**4. Results**

Since this work is about the methodology for determining time-dependent failure rates for individual failure modes and no field data from batteries are available that represent future load scenarios and failure criteria, the results are presented in normalised form. This section shows the results of the Weibull analysis performed with the field data derived from the *Report on Battery Failure Modes* by BCI 2015 as described in Section 3.

Nevertheless, the older database is used to demonstrate the methodology from Section 2, but the absolute failure rates *λ*(*t*) are not relevant for the reasons mentioned above. The Weibull parameters are presented, but the scale parameter *T* is normalised with respect to a typical vehicle lifetime. Subsequently, the calculated time-dependent failure rates *λ*(*t*) are also presented normalised. The time is normalised with respect to a typical vehicle lifetime *τvehicle*. The failure rates *λFailureMode*(*t*) of the individual failure modes from Section 3 are normalised to the failure rate *λS*(*t*) of the still functional batteries of the failure mode *Serviceable* after *τvehicle*,0.6 = 0.6 · *τvehicle* vehicle lifetimes according to the following Equation (8):

$$
\lambda\_{\text{FailureMode},normalised}(t) = \frac{\lambda\_{\text{FailureMode}}(t)}{\lambda\_S(\pi\_{\text{vehicle},0.6})} = \frac{\lambda\_{\text{FailureMode}}(t)}{\lambda\_{S,0.6}}\tag{8}
$$

The parameters determined by Weibull analyses are summarised in a table in the following section and the determined time-dependent failure rates *λ*(*t*) are listed individually.

#### *4.1. Estimated Weibull Parameters*

The results of the Weibull analyses are obtained by parameter estimation with maximum likelihood estimation (MLE) and the application of bias correction. Table 2 shows the results of the estimated Weibull parameters. For the available five failure modes, the determined shape parameters *b* and the normalised scale parameter *T* are listed.

**Table 2.** Determined Weibull parameters for the failure modes of the total sample. The table shows the estimated shape parameter *b* for each failure mode and the normalised scale parameter *T* by a typical vehicle lifetime *τvehicle*. (Analysis method: MLE; Rank method: Median ranks; Confidence bounds method: Fisher matrix; unbiasing parameters)


The parameters listed in Table 2 give an indication for the failure behaviour of the obtained component. In general, components can fail due to different reasons. The classic bathtub curve is divided into three areas that show different stages of failure behaviour and thus are linked to different magnitudes of the shape parameter *b*. Early failures with a decreasing failure rate *<sup>λ</sup>early* and a shape parameter *<sup>b</sup>* << 1 are often due to manufacturing defects, incorrect storage and long storage times. Random failures with a constant failure rate *λrandom* and a shape parameter *b* ≈ 1 are often encountered in electronic components. The third area of the bathtub curve is indicated as wear out or ageing failures with an increasing failure rate *<sup>λ</sup>ageing* and a shape parameter *<sup>b</sup>* >> 1. As batteries age with first use, their failure rate is expected to be continuously or exponentially increasing during use like *<sup>λ</sup>ageing* with a shape parameter *<sup>b</sup>* >> 1. The second parameter determined is the scale parameter *T* . This parameter indicates the time at which the probability of failure *F*(*T*) = 63.2% is reached and a corresponding amount of parts of the sample have failed.

In the following Section 4.2, the results of the Weibull analyses are presented as timedependent failure rates for the different failure modes listed in Table 2 and its parameters and failure behaviour are discussed.

#### *4.2. Failure Rates of Lead Battery Failure Modes*

The Weibull parameters listed in Table 2 are obtained by Weibull adaptions according to the methodology presented in Section 2 to lifetime values of the different failures modes as presented in Section 3. According to Section 2.3 the failure rate *λ*(*t*) can be calculated 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*. The five failure modes and their Weibull parameters from Table 2 are used to calculate these failure rates *λ*(*t*). In the following, this is shown individually with the Weibull parameters from Table 2 in normalised graphs by a typical vehicle lifetime *τvehicle*. All failure rates are displayed with their upper and lower confidence interval for a confidence level of 95% (on time, type 1).

The failure rates are normalised by failure rate *λS*,0.6 of the functional batteries of the failure mode *Serviceable* according to Equation (8). Consequently the still good batteries and their failure rate *λS*(*t*) is displayed at first in Figure 3.

**Figure 3.** The normalised failure rate *λS*(*t*) of the failure mode *Serviceable* with a shape parameter *b* = 1.239 and a normalised scale parameter *T* = 1.134 is shown. In addition, the upper and lower confidence limits for a confidence level of 95% are displayed.

