Remaining Useful Life Estimation Based on Gamma ...

Remaining Useful Life Estimation Based on Gamma Process Considered with Measurement Error Qidong Wei

Dan Xu

Science & Technology on Reliability & Environmental Engineering Laboratory School of Reliability and Systems Engineering, Beihang University Beijing, China [email protected]

Science & Technology on Reliability & Environmental Engineering Laboratory School of Reliability and Systems Engineering, Beihang University Beijing, China [email protected]

Abstract—For high reliability and long life components and systems, remaining useful life (RUL) estimation is the key of the prognostics and health management (PHM). RUL is important to verify reliability of components, and set a condition-based maintenance policy. With degradation data, RUL is able to be computed. For most products, degradation theoretically is a monotonically decreasing process. Because degrading process is an irreversible process for most components that they can only cumulate damage but not cure themselves. However, the actual degrading process described by the online monitoring data is usually not strictly monotonic. In most cases, the performance parameters are detected degrading with fluctuation in a little interval. Base on these facts, this paper proposes the monitored degrading process with fluctuation be made of a gamma process combined with random measurement error. A gamma process is utilized to model degrading process for its monotonicity. The measurement error is an inevitable error that no matter how accurate the measuring equipment is, the measurement result always deviates from the true value. In most cases, the measurement error fits normal distribution, whose mean and variance are related to the accuracy of the measuring equipment. With the approach proposed in this paper, the gamma process is able to fit the real degrading process and prognosticate life without measurement error. The method of moments is utilized to estimate model parameters. At last, a numerical example is used to illustrate the modeling method of utilizing a gamma process combined with random measurement error. The prognosticating result shows that the approach considered measurement error can give a shorter interval of prognosticating lifetime than the classical gamma process. Keywords-component; remaining useful life; gamma process; measurement error

ACRONYM PHM RUL

Prognostics and Health Management Remaining Useful Life NOTATION

y (t ) ytrue (t ) ε err (t )

Measured degradation data at time t True value of performance at time t Measured error at time t

Δw(t )

α λ σ2 E[ X k ] M (t )

Increment of degradation data at time t Shape parameter of the gamma distribution Scale parameter of the gamma distribution Variance of the Gaussian distribution The k-th raw moment The moment-generating function I. INTRODUCTION

In many fields such as aeronautics and industry, equipments are usually large and complex. Once they fail, it will bring huge economic losses even casualties. To ensure the health running of equipments, prognostics and health management (PHM) has been proposed and discussed. PHM improves the operational reliability by replacing the components which is about to fail. Estimation of remaining useful life (RUL) is the key to PHM, because accurate estimation is helpful to make rational condition-based maintenance applications. RUL prognostic approaches can be categorized into physical model approaches, data-driven approaches, and hybrid approaches. Sikorska[1] had discussed RUL prognostic modeling options. Physical model approaches need well understanding of failure physics of components. Once the physical model is built exactly, the estimation of RUL can be highly accuracy. A good review on machinery diagnostics and prognostics was given by Jardine[2]. The modeling on physics of failure is difficult, especially when the system contains lots of components. Data-driven approaches can prognosticate RUL by utilizing the data collected from sensors. A review on the statistical data driven approaches to estimate RUL was given by Si[3]. Gebraeel et al.[4] proposed a method to mathematically model degradation-based signals with stochastic models. Three main sources of information were considered: real-time degradation characteristics of component, the degradation characteristics of the component’s population, and the real-time status of the environmental conditions. Zio and Maio[5] presented a fuzzy approach that RUL is predicted by fuzzy similarity analysis. The evolution data are compared to a library of reference

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trajectory patterns to failure, and aggregating RUL in a weighted sum accounts for their similarity. Tian et al.[6] utilized both failure and suspension condition monitoring data as the input of artificial neural network to predict RUL. Medjaher et al[7] utilized the mixture of Gaussians Hidden Markov Models represented by Dynamic Bayesian Networks as a modeling tool to predict the RUL of bearings. Hu et al.[8] proposed an ensemble data-driven prognostic approach which combines multiple member algorithms with a weighted-sum formulation. Three weighting schemes were proposed: the accuracy-base weighting, the diversity-based weighting and the optimizationbased weighting.

