Thèse de doctorat de l’UTT Yingjun DENG Degradation Modeling Based on a Time-dependent Ornstein-Uhlenbeck Process and Prognosis of System Failures Spécialité : Optimisation et Sûreté des Systèmes 2015TROY0004 Année 2015 THESE pour l’obtention du grade de DOCTEUR de l’UNIVERSITE DE TECHNOLOGIE DE TROYES Spécialité : OPTIMISATION ET SURETE DES SYSTEMES présentée et soutenue par Yingjun DENG le 24 février 2015 Degradation Modeling Based on a Time-dependent Ornstein-Uhlenbeck Process and Prognosis of System Failures JURY M. I. NIKIFOROV PROFESSEUR DES UNIVERSITES Président Mme A. BARROS PROFESSEUR DES UNIVERSITES Directrice de thèse Mme A. GÉGOUT-PETIT PROFESSEUR DES UNIVERSITES Rapporteur M. A. GRALL PROFESSEUR DES UNIVERSITES Directeur de thèse M. M. PANDEY PROFESSOR Rapporteur M. C. PAROISSIN MAITRE DE CONFERENCES Examinateur M. M. ROUSSIGNOL PROFESSEUR EMERITE Examinateur Acknowledgments The initial motivation to bridge between mathematics and engineering problems drives me to the interesting topic proposed by my supervisors Prof. Anne Barros (currently at NTNU, Norway) and Prof. Antoine Grall. And here I appreciate them a lot for giving me the opportunity to do this PhD at Institut Charles Delaunay, UMR CNRS 6279 & Universit´ede Technologie de Troyes (UTT), also for their inspiration, patience and support in last 3 years. The financial support from China Scholarship Council is appreciated. It is my honor that Prof. Anne G´egout-Petit and Prof. Mahesh Pandey have accepted to review this thesis. I am also glad that Prof. Michel Roussignol, Prof. Igor Nikiforov, and Dr. Christian Paroissin have accepted to be my jury committee members. During my PhD period, I have been benefitted a lot from frequent discussions with Prof. Michel Roussignol at Universit´ede Paris-Est Marne-la-Vall´ee(UPEM). His insight and ri- gorousness in mathematical research influence me a lot, without whom I would be lost in problems and mistakes. I am grateful for those outside experience with Prof. Damien Lam- berton for stochastic analysis at UPEM, with Dr. Hans Van der Weide for first passage problems at TUDelft, Netherlands, and also in S´eminaireBachelier Paris at Institut Henri Poincar´e,Paris VI for financial mathematics. Many thanks are due to my sincere colleagues at UTT, especially to Igor Nikiforov, Mitra Fouladirad, Edith Grall-Ma¨es,Estelle Deloux, Malika Kharouf, Nicholas Lefebvre, Yann Dijoux, Yves Langeron, Tuan Huynh, Elias Khoury and Khanh Le Son. Sincere assistance on administrative issues from Marie-Jos´eRousselet, Veronique Banse and Bernadette Andre, makes my life much easier in France. And it is really a fantastic experience to work in the same office during last 3 years with Kim Anh Nguyen, Danh Ngoc Nguyen and Houda Ghamlouch. Technical Discussions with my friends help me a lot, especially those with Hao Jiang, Jiange Li, and Chengfa Wu. Also my Chinese friends in Troyes make me feel like at home, especially with Yugang Li, Xiaowei Lv, Hongchang Han, Hui Shang, Fei Zhu, Heping Li, Yaofu Cao etc.. Not accompanying with them in such a long period, I am really in debt to my mother, father, sister and my girlfriend Hanbing. Their understanding, encourage and support are essential for me to complete this thesis. Yingjun Deng Abstract This thesis consists of four parts : stochastic degradation modeling, prognosis of system failures, failure level estimation and maintenance optimization. These parts are connected and dedicated to three core issues of system failures : description, prediction and prevention. The first part about stochastic degradation modeling proposes a degradation model based on a time-dependent Ornstein-Uhlenbeck (OU) process. Such a process utilizes the inspection records to establish the dynamic description of degradation process. The time-dependent OU process is proved superior by its statistical properties on controllable mean, variance and correlation. Its mean-reverting property can be introduced to interpret temporary correlated fluctuations from an overall degrading trend in degradation records. Corresponding parameter estimation is proposed based on maximum likelihood estimation method. A case study is performed to test the model's fitting goodness based on a degradation data-set of a passive component in power plants. The second part about prognosis of system failures is discussed further based on the time- dependent OU process. And the first passage time is introduced as the system failure time, to a pre-set failure level. Later how to estimate the failure time is discussed based on two kinds of views : partial differential equation and integral equation. These two views lead to various estimation techniques from different concentrations, and they can be classified into 3 categories : analytical approximations, numerical algorithms and Monte-Carlo simulation methods. Simulation tests are done to calculate first passage density based on proposed methods. The third part about failure level estimation proposes some techniques to estimate failure