The pattern of the failure rate *λS*(*t*) of the failure mode *Serviceable* in Figure 3 clearly belongs to the category of random failures with a shape parameter very close to *b* = 1. This makes sense with the information about this failure mode according to Section 3.7. Since the batteries are still functional and there is no fault due to ageing, the battery changes in car workshops due to *Serviceable* are highly random and logically there is no progression, as expected with an ageing mechanism.

The failure rate *λOC*(*t*) of the first real failure mode *Open Circuit* is shown in Figure 4.

**Figure 4.** The normalised failure rate *λOC*(*t*) of the failure mode *Open Circuit* with the estimated shape parameter *b* = 1.819 and a normalised scale parameter *T* = 2.127 is shown. In addition, the upper and lower confidence limits for a confidence level of 95% are displayed.

The failure rate *λOC*(*t*) in Figure 4 shows a pattern that cannot be clearly assigned to either random or ageing mechanisms. The failure rate *λOC*(*t*) increases with time for the shape parameter *b* = 1.819 determined by Weibull analysis. However, no clear assignment to ageing effects with linear or exponential increase and shape parameter *b* ≥ 2 or to completely random failures with shape parameter *b* ≈ 1 is possible. This uncertainty is also shown by the wide confidence bounds. Only a few batteries failed with this failure mode, as can be seen in Figure 2. The small sample size of this failure mode results in very wide confidence bounds. Thus, on the one hand, the progression of the failure rates could be rather exponential, as shown by the upper confidence limit. On the other hand, it could be almost randomly distributed, as the lower confidence limit shows. This could be explained by the knowledge of the failure mode *Open Circuit* from Section 3.3. It can be assumed that the *Open Circuit* is mainly caused by a mechanical impact from the outside and only in rare cases, e.g., corrosion inside leads to such strong instabilities that a rupture can occur due to normal vibrations in the vehicle. Before an *Open Circuit* occurs, a failure of the battery is more likely due to its greatly increased internal resistance. This Weibull statistical analysis shows that for the *Open Circuit* failure mode, different ageing mechanisms overlap and the failure rate does not show a consistent failure behaviour. According to the failure description in Section 3.3, an assignment of the failure mode *Open Circuit* to the sudden failures would fit but cannot be done definitively with this database. For this purpose, a larger sample and a longer observation period would make sense in order to obtain more meaningful results.

The next failure mode according to Table 2 is *Plates and Grids* and the corresponding failure rate *λPG*(*t*) is shown in Figure 5.

The development of the failure rate *λPG*(*t*) in Figure 5 shows a moderate exponential increase. The shape parameter *b* = 2.812 clearly belongs to the ageing effects, which are characterised by a shape parameter *b* >> 1. With the knowledge of the ageing mechanism *Plates and Grids* from Section 3.4, the curve in Figure 5 is understandable. When the battery is not in use, only minor ageing takes place. Only with increasing time and use of the battery does the ageing effect become apparent and eventually lead to a failure of the battery due to increased internal resistance. The allocation of the failure mode *Plates and Grids* to the gradual faults is suitable for the curve according to Figure 5, resulting from the shape parameter *b*.

**Figure 5.** The normalised failure rate *λPG*(*t*) of the failure mode *Plates and Grids* with a shape parameter *b* = 2.812 and a normalised scale parameter *T* = 0.582 is shown. In addition, the upper and lower confidence limits for a confidence level of 95% are displayed.

The following Figure 6 shows the normalised failure rate *λWOA*(*t*) of the failure mode *Worn out and Abused*.

**Figure 6.** The normalised failure rate *λWOA*(*t*) of the failure mode *Worn out and Abused* with a shape parameter *b* = 2.255 and a normalised scale parameter *T* = 0.826 is shown. In addition, the upper and lower confidence limits for a confidence level of 95% are displayed.

The shape of the failure mode *Worn out and Abused* in Figure 6 is almost linear with a shape parameter close to *b* ≈ 2. This clearly indicates a wear failure or ageing mechanism. In addition, the scale parameter *T* shows that many batteries survive for a very long time of a typical vehicle lifetime until this failure mode becomes sufficiently dominant to cause the battery to fail. This can be well explained by Section 3.5, since capacity loss is decisively caused by cyclisation of the battery, which requires the battery to be used for a

long time. The clear assignment of the failure mode to the ageing effects fits well with the gradual faults.

