CN104778337A

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Info

Publication number: CN104778337A
Application number: CN201510217102.3A
Authority: CN
Inventors: 吕琛; 程玉杰; 王洋; 周博
Original assignee: Beihang University
Current assignee: Beijing Hengxing Yikang Technology Co Ltd
Priority date: 2015-04-30
Filing date: 2015-04-30
Publication date: 2015-07-15
Anticipated expiration: 2035-04-30
Also published as: CN104778337B

Abstract

本发明公开了一种基于函数型主成分分析与贝叶斯更新的锂电池剩余寿命预测方法，首先，利用函数型主成分分析(functional principal component analysis,FPCA)构建锂电池非参数退化模型，基于此退化模型对锂电池剩余寿命进行预测，并分析了不同数量的建模数据对预测精度的影响。为了实现锂电池退化模型的实时更新，本发明提出了经验贝叶斯方法。通过贝叶斯方法对锂电池退化模型进行实时修正，可以得到更精确的锂电池退化模型。在此基础之上，本发明采用参数bootstrap方法计算锂电池剩余寿命分布，并计算其置信区间。

The invention discloses a method for predicting the remaining life of a lithium battery based on functional principal component analysis and Bayesian update. First, a non-parametric degradation model of a lithium battery is constructed using functional principal component analysis (FPCA), based on This degradation model is used to predict the remaining life of lithium batteries, and the influence of different amounts of modeling data on the prediction accuracy is analyzed. In order to realize the real-time update of the lithium battery degradation model, the present invention proposes an empirical Bayesian method. A more accurate lithium battery degradation model can be obtained by correcting the lithium battery degradation model in real time through the Bayesian method. On this basis, the present invention adopts the parameter bootstrap method to calculate the remaining life distribution of the lithium battery, and calculates its confidence interval.

Description

Translated fromChinese

一种基于函数型主成分分析与贝叶斯更新的锂电池剩余寿命预测方法A Lithium Battery Remaining Life Prediction Method Based on Functional Principal Component Analysis and Bayesian Update

技术领域technical field

本发明涉及锂电池剩余寿命预测的技术领域，具体涉及一种基于函数型主成分分析与贝叶斯更新的锂电池剩余寿命预测方法。The invention relates to the technical field of lithium battery remaining life prediction, in particular to a lithium battery remaining life prediction method based on functional principal component analysis and Bayesian update.

背景技术Background technique

随着锂离子电池的发展，有关锂离子电池寿命预测的研究起始于20世纪80年代。有效的电池寿命预测不仅能够预测潜在的风险，从而为电池的使用提供有效的指导，并且可以减少电池失效引起的相关损失。With the development of lithium-ion batteries, research on life prediction of lithium-ion batteries began in the 1980s. Effective battery life prediction can not only predict potential risks, thereby providing effective guidance for battery use, but also reduce related losses caused by battery failure.

关于电池寿命预测目前有很多方法，其中大多都是基于模型的和基于数据驱动的预测方法。现有的基于模型的寿命预测方法可以分为基于电化学模型的方法，基于等效电路的方法，基于性能的方法和基于解析模型的方法。电化学模型是基于多孔电极理论和纳米力学建立的，该模型描述了电池极化过程和电池内部特定组件的物理变形。等效电路模型经常采用等效电路对电池进行建模，并采用许多技术估计模型参数。电化学模型和等效电路模型在一些特定的案例中具有良好的效果，然而，由于技术、设计和材料等原因，这些模型适用性较差。基于性能的模型需要建立退化因素与电池性能之间的关系，然而，考虑到诸多环境变量之间的相互影响，退化因素与电池性能之间的关系需要通过不同工作状况下的电池仿真与大量加速寿命试验才能得到。解析模型可以通过粒子滤波、无味卡尔曼滤波等方法建立。当模型建立准确时，该模型可以得到精确的预测结果。然而，准确的模型参数往往难以获取。相比这四种电池模型，基于数据驱动的方法不需要任何电池退化机理的先验知识以及物理化学方程。这种模型适用性强，并可以实现实时的寿命预测。近年来，基于数据驱动的电池寿命预测方法广受研究者关注。许多基于数据驱动的锂电池寿命预测方法已被提出。然而，目前大多基于数据驱动的锂电池寿命预测方法需要对电池数据进行特征提取，而不恰当的特征选取可能会显著影响电池寿命预测的精度。同时，大多方法只能给出电池寿命的点估计，不能给出其置信区间以及电池容量达到失效阈值时的剩余寿命分布。There are currently many methods for battery life prediction, most of which are model-based and data-driven prediction methods. Existing model-based lifetime prediction methods can be classified into electrochemical model-based methods, equivalent circuit-based methods, performance-based methods, and analytical model-based methods. The electrochemical model is based on porous electrode theory and nanomechanics, which describes the battery polarization process and the physical deformation of specific components inside the battery. Equivalent Circuit Models Batteries are often modeled using equivalent circuits and a number of techniques are used to estimate model parameters. Electrochemical models and equivalent circuit models work well in some specific cases, however, due to technical, design and material reasons, these models are less applicable. Performance-based models need to establish the relationship between degradation factors and battery performance. However, considering the interaction between many environmental variables, the relationship between degradation factors and battery performance needs to be simulated and accelerated through battery simulation under different operating conditions. Life test can only be obtained. The analytical model can be established by methods such as particle filter and tasteless Kalman filter. When the model is built accurately, the model can produce accurate prediction results. However, accurate model parameters are often difficult to obtain. Compared with these four battery models, the data-driven approach does not require any prior knowledge of battery degradation mechanisms and physicochemical equations. This model has strong applicability and can realize real-time life prediction. In recent years, data-driven battery life prediction methods have attracted widespread attention from researchers. Many data-driven lithium battery life prediction methods have been proposed. However, most current data-driven lithium battery life prediction methods require feature extraction from battery data, and inappropriate feature selection may significantly affect the accuracy of battery life prediction. At the same time, most methods can only give a point estimate of battery life, but cannot give its confidence interval and the remaining life distribution when the battery capacity reaches the failure threshold.

发明内容Contents of the invention

本发明提出了一种基于函数型主成分分析(Functional Principal Component Analysis,FPCA)与贝叶斯更新的锂电池寿命预测方法。该寿命预测方法无需提取任何电池退化特征，不仅可以实现锂电池寿命点估计，并且可以给出电池的剩余寿命分布。FPCA是一种函数型数据分析的统计方法，可以捕获隐藏在原始数据中的最突出特征。同时，大量研究表明，当数据集被视为连续函数时，FPCA方法所提供的信息比传统的离散数据分析法所提供的信息更有价值。因此，本发明采用FPCA方法建立电池的非参数退化模型。此外，为了实现锂电池退化模型的实时更新，本发明引入了贝叶斯方法，通过贝叶斯方法对锂电池退化模型进行实时修正，可以得到更精确的锂电池退化模型。在此基础之上，利用参数Bootstrap方法，实现锂电池剩余寿命分布和置信区间的计算。The present invention proposes a lithium battery life prediction method based on Functional Principal Component Analysis (FPCA) and Bayesian update. This life prediction method does not need to extract any battery degradation features, not only can realize the life point estimation of lithium battery, but also can give the remaining life distribution of the battery. FPCA is a statistical method of functional data analysis that captures the most salient features hidden in raw data. At the same time, a large number of studies have shown that when the data set is regarded as a continuous function, the information provided by the FPCA method is more valuable than that provided by the traditional discrete data analysis method. Therefore, the present invention uses the FPCA method to establish a non-parametric degradation model of the battery. In addition, in order to realize the real-time update of the lithium battery degradation model, the present invention introduces a Bayesian method, and a more accurate lithium battery degradation model can be obtained by correcting the lithium battery degradation model in real time through the Bayesian method. On this basis, the parameter Bootstrap method is used to realize the calculation of the remaining life distribution and confidence interval of lithium batteries.

