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Double amplitudes of the vibration displacements of #4 hydro-turbine-generator under different working conditions
Due to the frequently occurred adverse vibration of hydraulic structures, vibration risk assessment is significant for the water energy efficiency of hydropower station and the safety of people and structures. Recently, the abnormal vibration of hydro-turbine-generator in a large hydropower station occurred and the main influencing factors of vibration are analyzed based on the prototype data and engineering experience. Different from the deterministic variable features in traditional support vector domain description (SVDD) algorithms, the feature of vibration amplitude is actually a random variable so that the different target objects will be obtained at different confidence levels. In order to assess the vibration range and excessive vibration probability, the original SVDD boundary at relatively low confidence level is firstly calcu- lated. Then, the boundary extension operation with detailed theoretical deduction is performed and the extended boundary is further optimized inspired by path planning problem. The advantage of proposed approach is that it can improve the data fitting performance for single dimension (i.e. vibration amplitude) without leading to complex bound- ary which cannot be used for vibration risk assessment. By applying this approach to the practical vibration problem, the quantitative and slightly conservative assessment results are conveniently obtained, which indicate that this approach is reasonable and cost-effective. Keywords Vibration risk assessment, support vector domain description technique, boundary optimization, artificial potential field, vibration amplitude range, excessive vibration probability, vibration induced by high dam flood discharge Introduction Due to the important need for comprehensive utilization of water resources, many large water conservancy projects characterized by high head, high-speed flow and tremendous flood discharge have been built or are currently under construction worldwide, especially in Southwest China. Therefore, the tremendous energy dissi- pated in the flood discharge process may induce intense vibration of the hydraulic structures. In the past few years, Yellow River Engineering Consulting Co., Ltd, Zhengzhou, China State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin, China Sanjiangyuan Collaborative Innovation Center, Qinghai University, Xining, China Post-Doctoral Mobile Station of Water Conservancy Engineering, Hohai University, Nanjing, China Corresponding authors: Chao Liang, Post-Doctoral Research Center, 109 Jinshui Road Zhengzhou, Henan Province, Zhengzhou 450003, China. Email: liangchao_0016@sina.cn Yuansheng Zhang, State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, 135 Yaguan Road, Jinnan District, Tianjin, Tianjin 300350, China. Email: zhangyuanshengpp@foxmail.com Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (https:// creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage). 2 1310 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) the vibration problems generated by high dam flood discharge are studied and beneficial results are obtained by 1 2 researchers. Darbre et al. and Proulx et al. investigated the effects of water level on the dynamic characteristics of Mauvoisin arch dam and Emosson arch dam in Switzerland based on prototype dynamic test data. Li and Lian performed vibration prediction, safety evaluation and structural optimization design for a hydraulic radial gate using hydroelastic model experiment and numerical simulation method. In order to further investigate the vibration problem induced by flood discharge, Lian and Huokun proposed improved modal identification method for hydraulic structures based on ERA algorithm, developed the static-dynamic coupling analysis approach of counter-arched slab in plunge pool to evaluate its operation state, summarized the frequently occurred engineering problems and corresponding appropriate solutions during the discharge processes of hydro- 6,7 8 9,10 power projects, and clarified a new vibration mechanism for the spillway guide wall. He et al. proposed 11,12 improved theoretical calculation approach and sensor optimal arrangement method, respectively. Zhang et al. proposed novel methods to precisely calculate structural vibration modes. Moreover, the research on the dynam- ics of hydroelectric generation system has attracted a lot of attentions. Some useful achievements have been made 13 14 in the aspects of new approaches for system fault identification, optimal maintenance scheme for the system, 15,16 dynamic response analysis of the nonlinear system under adverse conditions and system dynamic stability evaluation. However, the influence of flood discharge from dam body on the dynamic characteristics of hydro- power units is not involved in the relevant researches. Furthermore, the near-field vibrations induced by flood discharge of several large hydropower engineering 17–20 projects are recently observed and have aroused significant attentions. If the vibrations of hydraulic structure and surrounding ground reach a certain magnitude, the crack propagation in structure and abnormal operation of equipment will be induced and adverse effects on safety of nearby buildings, physical and mental health of the hydropower station staff and surrounding residents will be generated. In order to effectively solve this problem, 17,18 18,21 19 the vibration source identification, vibration propagation law, vibration prediction, optimal operation 18,19 20 scheme and dynamic vibration damping method are studied by the researchers and engineers for the ground vibration problem induced by high dam flood discharge. Therefore, it is of great importance to predict, monitor and control the adverse vibrations of hydraulic structures and surrounding ground. In recent several years, significant progress has been made in the study of artificial intelligence which is rec- ognized as one of the most promising research fields. It is believed that the combination of conventional research and artificial intelligence is the inevitable trend of engineering technology development. Due to the limited knowl- edge on fluctuating pressure generation mechanism of high velocity flow, complex nonlinear dynamic character- istics of structures, flow-structure – electromagnetic coupling effects, the causes and mechanisms of abnormal vibration of hydropower station are sometimes difficult to clarify, and thus the statistical-based artificial intelli- gence algorithm should be applied to predict, monitor and control the structure vibration without understanding the detailed generation mechanism of vibration. Yuan et al. utilized the neural network approach to establish the flow-induced vibration of high arch dam. Xu et al. proposed a prediction method for powerhouse structure vibration based on flies optimization algorithm and generalized regression neural network. As a class of efficient 24 25 intelligence algorithm, support vector machine (SVM), support vector domain description (SVDD) and their modified versions have been widely studied, such as the multi-kernel learning support vector regression, bound- 27 28 ary optimized SVDD algorithm, combined method of SVM and particle swarm optimization algorithms, 29 30 combined method of SVM and fast messy genetic algorithms and modified least squares SVM. Moreover, the related methods are frequently used to solve the problems of seismic risk assessment, bearing fault detection 32 33 and diagnosis, flood hazard risk assessment and so on. Recently, the related methods are applied in hydraulic structure vibrations to perform damage diagnosis of spillway guide wall, time–frequency domain characteristic estimation for the vibration of bulb hydrogenerator hydropower house and prediction of flow- induced pipeline vibration. Aiming at the abnormal vibration problem of hydro-turbine-generator that is recently occurred in a large hydropower station, the SVDD-based post-processing approach is proposed to accurately and conveniently assess the vibration range and excessive vibration probability for the hydro-turbine-generator. The proposed approach can deal with the target objects with deterministic and random variable features and is able to improve the data fitting accuracy in single dimension (i.e. vibration amplitude) without leading to extremely complex boundary that is not applicable to risk assessment. The reminder of this article is organized as follows. The next section described the basic engineering information and practical vibration problem of the hydropower station and analyzed the spatial distribution and main influencing factors of the vibration. In the subsequent section, the theoretical basis, calculation procedure and numerical verification of the proposed approach based on Gauss kernel SVDD tech- niques are described in detail. Moreover, the proposed approach is applied to the practical vibration problem of Zhang et al. Zhang et al. 13113 the hydro-turbine-generator in the penultimate section and the results analysis and discussion are included. Finally, a conclusion is given. Prototype dynamic experiment for the hydropower station Vibration problem of the hydropower station Basic information of the hydropower station. The large hydropower station is located in a small county town in Sichuan Province. Electricity generation is the major task of this hydropower station and it can also improve comprehen- sive utilization of water resources in downstream. The dam is 69.5 m in height and 439.73 m in width, with a 3 3 reservoir capacity of 9.12 107 m and planned flood discharge of 18,300 m /s. Moreover, the water retaining dam section, hydropower house dam section, discharge sluice dam section, diversion channel dam section and water retaining dam section are arranged from the left bank to right bank in this water conservancy project. As shown in Figure 1, the general arrangement for the hydraulic structures of this project is illustrated. This hydropower station consists of four hydro-turbine-generators of which the single unit capacity is 150,000 kW. The structural components of the hydro-turbine-generator are illustrated in Figure 2. Abnormal vibration problem of #4 generator unit. On 18 July 2016, the #4 hydro-turbine-generator is started at 9 a.m. and the load of generator is increased to 100 WM after 5 min. Then, the hydro-generator operated in normal state until the intense vibrations of stator foundation, generator lower racks and turbine head cover occurred at 9:22 a.m. and the oscillation amplitude of hydraulic turbine bearing suddenly increased from 66 mm in x direction Figure 1. Hydraulic structure arrangement of this project. Figure 2. Structure components of hydro-turbine-generator in this hydropower station. 