The failure mode *Short Circuit* is shown in Figure 7.

**Figure 7.** The normalised failure rate *λSC*(*t*) of the failure mode *Short Circuit* with a shape parameter *b* = 2.637 and a normalised scale parameter *T* = 0.596 is shown. In addition, the upper and lower confidence limits for a confidence level of 95% are displayed.

The development of the failure rate *λSC*(*t*) in Figure 7 shows a slightly exponential increase with the shape parameter *b* = 2.637. This indicates a strongly use-dependent ageing mechanism. According to Section 3.6, the failure mode *Short Circuit* can occur for different reasons. It can only occur completely randomly due to misuse from the outside or extreme operation out of specifications with frequent severe deep discharges, so dendrites can grow at an early stage. In normal use, energy management should protect the battery from frequent deep discharge conditions so that dendrite growth, if it occurs at all, takes a long time to develop. The comparatively small scale parameter *T* indicates vehicles in which the energy management is poor from the start and the battery is thus exposed to frequent deep discharges early on.

The results of the Weibull adaptation and the Weibull parameters determined on the basis of the BCI 2015 *Report on Battery Failure Modes* field data and the visualised failure rates *λ*(*t*) for the individual failure modes are summarised and discussed in the following Section 5.

## **5. Conclusions**

Reliability analyses in vehicles are becoming increasingly important due to the complexity of vehicles. To ensure safe operation, proof of reliability must be provided for more components. In this study, an approach to determine failure rates for lead batteries from field data is presented. It has been shown that the statistical reliability analysis can also be applied to electrochemical components to derive appropriate failure rates. In particular, by knowing the reason for the failure of the lead battery, the developments of the failure modes over time as shown can be understood but also justified.

Batteries with the failure mode *Serviceable* show a progression of the failure rate over time, which clearly indicates random failures, as the failure description indicated. The failure mode *Open Circuit*, on the other hand, shows a failure behaviour over time that cannot be clearly assigned to random or ageing failures. The small sample size and the short observation period can have a negative influence on the accuracy of the Weibull analysis. For a longer observation period, a clearer result could emerge for the failure mode *Open Circuit* and a clearer assignment to sudden faults or gradual faults could be possible. The two failure modes, *Plates and Grids* as well as *Worn out and Abused*, show progressions of the failure rates over time, which can clearly be assigned to the ageing mechanisms. The last investigated failure mode *Short Circuit* can, according to the description in Section 3.6, arise due to dendrites slowly developing through deep discharges, also called soft short, or due to hard short circuits caused by external rather random influences. The determined failure rate for the failure mode *Short Circuit* shows an exponential progression over time and is rather to be assigned to the wear or ageing mechanisms for these data.

With the presented approach, ageing models for the investigated failure modes can be determined from field data, which are useful for the safety analysis in the power supply system and the entire vehicle. The field data examined are still based on conventional vehicles in which the lead battery is used as a typical starter battery. In future driving applications, however, a 12 V battery will be needed in the power supply system, which will not be used for starting the engine anymore but much more for the safe supply of energy in critical cases. This means that the 12 V battery must provide the energy and power for a so-called safe stop scenario, which brings the vehicle to a safe stop at the emergency lane by a double lane change. These requirements are of course different from those of a conventional, mainly manually driven vehicle. Thus, the battery is continuously monitored and operated in a favourable operating range for the battery. By monitoring the battery's condition and ageing progress, it is possible to replace the battery before it fails, so that critical failures can be avoided.

Nevertheless, this study used these old field data from outdated batteries just to show the approach for future field investigations on batteries to determine relevant failure rates for future driving applications. Therefore, in further investigations, the collected lead batteries could be tested with the new failure criterion of a defined safe stop scenario when collecting field data. Furthermore, the determination of the failure mode by tear down analysis is prone to subjective evaluations. One possibility would be to electrically test the collected batteries and evaluate their ageing progress with objective criteria. This would be useful for assessing the reliability of 12 V batteries such as lead batteries for future driving applications.

**Author Contributions:** Conceptualization, R.C., F.H. and K.P.B.; methodology, R.C. and F.H.; software, R.C.; validation, R.C. and F.H.; formal analysis, R.C. and F.H.; investigation, R.C. and F.H.; resources, R.C.; data curation, R.C.; writing—original draft preparation, R.C.; writing—review and editing, R.C.; visualization, R.C.; supervision, F.H. and K.P.B.; project administration, R.C. and F.H.; All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

