Providing Better Insights
Improved life analyses using degradation testing
by Necip Doganaksoy, Gerald J. Hahn and William Q. Meeker
In an earlier Statistics Roundtable column,1 we advocated for the use of degradation testing (that is, monitoring change in a quality or performance characteristic) instead of time to failure. Such testing is especially useful in (often accelerated) life testing when it is not possible or practical to conduct life tests until many, or possibly all, of the test units have failed.
In such cases, all that is known about the unfailed units, or so-called censored observations, is that their times to failure exceed their current test times. Censored observations often seriously weaken statistical analyses, leading analysts to turn instead to physically meaningful degradation measurements.
Over the years, degradation testing has become increasingly popular. Modern sensor technologies enable capture and storage of vast amounts of data on measurements related to product degradation over time.
In this column, we renew the argument for conducting meaningful degradation testing. Such testing can provide useful additional insights, give improved extrapolations and, in some situations, provide more precise analyses. An earlier article emphasized the first of these advantages and provided extensive technical detail.2
Battery performance study
Our case study concerns the development of a new battery consisting of electrochemical cells.3 A key application of the battery is to provide a backup source of power for uninterruptible power supplies. The battery was designed to carry the full load for up to 15 minutes to allow sufficient time to start the main backup power source (for example, a diesel generator).
Because a battery is subjected to repeated charge and discharge cycling, it gradually loses its ability to hold power and, ultimately, fails. A major goal of the study was to estimate the probability that the battery could sustain a 15-minute discharge at its use environment after 1,000, 5,000, 10,000 and 20,000 cycles. In addition, it was hoped that the test results would provide insights into failure mechanisms that would trigger product improvement.
An accelerated test (AT) was undertaken to investigate battery performance using two types of acceleration:
- Use rate acceleration. In laboratory testing, you can run discharge cycles at a considerably higher frequency than in real-life applications. Allowing for cooling between cycles and recharging the battery, it took about one hour to complete a 15-minute discharge cycle.
- Aging rate acceleration. Accelerated aging required testing at more severe conditions than the nominal use condition of a temperature of 175°C and rated power of 135 watts/cell.
The AT was designed as a two-level full-factorial experiment for temperature and power, plus an intermediate condition center point. Five batteries were tested at each of these five conditions. A battery was considered to reach end of life when it could no longer sustain a discharge cycle for 15 minutes.
Test results and traditional life data analysis
The test results were analyzed after 9,000 cycles and are shown in Figure 1. All but two of the 25 batteries (both at the low power and temperature test condition) had failed by that time.
Based on physical and empirical justifications, median log life was expressed as a function of temperature and power, incorporating the Arrhenius relationship for temperature and the power law for power. A lognormal distribution with constant shape parameter (standard deviation of log life) was used to model the battery lifetimes at each test condition.4
Table 1 shows the resulting estimated probability of failure at the nominal use condition (175°C and 135 watts/cell), as determined from the fitted model, after 1,000, 5,000, 10,000 and 20,000 cycles, and the associated (large sample theory) 95% confidence intervals around these estimates.
This traditional analysis provided a good fit to the data. Thus, there did not appear to be any incentive to develop a further model based on degradation data, especially because there was little censored data. In the remainder of this column, we show that meaningful degradation data and their appropriate analysis can, nevertheless, provide important added benefits.
Battery degradation data
Resistance during discharge measures opposition to electric current and thus can be used as the primary measure of battery performance degradation. As a battery ages, its internal resistance generally increases. Such increased resistance adversely affects the duration of the discharge cycle. This leads to a direct association between end of life (that is, the battery can no longer sustain a discharge at the rated power for 15 minutes) and resistance. A battery reaches end of life when its resistance exceeds a defined threshold.
Resistance measurements were obtained nondestructively via electrical instrumentation for each test battery using a standard discharge protocol after (approximately) every 100 cycles up to 1,000 cycles and every 500 cycles thereafter. Testing was continued on most batteries beyond their nominal failure times.
Figure 2 displays the observed resistance paths for each of the five batteries at each of the five experimental conditions, as well as the failure threshold (taken as 15 milliohms for conditions one, three and four and, for engineering purposes, slightly higher for conditions two and five).
Analysis of degradation data
Examination of the simple plots in Figure 2 conveys important information about degradation patterns and battery life. In particular, we note that:
- Initially, all batteries had a resistance of about 5 milliohms. The batteries at conditions one, three and four then experienced an initial improvement (that is, drop in resistance). This drop is most pronounced at condition one, at which it lasts for about 1,500 cycles, and least prominent at condition three. In contrast, the batteries at conditions two and five (both at the "high" power level of 180 watts/cell) did not experience an initial resistance drop.
