ARTICLES

Damage detection in generating units using time series analysis

F. Franzoni^†, S. Da Silva^† and G. Carvalho Brito Junior^‡

^† Western Paraná State University (UNIOESTE), Centro de Engenharias e Ciências Exatas (CECE), Campus de Foz do Iguaçu, Av. Tarquínio Joslin dos Santos n. 1300, ZIP Code 85870-900, Foz do Iguaçu, Paraná, Brasil, e-mails: felipe.franzoni2@gmail.com, samuel.silva@unioeste.br,
^‡ Itaipu Binacional, Av. Tancredo Neves, n., 6731, ZIP Code 85866-900, Foz do Iguaçu, Paraná, Brasil
gcbrito@itaipu.gov.br

 

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Abstract - The present paper describes a novelty approach to detect structural changes in the generating units of Itaipu power plant. The methodology is based solely on output time series measured in the rotating electric machine. The method deals with the application of time-series analysis through auto-regressive moving average models (ARMA) and statistical modeling for the linear prediction of damage diagnosis. In order to illustrate the procedure proposed is employed a simplified mathematical model regarding shaft radial vibrations in the generating unit 18A of Itaipu power plant. Several parameters variations are imposed in the system to simulate real damages conditions in the electrical machine. The approach is also evaluated with experimental data. The efficacy, advantages and drawbacks of the proposed method are demonstrated through these numerical and experimental tests.

Keywords - Damage Detection; ARMA Models; Electrical Machines; Statistical Models.

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I. INTRODUCTION

The study of algorithms for damage detection in mechanical system is an area of interest, mainly due to the demand for better performance, reliability and availability requirements in engineering structures. Particularly, the damage detection processes in the generating units of Itaipu power plant roles a strategic function, as this power plant provides 90% of the electrical energy consumed in Paraguay and more than 20% of the Brazilian electric grid demand. In the practice way, several conventional methods adopt the root means square values (RMS) and/or overall vibration values in the machine, as for example the DIN 45666, ISO 10816 or NBR 10082. Unfortunately, in the most cases, these features failed due to the unavoidable experimental errors, dynamics effects, variability, etc. So, to apply these methods is necessary the human advanced experience (Fugate et al., 2001).

In order to overcome the drawbacks above, an attractive procedure has successful application in structural damage examples (Sohn and Farrar, 2001; and Fugate et al., 2001). The approach is based on the employ of one-step-ahead error prediction of time-series model as damage-sensitive index. The main idea is to construct a reference model by using the signal data set in healthy condition and to use this reference model to filter the data in unknown condition a posteriori.

Many authors have shown different applications with procedures involving time-series analysis. Wang (2003) applied this paradigm using an auto-regressive (AR) model of vibration signal for gear fault detection comparing with wavelet analysis and resonance demodulation. Sohn and Farrar (2001) proposed an approach consisting of a two-stage prediction model AR-ARX, where ARX is an auto-regressive with exogenous input model. Lu and Gao (2005) proposed a different linear model written in ARX form without the excitation term. In this case, the acceleration response signal was used as the "input" to the ARX model. Nair et al. (2006) modeled the signal obtained from sensor as an ARMA model. They proposed a new damage-sensitive index, which was a function of the first three AR components and a hypothesis test involving the t-test to obtain a decision as to whether or not the structure was damaged.

Recently, statistical models algorithms has been applied to decide correctly if damages are present or not based on the error prediction from time-series models (AR, ARX, AR-ARX, ARMA or ARMAX and its non-linear variations). In the point of practical view, the statistical process control (SPC) is the most simple and effective. Fugate et al. (2001) employ with success this method in a concrete bridge column when the structural changes were progressively applied. Silva and Dias Jr. (2007) exemplified the application of AR-ARX models and SPC by using a vertical rotor with four degrees of freedom. The fuzzy clustering algorithms are other possible algorithms to confirm structural variations correspondent to damage. In this sense, Silva et al. (2007a; 2008a) compared the two most common fuzzy classifiers fuzzy c-means (FCM) and Gustafson-Kessel procedures. In the present paper, the SPC was used to evaluate the damage classification due to the simplicity explained before.

The goal in the present paper is to illustrate the stages necessaries to perform damage detection using an ARMA reference model and SPC with applications in the monitoring of structural healthy from a generating unit of Itaipu power plant. First, a simplified analytical model was used to simulate the shaft radial vibrations in the Itaipu electric machine and after, the tests are realized using experimental data from shaft vibration signals of the Itaipu generating unit. In the next sections are presented the methodology proposed, the results reached by numerical and experimental simulations, the discussions and the further directions of researches to provide the practical applicability of this method.

II - OVERVIEW OF THE METHODOLOGY

The basic procedure proposed in this paper consists of several steps, the same to experimental data and simulated signals. Firstly is presented the mathematical model used to simulate the generating unit radial vibrations. In the following, the ARMA damage-sensitive index feature is summarized. The section is concluded with the statistical process control (SPC) for direct linear prediction of damage diagnosis in the electric machine.

A. Simplified analytical model of the generating unit

This section describes a simplified mathematical model developed to analyze the dynamic behavior of Itaipu Power Plant generating units (18A), regarding shaft radial vibrations. Once validated, one can employ this model to evaluate the operating condition of the generating units and to detect and diagnose incipient failures. For instance, variations in natural frequencies caused by abnormal changes in the bearing stiffness coefficients (structure and bearings' oil film), can indicate modifications originated by damage.

Figure 1 and 2 show a simplified physical model for the generating unit, with 2 degrees of freedom, the radial displacement x(t)=x in the center of mass and the angular displacement q(t)= q.

Fig. 1: Detail of the degrees of freedom in the machine.

Fig. 2: Itaipu Power Plant generating unit physical model.

Obviously, this model has several simplifications. For instance, the model does not take into account the non-linear behavior of the dynamic coefficients of guide bearings' oil films, as well as the dynamic effects of the turbine labyrinths or the axial flexibility of the generator rotor. Another huge simplification is to consider that the generating unit oscillates in a plane. It is also a considerable simplification disregarding the shaft stiffness, which is usually much higher than the bearing stiffness for this kind of machine.

In order to illustrate the application proposed in the present paper, it was found that the following model is reasonable to simulate the fundamental dynamic behaviour of this machine. This model can be written by:

(1)

The inertial parameters in Eq. (1) are given by:

m₁₁ = m_Total, m₁₂ = m₂₁ = 0, m₂₂ = J (2)

where m_Total is the total mass of the machine (m_Total = 2780 ton.), composed by the mass of turbine rotor including the water (890 ton) and mass of generating rotor (1890 ton.), and J is the moment of inertia relative to the center of mass, given by J=1.36×10⁸ kg.m². On the other hand, the stiffness parameters in Eq. (1) are written such as:

	(3)
	(4)
	(5)

where k₀=0.83×10⁹ N/m is the radial stiffness of generator upper guide bearing; k₁=-0.6×10⁹ N/m is the generator radial magnetic stiffness; k₂=1.25×10⁹ N/m is the radial stiffness of generator lower guide bearing; k₃=1.43×10⁹ N/m is the radial stiffness of turbine guide bearing; k₅=32.0×10⁹ N/m is the thrust bearing axial stifness, z_0g=7.8 m, z_1g=4.8 m, z_2g=2.8 m, z_3g=-5.7 m are the distances relatives to center of mass, and R=2.1125 m is the radius in the thrust bearing pads pivoting point .

The applied force F(t) in the coordinate z_FG =4.8 m, corresponding an incipient magnetic unbalance, that cause the radial displacement x(t) and the angular displacement q(t) is composed by the following forces:

F = F_i + F₀ + F₁ + F₂ + F₃ (6)

where F_i is the inertial force amplitude, F₀ is the generator upper guide bearing radial reaction force, F₁ is the generator magnetic radial force, F₂ is the generator lower guide bearing reaction force and F₃ is the radial reaction force in the turbine guide bearing. It is very difficult to estimate the value of each component of force F. In this paper, the value used to represent the amplitude force F is chosen after a trial-error procedure based on the real displacement measurements in the machines. The order of this value is 10⁶~10⁷ N. Several harmonics of fundamental frequency Ω of the generating unit are excited. However, it was considered the excitation of 1 and ¼ of fundamental frequency Ω (excitation caused by partial load vortexes):

(7)

where the fundamental frequency to the machine operating in 60 Hz (brazilian system) is 92.3 RPM.

