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The planetary gearbox is a critical part of wind turbines, and has great significance for their safety and reliability. Intelligent fault diagnosis methods for these gearboxes have made some achievements based on the availability of large quantities of labeled data. However, the data collected from the diagnosed devices are always unlabeled, and the acquisition of fault data from real gearboxes is time-consuming and laborious. As some gearbox faults can be conveniently simulated by a relatively precise dynamic model, the data from dynamic simulation containing some features are related to those from the actual machines. As a potential tool, transfer learning adapts a network trained in a source domain to its application in a target domain. Therefore, a novel fault diagnosis method combining transfer learning with dynamic model is proposed to identify the health conditions of planetary gearboxes. In the method, a modified lumped-parameter dynamic model of a planetary gear train is established to simulate the resultant vibration signal, while an optimized deep transfer learning network based on a one-dimensional convolutional neural network is built to extract domain-invariant features from different domains to achieve fault classification. Various groups of transfer diagnosis experiments of planetary gearboxes are carried out, and the experimental results demonstrate the effectiveness and the reliability of both the dynamic model and the proposed method. Keywords: Wind turbine planetary gearbox, Lumped-parameter dynamic model, Intelligent fault diagnosis, Convolutional neural network, Transfer learning theory 1 Introduction accurate gearbox fault diagnosis is of great significance to Wind energy has become one of the vital energy sources improve the safety, reliability and economy of WTs [4]. in the world, while wind power generation systems In recent years, many intelligent methods have been have been widely studied and applied [1]. The planetary investigated for gearbox fault diagnosis [5–9], while gearbox is one of the critical components in the the proposed methods have two assumptions: (1) the transmission system of wind turbines (WTs) because of training and testing data are derived from the same its advantages of compact structure, high power density probability distribution; (2) enough labeled history data and desirable transmission efficiency [2]. However, in with fault information can be obtained [10]. However, operation, planetary gearboxes are prone to failure and in industrial applications, it is impractical to satisfy have high maintenance costs under dynamic load and those two assumptions because of operating condition frequently changing operating conditions [3]. Therefore, change, equipment wear degradation, and environmental noise interference, leading to differences of data in the probability distribution [11] and unlabeled data collected from the diagnosed devices [12]. *Correspondence: nihaozhaoyao@163.com Therefore, to solve the above two disadvantages, School of Electrical Engineering, Shanghai University of Electric Power, No. some studies have introduced transfer learning into 2103, Pingliang Road, Yangpu District, Shanghai 200090, China © The Author(s) 2022. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. Li et al. Protection and Control of Modern Power Systems (2022) 7:22 Page 2 of 14 fault diagnosis of mechanical equipment [13–16]. The the signal characteristics of the gearboxes, various transfer learning tasks consist of two datasets, one from dynamic models have been presented, including multi- the source domain and the other from the target domain. body models and lumped parameter models [26–28]. The data in the target domain is distributed differently Dynamic simulation of the planetary gearbox with from the data in the source domain but contains relevant different faults has been realized, and it proves that the knowledge. Thus, the goal of transfer learning is to dynamic model can show many features of the actual improve the property of the predictive model for the signals [29–33]. The above studies have a common target domain by using the common knowledge of the approach of introducing the influences of gear faults source and target domains. With the theoretical research into the dynamic models by changing the mesh stiffness of deep learning, deep hierarchical models are applied to function of the mesh pair. In order to get a more accurate learn transferable features from the cross-domain data vibration response, reference [32] constructs a vibration automatically [17–19]. signal model that can express the effect of transmission Hence, transfer learning can use the learned common path by using a modified Hamming function. knowledge from the source domain to solve a related task In this paper, a dynamic-model-based method for WT in the target domain [20–22]. Accordingly, some transfer- planetary gearbox fault diagnosis using a deep transfer learning-based methods that mainly concentrate on the learning network (DTLN) is proposed. A modified transfer tasks between different operation conditions are lumped-parameter dynamic model is established applied in [11, 23, 24]. Also, the transfer fault diagnoses to simulate the vibration signals of a planetary gear among different devices have been studied. Reference train, and the resultant vibration response is analysed [12] proposes a transfer learning method for bearing by considering the transmission path of the signals. fault diagnosis, and its effectiveness is verified by the Then, an optimized DTLN based on a one-dimensional datasets acquired from three different machines. In [25], deep convolutional neural network (1-D CNN) is a transfer method is presented and the health conditions built. The DTLN