4 1312 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Figure 3. Layout of dynamic displacement sensors. Figure 4. Photos of sensor installation. and 70 mm in y direction to 189 mm and 228 mm, respectively. Consequently, the conventional monitoring system gave an alarm because the structural vibration seriously exceeded the standard. It is noted that the water head of hydro-generator, discharge from dam body, openings of #1 to #4 gates and openings of #5 to #7 gates are 19 m, 3690 m /s, 4.7 m and 0 m, respectively, and the #1 to 3# hydro-turbine-generators are not put into operation when the abnormal vibration occurred. In order to ensure the safety of the hydropower station, the #4 generator is stopped and a series experiments are carried out to test its dynamic behaviors under different load conditions in dry season. However, no abnormal vibration occurred during the experiments, which means that the shaft structures can work normally and there are no imbalances of structural mass and magnetic force. Therefore, it is believed that the abnormal vibration is related to both generator operation state and flood discharge condition. In order to further investigate the dynamic behavior of the #4 hydro-turbine-generator, the dynamic prototype experiments are carried out in flood season. Prototype dynamic experiment Experiment scheme. As shown in Figure 3, the layout of dynamic displacement sensors installed on #4 hydro- turbine-generator is illustrated. The photograph of the installation for the dynamic displacement sensors in prototype is given in Figure 4. Comprehensively considering the requirements from hydropower station operation Zhang et al. Zhang et al. 13135 Figure 5. Dynamic testing system applied in prototype experiment. and experimental research, the working conditions in the prototype dynamic experiment are determined to ana- lyze the #4 unit vibration law with the variations of generator load, gate opening, flood discharge volume, hydro- generator water head and so on. Due to space limitations, the detailed description for the working conditions is given in Appendix 1. As shown in Figure 5, the dynamic testing system applied in prototype experiment is illustrated and the detailed information for the dynamic displacement sensor and Data Acquisition & Signal Processing system are listed. Experiment results analysis. As shown in Figure 6, the root mean square (RMS), double amplitude and dominant frequency for the vibration of #4 unit in some typical working conditions are illustrated. The detailed description for the working conditions is shown in Appendix 1. In order to comprehensively demonstrate the vibration condition of this hydro-turbine-generator under all the working conditions in prototype test, detailed vibration data analysis results, including RMS, double amplitude and dominant frequency are given in Appendix 2. It is noted that the vibrations measured by sensors H1 and H3 are more intense than the vibrations measured by other sensors in most working conditions, which indicates that the excessive vibration are most likely to be firstly generated on the turbine head cover. Therefore, the vibration signal measured by sensor H3 in vertical direction on turbine head cover is chosen to be the main object of study in subsequent analysis. In order to further investigate the vibration problem of #4 unit, the RMSs for the vertical vibration of turbine head cover measured by sensor H3 in different working conditions are illustrated in Figure 7. It is noted that the “condition group” shown in the legend of Figure 7 is consist of the working conditions with similar operating conditions and step-changed #4 unit loads, which can be seen in Appendix 1. For the working con- ditions 13 to 17 in condition group 5, the #4 unit loads are approximately 60, 80, 100, 120 and 140 MW, respec- tively. Moreover, the #4 unit loads for different wording conditions in other condition groups are approximately set to 80, 100 and 120 MW. From Figure 7, it is noted that the vibration RMSs measured in condition groups 5 to 7 are significantly larger than those measured in condition groups 1 to 4, and the vibration RMSs for the conditions with relatively high #4 unit load are obviously greater than those for the conditions with relatively low #4 unit load. Outwardly, the vibration amplification effect can be generated under both working conditions with large abandoned water flow and heavy load of #4 unit. In order to identify the operating parameters of this hydropower station that has significant influence on the turbine head cover vibration, the simple correlation analyses between vibration RMSs and operating parameters listed in Appendix 1 are carried out. It is noted that no single factor is able to have a linear dominant effect on the vibration amplification and the operating parameters, such as the openings of #1 to #4 gate, #4 unit load, abandoned water flow and discharge flow from #4 unit have relatively high correlations with the vibration RMSs of turbine head cover. Due to the space limitation, only the variation of vertical vibration RMSs with #4 unit loads is illustrated in Figure 8. Therefore, the water is mainly discharged from the diversion channel dam section instead of the discharge sluice section to reduce the #4 unit vibration when the total discharge volume is relatively small. When the total discharge volume is large, considerable flood is discharged from the discharge sluice dam section and #4 hydro- power unit stops operation due to the flexibility of the power grid dispatching rules. Obviously, this traditional 6 1314 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Zhang et al. 7 Figure 8. Vertical vibration RMSs of turbine head cover with different discharge flow from #4 unit. Analysis for the main influencing factors of #4 unit vibration Based on the test result from the manufacturer of #4 hydro-turbine-generator and the operating experience from operation and maintenance personnel, it is almost certainly that there are no imbalances of structural mass and magnetic force in the course of #4 unit operation. The abnormal excessive vibration is generated under specific working conditions with large discharge flow from #1 to #4 hydraulic gates and heavy load of #4 unit. According to the current analysis, it is considered that the abnormal vibration of #4 unit mainly includes three vibration components. Firstly, since the hydraulic structures (such as the gates and gate piers) in the discharge sluice dam section (shown in Figure 1) vibrate strongly when the water discharges from this dam section, the vibration of the hydraulic structure can propagate to #4 hydropower unit and contribute to its abnormal vibration. Secondly, the water flow discharged from the discharge sluice dam section may affect the flow pattern and hydraulic excitation in #4 hydropower unit from upstream side or downstream side, and this can induce the abnormal vibration component. Thirdly, it is inevitable that the vibration will be generated during the operation of hydro-turbine-generator. Although this vibration component is not very large, it cannot be ignored and will be superimposed with the vibration components caused by other reasons to form the abnormal vibration. Figure 6. The RMSs, double amplitudes and dominant frequencies of #4 unit vibrations in partial typical working conditions. (a) The vibration components caused by the above three mechanisms and their coupling effects are currently con- Vibration RMSs; (b) double amplitudes; (c) dominant frequencies. sidered to be the generation mechanism of this engineering problem. As shown in Figure 1, #4 hydropower unit is located closest to the discharge sluice dam section so that the aforementioned abnormal vibration components induced by the first two reasons can be quite large. It is obvious that the first two abnormal vibration components of other three hydropower units can be significantly weaker than those of #4 hydropower unit. This is considered to be the main reason why the other three same hydro- turbine-generators are not affected by the abnormal vibration. Based on the aforementioned possible generation mechanism of abnormal vibration, some potential engineer- ing measures can be applied to reduce the vibration of hydropower unit. In order to cut off the vibration prop- agation path from hydraulic structures to hydropower unit, the damping material (such as the shock absorption rubber) should be filled into construction joints between the hydraulic structure and hydropower unit. In order to reduce the influence of water flow discharged from dam body on the flow pattern and hydraulic excitation in hydropower unit, the guide walls in upstream and downstream sides should be extended as long as possible. It is worth pointing out that the relevant engineering measures should be considered in design stage. Once the hydro- power project has been completely finished (just as the project in this paper), most of the engineering measures (including the aforementioned two engineering measures) are hardly to be realized in practice due to economic consideration, uncertainty of actual effect and the concern of inducing other new problems. Since the generation Figure 7. RMSs for the vertical vibration of turbine head cover in different working conditions. mechanism of abnormal vibration is not the focus of this research and actually, it is still not very clear, the further analysis and discussion on the causes of abnormal vibration and the relevant engineering optimization measures operation scheme will decrease the efficiency of power generation and increase the burden of diversion channel. are not involved in this paper. Therefore, the vibration risk under different working conditions should be reasonably evaluated so that a com- For this engineering problem, it is very difficult to precisely clarify the detailed cause and mechanism for the promise operation scheme can be developed to alleviate the above problem. abnormal vibration and effectively reduce the vibration by precisely invalidating the vibration source and cutting Zhang et al. Zhang et al. 13157 Figure 8. Vertical vibration RMSs of turbine head cover with different discharge flow from #4 unit. Analysis for the main influencing factors of #4 unit vibration Based on the test result from the manufacturer of #4 hydro-turbine-generator and the operating experience from operation and maintenance personnel, it is almost certainly that there are no imbalances of structural mass and magnetic force in the course of #4 unit operation. The abnormal excessive vibration is generated under specific working conditions with large discharge flow from #1 to #4 hydraulic gates and heavy load of #4 unit. According to the current analysis, it is considered that the abnormal vibration of #4 unit mainly includes three vibration components. Firstly, since the hydraulic structures (such as the gates and gate piers) in the discharge sluice dam section (shown in Figure 1) vibrate strongly when the water discharges from this dam section, the vibration of the hydraulic structure can propagate to #4 hydropower unit and contribute to its abnormal vibration. Secondly, the water flow discharged from the discharge sluice dam section may affect the flow pattern and hydraulic excitation in #4 hydropower unit from upstream side or downstream side, and this can induce the abnormal vibration component. Thirdly, it is inevitable that the vibration will be generated during the operation of hydro-turbine-generator. Although this vibration component is not very large, it cannot be ignored and will be superimposed with the vibration components caused by other reasons to form the abnormal vibration. The vibration components caused by the above three mechanisms and their coupling effects are currently con- sidered to be the generation mechanism of this engineering problem. As shown in Figure 1, #4 hydropower unit is located closest to the discharge sluice dam section so that the aforementioned abnormal vibration components induced by the first two reasons can be quite large. It is obvious that the first two abnormal vibration components of other three hydropower units can be significantly weaker than those of #4 hydropower unit. This is considered to be the main reason why the other three same hydro- turbine-generators are not