CN112782603B - Lithium ion battery cycle life distribution fitting method based on interval truncation data - Google Patents

Lithium ion battery cycle life distribution fitting method based on interval truncation data Download PDF

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CN112782603B
CN112782603B CN202011602586.0A CN202011602586A CN112782603B CN 112782603 B CN112782603 B CN 112782603B CN 202011602586 A CN202011602586 A CN 202011602586A CN 112782603 B CN112782603 B CN 112782603B
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CN112782603A (en
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汪秋婷
沃奇中
戚伟
肖铎
刘泓
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Zhejiang Xingyao Lithium Battery Technology Co ltd
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Hangzhou City University
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
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    • G01R31/36—Arrangements for testing, measuring or monitoring the electrical condition of accumulators or electric batteries, e.g. capacity or state of charge [SoC]
    • G01R31/382—Arrangements for monitoring battery or accumulator variables, e.g. SoC
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
    • G01R31/00—Arrangements for testing electric properties; Arrangements for locating electric faults; Arrangements for electrical testing characterised by what is being tested not provided for elsewhere
    • G01R31/36—Arrangements for testing, measuring or monitoring the electrical condition of accumulators or electric batteries, e.g. capacity or state of charge [SoC]
    • G01R31/392—Determining battery ageing or deterioration, e.g. state of health
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Abstract

The invention relates to a lithium ion battery cycle life distribution fitting method based on interval truncation data, which comprises the following steps of: step 1, analyzing periodic cycle aging data: designing a periodic charge-discharge experiment, and repeatedly charging and discharging between a set minimum SOC value and a set maximum SOC value after measuring the initial capacity of each lithium battery; step 2, establishing a probability density function based on Weibull distribution and inverse Gaussian distribution; and 3, performing maximum likelihood estimation based on the full-area data. The invention has the beneficial effects that: calculating the maximum likelihood estimation of the failure time distribution based on the failure time characteristics of Weibull distribution and inverse Gaussian distribution; analyzing the influence degree of different EOL criteria on the estimation precision by using a simulation experiment, and minimizing the lost information under the condition of less detection time points; the lithium battery life distribution fitting method based on the interval truncation data has practical application significance.

Description

Lithium ion battery cycle life distribution fitting method based on interval truncation data
Technical Field
The invention belongs to the technical field of prediction of residual life (RUL) of lithium batteries, and particularly relates to a life model based on long-term cycle charge and discharge experimental data of a lithium battery and a life distribution fitting method based on Weibull distribution and inverse Gaussian distribution.
Background
The lithium battery has the advantages of high energy density, environmental protection and the like, and is widely applied to power supply of new energy electric vehicles. Prediction of the remaining life (RUL) of the lithium battery belongs to one of important components of a battery management system, and accurate prediction of the RUL of the lithium battery has a great influence on safe operation and economic development of an electric automobile. Currently, the commonly used estimation methods for battery RUL are mainly classified into three categories: a battery physical failure model method, a data driving method and a prediction method based on a fusion technology. The physical failure model method of the battery is easily interfered by the external environment and is difficult to dynamically track the degradation process of the battery. The data-driven rule requires a large amount of historical data as a basis, and when the training data amount is not large enough or the inclusion condition is insufficient, the prediction accuracy cannot be ensured. The life test is to randomly select some test products from a batch of products, place the products to be tested under test conditions which are actually concerned by people for testing, observe and record the failure time of the test products. In the life test, in order to terminate the test in advance and reduce the test time, a tail-end test is generally adopted. The so-called truncation test is to put a batch of products into the test and stop the test when partial products fail, and two types of common truncation tests are a timing truncation test and a fixed number truncation test respectively. In order to reduce the cost of the test and facilitate the data acquisition in the test, interval observation is generally adopted: equal interval observation, equal probability observation and statistical optimal observation. Part of the product that has not failed may be removed for some reason during the actual life test, resulting in missing test data. Therefore, how to estimate the service life distribution of the product by using the incomplete information becomes a research hotspot in the technical field of prediction of the residual service life of the lithium battery.
Disclosure of Invention
The invention aims to overcome the defects and provide a lithium ion battery cycle life distribution fitting method based on interval truncation data.
