A fuzzy diagnosis approach using dynamic fault trees

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Abstract

By incorporating digraph models, fault trees and fuzzy inference mechanisms in a unified framework, a novel approach for fault diagnosis is developed in this work. To relieve the on-line computation load, the fault origins considered in diagnosis are limited to the basic events in the cut sets of a given fault tree. The symptom occurrence order associated with each root cause is derived from system digraph with the qualitative simulation techniques. The implied candidate patterns are enumerated according to two proposed theorems and then encoded in the inference system with IF–THEN rules. The simulation results show that the proposed approach is not only feasible but also capable of identifying the most likely cause(s) of a hazardous event at the earliest possible time.

Introduction

Owing to the continuous increase in scale and complexity of the modern chemical processes, numerous computer-aided fault diagnosis methods have been developed in recent years to assist operator to locate fault origins which may lead to catastrophic consequences. A wide variety of different approaches were discussed in the literature, e.g. the expert systems (Rich & Venkatasubramanian, 1987; Petti, Kleni, & Dhurijati, 1991), the state observers (Chang, Mah, & Tsai, 1993; Chang & Chen, 1995; Chang and Hwang 1998a, Chang and Hwang 1998b) the neural networks (Venkatasubramanian & Chan, 1989; Watanabe, Matsuura, Kuboto, & Himmelblau, 1989; Venkatasubramanian, Vaidyanathan, & Yamamoto, 1990; Hoskins, Kalivur, & Himmeblau, 1991; Tsai, Chang, & Chen, 1996) and the signed directed graphs (SDG) (Iri, Aoki, O'Shima, & Matsuyama, 1979; Shiozaki, Matsuyama, O'Shima, & Iri, 1985; Kramer & Palowitch, 1987), etc. Ulerich and Powers (1988) reported the first attempt to perform fault diagnosis on the basis of fault trees. The most significant advantage of their approach is that the candidates of fault identification are restricted to only the causes of one or more given top events and, consequently, the diagnosis procedure can be greatly simplified. However, it should be noted that only a conceptual framework was proposed in this study. There is thus a need to develop systematic algorithms for its implementation. In addition, only the eventual symptoms in process measurements were utilized in Ulerich and Powers (1988). It is our belief that the occurrence order of these on-line symptoms should also be incorporated in an improved strategy for locating the fault origins.

In this work, the fuzzy set theory (Ross, 1995) is adopted to facilitate construction of a fault-tree based diagnosis system. This selection is mainly due to the following reasons:

  • The fault tree is basically a qualitative model of all failure mechanisms leading to an eventual top event. Therefore, the on-line symptoms observed in process measurements must be interpreted accordingly in a consistent manner. In a traditional fault tree, the abnormal process conditions are often classified with linguistic terms, e.g. “too high” or “too low”, and they can be appropriately characterized with the fuzzy membership functions.

  • As suggested by Ulerich and Powers (1988), the occurrence probabilities of top event and also its causes should be computed in real time. To accomplish these tasks, it is necessary to produce the on-line estimates of (1) the reliability (or availability) of each hardware item in the given system and (2) the frequency of every significant external disturbance. In the former case, although it is possible to extract some of the reliability parameters, e.g. the failure rate and mean time to repair, from statistical data (if available), their accuracy is highly questionable. On the other hand, the latter information can only be obtained from operation experience and its nature is often imprecise. Since a rigorous statistical treatment is really not practical under this circumstances, a set of fuzzy occurrence measures are adopted instead in this study as the conclusions of fault diagnosis.