comprises three modules: health of bearings used in actual devices are classified with condition recognition module, domain classifier and the help of the diagnosis knowledge from those used distribution discrepancy metrics. With the proposed in the laboratory. Based on such methods, the fault three modules, the DTLN can extract domain-invariant diagnosis model trained with labeled data obtained from features from the simulation data and the actual data, one machine can be generalized to the unlabeled data and the fault classification of actual datasets is realized. obtained from other similar machines. However, in the The introduction of simulation datasets makes up for fault diagnosis of WT gearboxes, the above methods will the possible influence of insufficient samples in fault encounter the following two problems: diagnosis models. Finally, multiple transfer diagnosis experiments are performed to verify the feasibility of the proposed method. (1) Labeled fault data from similar machines are hard The main insights and contributions of this paper are to obtain. The planetary gearboxes in WTs will not summarized as follows. be allowed to run to failure since such a fault could lead to the breakdown of a WT or even serious accidents. In addition, gearboxes often undergo a (1) A novel fault diagnosis method combining transfer long degradation process from normal to failure. learning with the dynamic model is proposed. This Therefore, the acquisition of fault data is time- aims to remove the difficulty in obtaining enough consuming and laborious. labeled fault samples in applications. The cross- (2) Experimental data acquisition of WT gearboxes domain-invariant features of the simulation signal and the actual signal are learned by a deep transfer is costly. WTs are usually large in size, so it is learning network, so as to realize the fault diagnosis expensive to build experimental platforms similar of the actual signals. to the actual ones, while in the laboratory, when (2) The optimized DTLN comprises three parts: health the type and extent of the faults are changed, new condition recognition module, domain classifier components are required and this is costly. and distribution discrepancy metrics. The health condition recognition module is based on a 1-D Such problems lead to insufficient samples in the CNN built to learn the deep features of the input actual fault diagnosis task. As a result, the performance data, while the domain classifier and distribution of the deep transfer learning models will deteriorate discrepancy metrics are applied to help the network and even fail to complete the diagnostic task. To solve learn more domain-invariant features. the problems, an easier method is needed to get signals containing actual fault features. To gain an insight into Li et al. Protection and Control of Modern Power Systems (2022) 7:22 Page 3 of 14 (3) The proposed diagnosis method is based on unsu - into the model by modifying the mesh stiffness function pervised transfer learning theory. The labeled of the mesh pairs. samples are not necessarily needed in the target domain. In practical applications of fault diagnosis, 2.1 Lumped‑parameter model for a single stage planetary the data obtained from the devices to be diagnosed gear train are always unlabeled. Therefore, it is appropriate for The lumped-parameter model is shown in Fig. 1. The actual real-time diagnostic scenarios. system consists of one ring gear ‘r’, one sun gear ‘s’, one (4) A model and data-driven approach is proposed. carrier ‘c’ and N equally spaced planet gears ‘p ’. Herein, The application of traditional artificial intelligence Oxy is the coordinate system rotating at the speed of methods relies on a large number of labeled sam- ω with the x axis going through the center of p . The c 1 ples of devices to be diagnosed. Compared with sun gear has three degrees of freedom, i.e., two lateral the traditional artificial intelligence method which motions (x, y) and one torsional motion (u). The other only relies on the data-driven, the proposed method components have only torsional motion (u). According requires only unlabeled samples to be diagnosed, to Newton’s second law, the motion of a planetary gear and the required number of samples in the target train can be written as several second-order differential domain is greatly reduced. Therefore, the require - equations: 2 m (x¨ − 2ω y˙ − ω x ) + k x + c x˙ − [k (t)x + c x˙ ] sin ψ = 0 s s c s s sx s sx s spn spn spn spn sn n=1 N 2 ˙ ˙ m (y¨ − 2ω x − ω y ) + k y + c y˙ + [k (t)x + c x ] cos ψ = 0 s s c s s sy s sy s spn spn spn spn sn c n=1 �� � � I r u¨ + k u + c u˙ + [k (t)x + c x˙ ] = T (t) r s s su s su s spn spn spn spn s s (1) n=1 �� � I r u¨ + k u + c u˙ + [k (t)x + c x˙ ] = 0 r r ru r ru r rpn rpn rpn rpn n=1 � � � I r u¨ + k (t)x + c x˙ − k (t)x − c x˙ = 0 pn pn spn spn spn spn rpn rpn rpn rpn pn N N �� � � � � I r u¨ + k u + c u˙ − [k (t)x + c x˙ ]+ [k (t)x + c x˙ ] =−T (t) r c c cu c cu c spn spn spn spn rpn rpn rpn rpn c c n=1 n=1 ment of the dataset is reduced and its value in prac- tical application is increased. The rest of this paper is organized as follows. The dynamic model is presented in Sect. 2, and the proposed fault diagnosis framework is described in Sect. 3. In Sect. 4, the proposed dynamic model is validated, and its feasibility is validated on various experimental scenarios. Section 5 draws the conclusions. 