本发明采用的技术方案为：一种基于函数型主成分分析与贝叶斯更新的锂电池剩余寿命预测方法，该方法步骤如下：The technical solution adopted in the present invention is: a method for predicting the remaining life of a lithium battery based on functional principal component analysis and Bayesian update, the steps of which are as follows:

步骤一、利用函数型主成分分析(functional principal component analysis,FPCA)构建锂电池非参数退化模型，基于此退化模型对锂电池剩余寿命进行预测，并分析了不同数量的建模数据对预测精度的影响；Step 1. Use functional principal component analysis (FPCA) to construct a non-parametric degradation model for lithium batteries. Based on this degradation model, predict the remaining life of lithium batteries, and analyze the impact of different amounts of modeling data on the prediction accuracy. Influence;

步骤二、为了实现锂电池非参数退化模型的实时更新，通过贝叶斯方法对锂电池退化模型进行实时修正，可以得到更精确的锂电池退化模型；Step 2. In order to realize the real-time update of the lithium battery non-parametric degradation model, the Bayesian method is used to correct the lithium battery degradation model in real time, so that a more accurate lithium battery degradation model can be obtained;

步骤三、采用参数bootstrap方法计算锂电池剩余寿命分布，并计算其置信区间。Step 3, using the parameter bootstrap method to calculate the distribution of the remaining life of the lithium battery, and calculate its confidence interval.

进一步的，步骤一中，基于FPCA的锂离子电池寿命预测模型从本质上来说是非参数模型，在预测过程中利用已知的同类型样本的退化数据和预测对象的现场数据，进行退化模型训练，并基于建立好的退化模型实现锂电池的寿命预测，具体过程如下：Further, in step 1, the FPCA-based lithium-ion battery life prediction model is essentially a non-parametric model. During the prediction process, the known degradation data of the same type of samples and the field data of the predicted object are used to perform degradation model training. And based on the established degradation model, the life prediction of lithium battery is realized. The specific process is as follows:

(1)假定预测起始点为t₀，验证对象在时刻t₀以前的所有容量数据为同类型其他锂电池的容量数据为将两部分数据结合，用于建立非参数退化模型；(1) Assuming that the starting point of prediction is t₀ , all the capacity data of the verification object before time t₀ are The capacity data of other lithium batteries of the same type are Combining the two parts of data is used to establish a non-parametric degradation model;

(2)利用结合后的数据集，采用FPCA方法建立非参数退化模型，并估计均值函数u(t)，主成分得分ξ_ik以及权重函数φ_k(t_ij)；(2) Using the combined data set, use the FPCA method to establish a non-parametric degradation model, and estimate the mean function u(t), the principal component score ξ_ik and the weight function φ_k (t_ij );

(3)根据建立的退化模型，给出未来的时间序列其相应时间点的容量值可以预测，如对于时刻t_l,d，该时刻的容量值S_l(t_l,d)可以按照如下方法进行估计：(3) According to the established degradation model, the future time series is given Its capacity value at the corresponding point in time It can be predicted, for example, for time t_l,d , the capacity value S_l (t_l,d ) at this time can be estimated as follows:

$E E. [[{S S}_{l l} (({t t}_{l l,, d d}))]] = = \overset{^^}{μ μ} (({t t}_{l l,, d d})) + + {Σ Σ}_{k k = = 11}^{K K} {\overset{^^}{ξ ξ}}_{k k} * * \overset{^^}{φ φ} (({t t}_{l l,, d d})) . .$

进一步的，步骤二中，采用贝叶斯方法实现退化模型的实时更新，假设在时刻t＝{t₁,t₂,…,t_m}获得了新的观测值，t_m是观测终止时间，假设主成分得分的先验分布为正态分布，均值函数u(t)与特征函数φ_k(t)，k＝1,2,...,K都是固定的，假设锂电池退化模型为先验分布ξ_ik服从ξ_ik～N(0,λ_k)，其中ξ₁,ξ₂,...,ξ_K不相关，ε(t)服从ε(t)～N(0,σ²)，其中σ²是固定的，那么主成分得分的后验分布可以按以下公式计算：Further, in step 2, the Bayesian method is used to update the degradation model in real time, assuming that new observations are obtained at time t={t₁ ,t₂ ,…,t_m }, t_m is the observation termination time, Assuming that the prior distribution of the principal component scores is a normal distribution, the mean function u(t) and the characteristic function φ_k (t), k=1,2,...,K are fixed, and the lithium battery degradation model is assumed to be The prior distribution ξ_ik obeys ξ_ik ~N(0,λ_k ), where ξ₁ ,ξ₂ ,...,ξ_K are irrelevant, and ε(t) obeys ε(t)~N(0,σ² ) , where^σ2 is fixed, then the posterior distribution of the principal component scores can be calculated as follows:

${(({ξ ξ}_{i i 11}^{* *},, {ξ ξ}_{i i 22}^{* *},, . . . . . .,, {ξ ξ}_{iK iK}^{* *}))}^{' '} ~ ~ N N ((Cd Cd,, C C))$

其中 $C = {(\frac{1}{σ^{2}} P {(t)}^{'} P (t) + Λ^{- 1})}^{- 1}$ 且 $d = \frac{1}{σ^{2}} P {(t)}^{'} (S (t) - u (t)),$ in $C = {(\frac{1}{σ^{2}} P {(t)}^{'} P (t) + Λ^{- 1})}^{- 1}$ and $d = \frac{1}{σ^{2}} P {(t)}^{'} (S (t) - u (t)),$

S(t)＝(S(t₁),S(t₂),…,S(t_m))′，u(t)＝(u(t₁),u(t₂),…,u(t_m))′，S(t)=(S(t₁ ),S(t₂ ),…,S(t_m ))’, u(t)=(u(t₁ ),u(t₂ ),…,u( t_m ))',

Λ＝diag(λ₁,λ₂,…,λ_K)，Λ=diag(λ₁ ,λ₂ ,…,λ_K ),

利用特征值λ_k，k＝1,2,...,K，可以计算主成分得分分布，而特征值λ_kk＝1,2,...,K及特征向量φ_k，k＝1,2,...,K可通过对历史退化容量退化数据进行FPCA得到，因此，更新的主成分得分k＝1,2,...,K其后验分布可以通过矩阵C和特征向量d获得，同时，更新的均值函数可以表示为：Using eigenvalues λ_k , k=1,2,...,K, the principal component score distribution can be calculated, while eigenvalues λ_k k=1,2,...,K and eigenvectors φ_k , k=1 ,2,...,K can be obtained by performing FPCA on the historical degradation capacity degradation data, therefore, the updated principal component score The posterior distribution of k=1,2,...,K can be obtained through the matrix C and the eigenvector d, and at the same time, the updated mean function can be expressed as:

${u u}^{* *} ((t t)) = = u u ((t t)) + + {Σ Σ}_{k k = = 11}^{K K} E E. [[{ξ ξ}_{k k}^{* *}]] \times \times {φ φ}_{k k} ((t t)) . .$

进一步的，步骤三中，在退化模型后验分布更新的基础上，采用参数bootstrap方法，计算电池在给定时间t^*的RLD，并计算置信度为α的置信区间，置信区间的估计过程如下：Further, in step 3, on the basis of updating the posterior distribution of the degradation model, the parameter bootstrap method is used to calculate the RLD of the battery at a given time t^* , and calculate the confidence interval with a confidence degree of α. The estimation process of the confidence interval is as follows :

(1)从更新的函数型主成分得分正态分布中进行采样(1) Sampling from the updated normal distribution of functional principal component scores

(2)对于b的每个取值，得到一个模拟信号如下：(2) For each value of b, an analog signal is obtained as follows:

${S S}_{b b} = = u u ((t t)) + + {Σ Σ}_{k k = = 11}^{K K} {ξ ξ}_{k k}^{b b} {φ φ}_{k k} ((t t)),, k k = = 11,, . . . . . .,, K K$

(3)取T_b＝inf_t∈[0,T]{S_b(t)≤D}(3) Take T_b =inf_t∈[0,T] {S_b (t)≤D}

通过获取样本值T₁,T₂,...,T_B，可以得到锂电池的RLD，置信区间为1-α的电池剩余寿命可以通过以下公式得出：By obtaining the sample values T₁ , T₂ ,...,_TB , the RLD of the lithium battery can be obtained, and the remaining life of the battery with a confidence interval of 1-α can be obtained by the following formula:

P(L≤T′-t^*≤U|S^*(·),T′≥t^*)＝1-αP(L≤T′-t^* ≤U|S^* ( ),T′≥t^* )＝1-α

考虑到L和U的计算比较困难，采用一个近似的1-α置信区间来进行估计，其中α/2和(1-α)/2分位点用采样值T₁,T₂,...,T_B进行计算。Considering that the calculation of L and U is difficult, an approximate 1-α confidence interval is used to estimate, where The α/2 and (1-α)/2 quantiles are calculated using sampled values T₁ , T₂ ,...,_TB .

本发明的优点与积极效果在于：Advantage and positive effect of the present invention are:

(1)本发明方法将电池容量数据看做关于循环次数的连续函数，并采用函数型数据分析的方法对锂电池容量数据进行分析，相比于传统的离散型数据分析方法能够提取更多的信息，所建立的电池退化模型更为准确；(1) The method of the present invention regards the battery capacity data as a continuous function about the number of cycles, and adopts the method of functional data analysis to analyze the lithium battery capacity data, which can extract more data than the traditional discrete data analysis method information, the established battery degradation model is more accurate;

(2)本发明方法无需对锂电池数据进行特征提取，避免了不恰当的特征选取对电池寿命预测精度造成的影响；(2) The method of the present invention does not need to carry out feature extraction to the lithium battery data, avoiding the impact of inappropriate feature selection on the accuracy of battery life prediction;

(3)本发明方法充分结合了总体样本的退化信息与个体样本退化信息，所建立的退化模型具有更强的适用性；(3) The method of the present invention fully combines the degradation information of the overall sample and the degradation information of individual samples, and the degradation model established has stronger applicability;

(4)本发明方法不需要预知电池复杂的退化机理及建立精确的数学模型；(4) The method of the present invention does not need to predict the complex degradation mechanism of the battery and establish an accurate mathematical model;

(5)本发明方法不仅可以给出锂电池剩余寿命的点估计，同时可以给出锂电池剩余寿命分布及置信区间；(5) The method of the present invention can not only provide a point estimate of the remaining life of the lithium battery, but also provide the distribution and confidence interval of the remaining life of the lithium battery;

附图说明Description of drawings

图1为4个锂电池的容量退化曲线；Fig. 1 is the capacity degradation curve of 4 lithium batteries;

图2为用非参数估计法估计的均函数与特征函数示意图，其中，图(a)为估计的均函数图(b)为估计的特征函数φ₁(t)；图(c)为估计的特征函数φ₂(t)；Figure 2 is a schematic diagram of the mean function and characteristic function estimated by the non-parametric estimation method, in which, figure (a) is the estimated mean function Figure (b) is the estimated characteristic function φ₁ (t); Figure (c) is the estimated characteristic function φ₂ (t);

图3为电池#5，#6，#7的实际测量值和重构数据图，其中，(a)为电池#5的实际测量值和重构数据图，(b)为电池#6的实际测量值和重构数据图，(c)为电池#7的实际测量值和重构数据图；Figure 3 is the actual measured value and reconstructed data diagram of batteries #5, #6, and #7, where (a) is the actual measured value and reconstructed data diagram of battery #5, and (b) is the actual measured value and reconstructed data diagram of battery #6. Measured value and reconstructed data diagram, (c) is the actual measured value and reconstructed data diagram of battery #7;

图4为电池#18的容量预测结果示意图；Fig. 4 is a schematic diagram of the capacity prediction result of battery #18;

图5为电池#5的容量预测结果示意图；Fig. 5 is a schematic diagram of the capacity prediction result of battery #5;

图6为电池#6的容量预测结果示意图；Fig. 6 is a schematic diagram of the capacity prediction result of battery #6;

图7为现场数据量不同时电池#5的预测结果示意图，其中，图(a)现场数据为第1～30循环容量数据，图(b)现场数据为第1～60循环容量数据，图(c)现场数据为1～90循环容量数据，图(d)现场数据为1～120循环容量数据；Figure 7 is a schematic diagram of the prediction results of battery #5 when the amount of on-site data is different, where the on-site data in figure (a) is the 1st to 30th cycle capacity data, and the figure (b) is the 1st to 60th cycle capacity data, and the figure ( c) The on-site data is 1-90 cycle capacity data, and the field data in Figure (d) is 1-120 cycle capacity data;

图8为现场数据不同时的电池预测RMSE和预测误差值，其中，图(a)为电池预测RMSE值，图(b)为电池预测误差值；Figure 8 shows the battery prediction RMSE and prediction error values when the field data is different, where, Figure (a) is the battery prediction RMSE value, and Figure (b) is the battery prediction error value;

图9为电池退化模型在第100周期、第110周期、第120周期进行更新预测的RLD值和经验累积分布函数示意图，其中，图(a)、图(b)分别为电池退化模型在第100周期进行更新预测的RLD值和经验累积分布函数，图(c)、图(d)分别为电池退化模型在第110周期进行更新预测的RLD值和经验累积分布函数，图(e)、图(f)分别为电池退化模型在第120周期进行更新预测的RLD值和经验累积分布函数；Figure 9 is a schematic diagram of the RLD value and the empirical cumulative distribution function of the update prediction of the battery degradation model in the 100th cycle, the 110th cycle, and the 120th cycle. The RLD value and empirical cumulative distribution function of the periodic update prediction, Figure (c) and Figure (d) are the RLD value and empirical cumulative distribution function of the battery degradation model updated and predicted in the 110th cycle, Figure (e), Figure ( f) are the RLD value and empirical cumulative distribution function of the battery degradation model updated and predicted in the 120th cycle, respectively;

图10为不同预测起始点的估计正态密度函数示意图。Figure 10 is a schematic diagram of the estimated normal density function for different prediction starting points.

