Luca Tiberi

Xavier Saloppé

Charles Rasson

T.-H Pham

Recognition of Emotional Body Posture Expressions in Forensic Inpatients Who Committed a Sexual Offense

f f f • • • • • • • • •


Lu Zhang

Claude Delpha

Demba Diallo

Effect of Fault Severities and Noise Levels on Fault Isolation in 7-Phase Electrical Machines

Keywords: Fault isolation, Stationary frame, Non-incipient faults, Incipient faults, Noise levels

This paper presents a fast and efficient fault isolation method in 7-phase electrical machines based on the phase currents projections in the stationary reference frames. The study considers both non incipient faults with 15% to 30% fault severities, and incipient ones whose severities vary 1% to 6%. The noise level effect on the fault isolation is also considered. The fault features are extracted from the transformed currents in the frequency domain. The features are processed with a multi-step classification methodology based on usual techniques (principal component analysis, linear discriminant analysis and support vector machine). The simulation results show that the fault classification under low noise level conditions is effective with an accuracy higher than 98%. However, when the noise level increases, the proposal fails to classify incipient faults.

I. INTRODUCTION

F Ault Detection and Diagnosis (FDD) can be decomposed in three steps: detection (detect if a change has occurred), isolation (isolate the fault location and type), and estimation (estimate the fault severity) [START_REF] Gertler | Fault detection and diagnosis in engineering systems[END_REF].

Once the fault is detected, it can be isolated and its severity estimated. Indeed, in a complex system the faults can affect any component (electrical machine, energy storage systems, cables, connectors, ...) [START_REF] Furse | Fault diagnosis for electrical systems and power networks: A review[END_REF]. The fault type identification and its location are important steps for an accurate fault estimation that is crucial for maintenance and reliability purposes. Therefore, this work will mainly focus on the isolation process. Undoubtedly, the wide range of fault types is a significant challenge to fault isolation [START_REF] Soliman | Power systems fault type identification based on park's transformation algorithm[END_REF]. In recent years, advanced machine learning techniques, such as deep learning and reinforcement learning, are actively investigated to improve fault isolation performance [START_REF] Tayeb | Transmission line faults detection, classification and location using artificial neural network[END_REF]- [START_REF] Wen | A new reinforcement learning based learning rate scheduler for convolutional neural network in fault classification[END_REF]. For example, in [START_REF] Tayeb | Transmission line faults detection, classification and location using artificial neural network[END_REF], the authors proposed applying a Back-Propagation (BP) neural network architecture as an alternative method for fault detection, classification, and isolation in a transmission line system. In [START_REF] Dalstein | Neural network approach to fault classification for high speed protective relaying[END_REF], the authors presented a novel fault classification method for high-speed relay protection systems based on multiple neural networks. Besides, there is a growing trend to combine the advantages of model-based and data-driven approaches for fault isolation, especially for non-linear systems [START_REF] Jung | Combining modelbased diagnosis and data-driven anomaly classifiers for fault isolation[END_REF]- [START_REF] Samantaray | Fault classification and location using hs-transform and radial basis function neural network[END_REF]. In [START_REF] Gomathy | Fault detection and classification with optimization techniques for a three-phase single-inverter circuit[END_REF], the authors proposed a combined approach that integrated Cuckoo Search Optimization (CSO) and Relevance Vector Machine (RVM) to

The authors would like to thank China Scholarship Council. classify open and short-circuit faults in three-phase inverter, which aimed to achieve higher fault classification accuracy while minimizing the training and testing durations. Reference [START_REF] Samantaray | Fault classification and location using hs-transform and radial basis function neural network[END_REF] proposed a method for classifying any-phase or multiphase faults on transmission lines using Radial Basis Function Neural Network (RBFNN) and Recursive Least Squares (RLS) algorithm. Although these advanced machine learning methods can handle complex high-dimensional data, they usually require huge amount of training data to achieve optimal performance. If not, the learned models may be biased or inaccurate. Additionally, preparing and collecting large, highquality datasets can be time-and resource-intensive. Therefore, the approach proposed in this paper relies only on fault features in the frequency domain and simple and popular approaches to isolate fault types, as the follow-up in [START_REF] Zhang | Current-based analytical model for fault detection and diagnosis in 7-phase machines[END_REF].

In [START_REF] Zhang | Current-based analytical model for fault detection and diagnosis in 7-phase machines[END_REF], the authors developed an analytical model of the phase currents in a 7-phase machine, with a focus on the transformed components in the stationary reference frames. This model assumed that most of the faults in the electrical drive affect the parameters (amplitude, bias and phase shift) of the phase currents. It was shown that the transformed currents in the fictitious four machines are effective fault features with significant fault signatures in the frequency domain. The DC and fundamental components are of particular interest. Consequently, the proposed fault classification method is based on these features. The performance of the classification will be analysed for non incipient and incipient faults with different fault severities, and under varying noise levels.

The contributions of this paper are: (1) Proposal of fault isolation features: the amplitude of 0F and 1F components of the transformed phase currents. (2) Fault isolation analysis for non incipient and incipient fault types with different fault severities. (3) Fault isolation performance under different noise levels.

The paper is organized as follows: Section II briefly introduces the analytical model and the dataset. Section III presents several classification approaches. Section IV discusses different scenarios (non incipient and incipient faults under different noise levels). The conclusion and perspectives are given in Section V.

II. DATA DESCRIPTION

The phase currents in 7-phase machines can be modelled in the natural, stationary (α, β), or synchronous (d, q) frame. The stationary and synchronous frames are based on a decomposition of the actual machine into four fictitious machines: a onephase Homopolar Machine (HM) and three independent and orthogonal coordinate fictitious machines: Principal Machine (PM), Secondary Machine (SM), and Tertiary Machine (TM), in α -β planes or d -q planes [START_REF] Zhang | Current-based analytical model for fault detection and diagnosis in 7-phase machines[END_REF]. In the following, only the (α, β) reference frame is considered due to space limitation.

The faulty currents in the natural frame can be expressed in [START_REF] Gertler | Fault detection and diagnosis in engineering systems[END_REF]. It is assumed that that no matter the fault type and evolution, the fault effect on the phase currents can be emulated with three parameters ( i j , φ j and γ j ). The faulty currents, projected into the stationary frame with the Clarke transformation, are expressed in [START_REF] Furse | Fault diagnosis for electrical systems and power networks: A review[END_REF].

i j(j=1...7)f = I √ 2(1 + i j )sin(θ + (1 -j)ϕ + φ j ) + γ j (1)
where, θ = ωt, ω is the angular frequency, t is the current time, ϕ is the natural phase shift equal to 2π 7 in 7-phase machine, I is the RMS value of the phase current. i j , φ j and γ j will represent the fault parameters resulting in a modification of the amplitude, initial phase shift, and bias, respectively.

Fault frequencies in the stationary frame are summarized in TABLE I. The spectrum of the faulty current in the stationary frame includes DC (0F ) and/or fundamental component (1F ).

H PM ipα × G PM ipα × i pβ × i pβ × SM isα SM isα × i sβ i sβ × (1) TM itα TM itα × i tβ i tβ × (1) HM i0 HM i0 × PS PM ipα × MV PM ipα × × i pβ × i pβ × (2) × SM isα × SM isα × i sβ × (1) i sβ × (3) TM itα × TM itα × i tβ × (1) i tβ × (3) HM i0 × HM i0 ×
H: healthy conditions; G: gain fault; PS: phase shift fault; MV: mean value fault. (1) For phase 1, no 1F component for i sβ and i tβ . (2) For phase 1, no 0F component for i pβ . (3) For phase 1, no 0F component for i sβ and i tβ .

The amplitudes of the transformed currents in the stationary frame will be used as features to isolate the faults. The dataset (TABLE II) is composed of:

• Three fault types: gain fault (g), phase shift fault (ps) and mean value fault (mv). , where, σ 2 s is the variance of the signal, and σ 2 υ the variance of the Gaussian distributed noise, υ ∼ N (0, σ 2 υ ). There are two components (0F and 1F ) for the seven currents in total, under study. Thus the data matrix Y [N ×M ] is expressed in [START_REF] Soliman | Power systems fault type identification based on park's transformation algorithm[END_REF], where, N is the number of samples, and M the number of variables equal to 14. In the following, the fault classification will be evaluated for both fault types (non incipient and incipient) with different fault severities and noise levels.