The proposed fault diagnosis procedure can be implemented in two stages: the off-line preparation stage and the on-line implementation stage (Fig. 1). In the former case, a SDG system model is first constructed and the fault trees corresponding to the given top events are then synthesized according to the Lapp-and-Powers algorithm (1977). The symptom occurrence order caused by the basic events in each cut set can be easily determined with the qualitative simulation techniques (Chang & Hwang, 1992) on the basis of the SDG model. In addition, two theorems are developed in this study to facilitate enumeration of all possible symptom patterns that may be observed at any time during operation. These patterns are then translated into a set of IF–THEN fuzzy inference rules for assessing the occurrence possibilities of the basic events in every cut set and also the top events. In the next stage, the on-line measurement data are first normalized. These normalized values are used as the inputs to a fuzzy inference system for computing the occurrence indices of top events and cut sets in real time. The Tennessee Eastman (TE) process simulation program (Downs & Vogel, 1993) is chosen as the platform for verifying the feasibility of the proposed approach. Extensive simulation results of the reactor section, the stripper section and also the entire process are presented at the end of this paper.

Section snippets

Qualitative system models

The products of the first three steps in the off-line preparation stage, i.e. the SDG, the fault trees and the minimal cut sets, can be considered as qualitative system models in different formats. It should be noted that the SDG construction methods have already been discussed extensively in the literature, e.g. Iri et al. (1979) This study follows basically the conventional approach to build digraph models. Before developing these digraphs, the scope of fault diagnosis (SCFD) must be

Symptom occurrence order

The effects of base event(s) in a cut set usually propagate throughout the entire system sequentially. In general, a series of intermediate events may occur before the top event. Since the performance of a diagnosis scheme should be evaluated not only in terms of its correctness but also its timeliness, it is the intention of this research to develop a fault identification procedure taking both the eventual symptoms and also their occurrence order into consideration. To identify this symptom

Enumeration of candidate patterns

If all symptoms in a SOO can be observed on-line, then it is certainly reasonable to confirm the existence of corresponding fault origin(s). However, it is also possible to find that these symptoms are only partially developed during the incipient period of an eventual system hazard and, further, their pattern may vary at different times during operation. To facilitate later discussions, let us define the collection of on-line symptoms at any time after the introduction of basic event(s) in a

Fuzzy inference system

In this work, the final product prepared off-line is a fuzzy inference system (FIS). A sketch of its framework is presented in Fig. 7. If this system is to be implemented on-line, the measurement data must be first converted to a set of normalized deviations with respect to the given reference values and, then, used as inputs to FIS. The core of FIS is a collection of IF–THEN rules, which can be further divided into three distinct classes. The outputs of FIS are the occurrence index of top

The on-line implementation procedure

During operation, the process measurements are taken continuously on-line and then converted to the normalized deviations according to Eq. (3). These values are then used in the fuzzy inference system to calculate the occurrence indices of cut sets (csis) and also the top event (OITE). The fuzzy inference algorithm proposed by Mamdani and Assilian (1975) is adopted in this work. The index OITE can be utilized as the basis for alarm generation and csis are essentially the measures for ranking

Case studies

Extensive simulation studies have been carried out in this work to demonstrate the effectiveness of the proposed fault diagnosis approach. As mentioned before, the test data were generated with the TE process simulator. To execute the simulator successfully, it is necessary to specify a proper control configuration for the entire process. Thus, the following five control loops are included in all simulation runs:

  • (1)

    reactor temperature control loop,

  • (2)

    recycle flow control loop,

  • (3)

    separator temperature

Conclusion

A novel fault diagnosis approach based on knowledge concerning all possible patterns in on-line symptoms is proposed in this paper. For simplification purpose, the candidates of fault identification is limited to the minimal cut sets of a given fault tree. The fault propagation paths (FPPs) and symptom occurrence order (SOO) associated with each cut set are generated from the system digraph with a systematic procedure. All possible symptom patterns of each fault origin can be enumerated

Notation

cskthe occurrence index of kth cut set
dithe normalized process deviation of ith measured variable
Fthe linguistic interpretation function
INOCthe location of upper interior corner of the membership function NOC
IOCRthe location of upper interior corner of the membership function OCR
Mthe set of all measured variables
mithe measurement value of ith measured variable
missthe value of mi at steady state
missthe average value of miss
NCPthe number of candidate patterns
NOCnot occur, linguistic value of

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