Journal of Korean Society of Coastal and Ocean Engineers (한국해안·해양공학회논문집)

Korean Society of Coastal and Ocean Engineers (한국해안·해양공학회)
권호 표지
DOI QR코드

DOI QR Code

Estimation of Residual Useful Life and Tracking of Real-time Damage Paths of Rubble-Mound Breakwaters Using Stochastic Wiener Process

추계학적 위너 확률과정을 이용한 경사제의 실시간 피해경로 추적과 잔류수명 추정

  • Lee, Cheol-Eung (Department of Architectural, Civil, and Environmental Engineering, Kangwon National University)
  • 이철응 (강원대학교 건축.토목.환경공학부)
  • Received : 2020.05.07
  • Accepted : 2020.06.04
  • Published : 2020.06.30
Download PDF
⟨ Previous Next ⟩

Abstract

A stochastic probabilistic model for harbor structures such as rubble-mound breakwater has been formulated by using the generalized Wiener process considering the nonlinearity of damage drift and its nonlinear uncertainty, by which the damage path with real-time can be tracked, the residual useful lifetime at some age can also be analyzed properly. The formulated stochastic model can easily calculate the probability of failure with the passage of time through the probability density function of cumulative damage. In particular, the probability density functions of residual useful lifetime of the existing harbor structures can be derived, which can take into account the current age, its present damage state and the future damage process to be occurred. By using the maximum likelihood method and the least square method together, the involved parameters in the stochastic model can be estimated. In the calibration of the stochastic model presented in this paper, the present results are very well similar with the results of MCS about tracking of the damage paths as well as evaluating of the density functions of the cumulative damage and the residual useful lifetime. MTTF and MRL are also evaluated exactly. Meanwhile, the stochastic probabilistic model has been applied to the rubble-mound breakwater. The related parameters can be estimated by using the experimental data of the cumulative damages of armor units measured as a function of time. The theoretical results about the probability density function of cumulative damage and the probability of failure are very well agreed with MCS results such that the density functions of the cumulative damage tend to move to rightward and the amounts of its uncertainty are increased as the elapsed time goes on. Thus, the probabilities of failure with the elapsed time are also increased sharply. Finally, the behaviors of residual useful lifetime have been investigated with the elapsed age. It is concluded for rubble-mound breakwaters that the probability density functions of residual useful lifetime tends to have a longer tail in the right side rather than the left side because of the gradual increases of cumulative damage of armor units. Therefore, its MRLs are sharply decreased after some age. In this paper, the special attentions are paid to the relationship of MTTF and MRL and the elapsed age of the existing structure. In spite of that the sum of the elapsed age and MRL must be equal to MTTF deterministically, the large difference has been shown as the elapsed age is increased which is due to the uncertainty of cumulative damage to be occurred in the future.

추계학적 WP을 이용하여 불확실성을 고려하면서 항만 구조물의 실시간에 따른 피해와 파괴확률 그리고 잔류수명을 해석할 수 있는 모형을 수립하였다. 과거부터 현재까지의 피해상태와 미래에 발생될 피해 진행 과정에 포함되는 불확실성을 고려할 수 있는 추계학적 확률모형이다. 피해경로를 추적할 수 있으며 누적피해의 밀도함수도 산정하여 파괴확률을 추정할 수 있다. 또한 구조물의 잔류수명에 대한 밀도함수도 구할 수 있다. 최소자승법과 최우도법을 이용하여 모형의 파라미터를 추정할 수 있는 방법도 제시하였다. 검증을 위해 시간의 진행에 따른 누적피해와 잔류수명에 대한 밀도함수를 산정하고 해석하였는데 이론적인 결과가 MCS 기법의 수치적인 결과와 매우 잘 일치하였다. 또한 내구수명이나 잔류수명에 대한 밀도함수의 거동과 MTTF와 MRL이 정량적으로 잘 일치하였다. 한편 본 연구에 수립된 모형을 경사제에 적용하기 위하여 피복재 피해에 대한 수리모형 실험자료를 활용하여 모형의 파라미터들을 추정하였다. 시간의 진행에 따른 피복재 누적피해의 밀도함수와 파괴확률을 산정하였는데 MCS의 결과와 이론적인 결과가 매우 잘 일치하였다. 경과시간이 클수록 밀도함수가 우측으로 이동하면서 불확실성이 커지면서 파괴확률이 급격하게 증가하였다. 또한 재령에 따른 잔류수명의 거동특성을 해석하였는데, 잔류수명의 분포함수에서 좌측보다는 우측 꼬리 부분이 길게 형성되어 MRL이 급격하게 감소하는 경향을 보였다. 이는 경사제 피복재의 피해가 완만하게 증가하는 현상을 반영한 것으로 판단된다. 특히 재령과 내구수명 그리고 잔류수명의 관계를 해석하였는데, 재령이 오래될수록 재령과 MRL의 합이 MTTF와 큰 차이를 보이고 있다. 이는 재령이 증가하면 잔류수명의 평균인 MRL이 불확실성에 의하여 급격히 감소하기 때문이다.

