature, this method can quantitatively characterize the nonlinear intensity of the degradation trajectory and dynamically match the optimal differential step size. Specifically, in the smooth stage with low curvature, the algorithm uses a large step size to smooth data fluctuations and suppress noise interference, while improving computational efficiency; while in the rapid degradation stage with high curvature, it switches to a small step size to ensure high-resolution sampling, thereby accurately capturing the key inflection point and evolution characteristics when capacity drops sharply. This dynamic adjustment strategy fundamentally balances the contradiction between prediction accuracy and computational efficiency, achieving accurate feature extraction of complex nonlinear degradation trajectories throughout the entire life cycle.

[0077] 2. This invention overcomes the inherent limitations of the single Gaussian distribution assumption. Addressing the complex multimodal characteristics of measurement noise and differential noise under actual working conditions, it introduces a Gaussian mixture model for refined modeling. This method decomposes the complex, non-normally distributed noise into a linearly weighted combination of several independent Gaussian components, achieving accurate fitting of the noise probability density function and effectively eliminating modeling biases introduced by distribution mismatch. This precise quantification of multimodal uncertainty ensures that the prediction results can fully cover real degradation fluctuations, thereby significantly enhancing the robustness and reliability of the cross-domain prediction system in complex dynamic environments.

[0078] 3. This invention constructs a cross-domain probabilistic prediction framework that integrates a multimodal uncertainty propagation mechanism, achieving high-precision migration prediction from individual cells to battery packs. This framework deeply integrates multiple independent noise components generated by a Gaussian mixture model into the unscented transformation prediction process, which can accurately capture and transmit the complex uncertainties inherent in cross-domain migration, measurement, and differential processes. It effectively overcomes the technical challenge of traditional methods in dealing with multi-source heterogeneous uncertainties. This mechanism significantly improves the accuracy and robustness of degradation trajectory prediction, ensuring the reliability and stability of prediction results in system-level cross-domain scenarios. Attached Figure Description

[0079] Figure 1 This is a flowchart view of the single-to-system cross-domain health prediction process of the present invention, which integrates adaptive step-size decomposition and noise uncertainty.

[0080] Figure 2 This is a graph showing the measurement noise fitting results of the present invention;

[0081] Figure 3 This is a graph showing the differential noise fitting results of the present invention;

[0082] Figure 4 This is a schematic diagram of the degradation trajectory probability distribution of the battery pack 1 throughout its entire life cycle according to the present invention;

[0083] Figure 5 This is a schematic diagram of the average percentage error (MPE) evaluation results of different prediction methods of the present invention;

[0084] Figure 6 This is a schematic diagram of the average relative accuracy (MRA) evaluation results of different prediction methods of the present invention. Detailed Implementation

[0085] The present application will now be described in further detail with reference to the accompanying drawings. It should be noted that the following specific embodiments are only used to further illustrate the present application and should not be construed as limiting the scope of protection of the present application. Those skilled in the art can make some non-essential improvements and adjustments to the present application based on the above application content.

[0086] Example 1: As Figure 1 - Figure 6 As shown, an adaptive cross-domain battery pack health state probability prediction method includes four core steps: S1, data preprocessing and normalization; S2, adaptive step-size differential modeling based on local curvature; S3, multimodal noise modeling; and S4, cross-domain migration probability prediction based on GMM-UT. The specific implementation steps are as follows:

[0087] S1. Data Preprocessing and Trajectory Normalization: In order to eliminate individual differences and extract uniform degradation features, the battery cell data is first processed. First, outliers are removed. If the absolute value of the capacity difference between adjacent cycles exceeds 10% of the battery's maximum capacity, it is identified as an outlier and removed. Second, trajectory normalization is performed using Euclidean distance. The Euclidean distance between each battery cell's degradation trajectory and all other cell trajectories is calculated. The trajectory with the smallest sum of distances to other trajectories is selected as the reference trajectory. Scaling factors are assigned to other cell trajectories. The Euclidean distance between them and the reference trajectory is minimized through scaling operations, thus completing the normalization process.