- Resistance increases at an approximately constant rate over time (as evidenced by the linearity of the plots) at conditions two and five from the outset, and at the other three conditions following the initial drop.
- The rate of degradation increases with increasing temperature, as can be seen by comparing the steepness of the plotted curves at conditions one and two versus conditions four and five, respectively.
- The rate of degradation varies from battery to battery even at the same test condition, as evidenced by the different slopes of the individual battery plots at each of the test conditions.
In addition, a statistical (nonlinear mixed effects) model was fitted to the data.5 This included both so-called fixed effect and random effect terms. The fixed effect terms related power, temperature and elapsed time to resistance. The random effect terms quantified variability attributable to differences in initial battery resistance, variability in degradation rates between batteries, and individual battery measurement and other variability. Although more complicated conceptually than the graphical analyses, the statistical analyses quantified the results and could be readily implemented using available statistical software.
Advantages of obtaining degradation data
Additional physical insights: The degradation data provided important insights into the mechanisms through which temperature and power affect performance over life.6 For example, the modeling of the impact of temperature on resistance yielded useful information about the effect of operating temperature excursions on degradation.
The resistance drop (that is, performance improvement) during the approximately first 1,500 cycles at three test conditions was unexpected and of particular interest. Exploring its mechanism led the product engineering team to a better understanding and a possible improvement in design (for example, extend the improvement period beyond 1,500 cycles).
Potential of improved precision: Under certain circumstances, the use of degradation measurements instead of life data will lead to improved precision in estimation.
This is the case, for example, when there is extensive censoring relative to the distribution characteristic that is to be estimated; for example, in estimating the 0.10 quantile of a life distribution from just three failures among 50 test units. We would not expect any appreciable improvement in precision from using the degradation data in the battery degradation example—especially in estimating relatively low failure probabilities—because all but two batteries ran beyond their failure definition threshold.
Improved extrapolation: Even though the traditional lifetime data analysis appeared to be quite reasonable in our example, the possibility of a change in degradation mechanism raises questions about its validity outside the region of experimentation. Extrapolation is always dangerous, and especially so when it is not based on a solid theoretical foundation. Appropriately selected degradation data that relate directly to the failure mechanism provides insurance against inappropriate extrapolation.
As meaningful degradation data become more readily accessible, the procurement and analysis of such data should be considered an integral part of product reliability modeling.7
References and Notes
- William Q. Meeker, Necip Doganaksoy and Gerald J. Hahn, "Using Degradation Data for Product Reliability Analysis," Quality Progress, June 2001, pp. 60-65.
- Necip Doganaksoy and D.B. Hall, "Gaining Physical Insights From Degradation Data: A Case Study," Journal of Quality Technology, Vol. 45, 2013, pp.188-199.
- Ibid. Greater detail on the battery performance study can be found in this article.
- Ibid. The data were analyzed by Doganaksoy and Hall using standard methods described in the literature. Also see: Wayne B. Nelson, Accelerated Testing: Statistical Models, Test Plans, and Data Analyses, Wiley, 1990; William Q. Meeker and Luis A. Escobar, Statistical Methods for Reliability Data, Wiley, 1998; and Gerald J. Hahn, William Q. Meeker and Necip Doganaksoy, "Speedier Reliability Analysis," Quality Progress, June 2003, pp. 58-64.
- Doganaksoy and Hall, "Gaining Physical Insights From Degradation Data: A Case Study," see reference 2.
- For further discussion, see: Wayne B. Nelson, Accelerated Testing: Statistical Models, Test Plans, and Data Analyses (chapter 11), Wiley, 1990: William Q. Meeker and Luis A. Escobar, Statistical Methods for Reliability Data (chapters 13 and 21), Wiley, 1998: and William Q. Meeker, Necip Doganaksoy and Gerald J. Hahn, "Using Degradation Data for Product Reliability Analysis," Quality Progress, June 2001, pp. 60-65.
Necip Doganaksoy is principal statistician at GlobalFoundries in Malta, NY. He has a doctorate in administrative and engineering systems from Union College in Schenectady. Doganaksoy is a fellow of ASQ and the American Statistical Association.
Gerald J. Hahn is a retired manager of statistics at the GE Global Research Center in Schenectady, NY. He has a doctorate in statistics and operations research from Rensselaer Polytechnic Institute in Troy, NY. Hahn is a fellow of ASQ and the American Statistical Association.
William Q. Meeker is professor of statistics and distinguished professor of liberal arts and sciences at Iowa State University in Ames. He has a doctorate in administrative and engineering systems from Union College in Schenectady, NY. Meeker is a fellow of ASQ and the American Statistical Association.