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"International Journal of Turbomachinery, Propulsion and Power"
, Volume 7 (3) – Jun 28, 2022

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International Journal of Turbomachinery Propulsion and Power Article Multi-Channel High-Dimensional Data Analysis with PARAFAC-GA-BP for Nonstationary Mechanical Fault Diagnosis 1 , 1 , 2 1 Hanxin Chen *, Shaoyi Li and Menglong Li Wuhan Institute of Technology, School of Mechanical and Electrical Engineering, Wuhan 430074, China; lisy@ncpu.edu.cn (S.L.); mlong95@163.com (M.L.) Nanchang Institute of Science and Technology, School of Artiﬁcial Intelligence, Nanchang 330108, China * Correspondence: pg01074075@ntu.edu.sg Abstract: Conventional signal processing methods such as Principle Component Analysis (PCA) focus on the decomposition of signals in the 2D time–frequency domain. Parallel factor analysis (PARAFAC) is a novel method used to decompose multi-dimensional arrays, which focuses on analyzing the relevant feature information by deleting the duplicated information among the multi- ple measurement points. In the paper, a novel hybrid intelligent algorithm for the fault diagnosis of a mechanical system was proposed to analyze the multiple vibration signals of the centrifugal pump system and multi-dimensional complex signals created by pressure and ﬂow information. The continuous wavelet transform was applied to analyze the high-dimensional multi-channel signals to construct the 3D tensor, which makes use of the advantages of the parallel factor decomposition to extract feature information of the complex system. The method was validated by diagnosing the nonstationary failure modes under the faulty conditions with impeller blade damage, impeller perforation damage and impeller edge damage. The correspondence between different fault char- acteristics of a centrifugal pump in a time and frequency information matrix was established. The characteristic frequency ranges of the fault modes are effectively presented. The optimization method for a PARAFAC-BP neural network is proposed using a genetic algorithm (GA) to signiﬁcantly Citation: Chen, H.; Li, S.; Li, M. improve the accuracy of the centrifugal pump fault diagnosis. Multi-Channel High-Dimensional Data Analysis with PARAFAC-GA-BP Keywords: parallel factor analysis; genetic algorithm; BP neural network; fault diagnosis for Nonstationary Mechanical Fault Diagnosis. Int. J. Turbomach. Propuls. Power 2022, 7, 19. https://doi.org/ 10.3390/ijtpp7030019 1. Introduction Academic Editor: Giorgio Pavesi Mechanical equipment plays a signiﬁcant role in the construction of the national econ- Received: 1 February 2022 omy and is an integral part of the entire industrial sector [1]. With the great developments Accepted: 3 May 2022 in the improvement of modern production that have taken place, the structures of modern Published: 28 June 2022 equipment are becoming much more complex. The mechanical equipment needs to remain resilient in severe working conditions. Due to the inﬂuence of many unavoidable severe Publisher’s Note: MDPI stays neutral environmental factors, machinery and equipment such as the centrifugal pumps, gearboxes, with regard to jurisdictional claims in engines, and other major components that work under a heavy load, high temperature published maps and institutional afﬁl- and high pressure experience a variety of the failures. Especially given the extension in iations. their working life, the mechanical components inevitably suffer aging, wear, tear, etc. If failure in the machinery and equipment is not handled promptly, minor damages progress to severe failures, which delays production, causes huge economic losses and serious Copyright: © 2022 by the authors. accidents that endanger the lives of staff [2]. The timely prevention of mechanical equip- Licensee MDPI, Basel, Switzerland. ment failure to maintain the safe operation of equipment in industrial production is of This article is an open access article paramount importance. distributed under the terms and Centrifugal pumps have excellent properties such as a simple structure, high efﬁciency conditions of the Creative Commons and stable performance. They are widely used in industrial production. It is necessary to Attribution (CC BY-NC-ND) license diagnose and monitor the running status of the centrifugal pumps during the complex (https://creativecommons.org/ industrial process such as in the oil industry, etc. [3]. The current mainstream vibration licenses/by-nc-nd/4.0/). Int. J. Turbomach. Propuls. Power 2022, 7, 19. https://doi.org/10.3390/ijtpp7030019 https://www.mdpi.com/journal/ijtpp Int. J. Turbomach. Propuls. Power 2022, 7, 19 2 of 21 signal-based centrifugal pump fault diagnosis method mainly relies on machine learning [4]. Reference [5] selects the time-domain characteristic signal of an electrical submersible pump using the decision tree algorithm and inputs it into a classiﬁer to realize fault separation. Studies in the literature [6] introduce the idea of the k-nearest neighbor algorithm into traditional Markov distance fault judgment to forecast three common centrifugal pump faults. Bordoloi D J et al. used support vector machines to effectively diagnose the blockage level and obstruction cavities at different pump speeds [7]. In the context of Industry 4.0, given the progress in computer science, sensors, cloud technology, big data, etc., the large scale of data collection and the storage of the complex industrial systems are becoming easier and the scale of data is becoming larger, which characterizes the structure of data as high-dimensional. Currently, the processing of high-dimensional data from the large scale industrial processes, which are used to mine valuable information, is a hot topic in the literature [8]. Traditional data-processing methods have a great capability of reducing the dimension of data, such as the Principal Component Analysis (PCA), Intrinsic Modal Analysis (EMD), Wavelet Packet Energy (WPE) and local characteristic analysis (LFA) [9]. Combining the above dimensionality reduction method with a neural network to process massive data and realize data mining has become the mainstream research direction in the research community [10]. C Cui constructed the PCA-BP-MSET model to achieve effective fault warning in an air compressor fault diagnosis system [11]. For abnormalities in the sensor system, Yu used EMD to process the data and PNN as a classiﬁer to achieve fault classi- ﬁcation [12]. Compared with the above algorithms, the parallel factorization processing tensor has the advantage of reducing data loss and computational complexity because the tensor represents the properties of the higher order data without damaging the intrinsic structure and underlying information of the data. One of the most promising theories of parallel factorization comes from Kruskal and the new concept of k-order [13]. The k-order for matrix A is the maximum that satisﬁes the condition that any k column vectors of the matrix A are uncorrelated linearly, which reveals sufﬁcient conditions for the application of the parallel factorization method and lays foundations for its applications in signal processing [14]. Zhang et al. applied PARAFAC decomposition for radar spatial-temporal signal processing to achieve the automatic angle and frequency matching [15]. Li et al. [16] used the parallel factor analysis to deal with the separation of multiple fault sources in the mechanical equipment and achieved the desired results. Sidiropoulos et al. used PARAFAC analysis for the recognition and identiﬁcation of multiple targets in MIMO radar systems [17]. Weis et al. used the PARAFAC algorithm in their EEG data analysis to determine the individual components of the correlation [18]. Yang et al. constructed tensor using wavelet transform and processed multi-dimensional fault signals with parallel factor theory to achieve effective classiﬁcation [19]. Genetic algorithm (GA) comes from the idea and mechanism of natural evolution as the optimal parallel search in the laws of biology. It is constructed by simulating the principle of “natural selection and survival of the ﬁttest” in the natural evolutionary process. GA provides a solution to complex nonlinear problems that are not easily solved by the traditional optimization methodology [20]. A genetic algorithm was proposed for combination with the support vector machine (SVM) to achieve the optimal algorithm for fault diagnosis of the rolling bearing machines [21]. The ICA algorithm was implemented for the feature extraction of the signal in the motor bearing, which is combined with GA to optimize the radial basis neural network for fault diagnosis. The diagnostic accuracy was signiﬁcantly improved [22]. Compared with the traditional neural network (NN), the optimized and improved NN has an optimal network structure and higher accuracy. This paper investigates the relevant theory relating to signal matrix decomposition and applies continuous wavelet transformation to multi-channel signal analysis