Product technical life prediction based on multi-modes and fractional Lévy stable motion

https://doi.org/10.1016/j.ymssp.2021.107974Get rights and content

Highlights

Abstract

Some equipment degradation processes have long-range dependence (LRD) and multi-modes characteristics. The multi-modes are caused by changes of the external environment, the operating conditions and the loads throughout the lifetime of the equipment. In the present paper, a multi-modal Fractional Lévy Stable Motion (FLSM) degradation model is developed to predict the product technical life or remaining useful life (RUL) of equipment. The advantage of FLSM lies in its LRD characteristics and its ability to describe multiple stochastic distributions as the tail parameter α changes. Multi-modes, switching points and modal categories are identified by change point detection and clustering algorithms, and a Markov state transition matrix describes the modes switching law. The probability density function (PDF) of RUL is established by Monte Carlo Simulation. The effectiveness of the prediction model is verified by a practical example of a blast furnace.

Introduction

The efficient prediction of the remaining useful life (RUL) of equipment can allow the effective scheduling of its maintenance, for high availability and prevention of major accidents due to equipment failure [1], [2]. In recent years, researchers have been proposing models of the degradation of equipment based on stochastic processes, such as the Gamma process [3], [4], Markov process [5], [6] and Wiener process [7], [8], [9]. However, these stochastic process models present two shortcomings. First, the incremental processes of these stochastic processes are all independent, and they can, thus, only rely on the current state, which leads to inaccurate prediction results due to the little dynamic information considered. To solve this problem, the LRD model of RUL is proposed in [10], [11], which comprehensively combines the past and current degradation states to predict the future RUL. Second, the above studies are based on historical data collected under controlled experimental conditions. However, in practice, the operating conditions, external environment and loads inevitably change. For example, the relaxation effect of a battery [12] and the slag skin effect of a blast furnace [13] lead to multi-modes degradation processes.

In engineering practice, there are many industrial equipment degradation processes with LRD characteristics, for example, in lithium batteries [14], turbofan engines [10]. For the RUL prediction under these conditions in practice, the Long Short-Term Memory (LSTM) Recurrent Neural Network (RNN) [15], [16], Generalized Cauchy Method [17] and a degradation model based on fractional Brownian motion (FBM) [18], [19] have been recently used. LSTM-RNN, like any neural network, needs sufficient data for training. The increment of the FBM and Generalized Cauchy Method obeys the Gaussian distribution, but in practice, the degradation of most equipment industrial equipment obeys non-Gaussian distributions [20]. To overcome these shortcomings, in this paper we adopt FLSM [21], [22] to account for the LRD characteristics of the degradation process. The FLSM is a non-Gaussian process with infinite variance [23], [24], which is particularly important to model degradation processes with sharp jumps. In addition, FBM is just a special case of FLSM [25], [26].

In this paper, we provide a detailed analysis of FLSM. Firstly, the integral expression of FLSM is discretized and according to the definition of stable distribution [27], it is obtained that FLSM obeys a Lévy stable distribution and has the advantage of describing also high jump in data. Secondly, based on the definition of self-similarity [28], [29], the self-similarity parameter of FLSM is derived and the Maruyama parameter [30] is used to obtain the specific distribution of the FLSM increment, which provides a way for estimating the parameters of the degradation model. Finally, the LRD characteristics of FLSM are discussed in three casesH=1/α,H<1/α and H>1/α. Then, the LRD condition of FLSM, H>1/α, is obtained and considered as a necessary condition for the establishment of the LRD degradation model.

It should be noticed that different operating conditions, maintenance activities, environmental and load changes lead to the generation of multi-modes of degradation in practice. At present, most of the research on multi-modes have focused on artificially setting the modes number and switching times [31], [32]. On the other hand, the recording of modal switching is not easy to achieve in practice. In the absence of mode labelling, the description of the mode switch during the degradation process faces various challenges, one of which is how to find the times of mode switching and the categories of modes. Segmentation and clustering algorithms have been proposed to identify the mode switching points and the categories of modes in degradation processes. Furthermore, the switching law between the modes should be provided. For this, in this paper the Markov state transition matrix [32], [33] is used to describe the multi-modes transition law, and the identified mode switching points and categories are used to estimate the elements of the matrix.

For practical applications, it is necessary to develop degradation models with both LRD and multi-modal characteristics to predict the RUL. FLSM is used as the driving term of the diffusion function in the degradation model to reflect the LRD and non-Gaussian characteristics of the degradation process, and the stochasticity of the drift function is used to represent the multi-modal characteristics in the degradation process. Due to the stochasticity of the drift function, the problem of coupling with the drift function will occur when the maximum likelihood estimation method is used. Therefore, the characteristic function method [34], [35] is used in this paper to avoid this problem. The established degradation model can be used to predict the RUL of equipment. In [36], [37], it is proposed to derive the degradation model by the weak convergence theorem and explicit calculation formula for the PDF of the RUL is obtained. However, the Lévy stable distribution does not have a fixed PDF format, which makes it difficult to derive the PDF of the RUL. In order to solve this problem, the Monte Carlo method [38] is used in this study to obtain the PDF of the RUL. The real degradation data of the No. 2 blast furnace wall temperature of the Guangxi Liuzhou Iron and Steel (Group) Company are used to verify the effectiveness of the multi-modal FLSM degradation model. Then, compared with the FBM-based multi-modal degradation model [39] and FBM-based nonlinear degradation model [40], the superiority of the multi-modal FLSM degradation model is discussed from two perspectives.

More specifically, according to the definition of the RUL and a series of transformations, the RUL becomes the first arrival time when Lévy stable motion reaches the time-varying and stochastic threshold. At this point, the main task is that of calculating the PDF of such the time-varying and stochastic threshold. Under the condition that the drift coefficient in each specific mode is constant, there are two cases depending on the categories of modes. When the categories of modes are equal to 2, the explicit calculation formula of the drift function PDF is obtained; when the categories of modes are greater than 2, the Markov Chain Monte Carlo process [41] is used to obtain the value distribution of the drift function under appropriate assumptions. However, the drift coefficient of each segment of each specific mode is different. In this paper, we make the assumption that the drift coefficient under each specific mode obeys a normal distribution. Therefore, the full probability formula [42] needs to be used, to extend the PDF of the drift function in the case where the drift coefficient is a stochastic variable.

In summary, for the RUL prediction of the non-Gaussian degradation process with LRD and multi-modes characteristics, a degradation model driven by a piecewise drift term and a FLSM is developed. The piecewise drift term describes the mode switching characterized by a Markov state transition matrix, and the FLSM is used to predict the LRD and non-Gaussian characteristic in the degradation. The RUL prediction process are as follows:

Step1: the prediction of future modes. A framework to identify the multi-modes in the degradation process is first needed. Based on the historical degradation data, the potential mode switching points are identified. Then, using a clustering method, the degradation segments divided by the potential mode switching points are clustered as several categories automatically; Since the modal switching often occurs is a random event with Markov characteristics in the degradation process, a Markov transfer matrix was constructed by the identified modal switching points and the categories for predicting the future modes.

Step 2: the prediction of future degradation. the degradation model was constructed by the drift term corresponding to the future modes and the FLSM driver term satisfying the LRD condition for predicting future degradation.

Step 3: the prediction of RUL. When the future degradation reaches the threshold for the first time, the equipment is considered to have failed. The RUL prediction value is the time difference between the failure time and the initial prediction time.

