CN107480440B - Residual life prediction method based on two-stage random degradation modeling - Google Patents

Residual life prediction method based on two-stage random degradation modeling Download PDF

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CN107480440B
CN107480440B CN201710658819.0A CN201710658819A CN107480440B CN 107480440 B CN107480440 B CN 107480440B CN 201710658819 A CN201710658819 A CN 201710658819A CN 107480440 B CN107480440 B CN 107480440B
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周东华
张峻峰
何潇
张建勋
张海峰
卢晓
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Shandong University of Science and Technology
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Abstract

The invention discloses a residual life prediction method based on two-stage random degradation modeling, which belongs to the field of industrial monitoring and fault diagnosis and mainly comprises the following steps: performing off-line modeling, updating on-line parameters and predicting the residual life; wherein the offline modeling process comprises: collecting historical degradation data; obtaining a variable point estimation value of each group of degradation by utilizing maximum likelihood estimation, and obtaining the distribution characteristic of the variable point by utilizing statistical analysis; identifying two-stage degradation model parameters off line based on an expected maximization algorithm; taking the parameter estimation value obtained offline and the statistical characteristics of the variable point distribution as the prior information of online parameter updating; the online parameter estimation and the residual life prediction comprise the following steps: collecting degradation data on line; updating on line based on Bayesian theory model parameters; and estimating the residual life of the current operation equipment based on the updated parameters. The invention can model the degradation data with two-stage characteristics and accurately predict the residual life.