III. ISOLATION TECHNIQUES

For this application, linear (Principal Component Analysis, Linear Discriminant Analysis), and non linear techniques (Support Vector Machine) are investigated. Indeed, these techniques are very popular and already successfully used in several applications.

A. Principal Component Analysis (PCA)

Principal Component Analysis, an unsupervised method, initially proposed by Pearson (1901) and later developed by Hotelling (1947), is a standard multivariate technique. Its main advantage is its ability to identify the underlying structure or patterns in high-dimensional data through a projection into a lower-dimensional space, while keeping the maximum information variance [START_REF] Wold | Principal component analysis[END_REF].

The PCA is done in the following steps: 1) Normalize the data to have zero mean and unit variance:

Y [N ×M ] = [ y pα(0F ) , • • • , y 0(0F ) , y pα(1F ) , • • • , y 0(1F ) ] =          y 1-pα(0F ) • • • y 1-pα(1F ) • • • . . . . . . . . . . . . y i-pα(0F ) • • • y i-pα(1F ) • • • . . . . . . . . . . . . y N -pα(0F ) • • • y N -pα(1F ) • • •          (4)
where, N is the number of samples, i ∈ [1, N ], and M the number of variables equal to 14.

Λ =         λ 1 . . . λ j . . . λ M         (6)
P = [p 1 • • • p j • • • p M ] (7)
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         i0 ipα i pβ itα i tβ isα i sβ          = 2 7          1 √ 2 1 √ 2 1 √ 2 1 √ 2 1 √ 2 1 √ 2 1 √ 2 1 cos(ϕ) cos (
                  i1 i2 i3 i4 i5 i6 i7          (2)
4) Sort the eigenvalues in descending order to determine the eigenvectors that span the principal and residual subspaces. Typically, the principal subspace is spanned with the eigenvectors whose Cumulative Percent of Variance (CPV) is higher than 95%.

CPV(l) = l j=1 λ j M j=1 λ j ≥ 95% (8)
5) Create the projection matrix whose columns are the eigenvectors. This matrix will be used to project the actual data into lower-dimensional subspaces (principal or residual).

Y [N ×l] = Y [N ×M ] P [M ×l] (9)

B. Linear Discriminant Analysis (LDA)

Linear Discriminant Analysis is a supervised machine learning algorithm that can discriminate classes that can be linearly separated. It is based on the simultaneous maximisation of inter-class and minimization intra-class variances [START_REF] Balakrishnama | Linear discriminant analysis-a brief tutorial[END_REF]. The key point is the computation of the inter-class matrix S b and intra-class matrix S w from which the eigenvalues and eigenvectors of the matrix S w -1 S b can be obtained.

S b = p k=1 N k (µ k -µ)(µ k -µ) T (10)
S w = p k=1 y∈ Y k ( y -µ k )( y -µ k ) T (11)
where,

N k (k = 1, 2, • • • , p) is the number of the k th class, Y k (k = 1, 2, • • • , p) is the set of the k th class, µ k (k = 1, 2, • • • , p)
is the mean value of the k th class, µ is the mean value of the whole samples. LDA exhibits higher classification performance than PCA when the main change between the sampled data affect the average values. LDA and PCA are optimal when the data follow a Gaussian distribution, which is the case in this study as shown in [START_REF] Zhang | Fault severity estimation in 7phase electrical machines in a noisy environment[END_REF]. Therefore, they can be used in the following.

C. Support Vector Machine (SVM)

Support Vector Machine is a supervised machine learning algorithm that can handle the separation of classes, which are linearly or non-linearly separable. The main idea behind SVM is to find an optimal hyperplane that best separates the data points of different classes in the feature space. In non-linear cases, the key is to choose an appropriate kernel function to map the input features into a higher dimensional space. Common kernel functions include linear, polynomial, Radial Basis Function (RBF), and sigmoid [START_REF] Hearst | Support vector machines[END_REF].

IV. NUMERICAL SIMULATIONS

In the following simulations, we consider three single non incipient faults [15% to 30%] and three single incipient faults [1% to 6%] under different noise levels (20 dB and 5 dB), to evaluate the classification performance.

A. Low noise level case

In this section, the noise level is considered as low with a SNR set to 20 dB. First, the data matrix Y is analyzed using PCA. The results are depicted in Fig. 1. From the initial fourteen variables, we can observe that the cumulative sum of the first three principal components exceeds 95% (Fig. 1-a). However, if the data is projected in the subspace defined by the first two principal components (representing almost 90% of the initial total information), the two classes (non incipient and incipient faults) can be linearly separated (Fig. 1-b). Therefore, LDA can be used for fault isolation. The LDA classification results are presented in (Fig. 2). It can be observed that after the first processing the non incipient faults are clearly separated. However, a second step is necessary to separate the data for incipient faults as displayed in (Fig. 3). But, there is still an overlap between the incipient gain fault and incipient phase shift fault. These two classes will be separated with SVM. The presented results are obtained with the following variables: y 0F and y 1F of i pα . The classification is evaluated successfully with other variables. A linear kernel is utilized. The results are illustrated in Fig. 4. It can be observed that the separation is highly satisfactory, with an accuracy higher than 98.8%, as displayed in the confusion matrix presented in [START_REF] Samantaray | Fault classification and location using hs-transform and radial basis function neural network[END_REF]. It can be concluded that when the measurement noise level is low, non incipient and incipient faults can be clearly separated with a linear classifier in a lower dimension feature space. However, in the case of incipient faults, a higher dimension is required with a linear SVM to handle the overlap between gain and phase shift faults. Finally, the flowchart of the classification is shown in Fig. 5. In this section, we assume that the phase current measurements are highly corrupted with the noise, with SNR set to 5dB. The PCA results are presented in Fig. 6. Fig. 6-b shows significant overlaps in the 2D frame spanned by the first two principal components. This is confirmed by the cumulative sum percentage displayed in Figure 6-a from which it can be observed that PCA will not be effective in this case. One can only expect a separation of the non incipient and incipient classes. The LDA results are displayed in Fig. 7. One can observe an improvement because the non incipient mean value fault can now be discriminated from all the other ones. Then, LDA is performed on the data matrix from which the mean value fault data are removed. The results are shown in Fig. 8, and the confusion matrix, in [START_REF] Zhang | Current-based analytical model for fault detection and diagnosis in 7-phase machines[END_REF]. There is a significant improvement as the remaining non incipient faults can be separated.

The classification of non incipient gain and phase shift faults can be improved with a linear SVM classifier as shown in Fig. 9, and the confusion matrix in [START_REF] Wold | Principal component analysis[END_REF]. Non incipient gain and phase shift faults can be separated with an accuracy higher than 98.5%. confusion matrix = 98.6% 1.4% 1.5% 98.5%

From the previous results, it can be concluded that PCA, LDA and even SVM with a non linear kernel failed to classify incipient faults when the phase currents measurements are highly corrupted with noise. Consequently, the classification flowchart under high noise level conditions is depicted in Fig. 10.

Authorized licensed use limited to: Centrale Supelec. Downloaded on January 17,2024 at 10:22:59 UTC from IEEE Xplore. Restrictions apply. The simulation results show that under low noise level, the multiple-step classification methodology is successful in separating non incipient faults from incipient ones, and also successful in discriminating the fault types. However, when the noise level increases, the proposal is only successful in discriminating non incipient faults from incipient ones, and separating non incipient fault types. Future works will be devoted to find more relevant incipient fault features. Mixed faults could also be considered as they may be representative of faults that may occur with the complex system.