The lithium ion battery cycle life distribution fitting method based on interval truncation data comprises the following steps:
step 1, analyzing periodic cycle aging data: designing a periodic charge-discharge experiment, and repeatedly charging and discharging between a set minimum SOC value and a set maximum SOC value after measuring the initial capacity of each lithium battery;
step 2, establishing a Probability Density Function (PDF) based on Weibull distribution and inverse Gaussian distribution;
and 3, performing maximum likelihood estimation based on the full-area data: when the detection data is full-area data (continuous value), let θ be (θ)1,θ2) Parameter vectors representing Weibull distribution and inverse Gaussian distribution, where θ1And theta2Mean and shape parameters, respectively; then the maximum likelihood estimate of the Weibull distribution is θW(λ, β), the maximum likelihood estimate of the inverse gaussian is θIG(v, η); obtaining an average parameter theta based on likelihood theory1And a shape parameter theta2Radix Ginseng (radix Ginseng)Number vector θ ═ θ1,θ2) Asymptotic normality of (a); setting the service life parameters of the n lithium batteries as T corresponding to the time point when the capacity of each battery reaches the preset end voltage1,...,Tn(ii) a For the observation sample t ═ t (t)1,...,tn)TThe log-likelihood function is:
Figure GDA0003575412070000021
in the above formula, f is the corresponding probability density function, and if it is Weibull distribution, f is
Figure GDA0003575412070000022
If it is an inverse Gaussian distribution, f is
Figure GDA0003575412070000023
Step 4, calculating failure probability: assume a probability vector of piSatisfy the following requirements
Figure GDA0003575412070000024
Wherein m isiThe total number of detection times of the ith lithium battery is shown, F (·;. theta.) is a probability density function of a lithium battery life parameter, and pijThe failure probability of the ith lithium battery in the jth detection interval is satisfied with pij＝F(xij；θ)-F(xi,j-1；θ)，j∈{1,...,mi},i∈{1,...,n}，xijThe j detection data of the ith lithium battery in the experiment are obtained;
step 5, establishing a cycle life model: assuming that all lithium batteries share the same detection time point, m is satisfiedi＝m，xij＝xjAnd j ∈ {1,..,. m }, i ∈ {1,..,. n }, where m ∈iThe total number of detection times of the ith lithium battery is represented, m is the number of equal-interval measurement time points, and n is the number of lithium batteries; then c isij＝cj，pij＝pjWherein c isij＝(xi,j-1,xij]The j-1 detection point and the j detection point of the ith lithium batteryData intervals between the measurement points; p is a radical ofijIs the failure probability, p, of the ith lithium battery in the jth detection intervaliIs a probability vector; deriving a vector piP, i ∈ {1,..., n }; a vector of sample points in the case of a common detection time point is represented as,
N＝(N+1，...，N+m，1-N++)-M(n，p) (13)
in the above formula, M (n, p) is a polynomial distribution parameter, n is the number of lithium batteries, M is the number of equally spaced measurement time points, pi＝p；N+jThe sum of j detection data of n lithium batteries satisfies the formula
Figure GDA0003575412070000031
N++The sum of all detection data of n lithium batteries meets the formula
Figure GDA0003575412070000032
Observed data n ═ n (n)1,...,nn) The polynomial log-likelihood function of (d) is:
Figure GDA0003575412070000033
in the above formula, θ ═ θ1,θ2) A parameter vector representing a Weibull distribution and an inverse gaussian distribution; n is an observation vector of N, NijThe j time detection data sum of the ith lithium battery is obtained; p is a radical ofijThe failure probability of the ith lithium battery in the jth detection interval is obtained; then the observed data n is equal to (n)1,...,nn) The polynomial log-likelihood function of (a) can be simplified to equation (15):
Figure GDA0003575412070000034
in the above formula, the first and second carbon atoms are,
Figure GDA0003575412070000035
n+jthe sum of j detection data of n lithium batteries, wherein m isThe number of time points is measured at equal intervals.
Preferably, step 1 specifically comprises the following steps:
step 1-1, performing 150-200 times of cyclic charge and discharge experiments on n lithium batteries, wherein n is more than 40; obtaining Approximate Failure Time (AFT) as real accurate failure time by utilizing a cubic spline interpolation function;
step 1-2, establishing a detection time point formula; assuming that the number of equally spaced measurement time points is m, the detection time point coordinates are defined as,
Figure GDA0003575412070000036
in the above formula, F is a lifetime function, according to a set end life (EOL), F is a cumulative distribution function of a true value, and the value of F is always fitted with a data distribution of an Approximate Failure Time (AFT); m is the number of time points measured at equal intervals;
step 1-3, fitting a single battery interpolation curve based on Weibull distribution: an equal probability detection interval (EP) is chosen, and the number of cycles of the degradation interval that are close to the approximate time to failure (AFT) is highlighted vertically with a thick solid line.
Preferably, step 2 specifically comprises the following steps:
step 2-1, assuming t is the sampling time, Weibull distribution with a mean parameter λ > 0 and a shape parameter β > 0, the probability density function of Weibull distribution being,
Figure GDA0003575412070000041
in the above formula, t is the sampling time, λ is the average parameter, and β is the shape parameter;
step 2-2, the probability density function of the inverse Gaussian distribution with the mean parameter v > 0 and the shape parameter η > 0 is:
Figure GDA0003575412070000042
in the above formula, t is the sampling time, v is the average parameter, and η is the shape parameter.
