Remaining useful lifetime estimation and noisy gamma deterioration process

Son, Khanh Le; Fouladirad, Mitra; Barros, Anne

doi:10.1016/j.ress.2015.12.016

Cited by 97 publications

(41 citation statements)

References 26 publications

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“…Hence, in the case { W ( t ); t ≥ 0} passes w max at or before t , RUL ( t ) is null. The Cdf of the RUL at time t is defined (see also the work of Le Son et al) as the conditional probability that the unit fails within t + τ , given Z ( t ) = z t , where Z ( t ) = { Z ( t k ); 0 ≤ k ≤ n , t k ≤ t } is the set of all the measurement collected up to t . Hence, because { W ( t ); t ≥ 0} is a monotonically increasing process, the Cdf of RUL ( t ) is given by

\begin{matrix} true \\ F_{RUL (t)} (τ| z_{t}) = \Pr \{R U L (t) \leq τ false| z_{t}\} & = 1 - \int_{0}^{w_{\max}} F_{W (t + τ) |W (t)} (w_{\max}| w_{t}) f_{W (t) |Z (t)} (w_{t}| z_{t}) d w_{t} \\ = 1 - F_{W (t + τ) |Z (t)} (w_{\max} |z_{t}), τ \geq 0, \end{matrix}

where F W ( t + τ )| W ( t ) ( w max | w t ) is the conditional Cdf of the actual degradation level W ( t + τ ), given the current level W ( t ) = w t , F W ( t + τ )| Z ( t ) ( w max | z t ) is the conditional Cdf of the actual degradation level W ( t + τ ), given Z ( t ) = z t , evaluated at w max .…”

Section: The Likelihood Function and The Rulmentioning

confidence: 99%

“…On the basis of the Assumptions (1)(2)(3)(4) in Section 2, the pdfs in the likelihood function (10) can be computed, for any i = 1, … , m and k = 1, … , n i , via the following set of recursive equations:…”

Section: The Likelihood Function and The Rulmentioning

confidence: 99%

“…In the papers dealing with noisy degradation data, the measurement error is generally assumed to have mean zero and to be independent of the actual degradation process (see, eg, other works [1][2][3][4][5][6][7][8] ). However, as argued in the work of Zhai and Ye, 9 there are many experimental situations where it is more realistic to assume that the measurement error depends on the actual degradation level.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A perturbed gamma degradation process with degradation dependent non‐Gaussian measurement errors

Giorgio

Mele

Pulcini

2018

Appl Stoch Models Bus & Ind

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This paper proposes and illustrates a new perturbed gamma degradation process where the measurement error is modeled as a non‐Gaussian random variable that depends stochastically on the actual degradation level. The expression of the likelihood function for a generic set of noisy degradation measurements is derived, and the expression of the remaining useful life distribution of a degrading unit that fails when its degradation level exceeds a given threshold limit is formulated. A particle filter method is suggested, which allows one to compute the likelihood function and to estimate the remaining useful life distribution in a quick yet efficient manner. In addition, a closed‐form approximation of the perturbed gamma process is proposed to use in the special, yet meaningful, case where the standard deviation of the measurement error depends linearly on the actual degradation level. Finally, an applicative example is discussed, where the parameters of the perturbed gamma process, the remaining useful life distribution, and the mean remaining useful life of the degrading units are estimated from a set of noisy real degradation data.

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\begin{matrix} true \\ F_{RUL (t)} (τ| z_{t}) = \Pr \{R U L (t) \leq τ false| z_{t}\} & = 1 - \int_{0}^{w_{\max}} F_{W (t + τ) |W (t)} (w_{\max}| w_{t}) f_{W (t) |Z (t)} (w_{t}| z_{t}) d w_{t} \\ = 1 - F_{W (t + τ) |Z (t)} (w_{\max} |z_{t}), τ \geq 0, \end{matrix}

Section: The Likelihood Function and The Rulmentioning

confidence: 99%

Section: The Likelihood Function and The Rulmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A perturbed gamma degradation process with degradation dependent non‐Gaussian measurement errors