• 2 s σ 2 υ

 22 Several fault severities: incipient faults [1% to 6%] and non incipient faults [15% to 30%]. • Different noise levels: 20 dB and 5 dB. The noise level is defined by considering the Signal to Noise Ratio (SNR), SN R = 10 × log 10 σ

2 )

 2 Compute the covariance matrix C [M ×M ] . 3) Compute the eigenvectors (P [M ×M ] ) and eigenvalues (Λ [M ×M ] ) of the covariance matrix C [M ×M ] . The eigenvectors represent the principal components, while the eigenvalues show the explained variance of each component. CP = ΛP (5)

Fig. 1 :

 1 Fig. 1: PCA results wtith SNR = 20 dB

Fig. 2 :Fig. 3 :Fig. 4 :Fig. 5 :

 2345 Fig. 2: Step 1: LDA classification when SNR = 20 dB

Fig. 6 :

 6 Fig. 6: PCA results with SNR = 5 dB

Fig. 7 :

 7 Fig. 7: Step 1: LDA classification when SNR = 5 dB

Fig. 8 :

 8 Fig. 8: Step 2: LDA classification results with SNR = 5 dB

Fig. 9 :Fig. 10 :

 910 Fig. 9: Step 3: Linear SVM classification results with SNR = 5 dB

TABLE I :

 I Fault frequencies in the stationary frame

	Case	Current	Harmonic 0F 1F	Case	Current	Harmonic 0F 1F

TABLE II :

 II Number of samples in the simulations

							Number of samples
	SNR	Fault
					incipient faults				non incipient faults
	(dB)	types
			1%	2%	3%	4%	5%	6%	15%	20%	25%	30%
		g	200	200	200	200	200	200	200	200	200	200
	20	ps	200	200	200	200	200	200	200	200	200	200
		mv	200	200	200	200	200	200	200	200	200	200
		g	200	200	200	200	200	200	200	200	200	200
	5	ps	200	200	200	200	200	200	200	200	200	200
		mv	200	200	200	200	200	200	200	200	200	200
			g: gain fault;	ps: phase shift fault;	mv: mean value fault.

Y [N ×M ] = [ y pα(0F ) , y pβ(0F ) , y sα(0F ) , y sβ(0F ) , y tα(0F ) , y tβ(0F ) , y 0(0F ) ,

y pα(1F ) , y pβ(1F ) , y sα(1F ) , y sβ(1F ) , y tα(1F ) , y tβ(1F ) , y 0(1F ) ]

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L A Tiberi

X Saloppé

C Jacquemyn

R Jaur

T H Pham

Multi-level emotions recognition among forensic sexual offenders: Preliminary results

INTRODUCTION

METHOD RESULTS

DISCUSSION

Participants

The sample is composed of 88 male participants, divided in three groups. The first two groups are forensic inpatients, interned in the High-Risk Security Forensic Hospital "Les Marronniers" (Belgium), under the Law for the internment of persons (2014). Based on their criminal records, they were either categorized into SO group (n = 20) if they committed at least one sexual infraction, or into NSO group (n = 17) if they did not. The NCP group is composed of men from the community (n = 51) through a call for participants published in some public places and on social networks.

No difference was found between SO and NSO on PANAS, UPPS-P and MC-SDS self-reports descriptive variables (Table 1). However, SO were significantly older (p ≤ .001), less educated (p ≤ .001), and exhibited lower score (p ≤ .001) on Perseverance subscale than NCP. NSO were only less educated (p ≤ .001) than NCP. Finally, on a criminal level, NSO committed more thefts (p ≤ .05) and more threats (p ≤ .05) than SO.

Sexual Offenders (SO) exhibit deficits in socio-affective functioning (Thornton, 2002), such as emotions management and regulation (Gillespie et al., 2012;Stinson et al., 2016). Clinicians and researchers acknowledge the emotions, identifying and regulation, as a key component in sexual offending, as important as cognition alone (Gannon & Ward, 2017). Accurate emotions recognition is fundamental in appropriate social interactions (Fernández-Dols et al., 2017). Despite a growing body of literature, intensifying these last years (Hudson et al., 1993;Gery et al., 2009;Gillespie et al., 2015Gillespie et al., , 2021;;Suchy et al., 2009;Wegrzyn et al., 2017), there is still little research on emotions recognition among SO. This paucity of literature is unexpected as emotions recognition has been theorized as the first step in the dynamic process of empathic behaviors (Marshall & Marshall, 2011;Marshall et al., 1995). Previous research on facial expressions of emotions yielded contradictory results (Tiberi et al., accepted), using diverse stimuli and methodology (e.g.: static mono-chromatic stimuli). In addition, to the authors knowledge, no study has yet been conducted on forensic SO inpatients.

Aim: Assessment of multi-level emotions (face, voice -prosody, voice -semantic and body posture) recognition competency among forensic SO inpatients, compared to forensic 'Non-Sexual Offender' inpatients (NSO) and 'Non-Clinical Population' (NCP).

Contact: luca.tiberi@umons.ac.be

Table 1 -Descriptive statistics of the three groups Procedure Voluntary forensic inpatients selection was undertaken in accordance with their psychologist, based on the following inclusion criteria: a) inpatients aged between 18 to 65 years old, b) whose mother tongue is French and c) with sufficient cognitive competencies and with a stable mental state to realize such tasks. Two sessions (min.) were organized with forensic inpatients. The first was conducted in the patient respective Care Unit to present the research, to sign consent sheet, and to complete the anamnesis and SRs (MC-SDS and UPPS-P). The second (and third if necessary) session was conducted in a research room, to complete the PANAS, followed by the emotions recognition tasks. A session duration was between 1h and 2h (pauses included), based on inpatients cognitive ability and tiredness.

For the NCP participants, one session was organized inside a research room at UMONS. A session duration was about 2h (pauses included). NCP participants received a 10€ Amazon gift card after the session completion.

Data Analysis

Descriptive analyses were performed to describe the samples. In the absence of data normality, we carried out Kruskal-Wallis's H analyses (Table 2), followed with post-hoc Mann-Whitney's U analyses (Bonferroni correction; p ≤ .016) for 2x2 groups comparisons. The majority of 2x2 groups comparisons between SO and NCP, and between NSO and NCP were significant (p ≤ .016), except for: 'FEE -Mean Difficulty' (p = .04), 'VEE -Prosody Mean Recognition Accuracy' (p = .52) and 'VEE -Semantic RT Difficulty' (p = .14).

Multi-level emotion recognition tasks

No significant difference was found between SO and NSO.

SO exhibit significant lower performances (e.g.: longer RT, lower mean recognition accuracy) than NCP, but unexpectedly not in comparison with NSO. These results tend to suggest that SO and NSO groups are more similar than expected, despite, for example, their psychopathological profile (psychosis).

Future perspectives are numerous. First, a deeper specific analysis (six basic emotions) will be undertaken to identify whether SO and/or NSO exhibit specific deficit in negative emotions, as hypothesized in the literature (Chapman et al., 2018). Second, a more precise analysis regarding the MD and PD could lead to a refined discrimination between these two groups, in line with RDoC framework (Insel, 2014). Third, the use of refined statistical indexes, sensitivity (d') and response bias (c) as retrieved in the Signal Detection Theory (Hautus et al., 2022), would be promising for a sharper analysis. Finally, the impact of supplementary stimuli variables as model ethnicity or model gender should be assessed through a Multivariate Analysis of Variance.