[0088] S2. Adaptive Step-Size Differential Modeling Based on Local Curvature: This is one of the core innovations of this invention, aiming to solve the problem that a fixed step size easily amplifies measurement noise in the stable early stage, while it is difficult to effectively respond to the rapidly evolving dynamic characteristics in the later stage of rapid capacity decline. This step dynamically adjusts the step size of the differential calculation by quantifying the nonlinear characteristics of the battery degradation trajectory. The specific implementation process is divided into the following three sub-steps:

[0089] (1) Initial decomposition of normalized trajectory

[0090] Due to the significant fluctuations in the degradation trajectory of individual battery cells, an initial decomposition is performed to separate the uncertainties introduced during the measurement process. Specifically, while constructing a stable degradation trajectory, smoothing splines are used to extract the dynamic characteristics of measurement noise. Simultaneously, measurement noise is separated and extracted from the original trajectory. The nonlinear dynamics of the initial decomposition process and the degradation trajectory can be expressed as follows:

[0091]

[0092] in, This represents the initial degradation trajectory during a charge-discharge cycle. This indicates a stable degradation trajectory. This represents the noise uncertainty in the measurement process, i.e. and The measurement noise extracted between them This indicates the uncertainty of state transitions. Indicates description from arrive The nonlinear state transition function; however, due to the small difference magnitude, modeling based solely on this equation cannot accurately capture the nonlinear trend in the battery degradation process. Therefore, differential decomposition is introduced to extract the original differential dynamic features from the battery degradation data by specifically modeling the differences between state values ​​and adjacent time steps.

[0093] (2) Adaptive step size adjustment mechanism

[0094] To accurately characterize the nonlinear features of battery degradation trajectories, a differential decomposition based on an adaptive step size adjustment mechanism using local curvature is employed. Curvature is closely related to the degradation trend, and local curvature can accurately output the nonlinear intensity of each data point. By selecting local curvature for step size adjustment, adaptive differential decomposition is achieved. Its core is to accurately quantify the nonlinear features of the degradation trajectory, enabling the step size to dynamically adapt to the changing patterns of different degradation stages. This fundamentally solves the shortcomings of traditional fixed step sizes. This method conforms to the actual degradation mechanism of batteries and balances prediction accuracy and computational efficiency.

[0095] First, the local curvature of the stable degenerate trajectory needs to be calculated to provide a quantitative basis for subsequent adaptive step size adjustment. Since it has good adaptability to equidistant sampling data points, the three-point curvature estimation method is chosen for calculating the local curvature. For any three consecutive data points, the local curvature is calculated as follows:

[0096]

[0097] in, , , It represents the battery capacity corresponding to three adjacent data points. This refers to the calculated local curvature. Furthermore, the curvature of all individual battery cells at the corresponding data points across all charge-discharge cycles can be expressed as:

[0098]

[0099] in, and These represent the m-th charge-discharge cycle and the corresponding capacity curvature of the n-th battery cell, respectively. It refers to a specific charge-discharge cycle of a single battery cell. The loop refers to The local curvature refers to This represents the m-th charge-discharge cycle of the n-th battery cell. The curvature representing the capacity of the nth battery cell in the mth charge-discharge cycle. It refers to "capability" and is used in expressions only as an identifier of "capacity".

[0100] Based on the calculated local curvature, the differential step size is adaptively adjusted. Low curvature and high curvature stages are selected according to the curvature values. The low curvature and high curvature stages correspond to the gentle and steep segments of the degenerate trajectory, respectively. Then, the step size is dynamically adjusted for these two stages. The adaptive step size adjustment is as follows:

[0101]

[0102] in, This represents the local curvature at a point on the trajectory. Indicates the low curvature threshold. and These are the maximum and minimum step sizes, respectively. It is an adjustment coefficient that controls the sensitivity of the step size to changes in curvature. Characterizing the degree of deviation between current curvatures, a low curvature threshold is used to quantify the difference between the current state and the ideal linear degradation process. The squared term enhances the sensitivity to high curvature, prevents the cancellation of positive and negative deviations, and thus highlights the nonlinear characteristics. It is a Gaussian decay function, which can form a smooth transition region in the vicinity to avoid abrupt changes in the step size. The dynamic decay part corresponding to the maximum step size can maintain a relatively large step size in the low curvature stage, thereby improving computational efficiency. The minimum unit of step size is set to 1 to ensure that there is still a basic sampling density in the high curvature stage, avoiding the omission of key mutations.

[0103] To ensure that the adaptive step size accurately reflects the dynamic degradation characteristics of the battery, the determination of each variable should be consistent with the physical degradation mechanism and the statistical characteristics of the experimental data.

[0104] First, the curvature is calculated based on the smoothed degenerate trajectory, and the curvature histogram is analyzed to determine the representative curvature threshold. The lower limit of curvature is defined as the 20th percentile of the curvature distribution.

[0105]

[0106] in This refers to the local curvature of the capacity. This refers to the minimum threshold of local curvature. This refers to taking the 20th percentile of the local curvature, which effectively covers most of the data points related to the flattening degradation stage. Based on this threshold, the set of sampling points for the flattening stage is identified as...

[0107]

[0108] Data points with local curvature less than a curvature threshold are grouped into a set and labeled as follows: That is, after extracting the flat phase, the set of sampling points in the flat phase is used to construct a sequence. To quantify the continuous segments satisfied by the sampling points, the length of each segment is calculated as follows.

[0109]

[0110] in Representing the The length of a continuous flat segment (the number of sampling points it contains). Represents the starting position index of the current "flat segment". This represents the "offset" or "steps" extending backward from the starting point. The "length" of a continuous flat segment;

[0111]

[0112] Indicator function Sampling points are used to identify and count continuous flat phases. To obtain a representative reference value for the flat phase step size, the maximum recommended step size is used. Determined as a segmented length sequence The 80th percentile,

[0113]

[0114] Represents the maximum step size. For the segment length sequence calculated above, This represents the 80th percentile of the sequence;

[0115] This data-driven approach fully leverages the statistical characteristics of the flat degradation phase while mitigating the impact of outlier sampling intervals. Furthermore, by normalizing the standard deviation of local curvature, an adjustment coefficient for adaptive step size updates is obtained, ensuring that the final step size strategy aligns with the inherent smoothness of capacity degradation and fully responds to changes in the degradation rate.

[0116]

[0117] in, It is the standard deviation of the local curvature. It is a constant to prevent division by zero (usually taken as 1e-6). This represents the baseline adjustment coefficient determined through offline training, when... A larger step size indicates larger curvature fluctuations, requiring a smaller step size to improve the response speed to curvature changes; otherwise, it indicates smaller curvature fluctuations, requiring a larger step size to make it smoother and avoid being too sensitive to noise.

[0118] (3) Construction of difference model

[0119] An adaptive step size is used in the difference decomposition to extract the nonlinear features of the degenerate trajectory.

[0120]

[0121] in Represents the difference result, Indicates adaptive step size, and These represent the nth charge-discharge cycle and the corresponding capacity, respectively.

[0122] By smoothing the difference results, the dynamics of the difference noise can be extracted.

[0123]

[0124] in, Indicates the result of smoothing. Represents the dynamics of differential noise.

[0125] By using differential decomposition, a stable degradation trajectory and degradation uncertainty (including measurement noise and differential noise) are obtained, which provides a general benchmark and uncertainty quantification for subsequent migration prediction between the source and target domains.

[0126] S3. Multimodal noise modeling

[0127] In the source domain scenario, Adaptive Step Size Differential Modeling Based on Local Curvature (ASDMD-LC) can accurately extract the degradation features of individual battery cells. However, migration prediction from battery cells to battery packs brings new challenges. Non-uniform degradation between batteries generates additional uncertainties at the system level, and there are significant differences in the degradation modes of battery packs and battery cells. To achieve battery pack health prediction, two key issues must be addressed. First, the interference caused by measurement noise and differential noise throughout the battery's life cycle is crucial for maintaining feature integrity. The second issue is the domain difference between battery cells and battery packs, which must be minimized through cross-domain transfer mechanisms. Then, a probabilistic method is used to predict the probability distribution of degradation trajectories. Therefore, the GMM-UT migration prediction framework is proposed. GMM is first applied to model the measurement noise of the initial decomposition and the differential noise of the secondary decomposition to quantify the uncertainty of noise. Based on this, a cross-domain transfer mechanism based on Euclidean distance is used to align the feature spaces of battery cells and battery packs. Finally, UT is used to generate the probability distribution prediction of battery pack degradation trajectories.

