Improved damage detection in Pelton turbines using optimized condition indicators and data-driven techniques

2021, Vol. 20(6) 3239 –3251 Original Article Structural Health Monitoring The Author(s) 2020 Article reuse guidelines: sagepub.com/journals-permissions DOI: 10.1177/1475921720981839 journals.sagepub.com/home/shm Improved damage detection in Pelton turbines using optimized condition indicators and data-driven techniques Weiqiang Zhao 1 , Mo `nica Egusquiza 1 , Aida Estevez 2 , Alexandre Presas 1 , Carme Valero 1 , David Valentı ´n 1 and Eduard Egusquiza 1 Abstract The health condition of hydraulic turbines is one of the most critical factors for the operation safety and financial benefits of a hydro power plant. After the massive entrance of intermittent renewable energies, hydropower units have to regulate their output much more frequently for the balancing of the power grid. Under these conditions, the components of the machine have to withstand harsher excitation forces, which are more likely to produce damage and eventual failure in the turbines. To ensure the reliability of these machines, improved condition monitoring techniques are increasingly demanded. In this article, the feasibility of upgrading condition monitoring of Pelton turbines using novel vibration indicators and data-driven techniques is discussed. The new indicators are selected after performing a detailed analysis of the dynamic behavior of the turbine using numerical models and field measurements. After that, factor analysis is carried out in order to assess which are the most informative indicators and to reduce the dimension of the input data. For the validation of the proposed method, monitoring data from an actual Pelton turbine that suffered from an important fatigue failure due to a crack propagation on the buckets have been used. The novel condition indicators as well as classical indicators based on the spectrum and harmonics levels have been obtained while the machine was in good operation, during different stages of damage and after repair. All of these have been used to train an artificial neural network model in order to predict the evolution of the crack until failure occurs. The results show that using the improved monitoring methodology enhances the ability to predict the appearance of damage in comparison to typical condition indicators. Keywords Condition monitoring, Pelton turbine, damage detection, condition indicator, factor analysis, principal component analysis Introduction As one of the most important renewable energies, hydropower converts the potential energy of water into electric energy. After its development since more than one century ago, hydropower has become a reliable energy resource. Over the last years, with the entrance of new renewable energies like solar photovoltaics and wind energy, whose output is random and hard to regulate, hydropower has become extremely significant for balancing the generation and consumption of electricity. Over the last decade, the installed capacity of wind and solar energies has increased from 13.2% and 2.1% to 24.0% and 20.7%, respectively. 1 This new scenario requires hydropower units to regulate their output much more frequently for the balancing of the power grid. Therefore, more flexibility is demanded and turbines have to operate many hours off-design. Due to the more severe operating conditions, components from Pelton turbine are more prone to suffer damage and/or failure. 2–4 Unexpected failures of the turbine may result in substantial economic losses, not only due to the costs related to the repair or replacement of the 1 Centre for Industrial Diagnostics and Fluid Dynamics (CDIF), Polytechnic University of Catalonia (UPC), Barcelona, Spain 2 Departamento de Ingenierı ´a meca ´nica y fabricacio ´n, Escuela Te ´cnica Superior de Ingenierı ´a, Universidad de Sevilla, Sevilla, Spain Corresponding author: Weiqiang Zhao, Centre for Industrial Diagnostics and Fluid Dynamics (CDIF), Polytechnic University of Catalonia (UPC), Av Diagonal, 647, 08028 Barcelona, Spain. Email: [email protected] 3240 Structural Health Monitoring 20(6) component but also to the downtime and production loss thus caused. As the most common action-type turbine, Pelton turbines are widely used in high-head hydropower plants. In the turbine, the high pressure of the water at the entrance is converted into a high-velocity jet using a nozzle. The jet impinges the runner as shown in Figure 1, converting the kinetic energy of the water into mechanical energy that is converted into electricity by the generator. During operation, strong pulsating forces are produced on the runner. The buckets that receive directly the high-speed jet of water resemble a cantilever beam and have to transmit the torque to the wheel. Therefore, one of the most common causes of failure in this type of turbines is the fatigue of the material on the bucket area. In order to ensure the availability of the machine, effective strategies such as periodic inspections and condition monitoring have been applied for protecting the machine and preventing serious failure. 