2 Dynamic model of planetary gearbox In this section, a modified lumped-parameter dynamic model is established. The model has the following char - acteristics: (1) the horizontal and vertical displacements of ring, planet and carrier that have limited influence on the resultant vibration response are ignored, (2) the lumped virtual spring-damping units are adopted in the model, (3) the effects of planet gear faults are introduced Fig. 1 Dynamic model of a single stage planetary gear train Li et al. Protection and Control of Modern Power Systems (2022) 7:22 Page 4 of 14 with: sources. Consequently, the mesh vibration acceleration signals of sun-planet and ring-planet mesh pairs are x =−x sin ψ + y cos ψ + u + u − e (t) spn s sn s sn s pn spn chosen to establish the resultant vibration signal model. (2) The transmission path of a vibration signal is composed x = u + u − e (t) rpn r pn rpn (3) of two parts [29]: the first part is from the meshing vibration sources to the case, while the second part is the case to the transducer location. The influence of the ψ = ψ − α sn n s (4) first part on the vibration signals can be modeled by an attenuation coefficient, while the second part can be ψ = 2π(n − 1)/N n (5) modeled by a modified Hamming function. Therefore, The second-order nonlinear differential equations of the resultant vibration signals of a planetary gearbox at motion can be solved by a fourth-order variable-step the sensor location can be described as: Runge–Kutta method after nondimensionalization. ξ( mod (ω t+ψ )−π) c n a(t) = exp W (S a cos(α − ω t − ψ ) n spn spn s c n 2.2 Timev ‑ arying mesh stiffness + S a cos(α + ω t + ψ )) Time-varying mesh stiffness is one of the main sources rpn rpn r c n (8) of vibration response in a dynamic system. When the gearbox is free of any defects, the meshing stiffness of where the Hamming function the gear is a function of its angular displacement and W = 0.54–0.46cos(ω t + ψ ). a and a are the n c n spn rpn can be approximated by a square waveform. If a tooth is acceleration signals of sun-planet and ring-planet mesh defective, partial contact loss will occur when the faulty pairs, respectively, and S and S are the attenuation spn rpn tooth engages, leading to a local reduction of the mesh coefficients from the mesh pairs to the case. ξ is used to stiffness function [27]. Four planetary gear conditions are control the bandwidth of the Hamming function. considered, including normal condition (NC), chipped tooth fault (CTF), surface wear fault (SWF) and missing 3 Proposed fault diagnosis framework tooth fault (MTF). The mesh stiffness losses denoted as In this section, the proposed fault diagnosis framework ΔK are different under diverse faults. As the meshing based on the dynamic model and DTLN are introduced stiffness is periodic, the meshing stiffness can be written as in detail. a Fourier series defined by (6) and (7), and the effects of the fault gears can then be introduced into the system. 3.1 Transfer learning problem definition In order to clearly describe the problem, some concepts (l) k (t) = k + k cos(l(ω t − 2πγ + φ )) spn spnm m spn ek are introduced as follows. We take the source domain as spna l=1 D = {(xS i,yS i)}, where xS i ∈ χ is a data sample and yS S S (l) i ∈ Y is its corresponding label, and the target domain +k sin(l(ω t − 2πγ + φ )) S m spn ek spnb as D = {(xT i)}, where xT i ∈ χ is a data sample. D and T T S (6) D are drawn from distribution P (X) and P (X), and T S T P (X) ≠ P (X) because of the domain bias. The same label (l) S T k (t) = k + k cos(l(ω t − 2πγ + φ )) rpn rpnm m rpn rpna ek space is used in different domains, i.e., Y = Y . In fault T S l=1 diagnosis, the goal of transfer learning is to improve the (l) +k sin(l(ω t − 2πγ + φ )) m rpn ek rpnb probabilistic prediction function of the domain D using (7) the knowledge that can be learned in the domain D . In this fault diagnosis, the target domain samples are Because of the partial reduction of meshing stiffness, the data obtained from the equipment to be diagnosed. the amplitude and phase modulation effects appear in the In practical application, these data are unlabeled. The vibration response spectrum in the form of sidebands, source domain samples are the available failure experi- whose frequency locations depend on the fault location mental data of similar equipment or the simulation data and fault type. These sidebands are also reflected in the from the simulation model of the equipment to be diag- vibration signals of actual planetary gearboxes. This is nosed. These are labeled. The target task can be described discussed in detail in Sect. 4. as realizing the condition recognition of the target domain samples, that is, adding condition labels to the 2.3 R esultant signal model samples to be diagnosed. In a planetary gearbox, the mesh vibrations along the torsional motion action lines are the main vibration Li et al. Protection and Control of Modern Power Systems (2022) 7:22 Page 5 of 14 Fig. 2 Structural illustration of the DTLN 3.2 S tructure and training process of the DTLN connected layer and a binary output layer. The binary As shown in Fig. 2, the optimized DTLN consists of three classifier setting with logistics regression is employed parts: health condition recognition module, domain clas- to distinguish between the source domain and target sifier and distribution discrepancy metrics. These are domain. The logistics regression is calculated as: briefly described below. d = (10) − (w ) f +b d d 1 + e 3.2.1 Health condition recognition module The health condition recognition module is based on a where w is the weight matrix of the classifier, b is the d d 1-D CNN, which has the function of feature extraction corresponding bias vector and f is the output of the layer and condition classification. In the 14-layer 1-D CNN, FC3. the first 13 layers are used for feature extraction and collation, and the last layer can be regarded as the 3.2.3 D istribution discrepancy metrics condition classifier. In the convolutional layer, feature In order to realize the extraction of domain-invariant extraction is carried out, where the rectified linear unit is features, a metric is required to