Keywords

References

  1. Ahmad, R. and Kamaruddin, S. (2012). An overview of time-based and condition-based maintenance in industrial application. Compu. Ind. Eng., 63(1), 135-149. https://doi.org/10.1016/j.cie.2012.02.002
  2. Bagdonavicius, V. and Nikulin, M. (2000). Estimation in degradation models with explanatory variables. Lifetime Data Analysis., 7, 85-103. https://doi.org/10.1023/A:1009629311100
  3. CEM (2006). Coastal Engineering Manual, U.S. Army Corps of Engineers, Washington D.C., USA.
  4. Chhikara, R.S. and Folks, J.L. (1977). The inverse Gaussian distribution as a lifetime model. Technometrics, 19, 461-468. https://doi.org/10.1080/00401706.1977.10489586
  5. Cox, D.R. and Miller, H.D. (1965). The theory stochastic processes, Methuen and Com., London.
  6. Crowder, M. and Lawless, J. (2007). On a scheme for predictive maintenance. European J. of Oper. Res., 176, 1713-1722. https://doi.org/10.1016/j.ejor.2005.10.051
  7. Cui, L.R., Loh, H.T. and Xie, M. (2004). Sequential inspection strategy for multiple systems under availability requirement. European J. of Oper. Res., 155, 170-177. https://doi.org/10.1016/S0377-2217(02)00822-6
  8. Dieulle, L., Berengguer, C., Grall, A. and Roussignol, M. (2003). Sequential condition-based maintenance scheduling for a deteriorating system. European J. of Oper. Res., 150, 451-461. https://doi.org/10.1016/S0377-2217(02)00593-3
  9. Doksum, K.A. and Hoyland, A. (1992). Models for variable-stress accelerated life testing experiments based on Wiener processes and the inverse Gaussian distribution. Technometrics, 34, 74-82. https://doi.org/10.2307/1269554
  10. Ebrahimi, N. (2009). The mean function of a repairable system that is subjected to an imperfect repair policy. IIE Transactions, 41, 57-64. https://doi.org/10.1080/07408170802322671
  11. Elwang, A. and Gebraeel, N.Z. (2009). Real-time estimation of mean remaining life using sensor-based degradation models. J. Manuf. Sci. Eng., 131, 51-59.
  12. Gebraeel, N.Z., Lawley, M.A., Li, R. and Ryan, J.K. (2005). Residual-life distribution from component degradation signals: A bayesian approach. IIE Transactions, 37, 543-557. https://doi.org/10.1080/07408170590929018
  13. Gasperin, M., Juricic, D, Baskoski, P. and Jozef, V. (2011). Modelbased prognostics of gear health using stochastic dynamic models. Mechanical Systems and Signal Processing, 25, 537-538. https://doi.org/10.1016/j.ymssp.2010.07.003
  14. Gorjian, N., Ma, L., Mittinty, M., Yarlagadda, P. and Sun, Y. (2009). A review on degradation models in reliability analysis, Proceeding of the 4th World Congress on Engineering Asset Management, Athens, Greece.
  15. Guo, C., Wang, W., Guo, B. and Si, A. (2013). A maintenance optimization model for mission-oriented systems based on Wiener degradation. Reliability Engineering and System Safety, 111, 183-194. https://doi.org/10.1016/j.ress.2012.10.015
  16. Jin, G., Matthews, D.E. and Zhou, Z. (2013). A bayesian framework for on-line degradation assessement and residual life prediction of secondary batteries in spacecraft. Reliability Engineering and System Safety, 113, 7-20. https://doi.org/10.1016/j.ress.2012.12.011
  17. Kuniewslci, S.P., van der Weide, J.A.M. and van Noortwijk, J.M. (2009). Sampling inspection for the evaluation of time-dependent reliability of deteriorating systems under imperfect defect detection. Reliability Engineering and System Safety, 94, 1480-1490. https://doi.org/10.1016/j.ress.2008.11.013
