Yujie Xu

Zhibin Zhou
email: zhibin.zhou@isen-ouest.yncrea.fr

Tianzhen Wang

Claude Delpha
email: claude.delpha@centralesupelec.fr

Tao Xie

An entropy clustering method for blades biofouling detection of marine current turbine under variable marine current speeds

Keywords: Marine current turbine, Blades biofouling detection, variable marine current speed, Entropy clustering method, Principal component analysis I

Marine current turbine (MCT) attracts extensive attention as a renewable energy power generation equipment. However, accurate and reliable blades biofouling detection for MCT remains a significant challenge, especially under variable marine current speed. This paper proposes an entropy clustering method to improve the detection accuracy of blades biofouling for MCT under variable marine current speed. Firstly, the stator current signal is decomposed based on variational mode decomposition (VMD) to enlarge the feature of blades biofouling. Secondly, to eliminate the influence of swell effect, fluctuating features caused by swell effect are divided into each subspace based on the proposed entropy clustering method. At the last stage, principal component analysis models (PCA) are established in each feature subspace to detect blades biofouling. To verify the effectiveness of the proposed method, experiments are carried out on a 230W MCT platform. The results show that the proposed method has satisfactory performance for blades biofouling detection under variable marine current speed.

INTRODUCTION

Tidal energy is sustainable and has advantages of clean, reliable, high energy density, and predictable [START_REF] Lewis | A standardised tidal-stream power curve, optimised for the global resource[END_REF]. Marine current turbine (MCT) has attracted more and more attentions. However, MCT commonly works in harsh surroundings, such as humidity, corrosion and heavy salt spray. Sediment and organisms are easily attached to blades [START_REF] Tao | Imbalance fault detection based on the integrated analysis strategy for marine current turbines under variable current speed[END_REF], which decrease the working efficiency of MCT, and destroy components of the system in severe cases. In addition, with the influence of variable marine current speed due to the swell effect [START_REF] Zhou | Power smoothing control in a grid-connected marine current turbine system for compensating swell effect[END_REF], the rotor speed of MCT may oscillate continuously under classical MPPT strategy [START_REF] Tian | Influence of surface waves on the hydrodynamic performance of a horizontal axis ocean current turbine[END_REF]. It further causes the irregular fluctuating amplitude and the frequencies feature of blades biofouling on stator current [START_REF] Gong | Imbalance fault detection of direct-drive wind turbines using generator current signals[END_REF], which increases the difficulty of blades biofouling detection, even leads to misdetection when the trained detection model is applied to variable marine current speed. Therefore, it is necessary to consider the difficulties of blades biofouling detection caused by variable marine current speed in practical applications.

At present, many fault detection methods for MCT have been proposed. Xie et al. [START_REF] Xie | A review of current issues of marine current turbine blade fault detection[END_REF] reviewed different blades biofouling detection methods for MCT in terms of sensorbased and sensorless-based methods. In [START_REF] Cao | Deterioration state diagnosis and wear evolution evaluation of planetary gearbox using vibration and wear debris analysis[END_REF], vibration sensors are installed to obtain vibration signals for detecting the faults. In [START_REF] Zheng | A sparse autoencoder and softmax regression based diagnosis method for the attachment on the blades of marine current turbine[END_REF], underwater cameras are installed to obtain image signals for detect blade imbalance fault of MCT. However, these sensors need to be impervious and robust to environmental interference. Some research are carried out by using current and voltage signals. Empirical Mode Decomposition (EMD) and variational mode decomposition (VMD) are used to detect blades biofouling in [START_REF] Zhang | An imbalance fault detection method based on data normalization and EMD for marine current turbines[END_REF][START_REF] Wei | A VMD Denoising-based Imbalance Fault Detection Method for Marine Current Turbine[END_REF]. However, there are still some problems such as the end effect and the modal confusion. Xie et al. [START_REF] Xie | An integration fault detection method using stator voltage for marine current turbines[END_REF] proposed a blades biofouling detection method based on principal components analysis (PCA) and frequency sequences subtraction (FSS). This method improves the accuracy of detection but traditional confidence limit is not applicable under variable marine current speed. All methods mentioned above have made significant contributions to blades biofouling detection under steady marine current speed, but the statistical features of blades biofouling are fluctuated due to swell effect, which lead to misdetection when the blades biofouling detection method under steady marine current speed directly applied to variable marine current speed.