[0128] The uncertainty in battery pack prediction mainly stems from two aspects: measurement noise and differential noise. Traditional methods typically assume that both types of noise follow a single Gaussian distribution. However, in reality, battery noise often exhibits complex multimodal characteristics. This overly simplistic assumption cannot represent the true distribution of noise, leading to a significant decrease in prognostic accuracy. To overcome this limitation, Gaussian Probability Matrix (GMM) is used to model these two types of noise. GMM assumes that the observed data is generated by a linear combination of multiple Gaussian probability density functions (PDFs), each representing a different component. GMM decomposes multimodal noise into several independent components, which can more accurately characterize the overall noise distribution and improve the reliability of uncertainty quantification. The PDF definition for GMM used to fit measurement noise and differential noise is as follows:

[0129]

[0130] in, The observed samples represent noise, and M is the number of Gaussian sub-components. Representative parameter set, including , , , Let represent the weight of the m-th Gaussian component, and satisfy .

[0131]

[0132] The weight of each Gaussian component should be between 0 and 1, and the sum should be 1.

[0133] The essence of applying Gaussian Mixture Model (GMM) to measurement or differential noise lies in estimating model parameters from noise data. The equations only illustrate the concept of fitting noise with Gaussian components; they do not directly generate parameter values ​​that represent the actual noise characteristics. Optimal parameter estimation is performed using the Log-Likelihood Function (LLF) and Expectation-Maximization (EM) algorithms. LLF transforms the matching degree between the Gaussian Mixture PDF and the observed noise data into a function maximization problem. This transformation provides a clear optimization objective for the EM algorithm. Then, the EM algorithm iteratively refines the parameter estimation by alternating between the expectation step (E-step) and the maximization step (M-step). Through this iterative optimization process, the algorithm eventually converges to the optimal parameter set, thereby achieving an accurate fit to the potential noise distribution. The detailed process of obtaining the optimal parameters is described below.

[0134] First, the likelihood function is defined as the product of the joint probability densities of all samples, as shown below:

[0135]

[0136] Likelihood function It indicates that in the parameter set Under these conditions, the joint probability density of the entire noise sample set was observed; The multiplication symbol indicates that the probability density values ​​of all samples are multiplied together. Represents the total number of samples. For sample index, For the first One observation sample, This represents the probability density of a single sample, i.e., a single noise point. In parameters The probability density of the following occurrences;

[0137] In a physical sense, the likelihood function represents the probability of observing the current noisy sample set given a specific set of parameters. A higher likelihood value indicates a better fit of the Gaussian Mixture Model (GMM) to the noisy data. However, directly calculating the likelihood function can easily lead to numerical underflow. Since the probability density value of a single sample is usually less than 1, when the sample size is large, multiplying these tiny values ​​together will cause the result to decay exponentially and approach zero, thus exceeding the effective representation precision of computer floating-point numbers. To solve the above problem, the natural logarithm of the likelihood function is used to form the LLF, which is expressed as follows:

[0138]

[0139] The parameter optimization problem of GMM can be transformed into finding the parameters that maximize LLF.

[0140]

[0141] After establishing LLF as the objective function for parameter optimization, the EM algorithm is used for iterative solution to maximize the function. First, the K-Means clustering algorithm is used for parameter initialization to obtain the initial values ​​of the model. Then, in step E, each sample is calculated. Posterior probability of belonging to the m-th Gaussian component The formula is as follows:

[0142]

[0143] The log-likelihood function transforms the product in formula (15) into a summation. This represents the summation of the log probabilities of all N samples. M is the number of Gaussian components. This represents the total probability density of each noise sample under the mixed distribution. The weight represents the m-th Gaussian component. Represents the Gaussian probability density function. This represents the mean and covariance of the m-th component;

[0144] Subsequently, in the M-step, the weights are updated using the posterior probabilities from the E-step. Mean vector Covariance Matrix To maximize LLF.

[0145]

[0146] This represents the updated weights, which are the new proportions of the m-th Gaussian component in the mixture model. The average contribution rate of this component in the overall data is calculated by summing the probabilities of all data points belonging to the m-th component and dividing by the total number of samples.