to construct a three-dimensional tensor. The parallel factor decomposition achieves the characteristic information extraction of the complex systems, which determine the frequency range of the faulty centrifugal pump. The effective feature frequency information extraction is Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 3 of 23 This paper investigates the relevant theory relating to signal matrix decomposition and applies continuous wavelet transformation to multi-channel signal analysis to con- struct a three-dimensional tensor. The parallel factor decomposition achieves the charac- teristic information extraction of the complex systems, which determine the frequency range of the faulty centrifugal pump. The effective feature frequency information extrac- Int. J. Turbomach. Propuls. Power 2022, 7, 19 3 of 21 tion is combined with the excellent adaptive updating ability and nonlinear characteristics of BP-NN. The BP-NN model is established to diagnose the fault modes of the centrifugal pump. In order to overcome the disadvantage of the slow convergence of the BP-NN, the combined with the excellent adaptive updating ability and nonlinear characteristics of optimization method based on GA is proposed to optimize the BP neural network model BP-NN. The BP-NN model is established to diagnose the fault modes of the centrifugal so that it finds the appropriate weights and thresholds at a quicker rate and rapidly pump. In order to overcome the disadvantage of the slow convergence of the BP-NN, the achieves fault classification. optimization method based on GA is proposed to optimize the BP neural network model so that it ﬁnds the appropriate weights and thresholds at a quicker rate and rapidly achieves 2. Principle of Parallel Factor Analysis fault classiﬁcation. Tensor is the high-dimensional form of data construction. The dimensionality of the 2. Principle of Parallel Factor Analysis data is called the order of the tensor and is considered the generalization of the matrix and vector T in ensor the is hi the gh-high-dimensional dimensional spaform tial con of data struction. constr uction. Tradition The al dimensionality methods such ofas the ICA, data is called the order of the tensor and is considered the generalization of the matrix and PCA, etc. used for processing data with high dimensionality generally spread the data vector in the high-dimensional spatial construction. Traditional methods such as ICA, PCA, into a two-dimensional matrix for processing to remove the structural data. The solution etc. used for processing data with high dimensionality generally spread the data into a often fails to achieve the expected results. PARAFAC is a common decomposition treat- two-dimensional matrix for processing to remove the structural data. The solution often ment in tensor decomposition. The core idea is to approximate the original tensor data by fails to achieve the expected results. PARAFAC is a common decomposition treatment in the sum of finite rank-1 tensors. tensor decomposition. The core idea is to approximate the original tensor data by the sum of ﬁnite rank-1 tensors. 2.1. Parallel Factor Model 2.1. Parallel Factor Model Tensor is a high-dimensional extension of the matrix. The order of the tensor repre- sents the Tensor dimis ensions a high-dimensional of the tensor extensi as shown on of inthe Fig matrix. ure 1. T The he vec order tor of form theed tensor by the rep- one- resents the dimensions of the tensor as shown in Figure 1. The vector formed by the dimensional time series of the vibration signal collected by the single-channel sensor is one-dimensional time series of the vibration signal collected by the single-channel sensor is the 1st order tensor. The matrix is the 2nd order tensor. The multi-dimensional array the 1st order tensor. The matrix is the 2nd order tensor. The multi-dimensional array above above the three-dimensional level is the high-order tensor. the three-dimensional level is the high-order tensor. Figure 1. Tensor. Figure 1. Tensor. In the two-dimensional matrix, the variable x generally is applied to indicate the p,q components of the two-dimensional matrix that the subscript denotes the x-axis and the In the two-dimensional matrix, the variable generally is applied to indicate the p ,q subscript q denotes the y-axis during the x—y 2D coordinate system. The variable x p,q,k components of the two-dimensional matrix that the subscript denotes the x-axis and the indicates the element of the three-dimensional matrix that the subscript p denotes x—axis, subscr subscipt ript𝑞 q denotes denotes yt— he a x yi- sax an is d d su ur bsing criptthe q d e 𝑥 n－ ot𝑦 es 2D z coo axisrd du inate ring tsh ys e x tem. —y— Th z e 3D var coia orb dle in ate p ,q,k system. The 2-D array of the 3D matrix constitutes the subarray of the 3D matrix. The indicates the element of the three-dimensional matrix that the subscript 𝑝 denotes 𝑥 − subarray is labeled as the slice of the 3D matrix in the axis. The low-rank decomposition of axis, subscript 𝑞 denotes 𝑦 －axis and subscript 𝑞 denotes z − axis during the 𝑥 －𝑦 －z the matrix is extended to construct the 3D matrix. Let the variable x be the elements of the p,q,k 3D coordinate system. The 2-D array of the 3D matrix constitutes the subarray of the 3D PQK three-dimensional matrix X 2 C , where p = 1, , P; q = 1, , Q; k = 1, , K. matrix. The subarray is labeled as the slice of the 3D matrix in the axis. The low-rank de- Three-dimensional matrices can be represented as vector outer product as follows: composition of the matrix is extended to construct the 3D matrix. Let the variable x p ,q,k PQK be the elements of the three-dimensional matrix XC , where X = a b c +, . . . , +a b c = a b c (1) 1 1 1 R R R å r r r r=1 k = 1,, K p= 1,,P; q= 1,,Q; . Three-dimensional matrices can be repre- sented as vector ou P ter pro Q duct as fo K llows: where a 2 C b 2 C c 2 C r = 1, 2, . . . , R. Equation (1) provides the low order r r r decomposition process of the 3D matrix. The orders of the 3D matrices X are R. The model for the low-rank decomposition of the 3D matrix as shown in Equation (1) is Parallel Factor Model. Figure 2 shows the procedure of the PARAFAC model. Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 4 of 23 X = a b c + ,..,+ a b c = a b c (1) 1 1 1 R R R r r r r=1 P Q K where a C b C c C r = 1,2,..., R . Equation (1) provides the low order de- r r r composition process of the 3D matrix. The orders of the 3D matrices are . The X R model for the low-rank decomposition of the 3D matrix as shown in Equation (1) is Par- Int. J. Turbomach. Propuls. Power 2022, 7, 19 4 of 21 allel Factor Model. Figure 2 shows the procedure of the PARAFAC model. Figure 2. Procedure for Parallel Factor Decomposition. Figure 2. Procedure for Parallel Factor Decomposition. Here, the deﬁnitions of the three matrices are as follows: Here, the definitions of the three matrices are as follows: A = [a , . . . , a ] 1 R BA==[b[ a , . . .,. ,.b.,a ] ] (2) 1 R 1 R C = [c , . . . , c ] 1 R B= [b ,...,b ] (2) 1 R The symbols A, B, and C are the three loading arrays in the PARAFAC model. Equation (2) C= [ c ,...,c ] 1 R shows that the components in the 3D array X are decomposed as the sum of the multiplication of R components. The symbols A, B, and C are the three loading arrays in the PARAFAC model. Equa- tion (2) shows that the components in the 3D array X are decomposed as the sum of the 2.2. Uniqueness of Parallel Factor Decomposition multiplication of R components. For a two-dimensional matrix, when the rank of the matrix is greater than 1, the two- dimensional matrix’s low-rank decomposition is not unique if there are no special structural 2.2. Uniqueness of Parallel Factor Decomposition constraints. For the matrix decomposition process X = AB , there exists another set of For a two-dimensional matrix, when the rank of the matrix is greater than 1, the two- matrices A, B that is X = AB . However, A 6= AP D , B 6= BP D , Here, the symbols A A B B dimensional matrix’s low-rank decomposition is not unique if there are no special struc- P and P are column swap matrices and the symbols D and D are the diagonal scale A B A B tural matrices. constraints. The uniqueness For the mat of the rix two-dimensional decomposition pr matrix ocess de Xcomposition = AB , there is ex illustrated ists anotby her set FF the converse method. Given any full-rank approach T 2 C with A A B B of matrices , that is . However, , , Here, A B X = AB A A B B T 1 T the symbols and are column swap matrices and the symbols and are A B X = AB = ATT B = AB A (3)B the diagonal scale matrices. The uniqueness of the two-dimensional matrix decomposition Among them FF is illustrated by the converse method. Given any full-rank approach TC with A = AT = [a , . . . , a ] (4) 1 F h i T T −1 T 1 (3) X B == AB B(T )= A =TTb , . .B . , b = AB (5) 1 F Among them where a and b are the column vectors of the arrays A and B. If the arrays A, B are full f f rank, A and B are also full rank matrices, then we have A= AT =a ,...,a (4) 1 F T T T X = AB = a b + a b + + a b (6) 1 2 F 1 2 F −1 T B= B (T ) =b ,...,b (5) 1 F The above formula satisﬁes the deﬁnition of the low order decomposition. However, T 6= PD. Therefore, the 2D matrix low-rank decomposition is not unique. where a and b are the column vectors of the arrays A and B. If the arrays A, B are The fundamental difference between the parallel factorization and the 2D matrix full decomposition rank, A and isB the are uniqueness also full of rank its decomposition, matrices, then w which e h is ave one of the reasons that the PARAFAC model is widely used in data analysis. The uniqueness theorem of PARAFAC T T T decomposition comes from the new concept of the k-order. The k-order for a matrix A is (6) X = AB = a b + a b ++ a b 1 1 2 2 F F the maximum order of k and satisﬁes the condition that any k column vectors of the array A are linearly uncorrelated, which reveals the sufﬁcient conditions for the uniqueness of the parallel factorization method for application in data analysis. Consider the sub-proﬁle matrix of the PARAFAC model along the X-axis. QK X = BD (A)C p = 1, 2, . . . , P (7) P Int. J. Turbomach. Propuls. Power 2022, 7, 19 5 of 21 PR QR KR Here, the matrix is A 2 R , B 2 R , C 2 R , if the following inequality is satisﬁed k + k + k 2(R + 1) (8) A B C The matrices A, B, and C are unique. 3. Hybrid Method with PARAFAC_GA_BP_NN 3.1. Algorithm on PARAFAC 3.1.1. Nuclear Consistency Estimation The PARAFAC algorithm is very sensitive to the pre-estimated factor F. When the parameter F is estimated as too low, no physically meaningful solution is obtained. If the parameter F is estimated as too high, it leads to an increase in the model error and makes the deviation between the calibration values and the true values larger. Therefore, a suitable value for factor F is very important for constructing the PARAFAC model. It is necessary to pre-estimate the number of factors. Since the ranks of the tensors are obtained asymptotically, different methods are usually used to evaluate the decomposition factor number from several perspectives. Here, Core Consistency estimation is an effective methodology for the estimation of the factors by calculating the level of the similarity between the super-diagonal array T and the core 3D data array G in the PARAFAC model. The calculation of Core Consistency (d) is deﬁned as follows: 0 1 F F F g t å å å de f de f B C d=1 e=1 f =1 B C d = 100 1 (9) B C @ F A where the parameter F is the factor number in the PARAFAC model, the parameter g de f is the element of the matrix G, the parameter t and is the element of T. For the ideal de f PARAFAC model, the superdiagonal arrays T and G should be very similar, at which point the kernel agreement value equals 100%. Usually, when the kernel agreement value is equal to or more than 60%, the model is considered to be close to trilinearity. However, when the kernel agreement value is lower than 60%, the model is considered to deviate from trilinearity. A much more accurate factor number is obtained according to the change in the kernel agreement value. 3.1.2. Trilinear Alternating Least Squares (TALS) X is an arbitrary three-dimensional data set. The two-dimensional matrices pqk deﬁned as X (Q K), X (P K) and X (P Q) that the corresponding elements satisfy p q the following conditions. X (q, k) = X (p, k) = X (p, q) = X (10) p q k pqk Then, the three-dimensional matrix is described as the joint cubic equation along the three different dimensions. 8 9 X = Bdiag(A(p, :))C , p = 1, 2, . . . , I < = X = Cdiag(B(q, :))A , q = 1, 2, . . . , J (11) : ; X = Adiag(C(k, :))B , k = 1, 2, . . . , K k Int. J. Turbomach. Propuls. Power 2022, 7, 19 6 of 21 where the variables X X and X denotes the slice of the three-dimensional matrix X in p q the three directions P, Q and K. The symbol diag(A(k, :)) denotes the square matrix after the diagonalization of the kth row elements of the matrix A and so on from Equation (11). 2 3 2 3 2 3 Bdiag(A(1, :))C Bdiag(A(1, :)) X i=1 6 7 6 7 6 7 Cdiag(B(2, :))C Cdiag(B(2, :)) X i=2 6 7 6 7 6 7 T PQK 6 7 = 6 7C = 6 7 = X (12) . . . . . . 4 5 4 5 4 5 . . . Adiag(C(I, :))C Adiag(C(I, :)) X i=I 2 3 Bdiag(A(1, :)) 6 7 Cdiag(B(2, :)) 6 7 6 7 = AB (13) 4 5 Adiag(C(I, :)) Then, the PARAFAC model is expressed in the form of the Khatri-Rao product. PQK X = A(BC) QKP (14) X = B(CA) KPQ X = C(AB) The basic idea of the TALS method is to update one array at one step by initializing a matrix and updating the remaining matrices using the Least Mean Square (LMS) Method. This step is repeated until the algorithm converges. The hypothetical 3D dataset X with the dimensions P Q K is represented by a trilinear model in the following form. x = a b c + e p = 1 . . . P q = 1 . . . Q k = 1 . . . K (15) p,q,k å p, f q, f k, f pqk f =1 Here, the symbol F denotes the number of components, the symbol a is the pth com- p, f ponent of the vector a , the symbol b is the qth component in the vector b , the symbol f q, f f c is the kth component in the vector c . The symbol x (p = 1, . . . , P, q = 1, . . . , Q, k, f f p,q,k k = 1, . . . , K). P Q K forms the three-dimensional space of the data set X. The symbol e (p = 1, . . . , P, q = 1, . . . , Q, k = 1, . . . , K) is the error, which forms the 3D error set E on pqk the P Q K coordinate system. The symbol A = [a , a , . . . , a ] is defined as a P F matrix. 1 2 P B = [b , b , . . . , b ] is a Q F matrix. The symbol C = [c , c , . . . , c ] is a K F matrix. 2 2 1 Q 1 K Matrix A is calculated as: 2 3 2 3 X BdiagC(1, :) ...1 6 7 6 7 X BdiagC(2, :) ...2 6 7 6 7 6 7 = 6 7A + E (16) . . . . 4 5 4 5 . . X BdiagC(K, :) ...K Here, X = Bdiag(C(k, :))A + E , k = 1, 2, . . . , K, E is the error. ...k ...k k The least mean square estimate of the matrix A is determined by the following equation. 2 3 2 3 BdiagC(1, :) X ...1 6 7 6 7 BdiagC(2, :) X ...2 6 7 6 7 A = (17) 6 7 6 7 . . . . 4 5 4 5 . . BdiagC(K, :) X ...K Here, [ ] is the generalized inverse. Int. J. Turbomach. Propuls. Power 2022, 7, 19 7 of 21 The matrix B is determined as: 2 3 2 3 Y CdiagA(1, :) ...1 6 7 6 7 Y CdiagA(2, :) ...2 6 7 6 7 6 7 = 6 7B + E (18) . . . . 4 5 4 5 . . Y CdiagA(P, :) ...P Here, Y = Cdiag(A(p, :))B + E , p = 1, 2, . . . , P, E is the error. ...p ...p The least mean square estimate of the matrix B is deﬁned as: 2 3 2 3 CdiagA(1, :) Y ...1 6 7 6 7 CdiagA(2, :) Y ...2 6 7 6 7 B = (19) 6 7 6 7 . . . . 4 5 4 5 . . CdiagA(P, :) Y ...P The matrix C is determined as follows. 2 3 2 3 Z AdiagB(1, :) ...1 6 7 6 7 Z AdiagB(2, :) ...2 6 7 6 7 = C + E (20) 6 7 6 7 . . Q . . 4 5 4 5 . . Z AdiagB(Q, :) ...Q Here, Z = Adiag(B(q, :))C + E , q = 1, 2, . . . , Q, E is the error. ...q ...q Q The least mean square estimate of the parameter C is deﬁned as: 2 3 2 3 AdiagB(1, :) Z ...1 6 7 6 7 AdiagB(2, :) Z ...2 6 7 6 7 C = 6 7 6 7 (21) . . . . 4 5 4 5 . . AdiagB(Q, :) Z ...Q Loop (1) to (3) are repeated, and the matrix is updated until convergence. 3.1.3. Algorithm Implementation of Parallel Factor Analysis PQK Each element X of the tensor X consists of a trilinear component model pqk as follows: x = a b c + e (22) pqk å p f q f k f pqk f =1 In signal processing, the parameter F contributes to the transient response signal, the variable a is the value of the f component related to the pth sample information, p f the variable b is the response value of the f th component related to the qth sample q f information, the variable c is the value of the f th component related to the kth sample k f information. The variables a , b and c are the components of the array A, B and C. The p f q f k f variable e is the measurement error. The above equation is in the form of the PARAFAC pqk model. It can be expressed in terms of three slice matrices that the trilinear model is expressed as in the following form, which is similar to the singular value decomposition in PCA. X (Q K) = Bdiag(a )C + E (J K), p = 1, 2, . . . , P p... p p... X (K P) = Cdiag(b )A + E (K I), q = 1, 2, . . . , Q (23) q... q q... X (P Q) = Adiag(c )B + E (P Q), k = 1, 2, . . . , K k... k k... Here, the parameters a , b and c are the pth row of the array A, the qth row of the p q k array B and the kth row of the array C. The symbols diag(a ), diag(b ) and diag(c ) are p q diagonal vectors of the F F matrix. The parameters a , b and c are the elements of i j k the diagonal vectors. The symbol “T” denotes the transpose of the matrix. The variables Int. J. Turbomach. Propuls. Power 2022, 7, 19 8 of 21 E (Q K), E (K P) and E (P Q) are three slices of the error array. Equation (22) p... q... k... is expressed as a matrix. (FFF) X = AT (C B) + E (24) (FFF) Here, the symbol is the Kronecker product, the array T is a two-dimensional matrix of the recombination of the core 3D data frame T. The variable T is a unit diagonal 3D-data array (also called a super diagonal array) with the matrix size (F F F) where the super diagonal element equals 1 and the remaining elements are zero. In the standard PARAFAC model, the sum of squared residuals (SSR) is the minimiza- tion of the loss function, which is deﬁned as: Q Q P K F P K SSR = x a b c = e (25) å å å pqk å p f q f k f å å å pqk p=1 q=1 k=1 f =1 p=1 q=1 k=1 PARAFAC decomposition can be implemented using Alternate Least Squares (ALS) with the following iterative process. Determining the number of the components F. Initialize arrays B and C Solve matrix A. h i T + T Solving the estimate a = diag B X (C ) p = 1, . . . , P of matrix A, which p... means the vector diag() obtains the elements on the main diagonal of the matrix. The + T T superscript “+” indicates the generalized inverse, B = (B B) B . The arrays B and C are estimated by the following equations. h i T + T b = diag C X (A ) , q = 1, . . . , Q (26) q... h i T + T c = diag A X (B ) , k = 1, . . . , K (27) k... Then, (3) and (4) are repeated until the SSR is less than the threshold, which is set by default as 1 10 . Based on the unique multi-decomposition in the PARAFAC model, the sub-arrays A, B and C are obtained, which represent the sample information, the response process information and sensing information. 