More details about this simplified mathematical model and validation tests can be found in the following paper (Brito Jr. et al., 2007).

B. Feature index through ARMA model

The first stage in the damage detection approach is devoted to the construction of an ARMA model based on the healthy states for each output discrete-time displacement x(k). The ARMA(na,nc) model can be written as (Ljung, 1998):

A(q^-1)x(k) = C(q^-1)e_x(k) (8)

where e_x(k) is the error between the measured signal and the output from ARMA prediction model. The polynomials in delay operator q^-1, A(q^-1) and C(q^-1), are written by the following equations:

(9)

This model is called reference base. The Burg method or prediction error method (PEM) can be used for estimating the polynomials A(q^-1) and C(q^-1) coefficients (Ljung, 1998). The most part is already implemented in the system identification toolbox for Matlab®. These routines can be employed to obtain the healthy ARMA model from Eq. (9).

The next step in this methodology is the investigation of the output data in unknown condition, namely y(k), using the reference model based on Eq. (9):

A(q^-1)y(k) = C(q^-1)e_y(k) (10)

where e_y(k) is the prediction error of the system in unknown condition (healthy or damaged). If the ARMA model in Eq. (8) is not a good prediction for the new output signal y(k), then the unknown residual error e_y(k) should change when compared with the reference error e_x(k). This would indicate a structural variation, possible associated with damage.

Several feature indexes can be employed, as for instance, to monitor the standard deviation of the unknown error and compare its value with the standard deviation of the healthy state. This ratio ? is given by (Silva et al., 2007):

(11)

It is assumed here that e_y(k) and e_x(k) are asymptotically normally distributed. It is worth noting that by comparing the value of the prediction error at each point representing one region of the structure or machine, most often the measuring point in the region of damage has a substantial change in its index. Thus, this index can be used effectively as damage location indicator, because an increase in this index value would indicate that the location of measurement is close to the damage.

If the index of Eq. (11) presents a non-Gaussian distribution, this approach must be modified, which can be done by using extreme value statistics (EVS) (Worden et al., 2002). EVS fits only the data distribution tail. But, if it is reasonable to assume that the set of data fall close to the normal distribution curve, this procedure is not used. However, in this paper the SPC methodology is used to extract an index performance to indicate healthy or unhealthy state.

C. Statistical process control

A current approach used in damage detection is the statistical process control (SPC). This method is based on a control chart, which is used for a continuous automatic monitoring of the machine condition (Silva and Dias Jr., 2007; Silva et al., 2008b). A control chart is composed by a centerline (CL) located at the mean value of the reference residual error e_x(k) and two additional horizontal lines, the upper and lower control limits (UCL and LCL) versus the sample number (Montgomery, 1996). The CL, UCL, and LCL are given by:

(12)

where is the sample mean value and S is the standard deviation, both relative to n observations in each sample. Z_a/2 is the percentage point of the normal distribution. The standard deviation S can be estimated by (Montgomery, 1996):

(13)

where Γ is the gamma function, m is the operator mean and std is the operator standard deviation. In general, when the system presents some damage, one can clearly observe a statistically significant number of outliers (samples outside the control limits) as well exemplified in Fugate et al. (2001) and Silva et al. (2008b).

III - RESULTS

The tests performed in order to illustrate the approach proposed are presented in this section.

A. Damage detection with simulated signals

The previously mathematical model is analyzed to simulate the shaft radial vibrations. Equation (1) is numerically solved by using a sampling rate of 199 Hz with 4096 time samples. Three different scenarios of undamaged state were considered in this work, each one obtained by the changes of the operational condition (10% of RMS noise added in the output time-series signal). The first one is utilized to obtain the reference ARMA model (case 1). The second one is employed to validate the ARMA model (case 2) and the last one is used to perform a false-positive test. Table 1 describes all cases investigated, where the cases from 4 to 10 correspond to numerical simulation performed in the damaged system (simulated as loosening different stiffness parameters values).

Table 1. Structural conditions evaluated.

The damages simulated considering only reduction of stiffness. There are damages that are related to the existence of preloaded applied and that can induce an increase in stiffness, but is a difficult situation depending on the operational condition. The results with the proposed index computed with the error prediction signal should vary associated directly with the structural change, independently of the type of damage.