affected by the abnormal vibration. Based on the aforementioned possible generation mechanism of abnormal vibration, some potential engineer- ing measures can be applied to reduce the vibration of hydropower unit. In order to cut off the vibration prop- agation path from hydraulic structures to hydropower unit, the damping material (such as the shock absorption rubber) should be filled into construction joints between the hydraulic structure and hydropower unit. In order to reduce the influence of water flow discharged from dam body on the flow pattern and hydraulic excitation in hydropower unit, the guide walls in upstream and downstream sides should be extended as long as possible. It is worth pointing out that the relevant engineering measures should be considered in design stage. Once the hydro- power project has been completely finished (just as the project in this paper), most of the engineering measures (including the aforementioned two engineering measures) are hardly to be realized in practice due to economic consideration, uncertainty of actual effect and the concern of inducing other new problems. Since the generation mechanism of abnormal vibration is not the focus of this research and actually, it is still not very clear, the further analysis and discussion on the causes of abnormal vibration and the relevant engineering optimization measures are not involved in this paper. For this engineering problem, it is very difficult to precisely clarify the detailed cause and mechanism for the abnormal vibration and effectively reduce the vibration by precisely invalidating the vibration source and cutting 8 1316 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Zhang et al. 9 n n X X off the vibration propagation path, due to the limited knowledge on fluid-solid interaction, damping mechanism, a ¼ 1; a ¼ a T ; C � a � c ¼ 0; 0 � a � C; c � 0 (4) i i i i i i i fluctuating pressure and vibration coherence between hydraulic structures on dam body and #4 unit. A feasible i¼1 i¼1 and effective method for avoiding abnormal excessive vibration of #4 unit in engineering practice is to develop an intelligence algorithm to accurately evaluate the vibration range and the risk of excessive vibration under different Substituting equation (4) into equation (3), the Lagrangian function can be rewritten and the original problem working conditions and to make #4 hydro-turbine-generator operate under the working conditions with low for calculating a hyper-sphere with minimum volume can be described as follows vibration risk. According to the aforementioned analysis, the increases of abandoned water and #4 unit load simultaneously n n X X contribute to the amplification effect for the turbine head cover vibration. Therefore, these two parameters are max a ðÞ T �T � a a T �T ðÞ a i i i i j i j considered as the features in the intelligent algorithm of SVDD for vibration risk assessment so that the effects i¼1 i;j¼1 (5) of flood discharge from #1 to #4 gates and #4 unit load on the vibration amplification of turbine head cover s:t: a ¼ 1; 0 � a � C 8i i i are both considered in the analysis. Although there is a positive correlation between the inflow volume and i¼1 vibration RMS based on prototype test data analysis, the inflow volume is very unlikely to contribute to the vibration amplification effect because the structural vibrations are always very slight during the reservoir stor- When a sphere is not a good fit for the boundary of data distribution, the inner product T �T can be ðÞ i j �� age process with large inflow volume and small discharge. Due to the positive correlation between the inflow generalized by a kernel function kðÞ T ; T ¼ /ðÞ T �/ðÞ T , where a mapping / of the data to a new feature space i j i j volume and abandoned water flow, as well as the abandoned water flow and vibration RMS, the is applied, and k is a positive definite kernel, or Mercer kernel. With such mapping, equation (6) can be deduced inflow volume is outwardly positive correlated to vibration RMS but in essence, the vibration RMS is inde- based on equation (5) pendent of inflow volume. It is noted that the discharge flow from #4 unit is almost linearly proportional to the n n X X #4 unit load because the unit load is theoretically equal to a constant (bulk density of water) multiplied by flow max a kðÞ T ; T � a a k T ; T ðÞ a i i i i j i j volume and water head, and the fluctuation of water head does not exceed 12% of the average value. i¼1 i;j¼1 (6) This implies that the influence of discharge flow from #4 unit on the unit vibration can be rationally repre- n n X X sented by the influence of #4 unit load. Therefore, although the discharge flow from #4 unit also shows a good s:t: a ¼ 1; a ¼ a /ðÞ T ; 0 � a � C 8i i i i i i¼1 i¼1 correlation with the vibration RMS, it is not involved in the feature set. Consequently, three features (i.e. #4 unit load, abandoned water flow and vibration amplitude at a confidence level) are involved in the vibration risk assessment using intelligent algorithm. Extension of the SVDD decision boundary The SVDD-based post-processing approach for vibration risk assessment In some practical problems, the features of the target objects are not all definite and some features should be regarded as random variables. As mentioned in ‘Analysis for the main influencing factors of #4 unit vibration’ Support vector domain description section, the feature set of target objects includes vibration amplitude, discharge volume and #4 unit load. It is In order to obtain the data description for a group of target objects, a hyper-sphere with minimum volume should noted that the vibration amplitude should be appropriately regarded as a random variable and the vibration RMS be found to enclose all or most of these target objects. The mathematical function for the aforementioned hyper- which is frequently used to describe the vibration intensity actually means that the vibration amplitude is dis- sphere can be expressed as follows tributed in the confidence interval [�RMS, RMS] with a certain probability. The probability that vibration amplitude at a certain moment exceeds the vibration RMS is approximately 32% if the distribution of structural ðÞ vibration is normal. Therefore, the RMS is far from representing the possible maximum vibration amplitude. The FR; a; n ¼ R þ C n (1) i i conventional analysis based on the vibration RMS is not able to reasonably estimate the most adverse vibration condition and definitely clarify the risk of excessive vibration. Consequently, a probability-based post-processing where F and n denote the hyper-sphere function and the number of the support vectors, a and R represent the approach can be presented to efficiently calculate the decision boundary to describe the domain of support vectors center and radius of the hyper-sphere, n denotes the slack variable, and the variable C gives the trade-off between of which the random variable feature values fall within the confidence intervals with different confidence levels. simplicity and the goodness of fit. For a test object Y, the distance from the center of the hyper-sphere is given by It is noted that the function FR ðÞ ; a; n satisfies the following constraint 2 2 kY � a k� R (7) 2 2 kT � a k� R þ n (2) i i In the following analysis, it is assumed that h-th feature for the target object T and test object Y, respectively ðhÞ ðhÞ ðhÞ where T denotes the i-th target object in target object set. denoted as x and y , are random variables of which the distributions are normal. If the confidence levels of x i i ðhÞ Incorporating equation (2) in equation (1), the Lagrangian function can be defined as and y change from s to r, the following relationship will be satisfied n n ðhÞ ðhÞ X X �� �� ðhÞ ðhÞ 2 2 2 2 y ¼ uy ; x ¼ ux 8i (8) s r i;s i;r LðÞ R; a; a ; n ¼ R þ C a R þ n � T � 2aT þ a � c n (3) i i i i i i i i i¼1 i¼1 where x and y represent the features of target and test objects, respectively; superscript h denotes the sequence where a � 0 and n � 0 are Lagrange multipliers. number of the random variable feature in the feature set; subscripts s and r represent the confidence levels of i i Assuming equation (3) reaches its minimum, the partial derivatives are equal to 0 and new constraints are random variable features; subscript i relate to the features of target object T ; u is a constant coefficient. obtained Therefore, equation (9) can be deduced based on equation (7) Zhang et al. Zhang et al. 13179 n n X X a ¼ 1; a ¼ a T ; C � a � c ¼ 0; 0 � a � C; c � 0 (4) i i i i i i i i¼1 i¼1 Substituting equation (4) into equation (3), the Lagrangian function can be rewritten and the original problem for calculating a hyper-sphere with minimum volume can be described as follows n n X X max a ðÞ T �T � a a T �T ðÞ a i i i i j i j i¼1 i;j¼1 (5) s:t: a ¼ 1; 0 � a � C 8i i i i¼1 When a sphere is not a good fit for the boundary of data distribution, the inner product T �T can be ðÞ i j �� generalized by a kernel function kðÞ T ; T ¼ /ðÞ T �/ðÞ T , where a mapping / of the data to a new feature space i j i j is applied, and k is a positive definite kernel, or Mercer kernel. With such mapping, equation (6) can be deduced based on equation (5) n n X X max a kðÞ T ; T � a a k T ; T ðÞ a i i i i j i j i¼1 i;j¼1 (6) n n X X s:t: a ¼ 1; a ¼ a /ðÞ T ; 0 � a � C 8i i i i i i¼1 i¼1 Extension of the SVDD decision boundary In some practical problems, the features of the target objects are not all definite and some features should be regarded as random variables. As mentioned in ‘Analysis for the main influencing factors of #4 unit vibration’ section, the feature set of target objects includes vibration amplitude, discharge volume and #4 unit load. It is noted that the vibration amplitude should be appropriately regarded as a random variable and the vibration RMS which is frequently used to describe the vibration intensity actually means that the vibration amplitude is dis- tributed in the confidence interval [�RMS, RMS] with a certain probability. The probability that vibration amplitude at a certain moment exceeds the vibration RMS is approximately 32% if the distribution of structural vibration is normal. Therefore, the RMS is far from representing the possible maximum vibration amplitude. The conventional analysis based on the vibration RMS is not able to reasonably estimate the most adverse vibration condition and definitely clarify the risk of excessive vibration. Consequently, a probability-based post-processing approach can be presented to efficiently calculate the decision boundary to describe the domain of support vectors of which the random variable feature values fall within the confidence intervals with different confidence levels. For a test object Y, the distance from the center of the hyper-sphere is given by 2 2 kY � a k� R (7) In the following analysis, it is assumed that h-th feature for the target object T and test object Y, respectively ðhÞ ðhÞ ðhÞ denoted as x and y , are random variables of which the distributions are normal. If the confidence levels of x i i ðhÞ and y change from s to r, the following relationship will be satisfied ðhÞ ðhÞ ðhÞ ðhÞ y ¼ uy ; x ¼ ux 8i (8) s r i;s i;r where x and y represent the features of target and test objects, respectively; superscript h denotes the sequence number of the random variable feature in the feature set; subscripts s and r represent the confidence levels of random variable features; subscript i relate to the features of target object T ; u is a constant coefficient. Therefore, equation (9) can be deduced based on equation (7) 10 1318 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Zhang