1 Introduction

In recent years, fault diagnosis (FD) has become important for safe operation of nuclear power plants (NPPs). Fuzzy and neuro-fuzzy approaches, support vector machines ^[1], K-nearest neighbors ^[2,3], decision tree (DT) and fuzzy decision tree (FDT) classification methods ^[4-9] are proposed and implemented to tackle the problem. FDT improves the DT capability by embedding fuzzy set (FS) theory in the FD and benefits of the advantages of both DT and FS. However, the not univocal correspondence between symptoms and fault classes might challenge the diagnosis task. Researches have been done in this area and FD systems like REACTOR, SINDBAD and OAS are developed ^[10-13]. All of these systems focus on the static state of NPP and method for prognostic is not mentioned. ISACS (integrated surveillance and control system) is developed in the OECD Halden Reactor Project with functions including identification of disturbance, diagnosis and prognostics ^[14-16]. This system mainly contains two parts: prior diagnosis, which is model-based; and on-line diagnosis, which is based on comparing the fault data with those in Process and Automatics (P&A) data base. Unfortunately, the prototype ISACS-1 is mainly utilized for extensive evaluation and further progression of the system has not been elaborated. In other FD systems, some are based on the complete model of the industrial systems ^[17-21], which is difficult to be applied on the systems whose complete model are hard to build. While some other systems are data-mining based ^[22,23], which is difficult for NPPs because it is often difficult to obtain fault data to train the classification model. J Ma and J. Jiang developed a FD system based on semisupervised classification scheme, which was considered as a promising tool. ^[24] However, the model are trained by the data of normal power plant, which may not diagnose the faults that owned only by NPPs.

Recent works on classic uncertain artificial intelligence methodology, such as rough set ^[25,26], artificial neural network (ANN) ^[27,28], fuzzy theory ^[29-31], grey relational analysis ^[32,33], petri nets ^[34,35] and support vector machine (SVM) ^[36,37], are introduced and combined for FD and prognostic in complex systems. Also, probabilistic graphical model has become new hotspot in this field, and the Bayesian networks (BN) model ^[38,39], Hidden Markov Models (HMM) ^[40-42], Latent Tree Models (LTM) ^[43] and cloud models ^[44] draw high attention. BN is especially typical. It is of solid theoretical foundation, with directed acyclic graph to express causal dependency and conditional probability table to quantify the uncertainty of causality. To improve the efficiency of FD, a graphical inference methodology named Dynamic Uncertain Causality Graph (DUCG) has been developed and successfully tested in many practical cases ^[45-52].

DUCG implements a causal-based expert system that represents the knowledge of risk experts in either a directed acyclic graph or a directed cyclic graph, with a probabilistic-based or evidence-based reasoning ^[45,46]. Nevertheless, the causal graphs construction of DUCG is complicate in some cases, turning out redundant on the symptoms needed by DUCG to correctly classify the occurring faults^[47-52]. In this paper, we propose a method to simplify the inference rules of DUCG based on a genetic algorithm (GA) that uses an FDT as surrogate model of the DUCG.

This paper is organized as follows. In Sect. 2, a brief introduction to the DUCG is given, including the inference algorithm used for fault diagnosis and the steps required for constructing the DUCG model. Section 3 illustrates the method for transforming the expert knowledge into a FDT by extracting a fuzzy rule base (FRB) from a first-tentative DUCG model that provides a non-optimal symptom matrix. In Sect. 4, GA is used for the optimization of the decision rules by performing a wrapper search around the FDT and results are discussed. In Sect.5, conclusions are drawn.

2 Brief introduction to DUCG

The methodology of DUCG is essential to apply the causal graphs to symbolize logical relationships in complex system in reality and to apply the virtual variables to express the causal uncertainties in that relationships, that is, the probabilities between child and parent variables. DUCG provides a compact representation of the existing causal logic among events that can occur in a process^[46,47], by resorting to causal graphs composed by variables (or events) (as illustrated in Table 1, where “V” is a generic variable and “N” and “k” are the variable number and state, respectively), connected by function F.

Table 1.Full-screen View

Variables (events) in DUCG (N,the variable number; k, state)

Variable	Symbol	Meaning
B		Failure type always with k=1
X		Symptom
Fn,k ;i,j		Uncertainty relationship between variable Vi,j in state j ($V\in \{B,X\}$, i=1,2,...,N) and variable Xn,k in state k, where i ≠n and j ≠ k

Show more

We can define F_n;i as a graph composed of elements connected by F_n,k;i,j^[46], as shown in Fig. 1(a), that can be defined as:

Fig.1Full-screen View

Examples of weighted functional elements (a) and a simple DUCG (b).

where, A_n,k;i,j accounts for the mechanism that V_i,j induces on X_n,k, without considering any other natural interaction with other mutual variables B_i,j, j' ≠ j; and (r_n;i/r_n) is a weighting factor of A_n,k;i,j, with $r_n\equiv {\displaystyle {\sum }_i{r}_{n;i}}$ and r_n;i being the causal relationship intensity between each V_i and X_n^[46-49].