Preferably, step 3 specifically comprises the following steps:
step 3-1, the maximum likelihood estimation formula based on Weibull distribution is as follows:
Figure GDA0003575412070000043
in the above formula, the first and second carbon atoms are,
Figure GDA0003575412070000044
is a Maximum Likelihood Estimate (MLE) of the shape parameter beta,
Figure GDA0003575412070000045
a maximum likelihood estimate for the mean parameter λ; solving a result based on a Nelder-Mead algorithm formula:
Figure GDA0003575412070000046
in the above formula, n is the number of lithium batteries,
Figure GDA0003575412070000047
is a Maximum Likelihood Estimate (MLE), t, of the shape parameter βiIs the ith observation sample; step 3-2, the maximum likelihood estimation formula based on the inverse Gaussian distribution is as follows,
Figure GDA0003575412070000048
Figure GDA0003575412070000049
in the above-mentioned formula, the compound has the following structure,
Figure GDA00035754120700000410
and
Figure GDA00035754120700000411
maximum likelihood estimates of v and η, respectively; for interval truncated data, assume xijFor the j-th detection data of the ith lithium battery in the experiment, i ═ 1,2,. and n, j ═ 1,2,. and miN is the number of lithium batteries, miRepresenting the total number of detection times of the ith lithium battery; the observed data of the ith lithium battery is not tiInstead, the vector formula (9), e.g. 1 st lithium battery, detects m altogether1Second, the 5 th lithium battery detects m5Secondly:
Figure GDA00035754120700000412
in the above formula, niThe observation data of the ith lithium battery are obtained; m isiThe total number of detection times of the ith lithium battery is represented; n isijThe j time detection data sum of the ith lithium battery is obtained; n isi+The sum of all detection data of the ith lithium battery; i (·) is an index function and meets the formula
Figure GDA0003575412070000051
cij＝(xi,j-1,xij]Assuming that x is the data interval between the j-1 detection point and the j detection point of the ith lithium battery i00; thus formula
Figure GDA0003575412070000052
And ni+The section truncation data can accurately represent the failure time of the ith lithium battery by being equal to 0.
Preferably, step 4 specifically comprises the following steps:
step 4-1, calculating the failure probability based on Weibull distribution:
Figure GDA0003575412070000053
in the above formula, pijThe failure probability of the ith lithium battery in the jth detection interval is obtained; i is the number of lithium batteries, j is the number of detection points, beta is a shape parameter, and lambda is an average value parameter;
4-2, calculating the failure probability based on inverse Gaussian distribution:
Figure GDA0003575412070000054
in the above formula, phi is the probability density function of the standard normal distribution, i is the number of lithium batteries, j is the number of detection points, and the parameter yijAnd wijSatisfies formula (12):
Figure GDA0003575412070000055
Figure GDA0003575412070000056
in the above formula, i is the number of lithium batteries, j is the number of detection points, and xijThe j detection data of the ith lithium battery in the experiment are shown, eta is a shape parameter, and v is an average value parameter.
Preferably, step 5 specifically comprises the following steps:
step 5-1, calculating cycle life based on Weibull distribution according to formula (16):
Figure GDA0003575412070000061
in the above formula, t is the sampling time, λ is the average parameter, and β is the shape parameter; n is the number of lithium batteries, m is the number of time points measured at equal intervals, xijThe j detection data of the ith lithium battery in the experiment are obtained; n isijThe j time detection data sum of the ith lithium battery is obtained; n isi+The data sum of the ith detection of the n lithium batteries is obtained;
the likelihood function of the Weibull distribution is transformed according to equation (15):
Figure GDA0003575412070000062
in the above formula, λ is an average parameter, and β is a shape parameter;
Figure GDA0003575412070000063
in N is the corresponding observation vector of the sampling point vector N, j is the number of the detection points, m is the number of the equal interval measurement time points, N+jAnd n++Definition of (A) and N+jAnd N++The same;
step 5-2, calculating cycle life based on inverse Gaussian distribution according to formulas (11) and (12):
Figure GDA0003575412070000064
in the above formula, v is an average parameter, η is a shape parameter, n is the number of lithium batteries,
Figure GDA0003575412070000065
wherein N is a corresponding observation vector of the sampling point vector N, N in the summation function is the number of lithium batteries, and the parameter yijAnd wijThe formula (12) is satisfied.
The invention has the beneficial effects that:
(1) the maximum likelihood estimation of the failure time distribution is calculated by utilizing the actual life interval truncated dataset of 48 single lithium batteries based on the failure time characteristics of Weibull distribution and inverse Gaussian distribution.
(2) The invention performs spline interpolation approximation on corresponding accurate failure time data (AFT).
(3) The invention utilizes simulation experiments to analyze the influence degree of different EOL criteria on the estimation precision, and can lose the least information under the condition of less detection time points.
(4) The lithium battery life distribution fitting method based on interval truncation data has practical application significance, supports experiment design and detection, and simulates the accuracy of experiment results by utilizing lognormal distribution.