et al. 11 () ! ! ! rffiffiffiffi 2 2 2 h�1 n n m n 2 X X X X X T 1 1 1 1 T i ðÞ j ðÞ j ðÞ ðÞ ðjÞ ðÞ j ðÞ h h h ðÞ j � y � a x þ y � a x þ y � a x � 1 (9) 2r ðÞ T ¼ e s ¼ 0; 1; 2; � � � (14) i;j i i s i i;s i i 2 2 2 2 R u R R s! r j¼1 i¼1 i¼1 j¼hþ1 i¼1 where m denotes the total number of the features in feature set. It is noted that the target object may contain several features and the nonlinear mapping actually maps each It is noted that the original SVDD decision boundary expressed by equation (7) is a hypersphere, while the feature of the target object into an infinite-dimensional space, respectively. The general term formula of the ðÞ modified SVDD decision boundary expressed by Equation (9) is a hyper-ellipsoid. The focal points of this hyper- elements generated by the nonlinear mapping of the random variable feature x is given as follows i;s ellipsoid can be expressed as () �� s �� rffiffiffiffi ðÞ h ðÞ h �� x i;s 1 i;s ðÞ h � F ¼ ðÞ a ; a ; � � �a ; þ f ; � � �a ; ; F ¼ ðÞ a ; a ; � � ; �a � f ; � � �a ; (10) 1 1 2 h c n 2 1 2 h c n 2r / x ¼ e s ¼ 0; 1; 2; � � � (15) i;s s! r ðhÞ where F and F represent the two focal points of the hyper-ellipsoid; a ,a , . . .,a denote the coordinates of the 1 2 1 2 n If the confidence level of x changes from s to r, the following relationship will be satisfied i;s original hypersphere center a; parameter f is given by the following equation ðÞ ðÞ h h pffiffiffiffiffiffiffiffiffiffiffiffiffiffi x ¼ ux 8i (16) i;s i;r f ¼ R u � 1 (11) Then, the following relationship can be obtained Consequently, equation (9) can be rewritten in a more concise form as () �� s � � rffiffiffiffi ðÞ h ðÞ 1 1 h �� � x �x u i;s kY � F k þkY � F k� 2uR (12) 1 u i;s 1 2 ðÞ h � 2r / x ¼ e s ¼ 0; 1; 2; � � � (17) i;r s! r �� �� As shown in Figure 9, the process of SVDD decision boundary extension due to the confidence level variation ðÞ ðÞ h h After a simple factorization operation, the relationship between / x and / x can be obtained i;r i;s of the random variable feature is illustrated. It is noted that this boundary extension is a simple linear variation by stretching the original hypersphere in the dimension related to the random variable feature and finally a hyper- �� no �� ðÞ ðÞ ðÞ h h h ellipsoid boundary is obtained. / x ¼ W / x s ¼ 0; 1; 2; � � � (18) i;r s i;s In order to improve the fitness precision of the original and modified SVDD decision boundary, kernel functions should be introduced into the calculation of the original decision boundary. In this section, the where Gauss kernel function is applied to obtain a better fit for the target objects. 8 9 For the target objects T and T , the Gauss kernel function can be defined as follows i j �� 2�� ðÞ < h 1 = x �1 no i;s 2 �� 1 ðÞ � 2 2 2r kT �T k W ¼ �e s ¼ 0; 1; 2; � � � (19) i j s : ; 2 u 2r k T ; T ¼ e (13) ðÞ i j Therefore, the vectors generated by the nonlinear mapping of T with and without considering the confidence where r denotes the width parameter. 0 level variation (denoted as / ðÞ T and /ðÞ T , respectively) satisfy the following relationship. i i The equivalent nonlinear mapping of target object T for Gauss kernel function, /ðÞ T , can be derived based on i i � � �� �� �� �� �� Taylor series expansion and simple formula transforms. It is noted that the original space will be mapped to �� no �� 0 ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ 1 2 h 1 2 h h / ðT Þ¼ ¼ ¼ D �/ðÞ T i / x ; / x ; � � �/ ; x ; � � � / x ; / x ; � � � ;W �/ x ; � � � i infinite-dimensional space by Gauss kernel function so that infinite elements are included in the vector /ðÞ T . The i i i i;r i i s i;s general term formula for the elements in vector /ðÞ T is expressed as follows (20) where hi no ðÞ D ¼ (21.1) 1; 1; � � � ;W ; � � � �� �� �� �� ðÞ 1 ðÞ 2 ðÞ h /ðÞ T ¼ (21.2) i / x ; / x ; � � ; �/ x ; � � � i i i;s Consequently, the decision boundary can be calculated by the following expression 0 2 k/ðYÞ � a k� R (22) Figure 9. Extension of the two-dimensional SVDD decision boundary. T denotes the target object vector; x denotes the feature of (1) where the expression for the hyper-sphere center a considering confidence level variation is given as follows target objects; superscripts 1 and 2 denote the sequence numbers of the features of target objects; x is a random variable with normal distribution; subscripts 1 and 2 relate to the features of target objects T and T , respectively; s and r denote the confidence 1 2 (1) (1) (1) (1) (1) (1) (1) (1) levels for confidence intervals [�x , x ] (or [�x , x ]) and [�x , x ] (or [�x , x ]). 1, s 1, s 2, s 2, s 1, r 1, r 2, r 2, r Zhang et al. Zhang et al. 1319 11 () rffiffiffiffi 1 T i 2r ðÞ T ¼ e s ¼ 0; 1; 2; � � � (14) s! r It is noted that the target object may contain several features and the nonlinear mapping actually maps each feature of the target object into an infinite-dimensional space, respectively. The general term formula of the ðÞ elements generated by the nonlinear mapping of the random variable feature x is given as follows i;s () �� s �� rffiffiffiffi ðÞ h ðÞ h �� x i;s 1 i;s ðÞ h � 2r / x ¼ e s ¼ 0; 1; 2; � � � (15) i;s s! r ðhÞ If the confidence level of x changes from s to r, the following relationship will be satisfied i;s ðÞ ðÞ h h x ¼ ux 8i (16) i;s i;r Then, the following relationship can be obtained () �� s � � rffiffiffiffi ðÞ h ðÞ 1 1 h �� � x �x u i;s 1 u i;s ðÞ h � 2r / x ¼ e s ¼ 0; 1; 2; � � � (17) i;r s! r �� �� ðÞ ðÞ h h After a simple factorization operation, the relationship between / x and / x can be obtained i;r i;s �� no �� ðÞ ðÞ ðÞ h h h / x ¼ W / x s ¼ 0; 1; 2; � � � (18) i;r s i;s where 8 9 �� 2�� ðÞ < h 1 = x �1 no i;s 2 ðÞ � 2r W ¼ �e s ¼ 0; 1; 2; � � � (19) s : ; Therefore, the vectors generated by the nonlinear mapping of T with and without considering the confidence level variation (denoted as / ðÞ T and /ðÞ T , respectively) satisfy the following relationship. i i � � �� �� �� �� �� �� no �� 0 ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ 1 2 h 1 2 h h / ðT Þ¼ ¼ ¼ D �/ðÞ T i / x ; / x ; � � �/ ; x ; � � � / x ; / x ; � � � ;W �/ x ; � � � i i i i;r i i s i;s (20) where hi no ðÞ D ¼ (21.1) 1; 1; � � � ;W ; � � � �� �� �� �� ðÞ 1 ðÞ 2 ðÞ h /ðÞ T ¼ (21.2) i / x ; / x ; � � ; �/ x ; � � � i i i;s Consequently, the decision boundary can be calculated by the following expression 0 2 k/ðYÞ � a k� R (22) where the expression for the hyper-sphere center a considering confidence level variation is given as follows 12 1320 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Zhang et al. 13 Figure 10. Decision boundaries calculated by Gauss kernel SVDD technique with different width parameters. (a) Width parameter r ¼ 80; (b) width parameter r ¼ 18; (c) width parameter r ¼ 13. n n X X Figure 11. Comparison of the original, extended and recalculated boundaries. 0 0 a ¼ a / ðT Þ¼ a � D � /ðÞ T (23) i i i i i¼1 i¼1 above two conditions very well, because the boundary curve is either not a single-valued function of abscissa in upper (or lower) half-plane or has a relatively low fitting degree. According to equation (22), the function of extended SVDD decision boundary can be immediately obtained when the confidence level of random variable feature changes from s to r as long as the decision boundary at the s Comparison of the original, extended and recalculated boundaries. In order to obtain an acceptable original boundary at confidence level of random variable feature is determined. It is noted that the decision boundary function is quite relatively low confidence level, the additional target objects should be added in the target object set to improve the complex and may be inconvenient for practical application especially in high dimensional cases. Actually, the boundary data fitting performance for the dimension of vibration amplitude. As shown in Figure 11, two target theoretical derivation and boundary function are presented to establish and consolidate the mathematical funda- objects with coordinates (150 MW, 7 mm) and (150 MW, 7 mm) are added and the distribution of relative position ments. In practice, we can modify the SVDD decision boundary originally calculated using Gauss kernel function of target objects is slightly changed so that the fitting accuracy of decision boundary is significantly improved. by just stretching the original boundary curve in the dimension related to the random variable feature of which the According to the engineering experience of hydro-turbine- generator vibration during operation, the vibration confidence level changes. Therefore, the boundary modification method is very convenient for application. amplitude will be obviously reduced and remains at a quite low level when the unit load reaches the rated load. Therefore, the addition of the two target objects is considered to be reasonable and realistic. Boundary optimization In Figure 11, the original decision boundary calculated by SVDD method, the modified boundary obtained by Disadvantage of traditional SVDD technique for vibration risk assessment. Due to the consideration of confidence level for the aforementioned boundary extension method and the recalculated boundary obtained by SVDD method are the random variable feature, it is expected that the decision boundary can enclose the target objects in an illustrated. It is noted that the target objects are symmetrically distributed on the two sides of the abscissa axis appropriate way with high fitness, so that the vibration risk can be accurately assessed. It is noted that increasing because the probabilistic distribution of the structural dynamic displacement under specific operating condition is the confidence level of vibration amplitude, instead of decreasing the fitting degree of decision boundary, is a more a normal distribution with zero mean. The original SVDD boundary represented by black solid-line is calculated reasonable and effective way to conservatively estimate the vibration. However, an ideal decision boundary is to enclose the target objects of which the features are #4 unit load and vibration displacement RMS, and the sometimes difficult to be obtained even if the Gauss kernel which shows a wonderful performance for data fitting penalty factor C and width parameter r are equal to 8.33 and 10, respectively. The vibration amplitude estimation is applied. As shown in Figure 10, the decision boundaries calculated by SVDD technique based on Gauss kernel when the #4 unit load is lower than 60 MW or larger than 150 MW is not involved in Figure 11 for the reasons with different width parameters at the 99.74% confidence level are illustrated (i.e. the three times vibration RMS that the #4 unit rarely operate with load lower than 60 MW or larger than 150 MW. Consequently, the original is selected as the vibration amplitude feature). boundary represented by black solid-line is considered as an appropriate boundary obtained by comprehensively It is noted that all the decision boundaries shown in Figure 10 do not enclose the target objects in the appro- considering the theoretical calculation results and engineering experience. priate way. The decision boundary in Figure 10(a) is underfitting, because the distances between the boundary However, the probability that vibration amplitude at a certain moment exceeds the vibration RMS is approx- curve and target objects are too large. In general, the relatively large distances may be acceptable due to the imately 32% for normal distribution so that the RMS is far from representing the possible maximum vibration conservative estimate for the vibration amplitude. Since the confidence level of the