Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 31181
A Bathtub Curve from Nonparametric Model

Authors: Eduardo C. Guardia, Jose W. M. Lima, Afonso H. M. Santos


This paper presents a nonparametric method to obtain the hazard rate “Bathtub curve” for power system components. The model is a mixture of the three known phases of a component life, the decreasing failure rate (DFR), the constant failure rate (CFR) and the increasing failure rate (IFR) represented by three parametric Weibull models. The parameters are obtained from a simultaneous fitting process of the model to the Kernel nonparametric hazard rate curve. From the Weibull parameters and failure rate curves the useful lifetime and the characteristic lifetime were defined. To demonstrate the model the historic time-to-failure of distribution transformers were used as an example. The resulted “Bathtub curve” shows the failure rate for the equipment lifetime which can be applied in economic and replacement decision models.

Keywords: Failure analysis, Parameter Estimation, Weibull distribution, bathtub curve, lifetime estimation

Digital Object Identifier (DOI):

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1882


[1] R. E. Brown and H. L. Willis, "The Economics os Aging Infrastructure," IEEE Power & Energy Magazine, pp. 36-43, May / June 2006.
[2] J. L. Velasquez-Contreras, M. A. Sanz-Bobi and S. G. Arellano, "General Asset Management Model in the Context of an Electric Utility: Application to Power Transformers," Electric Power Systems Research, vol. 81, pp. 2015-2037, 2011.
[3] L. Chmura, P. H. F. Morshuis, E. Gulski, J. J. Smit and A. Janssen, "Statistical analysis of subcomponent failurs in power transformers," Electrical Insulation Conference, 5 to 8 June 2011.
[4] A. E. B. Abu-Elanien, M. M. A. Salama and R. Bartnikas, "A Techno-Economic Method for Replacing Transformers," IEEE Transactions on Power Delivery, vol. 26, no. 2, pp. 817-829, 2011.
[5] W. Li, "Incorporatint Aging Failures in Power System Reliability Evaluation," IEEE Transactions on Power Systems, vol. 17, August 2002.
[6] H. Rinne, The Weibull Distribution, A Handbook, New York: CRC Press, 2008, pp. 27-31.
[7] W. Li, E. Vaahedi and P. Choudhury, "Power System Equipment Aging," IEEE Power & Energy Magazine, vol. 4, pp. 52-58, May/June 2006.
[8] H. L. Willis and R. R. Schrieber, Aging Power Delivery Infrastructures, Second Edition ed., Boca Raton, FL: CRC Press, 2013.
[9] Z. Li and J. Guo, "Wisdom About Age," IEEE Power & Energy Magazine, pp. 45-51, May/June 2006.
[10] M. M. A. El Aziz, D. K. Ibrahim and H. A. Kamel, "Estimation of the Lifetime of Electrical Components in Distribution Networks," The Online Journal on Electronics and Electrical Engineering, 2010.
[11] J. Endrenyi and G. J. Anders, "Aging, maintenance, and Reliability," IEEE Power & Energy Magazine, May/June 2006.
[12] X. Zhang, E. Gockenbach, V. Wasserberg and H. Borsi, "Estimation of the Lifetime of the Electrical Components in Distribution Networks," IEEE Transactions on Power Delivery, vol. 22, no. 1, January 2007.
[13] C. F. Joyce, "A Weibull Model to Characterize Lifetimes of Aluminun Alloy Electrical Wire Connections," Alcan International, 1989.
[14] E. A. Colosimo and S. R. Giolo, Análise de sobrevivência aplicada, São Paulo: Edgard Blucher, 2006.
[15] M. Hollander, D. A. Wolfe and E. Chicken, Nonparametrica Statistical Methods, Florida: John Wiley & Sons, 2013.
[16] R. Billinton and R. N. Allan, Reliability Evaluation of Engineering Systems: Concepts and Techniques, New York and London: Plenum Press, 1983
[17] R. B. Abernethy, The New Weibull Handbook, 5th ed., Florida: Robert B. Abernethy, 2006.
[18] ReliaWiki, Life Data Analysis Reference Book, Tucson:, 2013.
[19] Lifetime Reliability Solutions, "Do a Timeline Distribution Before doing a Weibull failure Analysis,” 2013. (Online). Available: