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Monitoring of trends and removal of undesired trends from operational/process parameters in wind turbines is important for their condition monitoring. This paper presents the homoscedastic nonlinear cointegration for the solution to this problem. The cointegration approach used leads to stable variances in cointegration residuals. The adapted Breusch-Pagan test procedure is developed to test for the presence of heteroscedasticity in cointegration residuals obtained from the nonlinear cointegration analysis. Examples using three different time series data sets—that is, one with a nonlinear quadratic deterministic trend, another with a nonlinear exponential deterministic trend, and experimental data from a wind turbine drivetrain—are used to illustrate the method and demonstrate possible practical applications. The results show that the proposed approach can be used for effective removal of nonlinear trends form various types of data, allowing for possible condition monitoring applications.

Recent forecasts show that renewable energy sources will be generating more than 25% of world’s electricity by 2035, with a quarter of this coming from wind [

It is well known that unexpected failures of turbine components (or subsystems)—such as gearboxes, generators, rotors, and electric systems—can lead to costly repair and often months of machine unavailability, thereby increasing operation/maintenance costs and subsequently cost of energy. Therefore, condition monitoring (CM) and fault diagnosis of WTs—in particular at the early stage of fault occurrence—is an essential problem in wind turbine engineering [

Many CM techniques have been developed to detect and diagnose abnormalities of WTs with the goal of improving gearbox reliability and increasing turbine availability, thereby reducing operation and maintenance costs, as reviewed in the literature [

The

The paper addresses the problem of trend removal/analysis of wind turbine operational data. A homoscedastic (or variance stabilizing nonlinear cointegration) nonlinear cointegration approach is proposed for this task. The objective is to demonstrate a new approach that could be potentially used for condition monitoring and fault detection of wind turbines in the presence of nonlinearity between operational parameters. It is important to note that—in the context of material presented—the homoscedasticity relates to the stable behavior of variance in cointegration residuals.

Previous approaches generally dealt with the existence of heteroscedasticity in the primitive or original data before performing any further analysis. However, in this paper, we coped with the existence of heteroscedasticity in cointegration residuals obtained from nonlinear cointegration process of time series data. In more detail, we have solved the problems of increasing (or unstable behavior) of the variance of cointegration residuals. To the best of the authors’ knowledge, the mentioned problems as well as heteroscedasticity in nonlinear cointegration in general have not been previously investigated in the literature.

The paper is structured as follows. Sections

For the sake of completeness this section briefly introduces the concept of linear cointegration. Firstly, stationarity and nonstationarity of time series are discussed.

In mathematics the concept of stationarity can be introduced using time series analysis. A given time series

Any time series

Equation (

Following this short introduction, the concept of linear cointegration can be introduced using a vector

In other words, the nonstationary

The stationary linear combinations

In essence, testing for linear cointegration is testing for the existence of long-run equilibriums (or stationary linear combinations) among all elements of

In general, the linear cointegration test consists of two steps.

The first step is to determine the existence of cointegration relationships and the number of linearly independent cointegrating vectors among multivariate (nonstationary) time series and to form the cointegration residuals.

The second step is to perform unit root tests on the cointegration residuals found to determine if they are stationary series (i.e., testing for stationarity).

For the first step, the Johansen cointegration method—developed in [

Linear cointegration has been successfully applied to remove unwanted environmental and/or operational variability in various damage detection SHM applications when data are linearly related and operational/environmental common trends are linear, as presented in [

It is well known that time series responses from engineering structure often exhibit nonlinear behavior. Moreover, operational and/or environmental common trends are typically believed to be nonlinearly related to response data used for damage detection. If this is the case, then the linear cointegration theory—described in Section

In the last twenty years, nonlinear cointegration has been studied in many different contexts, as discussed in [

A time series is said to be short memory if its information decays through time. In particular, a variable is short memory in mean (or in distribution) if the conditional mean (or conditional distribution) of the variable at time

Following this introduction, a general definition of nonlinear cointegration has been proposed in [

This simplified definition is still quite general to be fully operative, and, moreover, identification problems might arise in this general context [

Nonlinear cointegration has been recently proposed for SHM applications in [

It is well known that the variance—or volatility that is the square root of variance—of time series often changes over time [

A homoscedastic nonlinear cointegration method is proposed in this section. The method overcomes the heteroscedastic problem related to cointegration residuals by offering a variance stabilizing nonlinear cointegration.

Following the work presented in [

Substituting (

It is clear that for

The application of the first-degree Taylor approximation formula—defined as

Then, substituting (

The above equation can be approximated as

Equation (

It should be noted that the term

From (

Equation (

Finally, one obtains the transformed cointegration residual

Equation (

When (

This equation presents the

Various tests for heteroscedasticity can be used in practice [

The linear time series regression model for one independent variable

The Breusch-Pagan heteroscedasticity test is performed by regressing the squared residuals directly on the independent variables. In the linear time series regression model, one can assume that the mean of the residual

It should be noted that only one independent variable has been used in the current investigations. In general case—when more than one independent variable is employed—the test statistic equals

Because the original Breusch-Pagan test can only be used to test for heteroscedasticity in a linear regression model, hence the test has been adapted to be suitable for the work presented in this paper, that is, to test for the presence of heteroscedasticity in the cointegration residuals obtained from nonlinear cointegration analysis. In order to achieve this, the linear regression model in (

Following the same discussion as above, (

The hypotheses to be tested are the following.

More specifically, the null hypothesis is true (the cointegration residual

It should be noted that

Three examples that explain the homoscedastic nonlinear cointegration method and illustrate its application to nonlinear trend removal and a possible condition monitoring solution for wind turbines are presented in this section. These examples use three different time series data sets, that is, one piece of data with a nonlinear quadratic deterministic trend, another with a nonlinear exponential deterministic trend, and one more piece of experimental data from a wind turbine.

This section recalls the nonlinear cointegrating function

When the nonlinear cointegration form given by (

Similarly, the homoscedastic nonlinear cointegration—given by (

Figures

Nonlinear cointegration results obtained for the time series with a nonlinear quadratic deterministic trend

The data used in this example consist of two different time series variables

Following the homoscedastic nonlinear cointegration method—presented in Section

In the same way, when the homoscedastic nonlinear cointegration form—given by (

Figures

Nonlinear cointegration results obtained for the time series with a nonlinear exponential deterministic trend

Wind turbines are designed to operate in remote onshore and/or offshore areas, where strong winds are available. The WT converts wind kinetic energy into useful electrical energy. The main components of a typical utility-scale WT drivetrain consist of the gearbox, main shaft, main bearing, brake, generator shaft, and generator. The gearbox is placed between the hub and the generator and used to convert the low-speed high-torque power from the WT rotor to high-speed low-torque power used by the generator [

Wind turbine drivetrain used in the current study: (a) generator and phase marker; (b) gearbox and transmission systems.

Condition monitoring for the WT was continuously performed during thirty days in November 2012. Twelve different operational parameters were monitored. These parameters can be grouped into three categories as follows.

The SCADA data for the WT were collected at 10-minute intervals in a period of thirty days. As a result, 4320 data samples (or records) were acquired for each parameter for a variety of different operating conditions. The wind speed is a key operational parameter in wind energy systems [

Experimental wind turbine data showing the nonlinear relations between wind speed and other operational parameters: (a) generator speed versus wind speed; (b) generated power versus wind speed; (c) generator temperature (front part) versus wind speed; (d) gearbox temperature versus wind speed.

The nonlinear relation between the generated power (or the power output) and the wind speed.

The homoscedastic nonlinear cointegration method—presented in Section

Figures

Nonlinear cointegration results obtained for the experimental wind turbine data: (a) original cointegration residual

In summary, when the data from an intact wind turbine were analysed, the modified cointegration residual

Monitoring of trends and removal of undesired trends from operational/process parametric data in wind turbines has been addressed in this paper. The recently proposed homoscedastic nonlinear cointegration has been applied. The method has been illustrated using three different time series data sets, that is, one with a nonlinear quadratic deterministic trend, another with a nonlinear exponential deterministic trend, and one experimental data set from a wind turbine drivetrain.

The results show that the proposed method can effectively remove nonlinear trends from the analysed data and also remove heteroscedasticity from cointegration residuals. For the case study using experimental wind turbine data, the modified cointegration residuals have been shown to be homoscedastic for the data representing undamaged condition. It is expected that these modified cointegration residuals would instantly become heteroscedastic for the data from damaged wind turbines, thereby providing an effective condition monitoring and fault detection tool for wind turbines.

It is clear that further research work is required to test the method for different types of wind turbine data and trends. In particular, the proposed methodology should be investigated for operational data representing various types of faults in wind turbine drivetrains.

In this paper, the homoscedastic nonlinear cointegration method was effectively used for condition monitoring of wind turbines. However, because the proposed method is a general approach—which is simply based on the analysis of measurement data in terms of time series responses acquired from investigated processes or structures by sensors—the authors believe that this method can be properly applied to other engineering applications. It is also possible that researchers and practitioners from the field of Econometrics might benefit from employing our proposed method when they have problems with the existence of heteroscedasticity in economic and financial time series in general and particularly in residuals obtained from linear or nonlinear cointegration process.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The work presented in this paper was supported by funding from WELCOME Research Project no. 2010-3/2 sponsored by the Foundation for Polish Science (Innovative Economy, National Cohesion Programme, EU).