vibration intensity is consid- amplitude. If there are precision instrument in operation near #4 unit, the SVDD decision boundary should be ered in the calculation, the vibration amplitude should be conservatively estimated by increasing the confidence calculated on the basis of target objects of which the feature of dynamic displacement RMS should be multiplied level, instead of decreasing the fitting degree of decision boundary. Therefore, the decision boundary should be by a large amplification factor, thus the excessive vibration risk will be significantly reduced. Moreover, under the calculated with relatively high fitting degree. For the decision boundary in Figure 10(b), the fitting degree is general operating conditions with higher tolerance to vibration, the vibration amplitude feature of target objects obviously increased but the vibration amplitude which is greater (or less) than 0 mm is no longer a single-valued should be denoted as the product of dynamic displacement RMS and a relatively small amplification factor. function of #4 unit power. This leads to the uncertainty of the vibration amplitude range for the certain #4 unit Therefore, we attempt to develop an approach to obtain the modified decision boundary on the basis of original load and confidence level, thus this decision boundary cannot be applied to determine the magnitude and prob- boundary to reasonably estimate the excessive vibration risk and avoid time-consuming and complex calculation ability of excessive vibration. Moreover, the boundary shown in Figure 10(c) is divided into several isolated process. circles, which shows better boundary fitness but is also inapplicable to perform vibration risk assessment. Based on the aforementioned boundary extension operation, the modified boundary represented by green Actually, in order to obtain a decision boundary that can be used to reasonably describe the distribution range dashed-line in Figure 11 is calculated to enclose the target objects of which the vibration amplitude feature is of vibration amplitude under certain probability conditions, the boundary curve should be as close as possible to the product of constant 3 and displacement RMS. However, the deviation between the boundary and the target the target objects in the dimension of vibration amplitude and the semi-boundary for vibration amplitude larger objects is amplified by applying the aforementioned boundary extension method so that the extended boundary (or less) than 0 mm should be the single-valued function of #4 unit load. Although the Gauss kernel has a better represented by the green dashed-line is underfitting, although the original boundary before extension show good performance than most other kernels on data fitting, the calculated boundary function still cannot satisfy the data fitting performance in the dimension of vibration amplitude. The recalculated decision boundary represented Zhang et al. Zhang et al. 1321 13 Figure 11. Comparison of the original, extended and recalculated boundaries. above two conditions very well, because the boundary curve is either not a single-valued function of abscissa in upper (or lower) half-plane or has a relatively low fitting degree. Comparison of the original, extended and recalculated boundaries. In order to obtain an acceptable original boundary at relatively low confidence level, the additional target objects should be added in the target object set to improve the boundary data fitting performance for the dimension of vibration amplitude. As shown in Figure 11, two target objects with coordinates (150 MW, 7 mm) and (150 MW, 7 mm) are added and the distribution of relative position of target objects is slightly changed so that the fitting accuracy of decision boundary is significantly improved. According to the engineering experience of hydro-turbine- generator vibration during operation, the vibration amplitude will be obviously reduced and remains at a quite low level when the unit load reaches the rated load. Therefore, the addition of the two target objects is considered to be reasonable and realistic. In Figure 11, the original decision boundary calculated by SVDD method, the modified boundary obtained by the aforementioned boundary extension method and the recalculated boundary obtained by SVDD method are illustrated. It is noted that the target objects are symmetrically distributed on the two sides of the abscissa axis because the probabilistic distribution of the structural dynamic displacement under specific operating condition is a normal distribution with zero mean. The original SVDD boundary represented by black solid-line is calculated to enclose the target objects of which the features are #4 unit load and vibration displacement RMS, and the penalty factor C and width parameter r are equal to 8.33 and 10, respectively. The vibration amplitude estimation when the #4 unit load is lower than 60 MW or larger than 150 MW is not involved in Figure 11 for the reasons that the #4 unit rarely operate with load lower than 60 MW or larger than 150 MW. Consequently, the original boundary represented by black solid-line is considered as an appropriate boundary obtained by comprehensively considering the theoretical calculation results and engineering experience. However, the probability that vibration amplitude at a certain moment exceeds the vibration RMS is approx- imately 32% for normal distribution so that the RMS is far from representing the possible maximum vibration amplitude. If there are precision instrument in operation near #4 unit, the SVDD decision boundary should be calculated on the basis of target objects of which the feature of dynamic displacement RMS should be multiplied by a large amplification factor, thus the excessive vibration risk will be significantly reduced. Moreover, under the general operating conditions with higher tolerance to vibration, the vibration amplitude feature of target objects should be denoted as the product of dynamic displacement RMS and a relatively small amplification factor. Therefore, we attempt to develop an approach to obtain the modified decision boundary on the basis of original boundary to reasonably estimate the excessive vibration risk and avoid time-consuming and complex calculation process. Based on the aforementioned boundary extension operation, the modified boundary represented by green dashed-line in Figure 11 is calculated to enclose the target objects of which the vibration amplitude feature is the product of constant 3 and displacement RMS. However, the deviation between the boundary and the target objects is amplified by applying the aforementioned boundary extension method so that the extended boundary represented by the green dashed-line is underfitting, although the original boundary before extension show good data fitting performance in the dimension of vibration amplitude. The recalculated decision boundary represented 14 1322 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Zhang et al. 15 Figure 12. Boundary optimization process based on Gauss kernel SVDD. (a) APF of the computational domain; (b) contour map of APF for path search. Figure 13. The optimal boundary calculated by the proposed approach. by blue dot-line is obtained by SVDD algorithm with the same parameters as in the calculation of original boundary. It is noted that the recalculated boundary shows obviously better fitting degree, but the boundary curve is not a single-valued function of abscissa in upper (or lower) half-plane. Boundary optimization. It is noted that the extended boundary is always a correct classification boundary which encloses all of the target objects. Since the distance between the target objects and the decision boundary is amplified during the boundary extension process, the decision boundary should be optimized to enclose the target objects more closely in the dimension of vibration amplitude. This optimization problem can be equivalent to the path search problem and the obtained optimal path can be regarded as the new classification boundary that has better fitting accuracy. As shown in Figure 12(a), the computation domain surrounded by the original boundary section ACB and the extended boundary section ADB can be regarded as the computational domain. Moreover, the intersections of the original and extended boundaries A and B can be respectively selected as the origin and goal of the path (i.e. the optimal boundary) that we need to solve. The target objects in the compu- tational domain are regarded as obstacles that the path should bypass from above so that the optimal boundary will enclose all the objects. In order to calculate the optimal boundary in the present problem, the potential field is Figure 14. Three-dimensional spatial distribution for the target objects of which the features vibration amplitude, #4 unit load and defined for the computational domain, surrounding boundaries and obstacles (i.e. the target objects in compu- abandoned water flow are considered. (a) The general view; (b) the side view. tational domain) inspired by the artificial potential field (APF) method. Firstly, the gravitational field is defined over a relatively large domain with the smallest field function value at As shown in Figure 12, the calculated artificial potential field (APF) and the APF contour map for path search the goal point (i.e. point B). Thus, a boundary curve from starting point A to ending point B can be calculated are illustrated. The parameters K and C are equal to 0.1 and 1.0e10 , respectively. Moreover, the amplification along the direction of gravitational gradient. In order to keep the boundary curve within the computational factors X for different obstacles are illustrated in Figure 12(b). It is noted that the location of the optimal 0 0 0 0 domain and enclose the obstacles, the boundaries ACB and ADB and the obstacles T T , T T , T T , T T , 1 2 3 4 1 2 3 4 boundary can be adjusted by changing the parameters X (i¼ 1, 2, . . .. . .) for different boundaries and obstacles 0 0 0 T T , T T and T T are discretized into a series of isolated obstacle points and the repulsion fields are further 5 6 7 5 6 7 (i.e. the target objects) can be set to different values, and the smaller the value of X , the closer the optimal defined over them. Finally, the total artificial potential field is obtained by superimposing the gravitational field boundary is to the boundaries (or target objects) in computational domain. and repulsion fields and the optimal boundary can be calculated. As shown in Figure 13, the optimal boundary calculated by the proposed approach is given for the upper half- The gravitational field function and repulsion field function, denoted as F and F , can be expressed as the g r plane. Due to the symmetrical distribution of the target objects, the whole optimal decision boundary can be following formulations immediately obtained. It is noted that the fitting accuracy of the optimal decision boundary is significantly improved and the optimal boundary is a single-valued function of #4 unit load (abscissa) in upper (or lower) F z ¼ Kkz � Gk (24) ðÞ half-plane, which indicate the rationality and effectiveness of the proposed approach. 