II. MODELING The prognosticating RUL method proposed in this paper has two steps. First, the degradation described by measured data is modeled as a gamma process with noise. Second, the degrading process is extrapolated by the gamma process model without noise, and the RUL is estimated by computing first passage time. This paper proposed that the measured degradation data are made up of the true value of degrading performance and measurement error. It can be formulated as: y (t ) = ytrue (t ) + ε err (t )

Hybrid approaches attempt to take advantages of both physical model approaches and data-driven approaches, and estimate RUL by fusing the information from both approaches. Saha et al.[9] modeled electrochemical processes in the form of equivalent electric circuit combined with a statistical model of state transitions. For the difficulties of modelling the physical process, appropriate situations for hybrid approaches are rare.

(1)

where t is the measuring time, y(t) is the measured degradation data at time t, ytrue(t) is the true value of degrading performance at time t, and εerr(t) is the error of measurement at time t. In order to simplify the computing process, εerr(t) is treated as a Gaussian distribution, of which the mean value is 0. The simplification also fits characteristics of most science measurements. It should be noticed that ytrue(t) and εerr(t) are mutually independent, because degradation and measurement are two mutually independent events.

Gamma process, which is a stochastic process with independent non-negative increments, has been used to model the degrading process to estimate the RUL and compute the reliability. Noortwijk et al.[10] combined two stochastic processes to compute the time-dependent reliability of a structural component. The stochastic process of deteriorating resistance was modeled as a gamma process, and the stochastic process of loads was generated by a Poisson process. Lawless and Crowder[11] proposed a tractable gamma process model incorporating a random effect for the situation that individuals are observed degrade at different rates, even though no differences in treatment of environment. Park and Padgett[12, 13] provide a new accelerated degradation model based on a gamma process incorporating accelerated test variables. Tseng et al.[14] modeled the degrading process as a gamma process to optimize a step-stress accelerated degradation test plan. Kuniewski et al.[15] modeled the behavior of defects with a Poisson process and a gamma process. The random time of defects initiate were modeled as a non-homogeneous Poisson process, and the degradation was modeled as a non-decreasing time-dependent gamma process. All of them did not model the measurement error separated from the degrading process.

If a degrading process can be described as a path with nonnegative increments, the increments of degradation can be defined as: Δw(t ) = y (t + Δt ) − y (t )

(2)

where Δt is the interval time between two time-adjacent measurements. In this paper, Δt is simplified as a constant. In practice, it is a rational and common setting for most automatic measuring equipments. If the degrading process is a path with non-positive increments, the right of Eq. (2) should be inverted to ensure Δw(t) non-negative, because the gamma process need non-negative increments. Model as Eq. (2), define the true value of increment as: Δwtrue (t ) = ytrue (t + Δt ) − ytrue (t )

(3)

Substitution of Eq. (2) into Eq. (1) yields: Δw(t ) = Δwtrue (t ) + ε err (t + Δt ) − ε err (t )

(4)

where Δwtrue(t) fits gamma distribution, and εerr(t)~N(0,σ2). It should be noticed that εerr(t1) and εerr(t2) are mutually independent when t1≠t2 for the independence of two measurements. In this paper, it is hypothesized that degradation fits gamma process. Because the monotonicity of gamma process matches the characteristics of many products’ degradation. So in Eq. (4), Δwtrue(t)~Ga(α,1/λ), and εerr(t+Δt)εerr(t)~N(0,2σ2). Δw(t) can be computed from measured data, and parameters α, λ, σ2 need to be estimated.

This paper proposes a stochastic process based modeling method of separately building the degrading model and the measurement error model. The measured degrading data include true values of the degenerate performance and measurement errors. The true values of performance are modeled as a gamma process, and measurement errors follow the Gaussian distributions. The measured degrading data are used to estimate the parameters of the model. The degrading process is extrapolated by utilizing gamma process, and the time when the extrapolating degrading process reaches the failure threshold is the estimated value of RUL. Finally, the approach is applied to deal with the battery accelerated tests data.