represent the distribution used as an activation function. Then a maximum pooling difference between the features extracted from the source operation is introduced to reduce the feature dimension domain and those from the target domain. Here we use and enhance the feature robustly. The full connected the Wasserstein distance to measure the distribution layer and softmax regression are used at the end of the (S) discrepancy between the two datasets. Let P(f ) and network to perform classification tasks. In summary, the (T) (S) Q(f ) be the probability distributions where f and 2 2 output of the health condition recognition module can be (T) f are the features learned by 1-D CNN from the source defined as the output probability of the softmax function: domain and the target domain, respectively, according � � to the Kantorovich–Rubinstein dual theorem, The (w ) f +b 1 2 � � Wasserstein distance between the two distributions is (w ) f +b 2 2 e computed as: y = (9) � � w f +b . ( ) i 2 e � � (S) (T ) W = sup E G(f ) − E G(f ) P Q (w ) f +b 3 2 2 2 (11) i=1 �G� ≤1 where f is the output of the full connected layer FC2, w 2 i where ||G|| is the 1-Lipschitz function. th denotes the weight matrix that concatenates to the i For the three components of the DTLN introduced output neuron, b is the bias vector, and K is the number above, each corresponds to an optimization object. of health condition categories of the dataset. 3.2.4 Object 1 3.2.2 Domain classifier Minimize the health condition classification error of the The domain classifier is a binary classifier that dis - softmax classifier on source data. The objective function tinguishes source domains from target domains. As can be defined as the regression loss of a standard shown in Fig. 2, the domain classifier consists of a fully softmax classifier, as: Li et al. Protection and Control of Modern Power Systems (2022) 7:22 Page 6 of 14 Based on (16), in the back-propagation process, the � � m k (w ) f 2+b j parameters θ , θ , and θ are updated as: �� f c d 1 e L =− I[y = k] log c i � � m � T ∂L ∂L ∂J w ) f 2+b c d i=1 j=1 l θ ← θ − ε − + μ f f (17) ∂θ ∂θ ∂θ l=1 f f f (12) where m is the batch size of the data samples, k is the ∂L θ ← θ − ε c c (18) number of health condition categories, w denotes the ∂θ th weight matrix that concatenates to the i output neuron, and I[·] is an indicator function. ∂L θ ← θ − ε (19) d d ∂θ 3.2.5 Object 2 Maximize the domain classification error on the source where ε denotes the learning rate. and target domain datasets. The loss function of the binary After training, the classifier can recognize the unla - classifier can be represented as: beled samples from the target domain even if the learned domain-invariant features have equivocal domain catego- L = l log d(x + 1 − l log 1 − d x ) ( ) ( ( )) ries and domain discrepancy. As shown in Fig. 3, DTLN i i i i i=1 uses labeled samples from the source domain and unla- (13) beled samples from the target domain for training. The where l denotes the real domain label, and d(x ) is a invariant features of the domain are learned first, and i i function that represents whether x comes from the then the classifier determines the category based on the source domain or the target domain. The objective learned features. After the training, the trained network function can be written as: will be tested by the sample set from the target domain. n n s t 1 1 (S) (T ) L = L f + L f (14) 2i 2j n n s t 3.3 Proposed fault diagnosis framework i=1 j=1 The framework of the proposed method is illustrated in where f (S) i and f (T) j are the features learned from the Fig. 4. As shown, the method includes three parts, as 2 2 source domain and the target domain, respectively. introduced below. 3.2.6 Object 3 Minimize the Wasserstein distance between features 3.3.1 P art 1: data acquisition and preprocessing extracted from the source and target domain datasets. In this part, the source domain and target domain are Considering the gradient penalty item, the calculation constructed, where the target domain data samples are formula is given as: obtained from the gearbox to be diagnosed, and the source domain data samples are obtained by analyzing n n s t 1 1 (S) (T ) J = G(f ) − G(f ) − γ ∇ G(H) − 1 2i 2i H n n 2 s t i=1 i=1 (15) where γ is the tradeoff parameter, n and n are the s t respective numbers of training samples from the source domain and target domain, and H is a uniform sampling from the feature representations. In conclusion, in order to extract as many cross-domain- invariant features as possible, the final optimization object can be combined as: ∗ ∗ ∗ L θ , θ , θ = min L (θ , θ ) − L (θ , θ ) + μJ(θ ) c f c d f d f f c d θ ,θ ,θ f c d (16) where λ and μ are the hyperparameters, θ , θ , and θ are f c d the parameters of the feature extractor, health condition classifier, and domain classifier, respectively. Fig. 3 Flowchart of the proposed DTLN Li et al. Protection and Control of Modern Power Systems (2022) 7:22 Page 7 of 14 Fig. 4 Fault diagnosis framework of the proposed method the dynamic model. It is worth noting that the relevant Brake Motor controller controller parameters of the dynamic model are taken from the device to be diagnosed. After acquiring the vibration signal, the samples are processed and the frequency- 1 2 3 4 6 domain samples are used as the input of the DTLN. This is because frequency-domain samples are more robust Brake Parallel Planetary Motor Gearbox Gearbox to noise than time-domain samples and contain more (a) domain-invariant features. This will be demonstrated in detail in Sect. 4. 