6 Usually, sensors are installed on bearings where the vibrations generated by the turbine and the generator are transmitted. Vibration signals, as well as the operating parameters, are recorded by an acquisition system to be analyzed for diagnosis. Nowadays, remote online monitoring systems have been applied on hydropower plants so that it is possible to monitor the turbine in a diagnostics center far away from the power plant. 7 Accelerometers located in bearings are the sensors currently used to measure machine vibrations. Other types of sensors like acoustic emission sensors, strain gauges, and microphones have also been tested to monitor the condition of hydraulic turbines. 8–10 One of the most complicated tasks is to select alarm and trip levels which are able to detect incipient damage in the runner. 6 To begin with, the levels of each indicator have to be mapped for the whole operating range when the machine is in good condition. After that, the evolution of these indicators needs to be mapped when the machine suffers damage. Machine learning techniques are a good tool for that, 11 but they demand huge volumes of historic data which cover the evolution of the machine vibration throughout all the stages of damage. This type of machines never operates until they reach the end of their useful life, so the lack of data with the damaged machine ends up being a large hurdle for the monitoring of hydro turbines. These drawbacks can be partially overcome by developing numerical simulation models which can emulate synthetic damage. 12 Being different from the general machinery, the main components of hydro turbine units such as the runner and the generator are tailor-made pieces. The layout of turbine units also varies from one machine to another, which leads to different structural responses. Because of the above reasons, the diagnosis of a hydraulic turbine depends much on expertise and historical data. Some typical types of damage can be detected but many of them, especially the ones taking place in the runner, are not detectable before failure occurs. 13 This happens because the runner vibrations can hardly be detected by monitoring vibrations in the bearing; natural frequencies of the runner do not produce important deformations in the rotor. 14 In recent years, significant efforts have been made in application of data-driven and artificial intelligence (AI) for structural damage detection. Several researchers have applied artificial neural network (ANN) on the vibration pattern for damage or fault detection on various types of machines including helicopter pump, 15 wind turbine, 16 railway wheel, 17 and so on. However, there are a few studies on the implementation of AI on the experimental data of hydraulic turbines. With the development of computer techniques, more simulation data on hydraulic turbines are available for the training of ANN: RA Saeed et al. 18 used the principal component analysis (PCA) to extract features from the frequency response function (FRF) obtained from the simulation of a hydro turbine runner. The FRF features were used for training ANN in order to predict the crack length on the blade. With the years of accumulation of online condition monitoring data, it would become possible the application and optimization of data-driven and AI techniques for the monitoring of hydraulic turbines. In AI applications, a general way to deal with large training input is to reduce the size of the input pattern by PCA. 17 However, the calculation of the principal component (PC) depends on every input variable, 19–21 which means the PCs will change with the increase in the samples. In addition, there is no real meaning in the extracted PCs. 22 Factor analysis (FA) is an ideal tool to address this problem: it not only describes variables in terms of a lower number of factors that have different loading values regarding to the input variables, but also tries to find out one or more latent variables (common Figure 1. Schematic of the operation of the Pelton turbine bucket. 5 2Structural Health Monitoring 00(0) Zhao et al. 3241 factors) that exert causal influence (loadings) on these observed variables. 23 It was first proposed in 1904 for the explanation of psychological theories. 24 During its development of more than one century, many researchers have contributed to its theory including the number of factors to retain, 25 factor extraction algorithms, 26 and so on. Nowadays, FA has been applied into engineering applications: M Tripathi and SK Singal 27 introduced FA into weight determination to develop a novel water quality analysis. R Maskey et al. 28 provided explicit information on the appropriate application of exploratory FA on engineering problem. The engineering implements of FA have its potential to optimize monitoring by compressing the monitoring indicators into interpretable factors. In this article, a novel condition monitoring method for Pelton turbines is introduced and discussed. The first step consists in defining optimized condition monitoring indicators based on an extensive analysis of the dynamic behavior of a Pelton turbine. The main advantage of these new condition indicators is the ability of detecting runner vibrations and its variation with abnormal operation and damage. The