Bayesian network approach to fault diagnosis of a hydroelectric generation system

This study focuses on the fault diagnosis of a hydroelectric generation system with hydraulic‐mechanical‐electric structures. To achieve this analysis, a methodology combining Bayesian network approach and fault diagnosis expert system is presented, which enables the time‐based maintenance to transform to the condition‐based maintenance. First, fault types and the associated fault characteristics of the generation system are extensively analyzed to establish a precise Bayesian network. Then, the Noisy‐Or modeling approach is used to implement the fault diagnosis expert system, which not only reduces node computations without severe information loss but also eliminates the data dependency. Some typical applications are proposed to fully show the methodology capability of the fault diagnosis of the hydroelectric generation system.


| INTRODUCTION
2015 United Nations Climate Change Conference promised that the raise of global warming is almost 2°C compared to pre-industrial levels, which greatly promotes the electricity generation to turn to renewable energy such as hydropower generations. 1 China is leading to a hydropower boom, followed by India, Europe, the United States, and Japan. 2 Hydropower plants have been built in more than 160 countries, with a total number of 11 000 plants equipped with 27 000 hydro-turbine generator units at the end of 2017. 3 In China, the hydropower capacity is expected to increase to 380 gigawatts by 2020. 4 These hydropower plants are constructed at sites along rivers, including thirteen plants on the Salween or Nujiang and twenty plants along the Brahmaputra. 4 In Brazil, 375 small hydropower plants with the total capacity of 4799 MW are currently running, and another 1701 MW installed capacity will be constructed in the next 10 years. 5 Hydroelectric generation systems are under construction all over the world to ensure the enforcement of stricter energy and environmental policy. Obviously, the economic benefit and carbon dioxide mitigation of such hydroelectric generating systems are well known to the general public, [6][7][8][9][10][11] but the stability and safety impacts of themselves still require enough attentions.
Faults in the hydroelectric generation systems (HGS) inevitably result in unexpected safety accidents with enormous maintenance costs. [12][13][14] National Energy Administration issued that 80% of HGS' faults are caused by the vibration of the hydraulic-mechanic-electric components. 15,16 In general, the vibration in the HGS is defined as a drastic reciprocating motion caused by unbalanced forces and uncertain disturbances. 17,18 For instance, 60% of the vibration faults are attributable to the out-of-balance rotating bodies and the pressure pulsation of flow passage components in Japan. 19,20 The current study of the HGS's faults mainly focuses on the constituent components (eg, generators, hydro-turbines, and pipelines). [21][22][23] Additionally, the collection of the on-line monitoring data under the condition of fast information flow is another challenge for fault diagnosis of the HGS. 24,25 To adequately analyze the faults mechanism, to predict behavior of systems, to evaluate operating reliability, and to decrease maintenance costs, are the challenging tasks. Hence, it is of primary importance to provide the powerful methodology for the fault diagnosis of HGSs not only of systems but also of data available.
Some popular efficient approaches, combining monitoring data and expert experiences, are developed for the fault diagnosis such as fault tree analysis (FTA), event tree analysis (ETA), and Bayesian network (BN). [26][27][28] FTA and ETA are applied to evaluate the reliability of systems, whereas these approaches lack lateral linkages between nodes and also require high-quality experts to cope with complicated computations. 29 In light of this, BN is widely used to overcome the limitations of FTA and ETA since it successfully incorporates expert experiences by means of lateral linkages. [30][31][32] However, the modeling of BN in practical applications is still difficult and tedious, especially for the complicated systems. 33,34 Thus, it is emergent to present suitable approaches to reduce node computations without severe information loss.
This study aims to provide an efficient computational methodology for the fault diagnosis of the HGS. To establish a precise Bayesian network of the HGS, we fully analyze the complex fault types and their associated fault characteristics. The Noisy-Or modeling approach is used to eliminate the data dependency and to reduce node computations. The fault diagnosis expert system is proposed that is beneficial to the condition-based maintenance at the lowest cost. Finally, some typical applications are done to fully show the methodology capability of the fault diagnosis of the hydroelectric generation system. This study is structured as follows. Section 2 describes the global methodology of the BN fault diagnosis of the HGS. Section 3 presents the BN fault diagnosis model considering the hydraulic, mechanical, and electric factors. Section 4 performs the applications of the fault diagnosis model of the HGS. Conclusions and discussion in Section 5 summarize this study.

