추계학적 위너 확률과정을 이용한 경사제의 실시간 피해경로 추적과 잔류수명 추정 = Estimation of Residual Useful Life and Tracking of Real-time Damage Paths of Rubble-Mound Breakwaters Using Stochastic Wiener Process

다국어 초록 (Multilingual 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.

국문 초록 (Abstract)

추계학적 WP을 이용하여 불확실성을 고려하면서 항만 구조물의 실시간에 따른 피해와 파괴확률 그리고잔류수명을 해석할 수 있는 모형을 수립하였다. 과거부터 현재까지의 피해상태와 미래에 발생될 피해 진행 과정에포함되는 불확실성을 고려할 수 있는 추계학적 확률모형이다. 피해경로를 추적할 수 있으며 누적피해의 밀도함수도산정하여 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. 한편본 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 커지면서파괴확률이 급격하게 증가하였다. 또한 재령에 따른 잔류수명의 거동특성을 해석하였는데, 잔류수명의 분포함수에서좌측보다는 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.

참고문헌 (Reference)

1 이철응, "추계학적 확률과정을 이용한 경사제 피복재의 예방적 유지관리를 위한 조건기반모형" 한국해안,해양공학회 28 (28): 191-201, 2016

2 이철응, "추계학적 확률과정을 이용한 경사제 피복재의 시간에 따른 피해 경로 추정" 한국해안,해양공학회 27 (27): 246-257, 2015

3 이철응, "추계학적 감마 확률과정을 이용한 경사제의 기대 잔류유효수명 예측" 한국해안,해양공학회 31 (31): 158-169, 2019

4 이철응, "경사제 피복재의 누적피해를 이산시간 확률과정으로 고려한 조건기반 유지관리의 할인비용모형" 한국해안,해양공학회 29 (29): 109-120, 2017

5 Wang, X., "Wiener processes with random effects for degradation data" 101 : 340-351, 2010

6 Cox, D.R., "The theory stochastic processes"

7 Ebrahimi, N., "The mean function of a repairable system that is subjected to an imperfect repair policy" 41 : 57-64, 2009

8 Chhikara, R. S., "The inverse Gaussian distribution as a lifetime model" 19 : 461-468, 1977

9 Pandey, M. D., "The influence of temporal uncertainty of deterioration on life-cycle management of structures" 5 (5): 145-156, 2009

10 Singpurwalla, N. D., "Survival in dynamic environments" 10 (10): 86-103, 1995

11 Ross, S., "Stochastic processes, Wiley Series in Probability and Statistics" John Wiley & Sons, Inc 1996

12 Tseng, S., "Stochastic diffusion modelling of degradation data" 5 : 315-333, 2007

13 Cui, L. R., "Sequential inspection strategy for multiple systems under availability requirement" 155 : 170-177, 2004

14 Dieulle, L., "Sequential condition-based maintenance scheduling for a deteriorating system" 150 : 451-461, 2003

15 Kuniewslci, S. P., "Sampling inspection for the evaluation of time-dependent reliability of deteriorating systems under imperfect defect detection" 94 : 1480-1490, 2009

16 Gebraeel, N. Z., "Residual-life distribution from component degradation signals : A bayesian approach" 37 : 543-557, 2005

17 Wang, X., "Residual life estimation based on a generalized Wiener degradation process" 124 : 13-23, 2014

18 Si, X. -S., "Remaining useful lifetime estimation – A review on the statistical data driven approaches" 213 : 1-14, 2011

19 Li, N., "Remaining useful life prediction based on a general expression of stochastic process models" 64 : 5709-5718, 2017

20 Le Son, K., "Remaining useful life estimation based on stochastic deterioration models : A comparative study" 112 : 165-175, 2013

21 Si, X. S., "Remaining useful life estimation based on nonlinear diffusion degradation process" 61 : 50-67, 2012

22 Wang, X., "Real-time reliability evaluation with a general Wiener process-based degradation model" 30 : 205-220, 2013

23 Elwang, A., "Real-time estimation of mean remaining life using sensor-based degradation models" 131 : 51-59, 2009

24 Wang, W., "On the application of a model of condition-based maintenance" 51 : 1218-1227, 2000

25 Crowder, M., "On a scheme for predictive maintenance" 176 : 1713-1722, 2007

26 Moler, C., "Numerical computing with MATLAB"

27 Doksum, K. A., "Models for variable-stress accelerated life testing experiments based on Wiener processes and the inverse Gaussian distribution" 34 : 74-82, 1992

28 Nicolai, R. P., "Modelling and optimizing imperfect maintenance of coatings on steel structures" 37 : 234-244, 2009

29 Whitmore, G. A., "Modelling accelerated degradation data using Wiener diffusion with a time scale transformation" 3 : 27-45, 1997

30 Gasperin, M., "Modelbased prognostics of gear health using stochastic dynamic models" 25 : 537-538, 2011

31 Peng, C. Y., "Mis-specification analysis of linear degradation models" 58 : 44-455, 2009

32 Tsai, C. C., "Mis-specification analyses of gamma and Wiener degradation processes" 141 : 3725-3735, 2011

33 van Noortwijk, J. M., "Gamma processes and peak-over-threshold distributions for time-dependent reliability" 92 : 1651-1658, 2007

34 Bagdonavicius, V., "Estimation in degradation models with explanatory variables" 7 : 85-103, 2000

35 Whitmore, G. A., "Estimating degradation by a Wiener diffusion process subject to measurement error" 1 : 307-319, 1995

36 Tseng, S. T., "Determination of optimal burn-in parameters and residual life for highly reliable products" 50 : 1-14, 2003

37 Ye, Z., "Degradation-based burn-in with preventive mainten- ance" 221 : 360-367, 2012

38 Melby, J. A., "Damage progression on breakwaters" U. of Delaware 1999

39 CEM, "Coastal Engineering Manual"

40 PIANC, "Analysis of rubble mound breakwaters" 1992

41 Ahmad, R., "An overview of time-based and condition-based maintenance in industrial application" 63 (63): 135-149, 2012

42 Wang, W., "A two-stage prognosis model in condition based maintenance" 182 : 1177-1187, 2007

43 van Noortwijk, J. M., "A survey of the application of gamma processes in maintenance" 94 : 2-21, 2009

44 Gorjian, N., "A review on degradation models in reliability analysis" 2009

45 Shahraki, A. F., "A review on degradation modelling and its engineering applications" 13 (13): 299-314, 2017

46 Pecht, M., "A prognostic and health management roadmap for information and electronic-rich system" 50 : 317-323, 2010

47 Wang, W. B., "A model for residual life prediction based on brownian motion with an adaptive drift" 51 : 285-293, 2010

48 Guo, C., "A maintenance optimization model for mission-oriented systems based on Wiener degradation" 111 : 183-194, 2013

49 Jin, G., "A bayesian framework for on-line degradation assessement and residual life prediction of secondary batteries in spacecraft" 113 : 7-20, 2013

50 Li, N., "A Wiener-process-model-based method for remaining useful life prediction considering unit-to-unit variability" 66 : 2092-2101, 2019