3.2. Algorithm on GA GA is an evolutionary heuristic algorithm, which was developed from Darwin’s natural selection and biological evolution of genetics in 1975. It was originally created to handle large scale and complex optimization problems that could not be solved effectively by classical mathematical methods. The idea of GA is as follows: In a random initialized set, individuals are selected according to their ﬁtness size, and then crossover and mutation by genetics produce new sets that are better than the previous one and also relatively closer to the global optimal solution. When GA is used to solve a problem, the objective function and variables of the problem are determined ﬁrstly and the variables are encoded. The solution to the problem is represented by the strings of numbers in GA. The genetic operator operates directly on the strings. The encoding method is divided into binary encoding and real encoding. If the individual is represented by the binary encoding, the decoding formula for converting binary numbers to decimal numbers is deﬁned as: T R i i j1 F(x , x , . . . , x ) = R + x 2 (28) i1 i2 il å ij 2 1 j=1 Int. J. Turbomach. Propuls. Power 2022, 7, 19 9 of 21 Here, the parameters x , x , . . . , x are the ith string. The length of each string is l. Each i1 i2 il parameter is 0 or 1. The parameters T and R are the two endpoints of the ith string X . i i i The fundamental procedure of GA consists of selection, crossover and mutation operations. The new population is chosen from the old population with the probability threshold, which is determined by the ﬁtness values. The principle is that the better the ﬁtness value of an individual the higher the probability of a new population. The crossover operation consists of exchanging and combining two chromosomes to produce a new Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 10 of 23 superior individual. The mutation is to select any individual from the population and a point in the chromosome is chosen to b mutated to produce a better individual. In this paper, GA is used to optimize BP to improve the classiﬁcation diagnosis of centrifugal (2) Calculate the population fitness values from which the optimal individuals are iden- pumps. The basic implementation process is as follows: tified. (1) Random initialization of populations. (3) Select the chromosomes. (2) Calculate the population fitness values from which the optimal individuals are identified. (4) Crossover chromosomes. (3) Select the chromosomes. (5) mutation of chromosomes. (4) Crossover chromosomes. (6) Determine if the evolution is finished, if not, return to step 2. (5) mutation of chromosomes. (6) Determine if the evolution is ﬁnished, if not, return to step 2. 3.3. Principle on BP_NN 3.3. Principle on BP_NN Back Propagation is the multilayer feed-forward NN, which is trained according to the error. It has the broadest applications among NN at present. BP-NN is typical of the Back Propagation is the multilayer feed-forward NN, which is trained according forward network and has more than three layers without feedback. There is no intercon- to the error. It has the broadest applications among NN at present. BP-NN is typical nection within layers. Its structure is shown in Figure 3. The structure shows that the BP- of the forward network and has more than three layers without feedback. There is no NN neural network can realize the mapping from an n-dimensional input matrix to an m- interconnection within layers. Its structure is shown in Figure 3. The structure shows that dimensional output matrix by connecting the updated weight and threshold. In general, the BP-NN neural network can realize the mapping from an n-dimensional input matrix BP_NN uses the Sigmoid function or linear function as the transfer function. to an m-dimensional output matrix by connecting the updated weight and threshold. In general, BP_NN uses the Sigmoid function or linear function as the transfer function. f ( x )= (29) −x 1+e f (x) = (29) 1 + e Figure 3. Structure of BP_NN. Figure 3. Structure of BP_NN. In the BP_NN model, the node number of the hidden layer has a great inﬂuence on In the BP_NN model, the node number of the hidden layer has a great influence on diagnostic accuracy. A smaller number of nodes reduces the ability of the net to learn, diagnostic accuracy. A smaller number of nodes reduces the ability of the net to learn, which required an increase in the number of training cycles. Too many nodes makes which required an increase in the number of training cycles. Too many nodes makes the the training time longer, meaning that overﬁtting can easily occur. Reference [23] points training time longer, meaning that overfitting can easily occur. Reference [23] points out out that the optimal number of hidden layer nodes must exist. For the exploration of that the optimal number of hidden layer nodes must exist. For the exploration of this num- this number of nodes, many scholars have given various solutions [24–26], including ber of nodes, many scholars have given various solutions [24–26], including the use of the the use of the experimental method, the introduction of the hyperplane, dynamic full experimental method, the introduction of the hyperplane, dynamic full parameter self- parameter self-adjustment and so on. A series of empirical formulas are obtained. After adjustment and so on. A series of empirical formulas are obtained. After the summary, the optimal number of hidden layer nodes can be obtained. Refer to the following formula [24–26]: l m+ n+ a (30) l= log Here, the parameter n is the number of nodes on the input level, the variable 𝑙 is the number of nodes on the intermediate level, the variable m is the number of nodes on the output level and the variable a is a constant between 0 and 10. In the paper, the input nodes (n) equal 8, the output nodes (m) equal 4 and the nodes of intermediate level are set to be 3. Int. J. Turbomach. Propuls. Power 2022, 7, 19 10 of 21 the summary, the optimal number of hidden layer nodes can be obtained. Refer to the following formula [24–26]: l < m + n + a (30) l = log Here, the parameter n is the number of nodes on the input level, the variable l is the number of nodes on the intermediate level, the variable m is the number of nodes on the Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 11 of 23 output level and the variable a is a constant between 0 and 10. In the paper, the input nodes (n) equal 8, the output nodes (m) equal 4 and the nodes of intermediate level are set to be 3. 4. Experimental System of Centrifugal Pump 4. Experimental System of Centrifugal Pump The industrial experimental system of the slurry pump is shown in Figure 4. The The industrial experimental system of the slurry pump is shown in Figure 4. The model for the centrifugal pump in the experiment is Weir/Warman 3/2 CAH with a closed model for the centrifugal pump in the experiment is Weir/Warman 3/2 CAH with a closed impeller that is C2147. The diameter of the impeller is 8.5 inches. The centrifugal pump is impeller that is C2147. The diameter of the impeller is 8.5 inches. The centrifugal pump is driven by the motor. There is a V-belt drive between the motor and the centrifugal pump driven by the motor. There is a V-belt drive between the motor and the centrifugal pump with a transmission ratio of 13/6. The parameters of the motor are shown in Table 1. with a transmission ratio of 13/6. The parameters of the motor are shown in Table 1. Figure 4. Centrifugal pump experimental system. Figure 4. Centrifugal pump experimental system. T Table able 1. 