**1. Introduction**

The change in individual mobility is particularly evident in the electrification and automation of vehicles. The electrification of the powertrain leads to an increase in the weight of the vehicles. In addition to automation functions, this leads to higher safety requirements especially for electric braking and steering power assistance. Apart from these essential safety-relevant consumers steering (e.g., steer-by-wire) and braking (e.g., brake-by-wire), automated driving functions require sensors and actuators for position and environment detection as well as for control [1]. These components are supplied via the 12 V on-board power supply system (also known as powernet) of the vehicle and need a safe power supply to ensure a safe operation.

Therefore, knowing the failure behaviour and failure rates of the components and the power supply is crucial for the overall safety concept of the power supply system and the vehicle itself. Dominguez-Garcia et al. have already dealt with this increasing problem of safety-relevant consumers and their power supply in [2] by designing safe power supply system topologies. Subsequently, Kurita et al. also addressed the design of on-board power supply systems in [3] and placed the focus on the importance for future driving applications, which supports the actuality of the topic.

In such on-board power supply systems, the power supply is provided during normal operation by the alternator or DCDC converter in hybrid or electric vehicles. In addition,

**Citation:** Conradt, R.; Heidinger, F.; Birke, K.P. Methodology for Determining Time-Dependent Lead Battery Failure Rates from Field Data. *Batteries* **2021**, *7*, 39. https://doi.org/ 10.3390/batteries7020039

Academic Editors: Carlos Ziebert and Joeri Van Mierlo

Received: 30 March 2021 Accepted: 2 June 2021 Published: 15 June 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

the 12 V battery is a main component of the power supply system and, besides buffering peak loads and supplying the vehicle with standby power, e.g., when parking, it also acts as a backup power supply. This backup function of the 12 V battery in the on-board power supply system is now becoming very important and the search for a safe power supply at the 12 V level with proof of reliability must be conducted.

Modern vehicle architectures consist to a large extent of electrical and/or electronic (E/E) systems. These systems are crucial for the safe operation of the vehicle and must therefore meet functional safety standards. The vehicle power supply system, as an obligatory component for safe operation, must therefore be developed according to the ISO 26262 standard [4,5]. Batteries are not in the scope of the ISO 26262 standard although the battery is a main component of the power supply system and its failure rates need to be included. In contrast to electrical components with constant failure rates, electrochemical components such as batteries are subject to ageing and thus do not have constant failure rates over time. Nevertheless, this failure behaviour and the resulting increasing failure rates must be included in the functional safety concept. Therefore, a method for determining battery failure rates must be established and checked for plausibility with the failure mechanisms investigated.

The lead battery can be part of a safe power supply system architecture if its ageing mechanisms and faults are manageable. In an early stage Albers et al. identified in [6] the necessity for further investigations of batteries in the context of functional safety requirements in the changing environment of vehicles with new functions. Proving reliability is difficult for electrochemical components such as the lead battery due to their ageing behaviour over time. Previous investigations on the reliability of the lead battery, such as those carried out by Albers et al. in [7], examined the batteries by tear down analysis to obtain a probability distribution of the failures that occurred. With the failure distribution, they were able to calculate fixed failure rates for the lead battery by using the corresponding ADAC failure statistics [8]. However, the ageing behaviour and thus the failure rate of electrochemical elements such as the lead battery does not follow the bathtub curve like electronic components, which are dominated by random failures in terms of time. The ageing process of electrochemical components starts with first use and batteries can be considered as wear out components. This means that the ageing behaviour behaves like the third area of the bathtub curve with increasing failure rate and therefore a fixed failure rate is less appropriate for wear out components.

In [9], Mürken et al. chose the approach of using workshop data on the battery exchange of the vehicle battery and its lifetime values. With the help of the individual lifetime values, it was possible to determine an ageing model based on a Weibull distribution for the failure of the battery. This made it possible to calculate the reliability of the overall battery over time and thus time-dependent failure rates. A consideration of the different failure modes and thus corresponding failure rates could not be investigated in this study due to lack of knowledge about the occured failure mechanisms. Furthermore, many batteries are replaced without faults and thus these functioning batteries are included in the failure rates in this approach.

Varying ageing mechanisms lead to different ageing behaviour; this was already recognised by Kumar et al. in [10]. It was shown that lifetime estimates of batteries are possible with Weibull analyses and that the Weibull parameters determined differ for varying failure mechanisms. Thus, an assignment of the failure mechanisms to the gradual faults or the sudden faults was possible. However, it remains open how the failure rate for certain failure modes develops over time and whether this is plausible with the knowledge of the ageing mechanism.

Since recent studies by the ADAC in [11] again show that the battery is responsible for a large number of vehicle failures as well as increasing safety requirements for future driving applications, it is important to include specific battery failure rates. Therefore, this article presents a method to calculate time-dependent failure rates of different investigated battery failures to get a better understanding of the progressing of ageing mechanisms. For this purpose, the knowledge about the dominant failure mechanism is required in addition to the lifetime values of the failed batteries. With the knowledge gained about the development of the failure mechanisms over time, the diagnostic mechanisms of the electric battery sensor can be prioritised and further developments can be made to ensure a safe power supply from the battery.

The main focus of this study is on the methodology for determining time-dependent failure rates and not on absolute values of the failure rates. The reason for that is the data available so far from 12 V lead batteries come from conventional combustion vehicles and were partly operated without any energy management and therefore do not cover the future application of a safety battery in backup operation at all. The failure criterion in previous studies is the ability to start the vehicle by providing the high starting current, whereas the requirement for a safety battery is the safe supply in the event of a safe stop scenario. For future investigations, the current profile required in a safe stop scenario would therefore be more suitable as a failure criterion. Nevertheless, in order to present the methodology with real field data, the collected field data of the Battery Council International (BCI) are used, which, however, come from such conventional vehicles. The BCI publishes a *Report on Battery Failure Modes* every five years; the most recent one from 2020 [12] is only available for purchase, which is why the methodology in this paper is shown using the example of the study from 2015 [13].

This study begins with an introduction to the methodology for determining timedependent failure rates for different failure modes based on field data. Subsequently, a database is presented to demonstrate the methodology. Finally, the results of timedependent failure rates are presented in normalised format and then discussed.