3.3.2 Part 2: network training and fault classification In order to extract more domain-invariant features of the source domain and target domain, frequency domain samples are used to train the DTLN, and the trained network can be obtained. The training process (b) is based on (17–19). The trained network is tested by Fig. 5 Experimental setup for gearbox dataset. (a) DDS. (b) fault unlabeled testing samples from the target domain and gears outputs classified results. Table 1 Parameters of the dynamic model 3.3.3 Part 3: output of the diagnostic results Nomenclature s p r c The trained diagnostic model is applied to the fault Teeth number 20 40 100 – diagnosis of experimental equipment to output the Module (mm) 1 1 1 – diagnosis results. In order to show the feasibility of the Pressure angle (deg) 20 20 20 – proposed method, the above classification results are Teeth width (mm) 10 10 10 – analyzed visually. Mass (kg) 0.069 0.057 0.422 0.709 According to the above fault diagnosis framework, Inertia (kg·mm ) 6.415 11.791 1417.482 364.756 the dynamic model is used to construct the source Young’s modulus (G Pa) 206 206 206 206 domain in the transfer learning method. This is helpful Poisson’s ratio 0.3 0.3 0.3 0.3 for fault diagnosis of true devices. In the following, the Pitch diameter (mm) 20 40 100 – rationality and advantages of the proposed method are −1 Torsional bearing stiffness (N·mm ) 0 0 1000 0 verified. 4 Experimental results and comparisons 4.1 Validation and analysis of the simulation model In this section, similarities between the simulation signal 4.1.1 P lanetary gearbox fault experiment and the actual signal are analyzed. Multiple experiments The gearbox dataset is collected from the drivetrain are performed to validate the network and fault diagnosis dynamic simulator (DDS) shown in Fig. 5a. The planetary framework. Li et al. Protection and Control of Modern Power Systems (2022) 7:22 Page 8 of 14 gearbox has two stages, and the faults of planet gears in to the structure of DDS, S and S are set as 0.4 and spn rpn the first stage are studied. For vibration signal acquisi - 0.9, while ξ is −1. tion, an acceleration transducer is mounted. Experiments To simplify the analysis, the order spectra are are carried out on planet gears in four healthy conditions, represented by normalizing with rotational frequency of shown from left to right in Fig. 5b as NC, CTF, SWF and the carrier. The mesh order H = Z denotes the mesh m r MTF, respectively. The sampling frequency of the trans - frequency f (H = f /f = Z f /f = Z , where Z is the m m m c r c c r r ducer is set at 12 kHz. teeth of ring gear). The rotation period of the carrier T is equal to Z T , where T is the mesh period. For the r m m 4.1.2 Simulation parameters planetary gear train, Z = 100, f = 100f , T = 100T . r m c c m The basic design parameters are listed in Table 1. The With the above settings, the vibration response can be planetary gearbox has three planet gears (N = 3) with determined. a fixed ring gear. The mesh damping of sun-planet pair and ring-planet are set as 242.6 N·s/m and 410.3 N·s/m, 4.1.3 Sp ectrum analysis of simulation signal respectively. The bearing stiffness and damping of sun −1 −1 As described in Sect. 2, modulation effects appear in gear are assumed to be 15 N·mm and 9.2 N·s·m , vibration response because of the partial reduction respectively. The constant torque acting on the carrier is of meshing stiffness, in the form of sidebands in the 1.26 Nm, and the sampling frequency of the simulation vibration spectrum. Therefore, the frequency spectra are signal is 12 kHz. In the resultant signal model, according analyzed. Fig. 6 Frequency spectra of vibration signals from dynamic model (above) and DDS (below). (a) NC. (b) SWF. (c) CTF. (d) MTF Li et al. Protection and Control of Modern Power Systems (2022) 7:22 Page 9 of 14 (a) simulation and experiment. As shown in Fig. 6b, the 1.2 global fault causes the characteristic frequency f of the sun gear. The amplitudes of the order spectrum locate 0.6 at the f ± kf ± nf (k is an integer). As seen, the simula- m s c tion and experimental results are consistent. From the above analyses, it is verified that the simulation results of the dynamic model contain some features of the actual (b) signals. This is the basis for the applicability of trans - Training fer learning theory, i.e., the source domain and target Testing domain contain common diagnostic knowledge. In addi- tion, fault features can be detected from the order spec- trum. Therefore, the fault identification of frequency 0 1500 3000 domain signals can help the overall diagnosis decision. Epoch Fig. 7 Penalty parameter and loss of the proposed DTLN. (a) Penalty 4.2 Transfer fault diagnosis experiments parameter. (b) Training and testing loss The three datasets required for validation and 16 diagnosis experiments are described in this section. When planet gear faults occur, fault features will (1) A: Experimental Planetary Gearbox Dataset The appear near the meshing frequency, while the frequency dataset is collected from DDS, which contains four locations depend on the fault location and type. As working conditions with various motor speeds and shown in Fig. 6a, when working under NC, the sidebands a certain load: 1200r/min (A ), 1800r/min (A ). 