The content structure of this paper is as follows. The second section introduces the Lévy stable distribution and some characteristics of FLSM, among which the infinite variance of FLSM and LRD conditions are the key aspects. The third section provides an algorithm for identifying the mode switching points and the categories of modes in the degradation process and constructs the Markov state transition matrix representing the modes switching law. The RUL prediction model and the incremental distribution of FLSM are given in Section 4. Section 5 describes the parameter estimation of the degradation model. Section 6 uses the real degradation data of the No. 2 blast furnace wall temperature of the Guangxi Liuzhou Iron and Steel (Group) Company to verify the validity of the model. A comparison is made with the FBM-based multi-modal degradation model and the superiority of the multi-modal FLSM degradation model is highlighted. Finally, section 7 gives some conclusions.

Section snippets

Long-range dependence

The FLSM can be defined by the following stochastic integral [21], [22]LH,αt=-at-s+H-1α--s+H-1α+bt-s-H-1α--s-H-1αMdswhere aandbare non-zero constants, (t-s)+=maxt-s,0,(-s)-=maxs,0 and H is the self-similar parameter. M(ds) is a cluster of symmetric Lévy stable random variables, which are independent of each other. More often, the mutual independence of the sequence can be represented by the self-similar parameter H=1/2[43], [44]. So, the stochastic variable cluster M(ds) satisfies:Mcds=c12Mds

Identification and switching law of multi-modes

Fig. 1 shows the multi-modes degradation of the blast furnace wall temperature caused by the slag skin switching in different states. Slag skin has three states: formation, falling off and stability, which correspond to temperature drop, rise, and stability of blast furnace wall. The multi-modes show trend changes in the degradation process. To model such process, there are some challenges: one is how to find the modal switching points and modal categories; another is how to model the switching

Remaining useful life prediction base on the degradation model

Considering both LRD and multi-modes, in this section, a flexible degradation model is established to predict the RUL. Under the LRD condition of H>1/α, the diffusion term is driven by FLSM and the stochastic drift function0tμ[φs]ds is used to describe the multi-modes of the degradation process. The multi-modal FLSM degradation model is expressed by the following equation.Yt=Y0+0tμ[φs]ds+δaLH,αtwhere Y(0) is the initial value of degradation and φ(t) represents the mode at timet; μ[φt] is the

Parameter estimation

The estimation of the modal switch probability was given in the previous section. There are only two tasks left in the parameter estimation of the multi-modal degradation model. The first is the estimation of the parameters in the diffusion function and the second is the drift function. Since the self-similar parameter H describes the dependence of the degradation curve on time, it is not coupled with other parameters, so the self-similar parameter H can be estimated first. The

Remaining useful life prediction of blast furnace

During the production process in a blast furnace, the surface of the cooling wall maintains a certain thickness of slag skin, stably attached, so as to slow down the corrosion of the furnace lining and protect the cooling wall, which is necessary to ensure the longevity of the blast furnace. Since the thickness of the slag skin cannot be measured directly, it cannot be used as a real-time monitoring index for the safety of the blast furnace. When there is no slag skin or the slag skin is too

Conclusion

A multi-modal FLSM degradation model with both multi-modal and LRD characteristics is proposed in this paper, which is suitable for RUL prediction of industrial equipment with LRD characteristics under varying external environment and operating conditions. The FLSM that satisfies the LRD condition αH>1 is used as the driving term of the diffusion function, and the stochasticity of the drift function reflects the multi-modes switching of the degradation process. At the same time, the Markov

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

CRediT authorship contribution statement

Shouwu Duan: Software, Data curation, Visualization, Experiment, Writing - original draft, Writing - review & editing. Wanqing Song: Conceptualization, Resources, Methodology, Validation, Writing - review & editing. Enrico Zio: Formal analysis, Project administration, Writing - review & editing. Carlo Cattani: Supervision, Writing - review & editing. Ming Li: Funding acquisition, Administration. Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

First, I would like to thank Guangxi Liuzhou Iron and Steel (Group) Company for the data provided. Secondly, I would like to thank the Graduate School of Shanghai University of Engineering and Technology for its strong funding and Professor Song Wanqing's meticulous guidance.

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