1.Motor Motorparameters. parameters. Rated Rated Maximum Rated Rated Rated Maximum Rated Rated Rated Ambient Overload Motor Ambient Overload Motor Model Speed Speed Power Voltage (V) Temperature Factor Size Model Voltage Speed Speed Power (RPM) (RPM) (HP) Temperature Factor Size ( C) (V) (RPM) (RPM) (HP) (℃) 230/460 1200 1180 40 40 1.15 362 T 230/460 1200 1180 40 40 1.15 362 T The vibration signal acquisition system is shown in Figure 5, which mainly consists of The vibration signal acquisition system is shown in Figure 5, which mainly consists a signal analyzer and a laptop computer for storing data. The system acquires multiple of a signal analyzer and a laptop computer for storing data. The system acquires multiple channel signals including 3-axis vibration, acoustics, ﬂow, pressure and temperature. The channel signals including 3-axis vibration, acoustics, flow, pressure and temperature. The following conditions are satisﬁed for the acquisition of the experimental data. following conditions are satisfied for the acquisition of the experimental data. (1) Data collection does not begin until the centrifugal pump is running smoothly. (1) Data collection does not begin until the centrifugal pump is running smoothly. (2) The sampling frequency satisﬁes the sampling theorem. (2) The sampling frequency satisfies the sampling theorem. (3) Multiple sets of data are collected for experiments conducted in each state. (3) Multiple sets of data are collected for experiments conducted in each state. In order to collect nonlinear multi-fault-mode characteristic signals, when the cen- trifugal pump is running steadily, the motor speed is set to be 1200 rpm for data acquisi- tion. The data acquisition time of each group is 20 s. The sampling frequency is 9 kHz. The system synchronously collects online data on the vibration, acoustics, flow, pressure, etc. The nonlinear operation state of the machinery during the industrial process is simulated by controlling the flow rate and pressure of flow during the processing circuit, which con- sists of the nonlinear and nonstationary multi-failure mode. Int. Int. JJ.. T Turbomach. urbomach. Pr Prop opuls. uls. P Power ower 2022 2022,, 7 7,, 19 x FOR PEER REVIEW 12 11 of of 21 23 Figure 5. Data acquisition system. Figure 5. Data acquisition system. In order to collect nonlinear multi-fault-mode characteristic signals, when the centrifu- 5. Simulated Signal for PARAFAC Analysis gal pump is running steadily, the motor speed is set to be 1200 rpm for data acquisition. The Considering that the vibration signals acquired in the condition monitoring of me- data acquisition time of each group is 20 s. The sampling frequency is 9 kHz. The system chanical equipment in the practical industrial environment generally were corrupted by synchronously collects online data on the vibration, acoustics, ﬂow, pressure, etc. The the heavy noise signals, the typical numerical signal is generated to simulate the charac- nonlinear operation state of the machinery during the industrial process is simulated by teristic vibration information in the fault diagnosis of a mechanical system by using Equa- controlling the ﬂow rate and pressure of ﬂow during the processing circuit, which consists tion (31), which is used to assess the effectiveness of the proposed method based on PAFARAF of the nonlinear and nonstationary multi-failure mode. and continuous wavelet transform (CWT). The simulated signal consists of impulse signals when the fault occurs in the equipment and Gaussian White Noise (GWN) with 1 dB sig- 5. Simulated Signal for PARAFAC Analysis nal-to-noise ratio (SNR). Considering that the vibration signals acquired in the condition monitoring of me- x(t)= s(t)+ n(t) chanical equipment in the practical industrial environment generally were corrupted by the −700( t−i / f ) (31) s(t)= (1+ 0.2cos( 2* pi * f t))e cos( 2* pi * f (t−i / f )) heavy noise signals, the typical numerical signal is generated to simulate the characteristic r n i vibration information in the fault diagnosis of a mechanical system by using Equation (31), which is used to assess the effectiveness of the proposed method based on PAFARAF and Here, the function s(t) is the periodic shock signal, the symbols f and f are r i continuous wavelet transform (CWT). The simulated signal consists of impulse signals the rotation frequency and faulty frequency, which are 30 Hz and 200 Hz. The inherent when the fault occurs in the equipment and Gaussian White Noise (GWN) with 1 dB f ( ) frequency is 2000 Hz. The symbol n t denotes the noise signal. The faulty signal signal-to-noise ratio (SNR). is simulated to consist of the rotational frequency, faulty frequency and intrinsic fre- x(t) = s(t) + n(t) quency with noise corruption. The sampling frequency and analysis points are set as fol- 700(ti/ f ) (31) s(t) = (1 + 0.2 cos(2 pi f t))e cos(2 pi f (t i/ f )) lows: f =12000 å , N= 8192 . The tirme and frequency domains n of the simulated sig- nal are shown in Figure 6, and it can be found that the fault characteristics are correlated Here, the function s(t) is the periodic shock signal, the symbols f and f are the with both the inherent frequency of the system and the rotation frequency of the motor r i rotation frequency and faulty frequency, which are 30 Hz and 200 Hz. The inherent shaft. It can be seen that the frequency components related to the fault characteristics in- frequency f is 2000 Hz. The symbol n(t) denotes the noise signal. The faulty signal is clude harmonic frequencies 2 f , modulation frequencies f −nf , and other frequen- i n i simulated to consist of the rotational frequency, faulty frequency and intrinsic frequency cies. The key point for accurate fault identification is to extract the useful frequencies re- with noise corruption. The sampling frequency and analysis points are set as follows: lated to the faulty characteristics from the original noise signal. f = 12, 000, N = 8192. The time and frequency domains of the simulated signal are shown in Figure 6, and it can be found that the fault characteristics are correlated with both the inherent frequency of the system and the rotation frequency of the motor shaft. It can be seen that the frequency components related to the fault characteristics include harmonic frequencies 2 f , modulation frequencies f n f , and other frequencies. The key point i n i for accurate fault identiﬁcation is to extract the useful frequencies related to the faulty characteristics from the original noise signal. Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 13 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 13 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, 19 12 of 21 Figure 6. Corrupted simulation signal with noise. Figure 6. Corrupted simulation signal with noise. Figure 6. Corrupted simulation signal with noise. Although the frequency domain of the simulated signal in Figure 6 presents the fre- quency information related to the fault characteristics, it is buried by heavy noise and Although the frequency domain of the simulated signal in Figure 6 presents the Although the frequency domain of the simulated signal in Figure 6 presents the fre- inherent frequency-related components. The failure characteristics are buried and located frequency information related to the fault characteristics, it is buried by heavy noise quency information related to the fault characteristics, it is buried by heavy noise and and in heavy inherno ent ise. frequency-r It is necess elated ary for components. the corrupteThe d data failur to be e characteristics processed to ex arte ract buried the fand ault inherent frequency-related components. The failure characteristics are buried and located located characte in rist heavy ic frequenc noise. ies It is ac necessary curately. C for WT the is corr used upted to an data alyze to tbe he pr simula ocessed tedto imp extract ulse s the ig- in heavy noise. It is necessary for the corrupted data to be processed to extract the fault fault nal. Th characteristic e wavelet bas fris equencies function accurately is “comr3− .3CWT ”. The iscen used ter to freanalyze quency o the f the simulated wavelet f impulse unction characteristic frequencies accurately. CWT is used to analyze the simulated impulse sig- signal. is 3 Hz. The Figur wavelet e 7 shows basis the function CWT of is th“comr3 e simulated 3”. si The gnal. center Howev frequency er, the fault of the -relat wavelet ed fre- nal. The wavelet basis function is “comr3−3”. The center frequency of the wavelet function function quency com is 3p Hz. onen Figur ts are e 7 not shows filtered the ou CWT t, which of the ind simulated icates thsignal. at the tHowever raditional , the time fault- -fre- is 3 Hz. Figure 7 shows the CWT of the simulated signal. However, the fault-related fre- related frequency components are not ﬁltered out, which indicates that the traditional time- quency transformation is not effective enough to extract the weak fault characteristics of quency components are not filtered out, which indicates that the traditional time-fre- frequency transformation is not effective enough to extract the weak fault characteristics of the frequency components from the simulated complex noised signal. quency transformation is not effective enough to extract the weak fault characteristics of the frequency components from the simulated complex noised signal. the frequency components from the simulated complex noised signal. Figure 7. Wavelet transform of the simulated signal. Figure 7. Wavelet transform of the simulated signal. Figure 7. Wavelet transform of the simulated signal. PARAFAC