1 2 locate at f ± nf (n is an integer) because of the modu- m c Each health condition, i.e., NC, CTF, MTF and lation of the transmission path. After introducing fail- SWF, has 800 samples. Thus, this dataset has a ures of the planet, some impulsive signals appear. As a total of 3200 samples, each of which has 2000 data result, the spectrum contains some additional frequency points. components. As shown in Fig. 7c and d, when the planet (2) B: Planetary Gearbox Dataset Used by Another gear has local faults such as CTF and MTF, these side- Group The dataset is collected from a similar bands are at the locations of f ± mf ± nf (m is an inte- m p c planetary gearbox under different working ger), which also exist in the actual signal. This signifies conditions. This was provided by Yan’s group [34]. that the signals are modulated by the fault of planet gear Four healthy conditions in the dataset are selected, and the transmission path. It is worth noting that MTF and the working conditions are investigated with causes more characteristic frequencies than CTF in both the rotating speed system load set at 20 Hz–0 V (B ) and 30 Hz–2 V (B ). Similarly, 800 samples for 1 2 each condition are intercepted, and each sample contains 2,000 points. Table 2 Virous transfer task (3) C: Dataset Acquired by Simulation The dataset Class Transfer task Source Target Healthy is acquired by dynamic simulation, and rotation domain domain conditions frequencies of the sun gear are set at 20 Hz (C ) 1 A → B 1200 rpm 1200 rpm NC, 1 1, and 30 Hz (C ), each of which also has four healthy B → A CTF, 1 1 conditions. The conditions and properties of the MTF, A → B 1800 rpm 1800 rpm 2 2, samples are the same as those of datasets A and B. SWF B → A 2 2 2 A → B , 1200 rpm 1800 rpm 1 2 B → A Sixteen transfer fault diagnosis experiments are shown 1 2 B → A , 1800 rpm 1200 rpm in Table 2. Taking the task A → B for example, A 2 1 1 1 1 A → B 2 1 is the source domain, and B is the target domain. The → A , 1200 rpm 1200 rpm 3 C 1 1 standard assessment protocol for unsupervised transfer C → B 1 1 learning missions is adopted. In each transfer task, the C → A , 1800 rpm 1800 rpm 2 2 training dataset consists of all labeled data samples from C → B 2 2 the source domain and half of the unlabeled data sam- 4 C → A , 1200 rpm 1800 rpm 1 2 ples from the target domain, while the testing dataset is C → B 1 2 composed of the other half of samples from the target C → A , 1800 rpm 1200 rpm 2 1 C → B 2 2 domain. Among the 16 experiments, the groups of Class Loss Amplitude Li et al. Protection and Control of Modern Power Systems (2022) 7:22 Page 10 of 14 Table 3 Architecture of the 1-D CNN the learning rate is set as 0.001. The batch size is set as 512, and the epoch of the training is 3000. Taking the Layer Parameters Stride Output size experiment C → A for example, the loss function dur- 1 1 Input / / 2000 × 1 ing the training process is drawn in Fig. 7b. It is clear that C1 64 × 1 conv 1 16-[2000 × 1] the loss function converges after about 1500 steps. P1 2 × 1 max-pool 2 16-[1000 × 1] To reduce contingency and particularity of results, C2 16 × 1 conv 1 32-[1000 × 1] each transfer fault diagnosis experiment is carried out 10 P2 2 × 1 max-pool 2 32-[500 × 1] times, and the results are shown in Table 4. It is worth C3 16 × 1 conv 1 32-[500 × 1] noting that DTLN-T indicates that the input of the net- P3 2 × 1 max-pool 2 32-[250 × 1] work is time-domain samples, while DTLN-F indicates C4 5 × 1 conv 1 64-[250 × 1] that the input is frequency-domain samples. The fig - P4 2 × 1 max-pool 2 64-[125 × 1] ures in Table 4 represent the average accuracy rate and C5 5 × 1 conv 1 64-[125 × 1] standard deviation of 10 repeated classification experi - P5 2 × 1 max-pool 2 64-[62 × 1] ments. The accuracy rate reflects the reliability of the FC1 Flatten / 3968 × 1 method, and the standard deviation reflects the stability FC2 Fully-connected / 256 × 1 of the method. In the transfer experiments between dif- FO Softmax / 4 × 1 ferent devices, i.e., A → B, B → A, the diagnostic accura- cies of the proposed method are over 91%. In addition, in the transfer experiments from dynamic model to actual devices, i.e., C → A, C → B, the average diagnostic accu- 1 are the transfer diagnosis from one machine to another racy of Class 3 is 90.9%, which indicates that the pro- under similar working conditions, while those in Class posed method is feasible. 2 are under different working conditions. Class 3 is the transfer fault diagnosis between dynamic model and actual machines at the same speed, while Class 4 is the 4.3 Results comparison and visual analysis diagnostic experiments of the proposed method at differ - In order to demonstrate the effectiveness and the feasi - ent rotational speeds. bility of the proposed method, two other networks are The detailed parameters of the DTLN can be found in chosen for comparison. Among them, the basic convolu- Table 3, in which 64 × 1 conv denotes the size of the con- tional network has the same structure and parameters as volutional kernel, 2 × 1 max-pool stands for the size of the 1-D CNN introduced above, and uses source data for max-pooling operation, and 16-[2000 × 1] represents 16 training and then tries to classify target data. In addition, feature maps of size 2000 × 1. In order to restrain noise a domain adversarial neural network (DANN) which is a and extract useful knowledge, a wide kernel is used in C1. commonly used transfer learning method is also tested, As shown in Fig. 7a, the hyperparameters λ and μ in (16) and its parameter setting refers to [35]. The learning rate –4 are set to gradually increase from 0 to 1, and the calcula- of the CNN is 0.01, and is 2 × 10 for the DANN. The tion formula is 2/(1 + exp(-10 × p))−1, where p denotes Adam algorithm is used as optimization algorithm in the training progress. In order to minimize the loss func- both methods. It is worth noting that the inputs of the tion, the Adam is used as an optimization algorithm and CNN, DANN and DTLN-F are all frequency-domain Table 4 Recognition result of experiments Method CNN DANN DTLN‑ T DTLN‑F Method CNN DANN DTLN‑ T DTLN‑F A → B 0.413 ± 0.028 0.888 ± 0.012 0.962 ± 0.009 0.961 ± 0.009 A → B 0.314 ± 0.024 0.77 ± 0.022 0.919 ± 0.022 0.911 ± 0.011 1 1 1 2 B → A 0.411 ± 0.015 0.885 ± 0.020 0.953 ± 0.015 0.964 ± 0.008 B → A 0.337 ± 0.031 0.798 ± 0.042 0.915 ± 0.023 0.906 ± 0.030 1 1 1 2 A → B 0.410 ± 0.015 0.879 ± 0.018 0.952 ± 0.010 0.951 ± 0.007 B → A 0.332 ± 0.031 0.781 ± 0.025 0.889 ± 0.018 0.918 ± 0.006 2 2 2 1 B → A 0.406 ± 0.016 0.891 ± 0.010 0.959 ± 0.016 0.949 ± 0.011 A → B 0.339 ± 0.019 0.783 ± 0.027 0.890 ± 0.014 0.906 ± 0.008 2 2 2 1 Average 0.410 0.886 0.957 0.956 Average 0.331 0.783 0.903 0.910 C → A 0.318 ± 0.027 0.723 ± 0.042 0.826 ± 0.010 0.904 ± 0.013 C → A 0.285 ± 0.030 0.658 ± 0.049 0.767 ± 0.017 0.892 ± 0.016 1 1 1 2 C → B 0.313 ± 0.031 0.702 ± 0.045 0.836 ± 0.008 0.914 ± 0.013 C → B 0.293 ± 0.023 0.656 ± 0.047 0.737 ± 0.027 0.882 ± 0.019 1 1 1 2 C → A 0.336 ± 0.027 0.688 ± 0.032 0.818 ± 0.011 0.899 ± 0.018 C → A 0.287 ± 0.025 0.673 ± 0.049 0.768 ± 0.027 0.898 ± 0.021 2 2 2 1 C → B 0.336 ± 0.029 0.676 ± 0.025 0.820 ± 0.016 0.918 ± 0.008 C → B 0.288 ± 0.028 0.696 ± 0.049 0.748 ± 0.032 0.885 ± 0.013 2 2 2 2 Average 0.326 0.697 0.825 0.909 Average 0.288 0.671 0.755 0.889 Li et al. Protection and Control of Modern Power Systems (2022) 7:22 Page 11 of 14 validation on the subclass in Table 2 are shown in Fig. 8. CNN DTLN-T Taking the task C → A for example, the classification 1 1, DANN DTLN-F accuracy and standard deviation of various methods are compared in Fig. 9. By comparing the results, three observations can be made: (1) For the transfer fault diagnosis missions where unlabeled data are retrievable in the target domain, the networks based on transfer learning are superior. It suggests that transfer learning can be an effective instrument to facilitate the practical Class 1Class 2Class 3Class 4 Subclass of experiments application of intelligent diagnostics. Additionally, Fig. 8 Classification accuracy of average tenfold cross validation on compared with DANN, DTLN obtains higher the subclass of experiments classification accuracies and lower standard deviations, as shown in Fig. 10. It indicates that DTLN reduces the distribution discrepancy between different domains more effectively and is relatively stable. (2) Compared with experiments where the networks are trained by data from machines, the accuracies of experiments that replace the actual labeled data with the simulation data are reduced. This indicates that the fault signals of similar equipment contain more domain-invariant features, but the simulated signals contain fewer. However, compared with the time-domain samples, the classification accuracy CNNDANNDTLN-T DTLN-F obtained when the frequency-domain samples Classification method are used as the input is higher, as shown in Fig. 8. Fig. 9 Classification accuracy of average tenfold cross validation Therefore, it can be inferred that when the samples on the transfer experiments from C (source domain) to A1 (target are in the frequency domain, it is more favorable domain) for the DTLN to extract the domain-invariant features. Therefore, the method proposed in this paper adopts the frequency-domain samples as the input. As a result, the transfer experiments samples. The accuracies and standard deviations on the from simulation model to actual devices can realize 16 transfer fault diagnosis experiments are shown in relatively high accuracy. When the set speed of the Table 4. Classification accuracies of average tenfold cross Source_NC Source_CTF Source_MTF Source_SWF Target_NC Target_CTF Target_MTF Target_SWF (a)(b) (c) (d) Component 1Component 1Component 1 Component 1 Fig. 10 The visualization of the learned features on the dataset C (source domain) and the dataset A1 (target domain): (a) CNN, (b) DANN, and (c) DTLN-T, (d) DTLN-F Component 2 Classification accuracy (%) Classification accuracy (%) Component 2 Component 2 Component 2 Li et al. Protection and Control of Modern Power Systems (2022) 7:22 Page 12 of 14 dynamic model is the same as the actual speed, From Fig. 10a, the features learned by CNN have clear the average classification accuracy is 90.9%. Under distribution discrepancy. As a result, when CNN is different speed settings, the average classification trained with C , its recognition for A is close to surmise. 1 1 accuracy is 88.9%. This proves that the diagnosis As for DANN, the cross-domain distribution discrepancy method combining the dynamic model with is amended to a certain extent as shown in Fig. 10b, so a deep transfer learning network has practical the accuracy of DANN for A is much higher than that of value. In similar cross-domain transfer diagnosis CNN. From Fig. 10c and d, the proposed DTLN is able to experiments, an 84.32% recognition rate for bearing amend the distribution discrepancy between the learned faults in mechanical equipment is achieved in [25], features of different datasets. However, because of the while reference [12] achieves an 86.3% diagnosis difference in transferability between the subclass samples accuracy. Therefore, compared with the existing of the source domain and target domain, the distribution research results, the proposed method has certain discrepancy of the subclass samples is corrected asym- advantages. metrically. For example, in Fig. 10c, the distribution dis- (3) The accuracies obtained from the transfer learning crepancy of the cross-domain samples with CTF is still between datasets under similar working conditions severe after the correcting of DTLN-T. To illustrate with are higher. This is exemplified by the fact that the confusion matrices, shown in Fig. 11c and d, the classi- accuracy of Class 2 is lower than that of Class 1, fication effects of DTLN-T and DTLN-F are defective, and Class 4 is lower than that of Class 3. Therefore, whereas the classification of DTLN-F is better. the working condition is one of the important influencing factors of transfer learning in practical application. In order to improve the accuracy of 5 Conclusion diagnosis, the rotational speed of the dynamic In this paper, a dynamic-model-based transfer learning model can be adjusted to be the same as that of the fault diagnosis method for WT planetary gearboxes is actual equipment