1 1 1 Practical application and discussion F ðÞ z ¼ X � (25) r i kz � O kC k r kz � O k ik Practical application where z denotes an arbitrary point in the computational domain; G represents the goal point (i.e. point B); O It is noted that three features are involved in the vibration risk assessment of hydro-turbine- generator according denotes the i-th discrete obstacle point; X denotes the amplification factor for the i-th obstacle point in the to the analysis in ‘Analysis for the main influencing factors of #4 unit vibration’ subsection. In order to conve- repulsion field function; K and C are the constant parameters for the gravitational and repulsion field functions, niently and clearly describe the proposed approach, only two features (i.e. #4 unit load and vibration amplitude) respectively. are considered in ‘The SVDD-based post-processing approach for vibration risk assessment’ section. Due to the Zhang et al. Zhang et al. 1323 15 Figure 13. The optimal boundary calculated by the proposed approach. Figure 14. Three-dimensional spatial distribution for the target objects of which the features vibration amplitude, #4 unit load and abandoned water flow are considered. (a) The general view; (b) the side view. As shown in Figure 12, the calculated artificial potential field (APF) and the APF contour map for path search are illustrated. The parameters K and C are equal to 0.1 and 1.0e10 , respectively. Moreover, the amplification factors X for different obstacles are illustrated in Figure 12(b). It is noted that the location of the optimal boundary can be adjusted by changing the parameters X (i¼ 1, 2, . . .. . .) for different boundaries and obstacles (i.e. the target objects) can be set to different values, and the smaller the value of X , the closer the optimal boundary is to the boundaries (or target objects) in computational domain. As shown in Figure 13, the optimal boundary calculated by the proposed approach is given for the upper half- plane. Due to the symmetrical distribution of the target objects, the whole optimal decision boundary can be immediately obtained. It is noted that the fitting accuracy of the optimal decision boundary is significantly improved and the optimal boundary is a single-valued function of #4 unit load (abscissa) in upper (or lower) half-plane, which indicate the rationality and effectiveness of the proposed approach. Practical application and discussion Practical application It is noted that three features are involved in the vibration risk assessment of hydro-turbine- generator according to the analysis in ‘Analysis for the main influencing factors of #4 unit vibration’ subsection. In order to conve- niently and clearly describe the proposed approach, only two features (i.e. #4 unit load and vibration amplitude) are considered in ‘The SVDD-based post-processing approach for vibration risk assessment’ section. Due to the 16 1324 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Figure 15. The optimal boundaries for the three-dimensional target objects calculated by the proposed approach. (a) Boundaries for the target objects with relatively low abandoned water flow; (b) boundaries for the target objects with relatively high abandoned water flow. significant influence of the feature abandoned water flow on vibration intensity, the abandoned water flow is also taken into consideration in the following analysis so that the proposed approach can be appropriately applied in engineering practice. As shown in Figure 14, the spatial distribution for the target objects considering three features is illustrated. It is noted that the abandoned water flow feature for all the target objects is discretely distributed and the discrete 3 3 3 3 3 values for the abandoned water flow are approximately 3400 m /s, 2500 m /s, 3900 m /s, 4700 m /s, 5600 m /s and 6000 m /s as shown in Figure 14(b). Therefore, the proposed method can be applied to the target objects with similar abandoned water flow so that the above proposed two-dimensional method is applicable for three- dimensional problems and even higher dimensional problems if more features can be considered to be distributed in discrete ranges. Actually, the path planning problem in high dimensional space is a challenging problem which is studied a lot but is not appropriately solved. In such a case, the amplification effect of the deviation between decision boundary and target objects induced by boundary extension is more significant in a high dimensional situation, while the boundary optimization of the proposed approach is difficult to be performed. Thus, the approximate solution using two-dimensional method to deal with three-dimensional vibration risk assessment problem of hydro-turbine-generator in this large hydropower station is practical and feasible. Due to the limited data, a reasonable decision boundary cannot be obtained when the abandoned water flow range is too small to include enough target objects. Therefore, the abandoned water flow features of target objects are divided into two ranges. The target objects with the abandoned water flow lower than 4000 m are projected to a single two-dimensional plane of which only the dimensions of #4 unit load and vibration amplitude are con- sidered. Similarly, the target objects with the abandoned water flow higher than 4000 m are projected to another single two-dimensional plane. It is noted that the vibration amplitudes of the target objects with similar #4 unit loads and relatively low abandoned water flows (lower than 4000 m ) are close to each other, but the vibration amplitudes of the target objects with similar #4 unit load and relatively high abandoned water flows (higher than 4000 m ) show larger differences and are significantly amplified. Therefore, the classification for the target objects with different abandoned water flow is reasonable due to the obviously different vibration intensity characteristics. By applying the proposed approach, the original, extended and optimal boundaries can be conveniently obtained as shown in Figure 15. It is noted that the fitting accuracy of the optimal decision boundary is quite high and the optimal boundary is a single-valued function of #4 unit load (abscissa) in upper (or lower) half-plane, which indicate the rationality and effectiveness of the proposed approach. The high-accuracy fitting lays the foundation for the quantitative and accurate assessment results for the vibration range and excessive vibration probability at a certain confidence level. Due to space limitation, only the detailed description of the vibration risk assessment for the target objects in Figure 15(b) with relatively high abandoned water flow and vibration inten- sities is carried out in the following analysis. Zhang et al. Zhang et al. 1325 17 Table 1. The vibration amplitudes under different cases and confidence levels. Vibration amplitude (mm) At the 68.27% confidence level At the 95.45% confidence level At the 99.73% confidence level #4 unit Measured Original Measured Extended Optimal Measured Extended Optimal Cases load (MW) RMS boundary 2*RMS boundary boundary 3*RMS boundary boundary 13 61.7 10.23 16.35 20.47 32.69 21.20 30.70 49.04 32.00 14 79.1 9.54 18.59 19.07 37.18 25.00 28.61 55.77 38.50 15 101.3 11.26 22.40 22.53 44.80 29.40 33.79 67.20 43.80 16 119.6 14.42 40.47 28.84 80.95 32.60 43.26 121.42 49.00 17 138.7 27.23 32.21 54.47 64.41 55.00 81.70 96.62 83.00 18 80.9 7.25 18.54 14.49 37.08 15.00 21.74 55.62 23.00 19 98.9 9.64 21.70 19.28 43.41 34.00 28.93 65.11 51.00 20 119.8 15.22 40.42 30.44 80.84 31.00 45.65 121.25 47.00 21 77.9 12.19 18.62 24.38 37.23 25.00 36.57 55.85 38.00 22 99.8 15.86 21.96 31.72 43.92 32.40 47.57 65.89 49.00 23 116.5 37.24 40.76 74.49 81.52 75.30 111.73 122.28 113.00 Table 2. The differences and errors for the vibration amplitudes obtained in different cases. Vibration amplitude difference (mm) Vibration amplitude error #4 unit RMS-original 2*RMS-optimal 3*RMS-optimal RMS-original 2*RMS-optimal 3*RMS-optimal Cases load (MW) boundary boundary boundary boundary (%) boundary (%) boundary (%) 13 61.7 6.11 0.73 1.30 37.39 3.44 4.06 14 79.1 9.05 5.93 9.89 48.70 23.72 25.69 15 101.3 11.14 6.87 10.01 49.71 23.37 22.85 16 119.6 26.05 3.76 5.74 64.37 11.53 11.71 17 138.7 4.97 0.53 1.30 15.44 0.96 1.57 18 80.9 11.30 0.51 1.26 60.92 3.40 5.48 19 98.9 12.06 14.72 22.07 55.57 43.29 43.27 20 119.8 25.20 0.56 1.35 62.35 1.81 2.87 21 77.9 6.42 0.62 1.43 34.51 2.48 3.76 22 99.8 6.10 0.68 1.43 27.79 2.10 2.92 23 116.5 3.52 0.81 1.27 8.63 1.08 1.12 As shown in Table 1, the vibration amplitudes (ordinate values in Figure 15(b)) for the measured vibration displacement histories, original, extended and optimal boundaries with the same #4 unit loads (abscissa values) and similar abandoned water flows are given at the 68.27%, 95.45% and 99.73% confidence levels. It is noted that the differences between the vibration amplitudes of some target objects and the optimal boundaries with same abscissa values are significant. The reason is that these target objects are not support vectors in the decision boundary determination. Actually, the vibration amplitude of the optimal boundary represents the vibration intensity under the most adverse working condition with certain #4 unit load and abandoned water flow at the fixed confidence level. Because of the vibration amplification effect which is extremely complex and insufficiently studied, the significant deviation between the vibration RMSs of the measured displacement histories can be generated when the two main influencing factors (i.e. the #4 unit load and abandoned water flow) are fixed and other operation conditions are variable. With the increase of target objects, the vibration amplitude range deter- mined by the optimal decision boundary will be more representative. As shown in Table 2, the differences and errors between the vibration amplitude ranges calculated by the displacement histories and different boundaries are listed. It is noted that the vibration amplitudes of the optimal boundaries are always a little larger than the vibration amplitude obtained by displacement histories at different confidence levels, which makes the assessments of vibration amplitude range and excessive vibration probability are slightly more conservative than the actual situation. 18 1326 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Figure 16. Probabilities for the excessive vertical vibration of head cover considering different vibration amplitude ranges under Case 17. Table 3. Excessive vibration probability for different vibration amplitude ranges. Excessive vibration probability for different vibration amplitude ranges At the 68.27% confidence level At the 95.45% confidence level at the 99.73% confidence level Optimal #4 unit load Measured Original Measured 2* Extended Optimal Measured 3* Extended boundary Cases (MW) RMS (%) boundary (%) RMS (%) boundary (%) boundary (%) RMS (%) boundary (%) (%) 13 61.7 30.50 10.04 4.78 0.26 4.57 0.44 0.00 0.29 14 79.1 30.06 4.65 4.46 0.00 0.74 0.25 0.00 0.00 15 101.3 32.60 3.92 4.06 0.00 0.55 0.15 0.00 0.00 16 119.6 33.29 0.25 3.82 0.00 1.82 0.12 0.00 0.00 17 138.7 31.49 23.89 4.63 1.97 4.43 0.24 0.02 0.22 18 80.9 33.12 0.30 3.80 0.00 3.64 0.25 0.00 0.23 19 98.9 36.51 0.49 1.90 0.00 0.00 0.10 0.00 0.00 20 119.8 28.73 0.16 4.22 0.00 3.99 0.24 0.00 0.22 21 77.9 29.89 11.84 4.67 0.26 4.40 0.27 0.00 0.22 22 99.8 30.71 15.44 3.99 0.58 3.76 0.31 0.00 0.25 23 116.5 29.47 23.79 5.09 3.49 4.64 0.35 0.15 0.30 As shown in Figure 16, the analyses for the vibration amplitude range and excessive vibration probability based on the critical displacement history measured in Case 17 are illustrated. It is noted that the kurtosis and skewness of the data are equal to 2.9419 and 0.0259 (close to 3 and 0), respectively, which indicates that the probabilistic distribution of the vibration amplitude under the same working condition at different times can be reasonably regarded as normal distribution. As the vibration displacement approximately satisfies the normal distribution, the