is a tensor decomposition algorithm and the decomposition is unique. In PARAFAC is a tensor decomposition algorithm and the decomposition is unique. In the case that the tensor models the N-dimensional relationship well, the parallel factor the case that the tensor models the N-dimensional relationship well, the parallel factor PARAFAC is a tensor decomposition algorithm and the decomposition is unique. In decomposition retains the original characteristic signal to a large extent while the feature decomposition retains the original characteristic signal to a large extent while the feature the case that the tensor models the N-dimensional relationship well, the parallel factor caused by the failure component of the mechanical system is extracted effectively from the caused by the failure component of the mechanical system is extracted effectively from decomposition retains the original characteristic signal to a large extent while the feature original complex system information. Based on the advantage of PARAFAC, the wavelet the original complex system information. Based on the advantage of PARAFAC, the caused by the failure component of the mechanical system is extracted effectively from coefﬁcients of the simulated signal after continuous wavelet transform are obtained, which wavelet coefficients of the simulated signal after continuous wavelet transform are ob- the original complex system information. Based on the advantage of PARAFAC, the is applied to construct one 3rd-order tensor with the dimension 1 200 8192. The tensor tained, which is applied to construct one 3rd-order tensor with the dimension 1 * 200 * wavelet coefficients of the simulated signal after continuous wavelet transform are ob- is decomposed by the parallel factor analysis to extract multiple factor components, which tained, which is applied to construct one 3rd-order tensor with the dimension 1 * 200 * Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 14 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, 19 13 of 21 8192. The tensor is decomposed by the parallel factor analysis to extract multiple factor components, which contain the channel, time and frequency information of the high-di- mensional original signal. To build a correct parallel factor model, it is necessary to select contain the channel, time and frequency information of the high-dimensional original the appropriate factor group fraction. The simulated signal determines the factor number signal. To build a correct parallel factor model, it is necessary to select the appropriate F by considering the cross-validation and the kernel consistency method proposed in Sec- factor group fraction. The simulated signal determines the factor number F by considering tion 3. the cross-validation and the kernel consistency method proposed in Section 3. Figure 8 shows the cross-validation of the simulated signal. When the number of fac- Figure 8 shows the cross-validation of the simulated signal. When the number of tors F is set to be from 1 to 3, the parallel factor cross-validation of the simulation signal is factors F is set to be from 1 to 3, the parallel factor cross-validation of the simulation signal better in both the fitting group and the validation group. The values of the explanatory is better in both the ﬁtting group and the validation group. The values of the explanatory variables reach more than 80% and the kernel consistency of the parallel factor model variables reach more than 80% and the kernel consistency of the parallel factor model reaches 100%. In summary, it is considered that the number of factors F is chosen as 3 to reaches 100%. In summary, it is considered that the number of factors F is chosen as 3 to establish the parallel factor model for the tensor, which is constructed by the simulation establish the parallel factor model for the tensor, which is constructed by the simulation signal for the data analysis. signal for the data analysis. Figure 8. Cross-validation Figure 8. Crfor osssimulated -validation for signal. simulated signal. Figure 9 shows the three subspaces which are obtained after the parallel factor decom- Figure 9 shows the three subspaces which are obtained after the parallel factor de- position of the simulated fault signal with noise addition. The loading values correspond composition of the simulated fault signal with noise addition. The loading values corre- to the channel, time and frequency information of the original signal. The residual values spond to the channel, time and frequency information of the original signal. The residual of the model ﬁtting are obtained. The simulated signal is decomposed by PARAFAC values of the model fitting are obtained. The simulated signal is decomposed by PARA- into a frequency matrix, time matrix and time–frequency information. The amplitudes FAC into a frequency matrix, time matrix and time–frequency information. The ampli- corresponding to the simulated impulse signal in the frequency matrix have obvious peaks tudes corresponding to the simulated impulse signal in the frequency matrix have obvi- at frequencies of 2000 Hz and 0~100 Hz, which shows the disadvantage that the low- ous peaks at frequencies of 2000 Hz and 0~100 Hz, which shows the disadvantage that the frequency characteristics associated with the fault component are not clearly extracted. The low-frequency characteristics associated with the fault component are not clearly ex- time–information matrix obtained after the decomposition of the parallel factor is analyzed tracted. The time–information matrix obtained after the decomposition of the parallel fac- with the power spectrogram as shown in Figure 10. The comparison between Figure 7 and tor is analyzed with the power spectrogram as shown in Figure 10. The comparison be- the results in Figures 9 and 10 veriﬁes that the PARAFAC algorithm has a great advantage in tween Figure 7 and the results in Figures 9 and 10 verifies that the PARAFAC algorithm a more accurate and efﬁcient form of feature extraction of the complex corrupted vibration has a great advantage in a more accurate and efficient form of feature extraction of the signals in fault diagnosis as compared to the traditional time–frequency domain signal complex corrupted vibration signals in fault diagnosis as compared to the traditional processing methods. time–frequency domain signal processing methods. Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 15 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 15 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, 19 14 of 21 Figure 9. Parallel factor decomposition of simulated signal. Figure 9. Parallel factor decomposition of simulated signal. Figure 9. Parallel factor decomposition of simulated signal. Figure 10. Figure Power 10. spectr Powum er spect of decomposed rum of decomp time os domain ed time loading domain matrix loading with matrix wi PARAF th P AC. ARAFAC. Figure 10. Power spectrum of decomposed time domain loading matrix with PARAFAC. 6. Discussion 6. Discussion 6. Discussion Based on the simulated signal analysis as shown in Figures 9 and 10, it has been veriﬁed Based on the simulated signal analysis as shown in Figures 9 and 10, it has been ver- that the time and frequency feature matrices can accurately characterize the fault model ified that the time and frequency feature matrices can accurately characterize the fault Based on the simulated signal analysis as shown in Figures 9 and 10, it has been ver- information. The multiple dimensional data model can be constructed by containing the model information. The multiple dimensional data model can be constructed by contain- ified that the time and frequency feature matrices can accurately characterize the fault acquired data from the accelerometer, ﬂow sensor and pressure sensor, which is analyzed ing the acquired data from the accelerometer, flow sensor and pressure sensor, which is model information. The multiple dimensional data model can be constructed by contain- and processed by the PARAFAC algorithm. The time and frequency loading matrices analyzed and processed by the PARAFAC algorithm. The time and frequency loading ing the acquired data from the accelerometer, flow sensor and pressure sensor, which is are extracted as the characteristic signals. The forty sets of data are collected from the matrices are extracted as the characteristic signals. The forty sets of data are collected from analyzed and processed by the PARAFAC algorithm. The time and frequency loading centrifugal pump system under one of the four running states that are normal (F1), impeller the centrifugal pump system under one of the four running states that are normal (F1), matrices are extracted as the characteristic signals. The forty sets of data are collected from blade damage (F2), impeller edge damage (F3) and impeller perforation damage (F4), which impeller blade damage (F2), impeller edge damage (F3) and impeller perforation damage the centrifugal pump system under one of the four running states that are normal (F1), are used to analyze the operation status of the