Figures 3 and 4 show the output time signal simulated with the model assuming the case 3 (healthy) and case 6 (with damage), respectively. It worth to observe that is very difficult to detect structural changes by analyzing only these figures.

Fig. 3: Output time signal simulated with the model to healthy condition(Case 1).

Fig. 4: Output time signal simulated with the model to damaged condition (case 6).

A previous analysis by Akaike Information Criteria (AIC) led that an ARMA(12,6) model is enough for a suitable prediction. Figure 5 presents the cross correlation function between the output and the residual error from model. Therefore, the prediction error is close to white noise process. Figure 6 shows the prediction ability of the reference model by comparison of the measured output and one-step ahead predicted model output from reference ARMA(12,6) model.

Fig. 5: Residual analysis - correlation function of residuals from reference ARMA(12,6) model.

Fig. 6: Measured output and one-step ahead predicted model output from reference ARMA(12,6) model.

The ARMA(12,6) residual errors were computed by all cases and it is compared the respective probability density function (PDF) estimated through the kernel smoothing method (Bowman and Azzalini. 1997). If there is damage in the machine the PDF should change. Figures 7 and 8 illustrate this by comparing the PDF of the residual error from case 3 (healthy) and case 6 (with damage) with the reference signal error (case 1).

Fig. 7: Comparison of the probability density function of ARMA residual error. Unknown condition corresponds to healthy state (Case 3).

Fig. 8: Comparison of the probability density function of ARMA residual error. Unknown condition corresponds to damage state (case 6).

Figure 9 and 10 show the same examples, but by comparing the PDF of the output signal. It is very important to note that the PDF of ARMA residual error provided bigger information about the damage than the PDF of the output signal.

Fig. 9: Comparison of the probability density function of output signal. Healthy condition (Case 3).

Fig. 10: Comparison of the probability density function of output signal. Damage condition (case 6).

In order to classify the structural states based on the dataset with a more rigorous statistical criterion was employed the SPC technique explained before in this paper. The prediction error has 4096 points. The col-lected data were arranged in four groups of 1024 samples each to estimate the CL, UCL, and LCL. Z_a/2 was chosen 2.57, which corresponds to 99% of confidence. 11 samples (i.e., 1% of 1024 samples) were expected to be outsides the control limits, even for the machine without any damage. Therefore, the threshold value is set to 11 samples. Figure 11 shows the reference error with the control limits for case 3. One can observe 11 outliers that is an indication of healthy condition, where the outliers are marked by '*'. On the other hand, Fig. 12 shows that the number of outliers increased when a structural variation was introduced, case 6. In this example, the number of outliers (140 outliers) increase when the severity becomes larger. However, this result is not conclusive due to low number of data considered. Tests with larger amount of data must be performed in order to take information about the level of severity. Indeed, the goal of the present paper is to evaluate if the ARMA error and SPC charts are useful to detect damages in the machine. The damage quantification must be addressed in further works.

Fig. 11: ARMA prediction error with SPC limits. The number of outliers (11) shows normal condition (Case 3).

Fig. 12: ARMA prediction error with SPC limits. The numer of outliers (140) indicates the existence of damage (Case 6).

Figure 13 presents the ratio between the standard deviations of the residual errors given by Eq. (12) for various undamaged and damage source.

Fig.13: Feature index to indicate the healthy or damaged states ? ratio framework.

Figure 14 provides the number of outliers for all cases described in Tab. 1, one can observe that the SPC present a suitable detection for incipient damage.

Fig. 14: Feature index to indicate the healthy or damaged states. Outliers evolution.

B. Damage detection with experimental data

Now the procedure proposed is evaluated with experimental data. The signals are measured in three points of a generating unit of Itaipu: at generator upper guide bearing, at the generator lower guide bearing and at turbine guide bearing. Figure 15 shows that each plane has two sensors placed in orthogonal localization, called X (Brazil) and Y (upstream). Thus, a total of six signals are sampled. The sampling rate for all tests was 2.4 kHz storing 38018 samples in each sensor. It was collected in two sets of data for days 06/03/2009 and 06/29/2009.

Fig. 15: Unit 18A with the measured points.

After sampling, the signals were preprocessed to eliminate the frequency of the network and a decimated process was executed, reducing both sets in 3801 samples. The processed signals from (06/03/2009) were considered as standard condition and used as a reference filter to determine ARMA errors.