Giorgio

Mele

Pulcini

2018

Appl Stoch Models Bus & Ind

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“…The maintenance decision is based on the RUL estimation. For the role played by the pitch system in a WT, predictive maintenance is a good choice (Langeron et al (2016),Le Son et al (2016). Le Son et al (2016) propose an estimated-RUL-based maintenance policy which takes the estimation RUL into account.…”

Section: Rul Prediction and Maintenancementioning

confidence: 99%

“…For the role played by the pitch system in a WT, predictive maintenance is a good choice (Langeron et al (2016),Le Son et al (2016). Le Son et al (2016) propose an estimated-RUL-based maintenance policy which takes the estimation RUL into account. This policy defines a RUL min as a fixed threshold, comparing the RUL min with the estimated RUL E(RUL(i)) at each inspection time T i , ω i is the estimated natural frequency at time T i…”

Section: Rul Prediction and Maintenancementioning

confidence: 99%

A pitch control system deterioration model considering wind speed turbulence intensity

Ma¹,

Fouladirad²,

Grall³

2018

Proceedings of the 10th IMA International Conference on Modelling in Industrial Maintenance and Reliability

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As a consequence of high failure rate occurring to wind turbines (WTs), more and more asset owners begin to pay attention to WT's maintenance planning. To implement an efficient condition-based maintenance policy, the deterioration modeling is of primary importance.The inherence character of wind makes WT's operation full of uncertainties, hence, the pitch actuator movement is random and it can be modeled by a stochastic process. Meanwhile, the deterioration of the pitch control system related to the usage of the actuator is random. When the wind speed is higher than the rated wind speed (a design WT parameter, when wind speed is higher than the rated wind speed, the pitch system begin to operate) and the wind speed turbulence intensity (TI) is very important, the deterioration rate of the pitch control system increases significantly. For instance the pitch system has more failures in a windy season. Therefore, in order to propose an appropriate maintenance policy for pitch control system, it is more sensible to take into account the influence of wind speed in deterioration modelling.In this paper, we focus on the deterioration of hydraulic pitch control system. After studying the factors influencing the deterioration, a stochastic process considering wind speed as covariates is used to model the deterioration. A maintenance policy based on an alarm threshold is considered.

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Misspecification analysis of gamma‐ and inverse Gaussian‐based perturbed degradation processes

Esposito,

Mele,

Castanier

et al. 2023

Appl Stoch Models Bus & Ind

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Albeit not equivalent, in many applications the gamma and the inverse Gaussian processes are treated as if they were. This circumstance makes the misspecification problem of these models interesting and important, especially when data are affected by measurement errors, since noisy/perturbed data do not allow to verify whether the selected model is actually able to adequately fit the real (hidden) degradation process. Motivated by the above considerations, in this paper we conduct a large Monte Carlo study to evaluate whether and how the presence of measurement errors affects this misspecification issue. The study is performed considering as reference models a perturbed gamma process recently proposed in the literature and a new perturbed inverse Gaussian process that share the same non‐Gaussian distributed error term. As an alternative option, we also analyze the more classical case where the error term is Gaussian distributed. We consider both the situation where the true model is the perturbed gamma and the one where it is the perturbed inverse Gaussian. Model parameters are estimated from perturbed data using the maximum likelihood method. Estimates are retrieved by using a new sequential Monte Carlo EM algorithm, which use allows to hugely mitigate the severe numerical issues posed by the direct maximization of the likelihood. The risk of incurring in a misspecification is evaluated as percentage of times the Akaike information criterion leads to select the wrong model. The severity of a misspecification is evaluated in terms of its impact on maximum likelihood estimate of the mean remaining useful life.

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Remaining useful lifetime estimation and noisy gamma deterioration process

Cited by 97 publications

References 26 publications

A perturbed gamma degradation process with degradation dependent non‐Gaussian measurement errors

A perturbed gamma degradation process with degradation dependent non‐Gaussian measurement errors

A pitch control system deterioration model considering wind speed turbulence intensity

Misspecification analysis of gamma‐ and inverse Gaussian‐based perturbed degradation processes

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