second step consists in applying FA in order to optimize the new indicators and reduce their dimension. To validate this new procedure, monitoring data from an existing Pelton turbine have been used. The machine suffered a catastrophic failure in which one of the buckets broke off during operation. The inspection of the machine later revealed the failure was due to a deviated jet, which caused several cracks to appear and propagate on different buckets. To check the ability of the new methodology to predict the propagation of the cracks, the novel condition indicators have been extracted from the available data and used to define an artificial damage index (ADI) based on the typical crack propagation behavior. These have been used in an ANN model in order to predict the ADI until failure occurs. The same process has been followed for the typical condition indicators used normally in condition monitoring of Pelton turbines in order to compare the results with the new method. This article is organized as follows: section ‘‘Methodology for improved monitoring of Pelton turbines’’ introduces the novel condition indicators and the FA algorithm, section ‘‘Study on an actual Pelton turbine’’ presents the application of the proposed method and two comparative methods on the real case, in section ‘‘Results,’’ the failure prediction performance of the proposed indicators is compared with the conventional methods, and section ‘‘Conclusion’’ provides the concluding remarks. Methodology for improved monitoring of Pelton turbines Definition of new condition indicators based on dynamic analysis Typical vibration condition indicators are effective for detecting some types of damage, but in many cases, the symptoms are only perceptible when damage is at an advanced stage. 2,29,30 Thus, it is of high interest to upgrade the current monitoring procedures so that any variation in the behavior of the turbine is rapidly detected by the system and the potential effects on the lifetime of the structure are assessed. In Figure 2, a sketch of a typical Pelton turbine has been represented. The excitation force is applied to the runner and vibrations are measured in the bearings with accelerometers. As indicated above, the jet impinges directly on the runner buckets generating strong vibrations that depend on the turbine structural response. In this type of turbine, the natural frequencies more prone to be excited are the ones of the runner. One of the issues, as in many hydraulic turbines, is that runner vibrations can hardly be detected by monitoring vibrations in the bearing because natural frequencies of the runner do not produce important deformations on the rotor. In Figure 3, the deformations in a runner and rotor when excited by the jet have been represented. The numerical simulation was done with the finite element method (FEM) and checked experimentally. It can be observed that while the runner has large deformations, the rotor is barely affected. In case of damage or abnormal operation, the runner dynamics varies and these changes have to be detected by the monitoring system. Figure 2. Pelton sketch showing the excitation and the monitoring position. Zhao et al. 3 3242 Structural Health Monitoring 20(6) A more detailed dynamic analysis of Pelton runners 31 indicates that the runner response is quite complicated, with many natural frequencies and modeshapes (tangential, axial, and radial). They are close to each other with high modal density and covering a high-frequency range well above the rotor natural frequencies. In Figure 4, the first natural frequencies of a Pelton runner and associated mode-shapes detected during the start-up transient are indicated. The main forces affecting a Pelton turbine during operation are of mechanical, hydraulic, and electromagnetic origin. The main hydraulic force comes from the impingement of the water jet on the buckets. Every time a bucket is in front of an injector, the runner receives a strong impact. During the impact, runner natural frequencies are excited. In normal operating conditions, the tangential modes are basically the ones to be excited (jets hit the runner tangentially), but with abnormal operation or incipient damage, other types of natural frequencies are excited too. The emergence of incipient damage in a structure may lead to a change in its dynamic response, which subsequently can be reflected in the vibration behavior. 13,32 The point is to detect the change in the runner vibrations and associate them with abnormal operation and damage. For improving the condition monitoring, the natural frequencies of the rotor and of the runner have to be monitored. In the new condition indicators, besides the typical synchronous bands for unbalance, misalignment, and bucket passing frequencies, other bands related to the natural frequencies of the structure have been considered. Dimension reduction of indicators based on FA and PCA In this article, two different methods are used to reduce the dimension of the indicator sets. These methods are the so-called PCA and FA. The procedure of PCA is shown in Figure 5. By PCA, an indicator matrix is rotated into a new projection matrix. The projection vectors (each being a linear combination of the variables) are an uncorrelated orthogonal basis set. The first PC has the largest possible variance, and each succeeding component in turn has the highest variance possible under the constraint that it is orthogonal to the preceding components. Only the components with highest eigenvalues are retained for the analysis. 