| METHODOLOGY
This section is dedicated to the overall theoretical background of the methodology adopted in the present study. A brief description of BN, Noisy-Or model, and expert system is presented.

| Bayesian network
BN is a statistical graphical model that combines the probability theory with the graphic theory. 35  probability tables (CPTs), which is represented by a directed acyclic graph (DAG). The BN displays the cause and effect relationships between the network variables, as shown in Figure 1.
The implementation of BN relying on the Bayes' theorem is defined as the exhaustive event set B 1 ,B 2 ,...,B n and the event A exist in a sample space Ω, and they, respectively, meet the conditions of P(B i ) > 0 (i = 1,2,3,...,n) and P(A) > 0. Hence, we get 36,37 : To enable the inference analysis of the BN, Equation (1) is subject to the following conditional independence hypothesis: The variable nodes (X 1 , X 2 ,…X n ) in the BN are conditionally independent for their father nodes. This means that the variable nodes satisfy the joint probability in Equation (2).
where pa i denotes the father node set of X i .

| Noisy-Or model
The major work of BN is to determine the CPT, whereas the deduction of the joint probability is growing exponentially with the increase of variable nodes. For the BN with nth binary discrete nodes, it generally requires 2 n th conditional probabilities to describe the network model. To reduce node computations, Noisy-Or modeling approach is applied in the BN calculation. A typical Noisy-Or model 38,39 is expressed as where y is a safety accident, X P is the set of fault nodes expressed by X 1 ,X 2 ,...X n ; X T is the truth set of fault nodes; P i is the probability of y if or only if X i = True.

| Fault diagnosis expert system
Fault diagnosis expert system is an intelligent tool that integrates expert experiences and Bayesian inferences, and it has significant advantages of the comprehensive collection of expert knowledge, the accurate simulation of expert thinking and the precision of fault diagnosis. The schematic diagram of the fault diagnosis expert system is performed in Figure 2. The development of the efficient fault diagnosis expert system will be beneficial to the condition-based maintenance at the lowest cost.

| Global methodology
Based on the above descriptions, Figure 3 is plotted to show the global methodology of Bayesian fault diagnosis of the HGS. The calculation process plan is concluded in the following steps: , i = 1,2,3,...,n. (2) , F I G U R E 2 Schematic diagram of a fault diagnosis expert system 1. Using expert experiences and monitoring data to collect the hydraulic, mechanical, and electric fault types in the HGS and also to investigate their associated fault characteristics. Based on this, a fault diagnosis model of Bayesian network for the HGS is presented. 2. The expert system gives the prior probabilities of nodes, and the Noisy-Or modeling approach is employed to reduce the node computations.

| MODEL
To model a BN of fault diagnosis, the critical task is to analyze the complex fault types and their associated fault characteristics in the HGS. We extensively collect the faults data of the HGS from literatures, on-site visit, and expert advice. In general, the HGS's fault refers to that the system works abnormally with enormous vibrations and can even lead to accidental shutdown or component damage since about 80% of HGS's faults are caused by component vibrations. Statistically, the disturbing forces (ie, the rotational unbalanced force of rotors, the hydraulic unbalanced force, and the unbalanced magnetic pull) with different magnitudes, directions, and frequencies will influence the performance of vibrations. Based on the operating characteristic of the HGS, the disturbing forces are attributed to the hydraulic, mechanical, and electric factors. Hence, the fault types and the associated fault characteristics can be performed in the fault diagnosis BN of the HGS, as shown in Figure 4.  . conditional probability  table refers to the conditional probability  table. HGS refers to the hydroelectric generation system