For estimating RUL, the degrading process need to be extrapolated. Based on the true value of degrading process at time t0, the value of degrading process at time t0+nΔt can be computed by the equation: n

ytrue (t ) = ytrue (t0 ) + ∑ Δwtrue (t0 + i Δt ) i =1

646

(5)

Although the distribution of measurement error is given as N(0,σ2), a single true value such as ytrue(t0) is impossible to be computed. So y(t0) is chosen to substitute ytrue(t0), and Eq(5) is transformed into: n

ytrue (t ) = y (t0 ) + ∑ Δwtrue (t0 + i Δt )

dM Δw(t) (t ) dt

d 2 M Δw (t) (t )

(6)

dt

d 3 M Δw (t) (t )

The gamma distribution of Δwtrue(t) can be estimated, so series of future degradation points can be computed by Eq. (6), and the RUL is the time before the value reaching the failure threshold. Monte Carlo simulation is used to get the distribution of estimating lifetime. Notice that those above equations are based on the hypothesis that the degrading process is a path with non-negative increments. For the degrading process with non-positive increments, Eq. (2) should be transformed into: Δw( t ) = y (t ) − y (t + Δt )

dt

ytrue (t ) = y (t0 ) − ∑ Δwtrue (t0 + i Δt )

λ3

(α 3 + 3α 2 + 2α + 6αλ 2σ 2 )

(15)

α = E [Δw(t )] λ

(7)

1 1

(8)

(9)

λ3

(17)

(α 3 + 3α 2 + 2α + 6αλ 2σ 2 ) = E [Δw(t )3 ]

(18)

And the k-th raw moment can be estimated using the k-th raw sample moment:

i =1

E[ Δw(t )k ] =

Those transformations have been done to ensure Δw(t) positive, what is called for by gamma distribution.

(16)

(α 2 + α + 2λ 2α 2 ) = E [Δw(t ) 2 ]

λ2

The extrapolation model Eq. (6) should be transformed into: n

t =0

1

λ

2

According to the method of moments, set the first-order derivative, the second-order derivative, the third-order derivative equal to the corresponding k-th raw moments:

And Eq. (3) should be transformed into: Δwtrue (t ) = ytrue (t ) − ytrue (t + Δt )

=

3

1

(13)

(14)

t =0

i =1

t =0

α λ

(α 2 + α + 2λ 2α 2 )

=

2

=

1 m ∑ Δw(ti )k m i =1

(19)

Transform Eq. (16), Eq.(17) and Eq. (18), the parameters estimation can be written as:

III. ESTIMATION OF PARAMETERS To complete the model, parameters α, λ, σ2 need to be estimated. As it shows in the last section, in Eq. (4), Δwtrue(t)~Ga(α,1/λ), εerr(t+Δt)-εerr(t)~N(0,2σ2). So the probability density function of Δw(t) could be computed, and the method of maximum likelihood function will work. But the calculation is heavy. Parameters in this paper are estimated by utilizing the method of moments.

⎡ E[ Δw(t )3 ] 3 ⎤ λ=⎢ − E[ Δw(t ) 2 ] + E[ Δw(t )]2 ⎥ Δ E w t 2 [ ( )] 2 ⎣ ⎦

−

1 2

α = λ E[ Δw(t )] 1⎛

(21) 1

⎞ ⎠

σ 2 = ⎜ E[Δw(t )2 ] − E[Δw(t )]2 − E[ Δw(t )] ⎟ λ 2 ⎝

As Δwtrue(t)~Ga(α,1/λ), the moment-generating function of Δwtrue(t) can be written as:

(20)

(22)

IV. NUMERICAL EXAMPLE

α

⎛ λ ⎞ M Δwtrue (t ) = ⎜ ⎟ ⎝λ −t ⎠

(10)

A. The Battery Accelerated Test National Aeronautics and Space Administration has done the accelerated tests on Li-ion batteries. These batteries had experienced three different operational profiles: charge, discharge and impedance. The charge profile was carried out in a constant current 1.5A mode until voltage reached 4.2V, then turned into a constant voltage mode, and end the charge when the charge current dropped to 20mA. Discharge was carried out at a constant current level of 2A. The battery No.7 which is chosen to illustrate the approach discharged until the voltage fell to 2.2V.