calculated excessive vibration probability is approximately equal to the theoretical results. Based on the vibration amplitude ranges obtained by the vertical dynamic displacement response of head cover and optimal decision boundaries at different confidence levels, the excessive vibration probabilities calculated by the same method shown in Figure 16 are given in Table 3. Zhang et al. Zhang et al. 1327 19 Figure 17. Comparison for excessive vibration probabilities of different vibration amplitude ranges at different confidence levels. (a) At the 68.27% confidence level; (b) at the 95.45% confidence level; (c) at the 99.73% confidence level. In order to describe the calculation results more clearly, the comparison for the excessive vibration probabilities calculated based on the normal distribution theory, the vibration amplitude ranges of the displacement histories, the original, extended and optimal boundaries are illustrated in Figure 17. It is noted that the analysis results in Cases 14, 15 and 19 are given in Table 3, but are not involved in Figure 17. The reason is that the target objects with the vibration amplitudes obtained by the displacement histories measured in Cases 14, 15 and 19 are not the support vectors in the calculation of original boundary, thus the significant deviations will almost certainly be generated and this does not mean that the proposed approach is unreasonable or inaccurate. According to the analysis results illustrated in Figure 17, the excessive vibration probabilities calculated on the basis of the normal distribution theory, the vibration amplitude range of displacement history and the optimal boundary are close to each other. Therefore, the probability that the vibration amplitude exceeds the vibration range calculated by the displacement response under a certain working condition can be conservatively replaced by the probability that the vibration amplitude exceeds the vibration range determined by the optimal boundary, and can be quantitatively described by the probability calculated by the normal distribution theory. Discussion It is noted that three procedures, i.e. the original boundary determination by conventional Gauss kernel SVDD technique, the extended boundary calculation based on original boundary and the boundary optimization algo- rithm inspired by the path planning problem, are included in the proposed approach. In the calculation process for the first procedure, the single-valued original boundary with high-accuracy fitness is relatively easy to be obtained due to the small amplitude of vibration displacement and the dense distribution of target objects at a low confidence level. Moreover, the additional target objects can be added in the target object set (as demonstrated in ‘Practical application’ subsection) to change the distribution of the relative position of target objects, thus the appropriate original boundary can be almost always obtained. In the boundary optimization process, it is convenient to define an additional repulsion field for the region outside the computational domain (shown in Figures 11 and 12), so that the optimal boundary curve can be kept within the computation domain during the calculation process. In order to concisely describe the proposed approach and improve the readability of the figures, the additional repulsion field is not considered in Figure 12. Moreover, the vibration amplitude of the optimal boundary represents the vibration intensity under the most adverse working condition with certain #4 unit load and abandoned water flow at the fixed confidence level. It is noted that the optimal boundary is calculated on the basis of the existing target objects, thus the representativeness of the vibration amplitude range determined by the optimal decision boundary increases with the increasing target objects. At present, permanent vibration monitoring systems have been applied to different hydraulic structures (such as bottom slab of plunge pool, underground corridor, arch dam body, etc.) in many hydropower projects, which lay the foundation for the vibration risk assessment by the proposed approach. Once the optimal boundary is determined, the vibration amplitude range and the probability of vibration amplitude beyond the range under different working conditions can be estimated according to lots of test data, and in such a case all the vibration influencing factors and mechanisms which are not sufficiently studied are considered in the estimation. It is noted that the quantitative, accurate and slightly conservative assessment results can be obtained by the proposed approach. Therefore, the engineers can conveniently obtain the optimal 20 1328 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) hydropower station operation scheme to satisfy both the flood discharge and power generation demands and the environmental vibration requirements in different situations. The calculation effort of the proposed approach is significantly less than that of the time-consuming Gauss kernel SVDD method with complex parameter adjust- ment process especially under the condition of large number of target objects. However, it is noted that the vibration of #4 unit cannot be reduced to a very small range according to the proposed approach. In order to develop an operation scheme to reduce the vibration more effectively, the more comprehensive and long-term prototype tests should be carried out. The vibration condition of #4 unit should be tested and analyzed under enough working conditions. Based on the results of prototype experiments, the power distribution among four hydropower units and discharge distribution among seven gates on the two dam sections shown in Figure 1 (i.e. the discharge sluice dam section and the diversion channel dam section) should be com- prehensively and detailedly considered in the operation scheme to further reduce the vibration. Noted that the problem of improving the boundary fitting performance without leading to extremely complex curves still exists. Thus, the proposed approach should be improved to deal with the boundary optimization problem in higher dimensional space to avoid the reductions of calculation efficiency and accuracy when this two-dimensional approach is applied to deal with the high-dimensional problem. Moreover, it must be pointed out that the application of the current version of proposed approach is largely dependent on the dynamic response of which the vibration amplitude approximately satisfies the normal distribution. Concluding remarks In order to accurately and conveniently assess the vibration amplitude range and excessive vibration probability at different confidence levels, the approach including original boundary determination, extended boundary calcu- lation and boundary optimization is proposed in this paper, and then the proposed approach is applied to deal with the practical vibration problem of the hydro-turbine-generator. In summary, the main contributions of this paper are as follows: The prototype dynamic test is firstly carried out to obtain the spatial distribution of vibration amplitude for the hydro-turbine-generator in the hydropower station in different cases. Then, the #4 unit load and abandoned water flow are considered to be the main influencing factors for the vibration amplitude according to the analysis based on the dynamic response of the measuring point with maximum vibration amplitude and the engineering expe- rience for the vibration problem induced by high dam flood discharge. The original boundary for the target objects at a relatively low confidence level is calculated based on Gauss kernel SVDD technique considering the vibration amplitude as a random variable feature. Then, the boundary extension operation based on the original boundary is proposed and its theoretical fundamental is deduced in detail. Inspired by the path planning problem, the artificial potential fields for the computational domain, surrounding boundaries and obstacles are constructed and the optimal boundary is further obtained. Comparing with the traditional SVDD technique, the advantage of the proposed approach is that it is able to conveniently improve the fitting performance for single dimension (i.e. vibration amplitude) without leading to extremely complex boundary which cannot be used for vibration risk assessment. In the practical application of the proposed approach, only the working cases 13 to 23 are considered because the relatively intense vibrations are generated under these conditions. The average probability of excessive vibra- tion calculated by actual vibration data is 4.13% under these working conditions, when the confidence level is set to 95.45%. This average probability is slightly different from the theoretical value 4.55%, because the actual vibration approximately but not completely satisfies the normal distribution. According to the evaluation results of the traditional and optimal approaches, the average excessive vibration probabilities are 0.60% and 2.96%, respectively. Moreover, when the theoretical confidence level is 99.73%, the average excessive vibration proba- bilities calculated by actual vibration data, traditional vibration evaluation method and proposed optimal approach are 0.25% (0.27% for theoretical value), 0.02% and 0.16%, respectively. Obviously, the proposed optimal approach provides quantitative, much more authentic and slightly conservative prediction results, which is exactly what the engineering practice requires. Therefore, based on the optimal boundary calculated by the proposed approach, the engineers of hydropower station can control the structural vibration intensity by adjusting operation conditions to satisfy the environmental vibration requirements in different situations. Inevitably, there are some defects in this study that need to be considered and improved in the subsequent research. The proposed approach can improve the fitting performance with smooth boundary for only one dimension, which significantly limits the application of the approach in engineering practice. Therefore, the research should be carried out in future to effectively deal with the problem in high dimensional space. Zhang et al. Zhang et al. 1329 21 Moreover, the reason why the abnormal vibration is always generated under specific working conditions with large discharge flow from #1 to #4 hydraulic gates and heavy load of #4 unit is very interesting. The more comprehensively and carefully analysis for the detailed generation mechanism of this engineering problem should be performed in subsequent research. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) disclose receipt of the following financial support for the research, authorship, and publication of this article: This research was funded by National Key R&D Program of China (No. 2016YFC0401905), Science Fund for Creative Research Groups of the National Natural Science Foundation of China (No. 51621092) and National Natural Science Foundation of China (No. 51509185, No. 51569025, No. 51779167, No. 51809194, No. 51709124), China Postdoctoral Science Foundation (2019M652550), and Funds for Postdoctoral Scientific Research in Henan Province (in 2019). ORCID iD Chao Liang https://orcid.org/0000-0002-4063-6519 References 1. Darbre GR, Smet CAMD and Kraemer C. Natural frequencies measured from ambient vibration response of the arch dam of Mauvoisin. Earthquake Eng Struct Dyn 2000; 29: 577–586. 2. Proulx J, Paultre P, Rheault J, et al. An experimental investigation of water level effects on the dynamic behaviour of a large arch dam. 