22 In an FA model, each condition indicator can be represented as a linear combination of the common factors and specific factors 33 Xd3n=md3n+Ld3MFM3n+Ed3nð1Þ where Xis the indicator matrix with nindicators (n dimension) and dsamples, mis the mean value of each column of X, and Lis a constant d3Mmatrix of factor loadings. The (i,j)th element of the d3Mmatrix L is the coefficient, or loading, of the jth factor for the ith variable. Fis a matrix of Mcommon factors of the indicator matrix, and Eis a vector of independent specific factors which represents the portion that cannot be explained by the common factors. In the indicator matrix, all of the indicators share these Mcommon factors and each can be represented as a linear function of common factors and its specific factor Figure 3. Deformations of a Pelton runner. Figure 4. Excitation of the first runner natural frequencies during start-up of a Pelton turbine. Figure 5. Procedure of PCA. 4Structural Health Monitoring 00(0) Zhao et al. 3243 Xi=mi+ai1F1+ai2F2+���+aiM FM+ei,i=1,2,...,dðÞ ð2Þ The indicators are related to the common factors by the factor loadings and can be classified into different common factors according to the loadings. Moreover, each common factor usually has a physical explanation regarding the dynamic behavior of the machine, which is an important advance compared to PCA. The main steps of FA are the following: 1. Calculating the eigenvalue of the covariance matrix of the input samples. The eigenvalues are the key for deciding the number of the common factors. According to the literature reviewed, a scree test is the most common and accurate criterion: 34 retaining all the factors above (i.e. to the left of) the inflection point of the scree plot of the eigenvalues. 2. Loading matrix calculation. With damage, some condition indicators severely change their values, especially when the machine approaches failure. Therefore, principal FA is applied in the proposed method, which is suitable for corrupt data. 23 According to PC method, the loading matrix Lis computed by the following equation Ld3M=ffiffiffiffiffi l1 ph1,ffiffiffiffiffi l2 ph2,...,ffiffiffiffiffiffi lM phM �� ð3Þ where l 1 ˜l 2 ˜... ˜l M are the mlargest eigenvalues of the correlation coefficient matrix of X, and h 1 ,h 2 , ..., h M are the corresponding orthonormalized eigenvectors of the eigenvalues. 3. Factor rotation. The idea of factor rotation is to transform the loading matrix into a structure where each of the retained factors is ideally loaded on fewer variables, that is, to make sure that the projection of each variable on the rotated factor axis is the largest (or the lowest). By factor rotation, the new factor loading matrix can be simplified and the factors have more physical meaning or interpretability. Failure prediction by ANN regression As one of the most common regression tools, a supervised ANN is developed in order to build up the relation between condition indicators and damage. Since a single hidden layer ANN with sufficient number of neurons will be suitable to fit any continuous function, 35 a feedforward neural network with one hidden layer as shown in Figure 6 has been used. The input layer consists of neurons for the normalized monitoring indicators, and the neurons in the hidden layer are defined as sigmoid activation functions. The output layer contains one neuron which is equal to a parameter related to a specific damage such as crack length. The LM (Levenberg–Marquardt) backpropagation function is applied as the update function Dw=JTJ+lI½� �1JTeð4Þ where Jis the Jacobian matrix of the partial derivative of Ewith respect to vor w. I and erepresent an identity matrix and output error vector, respectively. lis large in the beginning of the learning phase, which increases the convergence speed and then decreases in order to make the approximation more accurate. The ANN model will be trained by the indicators and be used for damage detection. Study on an actual Pelton turbine Case description An existing Pelton turbine prototype has been investigated to verify the proposed method. It is a horizontal shaft machine consisting of two runners, one shaft, and a generator. The structure is supported by two bearings, located between each one of the runners and the generator. Both runners have 22 buckets and each one of them is operated by one jet. The main characteristics of the machine are listed in Table 1. The researched turbine had been monitored by a monitoring system for 13 years. A total of five Figure 6. Architecture of the ANN model. Table 1. Basic operation parameters of the turbine unit. Parameter Value Rated head 770 m Maximum power 34 MW Rotating speed, f f 600 r/min (10 Hz) Number of buckets, Z b 22 Bucket passing frequency, f b 220 Hz Zhao et al. 5 3244 Structural Health Monitoring 20(6) accelerometers were installed on both bearings in the radial and axial directions, as seen in Figure 7. Accelerometers A11 and A31 were installed horizontally in the same direction in which the jet impinges the runner. A14, A34, and A35 were installed in the vertical and axial directions, respectively. The data were collected when the machine was operating at 40% and 100% of the rated output. Therefore, the measurement points are named after sensor and output (e.g. A14-04 and A14-10 represent measurement point A14 at 40% and 100% rated output, respectively). For the analysis, a history case has been selected. On 29 August 2011, the system detected an abnormal root mean square (RMS) vibration value surpassing the alarm threshold and the machine was stopped. The following inspection revealed the cause of the increased vibration: a piece of one bucket broke off while in operation and, in addition, cracks were present in the same area of several buckets (Figure 8). The inspection of the runner indicated that the incident took place due to the fatigue of the material, for which it was assumed that the machine had not been operating properly for some time. Machine vibrations in all the monitoring locations were available with the machine in good condition, with incipient damage and with severe damage. 36,37 After that, the machine was repaired and put again into service. New condition monitoring method In order to implement the new condition indicators on the monitoring system, an extensive analysis of the dynamic behavior of the turbine was carried out. 31,38 First, the natural frequencies of the machine were determined by means of numerical and experimental analysis on the prototype. After that, the response (i.e. vibration) of the machine under the dynamic loading of the water jets was studied. These investigations provided a deeper understanding of the operation of the Pelton turbine and allowed defining new frequency bands related to the most excitable modes of the runner. Besides the typical synchronous bands for unbalance, misalignment, and bucket passing frequencies, Table 2. The frequency ranges of the spectral bands. Mode/frequency Mode description Abbreviation Frequency range f f and harmonics f f f f 9–11 Hz 2*f f 2f f 19–21 Hz 3*f f 3f f 29–31 Hz Rotor modes Rotor mode 1 RoM1 25–38 Hz Rotor mode 2 RoM2 102–115 Hz Rotor mode 3 RoM3 115–134 Hz f b and harmonics f b f b 219–221 Hz 2*f b 2f b 439–441 Hz 3*f b 3f b 659–661 Hz Runner modes Axial runner mode RuM1 510–550 Hz Tangential runner mode RuM2 600–670 Hz Rim runner mode RuM3 670–700 Hz Figure 7. Sketch of the Pelton turbine and layout of the vibration sensors. Figure 8. Damage in a bucket. 6Structural Health Monitoring 00(0) Zhao et al. 3245 other bands related to the natural frequencies of the structure were devised. The latter is divided into two main groups: the ones corresponding to the rotor’s natural frequencies and the ones belonging to the runner. The rotor modes involve the deformation of the whole turbine and are found at frequencies below 300 Hz. The runner modes are found over 500 Hz and only involve the deformation of the runner and/or the buckets. Only the most relevant harmonic bands are selected according to the main excitation forces on the Pelton turbine (f f ,2f f ,3f f ,f b ,2f b , and 3f b ) (Figure 9). The proposed new condition indicators are listed in Table 2. For each measuring position, 12 spectral bands are selected as damage indicators. After extracting the new condition indicators based on the dynamics of the turbine, FA technique is used to reduce the dimension of the database and select the most relevant indicators. FA is preferred over PCA for dimension reduction as the physical meaning of the indicators is retained. The measurement point A34-10 is taken as an example to describe the process. First, the number of common factors is determined by the scree test. 26 The relative eigenvalues, as well as their cumulative percentages, are displayed in Figure 10. From the scree plot, it can be seen that the inflection point is the third one, which indicates that three common factors are sufficient for retaining most of the information of the indicator matrix. After the retained number has been determined, FA is conducted on the indicator matrix. The loading matrix is calculated according to equation (3). The loadings of each indicator on the three common factors are listed in Table 3, as well as the module of each indicator, which is the Euclidean norm of the loadings. Taking advantage of three loading factors, the loading matrix of point A34-10 can be represented in a Figure 9. Vibration spectrum of the Pelton turbine and spectral bands. Table 3. Loading factor of each damage indicator of point A34-10. Indicator Factor 1 Factor 2 Factor 3 Module f f 20.4337 0.1769 0.3147 0.5643 2*f f 0.0387 20.0440 20.8386 0.8406 3*f f 20.3144 0.8485 0.3779 0.9806 Rotor mode 1 20.2751 0.9032 0.3218 0.9975 Rotor mode 2 0.0368 0.4883 0.0200 0.4901 Rotor mode 3 20.4351 0.2472 0.3843 0.6310 f b 20.0392 0.1362 0.2187 0.2606 2*f b 20.4199 0.5066 0.7337 0.9855 3*f b 0.9085 0.0224 20.0414 0.9097 Runner mode 1 20.4024 0.6606 0.5677 0.9595 Runner mode 2 0.8938 20.1856 20.2492 0.9463 Runner mode 3 0.8416 20.1578 20.0389 0.8571 Figure 10. Scree plot of the PCA result of point A34-10. Zhao et al. 7 3246 Structural Health Monitoring 20(6) three-dimensional (3D) plot, which is helpful for visualizing and comparing the loadings of the indicators. The loading vectors are shown in Figure 11. Thus, the direction and length (module) of each vector indicate its relevancy with the common factors. For example, there are four indicators with negative loadings on the first common factor and seven with positive loadings. The indicator 3*f b has the smallest angle with the axis and the largest loading value, which indicates 3*f b has the greatest dependency with the first common factor. On the contrary, the length of vector f b is 0.2606, which is too small to represent any common factor. The three common factors are calculated and displayed in Figure 12, as well as their corresponding indicators. The data before and after repair are separated by a dash line. According to Table 3 and Figure 11, the indicators 3f b and RuM2 and RuM3 have the highest loading value on the common factor 1, which means they have a similar trend with each other before and after failure. The evolution of these three indicators as well as the corresponding factor are shown in Figure 12(a). Before the failure, their values grow gradually, and after repair, they increase dramatically. In the whole range, their values are higher after repair than before failure. However, factor 2 (Figure 12(b)) shows an inverse trend to factor 1 since the value plummets after repair. Being similarly, it shows a slight increase before failure, as factor 1. The indicators 3f f , RoM1, and RoM2 have a high relevancy with this common factor. As for factor 3 (Figure 12(c)), the value is higher in good condition than damage condition. However, there is no significant trend during failure. The FA can be regarded as a modal decomposition process: the vibration behavior of the structure is a superposition of different modes. Therefore, the indicators with high loading on the common factors are the most significant indicators and the indicators which have low loading on every factor are regarded as redundant components. For the measurement points, the indicators with loading module larger than 0.5 are selected as the most significant indicators. The same process has been carried out on the indicators of every measurement point and the modules have been listed in Table 4. The indicators with modules lower than 0.5 (marked in bold) were eliminated from the indicator matrix. In this way, the dimension of the indicator set is reduced. Comparative condition indicators (conventional indicators) Two comparative sets of indicators are extracted from the raw signal and used for training the ANNs to Figure 11. Loading factors of each new indicator on the three common factors. Figure 12. Common factors and the corresponding damage indicators (a to c). 8Structural Health Monitoring 00(0) Zhao et al. 3247 compare their respective damage detection performance with the newly proposed method. In these two cases, the typical spectral bands of this type of machinery are selected as the condition indicators without considering the specific dynamic characteristics of the Pelton turbine studied. The first condition indicator is the spectrum level of the turbine. The whole vibration spectrum (magnitude of the spectrum after applying the fast Fourier transform) is considered in order to build this set of indicators. According to the configuration of our monitoring system, the frequency range is 3–800 Hz with a resolution of 0.5 Hz, so there are 1595 lines in each spectrum. For each measuring point, the input pattern contains 1595 columns. The second set encompasses the harmonics levels of the spectrum. The spectral bands of the rotating speed and its harmonics are excited by unbalance or misalignment of the rotor. Different types of damage can be reflected by changes in the harmonics of f f . 15,30,39 The energy of a spectral band with a width of 2 Hz and containing each harmonic is calculated according to equation (2). Since the range of the spectrum is 3;800 Hz, 78 indicators are extracted from the raw Table 4. Loading factor modules. Indicator A11-04 A11-10 A14-04 A14-10 A31-04 A31-10 A34-04 A34-10 A35-04 A35-10 f f 0.89 0.92 0.45 0.37 0.65 0.75 0.61 0.56 0.79 0.87 2f f 0.21 0.91 0.53 0.86 0.86 0.93 0.62 0.84 0.53 0.87 3f f 0.76 0.99 1.00 1.00 1.00 1.00 0.99 0.98 0.97 1.00 RoM1 0.77 0.96 0.91 0.89 1.00 1.00 1.00 1.00 1.00 0.99 RoM2 0.96 0.92 0.92 0.89 0.74 0.51 0.74 0.49 0.80 0.22 RoM3 0.78 0.93 0.91 0.95 0.95 0.22 0.92 0.63 0.94 0.54 f b 0.69 0.69 0.75 0.87 0.90 0.77 0.61 0.26 0.43 0.44 2f b 0.59 0.51 0.48 0.84 0.75 0.58 0.82 0.99 0.33 0.80 3f b 0.42 0.23 0.09 0.54 0.97 0.61 1.00 0.91 0.88 0.84 RuM1 0.97 0.96 0.94 0.98 0.86 0.94 0.91 0.96 0.91 0.93 RuM2 0.99 0.93 0.80 0.89 0.58 1.00 0.77 0.95 0.94 0.97 RuM3 0.70 0.44 0.67 0.52 0.98 0.92 0.97 0.86 1.00 0.95 Note. The modules lower than 0.5 are marked in bold. Figure 13. PCA scree plot of the ‘‘spectrum level’’ (a) and ‘‘harmonic level’’ (b) indicators’ sets on point A31-04. Figure 14. The curve of real crack size and the artificial damage index. Zhao et al. 9