Instruments

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•

  Positive And Negative Affect Scales -PANAS (Gaudreau et al., 2006) • Marlowe-Crowne Social Desirability Scale -MC-SDS (Crowne & Marlowe, 1960) • Urgency, Premeditation, Perseverance, Sensation seeking, Positive Urgency -UPPS-P (Billieux et al., 2012)

Material•

  SRBOX -RB-730 (Cedrus) • E-Prime 2.0 (Schneider et al., 2002) • HP ZBook15 (15.6 inches; 1920*1080; 60 Hz) Note: FEE = Facial Expressions of Emotions; VEE = Vocal Expressions of Emotions; BEE = Bodily Expressions of Emotions; KW H = Kruskal-Wallis H; *p ≤ .05; **p ≤..001

  Bänziger, T., Mortillaro, M., & Scherer, K. R. (2011). Introducing the Geneva Multimodal Expression Corpus for Experimental Research on Emotion Perception. Emotion 12(5), 1161-1179. doi: 10.1037/a0025827; Billieux, J., Rochat, L., Ceschi, G., Carré, A., Offerlin-Meyer, I., Defeldre, A. C., Khazaal, Y., Besche-Richard, C., & Van Der Linden, M. (2012). Validation of a short French version of the UPPS-P Impulsive Behavior Scale. Comprehensive Psychiatry, 55(5), 609-615. doi: 10.1016/j.comppsych.2011.09.001; Chapman, H., Gillespie, S. M., & Mitchell, I. J. (2018). Facial affect processing in incarcerated violent males : A systematic review. Aggression and Violent Behavior, 38(October 2017), 123-138. doi: 10.1016/j.avb.2017.10.006; Crowne, D. P., & Marlowe, D. (1960). A New Scale of Social Desirability Independent of Psychopathology. Journal of Consulting Psychology, 24(4), 349-354.; Fernández-Dols, J.-M., & Russell, J. A. (2017). The Science of Facial Expression. Oxford University Press.; Gaudreau, P., Sanchez, X., & Blondin, J.-P. (2006). Positive and Negative Affective States in a Performance-Related Setting : Testing the Factorial Structure of the PANAS Across Two Samples of French-Canadian Participants. European Journal of Psychological Assessment, 22(4), 240-249. doi: 10.1027/1015-5759.22.4.240; Gery, I., Miljkovitch, R., Berthoz, S., & Soussignan, R. (2009). Empathy and recognition of facial expressions of emotion in sex offenders, non-sex offenders and normal controls. Psychiatry Research, 165(3), 252-262. doi: 10.1016/j.psychres.2007.11.006; Gillespie, S. M., Mitchell, I. J., Beech, A. R., & Rotshtein, P. (2021). Processing of emotional faces in sexual offenders with and without child victims : An eye-tracking study with pupillometry. Biological Psychology, 163(December 2020), 108141. doi: 10.1016/j.biopsycho.2021.108141; Gillespie, S. M., Mitchell, I. J., Fisher, D., & Beech, A. R. (2012). Treating disturbed emotional regulation in sexual offenders : The potential applications of mindful self-regulation and controlled breathing techniques. Aggression and Violent Behavior, 17(4), 333-343. doi: 10.1016/j.avb.2012.03.005; Gillespie, S. M., Rotshtein, P., Satherley, R.-M., Beech, A. R., & Mitchell, I. J. (2015). Emotional expression recognition and attribution bias among sexual and violent offenders : A signal detection analysis. Frontiers in Psychology, 6(May), 1-12. doi: 10.3389/fpsyg.2015.00595; Hautus, M. J., Macmillan, N. A., & Creelman, C. D. (2021). Detection Theory : A User's Guide (3rd Ed.). Routledge. doi: 10.4324/9781003203636; Hudson, S. M., Marshall, W. L., Wales, D., McDonald, E., Bakker, L. W., & McLean, A. (1993). Emotional recognition skills of sex offenders. Annals of Sex Research, 6(3), 199-211.; Insel, T. R. (2014). The NIMH Research Domain Criteria (RDoC) Project : Precision Medicine for Psychiatry. American Journal of Psychiatry, 171(4), 395-397. doi: 10.1176/appi.ajp.2014.14020138; Marshall, L. E., & Marshall, W. L. (2011). Empathy and antisocial behaviour. Journal of Forensic Psychiatry & Psychology, 22(5), 742-759. doi: 10.1080/14789949.2011.617544; Marshall, W., Hudson, S. M., Jones, R., & Fernandez, Y. M. (1995). Empathy in Sex Offenders. Clinical Psychology Review, 15(2), 99-113.; Schneider, W., Eschman, A., & Zurccolotto, A. (2002). E-Prime 2 : User's guide. Psychology Software Tools Inc.; Stinson, J. D., Becker, J. V., & McVay, L. A. (2016). Multimodal Self-Regulation Theory of Sexual Offending. In A.R. Beech & T. Ward (Eds.) The Wiley Handbook on the Theories, Assessment and Treatment of Sexual Offending -Volume I (p. 103-122). doi: 10.1002/9781118574003.wattso005; Suchy, Y., Whittaker, W. J., Strassberg, D. S., & Eastvold, A. (2009). Facial and Prosodic Affect Recognition Among Pedophilic and Nonpedophilic Criminal Child Molesters. Sexual Abuse, 21(1), 93-110. doi: 10.1177/1079063208326930; Thoma, P., Soria Bauser, D., & Suchan, B. (2013). BESST (Bochum Emotional Stimulus Set)-A pilot validation study of a stimulus set containing emotional bodies and faces from frontal and averted views. Psychiatry Research, 209(1), 98-109. doi: 10.1016/j.psychres.2012.11.012; Thornton, D. (2002). Constructing and Testing a Framework for Dynamic Risk Assessment. Sexual Abuse: A Journal of Research and Treatment, 14(2), 139-153. doi: 10.1177/107906320201400205; Tiberi, L.A., Saloppé, X., Vicenzutto, A., Equeter, L., & Pham, T.H. (accepted). Recognition of Global and Specific (Fear and Happiness) Facial Expressions of Emotions among Sexual Offenders: A Meta-Analysis. Acta Psychiatrica Belgica; Tottenham, N., Tanaka, J. W., Leon, A. C., McCarry, T., Nurse, M., Hare, T. A., Marcus, D. J., Westerlund, A., Casey, B. J., & Nelson, C. (2009). The NimStim set of facial expressions : Judgments from untrained research participants. Psychiatry Research, 168(3), 242-249. doi: 10.1016/j.psychres.2008.05.006; Wegrzyn, M., Westphal, S., & Kissler, J. (2017). In your face : The biased judgement of fear-anger expressions in violent offenders. BMC Psychology, 5(1), 1-12. doi: 10.1186/s40359-017-0186-zPsychiatric Assessment• Mini International Neuropsychiatric Interview -MINI(Sheehan et al., 1998) • Structured Clinical Interview for DSM-IV Axis II Disorders -SCID-II(First et al., 1997)

Table 2 -

 2 Comparisons

	SO

(Kruskal-Wallis H)

of emotional variables between the three groups Note: PANAS -NA = Negative Affect; PANAS -PA = Positive Affect; K-W H = Kruskal-Wallis H; *p ≤ .05; **p ≤..005; ***p ≤ .001 1 Forensic

Psychology Department, University of Mons (UMONS), Belgium 2 Center of Research in Social Defense (CRSD), Belgium 3 SCALab, CNRS, UMR 9193 , University of Lille (ULille), France 4 Service de psychiatrie, Hôpital de Saint-Amand-les-Eaux, France 5 Philippe Pinel Institute, Montréal, Canada

NSO (92.9%) exhibited more Mental Disorders [MD] (MINI) than SO (36.80%) (p ≤ .001), and specifically more psychotic disorders (NSO = 35.7%; SO = 0.00%; p ≤ .05). No difference was found between SO and NSO regarding Personality Disorders [PD] Clusters (SCID-II).


Lu Zhang

Claude Delpha

Demba Diallo

Fault Severity Estimation in 7-Phase Electrical Machines in a Noisy Environment

Abstract-This work proposes a method for estimating fault severity in the presence of noise using the measured currents for a 7-phase electrical machine. The method is based on analytical models in stationary reference frames and analysis of the DC and fundamental components in the four fictitious machines. The simulation results show that the estimation errors in the tertiary and homopolar machines are less noise-sensitive than in the principal machine, regardless of the fault severity level. On the other hand, if the fault significantly affects the magnitude of the currents (fault > 10%), the estimation in the secondary machine is less sensitive to noise than in the main machine. If the fault results in a change in the mean value, the estimates using the main machine characteristics are less sensitive to noise. If the fault affects the phase shift, the estimates are less sensitive in the secondary machine than in the main one even if the noise is significant in the measurements.