Drawings
FIG. 1 is a graph of measured capacity values (dots), cubic spline interpolation curves (black lines) and AFT-based interval measurement data of a single lithium battery;
FIG. 2 is a graph of interpolation of a single lithium battery based on Weibull distribution, EoL > 50% in FIG. 2(a), 50% < EoL < 80% in FIG. 2 (b);
Detailed Description
The present invention will be further described with reference to the following examples. The following examples are set forth merely to provide an understanding of the invention. It should be noted that, for those skilled in the art, it is possible to make various improvements and modifications to the present invention without departing from the principle of the present invention, and those improvements and modifications also fall within the scope of the claims of the present invention.
According to the invention, a lithium battery service life distribution model based on interval truncation data is established, and the fitting effect of Weibull distribution and inverse Gaussian distribution on cyclic aging data is compared.
As an embodiment, the lithium ion battery cycle life distribution fitting method based on interval truncation data comprises the following steps:
step 1, analyzing periodic cycle aging data: designing a periodic charge-discharge experiment, and repeatedly charging and discharging between a set minimum SOC value and a set maximum SOC value after measuring the initial capacity of each lithium battery;
step 1-1, performing 150-200 times of cyclic charge and discharge experiments on n lithium batteries, wherein n is more than 40; obtaining Approximate Failure Time (AFT) as real accurate failure time by utilizing a cubic spline interpolation function; as shown in fig. 1, when the end-of-life value EOL is less than 80% (thick solid line), the cycle number range including the approximate failure time in the interval measurement data of the lithium battery is represented by hatching, and the approximate cycle number is represented by a bold vertical line; FIG. 1 illustrates:
(1) sampling 10-15 data at equal intervals, wherein the cycle number shows a descending trend, and the plateau period appears in the cycle life of 300-1300 times; (2) approximate failure time ranged from 1300 to 1450; (3) after 2500 times of cyclic charge and discharge, the available capacity of the lithium battery is reduced to be less than 0.5 Ah;
step 1-2, establishing a detection time point formula; assuming that the number of equally spaced measurement time points is m, the detection time point coordinates are defined as,
Figure GDA0003575412070000071
in the above formula, F is a life function, according to a set end life (EOL), F is a cumulative distribution function of real values, and the value of F is always fitted with data distribution of Approximate Failure Time (AFT); m is the number of time points measured at equal intervals;
step 1-3, fitting a single battery interpolation curve based on Weibull distribution: selecting an equal probability detection interval (EP), and vertically highlighting and marking the cycle times of the degradation interval close to the Approximate Failure Time (AFT) by using a thick solid line; FIG. 2 shows that:
(1) interval time sampling points are close to the value area of EOL; (2) the approximate failure time range is 1575 to 1620 times when EOL < 50%, 1080 to 1120 times when 50% < EOL < 80%; (3) the larger the EoL value is, the smaller the failure range is, and the better the lithium performance is; (4) the interpolation result based on Weibull distribution is more accurate than the cubic spline interpolation result;
step 2, establishing a Probability Density Function (PDF) based on Weibull distribution and inverse Gaussian distribution;
step 2-1, assuming t is the sampling time, Weibull distribution with a mean parameter λ > 0 and a shape parameter β > 0, the probability density function of Weibull distribution being,
Figure GDA0003575412070000081
in the above formula, t is the sampling time, λ is the average parameter, and β is the shape parameter;
step 2-2, the probability density function of the inverse Gaussian distribution with the mean parameter v > 0 and the shape parameter η > 0 is:
Figure GDA0003575412070000082
in the above formula, t is the sampling time, v is the average parameter, and η is the shape parameter;
and 3, performing maximum likelihood estimation based on the full-area data: when the detection data is full-area data (continuous value), let θ be (θ)1,θ2) Parameter vectors representing Weibull distribution and inverse Gaussian distribution, where θ1And theta2Mean and shape parameters, respectively; then the maximum likelihood estimate of the Weibull distribution is θWThe maximum likelihood of the inverse gaussian distribution is estimated as θ (λ, β)IG(v, η); obtaining an average parameter theta based on likelihood theory1And a shape parameter theta2Is (theta) is given as the parameter vector of (theta)1,θ2) Asymptotic normality of (a); setting the service life parameters of the n lithium batteries as T corresponding to the time point when the capacity of each battery reaches the preset end voltage1,...,Tn(ii) a For the observation sample t ═ t (t)1,...,tn)TThe log-likelihood function is:
Figure GDA0003575412070000083
in the formula, f is a corresponding probability density function, and if the probability density function is Weibull distribution, f is a function in the formula (2); if the distribution is inverse Gaussian distribution, f is a function in the formula (3);
step 3-1, the maximum likelihood estimation formula based on Weibull distribution is as follows:
Figure GDA0003575412070000084
in the above formula, the first and second carbon atoms are,
Figure GDA0003575412070000091
is a Maximum Likelihood Estimate (MLE) of the shape parameter beta,
Figure GDA0003575412070000092
a maximum likelihood estimate for the mean parameter λ; solving a result based on a Nelder-Mead algorithm formula:
Figure GDA0003575412070000093
in the above formula, n is the number of lithium batteries,
Figure GDA0003575412070000094
is a Maximum Likelihood Estimate (MLE), t, of the shape parameter βiIs the ith observation sample; step 3-2, the maximum likelihood estimation formula based on the inverse Gaussian distribution is as follows,
Figure GDA0003575412070000095
Figure GDA0003575412070000096
in the above formula, the first and second carbon atoms are,