  18. Lee, C.-E. (2015). Estimation of time-dependent damage paths of armors of rubble-mound breakwaters using stochastic processes. J. of Korean Society of Coastal and Ocean Engineers, 27(4), 246-257 (in Korean). https://doi.org/10.9765/KSCOE.2015.27.4.246
  19. Lee, C.-E. (2016). Condition-based model for preventive maintenance of armor units of rubble-mound breakwaters using stochastic process. J. of Korean Society of Coastal and Ocean Engineers, 28(4), 191-201 (in Korean). https://doi.org/10.9765/KSCOE.2016.28.4.191
  20. Lee, C.-E. and Park, D.H. (2017). Discounted cost model of condition-based maintenance regarding cumulative damage of armor units of rubble-mound breakwaters as a discrete-time stochastic processes. J. of Korean Society of Coastal and Ocean Engineers, 29(2), 109-120 (in Korean). https://doi.org/10.9765/KSCOE.2017.29.2.109
  21. Lee, C.-E. (2019). Prediction of expected residual useful life of rubble-mound breakwaters using stochastic gamma process. J. of Korean Society of Coastal and Ocean Engineers, 31(3), 158-169 (in Korean). https://doi.org/10.9765/KSCOE.2019.31.3.158
  22. Le Son, K., Fouladirad, M., Barros, A., Levrat, E. and Lung, B. (2013). Remaining useful life estimation based on stochastic deterioration models: A comparative study. Reliability Engineering and System Safety, 112, 165-175. https://doi.org/10.1016/j.ress.2012.11.022
  23. Li, N., Lei, Y., Guo, L., Yan, T. and Lin, J. (2017). Remaining useful life prediction based on a general expression of stochastic process models. IEEE Transactions on Indus. Elec., 64, 5709-5718. https://doi.org/10.1109/TIE.2017.2677334
  24. Li, N., Lei, Y., Yan, T., Li, N. and Han, T. (2019). A Wiener-process-model-based method for remaining useful life prediction considering unit-to-unit variability. IEEE Transactions on Indus. Elec., 66, 2092-2101. https://doi.org/10.1109/TIE.2018.2838078
  25. Melby, J.A. (1999). Damage progression on breakwaters, Ph.D. Thesis, Dept. of Civil Eng., U. of Delaware, USA.
  26. Moler, C. (2004). Numerical computing with MATLAB, Society for Industrial Mathematics.
  27. Nicolai, R.P., Frenk, B.G. and Dekker, R. (2009). Modelling and optimizing imperfect maintenance of coatings on steel structures. Structural Safety, 37, 234-244. https://doi.org/10.1016/j.strusafe.2008.06.015
  28. Pandey, M.D., Yuan, X.-X. and van Noortwijk, J.M. (2009). The influence of temporal uncertainty of deterioration on life-cycle management of structures. Structure and Infrastructure Engineering, 5(2), 145-156. https://doi.org/10.1080/15732470601012154
  29. Pecht, M. and Jaai, R. (2010). A prognostic and health management roadmap for information and electronic-rich system. Microelectronics Reliability, 50, 317-323. https://doi.org/10.1016/j.microrel.2010.01.006
  30. Peng, C.Y. and Tseng, S.T. (2009). Mis-specification analysis of linear degradation models. IEEE Transactions on Reliability, 58, 44-455.
  31. PIANC (1992). Analysis of rubble mound breakwaters, Supplement to Bull. No. 78/79, Brussels, Belgium.
  32. Ross, S. (1996). Stochastic processes, Wiley Series in Probability and Statistics, John Wiley & Sons, Inc.
  33. Shahraki, A.F., Yaadav, O.P. and Liao, H. (2017). A review on degradation modelling and its engineering applications. Int. J. of Performability Eng., 13(3) 299-314.