Index Terms-Fault severity estimation, [START_REF] Zhang | Simulation analysis of SVPWM based on seven-phase permanent magnet synchronous motor[END_REF] F Ault Detection and Diagnosis (FDD), which can be decomposed into fault detection, isolation and estimation has become more and more attractive in the last decades

The authors would like to thank China Scholarship Council for funding. in all industrial sectors [START_REF] Basseville | Detection of abrupt changes: theory and application[END_REF]. In particular, attention has been paid to the monitoring of electrical machines which are at the heart of, for example, wind turbines or electric vehicles powertrain [START_REF] Benbouzid | Signal Processing for Fault Detection and Diagnosis in Electric Machines and Systems[END_REF]. Although three-phase machines are still dominant, five-, seven-, nine-phase machines are increasingly being designed and used because of their inherent fault tolerance and power sharing capabilities [START_REF] Ward | Preliminary investigation of an invertor-fed 5phase induction motor[END_REF]- [START_REF] Semail | Vectorial formalism for analysis and design of polyphase synchronous machines[END_REF]. This work focuses on monitoring a seven-phase electrical machine [START_REF] Thavot | Modeling of a sevenphase series-connected three-motor drive system[END_REF], [START_REF] Zhang | Simulation analysis of SVPWM based on seven-phase permanent magnet synchronous motor[END_REF], which is attractive in high power applications where reliability and availability are concerns. Usually, fault detection, isolation and estimation is based on the extraction and analysis of fault signatures or features from measured or estimated variables or parameters [START_REF] Benbouzid | Signal Processing for Fault Detection and Diagnosis in Electric Machines and Systems[END_REF]. Those features can be extracted in time, frequency or time-frequency domain, depending on the fault types. In [START_REF] Delpha | Multiple incipient fault diagnosis in three-phase electrical systems using multivariate statistical signal processing[END_REF], the authors presented an approach in the time domain to detect and diagnose several faults in a threephase electrical machine. In [START_REF] Zhang | Current-based analytical model for fault detection and diagnosis in 7-phase machines[END_REF], the authors presented an analytical model of the currents flowing in a 7-phase machine, which can provide sensitive fault signatures in the time and frequency domains while considering that a fault effect leads to changes in mean, offset and phase of the electrical signal. Once the fault is detected and isolated, it is important for maintenance purposes or control reconfiguration to estimate the fault effect severity. The performance of the estimator depends on the accuracy of the model and the quality of the used measurements [START_REF] Kay | Fundamentals of statistical signal processing: estimation theory[END_REF], [START_REF] Stoustrup | Fault estimation -a standard problem approach[END_REF]. Indeed, due to the increasing complexity of industrial systems, it is challenging to develop models, which are accurate in a wide operating range. Besides, measurements are very often polluted with nuisances due to switching devices and noise. This paper is a continuation of [START_REF] Zhang | Current-based analytical model for fault detection and diagnosis in 7-phase machines[END_REF]. Most electrical or mechanical faults in electrical machines affect the spectrum or characteristics (amplitude, phase, mean value) of the currents flowing in the windings. This work does not deal with the control performance. Whatever the control structure, it assumes that the phase currents are affected by fault occurrence. The objectives are to detect the fault and estimate its severity with the highest performance. The main contributions are: [START_REF] Basseville | Detection of abrupt changes: theory and application[END_REF] The development of analytical models in stationary frames of machine phase-currents when affected by faults that change their amplitude, phase or mean value; [START_REF] Benbouzid | Signal Processing for Fault Detection and Diagnosis in Electric Machines and Systems[END_REF] The evaluation of the fault severity estimation performance for different noise levels.

The flowchart of the estimation procedure is displayed in Fig. 1. The rest of this paper is organized as follows: Section II (right side of the flowchart) describes how the theoretical relation between the fault severity and noise-free fault features is extracted. Section III (left side of the flowchart) presents how the fault severity is estimated from fault features under different noise levels. In section IV, the evaluation of the estimation performance in the different fictitious machines is provided based on the analysis of the normalized root mean square deviation. The conclusions are given in Section V.

A. Description of frames

The 7-phase machine can be modeled in three representation spaces, expanding the conventional definition of space vector which is utilized for 3-phase systems [START_REF] Semail | Vectorial formalism for analysis and design of polyphase synchronous machines[END_REF], [START_REF] Zhang | Current-based analytical model for fault detection and diagnosis in 7-phase machines[END_REF]:

• Natural frame: The variables for the 7 phases are denoted as x 1 to x 7 . • Stationary transformed (α, β) frame: it can be represented by a one-phase Homopolar Machine (HM) (variable is x 0 ) and three two-phase independent fictitious machines: a Principal Machine (PM) (variables are x pα and x pβ ), a Secondary Machine (SM) (variables are x sα and x sβ ), and a Tertiary Machine (TM) (variables are x tα and x tβ ). • Synchronous transformed (d, q) frame: it is composed of a one-phase HM and three dq orthogonal coordinate fictitious machines. In the following, the synchronous representation space is not considered. This study is limited to the analysis of the DC component (0F ) and/or fundamental component (1F ) of the phase currents. The healthy and faulty currents in the natural frame are expressed as follows, respectively [START_REF] Zhang | Current-based analytical model for fault detection and diagnosis in 7-phase machines[END_REF]:

i j(j=1...7)h = I √ 2sin(θ + (1 -j)ϕ) (1)
i j(j=1...7)f = I √ 2(1 + △i j )sin(θ + (1 -j)ϕ + φ j ) + γ j (2)
where, θ = ωt, ω is the angular frequency; t is the current time; ϕ = 2π 7 is the natural phase shift multiple of the 7-phase machine; I represents the RMS value of the phase current; a gain fault, resulting in a modification of the current amplitude, can be denoted by △i j ; a phase shift fault, which modifies the initial phase of the current, can be represented by φ j ; a bias in the mean value of the current can be represented by γ j . The index j denotes the phase number; and h and f denote the healthy and faulty conditions, respectively.

Faults in a multi-electricity system can be categorized based on their location or origin, such as Generator Fault (fault occurs within a generator or other rotating machinery), Line Fault (fault occurs on a transmission or distribution line), Transformer Fault, Bus Fault (occurs within a bus or other electrical component that distributes power to multiple circuits) and so on. A fault can also be classified as symmetric or asymmetric according to whether it affects identically the phases. Usual faults include short-circuits, overload, ground fault, voltage sag or swell. Faults can interact, and each fault's impact depends on various factors, including its location, the equipment involved and the means of protection and control. For example, a short circuit can cause a significant increase in current, resulting in a change in amplitude (gain). However, a short circuit can cause a voltage drop or phase shift, mainly if the short circuit occurs near the end of a transmission line or sensitive equipment, which could result in a phase shift or mean value fault. Then, it can be assumed that no matter the fault type and evolution, the fault effect on the phase currents can be emulated with three main parameters, as described in the equation [START_REF] Benbouzid | Signal Processing for Fault Detection and Diagnosis in Electric Machines and Systems[END_REF].

B. Analytical model for fault severity estimation

As discussed, the problem for a multi-dimensional system is to find the elementary independent 2-dimensional machines. The matrix derived in (3) can ensure the families of odd harmonics: 7n ± 1, 7n ± 2, and 7n ± 3, for PM, SM, and TM, respectively [START_REF] Semail | Vectorial formalism for analysis and design of polyphase synchronous machines[END_REF].