Therefore, this paper analyzes the stator current signal and proposes an entropy clustering (EC) method is proposed to divide fluctuating features for MCT under variable marine current speed. Specifically, the main contributions of this paper are as follows:

1) To eliminate the influence of swell effect, fluctuating features are divided into subspaces, and blades biofouling were detected in each feature subspace.

2) An entropy clustering method is proposed to divide fluctuating features. This method selects central sample of each subspace by mutual information entropy, which have satisfactory performance.

3) The experimental data under the varying marine current speed condition are collected from a 230W marine current turbine prototype to verify the effectiveness of the proposed method.

The rest of this paper is organized as follows. The problem of blades biofouling detection under variable marine current speed is described in Section 2. The proposed method is described in Section 3. A case study is implemented in Section 4. The conclusion is summarized in Section 5.

II. PROBLEM DESCRIPTION

A. Effect of blades biofouling for stator current signal

MCT captures part of tidal energy and convert it into output mechanical torque to drive PMSG for generating electrical power [START_REF] Toumi | Modeling and Simulation of a PMSG-based Marine Current Turbine System under Faulty Rectifier Conditions[END_REF]. The stator current under healthy state can be expressed as:

s t r ( ) sin( ) i t A t   =+ ()
where t A denotes the amplitude of stator current,  denotes the initial phase,  denotes the number of pole pairs, r  denotes the shaft rotation angular speed of MCT, and rr 2π f  = denotes the shaft rotating frequency.

The torque of MCT is distorted under the blades biofouling state, it can be equivalent to attaching a virtual mass m to one of the blades. As shown in Fig. 1, buoyancy b F and gravity mg of the mass produce an additional torque. The rotating speed of bearing changes with torque, the shaft rotation angular speed with blades biofouling can be expressed as [START_REF] Xie | An integration fault detection method using stator voltage for marine current turbines[END_REF]:

n r r cos( ) t      = - + ()
where  denotes the coefficient of blades biofouling: where J is the moment of inertia, u r denotes the distance between the mass and the centroid of the shaft. The stator current of MCT under blades biofouling state can be expressed as follows [START_REF] Tao | Imbalance fault detection based on the integrated analysis strategy for marine current turbines under variable current speed[END_REF]:

b b n b r r r ( ) sin d sin sin( ) i t A A t t         =    = - +      ()
where b A denotes the amplitude of stator current under blades biofouling state. The instant frequency of stator current under blades biofouling state can be expressed as follows:

n r r 1 ( ) = cos(2π ) 2π 2π f t d f f t t        = - +    ()
According to Eq [START_REF] Gong | Imbalance fault detection of direct-drive wind turbines using generator current signals[END_REF], an excitation of the same frequency as r f are contained in stator current signal under blades biofouling state [START_REF] Li | A Wavelet Threshold Denoising-Based Imbalance Fault Detection Method for Marine Current Turbines[END_REF], which is different from the stator current signal under healthy state. Therefore, the shaft rotation frequency r f can be used as the blades biofouling feature of MCT under steady marine current speed.

B. Effect of swell effect for stator current signal

MCT operates on the seabed, and its operating conditions are largely affected by the variable marine current speed. To simulate the operation of MCT on the seabed, Zhou et al. [START_REF] Cao | Deterioration state diagnosis and wear evolution evaluation of planetary gearbox using vibration and wear debris analysis[END_REF] calculated the total current velocity through the combination of predictable tidal speed tide V and swell effect swell

V as shown in Fig 2. The predictable tidal speed tide

V is a constant, and the swell effect swell V contains more than one frequency component, which makes the marine current speed changes variably.

Fig. 2. The simulated waveform of the total current velocity

It can be seen that the swell effect causes oscillations of marine current speed within a given depth, and the marine current speed affects the shaft rotating speed of MCT [START_REF] Cao | Deterioration state diagnosis and wear evolution evaluation of planetary gearbox using vibration and wear debris analysis[END_REF]. In this situation, the shaft rotating speed as follows:

( )

tide swell current rv r v RR i i VV V      + = = = +  ()
where R is the blade radius of MCT,  is the blade tip speed ratio, vi  is the shaft rotating speed fluctuating caused by the ith component of swell effect swell V .

It can be seen that the varying marine current speed leads to fluctuations of shaft rotating speed. According to Eq. ( 4), the stator current under variable marine current speed of MCT with blades biofouling can be expressed as:

( )

rv b rv rv rv ( ) sin sin i t A t t        = - +     ()
The instant frequency of stator current with blades biofouling under variable marine current speed can be expressed as:

( ) ( ) ( ) rv r v r v ( ) cos 2π 2π ii ii f t f f f f t   = + - + +  ()
where rv f is the instant frequency of stator current under variable current speed, and vi f is instant frequency fluctuating caused by the ith component of swell effect. To sum up, the blades biofouling frequency feature of MCT under steady marine current speed is r f , but under variable marine current speed is r

v i i ff +  .

A. Stator current pre-processing based on HT and VMD

The stator current in one phase of the generator under variable marine current speed collected from MCT is rv () it , and the instantaneous frequency rv () ftof rv () it can be obtained by HT [START_REF] Rangel-Magdaleno | Hilbert spectrum analysis of induction motors for the detection of incipient broken rotor bars[END_REF]. Then, the instantaneous frequency rv () ft is decomposed into intrinsic mode functions (IMFs) denoted as ()

q ut K ( 1,2,..., ) qK = with the number K K based on VMD.
They can be expressed as:

( ) ( ) ( ) ( ) cos q q q u t A t t  = ()
where ()

q ut denotes the th q IMF, () q At
denotes the instantaneous amplitude of the th q IMF, () q t  denotes the instantaneous frequency of the th q IMF. And then, the f IMF contained blades biofouling feature is selected by the maximum correlation coefficient [START_REF] Liu | Research on leak location method of water supply pipeline based on negative pressure wave technology and VMD algorithm[END_REF] between IMFs and instantaneous frequency rv () ft .

B. Fluctuating features division based on the proposed entropy clustering method

The 1 IMF obtained after VMD decomposition is windowed to obtain a sample matrix

f 1 2 [ , ,..., ] K f f f = f
, and f K is the number of samples. To ensure the frequency information is not lost, the length of the window is chosen to be 3000 in this paper, that is, samples contain 3000 sampling points. The number of feature subspaces ( )

( ) 2 () 1f arg min 1, 2,..., , 1, 2,...
, i ji f i K j K  = - = = γ ()
where, i . The mutual information entropy (MIE) [START_REF] Lu | HTDet: A clustering method using information entropy for hardware Trojan detection[END_REF][START_REF] Hu | Information distances versus entropy metric[END_REF] between samples () i a

 and () i b

 can be expressed as follows:

( ) ( ) ( ) () ( ) ( ) ( ) ( ) ( ) ( ) 2 ( ) ( ) ( , ) ( , ) ( , )log ( ) ( )
i i i i a b ii i i i i ab a b a b ii ab p Ip pP            =    γ γ ()
where  to be the subspace centroid.

The MIE of each sample can be expressed by summing, which expressed as follows:

( )

I K i i i a a b b I I a b    - = =  () 1 ( ) ( ) ( ) 1 ( ) ( , ),
And then, the selected centroid of subspace i () γ can be expressed as: Iterate the above process to the minimum value of the objective function (Eq.( 10)). The blades biofouling feature subspace

1 () (1) MIE MIE MIE K  =  γ γ γ
can be obtained, when the value of the objective function is extremely small. Then, the number of subspaces K is determined by the index of _ S Dbw [START_REF] Huang | Incorporating Load Fluctuation in Feature Importance Profile Clustering for Day-Ahead Aggregated Residential Load Forecasting[END_REF]: T statistic and SPE statistic in the th l subspace, respectively. If the statistics exceeds their confidence limit, we assume that the MCT blade is biofouling and imbalanced.