Figure GDA0003575412070000097
and
Figure GDA0003575412070000098
maximum likelihood estimates of v and η, respectively; for interval truncated data, assume xijFor the j-th detection data of the ith lithium battery in the experiment, i ═ 1,2,. and n, j ═ 1,2,. and miN is the number of lithium batteries, miRepresenting the total number of detection times of the ith lithium battery; the observed data of the ith lithium battery is not tiInstead, the vector formula (9), e.g. 1 st lithium battery, detects m altogether1Second, the 5 th lithium battery detects m5Secondly:
Figure GDA0003575412070000099
in the above formula, niThe observation data of the ith lithium battery are obtained; m isiRepresenting the total number of detection times of the ith lithium battery; n isijThe j time detection data sum of the ith lithium battery is obtained; n is a radical of an alkyl radicali+The sum of all detection data of the ith lithium battery; i (·) is an index function and meets the formula
Figure GDA00035754120700000910
cij＝(xi,j-1,xij]Assuming that x is the data interval between the j-1 detection point and the j detection point of the ith lithium battery i00; thus formula
Figure GDA00035754120700000911
And ni+The section truncation data can accurately represent the failure time of the ith lithium battery by being 0;
step 4, calculating failure probability: assume a probability vector of piSatisfy the following requirements
Figure GDA00035754120700000912
Wherein m isiThe total number of detection times of the ith lithium battery is represented, F (·; theta) is a probability density function of a lithium battery life parameter, pijThe failure probability of the ith lithium battery in the jth detection interval is satisfied with pij＝F(xij；θ)-F(xi,j-1；θ)(j∈{1,...,mi},i∈{1,...,n})，xijThe j detection data of the ith lithium battery in the experiment are obtained;
step 4-1, calculating the failure probability based on Weibull distribution:
Figure GDA0003575412070000101
in the above formula, pijThe failure probability of the ith lithium battery in the jth detection interval is obtained; i is the number of lithium batteries, j is the number of detection points, beta is a shape parameter, and lambda is an average value parameter;
4-2, calculating the failure probability based on inverse Gaussian distribution:
Figure GDA0003575412070000102
in the above formula, phi is the probability density function of the standard normal distribution, i is the number of lithium batteries, j is the number of detection points, and the parameter yijAnd wijSatisfies formula (12):
Figure GDA0003575412070000103
Figure GDA0003575412070000104
in the above formula, i is the number of lithium batteries, j is the number of detection points, and xijThe j detection data of the ith lithium battery in the experiment are shown, eta is a shape parameter, and v is an average value parameter;
step 5, establishing a cycle life model: assuming that all lithium batteries share the same detection time point, m is satisfiedi＝m，xij＝xjAnd j ∈ {1,..,. m } (i ∈ {1,..,. n }), wherein m ∈ {1,..,. n }, whereiThe total number of detection times of the ith lithium battery is represented, m is the number of equal-interval measurement time points, and n is the number of lithium batteries; then c isij＝cj，pij＝pjWherein c isij＝(xi,j-1,xij]A data interval between a j-1 detection point and a j detection point of an ith lithium battery is set; p is a radical ofijIs the failure probability, p, of the ith lithium battery in the jth detection intervaliIs a probability vector; deriving a vector piP (i ∈ { 1.., n }); a vector of sample points in the case of a common detection time point is represented as,
N＝(N+1，...，N+m，1-N++)′-M(n，p) (13)
in the above formula, M (n, p) is a polynomial distribution parameter, n is the number of lithium batteries, and M is measured at equal intervalsNumber of points, pi＝p；N+jThe sum of j detection data of n lithium batteries satisfies the formula
Figure GDA0003575412070000111
N++The sum of all detection data of n lithium batteries meets the formula
Figure GDA0003575412070000112
Observed data n ═ n (n)1,...,nn) The polynomial log-likelihood function of (d) is:
Figure GDA0003575412070000113
in the above formula, θ ═ θ1,θ2) A parameter vector representing a Weibull distribution and an inverse gaussian distribution; n is an observation vector of N, NijThe j time detection data sum of the ith lithium battery is obtained; p is a radical ofijThe failure probability of the ith lithium battery in the jth detection interval is obtained; then observed data n ═ n1,...,nn) The polynomial log-likelihood function of (a) can be simplified to equation (15):
Figure GDA0003575412070000114
in the above formula, the first and second carbon atoms are,
Figure GDA0003575412070000115
n+jthe j-th detection data of n lithium batteries are summed, and m is the number of time points measured at equal intervals;
step 5-1, calculating cycle life based on Weibull distribution according to formula (16):
Figure GDA0003575412070000116
in the above formula, t is the sampling time, λ is the average parameter, and β is the shape parameter; n is the number of lithium batteries, and m is the time point of measurement at equal intervalsNumber, xijThe j detection data of the ith lithium battery in the experiment are obtained; n isijThe j time detection data sum of the ith lithium battery is obtained; n isi+The data sum of the ith detection of the n lithium batteries is obtained;
the likelihood function of the Weibull distribution is transformed according to equation (15):
Figure GDA0003575412070000117
in the above formula, λ is an average parameter, and β is a shape parameter;
Figure GDA0003575412070000118
in N is the corresponding observation vector of the sampling point vector N, j is the number of the detection points, m is the number of the equal interval measurement time points, N+jAnd n++Definition of (A) and N+jAnd N++The same;
step 5-2, calculating cycle life based on inverse Gaussian distribution according to formulas (11) and (12):
Figure GDA0003575412070000121
in the above formula, v is an average parameter, η is a shape parameter, n is the number of lithium batteries,
Figure GDA0003575412070000122
wherein N is a corresponding observation vector of the sampling point vector N, N in the summation function is the number of lithium batteries, and the parameter yijAnd wijThe formula (12) is satisfied.
