Experience with bicoherence of electricial power for condition monitoring of wind turbine bl~ades

W.Q. Jeffries J .A. Chambers D.G.lnfield

Indexing terms: Monitoring, Wind turbines, Bicoherence

Abstract: The authors explore the application of the normalised bispectrum or bicoherence to the problem of condition monitoring of wind turbine blades. Background information is provided on this type of condition monitoring, how it differs from more conventional condition monitoring of turbo machinery, and the motivation for selecting bicoherence. Bicoherence is defined and compared with the power spectral density. Complications in collecting suitable data, and estimating the bicoherence from that data are investigated; including the requirements of very long stationary data sets for consistent estimates, and computational difficulties in handling such large data sets. Bicoherence is then applied to electrical power output data obtained from a 45kW wind turbine. The turbine is operated in three configurations to represent normal and fault conditions. A blade with less flapwise stiffness but identical outer dimensions to the matched set of blades was fitted to simulate a damaged blade. Comparison of the results from the power spectral density and bicoherence indicates how the bicoherence might be employed for condition monitoring purposes. Slices of the bicoherence with one frequency fixed at the rate of rotation show clear differences between the configurations and substantially reduce the computational effort required to calculate the estimate.

1 Introduction

1. I Why condition monitoring of wind turbine blades is desirable The number and value of wind turbines in the UK and worldwide has increased dramatically over the past few

0 IEE, 1998 IEE Proceedings online no. 19982013 Paper received 2nd September 1997 W.Q. Jeffries is with the Energy and Electromagnetics Section, Depart- ment of Electrical and Electronic Engineering, Imperial College of Science Technology and Medicine, Exhibition Road, London SW7 2BT, UK J.A. Chambers is with the Signal Processing Section, Department of Elec- trial and Electronic Engineering, Imperial College of Science Technology and Medicine, Exhibition Road, London SW7 2BT, UK D.G. Infield is with the Centre for Renewable Energy Systems Technol- ogy, Loughborough University, Loughborough, Leicestershire LE1 1 3TU, UK

IEE Proc.-Vis. Image Signal Process., Vol. 145, No. 3, June 1998

years as wind farms have been installed by developers and accepted by major utilities as a viable method of electric power generation. By the end of 1995, 180MW of wind farm capacity h,ad been installed in the UK with up to 491 MW additional capacity expected, resulting from the third round of the Non Fossil Fuel Obligation and the first round of the Scottish Renewa- bles Order. Wind energy capacity in Europe alone sur- passed 2000MW in 1996 iind is increasing rapidly [l].

This expansion has resu lted in many ageing turbines, in addition to increasing numbers of turbines with a rating of 300kW and above. The probability of failure due to fatigue and wear increases as the number of operation cycles increases. Additionally, the probability of failure due to design or construction flaws is greatest with relatively new machines, especially very large ones. Since the wind turbine industry is fairly young with limited operational experjence, the fatigue life of most machines is not well defined. Blades are a crucial and expensive component of a wind turbine. They also are one of the most likely coimponents to fail [2]. Further- more, failure of a blade causes significant down time or even machine loss, as well as negative publicity. With present large wind turbinles and the highly competitive electricity generation market, the loss of revenue from one machine due to a component failure can cause con- siderable financial stress to a wind farm operator. A low-cost tool that can predict component failure would minimise down time and provide the operators with a better understanding of the ‘health’ of the wind farm. It could also help wind turbine manufacturers by sup- plying additional feedback to improve machine design.

Currently, wind turbine manufacturers employ fairly simple sensors to detect faults in their wind turbines. These sensors detect the results of failure, i.e. excessive vibration from blade failure or overspeed from gearbox failure, and not the condition of components. Although condition monitoring has, been applied successfully in several industries (e.g. aerospace, large power genera- tion and maritime shipping), it has not yet been applied to wind turbines. This was, until recently, partly due to the economics dictated b y the relatively small number of wind generators, and also to the substantial techni- cal challenges presented by such a project. Larger machines now allow more scope for sophisticated mon- itoring, and advances in signal-processing hardware promise lower costs to implement sophisticated signal processing.

The blades of a wind turbine are a vital component, yet the loads that they experience are difficult to

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predict. As the machine size increases, there have been problems with blades due to construction faults, unexpected vibrations and under-predicted loads. To be most effective in a wind farm, monitoring must be continuous and automatic. It should detect known modes of failure reliably without false alarms. A condition monitoring package should be capable of being retrofitted onto existing wind turbines without requiring additional sensors and wiring on the machine.

1.2 Considerations of condition monitoring wind turbine blades A successful blade condition monitoring system must detect a structural fault in a blade with sufficient warn- ing to prevent increased damage and the possibility of catastrophic failure. Therefore, the fault in the blade must modify some characteristic of the wind turbine dynamics and this change has to be detected. Ideally, the chosen characteristic is sensitive to the condition of the blade structure. Conventional condition monitoring of bearings, gearboxes and other parts of rotating machinery typically employs vibration sensors, e.g. accelerometers, mounted on the part in question. On a wind turbine, however, monitoring involves mounting the sensors on a rotating frame and then transmitting the signal to a stationary remote point for processing. The electrical power output of the wind turbine, on the other hand, is accessible from the ground and may offer more comprehensive information on the state of the wind turbine.

An assumption in this type of condition monitoring is that damage in a blade results in a decrease in its stiffness. Fatigue tests of glass reinforced plastic cou- pons show a smooth and continuous decline in stiffness as the test progresses. Generally, stiffness declines slowly and then drops rapidly just before failure. In quasi-static tests to failure on full-size wind turbine blades at the US National Renewable Energy Labora- tory, stiffness decreased by approximately 10% before failure. Fatigue effects then became visible, and stiff- ness dropped quickly as the blade failed [3]. Defects in construction, such as voids in the resin, improper seams between halves, improperly cured resin and defective fasteners, are another possible source of a loss of stiffness. The effect of a defect on stiffness could appear immediately on commissioning or emerge as the inadequate part fatigues in service. The modal proper- ties (mode shapes and their associated modal frequen- cies) of a structure, such as a blade, are another way of describing the physical properties of mass and stiffness distribution. If the physical properties change, then modal frequencies, their relative phases or the modal shape may also be altered.

The coupling from blade vibration to power output stems from more than one interaction between the blades and the rest of the wind turbine. The stiffness of a blade in the flapwise direction (in and out of the rotor plane) is considerably less than that in the edge- wise direction (within the rotor plane), and the flapwise (thrust) loads are greater; therefore, flapwise vibrations are typically much larger than edgewise vibrations. Flapwise motion, particularly at the tip where it is greatest, adds to (or subtracts from) the wind velocity, changing angle of attack, and thereby affecting lift, drag and power. This coupling of flapwise blade motion to power output is strongest in low to moderate

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winds when the blades are not greatly stalled (assuming a stall regulated machine), and it weakens as stalling increases. Analysis of the coherence of flapwise blade acceleration to electrical power supports this reasoning.

The vibration modes of the rotor also interact non- linearly with the rest of the structure causing harmonics in the power output signal at multiples of the rota- tional rate, 1P. In an ETSU report, Azzam et al. pre- dicted that torsional vibrations in the drive train, and therefore variations in power, would show a clear change in pattern due to blade defects, and that the nature of the change could help to determine the type of fault [4]. Depending on the type of nonlinearity, the harmonics may be phase-coupled to the fundamental frequency, and this coupling may vary as the blade properties deteriorate.

With the introduction of commercial wind turbines with megawatt ratings, the possibility of unexpected aero-elastic effects such as blade flutter has emerged as a potentially serious problem. In a few notable cases, blade flutter has been strong enough to destroy a blade. Such large vibrations should be reflected in the power output and may also be detectable.

1.3 Some differences from conventional condition monitoring From a condition monitoring perspective, wind tur- bines differ from conventional steam turbines in many important aspects. (i) The main shaft of a wind turbine turns at low speed and is coupled to an induction or asynchronous gener- ator through a gearbox (some recent designs incorpo- rate low-speed generators to eliminate the need for a gearbox). The main shaft of a steam turbine turns at synchronous speed and is coupled directly to a syn- chronous generator. (ii) The primary modes of vibration in a wind turbine involve the blades and tower as well as drive train, instead of primarily the torsional modes of the drive shaft in a steam turbine. (iii) The primary modes of vibration in a wind turbine are above the rotational frequency of the main shaft, instead of below in the case of the steam turbine. In fact, an important role of some condition monitoring systems for steam turbines is to watch the transition through resonances as the turbine runs up to the oper- ating speed. (iv) The vibration of the component of interest (the blades of a wind turbine versus the drive shaft and bearing in a conventional steam turbine) cannot be eas- ily measured. Accelerometers on the blades would require either telemetry or signal slip-rings to transmit the signals to the remote monitoring point. (v) A wind turbine operates at constantly varying power output which may change rapidly, whereas the output of most steam turbines changes relatively slowly and predictably. (vi) Wind turbines are still relatively low in power (0.1 to 2MW), and so a condition monitoring package for wind turbines would be much less expensive than that applied to conventional steam turbines with power rat- ings of 100-600MW.

1.4 Why use bicoherence? For this type of condition monitoring to be successful, small physical changes in the machine must be detected

IEE Proc.-Vis. Image Signal Process, Vol. 145, No. 3, June 1998

from a very noisy signal. If some feature of the signal amplifies the change and provides a reliable signature, then detection is made easier. Presence or absence of phase coupling between frequency components of the signal may provide such a signature. Since the power spectral density (PSD) discards phase information, it cannot detect the presence of phase coupling. The bis- pectrum, a third-order spectral statistic, detects phase coupling, but its magnitude varies with the power in the signal and therefore is not convenient for detection purposes. Bicoherence, a normalised bispectrum, over- comes this problem.

2

2. I Power spectral density The PSD Sx(u) serves as a convenient starting point for defining the bispectrum and bicoherence, and how they may be estimated. The PSD is the Fourier transform of the autocorrelation function. The direct method for estimating the PSD is via the fast Fourier transform (FFT) of the data. To reduce the variance of the esti- mate, the data are divided into shorter, and perhaps overlapping, segments and the results of each segment are averaged [5]. The mean is removed from each seg- ment, and a time window, such as the Hanning win- dow, may be applied to each segment before computing the FFT, in order to overcome spectral leakage. Thus, using notation to indicate the discrete nature of the data sequence and the estimate, the estimate of the PSD $(k) is

Power spectral density and bicoherence

S z ( k ) = C E [ X k X J (1) where Xk is the FFT of the data of one segment, x(n), n = 1, ..., M , where M is the segment length; C is a con- stant that may include the sample period and a correc- tion for loss of energy due to time domain windowing; * denotes the complex conjugation; and E[ ] indicates statistical expectation, approximated by averaging.

2.2 Bispectrum The bispectrum is the two-dimensional Fourier trans- form of the third-order cumulant sequence of x(n) [6], and it may also be estimated directly from the FFT of data segments in very much the same way as the PSD [61:

B&, 1 ) = E [XkX&+,] (2) Note that the bispectrum is two-dimensional and com- plex. Nikias and Raghuveer [7] describe many of its properties; the more useful properties for condition monitoring are listed here. (i) If x(k) is a stationary zero-mean Gaussian process, then its bispectrum is identically zero. Unlike the PSD, the bispectrum is theoretically not affected by the pres- ence of Gaussian noise in the signal. (ii) The bispectrum detects quadratic phase coupling (QPC) of harmonic triplets. If three harmonic compo- nents of a process are related so that the frequency and phase of the third component is the sum of the fre- quencies and phases of the remaining two, i.e.

z(n) = cos(w, n + cpl ) + cos(w2n + cp2) + cos(w3n + (p3)

( 3 )

w3 = w 1 +w2 and (p3 = cpl +cp2 (4) where

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then the shape of the bispectrum is an impulse at the frequency pair, or bifrequericy (wl, w2)

2.3 Bicoherence Since the magnitude of the bispectrum varies according to the power spectrum of a signal, it can be useful to normalise the bispectrum by the power spectrum. This is variously called the ‘bisoherency’ [6], the ‘skewness function’ [8], or simply the ‘bicoherence’ [9], and is defined as

Sometimes a slightly diffierent denominator is used where

Sz(w1)Sz(w2) = E [x(wl)x”(~l)] E [X(w2)X*(w2)] (6)

is replaced with ~ \ X ( W , ) Xi(c0~)1~] [lo]. Often the magni- tude or magnitude squared of eqn. 5 is referred to as the bicoherence [ll]. Combining the estimates of the PSD and the bispectrum yields the estimate for the bicoherence b [6]

(7) The magnitude of the bicoherence indicates how much of the power in a frequency component is due to phase coupling with two other components. Noise in the sig- nal dilutes the phase coupling and reduces the value of the bicoherence.

2.4 Variance of the estimates One difficulty in applying higher order spectral esti- mates is the need for sufficient data to produce an esti- mate with an acceptably low variance. Even in ideal circumstances, for example: in the absence of noise, the underlying statistical properties of a random process cannot be determined exactly from a finite quantity of data; there is always some error in the estimate. Knowledge of the variance of spectral estimates is required to apply the estimators and to quantitatively evaluate results. Here the variance of an estimate is defined as [SI

where 6 is the estimate.

method is [5] The variance of a PSD estimated by the direct

(9)

if K non-overlapping independent segments, no time domain windowing and no frequency domain averaging are used. A tirne domain window, such as the Hanning window which reduces side-lobe leakage, suppresses the amount of information at the ends of segments, and thus increases the variance of the estimate. Overlapping segments by 50% recovers most of that information and reduces the variance of the estimate to about 10% greater than that of an estimate without overlapping talpered segments [SI. The following discussion does not further address the effects of windowing and overlapping on the variance of the estimates.

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The variance of the estimate of bicoherence is less straightforward to compute. Haubrich [ 121 argued that if the bicoherence is known to be zero, the variance of the estimate would vary inversely with the number of segments, and that in the limit of a large number of segments the estimate would be chi-square distributed with two degrees of freedom. Elgar and Guza [13] employed numerical simulation to show how non-zero bicoherence influences the variance of the estimate, and to find maximum likelihood functions to estimate true values of the bicoherence based on observed values.

3

3. I Experimental set-up to simulate normal and damaged blade conditions Experimental work was carried out on a three-bladed, upwind, stall-regulated turbine de-rated to 35 kW power at the Wind Test Site at the Rutherford Apple- ton Laboratory. The 8.2m blades were manufactured in glass fibre reinforced plastic by the LM Glasfiber Company. The tower was a lattice type 15.5m high.

In order to assess the efficacy of a damage detection system, some method of simulating damage is required. Ideally, it would be possible to weaken a blade, but practical considerations such as safety do not usually permit this. It was first proposed to add mass to a blade to alter the mass distribution, and therefore the modal properties of that blade. As this does not change the modal properties in the same way as a reduction in stiffness, it is not an ideal way to simulate damage. However, as it may also affect the interaction of the rotor with the rest of the machine, and thus alter the harmonics in the power output, it provides an interest- ing case to examine.

A superior simulation of damage arose during the project when a blade instrumented for surface pressure measurement (the ‘unmatched blade) was found to be significantly less stiff than the matched blade that it had replaced. The difference in stiffness is due to differ- ent internal structure, whereas the outside dimensions are virtually identical since they came from the same mould. This led to a good test, in which the power out- put of the turbine with the matched blades could be compared to that of the machine with one less stiff blade.

Examination of wind turbine power data

100 r

80 i P

0 2 4 6 a radius,m

Results of deflection tests with matched and unmatched blades Fig. 1 0 unmatched #+ matched (1294) + matched (1291)

Deflection tests were performed to quantify the dif- ferences in stiffness. Fig. 1 shows results normalised to the maximum weight applied to the tip of the blade for

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several series of tests. The lines follow the mean of sev- eral test results, one line for each blade tested. Blades 1291 and 1294 are from the same production run and are nominally the same. Although the data sets show scatter, it is clear that the unmatched blade deflects the most, and blades 1291 and 1294 deflect the least. Esti- mates of stiffness from the deflection data indicate that the unmatched blade is in the order of 10-30% less stiff than the matched blades, with the greatest difference near the root.

From January to September 1995, the machine was configured with the unmatched blades. Data were col- lected with and without a 5kg mass attached to the inside of the tip of the instrumented blade. When possi- ble, data were recorded immediately before and after the addition of the mass so that wind conditions were virtually unchanged. The sampling frequency was 1OOHz. The power transducer was specially ordered from Camille Bauer to have higher than typical response; it employs pulse width modulation and includes a two-pole RC-type low-pass output filter with -3dB point at 4.6Hz, fixed by the manufacturer.

In autumn 1995 the rotor was restored to a matched condition, and more data sets were collected. One of the matched blades (129 1) had instrumentation installed identical to that of the unmatched blade. In addition to power data, blade and hub flapwise acceler- ations, blade root strains, and a top dead centre mark were recorded for many of the data sets. The rotor data were collected with a PC-compatible based acqui- sition system located in the hub and then transferred via a fibre-optic serial link to the remote monitoring point.

3.2 Practical considerations when estimating bicoherence In applying spectral estimators including bicoherence to real data, several practical choices arise, often with conflicting requirements: one is the amount of data required to produce a reasonable estimate. Given a data set, the choice of segment length determines fre- quency resolution, the risk of the data being non-sta- tionary, the variance of the estimate and computational effort.

3.2. I Data length, segment length, frequency resolution and stationarity: With the assistance of a signal analyser, a sample rate of lOOHz was selected as the minimum possible to reduce the quantity of data required for the long time intervals needed, without the risk of significant aliasing. Data sets of up to 192K samples were recorded. The maximum practical seg- ment length on a PC for the computation of the bico- herence estimate is 2048 points, but using MATLAB [14] these estimates took over l h to compute on a 90MHz Pentium PC compatible with 40Mb of RAM. Each estimate based on a segment length of 2048 points yields a matrix of 1025 x 513 double precision complex results, which requires at least 24Mb to com- pute. However, computing only the inner triangle requires only half the number of data points, consider- ably reducing memory requirements. Doubling the seg- ment length quadruples the memory requirement. In this instance, hard disk-based virtual memory is consid- erably slower.

The more practical length of 1024 points per segment increases the number of segments in the estimate to

TEE Proc -Vis Image Signal Process Vol 145 No 3 June 1998

reduce the variance of the estimates, while still giving a frequency resolution of approximately 0.1 Hz. This is sufficient to resolve harmonics of the rotational rate of 0.64Hz. Experience with these estimates indicates that in the order of 100 segments produce sufficient averag- ing 1151; thus the composite data sets are made up to be at least lOOK samples long (with one exception in high winds when only 64K of data were available). The three blade conditions (matched, unmatched and unmatched with weight) were combined with three power levels to give nine separate cases.

3.2.2 Sorting data by segment: Wind turbine power is not stationary. Power output from the wind turbine can vary from nearly zero to rated power in only a few seconds, although a large rotor diameter helps reduce power variations by averaging wind speeds over a large blade area. There is also a strong variation of the wind speed at a period of about 100s. The operational characteristics of a wind turbine depend on the power output. Stall-regulated machines exploit aerodynamic stall in high wind speeds to limit power output, and pitch-regulated machines vary blade pitch to either feather or stall the blades in high winds. In either case, the aerodynamic characteristic changes with power output.

To counter this non-stationarity and to yield addi- tional information from the bicoherence analysis, the data sets were broken into segments of 1024 consecu- tive points, where the minimum value of the power in a segment was limited to 0.5kW in order to avoid peri- ods of power cut-out. Segments were sorted into three classes according to the mean value of the power: low power (0.5-12kW), moderate power (12-24kW) and high power (24-36kW). Long data sets were then con- structed by concatenating segments. Note that the pres- ence of segment boundaries means that overlapped processing is not permitted in calculating the PSD or bicoherence, and this increases somewhat the variance of the estimate.

In some cases, the data were adjusted to eliminate occurrences of zero power output but otherwise keep intact. This is particularly useful for PSD estimates, when looking at the variation of the estimates across different data sets with different wind and weather con- ditions.

3.3 Results In the following we present the estimations of PSD of blade tip acceleration and electrical power output, coherence between them, and bicoherence of power. The goal is to find reliable indicators of the condition of the blades, in this case represented by the three experimental configurations. The change in PSD and bicoherence from the matched to unmatched case is most important since this most closely resembles the type of change expected with a structural fault in a blade. The changes fall into two types: changes in modal frequencies of the structure and changes in phase coupling. The first changes are easily detected from blade accelerations when the turbine is stationary and excited randomly by the wind, but are more diffi- cult to detect while the turbine is operating; the second changes depend on the dynamics of rotation and can only be measured with the machine running.

3.3.1 Blade tip accelerations with the turbine stationary: If a wind turbine is not operating, then

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direct measurement of blade motion with accelerome- ters and analysis of the measurement via the PSD is straightforward. A modification to the structure could be expected to change the modal behaviour of the structure, which is measured by the accelerometers, and measurements confirm this.

. i! 3 0.6

0 1 2 3 4

frequency,Hz Power spectral density of Jlapwlse blade t p acceleration with Fig.2

rotor stationary, instrumented blade upwards

~ unmatched _ _ _ ~ unmatched plus weight

matched

Fig. 2 shows the PSD versus frequency of blade tip acceleration in the flapwise direction, with the instru- mented blade oriented upwards for the three blade con- ditions. Three groups of peaks are visible. The peak close to the 2Hz frequency is due primarily to the first tower mode, the central peak is due to the first asym- metrical rotor mode, and the near 3.5Hz frequency peak is due to the first symmetrical rotor mode of vibration [16]. Note the shifts in all peaks towards lower frequency with the unmatched and less stiff blade fitted. The addition of the weight to the tip of the unmatched blade makes only a slight difference to the tower modal frequency. With the turbine running, blade acceleration data are much more complicated and difficult to interpret, arid power data contain addi- tional information, complexity and noise.

0.8

8 0.6 E 8 0.4

C

S

0.2

0 0 1 2 3 4 5 6

frequency,Hz

Coherence of jlapwise t@ acceleration to electrical power output Fig.3 (matched blades)

3.3.2 Coherence of blade tip acceleration with power output: Coherence between blade tip motion and electrical power output gives the degree of linear relationship between the two as a function of fre- quency. The results of the estimate, shown in Fig. 3, show high coherence at thle rate of rotation, 0.64Hz,

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and two other broad peaks centred at 2 and 3.7Hz. Thus, a direct coupling of tip motion to power output is confirmed, at least at selected frequencies.

Note that the peaks do not coincide exactly with those of the stationary PSD of tip acceleration. The shift to the right of the symmetrical rotor mode is probably a result of centrifugal stiffening. However, many peaks of interest in the PSD of power are at fre- quencies of low coherence with flapwise tip motion, and therefore are not entirely due to linear coupling.

3.3.3 Changes in the power spectral density of power output: Since blade vibrations measure- ments are not usually available, the electrical power signal provides the next alternative. The power signal includes effects of dynamics from the entire machine, such as the drive train and generator, and so is both complex and noisy.

Fig. 4 shows the PSD of moderate power output for the matched and unmatched cases. The left and strong- est peak occurs at the rotational frequency of the blades, 1P or 0.64Hz. Imbalances between blades, such as unequal mass moments, pitch setting or twist, intro- duce variations in power at that frequency. A peak at twice that frequency is visible in the matched case, but not in the unmatched case. In low winds and when mass was added, the 2P peak is quite strong with the unmatched blades. At high power outputs, it is com- pletely masked by rising noise levels in all cases. Other peaks occur at harmonics of the rate of rotation, up to the twelfth harmonic 12P.

-4 10

0 2 4 6 a 10

frequency,Hz

Fig. 4 Lower trace: matched Upper trace: unmatched

Power spectral density of electrical power

O L 2.3 2.4 2.5 2.6

frequency,Hz

Fig.5 Close-u of the power spectral density of electricul power, at the Jirst assymetvica f ro tor mode

~ matched ~ unmatched ~ _ _ _ unmatched + weight

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The broad hump at just less than 2Hz (3P) is due to the wind field effects, such as turbulence and shear (rise in wind speed with height), as well as misalignment of the rotor with the wind direction due to yaw error and tilt of the rotor from vertical, and other factors such as tower shadow. The next peak, at 2.45Hz, coincides with the first asymmetric mode of the rotor, even though the coherence of flapwise tip motion to power output is very low at that frequency. Further examina- tion of that peak shows a small but discernible shift to the lower frequency with the unmatched blade. Fig. 5 shows PSDs of individual data sets; the bold lines are the PSDs with the unmatched blade. The shift is con- sistent across seven data sets with matched blades and four with unmatched blades, under a variety of wind and temperature conditions.

Another consistent shift in frequency is observed in a peak at 4.7Hz, which coincides with the first edgewise frequency of the rotor. An examination of PSDs of power with matched blades reveals movement of the peak as the season progressed from winter to late spring. When the frequency of the maximum value of the peak is plotted against the hub temperature recorded at the hub height, a clear trend of decreasing frequency with increasing temperature appears, as shown in Fig. 6. The 95% confidence limits contain at least half the points if the y value is independent, nor- mally distributed with constant variance [ 141.

4.8

N % 4.75 0

s cr 4.7 - 4.65

4.6 I 0 5 10 15 20 25

temperature,'C Frequency of maximum of p e d near 4.7Hz versus hub height Fig.6

temperature 0 matched blades % matched blades, no airbrake ~ linear fit . . . . . . . 95% confidence limits

3.3.4 Changes in bicoherence of power out- put: Estimates of bicoherence of electrical power yield results as a function of the bifrequency. The squared magnitude of the bicoherence may be presented as a surface or a contour map over a region (-z < w1 < n, -n < w2 < 76). Fig. 7 shows a portion of the region

region has many reflections: the region called the 'inner triangle' (marked IT) represents the most direct evalua- tion; all others are either identical or complex conju- gates of this. If this reduced set is presented, there is too much information for sensible interpretation. To meet the aim of finding easily discernible changes in pattern with change in blade structure, it is preferable if important information in the result can be extracted as a function of one variable.

Fixing one frequency and letting the other frequency vary has the effect of slicing the bicoherence along a

(cull wz both positive). The evaluation over the whole

IEE Proc-Vis. Image Signal Process., Vol. 145, No. 3, June 1998

plane parallel to one axis; this idea of using a ‘bicoher- ence slice’ is shown schematically in Fig. 7. Setting the fixed frequency at the rate of blade rotation 1P turns out to be an efficient and simple method of presenting the differences in the bicoherence between the various cases. Thus, all peaks that appear are at frequencies that are phase-coupled with the 1P component. This has an additional advantage of requiring considerably less processing time, which would allow larger data sets and/or higher resolution to be obtained.

0 xi4 Zl2 3x14 7 l

0 1 Fig. 7 The position of ‘slice’ is shown, evaluated for constant w2

Region of calculation of the bicoherence

a

b

0 5 10 15 20

normalised frequency,P

c Fi .8 Slices, at IP, o bicoherence of power for matched, wlmatched an! unmatched plus we$& c&s Low electrical power output a Unmatched h Unmatched + weight c Matched

Figs. 8 and 9 show the 1P slices for the three cases in low (Fig. 8) and moderate (Fig. 9) power categories.

IEE Proc.-Vis. Image S@al Process., Vol. 145, No 3, June 1998

Frequency is normalised by the rotational rate. Each plot contains at least two examples to indicate, at least roughly, how consistent the estimates are with different data sets. By the maximum likelihood functions of Elgar and Guza [13], an observed bicoherence of 0.2 produced with 64 independent segments corresponds 95% of the time to a true bicoherence of between 0.135 and 0.261. In all cases shlnwn in Figs. 8 and 9, at least 100 segments were used, and so this range is somewhat smaller. Nevertheless, the observed variations in bicoherence are largely {consistent with the expected variance of the estimate, especially in the unmatched cases. The variation in magnitude of peaks at harmonics between 6 and 12P for the matched case may be due to the non-stationarity within the moderate power category or to sensitivity to wind conditions. However, the pattern of peaks and valleys in the range of frequencies from 6P to 12P is consistent within the matched case.