centrifugal pump for the nonlinear multiple (F4), which are used to analyze the operation status of the centrifugal pump for the non- impeller blade damage (F2), impeller edge damage (F3) and impeller perforation damage fault diagnosis. linear multiple fault diagnosis. (F4), which are used to analyze the operation status of the centrifugal pump for the non- Based on Nyquist’s sampling theorem, the maximum frequency of the signal spectrum Based on Nyquist’s sampling theorem, the maximum frequency of the signal spec- linear multiple fault diagnosis. is half of the sample frequency of 4500 Hz. The time for data acquisition in each mode of trum is half of the sample frequency of 4500 Hz. The time for data acquisition in each Based on Nyquist’s sampling theorem, the maximum frequency of the signal spec- the impeller in the experiment is 20 s with eighteen thousand data points. A reduction in mode of the impeller in the experiment is 20 s with eighteen thousand data points. A re- trum the complexity is half of tof he data sampl processing e frequency for aof better 4500 comparison Hz. The time is requir for da ed, ta and acquis theiti pr on oposed in each duction in the complexity of data processing for a better comparison is required, and the mo PARAF de of AC thealgorithm impeller as in described the experiment in Section is 23 0 is s wi used th dir eight ectl een y to thou obtain sand data da points ta points for .the A re- proposed PARAFAC algorithm as described in Section 3 is used directly to obtain data four failure modes for feature extraction. duction in the complexity of data processing for a better comparison is required, and the points for the four failure modes for feature extraction. PARAFAC was used to process the test data. We considered the vibration signals, proposed PARAFAC algorithm as described in Section 3 is used directly to obtain data PARAFAC was used to process the test data. We considered the vibration signals, ﬂow signals and pressure signals from the multiple measurement points collected in the points for the four failure modes for feature extraction. flow signals and pressure signals from the multiple measurement points collected in the above experimental system for a total of the ﬁfteen channel signals. The purpose of choos- PARAFAC was used to process the test data. We considered the vibration signals, ing 15 data channels is that the 15 physical system variables are sufﬁcient as systematic flow signals and pressure signals from the multiple measurement points collected in the Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 16 of 23 above experimental system for a total of the fifteen channel signals. The purpose of choos- ing 15 data channels is that the 15 physical system variables are sufficient as systematic Int. J. Turbomach. Propuls. Power 2022, 7, 19 15 of 21 characteristics to control the nonstationary operation status. The operating status of the centrifugal pump is evaluated comprehensively from the multiple physical information facets, which makes the fault diagnosis of the centrifugal pump more reasonable and ef- characteristics to control the nonstationary operation status. The operating status of the cen- fective. trifugal pump is evaluated comprehensively from the multiple physical information facets, The number of factors of the PARAFAC model can be determined by choosing the which makes the fault diagnosis of the centrifugal pump more reasonable and effective. kernel consistent diagnosis method in Section 3. The number of factors ranges from 1 to The number of factors of the PARAFAC model can be determined by choosing the kernel 8. There are three groups of data to test the factors. Figure 10 shows nuclear consistency consistent diagnosis method in Section 3. The number of factors ranges from 1 to 8. There are estimation. three groups of data to test the factors. Figure 10 shows nuclear consistency estimation. As shown in Figure 11, when the number of factors is from 1 to 5, the kernel con- As shown in Figure 11, when the number of factors is from 1 to 5, the kernel consistency sistency values are above 60%. When the number of factors is greater than 5, the kernel values are above 60%. When the number of factors is greater than 5, the kernel consistency consistency values decrease rapidly by 60%. Therefore, the amount of factors in the PAR- values decrease rapidly by 60%. Therefore, the amount of factors in the PARFAC model FAC model is chosen to be 5. The PARAFAC algorithm is solved by the trilinear alternat- is chosen to be 5. The PARAFAC algorithm is solved by the trilinear alternating least ing least squares method. Figure 12 shows the signal analysis by PARAFAC used to obtain squares method. Figure 12 shows the signal analysis by PARAFAC used to obtain the ﬁve the five components in mode 2 under four operating states when the angular speed is 1200 components in mode 2 under four operating states when the angular speed is 1200 rpm. rpm. Mode 2 provides the frequency information. Figure 13 shows the signal analysis by Mode 2 provides the frequency information. Figure 13 shows the signal analysis by PARAFAC used to obtain the five components in mode 3 under four operating states PARAFAC used to obtain the ﬁve components in mode 3 under four operating states when the angular speed whenis the 1200 ang rpm. ular Model speed 3 ispr 1200 ovides rpm the . Mo time del 3 domain provide information. s the time domain information. Figure 11. Nuclear consistency estimation. Figure 11. Nuclear consistency estimation. The multi-channel complex signals are obtained from the centrifugal pump, which are analyzed by parallel factor decomposition to obtain the time and frequency information matrices. The time matrices as shown in Figure 13 are analyzed by Discrete Fourier Transform (DFT) to obtain the frequency domain information. DFT is deﬁned as follows: 2pjnk N 1 S(k) = x(kDtz)e , (n = 1, 2, . . . , N 1) (32) k = 0 Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 17 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 17 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, 19 16 of 21 Figure 12. The 1200 rpm centrifugal pump signal decomposition in Mode 2: (F1) for normal impeller, (F2) for blade damage, (F3) for impeller edge damage, (F4) for impeller perforation damage. Figure 12. The 1200 rpm centrifugal pump signal decomposition in Mode 2: (F1) for normal impeller, Figure 12. The 1200 rpm centrifugal pump signal decomposition in Mode 2: (F1) for normal impeller, (F2) for blade damage, (F3) for impeller edge damage, (F4) for impeller perforation damage. (F2) for blade damage, (F3) for impeller edge damage, (F4) for impeller perforation damage. Figure 13. 1200 rpm centrifugal pump signal decomposition in Mode3: (F1) for normal impeller, Figure 13. 1200 rpm centrifugal pump signal decomposition in Mode3: (F1) for normal impeller, (F2) for blade damage, (F3) for impeller edge damage, (F4) for impeller perforation damage. (F2) for blade damage, (F3) for impeller edge damage, (F4) for impeller perforation damage. Figure 13. 1200 rpm centrifugal pump signal decomposition in Mode3: (F1) for normal impeller, Figure 14 shows the spectra frequency of the fourth component under F1 and F2. The (F2) for blade damage, (F3) for impeller edge damage, (F4) for impeller perforation damage. The multi-channel complex signals are obtained from the centrifugal pump, which characteristic frequency under F2 is 250 Hz. Figure 15 shows the spectra frequency of the are analyzed by parallel factor decomposition to obtain the time and frequency infor- ﬁfth component under F1 and F3. The characteristic frequency under F3 is 184 Hz. Figure 16 The multi-channel complex signals are obtained from the centrifugal pump, which mation matrices. The time matrices as shown in Figure 13 are analyzed by Discrete Fourier shows the spectra frequency of the third component under F1 and F4. The characteristic are analyzed by parallel factor decomposition to obtain the time and frequency infor- Transform (DFT) to obtain the frequency domain information. DFT is defined as follows: frequency under F4 is 20 Hz. The rotation speed of the motor in this experiment was set to mation matrices. The time matrices as shown in Figure 13 are analyzed by Discrete Fourier 1200 rpm and the rotation frequency was 20 Hz. It is known that the fault characteristic Transform (DFT) to obtain the frequency domain information. DFT is defined as follows: frequency of the centrifugal pump impeller is generally related to the frequency component of the rotation frequency. The frequency of the impeller blade failure is expressed by the blade passing frequency, which is calculated by multiplying the rotation frequency by the number of blades, which was 20 10 = 200 Hz in this paper. Regarding the blade damage Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 18 of 23 −2jnk N −1 S (k )= x(ktz )e , (n= 1,2,, N −1) (32) k = 0 Figure 14 shows the spectra frequency of the fourth component under F1 and F2. The characteristic frequency under F2 is 250 Hz. Figure 15 shows the spectra frequency