Experimental result 1:
under the condition of all-region data, according to the detection interval time length, the influence of the quantity of different detection time points on the parameter estimation precision is researched; based on the number of experimental cells, 10000 samples were taken, each sample size being 43 (considering the common detection time point). According to the full-area experimental data, the parameter values of the three distribution functions are equal to the maximum likelihood estimation value; selecting equidistant detection interval time points as cycle times of every 80, 160 and 240, setting 2400 cycles as final detection time points, and respectively setting detection time intervals as 30, 15 and 10; meanwhile, simulating equal probability detection interval points by using interval truncation data, and generating a sample distribution function by using experimental data. The following table 1 is used for analyzing the influence of the interval length of the detection time points on the estimation result by comparing the maximum likelihood estimation empirical mean value and the standard deviation of the truncation method of different intervals, and simultaneously selecting 15 equal probability detection time point data as comparison data; the following table 1 shows that the interval truncated data with the EoL value of 50% -80% and the full-area data have the same estimation accuracy, but when the equidistant detection time point is 10, the maximum likelihood estimation error value and the accurate value deviation of the shape parameter are large; the results show that the improvement degree of the cycle life estimation precision is not large due to the increase of the number of the detection time points; in the embodiment, a simulation experiment (not shown in table 1) based on 90% EoL is performed, and the result shows that the method for estimating equidistant detection time points is only suitable for the case of a large number of detection time points, and the fitting effect of the method for estimating the equiprobable detection time points and accurate data is the best; meanwhile, under the current experiment setting condition and 15-time circulation intervals, the equal probability detection time point estimation method based on the Weibull distribution model can obtain an estimation result with higher precision.
TABLE 1 empirical mean and standard deviation table for maximum likelihood estimation of Weibull distribution parameters based on full-area and interval truncation data
Figure GDA0003575412070000123
Figure GDA0003575412070000131
In the table above, 30, 15 or 10 equidistant detection time points and 15 equal probability detection time points, the end life is 80% and 50%;
experimental results 2:
the following table 2 shows the empirical mean and standard deviation of the maximum likelihood estimation based on the inverse gaussian distribution parameters, and the simulation results show that, when the number of equidistant detection time points is 15 or 30, the mean of the parameter estimation under the condition of interval truncation data is equivalent to the mean of the parameter estimation under the condition of equal probability detection time point data, and the deviation value between the mean and the true value is small. However, when the number of equidistant detection time points is 10, there is a deviation of the mean value and a large standard deviation, while the estimated shape parameter μ is estimated using the true value, the estimated deviation value is not negligible. The above conclusion shows that the inverse gaussian life distribution model cannot be used for lithium battery cycle life estimation based on interval truncation data. Meanwhile, the estimation method based on the equidistant truncated data has the estimation precision equivalent to the true value only under the condition that the detection time point is 30. The number of detection time points is reduced, so that the estimation deviation value and the standard deviation are increased; table 2 below illustrates that when the number of equidistant detection time points is 10, a plurality of failure time points may occur within the same time interval, resulting in that the maximum likelihood estimation result of the distribution parameter is not usable.
TABLE 2 empirical mean and standard deviation data tables for maximum likelihood estimation of inverse Gaussian distribution parameters based on full-area and interval truncation data
Figure GDA0003575412070000132
In the above table, 30, 15 or 10 equidistant detection time points and 15 equi-probability detection time points, the end life is 80% and 50%.

