Chapter 5

Damage and Forecast Modeling ¹

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5.1. Introduction

The underlying issue that justifies a prognosis procedure is the implementation of maintenance policies or decision rules that take into account the history of the system, its environment, the diagnostic on its current state, and possibly the future operational conditions. Hence, we are placed straightaway in a context where online information must be integrated, and where decision rules are condition-based and/or dynamic with respect to this information. Thus, the aim is to exclude as much as possible decision-making from a mean behavior and to consider more delicate situations from the point of view of probabilistic approaches. Besides, we are often placed in a very operational approach, in the sense that we set a dynamic decision rule a priori and we look for tools enabling us to apply it by predicting the evolution of the system on a restricted horizon (by calculating the remaining useful lifetime in particular). In this framework, the use of models based on knowledge/learning makes sense and can happen to be better adapted and easier to implement than a probabilistic evolution model. On the contrary, on a more strategic level with an optimization point of view, it can be more useful to have at our disposal a mean performance criterion enabling us to optimize the possible decision rule(s), on an infinite horizon. A probabilistic approach will then certainly enable us to supply the required indicators. Hence, we have chosen to compare the modeling tools referred to as probabilistic — currently developed at Institut Charles Delauney (ICD) (lifetime models, ARMA models, Markov chains, stochastic processes, etc.) — and the non-probabilistic modeling tools currently developed at Centre de Recherche en Automatique de Nancy (CRAN) (neural networks and techniques based on artificial intelligence, etc.) at operational and strategic levels by favoring a priori the following two points:

– at the operational level, how are the two approaches complementary or competing?

– at the strategic level, how the results of the prognosis carried out from non-probabilistic approaches can be used to optimize decision rules with a probabilistic decision criterion?

5.1.1. Operational level

As shown in the literature review undertaken by Maxime Monnin [MON 09, BAN 09, CHI 99, LIA 06, MAZ 07], it turns out that in practice, the probabilistic models are used to the exclusion of non-probabilistic models (and vice versa), and it appeared interesting to us to challenge this split in DEPRADEM project. If this lack of interaction is explained partly by linking these tools to different scientific communities, it is also justified by the nature of the problems dealt with. The aim of the works developed within Khanh Le Son’s master and doctoral theses that started in October 2009 is to identify how the deterioration models developed at CRAN and those developed at ICD are comparable (in particular with respect to their use in maintenance decision-making) and how they can be complementary or competing. We are interested in the type of information that is used to build these models (online and offline in particular), in the kinds of components under consideration (size, structure, quantity of indicators characterizing their state), in the nature of covariables (present and future operational conditions in particular), and in the damage and decision indicators that can be supplied. The involvement of EDF (Electricité de France) in the project must favor the emergence of real study cases or, at least, must enable us to target the work on a few typical problems.

5.1.2. Strategic level

At this level, the main question is to know how the two approaches can be complementary. The tools used at CRAN to forecast failures enable us to build a data history regarded as offline feedback data. Based on this latter, we can propose a lifetime distribution or a stochastic process modeling the deterioration of the system under consideration. This modeling can be used to estimate the remaining useful lifetimes and to build prognosis-based maintenance policies.

During the first stage of Khanh Le Son’s thesis, we considered the operational level and implemented probabilistic approaches to predict a remaining useful life (RUL). These approaches are based on either the use of lifetime distributions that enable us to model the lifetime of a system or the use of stochastic processes that enable us to model the evolution of the system from the start to the failure. We gave more importance to the second approach that is more sophisticated. The advantage of this second approach depends on the number of states considered for the process (from two to infinity for a continuous process). It should be pointed out that the choice of the approach depends mainly on the type of data we deal with. The goal of this chapter is to give a general idea of these methods. To make a comparison with non-probabilistic approaches, we chose to work on data supplied by the International Conference on Prognostics and Health Management (PHM). The winners of the challenge [HEI 08, PEE 08, WAN 08] use non-probabilistic models based on neural networks or time series. To compare our method with the other non-probabilistic methods, we suggested a probabilistic method to model the deterioration of the system considered by the 2008 PHM data and to estimate the RUL.

The chapter is organized as follows. In section 5.2, the analysis of the sensor measurements of PHM data set is carried out. In section 5.3, a degradation indicator based on principal component analysis is proposed. In section 5.4, the prognostic method is applied with the new degradation indicator by using stochastic processes and the RUL estimation method is presented.

5.2. Preliminary study of data

5.2.1. Structure of the database

In the framework of the PHM challenge, two data sets are available: the training data set and the test data set. The PHM data set consists of data collected from 218 components of an unspecified system. There are three operational settings that have a substantial effect on the performance of the units. The data for each cycle of each unit include the unit ID, cycle index, three values for the operational settings, and 21 values for 21 sensor measurements connected to the state-of-the-units. The sensor data are noisy.

5.2.2. Performance criterion for the prognostic

The algorithm performance is evaluated by a function called score function, denoted by S, which evaluates the error in predicting the number of remaining time cycles and is defined as follows:

[5.1]

where

[5.2]

and

[5.3]

Other criteria are used in other articles, in particular the root mean square error (RMSE) is defined as follows:

[5.4]

The lower is the score, the better is the proposed RUL estimation algorithm. In the PHM Conference challenge, the following papers obtained the best results. Wang [WAN 08] obtained the total score of 5,636 with a similarity-based approach for estimating the RUL. In [PEE 08], an artificial neural networks mixed with a Kalman filter is used and the obtained mean square error is 984. Felix in [HEI 08] with a recurrent neural networks method achieved the mean square error of 519.8.

5.2.3. Definition of a deterioration indicator

In the framework of probabilistic prognosis methods, a stochastic process is used to model the deterioration phenomenon, i.e. an evolution from the operating state to the failure via intermediate states. For this purpose, we need to have a scalar indicator that models the deterioration. In the particular case of PHM data, among all the papers that obtained the best scores, we did not find any satisfactory deterioration indicator. This is due to the fact that they use either a neural network approach where a deterioration model is not necessary [HEI 08, PEE 08] or a very simplified model as in [WAN 08].

After an initial study of the training data set, the following features are pointed out:

– A degradation indicator cannot be directly deduced from the 21 sensor paths.

– The three measurements connected to operational settings can be represented by six clusters in a three-dimensional space. This second point is illustrated in Figure 5.1, where the variables OP1, OP2, and OP3 correspond to the measures of the three sensors devoted to operational settings. The six observed dots are actually six highly concentrated clusters. Hereafter, it is supposed that these clusters indicate six discrete operating conditions of the system. Hence, each cycle of a unit can be labeled by a regime ID from 1 to 6, replacing the original three variables of operational settings.

– For each unit, the changes of modes occur frequently and can represent very different sequences.

– If we select only the measurements corresponding to the same regime ID, some of the 21 sensor measurements connected to state-of-the-units will exhibit prominent trend along the life of a unit. In [WAN 08], for each mode, the plot of the collected data from one sensor is used to show the trend of the system’s degradation. Some sensors do not show a clear trend due to high noise or due to their low sensitivity to the degradation. Their inclusion in the analysis may lower the accuracy of prediction. According to [WAN 08], using a subset of the available sensors can help to improve the accuracy of RUL estimation; therefore, they propose to use a selection of seven sensors (2, 3, 4, 7, 11, 12 and 15). Hereafter, for the data analysis, we use this selection of seven sensors.

These first analyses brought out the following facts: the deterioration indicator cannot be defined directly from the measurements of 21 sensors, and the operational conditions corresponding to the six modes influence the trajectories of the 21 sensors and must be taken into account in the construction of the deterioration indicator. Furthermore, it is useless to take into account the 21 sensors, and the sensor selection suggested in [WAN 08] can be adopted.

5.3. Construction of the deterioration indicator

To build a deterioration indicator, we should identify a set of trajectories that can be considered as the realization of the same stochastic process and a real trend in these realizations that can be interpreted as the deterioration from the beginning of the lifecycle until the failure. The next sections are devoted to the construction of a deterioration indicator.

5.3.1. Study of the failure space with PCA

First, we apply PCA without taking into account the operational modes. We consider the data subset of the failure times for 218 units. Let X be the original data set where its rows indicate the information on individuals (at the failure time) and its columns indicate the information on variables (seven sensors). For the PCA corresponding to the chosen data set, the principal component’s coefficients and variances are the eigenvectors and eigenvalues of the covariance matrix of the centering matrix of X. The criterion to choose the principal component is based on the histogram of the variances of principal components. It can be seen in Figure 5.1(a) that the cumulative percentage of the total variation of the first two principal components is more than 98%. Figure 5.1(b) shows that the projections of failure time measurements on the two-dimensional principal component space are divided into six different groups. This could be due to the fact that the operational modes and the presence of six classes complicate the construction of a degradation indicator. To overcome this problem, we separate the operational modes of the failure times to build six data sets and we apply a PCA to data corresponding to each operational mode. Therefore, we obtain six different principal component spaces. The percentages of histogram for the first two principal components in Table 5.1 show that we can use the two-dimensional principal component space, called failure space, to build a degradation indicator for each unit in each mode. Moreover, the high concentration of the positions of different component failure states on the two principal components space depicted in Figure 5.2 leads us to consider only one failure place for each operational mode. We can consider the barycenter of the cluster of failure states on the two principal components space as the unique failure place.

Table 5.1. Contribution of the first three axes of PCA in each mode

5.3.2. Damage indicator defined as a distance

The degradation indicator can be calculated from this unique failure place at each time t, for each unit, as the distance between the seven selected sensor measurements projected in the failure space of the current unit’s mode and the unique failure place of the current mode. To improve the accuracy of the degradation indicator, the distance of each unique failure space is divided by the dispersion. The dispersion of each unique failure space is defined as the standard deviation of the barycenter of the failure places cluster in each mode.

The construction of the indicator can be formalized in the following way:

– let N_k be the number of individuals at the failure time in mode k,

– let be the barycenter of the projected measurements of the failure places inmode k,k =1,…,6,

– let , i = 1,…, N_k, be the ith projected failure place in mode k,

– let be the projected measurement of the component i at the arbitrary operating time j in mode k.

We then consider the following distance for component i, at time j, in mode k:

[5.5]

where Disper_k the dispersion of the failure places in mode k is defined in the following way:

[5.6]

for each unit, the degradation indicator is calculated by equation [5.5]. Thus, the damage indicator on [0,1,…, r_i] for r_i time steps is described by the following vector:

[5.7]

where k(j) refers to the operating mode at time j.

As an example, we give in Figure 5.4(a) the degradation indicator for component 1, then for all the components (Figure 5.4(b)). We can see that in Figure 5.4(b) there is an important variation between the degradation indicators of different units and there is a general nonlinear and decreasing trend (at the beginning of the cycles, the indicator value tends to be around 1, and at the end of the cycles, at the failure times, it tends to zero). We model then the trend of this indicator by considering this characteristic and then we propose the criterion of RUL estimation in the next section.

5.4. Estimation of the residual life span (RUL)

5.4.1. Simple approach based on the life span

As a first step, we apply the most basic probabilistic method for the prediction of RUL. This method consists of a simple lifetime approach. This means that we only consider the failure times of the units without taking into account the progressive degradation phenomena that lead us from the initial state to the failure.

Let T be the lifetime of a unit. While treating the failure times (or lifetime), the RUL at time t of each unit can be considered as the mean residual lifetime at time t presented as follows:

[5.8]

where R(x) = P(T > x) is the reliability function.

According to the histogram of the failure times of 218 units presented in Figure 5.5, it seems reasonable to propose a Weibull distribution to model the lifetime distribution. For the lifetime distribution modeling and the mean residual life calculation, a three-parameter Weibull distribution is considered. Since the failure time does not start from zero, it is reasonable to consider a non-zero location parameter. The Weibull density function with three parameters (μ, λ, θ) is defined as follows:

[5.9]

By a nonlinear regression, the estimated parameters are obtained, respectively:

The RUL is then estimated by using equation [5.8] on the test data set. The score obtained is S = 9,450. This score is not very convincing in comparison with other non-probabilistic methods where the obtained scores are around 5,636.

5.4.2. Stochastic deterioration model

We suggest here using a Wiener process to model the deterioration and to estimate theRUL.

5.4.2.1. Estimation of parameters of Wiener process

Let D_i,j be the deterioration indicator of component i at time j. We assume that each is a realization of a Wiener process with three parameters (η, σ, α) and therefore satisfies the following conditions:

1) the increments D_−,j − D_−,k, j ≠ k, are independent,

2) for all 0 < s < j, D _j+s − D_−,j is normally distributed with mean η((j + s)^α − j^α) and variance σ²((j + s) − j) = σ²s,

where D_−,j is the value of the process at time j. The deterioration increments ∆d_i,j = D_i,j+1 − D_i,j , i = 1,…, N, for each component follow a Gaussian distribution N(η((j +1)^α − j^α), σ²). We also write observations ∆d_i = (∆d_i,1,…, ∆d_i,ri), for i = 1,…, N, where r_i is the date of the last recording before failure of component i. We then estimate the parameters of Wiener process (η, σ, α) by maximizing the likelihood function [MOL 04]:

where N is the number of units, f_η,σ,α is the probability density function of all increments, and g_η,σ,α is the probability density function of all increments corresponding to each unit. The estimated parameters are obtained as follows:

[5.10]

These parameters will be applied to the test data set in the order to estimate the RUL for each unit. The next section discusses the RUL estimation method by Monte Carlo simulations.

5.4.2.2. RUL estimation by simulation of a Wiener process

The last cycle of the test data set is the time upon which we calculate the RULs. The estimation is carried out as follows: let r_i′ be the last recording time in the test data set for component the deterioration indicator at time r_i′, and L a fixed deterioration level. If we assume that L is the failure threshold, then L = 0. The Wiener process is non-monotonous and therefore can cross, several times, the same threshold (see Figure 5.6). Therefore, considering the failure time as the first passage time or as the last passage time to L might overestimate or underestimate the RUL. Here, after experiments on the test data set, we chose to consider that RUL at time r_i′ is the time that elapses between r_i′ and the mean of the first and last times of the overrun of level L written, respectively, as min and max (Figure 5.6). We thus define, for a simulated trajectory of Wiener process, the following:

[5.11]

where and are, respectively, the first and last times when threshold L is crossed. After n = 10,000 simulations, the estimated RUL, denoted by , is calculated by the empirical mean. For example, the estimated RUL of two units are given in Table 5.2.

Table 5.2. Estimated RUL and associated variance

By using the performance criteria from the PHM Conference, the obtained score S is 5,520, and the RMSE = 438. The performance of our method is thus better than those suggested by the winners of the 2008 PHM Conference data challenge.

5.5. Conclusion

The scores obtained with our method based on the use of Wiener process are, respectively, equal to S = 5,520 and RMSE = 438. These results show that a probabilistic approach of our probabilistic method based on the stochastic process can be more appropriate than other methods. These results tend to show that the procedure that consists of building a damage indicator is meaningful and that the use of a stochastic process can happen to be beneficial to the quality of the prognostic.

We can compare our result with some results obtained as follows.

– Non-probabilistic model with the similarity-based prognostic method of Wang in the 2008 PHM Conference [WAN 08]: the score is 5,636.

– Non-probabilistic model based on the neural networks of Peel [PEE 08] and Heimes [HEI 08] shows, respectively, the mean square error 519.8 and 984.

These results and our obtained score lead us to justify the accuracy of our probabilistic method.

Thus, the probabilistic approaches similar to what we present here seem to be promising. They are generally based on the construction of a deterioration indicator that makes sense with respect to the evolution of a system from a new state to the failure. Furthermore, the use of a stochastic process enables us to finely model the uncertainty observed at the level of this indicator. In contrast to the non-probabilistic approaches like neural networks, the stochastic processes enable us to take into account uncertainties and to give a distribution of RUL, which can be interesting as information to bring to a decision support process.

5.6. Bibliography

[BAN 09] BANJEVIC D., “Remaining useful life in theory and practice”, Metrika, Springer, vol. 69, no. 2, pp. 337–349, 2009.

[CHI 99] CHINNAM R., “On-line reliability estimation of individual components, using degradation signals”, IEEE Transactions on Reliability, vol. 48, no. 4, pp. 403–412, 1999.

[HEI 08] HEIMES F., “Recurrent neural networks for remaining useful life estimation”, International Conference on Prognostics and Health Management, PHM 2008, pp. 1–6, 2008.

[LIA 06] LIAO H., ZHAO W., GUO H., “Predicting remaining useful life of an individual unit using proportional hazards model and logistic regression model”, Proceedings of the Reliability and Maintainability Symposium (RAMS), pp. 127–132, 2006.

[MAZ 07] MAZHAR M., KARA S., KAEBERNICK H., “Remaining life estimation of used components in consumer products: life cycle data analysis by Weibull and artificial neural networks”, Journal of Operations Management, vol. 25, no. 6, pp. 1184–1193, 2007.

[MOL 04] MOLER C., “Numerical computing with MATLAB”, Society for Industrial Mathematics, 2004.

[MON 09] MONNIN M., Projet DEPRADEM — Modélisation des Dégradation et du Pronostic pour l’Aide à la Décision en Maintenance — Synthèse Bibliographique, Rapport de fin de post doctorat — CRAN — Henri Poincaré University Nancy 1, 2009.

[PEE 08] PEEL L., “Data driven prognostics using a kalman filter ensemble of neural network models”, International Conference on Prognostics and Health Management, PHM 2008, pp. 1–6, 2008.

[WAN 08] WANG T., YU J., SIEGEL D., LEE J., “A similarity-based prognostics approach for remaining useful life estimation of engineered systems”, International Conference on Prognostics and Health Management, PHM 2008, pp. 1–6, 2008.

 

 

1 Chapter written by Anne BARROS, Eric LEVRAT, Mitra FOULADIRAD, Khanh LE SON, Thomas RUIN, Benoît IUNG, Alexandre VOISIN, Maxime MONNIN, Antoine DESPUJOLS, Emmanuel REMY and Ludovic BENETRIX.