Description

Residual life prediction method based on two-stage random degradation modeling
Technical Field
The invention belongs to the field of industrial monitoring and fault diagnosis, and particularly relates to a residual life prediction method based on two-stage random degradation modeling.
Background
The residual life prediction method is used for estimating and predicting the residual operation time of the equipment by using historical and current operation data. The method can provide theoretical basis for maintenance decision and ensure safe and reliable operation of equipment, so the method is a key problem of prediction and health management technology and has been widely concerned and deeply researched in recent years.
Due to the switching of the external environmental stress and the change of the intrinsic degradation mechanism, the degradation rate and the fluctuation amplitude of the equipment are difficult to keep consistent all the time in the operation process. The existing single-stage approach is therefore no longer applicable and it is necessary to predict the remaining life of the device by means of a two-stage degradation model. In engineering practice, the degradation data is usually non-monotonous, so that the degradation model established based on the two-stage Wiener process is more reasonable. However, the existing two-step Wiener process degradation model does not provide an analytic life probability density function and a residual life probability density function in the first reaching sense, so that the residual life is difficult to predict on line.
Disclosure of Invention
Aiming at the technical problems in the prior art, the invention provides a residual life prediction method based on two-stage degradation modeling, which is reasonable in design, overcomes the defects of the prior art and has a good effect.
In order to achieve the purpose, the invention adopts the following technical scheme:
a residual life prediction method based on two-stage random degradation modeling comprises two parts of off-line modeling and on-line parameter updating and residual life prediction, and specifically comprises the following steps:
step 1: the off-line modeling process specifically comprises the following steps:
step 1.1: collecting n battery historical degradation data, and establishing a training data set X ═ X1,X2,...,XnTherein ofIndicates the ith battery at the moment
Figure GDA0002251562350000012
Has m at alliA plurality of degradation data; let Δ t be t, assuming all monitoring is at equal intervalsi,j-ti,j-1；
Step 1.2: the two-stage degradation model is defined as follows:
Figure GDA0002251562350000013
wherein, mu1And mu2The drift coefficient, σ, for a two-stage Wiener process model1And σ2For diffusion parameters, to describe the differences between different samples, let
Figure GDA0002251562350000014
And
Figure GDA0002251562350000015
Figure GDA0002251562350000016
and
Figure GDA0002251562350000017
is normally distributed expectation and variance, but for each individual sample, mu1And mu2Is a fixed constant;
step 1.3: for each set of degraded data X ═ X1,X2,...,XnRespectively constructing log-likelihood functions shown in formula (2), and obtaining a variable point tau of each group of degradation data through maximum likelihood estimationiAs shown in equation (3):
wherein, mui,σiRepresenting the drift and diffusion coefficients of the i-th degradation model, Xi∈{X1,X2,...,Xn}，xi,jRepresenting the jth degradation value in the ith set of degradations,
Figure GDA0002251562350000022
wherein,
Figure GDA0002251562350000024
denotes τiAnd obtaining a probability distribution function p (tau) of the change point by statistical analysis;
step 1.4: based on the discrete estimation parameters of the EM algorithm, the relationship between the iterative update of the parameters in the (k + 1) th step and the estimated value in the k < th > step is obtained as follows:
Figure GDA0002251562350000025
Figure GDA0002251562350000027
Figure GDA0002251562350000028
Figure GDA0002251562350000029
Figure GDA00022515623500000210
wherein,
Figure GDA00022515623500000211
respectively represent mu1p，σ1p，μ2p，σ2p，σ1，σ2At the estimate of the (k + 1) th iteration,
wherein
Figure GDA0002251562350000031
Wherein,
Figure GDA0002251562350000032
indicates that the centering brackets are expected;
step 1.5: continuously iterating the formula (4) and the formula (5) until all parameters are converged, wherein the parameter estimation value during convergence is the model parameter estimation value obtained offline, and the estimation value is used as the prior information updated online later;
step 2: updating an online model and predicting the residual life, specifically comprising the following steps:
step 2.1: collecting online degradation data of the operating equipment, and assuming that the current time is tκWith corresponding degraded data of X0:κ＝{x0,x1,...,xκA total of κ +1 datum;
step 2.2: the method for detecting the change point based on the maximum likelihood estimation comprises the following steps:
wherein x isi∈{x0,x1,...,xκ}，
Figure GDA0002251562350000034
Is an estimated value of the time of the change point tau;
if estimated to be
Figure GDA0002251562350000035
Then explain to tκExecuting step 2.3 if the change point does not appear until the moment; otherwise, the change point appears and is
Figure GDA0002251562350000036
Then step 2.5 is executed;
step 2.3: updating the first-stage model parameters and the variable point probability distribution by using prior information based on the Bayesian theory as follows:
wherein Pr (. cndot.) represents the probability of occurrence of an event in parentheses,. mu.1p,0Represents μ1Expectation of a priori distribution, σ1p,0Represents μ1Standard deviation of prior distribution;
step 2.4: the online residual life prediction result based on the two-stage degradation model under the random change point is as follows:
wherein,is shown at tκResidual life of lκP (τ) represents a probability distribution function of the change point;
wherein,
Figure GDA0002251562350000043
Figure GDA0002251562350000044
Figure GDA0002251562350000045
μa4＝μ2p(lκ-τ+tκ),μb4＝ξ-xκ-μ1p(τ-tκ),
Figure GDA0002251562350000046
Figure GDA0002251562350000047
step 2.5: updating the second-stage model parameters by using prior information based on Bayesian theory as follows:
Figure GDA0002251562350000048
wherein, mu2p,0Represents μ2Expectation of a priori distribution, σ2p,0Represents μ2Standard deviation of prior distribution;
step 2.6: obtaining a residual life prediction result according to a single-stage degradation modeling method, as follows:
Figure GDA0002251562350000049
the core idea and principle of the invention are as follows:
the invention provides a degradation model based on a two-stage Wiener process, and life and residual life analytical representation based on the first meaning of the model are obtained; meanwhile, an off-line model identification and on-line parameter updating method is provided by using an EM algorithm and a Bayesian theory.
The invention has the following beneficial technical effects:
compared with the traditional single-stage degradation model, the two-stage model provided by the invention can better describe the degradation data of the two-stage degradation characteristics; secondly, compared with the existing two-stage Wiener process degradation model, the method obtains the analyzed residual life prediction expression, and is more convenient for on-line calculation; and thirdly, an off-line model identification and on-line parameter estimation method is respectively provided based on an EM algorithm and a Bayesian theory, so that on-line parameter updating can be guaranteed while historical information is fully utilized, the calculation complexity is reduced, and the on-line calculation efficiency is improved.
Drawings
FIG. 1 is a flow chart of offline model identification.
FIG. 2 is a flow chart of online parameter updating and remaining life prediction.
Fig. 3 is a diagram illustrating actual data of battery degradation.
Fig. 4 is a diagram illustrating a parameter update result.
Fig. 5 is a schematic diagram of remaining life prediction.
FIG. 6 is a schematic diagram of a predicted expected and actual remaining life comparison.
Detailed Description
The invention is described in further detail below with reference to the following figures and detailed description:
to assist in understanding the present invention and to demonstrate the effectiveness of fault detection thereof, an example is described in detail below. This example illustrates the present invention based on the MATLAB tool using actual battery degradation data and demonstrates the effects of the present invention in conjunction with the figures.
1. The flow of the offline modeling process is shown in fig. 1, and the specific steps of the example are as follows:
step 1.1: collecting four groups of battery degradation data as shown in fig. 3, selecting three groups (CS2-35, CS2-37, CS2-38) for offline model identification;
step 1.2: the two-stage degradation model is defined as follows:
Figure GDA0002251562350000051
wherein, in order to describe the difference between different samples, letAnd
Figure GDA0002251562350000053
but for each individual sample, mu1And mu2Is a fixed constant.
Step 1.3: for each set of degraded data X ═ X1,X2,...,XnRespectively constructing log-likelihood functions shown in formula (2), and obtaining a variable point tau of each group of degradation data through maximum likelihood estimationiAs shown in equation (3):
Figure GDA0002251562350000061
wherein
Figure GDA0002251562350000062
The maximum likelihood function is then as follows:
Figure GDA0002251562350000063
this gives rise to estimates of the change points 623, 736, 753 (times), respectively. Assuming that the variable points obey the poisson distribution, obtaining a density parameter lambda of the poisson distribution as 704 according to statistical analysis;
step 1.4: based on the EM algorithm, iteration is repeated until the parameter estimation value is converged, and the final result is obtained as follows:
Figure GDA0002251562350000064
Figure GDA0002251562350000065
and the result is used as prior information for the later online method.
2. The flow of the online model updating and the residual life prediction is shown in FIG. 2;
step 2.1: collecting battery degradation data CS 2-36;
step 2.2: the maximum likelihood estimation-based variable point detection method comprises the following steps:
Figure GDA0002251562350000066
wherein x isi∈{x0,x1,...,xκ},
Figure GDA0002251562350000067
Is an estimate of the time of change point tau.
If estimated to be
Figure GDA0002251562350000068
Then explain to tκExecuting step 2.3 if the change point does not appear until the moment; otherwise, the change point appears and isThen step 2.5 is executed;
step 2.3: updating the first-stage model parameters and the variable point probability distribution based on the Bayesian theory as follows:
Figure GDA00022515623500000610
where Pr (-) represents the probability of occurrence of an event in parentheses. Mu.s1p,0Represents μ1Expectation of a priori distribution, σ1p,0Represents μ1Standard deviation of prior distribution.
Step 2.4: the online residual life prediction based on the two-stage degradation model under the random change point has the following results:
Figure GDA0002251562350000071
wherein,
Figure GDA0002251562350000072
is shown at tκResidual life of lκP (τ) represents the probability distribution function of the change point.
Wherein
Figure GDA0002251562350000073
Figure GDA0002251562350000075
μa4＝μ2p(lκ-τ+tκ),μb4＝ξ-xκ-μ1p(τ-tκ),
Figure GDA0002251562350000077
Step 2.5: the model parameters for the second stage are updated as follows:
Figure GDA0002251562350000078
wherein, mu2p,0Represents μ2Expectation of a priori distribution, σ2p,0Represents μ2Standard deviation of prior distribution.
Step 2.6: the residual life prediction result obtained according to the single-stage degradation modeling method is as follows:
Figure GDA0002251562350000079
according to the change point detection method described in step 2.2 and the parameter updating algorithm described in steps 2.3 and 2.5, the parameter updating result obtained when the change point discovery time is 681 (times) is shown in fig. 4;
the failure prediction is given to be 45% of the initial value, the residual life prediction probability density function is obtained by combining the parameter updating result and the residual life prediction method in the steps 2.4 and 2.6, as shown in fig. 5, and fig. 6 is a comparison of the residual life prediction expectation obtained by the method of the invention, the actual life and the result based on the single-stage linear degradation model and the exponential degradation model.
According to the life prediction result, the method can accurately predict the residual life of the battery.
It is to be understood that the above description is not intended to limit the present invention, and the present invention is not limited to the above examples, and those skilled in the art may make modifications, alterations, additions or substitutions within the spirit and scope of the present invention.