Claims (5)

1. A lithium ion battery cycle life distribution fitting method based on interval truncation data is characterized by comprising the following steps:
step 1, analyzing periodic cycle aging data: designing a periodic charge-discharge experiment, and repeatedly charging and discharging between a set minimum SOC value and a set maximum SOC value after measuring the initial capacity of each lithium battery;
step 2, establishing a probability density function based on Weibull distribution and inverse Gaussian distribution;
step 2-1, assuming t is the sampling time, Weibull distribution with a mean parameter λ > 0 and a shape parameter β > 0, the probability density function of Weibull distribution being,
Figure FDA0003575412060000011
in the above formula, t is the sampling time, λ is the average parameter, and β is the shape parameter;
step 2-2, the probability density function of the inverse Gaussian distribution with the mean parameter v > 0 and the shape parameter η > 0 is:
Figure FDA0003575412060000012
in the above formula, t is the sampling time, v is the average parameter, and η is the shape parameter;
and 3, performing maximum likelihood estimation based on the full-area data: when the detection data is full-area data, let θ be (θ)1,θ2) Parameter vectors representing Weibull distribution and inverse Gaussian distribution, where θ1And theta2Mean and shape parameters, respectively; then the maximum likelihood estimate of the Weibull distribution is θWThe maximum likelihood of the inverse gaussian distribution is estimated as θ (λ, β)IG(v, η); obtaining an average parameter theta based on likelihood theory1And a shape parameter theta2Is (theta) is given as the parameter vector of (theta)1,θ2) Asymptotic normality of (a); setting the service life parameters of the n lithium batteries as T corresponding to the time point when the capacity of each battery reaches the preset end voltage1,...,Tn(ii) a For the observation sample t ═ t (t)1,...,tn)TThe log-likelihood function is:
Figure FDA0003575412060000013
in the above formula, f is the corresponding probability density function, and if it is Weibull distribution, f is
Figure FDA0003575412060000014
If it is an inverse Gaussian distribution, f is
Figure FDA0003575412060000015
Step 4, calculating failure probability: assume a probability vector of piSatisfy the following requirements
Figure FDA0003575412060000016
Wherein m isiThe total number of detection times of the ith lithium battery is represented, F (·; theta) is a probability density function of a lithium battery life parameter, pijThe failure probability of the ith lithium battery in the jth detection interval is satisfied with pij＝F(xij；θ)-F(xi,j-1；θ)，j∈{1,...,mi},i∈{1,...,n}，xijThe j detection data of the ith lithium battery in the experiment are obtained;
step 5, establishing a cycle life model: assuming that all lithium batteries share the same detection time point, m is satisfiedi＝m，xij＝xjAnd j ∈ {1,..,. m }, i ∈ {1,..,. n }, where m ∈iThe total number of detection times of the ith lithium battery is represented, m is the number of equal-interval measurement time points, and n is the number of lithium batteries; then c isij＝cj，pij＝pjWherein c isij＝(xi,j-1,xij]A data interval between a j-1 detection point and a j detection point of an ith lithium battery is set; p is a radical ofijIs the failure probability, p, of the ith lithium battery in the jth detection intervaliIs a probability vector; deriving a vector piP, i ∈ {1,..., n }; a vector of sample points in the case of a common detection time point is represented as,
N＝(N+1，...，N+m，1-N++)-M(n，p) (13)
in the above formula, M (n, p) is a polynomial distribution parameter, n is the number of lithium batteries, M is the number of equally spaced measurement time points, pi＝p；N+jThe sum of j detection data of n lithium batteries satisfies the formula
Figure FDA0003575412060000021
j∈{1,...,m}；N++The sum of all detection data of n lithium batteries meets the formula
Figure FDA0003575412060000022
Observed data n ═ n (n)1,...,nn) The polynomial log-likelihood function of (d) is:
Figure FDA0003575412060000023
in the above formula, θ ═ θ1,θ2) A parameter vector representing a Weibull distribution and an inverse gaussian distribution; n is an observation vector of N, NijThe j time detection data sum of the ith lithium battery is obtained; p is a radical ofijThe failure probability of the ith lithium battery in the jth detection interval is obtained; then the observed data n is equal to (n)1,...,nn) Reduces the polynomial log-likelihood function of (a) to equation (15):
Figure FDA0003575412060000024
in the above formula, the first and second carbon atoms are,
Figure FDA0003575412060000025
n+jthe j-th detection data of the n lithium batteries are summed, and m is the number of the time points measured at equal intervals.