| CASE STUDY
The mechanical fault, as the most important influence factor on the safety of the HGS, is selected as a case study for the application of the BN proposed in this work. The typical mechanical fault (ie, the rubbing fault MF2, the misalignment fault of rotor MF3, and the mechanical axial crack MF4) and their associated fault characteristics (ie, the vibration with doubled frequency F2F0 and the vibration with third frequency F3F0) are finally modeled a studied BN, as shown in Figure 5. In the actual operation of hydropower stations, the rubbing fault (MF2) is triggered by improper assembly, shafting bend, rotor imbalance, and mechanical looseness, resulting in enormous vibrations and noises. The misalignment fault of rotor (MF3) generally leads to the deformation of shaft and rotor swing, which significantly reduces the operating efficiency of the HGS. The mechanical axial crack (MF4) has obvious adverse effects on the stiffness of shaft, which can cause unexpected shaft broken accidents with the increase of load and turbine speed. For the HGS's BN with critical mechanical faults performed in Figure 5, the possible working states of the fault nodes are "normal" and "trouble," as well as the fault frequencies for their associated fault characteristics nodes include "high" and "low." For the Noisy-Or model (3), the matrix of X P = X 1 = normal,X 2 = trouble,X 3 = trouble . Substituting the above probabilities into the Noisy-Or model (3-1), we obtain Based on the Noisy-Or model  and Equation (4), it can be obtained as.
Therefore, the CPT of node F2F0 is listed in Table 1.

CPT of node F3F0
Based on expert experiences, the probabilities are obtained as follows: Then, based on the Noisy-Or model (3), we can get: where the fault nodes set X P = { X 1 = normal, X 2 = trouble, X 3 = trouble} in Equation (7-1), X P = {X 1 = trouble, X 2 = normal, X 3 = trouble} in Equation (7-2), X P = {X 1 = trouble, X 2 = trouble, X 3 = normal} in Equation , and X P = {X 1 = trouble, X 2 = trouble, X 3 = trouble} in Equation . Thus, the CPT of node F3F0 is listed in Table 2. Using Bayes' theory presented in the methodology section, we establish the fault diagnosis expert system of the HGS that integrates expert experiences and Bayesian inferences. The BN inference is utilized to give some typical applications of the BN-based fault diagnosis of the HGS. Six cases are performed as follows.
• Case 1: Assuming the fact is the increasing vibration with doubled frequency. That is, the probability of the fault characteristic node F2F0 in "high" state is 1. Using the Bayesian diagnosis inference (the definition is revealed in the literature 40   From the analysis of cases 3 and 6, when the HGS shows the same fault characteristic except for the mechanical axial crack, the occurrence probability of the rubbing fault, and the misalignment fault of rotor will decrease.

DISCUSSION
In this work, the fault diagnosis method for the hydroelectric generation system coupling with hydraulic, mechanical, and electric factors is presented. The methodology adopted in this work is based on the Bayesian networks approach and the expert system. Herein, a complete Bayesian network fault diagnosis model of the generating system is implemented that takes into consideration the comprehensive knowledge of the vibration fault types and the associated fault characteristics. The Noisy-Or modeling approach is used to calculate the CPT of the presented Bayesian network to overcome the limitation of the complicated node computations and data dependency in current approaches. The final implementation of the fault diagnosis expert system realizes the combination of expert experiences and Bayesian inferences. The obtained results allow to develop the time-based maintenance to the condition-based maintenance, which achieves the goal of the reduction of the maintenance costs in hydropower stations. In addition, historical data collected from a hydropower station are a good method to improve the accuracy of the diagnosis, while it is extremely difficult to obtain diagnosis from manufacturers since such data are confidential. To propel the future study of historical data parameter learning or other data-based methods, we are attempting to cooperate with potential hydropower stations to carry out some experiments of the generating system. The above illustrations have been added to the manuscript to guide our future work. Moreover, the future work is designed to the extraction of the common fault characteristics to improve the coupling relationship of the electric faults with the mechanical hydraulic fault network.