The analytical expressions of the phase currents in the stationary frames can be deduced by combining (2) and the Clarke transformation in [START_REF] Ward | Preliminary investigation of an invertor-fed 5phase induction motor[END_REF]. The detailed derivation process and results can be found in [START_REF] Zhang | Current-based analytical model for fault detection and diagnosis in 7-phase machines[END_REF]. The general expressions of faulty currents under the three single faults are displayed in

ipα = √ 7 * I * sinθ + 2 7 * cos(j -1)ϕ * γj i pβ = - √ 7 * I * cosθ - 2 7 * sin(j -1)ϕ * γj isα = 2 7 * cos(j -1)3ϕ * γj i sβ = 2 7 * sin(j -1)3ϕ * γj itα = 2 7 * cos(j -1)2ϕ * γj i tβ =
        i0 ipα i pβ itα i tβ isα i sβ          = 2 7          1 √ 2 1 √ 2 1 √ 2 1 √ 2 1 √ 2 1 √ 2 1 √ 2 1 cos(ϕ) cos (
                  i1 i2 i3 i4 i5 i6 i7          (3)

TABLE II: Current under gain fault

Analytical model, △ij ∈ (0, 1)

ipα = 1 7 * 2 * △ij * (△ij + 7) * (1 + cos(j -1)2ϕ) + 49 * I * sin(θ -θ ′ pα ), tanθ ′ pα = △i j * sin(j-1)2ϕ 7+△i j +△i j * cos(j-1)2ϕ i pβ = 1 7 * 2 * △ij * (△ij + 7) * (1 -cos(j -1)2ϕ) + 49 * I * sin(θ -θ ′ pβ ), tanθ ′ pβ = 7+△i j -△i j * cos(j-1)2ϕ △i j * sin(j-1)2ϕ isα = 2 7 * △ij * 1 + cos(j -1)6ϕ * I * sin(θ + θ ′ sα ), tanθ ′ sα =
sin(j-1)2ϕ-sin(j-1)4ϕ cos(j-1)2ϕ+cos(j-1)4ϕ

i sβ = 2 7 * △ij * 1 -cos(j -1)6ϕ * I * sin(θ -θ ′ sβ ), tanθ ′ sβ = cos(j-1)2ϕ-cos(j-1)4ϕ sin(j-1)2ϕ+sin(j-1)4ϕ itα = 2 7 * △ij * 1 + cos(j -1)4ϕ * I * sin(θ + θ ′ tα ), tanθ ′ tα =
sin(j-1)ϕ-sin(j-1)3ϕ cos(j-1)ϕ+cos(j-1)3ϕ

i tβ = 2 7 * △ij * 1 -cos(j -1)4ϕ * I * sin(θ + θ ′ tβ ), tanθ ′ tβ = cos(j-1)ϕ-cos(j-1)3ϕ sin(j-1)ϕ+sin(j-1)3ϕ i0 = 2 7 * △ij * I * sin(θ -(j -1)ϕ)
-(j -1)2ϕ) -2 * cos(φj + (j -1)2ϕ) * I * sin(θ + θ ′ pα ), tanθ ′ pα = sinφ j +sin(j-1)2ϕ+sin(φ j -(j-1)2ϕ) 6+cosφ j -cos(j-1)2ϕ+cos(φ j -(j-1)2ϕ) i pβ = 1 7 * 39 + 10 * cosφj + 10 * cos(j -1)2ϕ -12 * cos(φj -(j -1)2ϕ) + 2 * cos(φj + (j -1)2ϕ) * I * sin(θ -θ ′ pβ ), tanθ ′ pβ = 6+cosφ j +cos(j-1)2ϕ-cos(φ j -(j-1)2ϕ) sinφ j -sin(j-1)2ϕ-sin(φ j -(j-1)2ϕ) isα = 1 7 * 4 * (1 + cos(j -1)6ϕ -cosφj ) -2 * (cos(φj -(j -1)6ϕ) + cos(φj + (j -1)6ϕ)) * I * sin(θ + θ ′ sα ), tanθ ′ sα =
sin(φ j +(j-1)2ϕ)+sin(φ j -(j-1)4ϕ)-sin(j-1)2ϕ+sin(j-1)4ϕ cos(φ j +(j-1)2ϕ)+cos(φ j -(j-1)4ϕ)-cos(j-1)2ϕ-cos(j-1)4ϕ

i sβ = 1 7 * 4 * (1 -cos(j -1)6ϕ -cosφj ) + 2 * (cos(φj -(j -1)6ϕ) + cos(φj + (j -1)6ϕ)) * I * sin(θ -θ ′ sβ ), tanθ ′ sβ =
cos(φ j +(j-1)2ϕ)-cos(φ j -(j-1)4ϕ)-cos(j-1)2ϕ+cos(j-1)4ϕ sin(φ j +(j-1)2ϕ)-sin(φ j -(j-1)4ϕ)-sin(j-1)2ϕ-sin(j-1)4ϕ itα = sin(φ j +(j-1)ϕ)+sin(φ j -(j-1)3ϕ)-sin(j-1)ϕ+sin(j-1)3ϕ cos(φ j +(j-1)ϕ)+cos(φ j -(j-1)3ϕ)-cos(j-1)ϕ-cos(j-1)3ϕ

i tβ = cos(φ j +(j-1)ϕ)-cos(φ j -(j-1)3ϕ)-cos(j-1)ϕ+cos(j-1)3ϕ sin(φ j +(j-1)ϕ)-sin(φ j -(j-1)3ϕ)-sin(j-1)ϕ-sin(j-1)3ϕ

i0 = 2 * 1 7 * 1 -cosφj * I * sin(θ + θ ′ 0 ), tanθ ′ 0 =
sin(φ j -(j-1)ϕ)+sin(j-1)ϕ cos(φ j -(j-1)ϕ)-cos(j-1)ϕ

Under healthy conditions, there is no current flowing in the SM, TM, and HM. However, when the system is faulty, new components and/or frequency distortions will appear in the actual phase currents [START_REF] Zhang | Current-based analytical model for fault detection and diagnosis in 7-phase machines[END_REF]. The frequency analysis is summarized in TABLE IV. It can be noticed that the spectrum of the faulty current in the stationary frames includes DC (0F ) and/or fundamental component (1F ). It can be observed that only the mean value fault induces the presence of DC (0F ) component in the four fictitious machines [START_REF] Zhang | Current-based analytical model for fault detection and diagnosis in 7-phase machines[END_REF].

For gain, phase shift, and mean value faults, the fault severity x ∈ [0, 1] corresponds to the modification of parameters △i j , φ j and γ j , respectively. The relations between the amplitude of the components y for 0F and 1F and the fault severity x can be expressed by a function y = ℧(x), as it can be seen in TABLE V to TABLE VII. These functions, retrieved under noise-free conditions are then used to estimate the fault parameter from the noisy fault features (Fig. 1).

H PM ipα × G PM ipα × i pβ × i pβ × SM isα SM isα × i sβ i sβ × (1) TM itα TM itα × i tβ i tβ × (1) HM i0 HM i0 × PS PM ipα × MV PM ipα × × i pβ × i pβ × (2) × SM isα × SM isα × i sβ × (1) i sβ × (3) TM itα × TM itα × i tβ × (1) i tβ × (3) HM i0 × HM i0 ×
H: healthy; G: gain fault; PS: phase shift fault; MV: mean value fault. (1) For phase 1, no 1F component for i sβ and i tβ . (2) For phase 1, no 0F component for i pβ . (3) For phase 1, no 0F component for i sβ and i tβ .

i 0F 1F ipα / y = 1 2 * 1 7 * 2x(x + 7)(1 + cos(j -1)2ϕ) + 49 * I i pβ / y = 1 2 * 1 7 * 2x(x + 7)(1 -cos(j -1)2ϕ) + 49 * I isα / y = kx, k = 1 2 * 2 7 * 1 + cos(j -1)6ϕ * I i sβ / y = kx, k = 1 2 * 2 7 * 1 -cos(j -1)6ϕ * I itα / y = kx, k = 1 2 * 2 7 * 1 + cos(j -1)4ϕ * I i tβ / y = kx, k = 1 2 * 2 7 * 1 -cos(j -1)4ϕ * I i0 / y = kx, k = 1 2 * 2 7 * I

III. NOISY PHASE CURRENTS ANALYSIS

A. Settings for the numerical model

A white Gaussian noise is added to the phase currents to simulate realistic measurements. To define the noise level, the Signal to Noise Ratio (SNR) is considered. As reminder SN R = 10 * log 10

σ 2 s σ 2 υ
, where, σ 2 s is the variance of the signal, and σ 2 υ is the variance of the Gaussian distributed noise υ ∼ N (0, σ 2 υ ). The following simulation studies are coded in MATLAB 2021a, and are carried out on a personal computer with an Intel Core i7 vPro 10th generation processor and 32 GB of memory. In the following, the SNR is set to 20, 15, 10 and 5 dB, respectively. When the SNR decreases, the noise level increases. The fault levels are set to [5% to 30%] for the three cases: gain fault varies from 0.05 to 0.3; phase shift varies from 0.05ϕ to 0.3ϕ, and mean value fault varies from 0.05I √ 2 to 0.3I √ 2. Two hundred simulations are run for the healthy conditions, but also each faulty severity case and each noise level (three single fault types, and four SNR values). Without losing generality, the RMS value of the phase current is assumed to be 4A, and only the results for the faults in the fourth phase of the machine are presented.