[0147]

[0148] This represents the updated mean vector, indicating the center position of the m-th Gaussian distribution. Let i be the i-th observation sample.

[0149] Represents the weighting coefficient, if The sample size is large, indicating that the sample size is large. This has a significant impact on the position of the mean of this component; if Very small, indicating that the sample size is small. It has almost no effect on this component. This represents the sum of the posterior probabilities of that component for all samples.

[0150]

[0151] This represents the updated covariance matrix. As a bias vector, sample points Compared with the newly calculated mean The distance between them This represents the square of the deviation, used to calculate the variance. These are the weighting coefficients. This is a weighted variance calculation process. It measures the data points belonging to the m-th component around the new mean. The degree of looseness in the distribution.

[0152] The E-step and M-step are executed alternately, and the LLF value is evaluated after each iteration. When the difference between the LLF values ​​of two consecutive iterations is lower than a preset threshold, the algorithm is considered to have converged. The parameter set obtained at this time is the optimal solution. The GMM model finally constructed achieves accurate fitting of the multimodal noise distribution.

[0153] like Figure 2 and 3As shown, the multimodal fitting results for measurement noise and differential noise are presented respectively. Both noise types are modeled as a weighted combination of three Gaussian components. The model can also adaptively adjust the number of Gaussian components according to the complexity of the noise characteristics. The histogram depicts the probability density function (PDF) of the actual measurement and differential noise, reflecting their true statistical characteristics. The three signed curves correspond to the three individual Gaussian components, while the bold curve represents the overall GMM fitting obtained by weighted superposition of these components. The high degree of agreement between the GMM fitting curve and the actual probability density distribution strongly confirms that measurement noise and differential noise exhibit significant multimodal characteristics, which cannot be captured by a single Gaussian assumption. This refined noise modeling lays a solid statistical foundation for subsequent degradation prediction.

[0154] S4. Cross-domain migration probability prediction based on GMM-UT

[0155] To achieve system-level capacity prediction, a cross-domain transmission method is first used to reduce the difference between the source and target domains. Then, a UT-based method is used to predict the probabilistic trajectory of degradation. During the prediction process, the noise statistical characteristics of the GMM fitting are fused and propagated to significantly improve the prediction accuracy.

[0156] The core objective of this mechanism is to quantify and minimize the domain difference between the target domain trajectory and the source domain reference trajectory, thereby aligning their spatial distributions. Specifically, the domain difference is defined as the average of the minimum Euclidean distances between each data point on the target domain trajectory and all points on the source domain reference trajectory. To reduce the difference, a scaling factor S is introduced to adjust the target domain trajectory. The domain difference is minimized by optimizing S, which is constructed as a minimization problem with the constraint S>0 inequality. The Lagrange multiplier method is used to solve this problem by transforming the constrained optimization problem into an unconstrained Lagrange extremum problem and iteratively updating S using Newton's method. The target domain trajectory is adaptively scaled to match the degradation trend of the source domain. This transfer mechanism effectively eliminates the domain difference while ensuring that the optimized S is consistent with the actual degradation characteristics of the battery pack. Therefore, the source domain model derived from ASDMD-LC can be directly applied to the target domain, laying a solid foundation for accurate system-level predictions.

[0157] Although cross-domain migration achieves alignment between the source and target domains, probabilistic prediction is needed to address the inherent uncertainties of battery pack degradation. This invention utilizes Unit-Time (UT) to process nonlinear state estimation and constructs an improved UT algorithm by integrating the multimodal noise statistical characteristics of Gaussian Mixture Models (GMMs) to characterize the multimodal characteristics of system-level degradation. First, for each Gaussian noise component in the GMM, a corresponding Sigma point set is independently generated by combining the current state estimate with the specific noise covariance of that component. Second, during the nonlinear propagation phase, a strict modal correspondence mechanism is executed, i.e., the Sigma point set of the m-th GMM component is propagated using only the noise statistical characteristics of the m-th component through the state transition function. Subsequently, based on the propagated Sigma point set, the predicted state mean and covariance under each noise mode are calculated. Finally, the statistics of each mode are weighted and fused using the hybrid weights of the GMM to reconstruct the global multimodal probability distribution of the battery pack degradation trajectory.