图11为一种基于函数型主成分分析与贝叶斯更新的锂电池剩余寿命预测方法的流程图示意图。FIG. 11 is a schematic flowchart of a method for predicting the remaining life of a lithium battery based on functional principal component analysis and Bayesian updating.

具体实施方式Detailed ways

下面结合附图以及具体实施例进一步说明本发明。The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments.

1.本发明提供的一种基于函数型主成分分析与贝叶斯更新的锂电池剩余寿命预测方法，具体步骤如下：1. A method for predicting the remaining life of a lithium battery based on functional principal component analysis and Bayesian updating provided by the present invention, the specific steps are as follows:

1.1基于FPCA的的锂离子电池退化模型1.1 FPCA-based lithium-ion battery degradation model

1.1.1容量退化模型1.1.1 Capacity degradation model

锂离子电池性能下降过程可视为一个潜在的随机过程。同时，电池的寿命周期可看作关于时间的函数。首先利用现有的容量数据建立锂电池退化模型以实现电池的寿命预测。由于函数型主成分分析(FPCA)是传统的主成分分析(PCA)的扩展，可以提供更稳定的估计值并避免引入高维协方差矩阵，同时，针对给定过程的不同数据集，FPCA能够提取包含他们共同特征的更多信息。因此本发明采用FPCA构建锂电池退化模型。The performance degradation process of lithium-ion batteries can be regarded as a potentially stochastic process. At the same time, the life cycle of the battery can be viewed as a function of time. Firstly, the lithium battery degradation model is established by using the existing capacity data to realize the life prediction of the battery. Since functional principal component analysis (FPCA) is an extension of traditional principal component analysis (PCA), it can provide more stable estimates and avoid the introduction of high-dimensional covariance matrix. At the same time, for different data sets of a given process, FPCA can extract More information on their common characteristics. Therefore, the present invention uses FPCA to construct a lithium battery degradation model.

令S_i(t_ij)为第i块电池在时间的容量观测值，tij是第i块电池的循环周期。考虑到观测值的观测误差，S_i(t_ij)可以用以下方式表达：Let S_i (t_ij ) be the capacity observation of the i-th battery at time, and tij be the cycle period of the i-th battery. Considering the observation error of the observed value, S_i (t_ij ) can be expressed in the following way:

S_i(t_ij)＝X_i(t_ij)+ε_ij(t),0≤t_ij≤T；j＝1,2,...,m_iS_i (t_ij )=X_i (t_ij )+ε_ij (t),0≤t_ij ≤T; j=1,2,...,m_i

X_ij是原始信号，u(t)为均值函数，cov(X(s),X(t))＝G(s,t)为协方差函数，T为最长失效时间，m_i为第i块电池的观测点数。假设ε_ij为独立同分布的随机误差，服从标准正态分布N(0,σ²)。G(s,t)可以依据特征向量φ_k和特征值λ_k按照以下公式表达：X_ij is the original signal, u(t) is the mean function, cov(X(s),X(t))=G(s,t) is the covariance function, T is the longest failure time, m_i is the i-th The number of observation points for a battery. Assume that ε_ij is an independent and identically distributed random error, which obeys the standard normal distribution N(0,σ² ). G(s,t) can be expressed according to the following formula according to the eigenvector φ_k and eigenvalue λ_k :

G(s,t)＝∑λ_kφ_k(s)φ_k(t),0≤s,t≤TG(s,t)＝∑λ_k φ_k (s)φ_k (t),0≤s,t≤T

在经典FPCA法中，我们假设第i条曲线X_i(t_ij)可以按照如下公式描述：In the classic FPCA method, we assume that the i-th curve_Xi (t_ij ) can be described by the following formula:

X_i(t_ij)＝u(t_ij)+∑ξ_ikφ_k(t_ij)X_i (t_ij )＝u(t_ij )+∑ξ_ik φ_k (t_ij )

其中u(t_ij)为平滑函数，φ_k(t_ij)为第k个特征函数。ξ_ik为主成分得分，并具有以下性质：Among them, u(t_ij ) is a smooth function, and φ_k (t_ij ) is the kth characteristic function._ξik is the principal component score and has the following properties:

E[ξ_ik]＝0； $E [ξ_{ik}^{2}] = λ_{k}$ E[ξ_ik ]=0; $E. [ξ_{ik}^{2}] = λ_{k}$

在实际应用中，经常选择前K个主成分表示X_i(t_ij)。因此X_i(t_ij)可以表示为：In practical applications, the first K principal components are often selected to represent_Xi (t_ij ). Therefore X_i (t_ij ) can be expressed as:

${X x}_{i i} (({t t}_{ij ij})) = = u u (({t t}_{ij ij})) + + {Σ Σ}_{k k = = 11}^{K K} {ξ ξ}_{ik ik} {φ φ}_{k k} (({t t}_{ij ij}))$

因此，锂离子电池的退化模型可以表示为：Therefore, the degradation model of lithium-ion batteries can be expressed as:

${S S}_{i i} (({t t}_{ij ij})) = = u u (({t t}_{ij ij})) + + {Σ Σ}_{k k = = 11}^{K K} {ξ ξ}_{ik ik} {φ φ}_{k k} (({t t}_{ij ij})) + + {ϵ ϵ}_{ij ij} ((t t)),, 00 \leq \leq {t t}_{ij ij} \leq \leq T T;; j j = = 1,2 1,2,, . . . . . .,, {m m}_{i i}$

1.1.2锂离子退化模型的非参数估计1.1.2 Non-parametric estimation of lithium-ion degradation model

大多数电池剩余寿命预测模型都是参数模型，需要对其参数形式进行假设。在该建模方法中，假设均值函数u(t)和协方差函数cov(X(s),X(t))是非参数化的，即我们对其参数形式不做任何假设。本部分内容主要u(t)和协方差函数cov(X(s),X(t))的估计法。Most remaining battery life prediction models are parametric models that require assumptions about their parametric form. In this modeling method, it is assumed that the mean function u(t) and the covariance function cov(X(s),X(t)) are non-parametric, that is, we do not make any assumptions about their parametric forms. This part mainly focuses on the estimation method of u(t) and covariance function cov(X(s),X(t)).