B. Estimated fault severity

The fault severity estimation under noisy conditions is done in two steps: (i) under noise-free conditions, the function ℧ -1 (.) is obtained through the analytical models, from the fault features (amplitude of DC and 1F components) and actual fault severity. (ii), the fault severity is estimated with the fault features extracted from noisy measurements such as x = ℧ -1 ( y).

Fig. 2 represents the estimated fault severity versus the actual one when using i pα under gain fault case when SNR equals to 20 and 5 dB respectively. For each severity, the simulation is repeated 200 times. The red dots are the average values, which are close to the actual fault level. However, we can observe an increase of the variance with the noise level. This is confirmed with the histograms displayed in Fig. 3, which show that the estimation follows, as expected, the Gaussian distribution. The same computations are done for all the phase current components of the four fictitious machines in the stationary representation frames. To evaluate the fault estimation performance, first the relative error is defined as ζ = x-x

x , where the actual fault severity is x and the estimated one x. Its averaged value (for the 200 simulations) is displayed in Fig. 4 to Fig. 6 for the three single faults and the considered current component in the stationary frame. It can be concluded that: [START_REF] Basseville | Detection of abrupt changes: theory and application[END_REF] For the same SNR level, the average error becomes negligible when the fault severity increases; (2) For the same fault severity, the noise degrades the estimation; (3) The noise effect is higher for gain and phase shift fault than mean value fault; (4) The component

+ (j -1)2ϕ) * I isα / y = 1 2 * 1 7 * 4 * (1 + cos(j -1)6ϕ -cosϕx) -2 * (cos(ϕx -(j -1)6ϕ) + cos(ϕx + (j -1)6ϕ)) * I i sβ / y = 1 2 * 1 7 * 4 * (1 -cos(j -1)6ϕ -cosϕx) + 2 * (cos(ϕx -(j -1)6ϕ) + cos(ϕx + (j -1)6ϕ)) * I itα / y = 1 2 * 1 7 * 4 * (1 + cos(j -1)4ϕ -cosϕx) -2 * (cos(ϕx -(j -1)4ϕ) + cos(ϕx + (j -1)4ϕ)) * I i tβ / y = 1 2 * 1 7 * 4 * (1 -cos(j -1)4ϕ -cosϕx) + 2 * (cos(ϕx -(j -1)4ϕ) + cos(ϕx + (j -1)4ϕ)) * I i0 / y =
N RM SD = N n=1 (x n -x) 2 N /µ x (4)
where, xn represents the estimated fault severity; x denotes the real fault severity; µ x is the average estimated fault severity; N is the number of estimated values (N = 200).

The NRMSD values for the estimation using the α and β axis information can be considered simultaneously, so the NRMSD value for each fictitious machine PM, SM, TM, and HM can be expressed as follows, respectively:

N RM SD P = N RM SD 2 pα + N RM SD 2 p β (5)
N RM SD S = N RM SD 2 sα + N RM SD 2 s β (6)
N RM SD T = N RM SD 2 tα + N RM SD 2 t β (7)
N RM SD H = N RM SD h (8)
In order to compare the estimation efficiency between the fictitious machines, if we consider the principal machine as the reference, we can compute the NRMSD ratio R such as: R s/p = N RM SD S / N RM SD P (9)

R t/p = N RM SD T / N RM SD P (10)
R h/p = N RM SD H / N RM SD P (11)
The results are displayed in Fig.s 7 to 10. It can be concluded that: (1) For gain fault, the estimation is more sensitive to the noise in the PM. (2) For phase shift fault, whatever the fault level, the estimation is less sensitive to the noise in the tertiary and homopolar machines. Under low noise level conditions, the PM is more suitable for fault estimation than the SM, particularly for high fault severities. (3) For mean value fault, the PM is less sensitive to the noise than the SM for fault estimation. The tertiary and homopolar ones exhibit the highest robustness. (4) We can notice that the fault estimation error ratio is almost constant for all fault types and severities. This paper evaluates the performance of a fault estimator to the fault severity and noise levels. The case study is a seven-phase electrical machine for which it is assumed that most of the faults have an impact on the magnitude, phase or mean value of the phase currents. An analytical model is developed from which the relations between the fault severity and the fault features (DC and 1F components) are extracted under noise-free conditions. These relations are then used to estimate the fault severity from noisy features. The relative error is used to evaluate the estimation performance.

From the Monte-Carlo simulations, the results showed that for different fault severities (0 to 30%), and different noise levels (20 to 5dB), the current components of the tertiary and homopolar fictitious machines, in the stationary representation space, are the most robust to the nuisances. Future works will further analyze the estimation robustness performance, and its application to experimental data. The estimation in the synchronous representation spaces will also be investigated because the current components are also usually available within the electrical drive.

2 7 *

 7 sin(j -1)2ϕ * γj i0 = 1 7 * γj Authorized licensed use limited to: Centrale Supelec. Downloaded on January 17,2024 at 10:35:27 UTC from IEEE Xplore. Restrictions apply.



1 7 * 4 *

 14 (1 + cos(j -1)4ϕ -cosφj ) -2 * (cos(φj -(j -1)4ϕ) + cos(φj + (j -1)4ϕ)) * I * sin(θ + θ ′ tα ),tanθ ′ tα =

1 7 * 4 * ( 1 -

 141 cos(j -1)4ϕ -cosφj ) + 2 * (cos(φj -(j -1)4ϕ) + cos(φj + (j -1)4ϕ)) * I * sin(θ -θ ′ tβ ), tanθ ′ tβ =

  Evolution of fault severity under gain fault

Fig. 2 :Fig. 3 :

 23 Fig. 2: Fault severity estimation with i pα under gain fault

5 (g) i 0 Fig. 4 :

 504 Fig. 4: Average fault estimation error under gain fault

5 (g) i 0 Fig. 5 :

 505 Fig. 5: Average fault estimation error under phase shift fault

5 (g) i 0 Fig. 6 :

 506 Fig. 6: Average fault estimation error under mean value fault

Fig. 9 :

 9 Fig. 9: Estimation errors ratio for phase shift fault

Fig. 10 :

 10 Fig. 10: Estimation errors for mean value fault

  -phase machines, Stationary frames, Frequency domain, Noisy environment

		NOMENCLATURE
	θ	Electrical angle
	ω	Angular frequency
	I	RMS value of the phase current
	△i j φ j	Gain shift of j th phase Phase shift of j th phase
	γ j	Mean shift of j th phase
	ϕ	Natural phase shift of the 7-phase machine phases
	i 0	Current in the Homopolar Machine
	i pα	Current α axis in the Principal Machine
	i pβ	Current β axis in the Principal Machine
	i sα	Current α axis in the Secondary Machine
	i sβ	Current β axis in the Secondary Machine
	i tα	Current α axis in the Tertiary Machine
	i tβ	Current β axis in the Tertiary Machine
	y	Amplitude of noise-free harmonics
	y	Amplitude of noisy harmonics
	x	Actual fault severity
	x	Estimated fault severity
	ζ	Error criterion
		I. INTRODUCTION

TABLE I :

 I Current under mean value fault

	Analytical model, γj ∈ (0, 1)

TABLE III :

 III Current under phase shift fault

Analytical model, φj ∈ (0, ϕ) ipα = 1 7 * 39 + 10 * cosφj -10 * cos(j -1)2ϕ + 12 * cos(φj