The procedures for building a DUCG can be summarized as follows:

1) A detailed analysis of the system to identify all B type variables (i.e., fault types);

2) To determine the related X type variables for each of the B type variables;

3) To quantify the causal relationships. i.e. the F type variables.

For example, in Fig. 1(b), B_33,1 indicates the leakage in the steam pipeline A, variable X₄₇, X₁₇₃ and X₁₇₄ represent the steam flow in pipelines A/B/C of a NPP, respectively; and, X₄₈ indicates the pressure in the steam pipeline A, where the normal, low and high pressure states are X_48,0, X_48,1 and X_48,2, respectively. The relationship, for example, between X₄₈ and X₄₇ can be modeled by F_48;47=(r_48;47/∑r_48;i)A_48;47, where, ∑r_48;i=r_48;47+r_48;173+r_48;174 (with r_48;47=r_48;173=r_48;174=1), and A_48;47 (the uncertain mechanism that links the three states of X₄₇ to the three states of X₄₈) is

$A_{48;47}=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0.1& 0.7\\ 0& 0.9& 0.3\end{array}\right)$,

which means that when X₄₇ is in its “1” state X_47,1, the probabilities of the X₄₈ in the “0” state X_48,0 is 0, in the “1” state X_48,1 is 0.1 and in the “2” state X_48,2 is 0.9. Similarly, when X₄₇ is in its “2” state X_47,2, the probabilities of the X_48,0, X_48,1 and X_48,2 are 0, 0.7 and 0.3, respectively.

By utilizing this modeling method, the DUCG of the secondary loop of Unit-1 of Ningde NPP is shown in Fig. 2. The causal graph is of great complexity, including 151{B,X} type variables and 976 functional relationships, of which 23 variables of type B (fault classes) are given in Table 2.

Fig.2Full-screen View

DUCG of the secondary loop of Unit-1 of Ningde NPP.

After construction of the DUCG model based on the expert knowledge regarding the expected progression of an accidental scenario, that allows setting the functional relationship among the variables, the diagnosis can be carried out by monitoring the signals behavior during the developing accidental scenario.

For better illustration, let us focus on the types and states of the variables shown in Fig. 3. The fault B_34,1 can be diagnosed as “turbine bypass valve GCT131VV wrongly opened during NPP normal operation” because it initiates the flow decrease in Steam pipelines A, B and C (X_1,1, X_2,1, X_3,1), which, subsequently, decreases the pressure in the steam manifolds VVP024MP and VVP025MP (X_4,1, X_5,1), and increases the feed-water flow in the stream generators SG-1, SG-2 and SG-3 (X_6,2, X_7,2, X_8,2). The reduction of the feed-water flow can further induce the average temperature to decrease in the first loop (X_9,1) and, eventually, cause the increase of the reactor power (X_10,1).

Fig.3Full-screen View

One fault in DUCG model.

Therefore, a specific state of a B-type variable corresponds to a set of X-type variables, which are all abnormal signals received from the sensor measurements. In other words, when a generic fault of class Cj, j=1,2,… occurs, a set of representative symptoms emerge that might not be univocal, as in this case.

3 From DUCG to fuzzy decision tree

The FDT methodology is selected to speed up the optimization of the DUCG model in the case of non-univocal symptoms. The FDT is used as surrogate model of the DUCG, whose fuzzy rules are to be extracted from the DUCG model to build the FRB knowledge. A first-tentative prior FRB is chosen to represent the classification reasoning. The FRB is composed of several fuzzy rules, each one related to a specific FS for each fault class B_i,j. As an example, with reference to Fig. 3, the fuzzy rule defining B_34,1 can be written as: “IF the flow in steam pipeline A, B and C is low (X_1,1, X_2,1, X_3,1); the pressure in steam manifolds VVP024MP and VVP025MP is low (X_4,1, X_5,1); the feed water flow in stream generators SG-1, SG-2 and SG-3 is high (X_6,2, X_7,2, X_8,2); the average temperature in the first loop is low (X_9,1) and the reactor power is high (X_10,1), THEN the turbine bypass valve GCT131VV has been wrongly opened (B_1,1)”. Symptoms of X-type variables mentioned in the fuzzy rule can be summarized into an observation vector σ =(X_1,1, X_2,1, X_3,1, X_4,1, X_5,1, X_6,2, X_7,2, X_8,2, X_9,1, X_10,1). All the fuzzy rules defining the relationships between symptoms and the 23 fault classes can be resumed in the symptom matrix (shown in Table 3 in a reduced form), in which the rows and columns are fault classes and symptoms, respectively, where “1” means that the symptom can be observed for the fault considered, and “0”, otherwise. We note that the relationship between symptoms and fault classes is not univocal, that is, one symptom may lead several faults and one fault can be inferred only when several symptoms are simultaneously observed. This is mainly due to the complex non-linear relationship of the events, but also to the partial and sometimes overwhelming expert knowledge at the basis of the DUCG.