(1)均值函数估计(1) Mean function estimation

我们利用核估计方法估计均值函数u(t_ij)，利用所有个体合并后的数据来估计均值函数u(t_ij)，即求解下面这个最小化问题：We use the kernel estimation method to estimate the mean function u(t_ij ), and use the combined data of all individuals to estimate the mean function u(t_ij ), that is, to solve the following minimization problem:

$min min {Σ Σ}_{i i = = 11}^{n no} {Σ Σ}_{j j = = 11}^{{n no}_{i i}} {k k}_{11} ((\frac{{T T}_{ij ij} - - t t}{{h h}_{u u}})) {{{S S}_{ij ij} - - {β β}_{00} - - {β β}_{11} (({t t - - T T}_{ij ij}))}}^{22}$

其中为核函数，通常选为高斯核函数，h_u表示估计u(t_ij)所选用的窗宽，则u(t)的估计值为 $\hat{u} (t) = {\hat{β}}_{0} (t) .$ in is the kernel function, usually selected as Gaussian kernel function, h_u represents the window width selected for estimating u(t_ij ), then the estimated value of u(t) is $\hat{u} (t) = {\hat{β}}_{0} (t) .$

$\overset{^^}{μ μ} ((t t)) = = {\overset{^^}{β β}}_{00} ((t t)) = = \frac{{J J}_{00} {I I}_{11} - - {J J}_{11} {I I}_{22}}{{I I}_{11}^{22} - - {I I}_{00} {I I}_{22}}$

其中：in:

${I I}_{m m} = = {Σ Σ}_{i i = = 11}^{n no} {Σ Σ}_{j j = = 11}^{{n no}_{i i}} {k k}_{11} ((\frac{{T T}_{ij ij} - - t t}{{h h}_{μ μ}})) {(({t t - - T T}_{ij ij}))}^{P P},, p p = = 0,1,2 0,1,2$

${J J}_{m m} = = {Σ Σ}_{i i = = 11}^{n no} {Σ Σ}_{j j = = 11}^{{n no}_{i i}} {k k}_{11} ((\frac{{T T}_{ij ij} - - t t}{{h h}_{μ μ}})) {Y Y}_{ij ij} {((t t - - {T T}_{ij ij}))}^{P P},, p p = = 0,1 0,1$

核函数估计法对均值函数的平滑估计效果的好坏与选取的窗宽大小有关，我们可以利用leave-out-one-curve的交叉核实来挑选合适的窗宽。通常情况下，根据经验决定的窗宽大小就可以保证理想的效果。The effect of the kernel function estimation method on the smooth estimation of the mean function is related to the size of the selected window width. We can use the leave-out-one-curve cross-validation to select an appropriate window width. Usually, the window width determined by experience can guarantee the ideal effect.

(2)协方差函数的估计(2) Estimation of covariance function

假设协方差函数C_X(s,t)＝cov(X(s),X(t))＝G(s,t),0≤s,t≤T。注意到cov(S_ij,S_il|T_ij,T_il)＝cov(X(T_ij),X(T_il))+δ_jlσ²，若j＝l则δ_jl＝1，反之δ_jl＝0。令则E[G_i(T_ij,T_il)|T_ij,T_il]≈cov(X(T_ij),X(T_il))+δ_jlσ²。只有当j≠l时，G_i(T_ij,T_il)才可以被用来估计协方差函数。当s≠t时，G(s,t)可通过下面最小化问题得到：Suppose the covariance function C_X (s, t) = cov (X (s), X (t)) = G (s, t), 0≤s, t≤T. Note that cov(S_ij ,S_il |T_ij ,T_il )=cov(X(T_ij ),X(T_il ))+δ_jl σ² , if j=l then δ_jl =1, otherwise δ_jl =0. make Then E[G_i (T_ij ,T_il )|T_ij ,T_il ]≈cov(X(T_ij ),X(T_il ))+δ_jl σ² . Only when j≠l, G_i (T_ij ,T_il ) can be used to estimate the covariance function. When s≠t, G(s,t) can be obtained by the following minimization problem:

$min min {Σ Σ}_{i i = = 11}^{n no} \underset{11 \leq \leq j j &NotEqual; &NotEqual; l l \leq \leq {N N}_{i i}}{Σ Σ} {k k}_{22} ((\frac{{T T}_{ij ij} - - s the s}{{h h}_{G G}},, \frac{{T T}_{it it} - - t t}{{h h}_{G G}})) {{{G G}_{i i} (({T T}_{ij ij},, {T T}_{il il})) - - {β β}_{00} - - {β β}_{1111} ((s the s - - {T T}_{ij ij})) - - {β β}_{1212} (({t t - - T T}_{il il}))}}^{22}$

其中表示某种二元核函数，h_G表示估计G(s,t)所用的窗宽。因此，对于s≠t，G(s,t)的估计值为：in Represents a binary kernel function, h_G represents the window width used to estimate G(s,t). Therefore, for s≠t, the estimate of G(s,t) is:

$\overset{^^}{G G} ((s the s,, t t)) = = {\overset{^^}{β β}}_{00} ((s the s,, t t)) = = \frac{(({J J}_{0000} {I I}_{1111} - - {J J}_{1010} {I I}_{0101})) (({J J}_{2020} {I I}_{0202} - - {I I}_{1111}^{22})) - - (({J J}_{1010} {I I}_{0202} - - {J J}_{0101} {I I}_{1111})) (({J J}_{1010} {I I}_{1111} - - {I I}_{2020} {I I}_{0101}))}{(({J J}_{0000} {I I}_{1111} - - {I I}_{1010} {I I}_{0101})) (({J J}_{2020} {I I}_{0202} - - {I I}_{1111}^{22})) - - (({J J}_{1010} {I I}_{0202} - - {I I}_{0101} {I I}_{1111})) (({I I}_{1010} {I I}_{1111} - - {I I}_{2020} {I I}_{0101}))}$

其中：in:

${I I}_{pq pq} = = {Σ Σ}_{i i = = 11}^{n no} {Σ Σ}_{11 \leq \leq j j &NotEqual; &NotEqual; 11 \leq \leq {n no}_{i i}}^{{n no}_{i i}} {k k}_{22} ((\frac{{T T}_{ij ij} - - t t}{{h h}_{G G}},, \frac{{T T}_{il il} - - t t}{{h h}_{G G}})) {(({s the s - - T T}_{ij ij}))}^{P P} {(({t t - - T T}_{il il}))}^{q q},, p p,, q q = = 0,1,2 0,1,2$

${J J}_{pq pq} = = {Σ Σ}_{i i = = 11}^{n no} {Σ Σ}_{11 \leq \leq j j &NotEqual; &NotEqual; 11 \leq \leq {n no}_{i i}}^{{n no}_{i i}} {k k}_{22} ((\frac{{T T}_{ij ij} - - t t}{{h h}_{G G}},, \frac{{T T}_{il il} - - t t}{{h h}_{G G}})) {G G}_{i i} (({T T}_{ij ij},, {T T}_{il il})) {(({s the s - - T T}_{ij ij}))}^{P P} {(({t t - - T T}_{il il}))}^{q q},, p p,, q q = = 0,1 0,1$

同样，窗宽h_G可利用leave-out-one-curve的交叉核实来挑选合适值，以获得更好的估计效果。Similarly, the window width h_G can use leave-out-one-curve cross-validation to select an appropriate value to obtain better estimation results.