TABLE IV :

 IV Fault frequencies in the stationary frames

	Case	Current	Harmonic 0F 1F	Case	Current	Harmonic 0F 1F

TABLE V :

 V Evolution of fault severity under mean value fault

TABLE VII :

 VII Evolution of fault severity under phase shift fault * 39 + 10 * cosϕx -10 * cos(j -1)2ϕ + 12 * cos(ϕx -(j -1)2ϕ) -2 * cos(ϕx + (j -1)2ϕ) * I * 39 + 10 * cosϕx + 10 * cos(j -1)2ϕ -12 * cos(ϕx -(j -1)2ϕ) + 2 * cos(ϕx

	i	0F	1F
	ipα 7 i pβ / y = 1 2 * 1 / y = 1 2 * 1 7

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Mebarek Belkadi

Anne Sergent
email: anne.sergent@limsi.fr

Yann Fraigneau

Bérengère Podvin

STATISTICAL PROPERTIES OF THERMAL PLUMES IN TURBULENT RAYLEIGH-BENARD CONVECTION WITH A ROUGH PLATE

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

MOTIVATION

In a real thermal system, the underlying surfaces involve specific topographies or small-scale roughness which have a significant impact on the heat transfer. Particularly, in the case of turbulent convection, the addition of wall roughness results in a global heat transfer enhancement and in some cases, in an increase of the exponent of the heat transfer scaling law (N u ∼ Ra β , N u Nusselt number, Ra Rayleigh number) [START_REF] Xie | Turbulent thermal convection over rough plates with varying roughness geometries[END_REF][START_REF] Rusaouën | Thermal transfer in Rayleigh-Bénard cell with smooth or rough boundaries[END_REF]. It is commonly admitted that the heat transfer enhancement originates from an intensification of the thermal plume emission at the tips of roughness elements [START_REF] Du | Turbulent thermal convection in a cell with ordered rough boundaries[END_REF].

In classic turbulent Rayleigh-Bénard convection, previous studies have shown the role of thermal plumes in the heat transport within the cavity [START_REF] Shang | Measured local heat transport in turbulent Rayleigh-Bénard convection[END_REF]. Several techniques of plumes description have been proposed to characterize plumes in terms of geometry, number, intensity or velocity. They can be based on the post-processing of a temperature field determined from thermochromic liquid crystal microspheres [START_REF] Zhou | Morphological evolution of thermal plumes in turbulent Rayleigh-Bénard convection[END_REF], on the identification of temperature threshold in numerical snapshots [START_REF] Shishkina | Analysis of sheet-like thermal plumes in turbulent Rayleigh-Bénard convection[END_REF], on the analysis of cliff-ramp structures in local time series [START_REF] Zhou | Statistical characterization of thermal plumes in turbulent thermal convection[END_REF], or on shadowgraph treatment [START_REF] Belkadi | Experimental and numerical shadowgraph in turbulent Rayleigh-Bénard convection with a rough boundary: investigation of plumes[END_REF].

In the present numerical study, we consider convection over a regularly roughened plate in a Rayleigh-Bénard cell. The three successive heat transfer regimes, previously observed in this kind of configuration (see for example [START_REF] Xie | Turbulent thermal convection over rough plates with varying roughness geometries[END_REF]), are modelled from inactive roughness (regime I) to a regime (called III) where the relative heat transfer increase is larger than the relative surface increase induced by roughness addition. The objective of this work is to identify how roughness changes the plume features and their dynamics, depending on the regime.

NUMERICAL SET-UP

We study the fluid flow occurring in an asymmetric 3D Rayleigh-Bénard rectangular cell filled with water. The cell is heated from below by an isothermal rough plate, modeled by a set of square-based obstacles as sketched on figure 1. Sizes of the roughness elements share some common geometric features with the Lyon's experiments [START_REF] Liot | Velocity fluctuations and boundary layer structure in a rough Rayleigh-Bénard cell filled with water[END_REF][START_REF] Rusaouën | Thermal transfer in Rayleigh-Bénard cell with smooth or rough boundaries[END_REF] such as aspect ratios or distribution, but we retain a larger roughness height (H r /H = 0.03, H cavity height) in order to trigger the first transition between the heat transfer regimes I and II to a lower Ra.

The Navier-Stokes equations are solved under the Boussinesq approximation. The parallel numerical solver (SUN-FLUIDH code) uses a finite volume discretization on staggered grids. The time marching is treated by a semi-implicit scheme combined with a prediction -correction scheme for the velocity -pressure coupling, with a global second-order accuracy. The code has been validated for the classic turbulent Rayleigh-Bénard convection by comparison with the DNS data [START_REF] Kaczorowski | Turbulent flow in the bulk of Rayleigh-Bénard convection: aspect-ratio dependence of the small-scale properties[END_REF]. Numerical simulations are performed for a large range of Rayleigh numbers in order to cover the three heat transfer regimes (10 5 ≤ Ra ≤ 10 10 ). We use meshes from 1.3 • 10 6 to 5 • 10 8 cells in order to ensure a sufficient space resolution. The computational domain is decomposed into 256 to 2048 subdomains depending on the considered Ra number. The convergence in space and time has been a posteriori checked by testing the convergence of several formulations of the time-averaged Nusselt number.

RESULTS

Taking advantage of the cavity asymmetry, we conduct a comparative study of the plume dynamics near the rough and smooth horizontal plates within the same Rayleigh-Bénard cavity for the three regimes of heat transfer.

First, we consider more specifically three particular Ra numbers, each belonging to a different regime (figure 1). The time monitoring of a set of probes recording temperature or vertical velocity, and located near the rough and smooth plates, shows how roughness alters the statistical properties of thermal plumes. In agreement with the experiments (see for example [START_REF] Du | Turbulent thermal convection in a cell with ordered rough boundaries[END_REF][START_REF] Shang | Measured local heat transport in turbulent Rayleigh-Bénard convection[END_REF][START_REF] Liot | Velocity fluctuations and boundary layer structure in a rough Rayleigh-Bénard cell filled with water[END_REF]), intense temperature fluctuations are observed more frequently near the rough plate than near the smooth plate, especially in the intermediate regime II. Concurrently, the largest vertical velocity fluctuations are found near the smooth plate rather than around the rough one. In addition, instantaneous characteristics of thermal plumes such as amplitude (in terms of local increment) or width (equivalent to a transit time) can also be calculated from the time series, following the [START_REF] Zhou | Statistical characterization of thermal plumes in turbulent thermal convection[END_REF] methodology. As expected, log normal distributions are generally recognized in the PDFs of the plume features. However, a noticeable change is observed in regime II, with a more frequent detachment of high energy plumes close to the rough plate, both in terms of potential and kinetic energy.

Finally, we make use of space-time diagrams [START_REF] Belkadi | Experimental and numerical shadowgraph in turbulent Rayleigh-Bénard convection with a rough boundary: investigation of plumes[END_REF] to quantify the spatio-temporal dynamics of the thermal plumes. An increase of the plume number with Ra is observed on both sides of the cavity for all regimes. Although a similar plume distribution near the smooth and rough plates is found in regime I and III, we show that a strong asymmetry between the two horizontal plates is established in regime II with a significant increase of the number of plumes close to the roughness. Moreover, we note in this regime the presence of highly dissipative structures not only near roughness but also within the bulk of the cavity (as seen on figure 1).

Figure 1 :

 1 Figure 1: Instantaneous flow vizualisation at Ra = 10 6 , 5 • 10 7 and 2 • 10 9 , respectively belonging to regimes I, II and III. Colour indicates the thermal dissipation field.

ACKNOWLEDGMENTS

This work was granted access to the HPC resources of GENCI-IDRIS under allocation 2a0326 made by GENCI. This work has benefited from the financial support of the INSIS CNRS (PEPS Energie 2018 CORINThe).

STATISTICAL PROPERTIES OF THERMAL PLUMES IN TURBULENT RAYLEIGH-BENARD CONVECTION WITH A ROUGH PLATE