1 1 1 __ K K K S Dbw
( )T 1 T ( ) 2 MIE MIE l l l l l l T T  - =  fW Λ Wf () ( )T T T T ( ) 2 MIE MIE (1 ) (1 ) ll l l l l l l SPE   = - -  f W W W W f (

D. Implementation procedures of the blades biofouling detection method

For blades biofouling detection using stator current signal of MCT under variable marine current speed, a specific method is proposed. The flow chart of the blades biofouling detection method is shown in Fig. 3. The stator current signal under variable marine current speed is collected from the MCT experiment platform. And the proposed method contains 3 steps as follows:

Offline training

Step1: To enlarge the feature of blades biofouling, the stator current signal is preprocessed based on HT and VMD.

Step2: The fluctuating features of Step3: Establish PCA models in each subspace to detect blades biofouling for MCT by using frequency domain feature of f IMF . As eq.( 16) and eq.( 17), if the statistics exceed the corresponding confidence limit, it indicates that the biofouling of blades for MCT has been detected.

IV. EXPERIMENTS ANALYSIS

A. Experiment platform

To verify the effectiveness of the proposed method, the stator current signal was collected on a 230W direct-drive permanent magnet synchronous MCT experimental platform. As shown in Fig. 4, the experimental platform contains four parts: turbine, direct-drive permanent magnet synchronous generator, data acquisition and states monitoring system, with the water pump system to control the water flow speed and for emulating swell effect. The sampling frequency of the data acquisition and states monitoring system is 1kHz, and the specific parameters of the experimental system are described in [START_REF] Xie | An integration fault detection method using stator voltage for marine current turbines[END_REF].

B. Experimental results and analysis

To verify the effectiveness and the applicability of the proposed method, this paper randomly selects stator current signal under variable current speed (Shaft rotation speed is 15.9, 16.8 and 17.8 rpm). The collected stator current signal is preprocessed based on Hilbert transform and VMD to enlarge the feature of blades biofouling and improve the accuracy of blades biofouling detection. The stator current signal under healthy state and blades biofouling state is transformed to the time-frequency domain, time-frequency signal under healthy state shown in Fig. 6. And then, the instantaneous frequency signal is decomposed into 5 components (IMF1, IMF2, …, IMF5) by VMD, as shown in Fig. 7. The IMF1 obtained by VMD decomposition is windowed to obtain sample matrix, the length of the window is chosen to be 3000 in this paper, that is, samples contain 3000 sampling points. Then all samples are divided into each subspace based on the proposed entropy clustering method. The centroids of each subspace are selected based on MIE, which is shown in Fig. 8. And the values of the _ S Dbw with different number of subspaces are shown in Table Ⅰ. The index of _ S Dbw is the smallest one, when the number of subspaces is 3. The fluctuating features division results are obtained and the visualization is shown in Fig. 9. Table Ⅱ shows the accuracy of blades biofouling detection based on different methods. The accuracy of blades biofouling detection by using the proposed entropy clustering (EC) method is 96.70%. The accuracy of blades biofouling detection by using PCA without feature division is 62.31%. The detection based on the PCA method without dividing features is unsatisfactory when the MCT works under variable marine current speed. The blades biofouling detection by using the combination of PCA with KNN and K-means method for fluctuating feature division still need to be improved in detection accuracy and efficiency. Compared to the methods based on KNN and K-means, the accuracy of the proposed method has been improved by 9.64% and 13.45%, respectively. In addition, the proposed method has greatly reduced the false alarm rate and missed alarm rate, and it takes less time.

V. CONCLUSIONS

To solve the problem that blades biofouling is difficult to be detected under variable marine current speed due to the fluctuating blades biofouling features. This paper proposes an entropy clustering method to divide fluctuating features. The advantages of the proposed method as follows:

(1) The proposed method detect blades biofouling by using stator current signal of MCT without additional sensors, which has satisfactory performance on stability.

(2) The proposed entropy clustering method selects central sample of each feature subspace by mutual information entropy, which demonstrates satisfactory performance.

(3) It has solid universal applicability in fluctuating features division and blades biofouling detection under variable marine current speed, it can be applied to other fault diagnosis fields with fluctuating features.