of the fifth component under F1 and F3. The characteristic frequency under F3 is 184 Hz. Figure 16 shows the spectra frequency of the third component under F1 and F4. The characteristic frequency under F4 is 20 Hz. The rotation speed of the motor in this experiment was set to 1200 rpm and the rotation frequency was 20 Hz. It is known that the fault characteristic frequency of the centrifugal pump impeller is generally related to the frequency compo- nent of the rotation frequency. The frequency of the impeller blade failure is expressed by Int. J. Turbomach. Propuls. Power 2022, 7, 19 17 of 21 the blade passing frequency, which is calculated by multiplying the rotation frequency by the number of blades, which was 20 × 10 = 200 Hz in this paper. Regarding the blade damage and impeller edge damage mode, the characteristic frequency is approximately and impeller edge damage mode, the characteristic frequency is approximately distributed distributed around 200 Hz. The characteristic frequency of the impeller perforation dam- around 200 Hz. The characteristic frequency of the impeller perforation damage is 12 Hz, age is 12 Hz, which is about 1/2 of the rotation frequency. Based on the above analysis, it which is about 1/2 of the rotation frequency. Based on the above analysis, it has been has been verified that the parallel factor algorithm is more effective for the characteristic veriﬁed that the parallel factor algorithm is more effective for the characteristic processing processing of the centrifugal pump multidimensional signal. of the centrifugal pump multidimensional signal. Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 19 of 23 Figure 14. Spectra analysis of fourth component under F1 and F2. Figure 14. Spectra analysis of fourth component under F1 and F2. Figure 15. Spectra analysis of ﬁfth component under F1 and F3. Figure 15. Spectra analysis of fifth component under F1 and F3. The time–frequency features extracted from the multi-source signals by PARAFAC decomposition are inputted to the BP model as features. The classiﬁcation accuracy of the model was calculated. The output of the BP model and the corresponding state of the centrifugal pump are shown in Table 2. Figure 16. Spectra analysis of third component under F1 and F4. The time–frequency features extracted from the multi-source signals by PARAFAC decomposition are inputted to the BP model as features. The classification accuracy of the model was calculated. The output of the BP model and the corresponding state of the centrifugal pump are shown in Table 2. Table 2. Expected output of neural network corresponding to each state of centrifugal pump. Output Label 1 2 3 4 F1 1 0 0 0 F2 0 1 0 0 F3 0 0 1 0 F4 0 0 0 1 Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 19 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, 19 18 of 21 Figure 15. Spectra analysis of fifth component under F1 and F3. Figure 16. Spectra analysis of third component under F1 and F4. Figure 16. Spectra analysis of third component under F1 and F4. Table 2. Expected output of neural network corresponding to each state of centrifugal pump. The time–frequency features extracted from the multi-source signals by PARAFAC Output Label 1 2 3 4 decomposition are inputted to the BP model as features. The classification accuracy of the F1 model was calcula1 ted. The outpu0 t of the BP mode 0 l and the corr 0esponding state of the F2 0 1 0 0 centrifugal pump are shown in Table 2. F3 0 0 1 0 F4 0 0 0 1 Table 2. Expected output of neural network corresponding to each state of centrifugal pump. Output Label 1 2 3 4 The eight statistics of each component constitute the feature vectors for subsequent F1 1 0 0 0 NP_NN and GA_BP_NN classiﬁcation. The time and frequency domain statistics of each component are calculated F2 as follows: 0 1 0 0 (1) Center of gravity frequency: F3 0 0 1 0 F4 0 0 0 1 å f S(k) k=1 F = (33) S(k) k=1 (2) Root Mean Square (RMS) of spectrum " # K K 1 1 F = S(k) S(k) (34) 2 å å K 1 K k=1 k=1 (3) Frequency of root mean square (RMS) f S(k) k=1 F = (35) S(k) k=1 Int. J. Turbomach. Propuls. Power 2022, 7, 19 19 of 21 (4) Peak Factor maxjx(i)j PF = s (36) x(i) i=1 Clearance Factor maxjx(i)j CLF = (37) N p ( jx(i)j) i=1 Waveform Factor 1 2 N i i=1 WF = (38) Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 21 of 23 jx(i)j i=1 Impulse Factor maxjx(i)j IF = (39) jx(i)j å (x(i)− x) N i=1 i=1 KF = Kurtosis Factor 4 (40) ( ) x(i)− x (x(i) x) N N i=1 i=1 x KF = (40) s i 0 14 N N N s ! i=1 å (x(i)x) B C i=1 1 2 @ A å x N N i i=1 Figure 17 shows the diagnostic correction of centrifugal pump features in the BP_NN Figure 17 shows the diagnostic correction of centrifugal pump features in the BP_NN classification model. There is one mistake between the actual value and predicted value classiﬁcation model. There is one mistake between the actual value and predicted value for for F1 and five mistakes between the actual value and predicted value for F2. We im- F1 and ﬁve mistakes between the actual value and predicted value for F2. We improved the proved the identification correction accuracy of fault status, as GA was applied to opti- identiﬁcation correction accuracy of fault status, as GA was applied to optimize the weights mize the weights and thresholds between each connection layer of the BP_NN model. and thresholds between each connection layer of the BP_NN model. Figure 18 shows the Figure 18 shows the diagnostic correction of centrifugal pump features in the GA_BP_NN diagnostic correction of centrifugal pump features in the GA_BP_NN classiﬁcation model. classification model. There is just one mistake between actual value and predicted value There is just one mistake between actual value and predicted value for F3, which is much for F3, which is much better than that for BP_NN without GA optimization. better than that for BP_NN without GA optimization. Figure 17. Comparison Figure 17. Com between pariso the n bet actual ween t and he ac predicted tual and values predi by cted BP_NN. values by BP_NN. Figure 18. Comparison between the actual and predicted values by GA_BP_NN. Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 21 of 23 ( ) x(i)− x N i=1 KF = 4 (40) (x(i)− x) i=1 x N N i=1 Figure 17 shows the diagnostic correction of centrifugal pump features in the BP_NN classification model. There is one mistake between the actual value and predicted value for F1 and five mistakes between the actual value and predicted value for F2. We im- proved the identification correction accuracy of fault status, as GA was applied to opti- mize the weights and thresholds between each connection layer of the BP_NN model. Figure 18 shows the diagnostic correction of centrifugal pump features in the GA_BP_NN classification model. There is just one mistake between actual value and predicted value for F3, which is much better than that for BP_NN without GA optimization. Int. J. Turbomach. Propuls. Power 2022, 7, 19 20 of 21 Figure 17. Comparison between the actual and predicted values by BP_NN. Figure 18. Comparison between the actual and predicted values by GA_BP_NN. Figure 18. Comparison between the actual and predicted values by GA_BP_NN. 7. Conclusions Aiming at resolving the multiple failure modes of the centrifugal pump impeller, an experimental system of a centrifugal pump was developed to collect multi-channel complex fault signals for its operation state. The continuous wavelet transform was applied to analyze the multi-channel signals to construct 3D tensors. The hybrid method of the multiple-dimensional-data fusion was proposed based on PARAFAC, BP_NN and GA. The multi-dimensional signal analysis of the complex systems accurately located the fault frequency range of the centrifugal pump. An improvement in the diagnostic accuracy was achieved. Due to the limitations of experimental conditions, this paper only investigated the accuracy of the single fault classiﬁcation of multi-channel signals, while the failure modes of mechanical equipment in actual production would be more complex, which also provides direction for future research. Author Contributions: Conceptualization, S.L.; methodology, H.C.; software, M.L.; validation, S.L.; formal analysis, S.L.; investigation, S.L. and M.L.; resources, H.C.; data curation, M.L.; writing—original draft preparation, S.L.; writing—review and editing, S.L. and H.C.; visualization, M.L.; supervision, H.C.; project administration, H.C.; funding acquisition, H.C. All authors have read and agreed to the published version of the manuscript. Funding: The National Natural Science Foundation of China (Grant 51775390). Informed Consent Statement: Informed consent was obtained from all subjects involved in the study. Data Availability Statement: Not applicable. 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"International Journal of Turbomachinery, Propulsion and Power" – Multidisciplinary Digital Publishing Institute

**Published: ** Jun 28, 2022

**Keywords: **parallel factor analysis; genetic algorithm; BP neural network; fault diagnosis

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