0.4 I 1

0.4 5

z o

a G) 0.2 U

C 0)

c ._

b

0.4 I ,I 0.2

0 0 5 10 15 20

C

normalised frequency,P

of’ power for, matched, unmatched

Moderate electrical power output a Unmatched b Unmatched + weight c Matched

The difference between the matched and unmatched cases is apparent. The matched case exhibits phase cou- pled harmonics from 6P to 12P, whereas the unmatched case is much less regular. In addition, clear peaks at 3.3P (2.13Hz) and 12.8P (8.26Hz) appear only in the unmatched and unmatched plus weight cases. At moderate and high power, a peak at 7.8P (4.7Hz) dominates the Unmatched cases. This is the fre- quency of the first edgewise rotor mode, as measured by blade root strain gauges while the turbine operated.

At high power, the squared magnitude of the 1P slice of bicoherence for the matched case is nearly uniformly low and below a value of 0.12 for the two data sets available. The unmatched cases, with and without the

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added weight (Fig. lo), show a marked difference at 13.35P (8.6Hz), especially considering the similarity everywhere else.

a, 0.5

?2

8 0.4 .K

I) ._

P

t: $ 0.2

0.3 3

2

C ul c ._

g 0.1

0 0 5 10 15 20

normalised frequency,P

Fig.10 unmatched plus weight cases High electrical power output

~ unmatched ~ ~ _ _ unmatched + weight

Slices, ut IP , of bicoherence of power for, unmatched and

4 Discussion and conclusions

The changes in PSD suggest that both a reduction in stiffness in one blade and an equal reduction of stiff- ness in all blades are possible to detect via the PSD approach. These faults affect peaks at 2.45Hz and 4.7Hz, respectively, as well as the number and sharp- ness of peaks at harmonics of the rate of rotation. Since the 4.7Hz peak coincides with the first edgewise mode of the rotor, it is likely that it is due primarily to a change in the edgewise stiffness. The observed shifts in frequency, although significant, are small. Basing a fault detection system on such a small change in the PSD would be difficult, especially since the change in stiffness employed here is relatively large compared with that expected in real trials.

On the other hand, slices of bicoherence with one fre- quency held at the rate of rotation and the other allowed to vary exhibit a clearly discernible change with the different configurations. The regularity of peaks at 6P to 12P in the matched case contrasts well with the less regular and generally higher magnitude of bicoherence in the unmatched cases. In the unmatched cases, the first edgewise rotor mode at 7.3P is clearly phase-coupled to 1P. An algorithm that calculates the area under the harmonic peaks and normalises it with the area between the peaks could exploit both of these differences. For example, simply summing two values of bicoherence located around each harmonic from 6P to 12P, and dividing that by the sum of two values at each half harmonic from 6.5P to 12.5P, yields an index of the regularity of the phase coupling at harmonics of 1P. Excluding data from the high power category, the index for unmatched cases is below 2.02, whereas for the matched cases it exceeds 8.74. The addition of mass to the unmatched blades is also clearly reflected by the appearance of a large peak at 8.6Hz in the bicohereiice of power data from the high power category. Employ-

ing only a slice of the bicoherence has the additional advantage of greatly reducing the computation required for the estimate, and thus makes it much more practical to apply.

For condition monitoring of wind turbine blades to become a reality, the results shown here for a 45kW wind turbine must be extended to commercially availa- ble machines with ratings of 0.5MW or more. Such work would include developing a comprehensive ana- lytical and computer simulation model in order to establish the coupling between blade stiffness and power output, and measuring power output from wind turbines in commercial wind farms.

5 Acknowledgments

This work stems mainly from the results of a two year project funded by the Engineering and Physical Science Research Council. The authors would like to thank the Rutherford Appleton Laboratory (RAL) for providing the wind turbine test facilities; members of the Energy Research Unit at RAL for their help and support; and Alun Coonick and Leon Freris from the Department of Electronic and Electrical Engineering, Imperial College, who provided additional support for Will Jeffries dur- ing summer 1996.

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