As εerr(t+Δt)-εerr(t)~N(0,2σ2), the moment-generating function of εerr(t+Δt)-εerr(t) can be written as: M ε err ( t +Δt ) −ε err ( t ) (t ) = exp(σ 2t 2 )

(11)

For Eq. (4) is true, the moment-generating function of Δw(t) can be written as:

M Δw( t ) ( t ) = M Δwtrue ( t ) (t ) M ε err ( t +Δt ) −ε err ( t ) (t ) α

⎛ λ ⎞ 2 2 =⎜ ⎟ exp(σ t ) ⎝λ −t ⎠

(12)

B. Battery degradation data

Take the first-order derivative, the second-order derivative, and the third-order derivative of Eq. (12) with respect to t, and set t equal to 0:

In this paper, the capacity which was measured in discharge profile is chosen as the characteristic of battery degradation. And the failure threshold is set at the level of 20% fade from

647

the initial state. The degradation data of battery No.7 has been shown in Fig. 1. C. Estimating RUL It shows in Fig. 1 that the degrading process jumps suddenly sometimes for unknown reasons. It obviously affects the accuracy of estimation. As Δw(t) is used for estimating parameters, data should be preprocessed. The sample data of Δw(t) should be rejected if its value obviously deviates from other values. Fig. 2. shows that most points stay in or around the interval from 0 to 0.02. Here the point is rejected if its absolute value is equal to or greater than 0.02. The cycle when capacity 10% fades is set as the start point of prognostics. Because the historical data are enough to estimate parameters and it is far from the failure threshold. Then take the points from the 1st cycle to the start point to estimate parameters. Sample moments of Δw(t) is easy to compute, and the results of parameters' estimation are shown in Tab. 1.

Figure 3. The estimating life with processing measurement error.

Before moving on to Monte Carlo Simulation, it should be noticed that the failure threshold set up for simulation should not be a constant. For the existence of measurement error, the battery may be detected reaching the failure threshold when the true value just comes near to the failure threshold. So the failure threshold is set as a constant failure threshold combined with a random variable which follows N(0,σ2).

Figure 4. The estimating life without processing measurement error.

Then the Monte Carlo Simulation with the model of Eq. (9) is done by 10,000 times. The result of estimating life is shown in Fig. 3. The method of modelling degradation as a classical gamma process which does not consider measurement error is also utilized to estimate life of battery No.7. The result is shown in Fig. 4. The results presented in Fig. (3) and Fig. (4) shows that estimation of life with the modified gamma process approach is much more concentrate around mid-value. It means that the modified gamma process approach can give a short interval of lifetime than the classical approach. But it should be noticed that the real life of battery No.7 is about 116 cycles. The reason why the results of both approaches obviously bias the real life is that there are several jumps between the start point of estimation and the failure point, especially at 89th and 90th cycle. Those great jumps seriously affected the accuracy of estimation because the gamma process lacks the ability of prognosticating jumps.

Figure 1. The degradation data of battery No.7.

V. DISCUSSION AND CONCLUSIONS In this paper, the method of modelling measured degradation as a gamma process with measurement error has been discussed. The result shows that it will be helpful to estimate a shorter interval of useful life, because the influence of measurement error is weakened. However, the method which considered measurement errors cannot prognosticate jumps during degradation as the classical gamma process. It still needs some more work. First, the distribution of measurement error may fit other distributions. Second, the parameters of the gamma process may be time-variable for

Figure 2. The increment data of degradation.