To determine the order of the reference ARMA model, an iterative algorithm based on AIC was implemented as described previously in this paper. Through this, it was obtained that an ARMA(12,6) model is enough for a suitable prediction. The procedures for validating models and prediction errors computing were the same executed in the simulated signals, as can be seen in the previous section of this paper. Thus vectors of prediction errors were sampled for each signal in reference and unknown condition. Figures 16 and 17 show the validation procedures based on model residuals associated with one of the cases (upper guide bearing in Y direction). The residual analysis presents a good correlation between the output and the residual error from ARMA model. Therefore, the prediction error is close to white noise process, as can see in Fig. 16. Figure 17 shows the measured output and one-step ahead predicted model output from reference ARMA(12,6) model that validates the prediction ability of the reference model with 91,41% of correct fitting. Now, a comparison between the prediction errors is executed making the respective probability density function (PDF). Figure 18 illustrates this by comparing the PDF of the residual error from signals of upper guide bearing in Y direction.

Fig. 16: Residual analysis - correlation function of residuals from ARMA(12,6) (upper guide bearing in Y direction).

Fig. 17: Measured output and one-step ahead predicted model output from ARMA(12,6) (upper guide bearing in Y direction).

Fig. 18: Comparison of the probability density function of ARMA residual error. Both sets correspond to upper guide bearing in Y direction being 06/03/09 the reference condition and 06/29/09 the unknown condition.

Again, the comparison with PDF is not enough to determine the damage state, thus a SPC procedure is required. In this case to estimate the CL, UCL, and LCL was used the set of reference signals of 06/03/09 and Z_a/2 was chosen 2.57, which corresponds to 99% of confidence. Therefore, the prediction error has 1024 points and the collected data were arranged in four groups of 256 samples each to the threshold value is set to 3 samples thus 3 samples (i.e., 1% of 256 samples) were expected to be outsides the control limits, even for the machine without any damage. Figure 19 shows the reference error with the control limits for upper guide bearing in Y direction signals, one can observe 3 outliers, that is an indication of healthy condition.

Fig. 19: ARMA prediction error with SPC limits (upper guide bearing in Y direction signal). The number of outliers (3) shows normal condition.

Figure 20 presents the ratio between the standard deviations of the residual errors given by Eq. (12) for all analyzed signals, one can observe all feature index close to threshold value (1), which indicated a healthy condition.

Fig. 20: Feature index to indicate the healthy or damaged states ? ratio framework.

IV. FINAL REMARKS

In this paper was proposed a methodology to detect damages and to avoid false alarms during the condition monitoring of the Itaipu Power Plant generating units (18A). A threshold value using the statistical process control was defined by using the number of outliers. The index feature involved the error prediction extracted through ARMA model reference constructed using the output data set in healthy condition. The authors point out three positives aspects in this approach for practical purposes in real-world structures: (1) The rate of false alarms can be reduced comparing with conventional methods actually used in the condition monitoring at Itaipu; (2) This method would be useful for detecting faults in the machine when the effect is small enough to be masked by noises in the system; and (3) The combination of this approach with wireless sensing system is very attractive, because it allows conducting an automatic monitoring without human supervision by using digital filters implemented in a DSP board.

The next step in this research work should be evaluate more tests by using the data set obtained by experimental measures to show the applicability of the proposal in real condition and to realize tests in order to locate the damage. In further researches the authors also intend to include quantification of damage, i.e., prognosis analysis. Approaches not based on mathematical model should consider the use of these features, i.e., outliers, error predictions, and other damage metric index (RMS, overall, etc.) for system with different damage levels. The inclusion of these features allows driving a supervised learning algorithm, as for instance, by using classical neural network to obtain the correlation between outliers and damage sources. Another question is the proposition of an index based on the total number of outliers in order to have a simpler relation to practical engineers in the field.

ACKNOWLEDGEMENTS
The authors would like to thank the support provided by Itaipu Binacional, Fundação Parque Tecnológico Itaipu (PTI) and National Council for Scientific and Technological Development (CNPq/Brazil). Felipe Franzoni is thankful the PIBIC/CNPq for his scholarship.

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Received: June 2, 2010.
Accepted: March 31, 2011.
Recommended by Subject Editor Jorge Solsona.