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Zhang et al. 1331 23 Appendix 1. Working conditions in prototype dynamic experiment. Working Gate opening (m) Unit load (MW) Upstream Downstream Water level Inflow Abandoned Power Outflow Discharge flow condition water water difference volume water flow generation volume from #4 3 3 3 3 3 No. group 1# 2# 3# 4# #1 #2 #3 #4 level (m) level (m) (m) (m ) (m ) flow (m ) (m ) unit (m ) 1 1 0110 141.00 139.00 140.00 81.00 1014.31 992.93 21.38 3420 484.04 2629.22 3814 540.30 2 0110 136.00 132.00 136.00 102.00 1014.05 993.31 20.74 3420 484.04 3261.04 3814 695.35 3 0110 130.00 121.00 132.00 119.00 1014.31 992.93 21.38 3368 484.04 2521.40 3496 816.35 4 2 1001 89.80 99.30 89.80 79.80 1012.22 992.19 20.03 2522 408.56 1921.56 3142 543.52 5 1001 88.90 80.50 89.50 101.50 1012.22 992.19 20.03 2522 408.32 2012.83 3142 715.15 6 1001 81.10 80.00 81.00 120.70 1012.22 992.19 20.03 2522 877.34 2051.83 3142 886.22 7 3 1111 92.20 92.10 92.20 80.10 1012.22 992.60 19.62 2650 652.99 2068.23 3119 552.66 8 1111 84.80 85.60 91.90 98.50 1012.22 992.19 20.03 2522 881.34 1956.84 3142 702.90 9 1111 80.50 79.50 81.50 119.70 1012.22 992.19 20.03 2522 877.92 2032.31 3142 888.61 10 4 3333 0.20 49.70 73.20 81.10 1014.55 993.30 21.25 3910 529.58 2900.88 2776 535.67 11 3333 0.20 49.60 51.50 99.90 1014.70 993.30 21.40 3910 529.58 2900.88 2776 687.24 12 3333 0.20 79.00 0.00 119.90 1014.70 993.00 21.70 3910 529.58 2900.88 2776 824.67 13 5 4.7 4.7 4.7 4.7 0.20 99.50 110.20 61.70 1013.71 994.82 18.89 4723 3720.95 1547.19 4918 515.18 14 4.7 4.7 4.7 4.7 0.20 100.30 88.90 79.10 1013.71 994.82 18.89 4723 3720.95 1547.19 4918 689.02 15 4.7 4.7 4.7 4.7 0.20 100.70 69.00 101.30 1013.91 994.31 19.60 4723 3720.95 1469.98 4918 869.43 16 4.7 4.7 4.7 4.7 0.20 90.50 0.10 119.60 1013.91 994.31 19.60 4723 3955.64 1168.50 4918 1026.75 17 4.7 4.7 4.7 4.7 0.20 70.50 0.10 138.70 1013.91 994.31 19.60 4723 3720.95 1187.84 4918 1122.17 18 6 5555 79.30 71.40 69.00 80.90 1013.89 995.08 18.81 5602 3694.56 1798.94 5494 651.01 19 5555 79.20 71.60 50.90 98.90 1013.89 995.08 18.81 5602 3925.57 1787.64 5494 814.30 20 5555 79.50 56.00 50.50 119.70 1013.89 995.08 18.81 5602 3925.57 1793.21 5494 1008.99 21 7 6666 81.00 78.90 80.50 77.90 1013.74 995.86 17.88 6077 4243.69 1934.41 5647 673.10 22 6666 80.60 78.40 59.50 99.80 1013.53 995.86 17.67 6077 4560.52 1970.13 5647 891.53 23 6666 80.60 61.30 58.20 116.50 1013.53 996.25 17.28 6077 4564.08 1998.27 5647 961.31 24 1332 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Appendix 2. RMSs of the vibration displacements of #4 hydro-turbine-generator under different working conditions. Sensor number Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 A4 2.0186 2.1185 1.7723 1.645 1.8247 1.8398 1.608 1.8149 B1 0.5391 0.5578 0.5185 0.5461 0.5251 0.5599 0.5211 0.4921 B2 0.4562 0.4305 1.0065 0.4299 0.5498 0.9271 0.1786 1.6615 B3 2.8198 3.0086 2.9302 3.2483 3.2707 3.2548 3.2117 3.0933 C1 0.7881 0.7946 0.8 0.7245 0.7334 0.8555 0.7333 0.7274 C2 1.7624 1.8859 2.0763 1.8537 1.9817 2.2833 1.8719 1.9818 C3 0.6218 0.6188 0.6475 0.6158 0.6349 0.6881 0.6589 0.6862 D1 0.4255 0.4203 0.4159 0.4473 0.4725 0.4546 0.4715 0.4753 D2 0.3953 0.4191 0.401 0.3639 0.3654 0.3723 0.3743 0.3683 D3 3.3094 3.0572 3.1143 2.7232 2.7444 2.881 3.4503 2.7737 H1 7.3802 9.2062 11.1987 3.0787 3.8323 4.958 3.1568 3.887 H2 7.3201 9.3317 11.6163 7.5484 9.7189 12.6832 7.8222 9.9072 H3 5.4342 7.6709 9.6661 5.7286 8.0663 10.8527 6.137 8.4462 M3 0.1852 0.1863 0.1862 0.2039 0.223 0.2481 0.208 0.2226 N3 2.7484 3.1474 2.4523 2.9887 2.964 2.8885 3.3214 3.505 Sensor number Case 9 Case 10 Case 11 Case 12 Case 13 Case 14 Case 15 Case 16 A4 1.6475 2.0473 2.1286 1.7174 1.8111 1.8201 2.2279 1.7427 B1 0.5227 1.3515 1.3626 1.3121 1.9362 1.9093 1.9753 2.0593 B2 3.0828 0.4074 1.3448 0.3788 0.8277 0.8281 0.8345 0.8926 B3 2.8978 3.0806 3.4082 3.0303 3.1457 3.1173 3.0744 3.6689 C1 0.8118 1.2236 1.2404 1.2358 1.5803 1.5452 1.553 1.6022 C2 2.3211 1.6618 1.7471 1.9483 1.9744 1.9973 2.1083 2.381 C3 0.7275 0.8902 0.9382 0.9384 1.1439 1.1747 1.1819 1.1897 D1 0.4642 1.0142 1.0184 0.9783 1.3908 1.3495 1.3474 1.3657 D2 0.3923 0.5885 0.5766 0.5844 0.7289 0.7615 0.7331 0.7639 D3 2.7754 2.8298 2.8449 2.628 3.2202 2.8908 3.6332 3.074 H1 5.1909 7.5156 9.1029 11.2272 12.7194 12.9082 15.2068 19.8838 H2 13.283 7.1472 9.0137 11.465 7.8271 8.5517 10.5262 13.8068 H3 11.4699 5.3839 7.4204 9.9126 10.2337 9.5355 11.2638 14.4204 M3 0.2555 0.2074 0.2077 0.2069 0.1722 0.1734 0.1735 0.1437 N3 2.6264 3.1966 2.8877 2.7408 2.9319 3.0044 2.9646 3.2871 Sensor number Case 17 Case 18 Case 19 Case 20 Case 21 Case 22 Case 23 A4 1.8421 1.9521 1.7149 1.7844 1.5199 1.7704 2.0592 B1 2.9051 1.9465 2.0532 2.2039 2.2811 2.4434 2.5731 B2 1.2469 0.8394 0.8207 0.8911 0.9665 0.9966 1.1553 B3 3.2013 3.4288 3.3692 3.0191 2.7704 3.4712 3.687 C1 1.8952 1.6161 1.6808 1.7892 1.8514 2.026 2.242 C2 2.9524 1.9879 2.1594 2.6616 2.1888 2.4932 3.1768 C3 1.6508 1.141 1.1558 1.2753 1.3057 1.4304 1.6129 D1 1.3752 1.4741 1.5373 1.5506 1.7161 1.8309 1.7275 D2 0.8692 0.7512 0.7654 0.7982 0.8594 0.8975 0.9561 D3 2.9451 3.2089 3.083 3.4857 3.3576 3.0762 3.0955 H1 40.3841 9.3647 11.5218 16.6583 4.6959 6.7105 9.3315 H2 19.3857 9.1188 11.3528 16.3144 9.8614 13.7516 20.7522 H3 27.2333 7.2455 9.6421 15.2175 12.1905 15.8582 37.2441 M3 0.1438 0.1737 0.1738 0.1748 0.2406 0.2849 0.3435 N3 3.4067 2.7549 2.7926 3.3564 3.2511 3.0207 3.1789 Zhang et al. Zhang et al. 1333 25 Double amplitudes of the vibration displacements of #4 hydro-turbine-generator under different working conditions Sensor number Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 A4 11.8256 15.1825 12.7411 11.3678 11.5204 12.4741 10.5667 11.8637 B1 4.9591 6.663 4.5522 4.3233 4.247 4.3742 4.2725 4.1453 B2 3.7003 4.3869 9.1171 5.8813 4.4632 9.488 6.609 7.6069 B3 19.0734 17.8909 15.4877 23.0788 19.722 19.8745 21.3622 19.1116 C1 6.4087 6.485 7.4387 5.455 5.7983 6.7139 5.3024 5.6076 C2 8.8501 9.2697 9.7656 8.8119 8.6975 10.8337 9.1553 8.6593 C3 4.8828 5.6076 5.2261 5.455 4.9973 7.5149 5.9128 7.3624 D1 3.5095 3.2806 3.5858 3.5858 3.8528 3.891 4.2343 4.0054 D2 3.1662 3.4332 3.128 2.9373 3.0899 3.0518 2.9755 3.1662 D3 18.5394 19.493 16.3268 15.6021 18.3867 15.068 22.1252 16.7083 H1 40.5119 55.7708 54.55 20.1034 18.8827 23.0407 18.0434 20.1415 H2 31.0515 46.7426 44.9625 41.0206 38.2995 52.5665 40.9698 41.453 H3 35.8199 75.9886 52.452 46.1959 42.5719 60.5773 56.9152 54.5501 M3 1.9836 2.0981 2.1744 2.0981 2.2507 2.1744 2.2888 1.9836 N3 19.1498 18.8064 14.1525 21.8582 18.959 19.1498 17.3568 22.5067 Sensor number Case 9 Case 10 Case 11 Case 12 Case 13 Case 14 Case 15 Case 16 A4 10.5286 13.1607 13.6566 12.7029 12.7411 12.3215 12.8555 11.1389 B1 4.2725 10.0454 11.851 11.1643 13.5295 16.0471 14.7756 13.4532 B2 19.9889 3.5095 17.4713 3.2043 6.4468 6.4087 6.5613 6.9427 B3 18.0053 20.8663 26.2069 19.6457 17.0898 17.7383 18.425 21.3241 C1 6.4087 10.1852 10.91 9.8419 12.97 12.7029 11.2152 14.0762 C2 9.9945 9.4986 9.9564 11.0626 10.4904 11.4822 12.3977 11.9019 C3 6.4468 7.8583 7.7438 7.5149 8.4305 9.3078 9.5749 9.1171 D1 3.9673 8.049 8.316 9.079 10.643 9.9182 10.376 11.0626 D2 3.3569 4.4632 4.9209 4.3487 6.5994 5.6839 7.0572 5.9891 D3 15.0299 16.0979 16.2124 15.9073 21.9345 19.6075 22.9644 19.5312 H1 25.0244 48.027 48.6373 57.9452 82.55 107.1929 107.4599 119.7432 H2 51.422 30.899 36.5447 45.7763 35.8326 38.9353 43.8689 58.6191 H3 63.7435 39.1005 44.3648 55.6181 77.7434 72.708 83.8469 96.3592 M3 2.2125 2.2888 2.327 2.1744 2.2888 1.9455 2.0599 1.4877 N3 16.7083 19.9508 19.226 19.989 19.8364 18.9208 21.5529 20.2178 Sensor number Case 17 Case 18 Case 19 Case 20 Case 21 Case 22 Case 23 A4 11.3296 11.5204 11.1008 11.5204 10.1089 10.1089 14.8773 B1 22.5321 13.682 16.5049 14.7502 15.92 19.1243 21.7945 B2 10.0708 6.1798 7.4387 7.9346 7.7438 8.2016 9.613 B3 20.0271 21.5911 23.2314 17.6239 20.4086 22.6211 22.4304 C1 16.5558 13.2751 13.0463 15.2588 15.0299 16.861 19.4168 C2 15.7165 12.4741 11.177 13.8473 12.5885 13.504 17.5858 C3 12.207 8.049 9.1934 9.7656 10.91 13.6947 12.3215 D1 10.376 11.94 12.4359 13.0463 13.3133 13.9999 15.4114 D2 6.5613 5.8746 5.9509 6.3324 6.9427 7.7057 7.4387 D3 20.1415 21.4385 21.057 20.79 17.8909 22.6211 19.6075 H1 322.264 50.9643 55.7326 89.9886 90.9042 124.4735 330 H2 86.975 36.9262 42.2668 64.8243 43.9452 59.5855 90.6371 H3 204.581 49.095 58.5173 99.9831 90.9042 124.4735 283.469 M3 1.4877 2.0218 1.9073 1.9073 2.327 2.4032 2.7466 N3 19.4549 16.8991 18.959 24.9099 19.9889 20.4849 23.7655 26 1334 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Dominant frequencies of the vibration displacements of #4 hydro-turbine-generator under different working conditions Sensor number Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 A4 0.0625 0.0656 0.06 0.0467 0.0833 0.0933 0.05 0.0667 B1 0.3438 0.1625 0.1567 1.1133 1.1133 0.1467 1.1133 1.1133 B2 0.0875 0.1594 0.05 0.13 0.1333 0.1233 0.2467 0.0833 B3 0.0656 0.0375 0.03 0.0367 0.0533 0.0333 0.07 0.05 C1 1.1125 1.1156 1.1133 1.1133 1.1133 1.1133 1.1133 1.1133 C2 1.1125 1.1156 1.1133 1.1133 1.1133 1.1133 1.1133 1.1133 C3 1.1125 1.1156 1.1133 1.1133 1.1133 1.1133 1.1133 1.1133 D1 0.05 0.0844 0.0733 0.1167 0.1267 1.1133 0.07 0.0867 D2 0.1219 0.1219 5.5567 0.1333 1.1133 1.1133 1.1133 1.1133 D3 0.0531 0.0625 0.03 0.05 0.04 0.04 0.0567 0.0633 H1 1.1125 1.1156 1.1133 1.1133 1.1133 1.1133 1.1133 1.1133 H2 1.1125 1.1156 1.1133 1.1133 1.1133 1.1133 1.1133 1.1133 H3 1.1125 1.1156 1.1133 1.1133 1.1133 1.1133 1.1133 1.1133 M3 40.5938 24.5156 17.9867 1.1133 1.1133 1.1133 1.1133 1.1133 N3 0.0281 0.0469 0.07 0.0633 0.0567 0.02 0.0567 0.07 Sensor number Case 9 Case 10 Case 11 Case 12 Case 13 Case 14 Case 15 Case 16 A4 0.0567 0.0633 0.0633 0.0567 0.0567 0.1033 0.0867 0.0433 B1 0.2633 0.8433 0.9533 0.8767 0.8433 0.8467 0.88 0.8433 B2 0.15 0.1067 0.09 0.1067 5.56 0.3133 1.1133 1.1133 B3 0.03 0.0233 0.05 0.06 0.03 0.0333 0.0567 0.0867 C1 1.1133 1.1133 1.1133 1.1133 1.1133 1.1133 1.1133 1.1133 C2 1.1133 1.1133 1.1133 1.1133 1.1133 1.1133 1.1133 1.1133 C3 1.1133 1.1133 5.56 5.5567 1.1133 1.1133 5.56 1.1133 D1 1.1133 0.8433 0.9433 0.8833 0.8433 0.8467 0.88 0.8433 D2 1.1133 0.06 0.1 5.5567 5.56 0.8467 0.88 0.2167 D3 0.0233 0.0367 0.0433 0.0633 0.0267 0.0467 0.03 0.0333 H1 1.1133 1.1133 1.1133 1.1133 1.1133 1.1133 1.1133 1.1133 H2 1.1133 1.1133 1.1133 1.1133 1.1133 1.1133 1.1133 1.1133 H3 1.1133 1.1133 1.1133 1.1133 1.1133 1.1133 1.1133 1.1133 M3 1.1133 24.01 45.8833 79.8233 31.3233 77.8633 30.5067 49.4133 N3 0.0467 0.0467 0.0233 0.04 0.03 0.0167 0.06 0.0433 Sensor number Case 17 Case 18 Case 19 Case 20 Case 21 Case 22 Case 23 A4 0.05 0.0333 0.0567 0.0625 0.0767 0.0533 0.07 B1 0.2033 0.88 0.8833 0.8708 0.9233 0.8633 0.8367 B2 0.2033 1.1133 1.1133 1.1167 1.1133 0.3367 1.1133 B3 0.07 0.06 0.0633 0.0625 0.0367 0.0767 0.0667 C1 1.1133 1.1133 0.8833 1.1167 1.1133 1.1133 1.1133 C2 1.1133 1.1133 1.1133 1.1167 1.1133 1.1133 1.1133 C3 1.1133 1.1133 1.1133 5.5625 1.1133 1.1133 1.1133 D1 1.1133 0.8133 0.8833 0.8708 0.9233 0.8633 0.8367 D2 0.36 0.4767 0.3933 0.8708 0.81 0.8633 0.2933 D3 0.0233 0.0267 0.04 0.0292 0.0233 0.0667 0.05 H1 1.1133 1.1133 1.1133 1.1167 1.1133 1.1133 1.1133 H2 1.1133 1.1133 1.1133 1.1167 1.1133 1.1133 1.1133 H3 1.1133 1.1133 1.1133 1.1167 1.1133 1.1133 1.1133 M3 3.0167 69.6833 64.84 51.6875 1.1133 1.1133 1.1133 N3 0.0567 0.0733 0.0433 0.05 0.03 0.0733 0.11
"Journal of Low Frequency Noise, Vibration and Active Control" – SAGE
Published: Oct 20, 2020
Keywords: Vibration risk assessment; support vector domain description technique; boundary optimization; artificial potential field; vibration amplitude range; excessive vibration probability; vibration induced by high dam flood discharge
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