TABLE I. RESULT OF ESTIMATING PARAMETERS Start point 66th cycle (10% fade)

Parameter values α

λ

σ2

3.5775

780.5590

9.777e-6

648

fitting degradation. Third, some other stochastic processes such as non-homogeneous Poisson process could be added to the model for possible jumps during degradation. In general, the discussion in this paper shows a way to reduce the influence of measurement error on estimating RUL to get a shorter RUL interval.

[7]

[8]

[9]

REFERENCES [1]

[2]

[3]

[4]

[5]

[6]

J. Z. Sikorska, M. Hodkiewicz, and L Ma, “Prognostic modelling options for remaining useful life estimation by industry,” Mechanical Systems and Signal Processing, vol. 25, no. 5, pp.1803-1836, 2011. A. K. Z. Jardine, D. Lin, and D. Banjevic, “A review on machinery diagnostics and prognostics implementing condition-based maintenance,” Mechanical systems and signal processing, vol. 20, no. 7, pp.1483-1510, 2006. X. S. Si, W. Wang, C. H. Hu, and D.H. Zhou, “Remaining useful life estimation–A review on the statistical data driven approaches,” European Journal of Operational Research, vol. 213, no. 1, pp.1-14, 2011. N. Gebraeel, and J. Pan, “Prognostic degradation models for computing and updating residual life distributions in a time-varying environment,” IEEE Transactions on Reliability, vol. 57, no. 4, pp.539-550, 2008. E. Zio, and F. D. Maio, ‘‘A data-driven fuzzy approach for predicting the remaining useful life in dynamic failure scenarios of a nuclear system,’’ Reliability Engineering & System Safety, vol. 95, no. 1, pp.4957, 2010. Z. Tian, L. Wong, and N. Safaei, ‘‘A neural network approach for remaining useful life prediction utilizing both failure and suspension

[10]

[11]

[12]

[13]

[14]

[15]

649

histories,’’ Mechanical Systems and Signal Processing, vol. 24, no. 5, pp.1542-1555, 2010. H. Z. Huang, J. Qu, and M. J. Zuo, ‘‘Remaining useful life estimation of critical components with application to beratings,’’ IEEE Transactions on Reliability, vol. 61, no. 2, pp.292-302, 2012. C. Hu, B.D. Youn, P. Wang, and J. T. Yoon, ‘‘Ensemble of data-driven prognostic algorithms for robust prediction of remaining useful life,’’ Reliability Engineering & System, vol. 103, pp.120-135, 2012. B. Saha, K. Goebel, S. Poll, and J. Christophersen, ‘‘Prognostics methods for battery health monitoring using a Bayesian framework,’’ IEEE Transactions on Reliability, vol. 58, no. 2, pp.291-296, 2009. J. M. van Noorwijk, J. A. M. van der Weide, M. J. Kallen, and M. D. Pandey, ‘‘Gamma processes and peaks-over-threshold distributions for time-dependent reliability,’’ Reliability Engineering & System Safety, vol. 92, no. 12, pp.1651-1658, 2005. J. Lawless, and M. Crowder, ‘‘Covariates and random effects in a gamma process model with application to degradation and failure,’’ Lifetime Data Analysis, vol. 10, no. 3, pp.213-227, 2004. C. Park, and W. J. Padgett, ‘‘Accelerated degradation models for failure based on geometric Brownian motion and gamma processes,’’ Lifetime Data Analysis, vol. 11, no. 4, pp.511-527, 2005. C. Park, and W. J. Padgett, ‘‘Stochastic Degradation Models With Several Accelerating Variables,’’ IEEE Transactions on Reliability, vol. 55, no. 2, pp.379-390, 2006. S.T. Tseng, N Balakrishnan, and C. C. Tsao, ‘‘Optimal step-stress accelerated degradation test plan for gamma degradation processes,’’ IEEE Transactions on Reliability, vol. 58, no. 4, pp.611-618, 2009. S. P. Kuniewski, J. A. M. van der Weide, J, M. van Noorwijk,‘‘Sampling inspection for the evaluation of time-dependent reliability of deteriorating systems under imperfect defect detection,’’ Reliability Engineering & System Safety, vol. 94, no. 9, pp.1480-1490, 2009.