以上的方法是对协方差函数在s≠t情况下进行估计的。因为协方差函数在对角线方向上达到最大，且在垂直对角线方向上使用局部二次拟合比局部线性拟合出的近似形状要好，所以在对角线方向上采用局部线性拟合，垂直对角线方向使用二次拟合。注意在具体实施拟合的时候，需将坐标轴顺时针旋转π/4,然后再求解最小化问题：The above method is to estimate the covariance function in the case of s≠t. Because the covariance function reaches the maximum in the diagonal direction, and the approximate shape obtained by using a local quadratic fit in the vertical diagonal direction is better than a local linear fit, a local linear fit is used in the diagonal direction , using a quadratic fit in the vertical diagonal direction. Note that when implementing the fitting, you need to rotate the coordinate axis clockwise by π/4, and then solve the minimization problem:

$\underset{{γ γ}_{00},, {γ γ}_{11},, {γ γ}_{22}}{min min} {Σ Σ}_{i i = = 11}^{n no} \underset{11 \leq \leq j j &NotEqual; &NotEqual; l l \leq \leq {N N}_{i i}}{Σ Σ} {k k}_{22} ((\frac{{T T}_{ij ij}^{* *} - - s the s}{{h h}_{G G}},, \frac{{T T}_{il il}^{* *} - - t t}{{h h}_{G G}})) {{{G G}_{i i} (({T T}_{ij ij}^{* *},, {T T}_{il il}^{* *})) - - {γ γ}_{00} - - {γ γ}_{11} (({s the s - - T T}_{ij ij}^{* *})) - - {γ γ}_{22} (({t t - - T T}_{il il}^{* *}))}}^{22}$

其中：in:

$[\begin{matrix} {T T}_{ij ij}^{* *} \\ {T T}_{il il}^{* *} \end{matrix}] = = [\begin{matrix} \sqrt{22} / / 22 & 22 \sqrt{22} / / 22 \\ - - \sqrt{22} / / 22 & \sqrt{22} / / 22 \end{matrix}] [\begin{matrix} {T T}_{ij ij} \\ {T T}_{il il} \end{matrix}]$

$[\begin{matrix} {S S}_{ij ij}^{* *} \\ {S S}_{ik ik}^{* *} \end{matrix}] = = [\begin{matrix} \sqrt{22} / / 22 & 22 \sqrt{22} / / 22 \\ - - \sqrt{22} / / 22 & \sqrt{22} / / 22 \end{matrix}] [\begin{matrix} {S S}_{ij ij} \\ {S S}_{ik ik} \end{matrix}]$

${G G}_{i i} (({T T}_{ij ij}^{* *},, {T T}_{il il}^{* *})) = = (({S S}_{ij ij}^{* *} - - {\overset{^^}{μ μ}}^{* *} (({T T}_{ij ij}^{* *})))) (({S S}_{ij ij}^{* *} - - {\overset{^^}{μ μ}}^{* *} (({T T}_{il il}^{* *}))))$

$\overset{^^}{μ μ} ((t t)) = = \frac{{J J}_{00} {I I}_{11} - - {J J}_{11} {I I}_{22}}{{I I}_{11}^{22} - - {I I}_{00} {I I}_{22}}$

${I I}_{m m} = = {Σ Σ}_{i i = = 11}^{n no} {Σ Σ}_{j j = = 11}^{{n no}_{i i}} {k k}_{22} ((\frac{{T T}_{ij ij}^{* *} - - t t}{{h h}_{μ μ}})) {(({t t - - T T}_{ij ij}^{* *}))}^{P P},, p p = = 0,1,2 0,1,2$

${I I}_{m m} = = {Σ Σ}_{i i = = 11}^{n no} {Σ Σ}_{j j = = 11}^{{n no}_{i i}} {k k}_{22} ((\frac{{T T}_{ij ij}^{* *} - - t t}{{h h}_{μ μ}})) {{Y Y}_{ij ij}^{* *} (({t t - - T T}_{ij ij}^{* *}))}^{P P},, p p = = 0,1 0,1$

那么得到：then get:

$\overset{&OverBar; &OverBar;}{G G} ((s the s,, t t)) = = {\overset{^^}{γ γ}}_{00} ((s the s,, t t)),, \overset{^^}{G G} ((t t,, t t)) = = \overset{&OverBar; &OverBar;}{G G} ((00,, t t / / \sqrt{22}))$

显然此时不包含随机误差的信息，G_i(T_ij,T_il)作为原始数据，利用局部线性拟合，得到显然不仅仅包含协方差函数G(t,t)，还包含随机误差的方差σ²，可以借助来得到方差σ²的估计。为了消除边界效应,我们去除区间的两个端点。令|T|＝T，T₁＝[inf{x:x∈T}+|T|/4,sup{x:x∈T}-|T|/4]，则Obviously at this time Does not contain random error information, G_i (T_ij ,T_il ) as the original data, using local linear fitting, get obviously It not only contains the covariance function G(t,t), but also contains the variance σ² of the random error, which can be used to get an estimate of the variance σ² . To eliminate boundary effects, we remove the two endpoints of the interval. Let |T|=T, T₁ ＝[inf{x:x∈T}+|T|/4, sup{x:x∈T}-|T|/4], then

对于本征函数和特征根我们通过求解特征方程：For the eigenfunction and characteristic roots We do this by solving the characteristic equation:

${&Integral; &Integral;}_{T T} \overset{^^}{G G} ((s the s,, t t)) {\overset{^^}{φ φ}}_{k k} ((s the s)) ds ds = = {\overset{^^}{λ λ}}_{k k} {\overset{^^}{φ φ}}_{k k} ((t t)),, 00 \leq \leq t t \leq \leq T T$

其中 $\underset{T}{&Integral;} {\hat{φ}}_{k} {(t)}^{2} dt = 1,$ 且 $\underset{T}{&Integral;} {\hat{φ}}_{k} (t) {\hat{φ}}_{m} (t) dt = 0, k &NotEqual; m .$ 我们先将离散化，再用多元分析的方法得到in $\underset{T}{&Integral;} {\hat{φ}}_{k} {(t)}^{2} dt = 1,$ and $\underset{T}{&Integral;} {\hat{φ}}_{k} (t) {\hat{φ}}_{m} (t) dt = 0, k &NotEqual; m .$ We will first Discretization, and then use the method of multivariate analysis to get

1.2基于退化模型的容量预测1.2 Capacity prediction based on degradation model

基于FPCA的锂离子电池寿命预测模型从本质上来说是非参数模型，在预测过程中利用已知的同类型样本的退化数据和预测对象的现场数据，进行退化模型训练，过程如下：The FPCA-based lithium-ion battery life prediction model is essentially a non-parametric model. In the prediction process, the known degradation data of the same type of samples and the field data of the predicted object are used to train the degradation model. The process is as follows:

(1)假定预测起始点为t₀，验证对象在时刻t₀以前的所有容量数据为同类型其他锂电池的容量数据为将两部分数据结合，用于建立非参数退化模型。(1) Assuming that the starting point of prediction is t₀ , all the capacity data of the verification object before time t₀ are The capacity data of other lithium batteries of the same type are The two parts of data are combined to build a non-parametric degradation model.

(2)利用结合后的数据集，采用FPCA方法建立非参数退化模型，并估计均值函数u(t)，主成分得分ξ_ik以及权重函数φ_k(t_ij)。(2) Using the combined data sets, use the FPCA method to establish a non-parametric degradation model, and estimate the mean function u(t), the principal component score ξ_ik and the weight function φ_k (t_ij ).