An experiment has been performed on an 230W MCT experimental platform to verify the effectiveness of the proposed method. The experimental results show that the proposed method achieved satisfactory performance for blades biofouling detection of MCT under variable marine current speed.

  b

Fig. 1 .

 1 Fig. 1. Equivalent diagram of blades biofouling



  under variable marine current speed, which increases the difficulty of blades biofouling detection. III. THE PROPOSED BLADES BIOFOULING DETECTION METHOD The proposed method is composed of three parts: (1) To enlarge the feature of blades biofouling, the stator current signal is decomposed based on Hilbert transform (HT) and VMD. (2) To eliminate effects of variable marine current speed, the fluctuating features are divided based on the proposed entropy clustering method. (3) PCA models are established in each subspace to detect blades biofouling. Details are elaborated in the following paragraphs.

  selected as the initial central samples of each feature subspace. And then, divide all samples into their subspace according to the principle that the distance between each sample and the randomly selected central sample of the subspace is minimum:

  central samples of each subspace are selected based on Mutual Information Entropy (MIE). Denote ()

Fig. 3 .

 3 Fig. 3. The flow chart of the blades biofouling detection method

  in K subspaces based on the proposed entropy clustering method. The proposed entropy clustering method selects central samples of each subspace based on mutual information entropy and the number of subspaces is determined by the index of _ S Dbw .

Fig. 4 .Fig. 5 .

 45 Fig. 4. The structure of the experimental platform In the experiment, the blades biofouling is emulated by winding ropes with 80g at the root of the blade. As shown in the Fig. 9, Fig. 9 (a) is the MCT in healthy state, and Fig. 9 (b) is the MCT in the simulated blades biofouling state.Healthy stateBlades biofouling (emulation)

Fig. 6 .

 6 Fig. 6. The instantaneous frequency signal

Fig. 8 .

 8 Fig. 8. The central samples of 3 feature subspace

Fig. 9 . 3 Fig. 10 .

 9310 Fig. 9. The visualization of 3 feature subspace divisionAfter fluctuating features division, PCA models are established in subspace 1-3 to detect blades biofouling. Fig.10shows the detection results in subspace 1-3. The training set of each PCA model is 100 healthy samples, and in test set, the samples 1-100 are under healthy state and the samples 101-200 are under blades biofouling state. This method makes the confidence limit and statistic correspond entirely in each feature subspace, and dramatically improves the accuracy of blades biofouling detection under variable marine current speed.

TABLE I .

 I THE INDEX OF DIFFERENT NUMBER OF SUBSPACES

	The number of feature subspaces	The index of _ S Dbw
	2	1.3372
	3	1.1759
	4	Null
	5	Null

TABLE II

 II

	.	THE ACCURACY OF BLADES BIOFOULING DETECTION
		UNDER VARIABLE MARINE CURRENT SPEED
	Method	Detection accuracy	Missed alarm rate	False alarm rate	Computati onal time
	VMD+PCA	62.31%	0%	75.38%	2.0279s
	VMD+KNN +PCA	87.06%	17.14%	8.74%	6.7152s
	VMD+K-means+PCA	83.25%	0.67%	32.83%	2.1329s
	VMD+EC+ PCA	96.70%	0.70%	5.90%	3.1952s

CAA Symposium on Fault Detection, Supervision and Safety for Technical Processes (SAFEPROCESS) | 979-8-3503-3775-4/23/$31.00 ©2023 IEEE | DOI: 10.1109/SAFEPROCESS58597.2023.10295924 Authorized licensed use limited to: Centrale Supelec. Downloaded on January 11,2024 at 12:59:07 UTC from IEEE Xplore. Restrictions apply.

Authorized licensed use limited to: Centrale Supelec. Downloaded on January 11,2024 at 12:59:07 UTC from IEEE Xplore. Restrictions apply.

Authorized licensed use to: Centrale Supelec. Downloaded on January 11,2024 at 12:59:07 UTC from IEEE Xplore. Restrictions apply.

ACKNOWLEDGMENT

This work was supported by the National Nature Science Foundation of China (grant numbers 61673260).