Methods for Reliability Assessment under Irregular Time-Varying-Stress Degradation Testing

LIU Tian-yu(刘天宇)，SUN Quan(孙 权)，2* ，PAN Zheng-qiang(潘正强)，FENG Jing(冯 静)

1 College of Information System and Management，National University of Defense Technology，Changsha 410073，China

2 State Key Laboratory of High Performance Computing，National University of Defense Technology，Changsha 410073，China

Introduction

With the development of manufacturing technology and materials science，more and more high-reliability products are emerging.Traditional life test is no longer suitable for reliability assessment of the products with long life and small samples.Instead degradation tests are commonly used in this situation，in which much more information can be obtained.Since time is often limited before launching a new product，engineers often raise the levels of certain stress in the accelerated degradation test (ADT).Three accelerated stresses commonly adopted are constant，step，and progressive stresses，and the corresponding methods of data processing and reliability assessment are studied in Refs.［1 -3］.However，in practical applications，some stresses are affected by many factors，leading to the stresses so difficult to control and vary with time irregularly，which are defined as irregular time-varying-stress (ITVS).

Most previous studies focus on the degradation modeling with degradation data under constant stress，for example，degradation path modeling［4］，stochastic process modeling［5-6］and degradation distribution modeling［7］.There are also some helpful techniques dealing with step-stress ADT［2］and progressive ADT［3］.However，few researches on reliability assessment methods for degradation test under ITVS are mentioned before.

1 Degradation Model under ITVS

1.1 Conventional degradation path model

The conventional degradation path model is a common method processing degradation data in constant-stress degradation test (CSDT).

It assumes that the degradation path follows a certain function which is determined by its model parameter vector.

where N is the number of units，miis the number of measurements on unit i，yijis the measurement of degradation variable at measured time tijon unit i，θiis the parameter vector of g(·)for unit i and εijdenotes the random error which follows a normal distribution，namely εij～(0，σ2).Several extensive degradation path models are listed in Table 1.

Table 1 Common degradation path models

Then，the method to obtain the failure-time distribution is shown as follows.

(1)With the degradation data measured，we can get the estimator of the parameter vector，separately.

(2)The degradation variable's first passage time to the threshold Dffor unit i can be calculated as

(3)With each sample's pseudo life L1，L2，…，LN，we can obtain the best fitted distribution of failure-time through goodness-of-fit test.

(4)Reliability of the product can be assessed through the failure-time's distribution.

1.2 Improved degradation path model for ITVS

As discussed above，the conventional degradation path model ignores the effect of stress on degradation variables and applies only to the CSDT.To overcome these difficulties，here we propose an improved degradation path model.Concerning the impact of stress on performance degradation，two hypotheses are given:

(1)measurement of degradation variables changes under different stresses;

(2)higher stress level which accelerates the degradation.

Based on the hypothesis above，the improved degradation path model under stress S can be expressed as

where g (·)is the degradation path function taking into account the influence of different stress on path function and h(·)is the compensation function of the degradation variable under stress S.θ and φ are the parameter vector of g(·)and h(·)，respectively.The compensation functions generally are empirical formulas acquired through experiments.θ (S)contains acceleration fuction to describe the relaitonship between stress S and some parameters in θ.For example，the Arrhenius function can reflect the accelerating effect of temperature on parameter β (in Table 1):

where E0is the activation energy，R is the gas constant，R =8.617 ×10-5eV/℃，and A is unknown parameter.

According to the cumulative exposure model［8］，we assume that the degradation rates only depend on the current stress and not on the history of the process.Let Δ (tij)denote the degradation value of g(·)between tijand ti，j-1for unit i:

where ti0=0 is the initial time of the degradation test，Sijis the average stress for unit i between tijand ti，j-1，θiand φiare the parameter vector of g(·)and h(·)for unit i.From Eq.(4)，we can find that the degradation varibable degrades over time through the cumulative exposure mode， and the measurement of the degradation varibable which is effected by stress is compensated through the item h(·).

With the degradation data，the nonlinear least square method is used to estimate parameters.The residual sum of squares for unit i is given as

After minimizing Ri，we can obtain the estimators of the parametersandfor unit i.

2 Reliabilty Assessment

With estimators of parameters and the threshold Df，we can extrapolate pseudo life Li，Sunder any given stress through Eq.(2).

Here we divide the stress into M levels between the peak and the valley value in the test.

Goodness-of-fit tests are performed on the pseudo life to compare the fit of different distributions.Generally a location-scale distribution，such as Weibull，normal，and lognormal，is selected to describe the distribution of failure-time.Now we take the lognormal for example to see how the parameters of the locationscale distribution are estimated.The probability density function of the lognormal distribution under stress Spis given as follows:

where p is the index of stress level，σpis the scale parameter，and μpthe location parameter.Since the failure mechanism can be assumed unchanged，scale parameter σsis constant［9］.According to the assumption，the total likelihood function under ITVS is

Then maximize Eq.(7)with the pseudo life to estimate the parameters，and.With the estimators of lognormal distribution，we can calculate the reliability indices such as MTTF，q-percentile life Tq，and draw the reliability curves under any given stress.

3 Numerical Example

3.1 Background Information

To establish an accurate life prediction model for Li-ion cells，five LiFePO418 650 cells are used to conduct the chargedischarge cycle life test.As the test is exposed to the ambient condition，the cell temperature，varying between 20-30 ℃，is hard to be controlled preciously and can be viewed as ITVS.In the test，the discharge capacity and average temperature of each cycle for each cell are measured respectively.Capacity degradation is often used to measure the deterioration of cell performance.As Li-ion cell capacity is extremely sensitive to temperature，it experiences fluctuated degradation in the cycle life test under ITVS (shown in Fig.1).

3.2 Degradation modeling

Fig.1 Li-ion cell capacity degradation curves under ITVS

Temperature has a double effect on Li-ion cells capacity［10］.On one hand，as temperature increases，the activity of Li-ion is enhanced and the effective internal resistance decreases，which results in the increment of cell capacity.On the other hand，higher temperature causes faster side reactions that bring permanent damage to a cell，and especially it will accelerate capacity degradation and shorten cell life.

According to the studies on the temperature characteristic of LiFePO4Li-ion cells［11］，Li-ion cell capacity is a function of temperature following the negative exponent law，namely

According to Ref.［12］，Li-ion cell capacity degrades following the power law path y = α + βtγover cycles，and the relationship between β and temperature S follows the Arrhenius acceleration function，namely

Substituting Eqs.(8)-(9)into Eq.(4)，we can get the improved degradation path model under ITVS after the merger of similar items:

The estimators of parameters for Eq.(10)are shown in Table 2.Taking Cell 1 for example，Fig.2 compares the tested capacity and the calculated values.

Table 2 Estimators of parameters of the capacity degradation model

3.3 Reliability assessment of cell capacity

Using the estimated parameters allows us to predict the failure-time of Li-ion cells at a certain temperature.Generally，the failure-time of a Li-ion cell is defined as the number of charge/discharge cycles the cell is capable of delivering before its capacity falls below 80% of its initial value［13］.Here we define the failure-time as g(·)falls below 80% of the average of g(0)，namely.

Fig.2 Tested capacity ＆ calculated value of Cell 1

Then we divide the stress for failure-time prediction into six levels between 20 ℃and 30 ℃and extrapolate the pseudo life of each cell under six stress levels through Eq.(5)，respectively (shown in Table 3).

Table 3 The pseudo life of five cells under six stress levels

Which distribution the pseudo life follows should be known before reliability assessment.The results of Kolmogorov-Smirnov goodness-of-fit test show that lognormal distribution is appropriate.Then maximize Eq.(7)with the pseudo life in Table 3 and we can obtain the parameter values= 5.619 4，= 5.553 1，= 5.487 6，= 5.422 9，= 5.359 1，= 5.296 2，and= 0.074 3.With the estimators of failuretime distribution，we can calculate the reliability indices.Here，the point estimate for MTTF and q-percentile life Tq(q=0.9，0.8，and 0.5)under six stress levels are shown in Table 4.Figure 3 illustrates the reliability curves of Li-ion cells.

Table 4 Point estimate and Interval estimate of reliability indices of Liion cells

4 Conclusions

Fig.3 Reliability curves for Li-ion cells

Degradation data under ITVS degradation test contains abundant information about product life and the relationship between life and stress.This paper investigates a novel method to solve this problem by improving the conventional degradation path model.With taking into account the influences of stress on the degradation variable measurements and degradation rates，a more precious degradation model under ITVS is established and the methods for reliability assessment are proposed.From the example，we find that the proposed model agrees quite well with the measured value.As the stress in an ITVS degradation doesn't need to be controlled preciously by expensive equipment，the method can be popularized with the advantage of low in cost.

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