to be diagnosed, so as to obtain proposed. This method introduces a dynamic simulation more accurate fault discrimination. dataset into the application of transfer learning and produces a diagnosis of unlabeled fault data obtained from actual machines. To verify the feasibility of the In order to intuitively show the classification effect, a proposed method, spectrum analysis of the simulated t-distributed stochastic neighbor embedding (t-SNE) and experimental signals is carried out, and 16 groups of algorithm is introduced. This can map the high-dimen - transfer fault diagnosis experiments are completed. From sional features into 2-D space and the distribution of the results, the following conclusions can be drawn. features can be plotted directly. Taking the task C → A 1 1 for example, the transferable features learned by CNN, (1) Through the spectrum analysis, the vibration DANN, DTLN-T and DTLN-F are shown in Fig. 10 via response solved by the dynamic model contains t-SNE. In addition, the confusion matrices for transfer some features of the actual fault vibration signal, results on dataset A can be explored. These are shown i.e., domain-invariant features required by transfer in Fig. 11. learning. (a)(b) (c) (d) NC NC NC NC CTF CTF CTF CTF MTF MTF MTF MTF SWF SWF SWF SWF NC CTFMTF SWF NC CTFMTF SWF NC CTFMTF SWF NC CTFMTF SWF Predicted label Fig. 11 The confusion matrices for the transfer results of the dataset A : (a) CNN, (b) DANN, and (c) DTLN-T, (d) DTLN-F True label Li et al. Protection and Control of Modern Power Systems (2022) 7:22 Page 13 of 14 Author’s information (2) The proposed DTLN can effectively realize the Dongdong Li received his B.S. and Ph.D. degrees from Zhejiang University recognition of unlabeled fault data from the target and Shanghai Jiao Tong University both in electrical engineering in 1998 and domain. In the application of transfer fault diagno- 2005, respectively. He is currently a professor and dean of College ofElectric Engineering in Shanghai University of Electric Power, Shanghai, China. His sis, the classification accuracy and stability of the current research interests include analysis of electric power system, renewable DTLN are better than those of the DANN. energy system, smart grid and power electronization of power system. (3) The proposed method combining a dynamic model Yang Zhao received his B.S. degree from Shanghai University of Electric Power in 2019. He is currently pursuing the M.S. degree in electrical engineering at with the deep transfer learning network can identify Shanghai University of Electric Power. His current research interests include four kinds of faults of the planetary gearbox. When artificial intelligence algorithm, fault diagnosis of wind turbine planetary the set speed of the dynamic model is the same as gearbox. Yao Zhao received the B.S. degree in automation from Anhui University, the actual speed, the average classification accuracy Hefei, China, in 2009, the M.S. degree in electrical engineering from Shanghai is 90.9%. Maritime University, Shanghai, China, in 2011, and the and Ph.D. degree in from the Nanjing University of Aeronautics and Astronautics, Nanjing, China, in 2016. He is currently a Lecturer with the Shanghai University of Electric Power, The results indicate that the proposed method that Shanghai, China. His main research interests include electrical machines and combines the transfer learning theory with dynamic power electronization of power system. model is feasible, whereas the dynamic model proposed can be further optimized. After introducing Author contributions the dynamic model, varied labeled fault data can be Not applicable. obtained, and the fault setting is more independent and Funding convenient. This leads to practical application value. Natural Science Foundation of Shanghai (21ZR1425400), Shanghai Rising- There has been rapid development of artificial Star Program (21QC1400200), National Natural Science Foundation of China intelligence algorithms. However, because of the (51977128), Shanghai Science and Technology Project (20142202600). operating environment, working condition, data Availability of data and materials acquisition difficulty etc., artificial intelligence methods Data and materials are not public, but can be uploaded and made public if in the field of fault diagnosis are developing slowly. necessary. Therefore, how to combine artificial intelligence methods with practical applications of condition Declarations recognition to achieve higher accuracy is the direction Competing interests of future research. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work List of symbols reported in this paper. m : Mass of the sun gear.; x , y : Translational displacement of x-axis and y-axis s s s for sun gear.; u : Torsional displacements for each component (i = s,r,c,p ).; k , c : Received: 28 December 2021 Accepted: 28 May 2022 i n ij ij Supporting stiffness and damping for each component at different motions (j = x,y,u).; k (t), k (t): Stiffness function between sun-planet and ring-planet spn rpn meshing pairs (n = 1,···,N).; c , c : Constant damping between sun-planet spn rpn and ring-planet meshing pairs.; e (t), e (t): Transmission errors of the nth spn rpn sun-planet and ring-planet pairs.; x , x : Relative displacements along the References spn rpn torsional motion action lines of sun-planet and ring-planet; I : Inertias of each 1. Nadour, M., Essadki, A., & Nasser, T. (2020). 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Protection and Control of Modern Power Systems – Springer Journals
Published: Dec 1, 2022
Keywords: Wind turbine planetary gearbox; Lumped-parameter dynamic model; Intelligent fault diagnosis; Convolutional neural network; Transfer learning theory
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