2. The lithium ion battery cycle life distribution fitting method based on interval truncation data according to claim 1, wherein the step 1 specifically comprises the following steps:
step 1-1, performing 150-200 times of cyclic charge and discharge experiments on n lithium batteries, wherein n is more than 40; obtaining approximate failure time as real accurate failure time by utilizing a cubic spline interpolation function;
step 1-2, establishing a detection time point formula; assuming that the number of equally spaced measurement time points is m, the detection time point coordinates are defined as,
Figure FDA0003575412060000031
in the formula, F is a life function, and the value of F is always fitted with the data distribution of the approximate failure time according to the cumulative distribution function of which the set end life F is a true value; m is the number of time points measured at equal intervals;
step 1-3, fitting a single battery interpolation curve based on Weibull distribution: equal probability detection intervals are selected, and the cycle times of the degradation intervals close to the approximate failure time are marked by the vertical highlighting of the thick solid line.
3. The lithium ion battery cycle life distribution fitting method based on interval truncation data according to claim 1, wherein the step 3 specifically comprises the following steps:
step 3-1, the maximum likelihood estimation formula based on Weibull distribution is as follows:
Figure FDA0003575412060000032
in the above formula, the first and second carbon atoms are,
Figure FDA0003575412060000033
is a maximum likelihood estimate of the shape parameter beta,
Figure FDA0003575412060000034
a maximum likelihood estimate for the mean parameter λ; solving a result based on a Nelder-Mead algorithm formula:
Figure FDA0003575412060000035
in the above formula, n is the number of lithium batteries,
Figure FDA0003575412060000036
for maximum likelihood estimation of the shape parameter beta, tiIs the ith observation sample;
step 3-2, the maximum likelihood estimation formula based on the inverse Gaussian distribution is as follows,
Figure FDA0003575412060000037
Figure FDA0003575412060000038
in the above formula, the first and second carbon atoms are,
Figure FDA0003575412060000039
and
Figure FDA00035754120600000310
maximum likelihood estimates of v and η, respectively; for interval truncated data, assume xijFor the j-th detection data of the ith lithium battery in the experiment, i ═ 1,2,. and n, j ═ 1,2,. and miN is the number of lithium batteries, miRepresenting the total number of detection times of the ith lithium battery; the observed data of the ith lithium battery is not tiInstead, vector formula (9):
Figure FDA0003575412060000041
in the above formula, niThe observation data of the ith lithium battery are obtained; m isiRepresenting the total number of detection times of the ith lithium battery; n isijThe j time detection data sum of the ith lithium battery is obtained; n is a radical of an alkyl radicali+The sum of all detection data of the ith lithium battery; i (·) is an index function and meets the formula
Figure FDA0003575412060000042
cij＝(xi,j-1,xij]Assuming that x is the data interval between the j-1 detection point and the j detection point of the ith lithium batteryi00; thus formula
Figure FDA0003575412060000043
And ni+The section truncation data indicates the failure time of the ith lithium battery as 0.
4. The lithium ion battery cycle life distribution fitting method based on interval truncation data according to claim 1, wherein the step 4 specifically comprises the following steps:
step 4-1, calculating the failure probability based on Weibull distribution:
Figure FDA0003575412060000044
in the above formula, pijThe failure probability of the ith lithium battery in the jth detection interval is obtained; i is the number of lithium batteries, j is the number of detection points, beta is a shape parameter, and lambda is an average value parameter;
4-2, calculating the failure probability based on inverse Gaussian distribution:
Figure FDA0003575412060000045
in the above formula, phi is the probability density function of the standard normal distribution, i is the number of lithium batteries, j is the number of detection points, and the parameter yijAnd wijSatisfies formula (12):
Figure FDA0003575412060000046
Figure FDA0003575412060000047
in the above formula, i is the number of lithium batteries, j is the number of detection points, and xijThe j detection data of the ith lithium battery in the experiment are shown, eta is a shape parameter, and v is an average value parameter.
5. The lithium ion battery cycle life distribution fitting method based on interval truncation data according to claim 1 or 4, wherein the step 5 specifically comprises the following steps:
step 5-1, calculating cycle life based on Weibull distribution according to formula (16):
Figure FDA0003575412060000051
in the above formula, t is the sampling time, λ is the average parameter, and β is the shape parameter; n is the number of lithium batteries, m is the number of time points measured at equal intervals, xijThe j detection data of the ith lithium battery in the experiment are obtained; n isijThe j time detection data sum of the ith lithium battery is obtained; n isi+The data sum of the ith detection of the n lithium batteries is obtained;
the likelihood function of the Weibull distribution is transformed according to equation (15):
Figure FDA0003575412060000052
in the above formula, λ is an average parameter, and β is a shape parameter; lWN in (lambda, beta; N) is the corresponding observation vector of sampling point vector N, j is the number of detection points, m is the number of equal-interval measurement time points, N+jAnd n++Definition of (A) and N+jAnd N++The same;
step 5-2, calculating cycle life based on inverse Gaussian distribution according to formulas (11) and (12):
Figure FDA0003575412060000053
in the above formula, v is the average parameter, η is the shape parameter, n is the number of lithium batteries, lIGN in (v, eta; N) is the corresponding observation vector of sampling point vector N, N in summation function is the number of lithium batteries, and parameter yijAnd wijThe formula (12) is satisfied.
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