levels based on inverse first passage problems. In previous literature the failure level is gene- rally treated as physical barriers or experts' opinions, based on which failure prognosis from first passage failures can hardly fit existing failure records. Therefore the effort in this part is paid to make up the gap between failure records and inspection records under the definition of first passage failure based on inverse first passage problems. When the lifetime distribution is given or estimated from failure records, we emphasizes on numerically reproducing the failure level under which the first passage time of the given stochastic process can have the same lifetime distribution with the given lifetime distribution. The fourth part about maintenance optimization investigates how to optimize mainte- nance policies based on the time-dependent OU process. Based on monitored system condi- tions and prognosis of system failures, condition-based maintenance is adopted to introduce preventive maintenance such that the balance can be achieved between operation costs and disastrous results caused by system failures. In this part, corresponding maintenance optimi- zation problems are discussed based on the time-dependent OU process and the hypothesis of continuously monitored system. Due to the unexplicit expression for prognosis of system failures, classical heuristic optimization procedures cannot be fulfilled. Therefore approximate first passage density is introduced to fulfill the maintenance optimization. R´esum´e Cette th`eseest organis´een quatre parties : 1. la mod´elisationstochastique de la d´egradation, 2. le pronostic de l'instant de d´efaillancedu syst`eme, 3. l'estimation du niveau d´egradationassoci´e`ala d´efaillance, 4. l'optimisation de la maintenance. Ces diff´erentes parties sont li´eesentre elles et tentent de d´ecrire,pr´evoir et pr´evenir la d´efaillancedu syst`eme. Dans la premi`erepartie, un mod`elestochastique de la d´egradations'appuyant sur un processus d'Ornstein-Uhlenbeck (OU) d´ependant du temps et sur l'exploitation conjointe des donn´esd'inspection est propos´e.Les qualit´esde ce mod`elesont d´emontr´ees au travers de ses propri´et´esstatistiques qui permettent d'ajuster de mani`ereind´ependante la moyenne, la va- riance et la corr´elation.Une propri´et´ede "convergence" vers la moyenne est ensuite exploit´ee pour interpr´eter la corr´elationtemporelle des fluctuations autour d'une tendance globale de d´egradation.Puis, s'appuyant sur une technique de maximisation de la vraisemblance, une m´ethode d'estimation des param`etres de ce mod`eleest propos´ee.Enfin, un cas d'application portant sur l'´etudede la d´egradationd'un composant passif de central ´electriqueest trait´e. La deuxi`emepartie de la th`eseest consacr´eeau pronostic de l'instant de d´efaillancedu syst`emeen s'appuyant sur un processus OU d´ependant du temps. Cet instant de d´efaillance est d´efinicomme le premier temps d'atteinte d'un ´etatde d´egradation,critique i.e. d'un ´etat de sant´einacceptable.. L'estimation de cet instant de d´efaillanceest abord´eselon deux ap- proches : i) ´equationsaux d´eriv´espartielles, ii) ´equationsint´egrales.Ces approches conduisent `adiff´erentes techniques d'estimation qui peuvent ^etre class´eesselon le sch´ema suivant : ◦ les techniques d'approximation analytique, ◦ les techniques d'approximation num´erique, ◦ les techniques de simulation de Mont´e-Carlo. Des essais num´eriquesdestin´esau calcul de la densit´ede l'instant de d´efaillanceet permettant la confrontation de ces diff´erentes techniques, concluent cette seconde partie de la th`ese. L'estimation du niveau d´egradation associ´e`ala d´efaillance, que l'on appelle dans la suite par commodit´eniveau de d´efaillance,est l'objet de la troisi`emepartie du document. Classi- quement, ce niveau de d´efaillanceest d´etermin´esur la base de caract´eristiques physiques ou d'avis d'experts. Cependant ce niveau de d´efaillance"th´eorique"n'est pas toujours coh´erent avec les donn´eesassoci´ees`ades d´efaillancesr´eelles.L'accent est donc mis sur la r´eductionde cet ´ecart.Pour ce faire, la loi de la dur´eede vie est suppos´eeconnue ou, tout au moins, estim´ee sur la base de donn´eesde d´efaillances.Le niveau de d´efaillancepeut alors ^etred´etermin´ede telle sorte que le processus stochastique de d´egradationconsid´er´econduise `aune distribution du premier temps d'atteinte du niveau de d´efaillancequi corresponde `ala densit´eestim´ee. La quatri`emepartie est d´edi´ee`al'optimisation de la maintenance lorsque le processus de d´egradationconsid´er´eest un processus OU d´ependant du temps. S'appuyant sur les donn´ees de surveillance continue du syst`eme et sur le pronostic de l'instant de d´efaillance, un com- promis peut ^etretrouv´eentre les co^utsde maintenance pr´eventive et ceux associ´es`aune x RESUM´ E´ d´efaillancedu syst`eme.Dans ce contexte, la formulation non explicite du pronostic de l'ins- tant de d´efaillancene permet pas d'exploiter les techniques d'optimisation classiques.