[0158] To verify the effectiveness of the technical solution proposed in this invention in practical applications, this embodiment conducted detailed experimental tests and comparative analyses based on real battery aging data. The capacity of the target domain battery pack was predicted using the method of this invention, and a probability distribution map of the degradation trajectory of a battery pack throughout its entire life cycle was plotted, as shown below. Figure 4 The results show that the generated prediction curves are in high agreement with the actual capacity degradation data, and the confidence interval formed by the output probability density distribution effectively covers the actual degradation fluctuation range, which intuitively confirms the robustness of the present invention in handling cross-domain migration and nonlinear degradation characteristics. In order to further quantify the performance advantages of the present invention, the method of the present invention was compared with three benchmark methods with different configurations.

[0159] like Figure 5 , Figure 6 As shown, the method proposed in this invention achieves significant reductions in key indicators such as mean percentage error (MPE) and mean relative accuracy (MRA), and has the highest prediction accuracy.

[0160] The embodiments described above are merely examples of several implementations of the present invention, and while the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the scope of protection of the present invention.

Claims

1. An adaptive cross-domain battery pack health state probability prediction method, characterized in that; Includes the following steps: S1. Data preprocessing and trajectory normalization: Remove outliers from the battery cell degradation trajectory, calculate the Euclidean distance between each battery cell degradation trajectory and all other cell trajectories, select the trajectory with the smallest sum of distances to other trajectories as the reference trajectory, assign scaling factors to the remaining cell trajectories and minimize their Euclidean distance with the reference trajectory through scaling operations, and complete the normalization process. S2. Adaptive step-size differential modeling based on local curvature: The normalized trajectory is initially decomposed to separate the stable degenerate trajectory from the measurement noise; the local curvature of the stable degenerate trajectory is calculated using the three-point curvature estimation method; the low curvature stage and the high curvature stage are selected based on the local curvature; the differential step size is dynamically adjusted through an adaptive step-size adjustment mechanism; the differential decomposition is performed using the adjusted adaptive step size to extract the nonlinear features of the degenerate trajectory and separate the differential noise. S3. Multimodal noise modeling: Gaussian mixture model (GMM) is used to model the measurement noise and differential noise obtained in step S2. The model parameters are initialized by K-Means clustering algorithm, and the model parameters are iteratively optimized by combining log-likelihood function (LLF) and expectation-maximization (EM) algorithm to achieve accurate fitting of multimodal noise distribution. S4. Cross-domain migration probability prediction based on GMM-UT: A scaling factor is introduced to adjust the battery pack degradation trajectory. The scaling factor is optimized by the Lagrange multiplier method and Newton's method to minimize the domain difference between the battery pack trajectory and the source domain reference trajectory. For each Gaussian noise component in GMM, a Sigma point set is generated by combining the current state estimate and the noise covariance of the corresponding component. Each Sigma point set is propagated through the state transition function and the mean and covariance of the predicted state under the corresponding mode are calculated. The mixed weights of GMM are used to perform weighted fusion of the statistics of each mode to output the multimodal probability distribution of the battery pack degradation trajectory.

2. The adaptive cross-domain battery pack health state probability prediction method according to claim 1, characterized in that: In step S1, the criteria for determining outliers are: if the absolute value of the capacity difference between adjacent cycles exceeds 10% of the battery's maximum capacity, it is determined to be an outlier and removed.

3. The adaptive cross-domain battery pack health state probability prediction method according to claim 1, characterized in that: In step S2, the initial decomposition process extracts the dynamic characteristics of measurement noise through smoothed splines to separate the stable degenerate trajectory from the measurement noise. In step S2, the local curvature of the stable degenerate trajectory is calculated using the three-point curvature estimation method. The nonlinear intensity of each data point is quantified using the coordinate information of three consecutive data points. In step S2, the differential decomposition uses an adaptive step size to calculate the differential result of the stable degenerate trajectory. After smoothing the differential result, the differential noise is separated. The low curvature threshold is the 20th percentile of the curvature distribution, and the maximum step size is the 80th percentile of the flat phase segment length sequence. The adjustment coefficient is obtained by normalizing the standard deviation of the local curvature. A constant to prevent division by zero is introduced during the normalization process. In step S2, the differential decomposition uses an adaptive step size to calculate the differential result of the stable degenerate trajectory. After smoothing the differential result, the differential noise is separated.

4. The adaptive cross-domain battery pack health state probability prediction method according to claim 3, characterized in that: The initial decomposition process of the normalized trajectory and the nonlinear dynamics of the degenerate trajectory can be expressed as follows: in, This represents the initial degradation trajectory during a charge-discharge cycle. This indicates a stable degradation trajectory. This represents the noise uncertainty in the measurement process, i.e. and The measurement noise extracted between them This indicates the uncertainty of state transitions. Indicates description from arrive The nonlinear state transition function.

5. The adaptive cross-domain battery pack health state probability prediction method according to claim 1, characterized in that: In step S2, the adaptive step size adjustment mechanism dynamically adjusts the step size based on local curvature, low curvature threshold, maximum step size, minimum step size, adjustment coefficient, and local curvature standard deviation.

6. The adaptive cross-domain battery pack health state probability prediction method according to claim 5, characterized in that: The adaptive step size adjustment mechanism is based on local curvature calculation as follows: in, This indicates that the adaptive step size adjustment mechanism is based on local curvature. , , It represents the battery capacity corresponding to three adjacent data points. This refers to the calculated local curvature. Furthermore, the curvature of all individual battery cells at the corresponding data points across all charge-discharge cycles can be expressed as: in, This represents the curvature of all individual battery cells at the corresponding data points across all charge-discharge cycles. and These represent the m-th charge-discharge cycle and the corresponding capacity curvature of the n-th battery cell, respectively. It refers to a specific charge-discharge cycle of a single battery cell. The loop refers to The local curvature refers to This represents the m-th charge-discharge cycle of the n-th battery cell. The curvature representing the capacity of the nth battery cell in the mth charge-discharge cycle. It refers to "capability," and in expressions, it is used solely as an identifier for "capacity." Based on the calculated local curvature, the differential step size is adaptively adjusted. Low curvature and high curvature stages are selected according to the curvature values. The low curvature and high curvature stages correspond to the gentle and steep segments of the degenerate trajectory, respectively. Then, the step size is dynamically adjusted for these two stages. The adaptive step size adjustment is as follows: in, This represents the local curvature at a point on the trajectory. Indicates the low curvature threshold. and These are the maximum and minimum step sizes, respectively. It is an adjustment coefficient that controls the sensitivity of the step size to changes in curvature. Characterizes the degree of deviation between current curvatures. It is a Gaussian decay function. The dynamic decay portion corresponding to the maximum step size; To ensure that the adaptive step size accurately reflects the dynamic degradation characteristics of the battery, the determination method of each variable should be consistent with the physical degradation mechanism and the statistical characteristics of the experimental data; First, the curvature is calculated based on the smoothed degenerate trajectory, and the curvature histogram is analyzed to determine the representative curvature threshold. The lower limit of curvature is defined as the 20th percentile of the curvature distribution. in This refers to the local curvature of the capacity. This refers to the minimum threshold of local curvature. This refers to taking the 20th percentile of the local curvature, which effectively covers most of the data points related to the flattening degradation stage. Based on this threshold, the set of sampling points for the flattening stage is identified as: Data points with local curvature less than a curvature threshold are grouped into a set and labeled as follows: This refers to the set of sampling points for the flat phase. After extracting the flat phase, a sequence is constructed. To quantify the continuous segments satisfied by the sampling points, the length of each segment is calculated as follows: in Representing the The length of a continuous flat segment (the number of sampling points it contains). Represents the starting position index of the current "flat segment". This represents the "offset" or "steps" extending backward from the starting point. The "length" represents a continuous flat segment. The symbol for multiplication; Indicator function Sampling points used to identify and count continuous flat phases; This is a standard logical judgment function. If the condition is true, the result is 1; if the condition is false, the result is 0. It is used to identify and count sampling points in continuous flat segments. For a given sampling point n, if it belongs to the k-th continuous flat segment, the function value is 1; otherwise, it is 0. It is a tool for counting in the formula. To obtain a representative reference value for the flat segment step size, the maximum recommended step size is determined as the segment length sequence. The 80th percentile; Represents the maximum step size. For the segment length sequence calculated above, This represents taking the 80th percentile of the sequence. ; in, It is the standard deviation of the local curvature. It is a constant to prevent division by zero (usually taken as 1e-6). This represents the baseline adjustment coefficient determined through offline training; Difference model construction An adaptive step size is used in the difference decomposition to extract the nonlinear features of the degenerate trajectory. in Represents the difference result, Indicates adaptive step size, and These represent the nth charge-discharge cycle and the corresponding capacity, respectively; the differential noise dynamics are extracted by smoothing the differential results. in, Indicates the result of smoothing. This represents the dynamics of differential noise.

