*2.1. Lifetime Distribution*

The raw data of lifetime values are the basis for lifetime analyses. For statistical estimations, these empirical data need to be represented by functions. The failure probability *F(t)* can represent such empirical lifetime values in a continuous matter. There are different types of lifetime distributions to properly represent various failure behaviours. A widely used distribution function that can be adapted to a large variety of shapes is the Weibull distribution. The Weibull distribution as the 2-parametric version has a shape parameter *b* and a scale parameter *T* to vary its behavior. The distribution also has the statistical variable *t*, which is in this case the lifetime value [14,15].

The failure probability *F(t)* of the 2-parametric Weibull distribution is given by:

$$F(t) = 1 - e^{-\left(\frac{t}{T}\right)^b} \tag{1}$$

The derivative of the failure probability *F(t)* of the 2-parametric Weibull is the probability density function *f(t)*:

$$f(t) = \frac{b}{T} \cdot \left(\frac{t}{T}\right)^{b-1} \cdot e^{-\left(\frac{t}{T}\right)^b} \tag{2}$$

Both the shape parameter *b* and the scale parameter *T* have to be estimated so that the Weibull distribution represents the collected lifetime values well. A widely used robust analysis method is the maximum likelihood estimation (MLE). The likelihood function is maximised to find the most likely values of the distribution parameters for a given data set. The logarithmic likelihood function, which is easier to maximize, is based on the probability density function *f*(*t*) for a chosen lifetime distribution, in this case the 2-parametric Weibull distribution from Equation (2) with the parameters *b* and *T* to be estimated:

$$\Lambda = \ln L = \sum\_{i=1}^{n} \ln f(t\_i; b\_\prime \, T) \tag{3}$$

Subsequently, the values for the parameters must be found, which result in the highest value for this function. The partial derivatives of the likelihood function are set equal to zero for each parameter:

$$\frac{\partial \Lambda}{\partial b} = 0 \; \& \; \frac{\partial \Lambda}{\partial T} = 0 \tag{4}$$

These determined Weibull parameters represent the lifetime behaviour for the given data set [16].

In a previous work, Mürken et al. already investigated the failure behaviour of the entire battery based on workshop data on battery replacement. In this approach, however, there was no information about the reason for the failure and consequently aged batteries in a still functional state were also included [9].

The approach presented in this paper allows individual failure modes to be investigated separately but in relation to the entire sample. For this purpose, a methodology is presented to determine ageing models for individual ageing mechanisms from individual lifetime values of batteries, which are verified with the knowledge of the ageing mechanisms.

In order to perform separate Weibull analyses for different failure modes, the data must be censored with respect to the different failure modes. Furthermore, the approach is shown on a data set that shows the details of the individual failure modes only for a certain period of time, so additional censoring with respect to time must be applied. These adjustments are explained in the following section.

## *2.2. Censoring Data*

This approach is shown with lifetime values of batteries with different failure modes; therefore, censoring according to the investigated failure modes is necessary. This censoring according to individual failure modes is shown in Table 1; the Failure mode A, B as well as Functional are listed next to the lifetime values in hours. The "1" indicates which lifetime values are to be assigned to the failure mode and accordingly the "0" indicates which lifetime values are censored for the specific reliability analysis. This approach is shown on lifetime values based on the BCI *Report on Battery Failure Modes* 2015 in which the breakdown into individual failure modes covers only a period up to 4 years, although the entire sample contains batteries that survived beyond these 4 years up to 12 years. Therefore, the BCI study, used to present this approach, is considered to be a study performed over a certain period of time, in this case for 4 years, and the batteries that survive this period are censored. Thus a censoring according to time has to be applied, which is known as a right censoring of type 1. The censoring according to time is applied for lifetime values from 4 years onwards, thus all remaining lifetime values of the batteries that survived longer than these 4 years are assigned a lifetime value of *tcensored* = 4 years = 35,040 h as shown in Table 1.


**Table 1.** Example of a table to censor data according to the failure mode and time.

With both censoring methods the Table 1 is an example of a censoring table as used for reliability analysis and ageing model building in this work.

The censoring of the data with regard to different failure modes may result in only a small sample number for rarely occurring failure mechanisms. For such small subsamples, the accuracy of the Weibull distribution fitting suffers and the Weibull parameters determined in the reliability analysis are overestimated. To avoid this, a bias correction is applied to the reliability analyses carried out in this work. The unbiased shape parameter *bu* is obtained by multiplying the shape parameter *b* determined by MLE with the bias correction factor *U*:

$$b\_u = b \cdot \mathcal{U} \tag{5}$$

Since multiple failure modes are investigated, censored data are examined; the bias correction factor *U* according to Zhang et al. [17] is given by:

$$\mathcal{U} = \frac{1}{1 + \frac{1.3\mathcal{T}}{r - 1.92} \sqrt{\frac{N}{r}}} \tag{6}$$

The calculated bias correction factor *U* varies with the number of failures considered *r* and the total sample size *N*. If a bias correction is carried out, first the biased shape parameter *b* determined in Equation (4) is unbiased by Equation (5). Then the corresponding scale parameter *T* is calculated using Equation (4). According to Tevetoglu et al. this provides valid results for reliability analysis even for small sample sizes and recommends the use of bias correction methods [18]. However, even with larger samples it is advisable to apply a bias correction to increase accuracy [19].

The techniques presented in this section can be used to determine characteristic Weibull parameters for individual failure modes using the Weibull distribution from Section 2.1. In Section 2.3, time-dependent failure rates can be calculated with these determined shape parameters *b* and scale parameters *T*.

#### *2.3. Derivation of Time-Dependet Failure Rates*

Since Weibull analyses have been carried out and the specific shape parameters *b* and scale parameter *T* have been determined, time-dependent failure rates can now be calculated. With the probability density function *f(t)* in Equation (2) and the failure probability *F(t)* in Equation (1) the failure rate *λ*(*t*) is obtained with Equation (7). By using the determined shape parameters *b* and scale parameter *T*, the failure rate *λ* can be calculated for different times *t*. This also applies to Weibull parameters, which were determined using bias correction as described in Section 2.2.

$$
\lambda(t) = \frac{f(t)}{1 - F(t)} = \frac{b}{T} \cdot \left(\frac{t}{T}\right)^{b-1} \tag{7}
$$

In addition to the shape parameter *b*, the development of the failure rate over time gives an indication of the ageing and failure behaviour due to different failure modes of a component. In order to be able to show the presented methodology for batteries with different failure modes, it is applied to lifetime values derived from the field investigations of the BCI with the *Report on Battery Failure Modes* 2015. Therefore, in the following Section 3, this database is presented and the individual failure modes are explained for a better understanding in relation to the development of the failure rate over time.

#### **3. Database of Lifetime Values**

The presented methodology in Section 2 requires a suitable database to prove its validity. The *Battery Council International* presented a field investigation of lead batteries in *Report on Battery Failure Modes* in 2015 [13]. This study is used to prove the presented approach from Section 2 with real battery data and thus the applicability to electrochemical elements and their different ageing mechanisms. Therefore, the database and the major battery failure modes are presented in this section.