  34. Si, X.-S., Wang, W., Hu, C.H. and Zhou, D.H. (2011). Remaining useful lifetime estimation - A review on the statistical data driven approaches. European J. of Oper. Res., 213, 1-14. https://doi.org/10.1016/j.ejor.2010.11.018
  35. Si, X.S., Wang, W., Hu, C.H. and Zhou, D.H. (2012). Remaining useful life estimation based on nonlinear diffusion degradation process. IEEE Transactions on Reliability, 61, 50-67. https://doi.org/10.1109/TR.2011.2182221
  36. Singpurwalla, N.D. (1995). Survival in dynamic environments. Statistical Science, 10(1), 86-103. https://doi.org/10.1214/ss/1177010132
  37. Tsai, C.C., Tseng, S.T. and Balakrishnam, N. (2011). Mis-specification analyses of gamma and Wiener degradation processes. J. Stat. Plan Inference, 141, 3725-3735. https://doi.org/10.1016/j.jspi.2011.06.008
  38. Tseng, S.T., Tang, J. and Ku, L.H. (2003). Determination of optimal burn-in parameters and residual life for highly reliable products. Naval Re. Logisics, 50, 1-14. https://doi.org/10.1002/nav.10042
  39. Tseng, S. and Peng, C. (2007). Stochastic diffusion modelling of degradation data. J. Data Sci., 5, 315-333. https://doi.org/10.6339/JDS.2007.05(3).351
  40. van Noortwijk, J.M. (2009). A survey of the application of gamma processes in maintenance. Reliability Engineering and System Safety, 94, 2-21. https://doi.org/10.1016/j.ress.2007.03.019
  41. van Noortwijk, J.M., van der Weide, J.A.M., Kallena, M.J. and Pandey, M.D. (2007). Gamma processes and peak-over-threshold distributions for time-dependent reliability. Reliability Engineering and System Safety, 92, 1651-1658. https://doi.org/10.1016/j.ress.2006.11.003
  42. Wang, W. (2007). A two-stage prognosis model in condition based maintenance. European J. of Oper. Res., 182, 1177-1187. https://doi.org/10.1016/j.ejor.2006.08.047
  43. Wang, X. (2010). Wiener processes with random effects for degradation data. J. Multivar. Anal., 101, 340-351. https://doi.org/10.1016/j.jmva.2008.12.007
  44. Wang, W., Scarf, P.A. and Smith, M.A.J. (2000). On the application of a model of condition-based maintenance. European J. of Oper. Res., 51, 1218-1227.
  45. Wang, W.B., Carr, M., Xu, W.J. and Kobbacy, A.K. (2010). A model for residual life prediction based on brownian motion with an adaptive drift. Microelectron Reliab., 51, 285-293. https://doi.org/10.1016/j.microrel.2010.09.013
  46. Wang, X., Jiang, P., Guo, B. and Cheng, Z. (2013). Real-time reliability evaluation with a general Wiener process-based degradation model. Quality and Reliability Eng. Int., 30, 205-220. https://doi.org/10.1002/qre.1489
  47. Wang, X., Balakrishnan, N. and Guo, B. (2014). Residual life estimation based on a generalized Wiener degradation process. Reliability Engineering and System Safety, 124, 13-23. https://doi.org/10.1016/j.ress.2013.11.011
  48. Whitmore, G.A. (1995). Estimating degradation by a Wiener diffusion process subject to measurement error. Lifetime Data Analysis, 1, 307-319. https://doi.org/10.1007/BF00985762
  49. Whitmore, G.A. and Schenkelberg, F. (1997). Modelling accelerated degradation data using Wiener diffusion with a time scale transformation. Lifetime Data Analysis, 3, 27-45. https://doi.org/10.1023/A:1009664101413
  50. Ye, Z., Shen, Y. and Xie, M. (2012). Degradation-based burn-in with preventive maintenance. European J. of Oper. Res., 221, 360-367. https://doi.org/10.1016/j.ejor.2012.03.028