(3)根据建立的退化模型，给出未来的时间序列其相应时间点的容量值可以预测。如对于时刻t_l,d，该时刻的容量值S_l(t_l,d)可以按照如下方法进行估计：(3) According to the established degradation model, the future time series is given Its capacity value at the corresponding point in time predictable. For example, for time t_l,d , the capacity value S_l (t_l,d ) at this time can be estimated as follows:

1.3基于贝叶斯更新的剩余寿命估计1.3 Remaining life estimation based on Bayesian update

1.3.1退化模型更新1.3.1 Degradation model update

为了实现退化模型的实时更新，本发明提出了贝叶斯方法。假设在时刻t＝{t₁,t₂,…,t_m}获得了新的观测值，t_m是观测终止时间。我们假设主成分得分的先验分布为正态分布，均值函数u(t)与特征函数φ_k(t)，k＝1,2,...,K都是固定的。假设锂电池退化模型为先验分布ξ_ik服从ξ_ik～N(0,λ_k)，其中ξ₁,ξ₂,...,ξ_K不相关，ε(t)服从ε(t)～N(0,σ²)，其中σ²是固定的。那么主成分得分的后验分布可以按以下公式计算：In order to realize the real-time update of the degradation model, the present invention proposes a Bayesian method. Assume that a new observation value is obtained at time t={t₁ ,t₂ ,...,t_m }, where t_m is the observation termination time. We assume that the prior distribution of the principal component scores is a normal distribution, the mean function u(t) and the characteristic function φ_k (t), k=1,2,...,K are all fixed. Suppose the lithium battery degradation model is The prior distribution ξ_ik obeys ξ_ik ~N(0,λ_k ), where ξ₁ ,ξ₂ ,...,ξ_K are irrelevant, and ε(t) obeys ε(t)~N(0,σ² ) , where^σ2 is fixed. Then the posterior distribution of the principal component scores can be calculated as follows:

Λ＝diag(λ₁,λ₂,…,λ_K)，Λ=diag(λ₁ ,λ₂ ,…,λ_K ),

利用特征值λ_k，k＝1,2,...,K，可以计算主成分得分分布，而特征值λ_kk＝1,2,...,K及特征向量φ_k，k＝1,2,...,K可通过对历史退化容量退化数据进行FPCA得到。因此，更新的主成分得分k＝1,2,...,K其后验分布可以通过矩阵C和特征向量d获得。同时，更新的均值函数可以表示为：Using eigenvalues λ_k , k=1,2,...,K, the principal component score distribution can be calculated, while eigenvalues λ_k k=1,2,...,K and eigenvectors φ_k , k=1 ,2,...,K can be obtained by performing FPCA on the historical degradation capacity degradation data. Therefore, the updated principal component scores The posterior distribution of k=1,2,...,K can be obtained through matrix C and eigenvector d. Meanwhile, the updated mean function can be expressed as:

${u u}^{* *} ((t t)) = = u u ((t t)) + + {Σ Σ}_{k k = = 11}^{K K} E E. [[{ξ ξ}_{k k}^{* *}]] \times \times {φ φ}_{k k} ((t t))$

1.3.2剩余寿命分布估计1.3.2 Estimation of remaining life distribution

大多数锂电池寿命预测方法仅能提供一个点估计。然而，在实际应用中锂电池的RLD更为重要。本部分发明内容在退化模型后验分布更新的基础上，提出了一个基于参数bootstrap方法，计算电池在给定时间t^*的RLD，并计算置信度为α的置信区间。置信区间的估计过程如下：Most lithium battery life prediction methods can only provide a point estimate. However, the RLD of lithium batteries is more important in practical applications. In this part of the invention, based on the update of the posterior distribution of the degradation model, a parameter-based bootstrap method is proposed to calculate the RLD of the battery at a given time t^* , and calculate the confidence interval with a confidence degree of α. The process of estimating the confidence interval is as follows:

(3)取T_b＝inf_t∈[0,T]{S_b(t)≤D}(3) Take T_b =inf_t∈[0,T] {S_b (t)≤D}

考虑到L和U的计算比较困难，我们采用一个近似的1-α置信区间来进行估计，其中α/2和(1-α)/2分位点用采样值T₁,T₂,...,T_B进行计算。Considering that the calculation of L and U is difficult, we use an approximate 1-α confidence interval to estimate, where The α/2 and (1-α)/2 quantiles are calculated using sampled values T₁ , T₂ ,...,_TB .

2.案例分析2. Case Analysis

2.1 NASA锂离子电池实验数据的描述2.1 Description of NASA lithium-ion battery experiment data

本研究采用NASA PCoE研究中心的锂离子电池寿命循环实验数据。本研究选择电池#5，#6，#7和#18验证该方法的有效性。如表1，AT，CC，DC，EOD和EOLC分别代表环境温度、充电电流、放电电流、放电终止电压、放电终止准则。This study uses the data from the life cycle experiments of lithium-ion batteries from the NASA PCoE Research Center. This study selected batteries #5, #6, #7 and #18 to verify the effectiveness of the method. As shown in Table 1, AT, CC, DC, EOD and EOLC respectively represent ambient temperature, charging current, discharging current, discharge termination voltage and discharge termination criterion.

锂离子电池首先以1.5A恒流充电，当电池电压达到4.2V之后，继续以恒压充电直到充电电流降到20mA。放电阶段锂电池以2A恒流放电直到电池电压达到EOD时实验停止。在该实验中，EOLC指的是电池最大充电电量(MCC)降到额定容量的70％。因此，锂电池失效阀值是1.38Ah。图1显示了4个锂离子电池的容量退化曲线。考虑到电池#7的容量总是大于预设的故障阀值1.38Ah，因此在后续验证中不采用电池#7作为验证对象。The lithium-ion battery is first charged with a constant current of 1.5A. When the battery voltage reaches 4.2V, it continues to be charged with a constant voltage until the charging current drops to 20mA. During the discharge phase, the lithium battery was discharged at a constant current of 2A until the battery voltage reached EOD and the experiment was stopped. In this experiment, EOLC refers to the maximum charge capacity (MCC) of the battery falling to 70% of the rated capacity. Therefore, the lithium battery failure threshold is 1.38Ah. Figure 1 shows the capacity degradation curves of four Li-ion batteries. Considering that the capacity of battery #7 is always greater than the preset fault threshold of 1.38Ah, battery #7 is not used as the verification object in the subsequent verification.

表1不同工况下的典型数据Table 1 Typical data under different working conditions

2.2非参数退化模型2.2 Non-parametric degradation model

我们把电池#18作为验证对象，并假定电池#18的现场数据是前70个循环的容量数据。然后，我们利用电池#5,#6,#7的全部容量退化数据和电池#18的现场数据构建非参数退化模型，并用非参数估计法估计模型参数。所估计均函数与特征函数如图2所示。We take battery #18 as the verification object, and assume that the field data of battery #18 is the capacity data of the first 70 cycles. Then, we construct a non-parametric degradation model using the full capacity degradation data of batteries #5, #6, and #7 and field data of battery #18, and estimate the model parameters with a non-parametric estimation method. The estimated mean function and characteristic function are shown in Figure 2.

为了验证锂电池退化模型的准确性，利用估计的均函数主成分得分{ξ₁,ξ₂}、特征函数φ₁(t)、φ₂(t)重构电池退化数据，并用均方根误差验证退化模型的准确性。图3显示了电池#5,#6,#7的实际容量曲线与重构曲线。表2中给出了相应的电池重构RMSE值。从表中可以看出，RMSE值都很小，这也证明了非参数退化模型的高度准确性。In order to verify the accuracy of the lithium battery degradation model, the estimated mean function Principal component scores {ξ₁ ,ξ₂ }, characteristic functions φ₁ (t), φ₂ (t) reconstruct battery degradation data, and use root mean square error to verify the accuracy of the degradation model. Figure 3 shows the actual capacity curves and reconstruction curves of batteries #5, #6, and #7. The corresponding battery reconstitution RMSE values are given in Table 2. It can be seen from the table that the RMSE values are all small, which also proves the high accuracy of the non-parametric degradation model.

表2电池#5，#6，#7重构RMSETable 2 Reconstructed RMSE of batteries #5, #6, #7

2.3基于退化模型的容量预测2.3 Capacity prediction based on degradation model

在建立非参数退化模型之后，选取电池#18作为验证对象，预测其未来循环的最大充电电量及置信区间。图4为电池#18的预测结果，置信度为1-α＝0.95。After establishing the non-parametric degradation model, battery #18 was selected as the verification object, and the maximum charging capacity and confidence interval of its future cycle were predicted. Fig. 4 is the prediction result of battery #18, the confidence level is 1-α=0.95.

同样，将电池#5和电池#6作为验证对象。现场数据假设为电池的前70个循环的容量数据。利用验证对象现场数据和其他电池的所有能量数据构建电池的退化模型。图5和图6为容量预测结果，置信区间为0.5到0.95。Likewise, battery #5 and battery #6 are used as objects of verification. Field data is assumed to be capacity data for the first 70 cycles of the battery. A degradation model of the battery is constructed using the field data of the verification object and all the energy data of other batteries. Figure 5 and Figure 6 are the capacity prediction results, the confidence interval is 0.5 to 0.95.

2.4基于给定失效阀值的锂电池寿命预测2.4 Lithium battery life prediction based on a given failure threshold

基于所估计的锂电池最大放电电量值，给定失效阈值，可以得到锂电池剩余使用寿命。在本研究中，故障阈值为1.38Ah，即额定容量的70％。同时，锂电池寿命置信区间设为1-α＝0.95。分别采用第1～10循环，第1～20循环……第1～110循环，第1～120循环的容量退化数据作为现场数据对锂电池寿命进行预测，以验证不同数量的现场数据对电池退化模型准确性的影响。本研究采用电池#5作为验证对象。图7显示了利用电池#5第1～30，第1～60，第1～90，第1～120循环的容量数据作为现场数据的预测结果。电池容量RMSE值及预测误差用来估计本研究所提方法的准确性，如表3和图8所示。Based on the estimated maximum discharge capacity of the lithium battery and given the failure threshold, the remaining service life of the lithium battery can be obtained. In this study, the failure threshold is 1.38Ah, which is 70% of the rated capacity. At the same time, the confidence interval of lithium battery life is set to 1-α=0.95. The capacity degradation data of the 1st to 10th cycle, the 1st to 20th cycle, the 1st to 110th cycle, and the 1st to 120th cycle are used as field data to predict the life of lithium batteries, so as to verify the impact of different quantities of field data on battery degradation. impact on model accuracy. In this study, battery #5 was used as the verification object. Fig. 7 shows the prediction results using the capacity data of the 1st to 30th, 1st to 60th, 1st to 90th, and 1st to 120th cycles of battery #5 as field data. The battery capacity RMSE value and prediction error are used to estimate the accuracy of the method proposed in this study, as shown in Table 3 and Figure 8.

表3现场数据量不同时电池#5的预测结果Table 3 Prediction results of battery #5 when the amount of on-site data is different

如表3和图8所示，随着现场数据数量的增长，容量数据的RMSE值表现出下降的趋势，预测寿命越来越接近于实际寿命，利用更多的现场数据训练模型可以使非参数退化模型有着更高的准确性。从表3中可以看出，最佳的预测寿命结果为第133循环，最小预测误差为6个循环。As shown in Table 3 and Figure 8, with the increase of the field data, the RMSE value of the capacity data shows a downward trend, and the predicted life is getting closer to the actual life. Using more field data to train the model can make the non-parametric The degradation model has higher accuracy. It can be seen from Table 3 that the best predicted life result is the 133rd cycle, and the minimum prediction error is 6 cycles.

2.5基于贝叶斯更新的RLD计算2.5 RLD calculation based on Bayesian update

本部分内容采用经验贝叶斯方法实现锂电池退化模型实时更新，并用bootstrap方法计算锂离子电池的剩余寿命分布与置信区间。该部分内容仍采用电池#5作为验证对象。假设电池#5的现场数据为前80个周期的容量数据，用现场数据及其他电池的容量退化数据建立电池退化模型。Bootstrap方法的采样次数设为B＝1000。每当获得新的容量数据，便对电池退化模型中的参数进行更新，从而得到更新的锂电池剩余寿命分布。从第81到第120循环，每测得一个新数据便执行一次更新，总计进行40次。图9显示了电池退化模型在第100、110、120循环进行更新所得到的剩余寿命分布及相应的经验累积分布函数(CDF)，图中曲线为估计的正态密度函数。为了便于比较，将三条曲线放入一个图中，如图10所示。This part uses the empirical Bayesian method to update the lithium battery degradation model in real time, and uses the bootstrap method to calculate the remaining life distribution and confidence interval of the lithium-ion battery. This part of the content still uses battery #5 as the verification object. Assuming that the field data of battery #5 is the capacity data of the first 80 cycles, the battery degradation model is established using the field data and the capacity degradation data of other batteries. The sampling frequency of the Bootstrap method is set to B=1000. Whenever new capacity data is obtained, the parameters in the battery degradation model are updated to obtain an updated remaining life distribution of lithium batteries. From the 81st to the 120th cycle, an update is performed every time a new data is measured, and a total of 40 times are performed. Figure 9 shows the remaining life distribution and the corresponding empirical cumulative distribution function (CDF) obtained by updating the battery degradation model at the 100th, 110th, and 120th cycle. The curve in the figure is the estimated normal density function. For ease of comparison, the three curves are put into one plot, as shown in Figure 10.

表4基于贝叶斯更新的预测结果Table 4 Prediction results based on Bayesian update

从图9和图10可以看出，随着更新时间点的后移，RLD的中心越来越接近实际的电池寿命。同时，时间轴上的分布区间变狭窄，峰值变高，表明了预测结果的不确定性变小，准确度提高，这是因为随着用于贝叶斯更新的数据增多，退化模型包含了更多的个体信息，因此可以获得更准确的预测模型。It can be seen from Figure 9 and Figure 10 that as the update time point moves later, the center of the RLD is getting closer to the actual battery life. At the same time, the distribution interval on the time axis becomes narrower and the peak value becomes higher, indicating that the uncertainty of the prediction results is reduced and the accuracy is improved. This is because as the data used for Bayesian update increases, the degradation model contains more Individual information, so a more accurate prediction model can be obtained.