7. The adaptive cross-domain battery pack health state probability prediction method according to claim 1, characterized in that: In step S3, the Gaussian mixture model fits the noise distribution by a linear weighted combination of multiple Gaussian sub-components. Each Gaussian sub-component includes weights, mean vectors, and covariance matrix parameters.

8. The adaptive cross-domain battery pack health state probability prediction method according to claim 7, characterized in that: The PDF definition of the GMM used to fit measurement noise and difference noise is as follows: in, The observed samples represent noise, and M is the number of Gaussian sub-components. Representative parameter set, including , , , Let represent the weight of the m-th Gaussian component, and satisfy . The weights of each Gaussian component should be between 0 and 1, and sum to 1. The likelihood function is defined as the product of the joint probability densities of all samples, as shown below: Likelihood function It indicates that in the parameter set Under these conditions, the joint probability density of the entire noise sample set was observed; The multiplication symbol indicates that the probability density values ​​of all samples are multiplied together. Represents the total number of samples. For sample index, For the first One observation sample, This represents the probability density of a single sample, i.e., a single noise point. In parameters The probability density of the following occurrences; The LLF is formed using the natural logarithm of the likelihood function, and it is expressed as follows: The parameter optimization problem of GMM can be transformed into finding the parameters that maximize LLF. After establishing LLF as the objective function for parameter optimization, the EM algorithm is used for iterative solution to maximize the function. First, the K-Means clustering algorithm is used for parameter initialization to obtain the initial values ​​of the model. Then, in step E, each sample is calculated. Posterior probability of belonging to the m-th Gaussian component The formula is as follows: The log-likelihood function transforms the product in the formula into a summation. This represents the summation of the log probabilities of all N samples, where M is the number of Gaussian components. This represents the total probability density of each noise sample under the mixed distribution. The weight representing the m-th Gaussian component. Represents the Gaussian probability density function. This represents the mean and covariance of the m-th component; Subsequently, in the M-step, the weights are updated using the posterior probabilities from the E-step. Mean vector Covariance Matrix To maximize LLF; This represents the updated weight, which is the new proportion of the m-th Gaussian component in the mixture model. The probability of all data points belonging to the m-th component is added together and divided by the total number of samples to calculate the average contribution rate of that component in the overall data. The vector represents the updated mean vector, indicating the center position of the m-th Gaussian distribution. For the i-th observation sample; Represents the weighting coefficient, if The sample size is large, indicating that the sample size is large. This has a significant impact on the position of the mean of this component; if Very small, indicating that the sample size is small. It has almost no effect on this component. This represents the sum of the posterior probabilities of that component for all samples; This represents the updated covariance matrix. As a bias vector, sample points Compared with the newly calculated mean The distance between them The square of the deviation is used to calculate the variance. These are the weighting coefficients, which are part of a weighted variance calculation process. They measure the weighting of data points belonging to the m-th component around the new mean. The degree of looseness of the distribution; The E-step and M-step are executed alternately, and the LLF value is evaluated after each iteration. When the difference between the LLF values ​​of two consecutive iterations is lower than a preset threshold, the algorithm is considered to have converged; the parameter set obtained at this time is the optimal solution.

9. The adaptive cross-domain battery pack health state probability prediction method according to claim 1, characterized in that: In step S4, the domain difference is defined as the average of the minimum Euclidean distances between each data point on the target domain trajectory and all points on the source domain reference trajectory. The optimization process of the scaling factor is a minimization problem with the constraint of S>0 inequality. It is transformed into an unconstrained extremum problem by using the Lagrange multiplier method, and S is iteratively updated using Newton's method.

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