Claims (1)

1. A residual life prediction method based on two-stage random degradation modeling is characterized in that: the method comprises two parts of off-line modeling, on-line parameter updating and residual life prediction, and specifically comprises the following steps:
step 1: the off-line modeling process specifically comprises the following steps:
step 1.1: collecting n battery historical degradation data, wherein the degradation data refers to the electric capacity of the battery, and establishing a training data set X ═ X1,X2,...,XnTherein of
Figure FDA0002272330930000011
Indicates the ith battery at the moment
Figure FDA0002272330930000012
Has m at alliA plurality of degradation data; let Δ t be t, assuming all monitoring is at equal intervalsi,j-ti,j-1；
Step 1.2: the two-stage degradation model is defined as follows:
Figure FDA0002272330930000013
wherein, mu1And mu2The drift coefficient, σ, for a two-stage Wiener process model1And σ2For diffusion parameters, to describe the differences between different samples, let
Figure FDA0002272330930000014
And
Figure FDA0002272330930000015
and
Figure FDA0002272330930000016
is normally distributed expectation and variance, but for each individual sample, mu1And mu2Is a fixed constant;
step 1.3: for each set of degraded data X ═ X1,X2,...,XnRespectively constructing log-likelihood functions shown in formula (2), and obtaining a variable point tau of each group of degradation data through maximum likelihood estimationiAs shown in equation (3):
Figure FDA0002272330930000017
wherein, mui,σiRepresenting the drift and diffusion coefficients of the i-th degradation model, Xi∈{X1,X2,...,Xn}，xi,jRepresenting the jth degradation value in the ith set of degradations,
Figure FDA0002272330930000018
Figure FDA0002272330930000019
wherein,
Figure FDA00022723309300000110
denotes τiAnd obtaining a probability distribution function p (tau) of the change point by statistical analysis;
step 1.4: based on the discrete estimation parameters of the EM algorithm, the relationship between the iterative update of the parameters in the (k + 1) th step and the estimated value in the k < th > step is obtained as follows:
Figure FDA0002272330930000021
Figure FDA0002272330930000022
Figure FDA0002272330930000023
Figure FDA0002272330930000024
Figure FDA0002272330930000025
Figure FDA0002272330930000026
wherein,
Figure FDA0002272330930000027
respectively represent mu1p，σ1p，μ2p，σ2p，σ1，σ2At the estimate of the (k + 1) th iteration,
Figure FDA0002272330930000028
wherein
Figure FDA0002272330930000029
Wherein,
Figure FDA00022723309300000210
indicates that the centering brackets are expected;
step 1.5: continuously iterating the formula (4) and the formula (5) until all parameters are converged, wherein the parameter estimation value during convergence is the model parameter estimation value obtained offline, and the estimation value is used as the prior information updated online later;
step 2: updating an online model and predicting the residual life, specifically comprising the following steps:
step 2.1: collecting the online degradation data of the battery, and assuming that the current time is tκWith corresponding degraded data of X0:κ＝{x0,x1,...,xκA total of κ +1 datum;
step 2.2: the method for detecting the change point based on the maximum likelihood estimation comprises the following steps:
Figure FDA0002272330930000031
wherein x isi∈{x0,x1,...,xκ}，
Figure FDA0002272330930000032
Is an estimated value of the time of the change point tau;
if estimated to beThen explain to tκExecuting step 2.3 if the change point does not appear until the moment; otherwise, the change point appears and is
Figure FDA0002272330930000034
Then step 2.5 is executed;
step 2.3: updating the first-stage model parameters and the variable point probability distribution by using prior information based on the Bayesian theory as follows:
Figure FDA0002272330930000035
wherein Pr (. cndot.) represents the probability of occurrence of an event in parentheses,. mu.1p,0Represents μ1Expectation of a priori distribution, σ1p,0Represents μ1Standard deviation of prior distribution;
step 2.4: the online residual life prediction result based on the two-stage degradation model under the random change point is as follows:
Figure FDA0002272330930000036
wherein,
Figure FDA0002272330930000037
is shown at tκResidual life of lκP (τ) represents a probability distribution function of the change point;
wherein,
Figure FDA0002272330930000043
μa4＝μ2p(lκ-τ+tκ),μb4＝ξ-xκ-μ1p(τ-tκ),
Figure FDA0002272330930000044
Figure FDA0002272330930000045
step 2.5: updating the second-stage model parameters by using prior information based on Bayesian theory as follows:
Figure FDA0002272330930000046
wherein, mu2p,0Represents μ2Expectation of a priori distribution, σ2p,0Represents μ2Standard deviation of prior distribution;
step 2.6: obtaining a residual life prediction result according to a single-stage degradation modeling method, as follows:
Figure FDA0002272330930000047
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