Sensor errors and ambiguous deviation ranges (that is, uncertainty in the symptoms values) can be accommodated in the FRB by defining a fuzzy observation vector $\varsigma =({\mu }_{S_1},{\mu }_{S_2},\mathrm{...},{\mu }_{S_n})$, in which each element ${\mu }_{S_n}$ is the degree of membership of a symptom S_n to the FS “symptom occurrence”.

The construction of the FDT aimed at substituting the DUCG within the GA-based optimization of the FRs can be done as shown in Fig. 4(a); in particular:

Fig.4Full-screen View

Steps for building a FDT (a) and example of a FRB-FDT (b)

1) Choose one fault class of the DUCG;

2) Select one symptom S_n;

3) Branch the tree based on presence or absence of the symptom for the fault class considered by accounting for its membership degree ${\mu }_{S_n}$;

4) If the symptom can be attributed to only one fault class, the associated node becomes a terminal leaf, otherwise, a new node is added with membership degree $(1-{\mu }_{S_n})$ and procedure starts again from 2).

When all symptoms are screened out and placed in related branches, the tree is terminated. Fig. 4(b) shows a general example of a FRB-FDT.

With the method illustrated above, the symptom matrix of Table 3 of the DUCG model in Fig. 2(b) is learnt by a FRB-FDT with 23 fault classes and 262 symptoms. The developed FRB-FDT is tested with respect to 800 data synthetically simulated by the China Nuclear Power Simulation Technology Co, Ltd (CNPSC). The results indicate that 58.9% of the anomalous behaviors are correctly classified with respect to the 23 classes, 29.7% are treated as ambiguous and 11.4% are wrongly classified due to the ambiguous assignment between symptoms and fault classes of the initial structure of DUCG based FRB-FDT (for the classification rate quantification, we assume the assignment to be correct when the membership grade to a class is greater than 0.8, ambiguous when is between 0.8 and 0.2 and wrong when it is lower than 0.2).

4 GA optimization of the inference rules

To improve the classification rate of the DUCG and lower its computational burden, thanks to a simpler symptoms-fault relationship, a single objective GA-based optimization is conducted for finding the most representative symptom matrix that would best describe the system behavior. We resort to the wrapper search scheme shown schematically in Fig. 5.

Fig.5Full-screen View

Wrapper research with GA.

A symptoms observation vector σ becomes a chromosome of the GA, and the fitness function to evaluate the performance of a set of symptoms in classifying the faults is defined as:

where a(1−λ) accounts for the classification rate λ of the FDT, and bα accounts for the number α of symptoms needed for classifying as many as possible fault classes, since the optimization must obey a parsimony principle (i.e., low α is preferable) to guarantee a low computational complexity of DUCG and a large classification rate, i.e., low (1−λ). The objective (fitness) Φ must be minimized. The parameters a and b are user-defined weight, and here we set a =80 and b =1, so as to balance the different magnitudes of (1−λ) and α, and normalize their mutual contribution to Φ. The main parameters of the GA used are listed in Table 4.

GA can better select symptoms for each fault class, hence the improvement of performance. The results of optimization are shown in Fig. 6. In Fig.6(a), the rate of correct classification increases with the number of generations, while the rates of ambiguous and wrong classifications decrease. The three rates reach steadiness at the 90^th generation, being 93.8%, 5.9% and 0.4%, respectively. In Fig. 6(b), the value of objective function decreases with increasing number of generations. This indicates that the new set of selected chromosomes can neglect the contradictory assignments between symptoms and fault classes, but, rather, link them without ambiguity. Therefore, the GA optimization of the symptom matrix is beneficial to the FDT and, thus, to the DUCG that would be built on the basis of these results.

Fig.6Full-screen View

Results of optimization.

To be more accurate, one experiment based on 150 fault sequences is carried out to verify the reduction in computation complexity. Fig. 7 shows the results by initial DUCG inference model and the 100^th generation of FDT-based GA optimization. The number of symptoms needed for classification of maximum fault classes, that is, α in Eq.(2), is used to measure the computation complexity. It is shown that the computational complexity can be significantly reduced after the optimization and the degrees of computation reduction are all above 70.5%, maximum to 84.95%.

Fig.7Full-screen View

Results in computation complexity

5 Conclusion

DUCG is an interesting method for fault diagnosis for industrial systems. However, causal graphs that are built by DUCG for inferring the fault classes can become huge for complex systems (as for the example of the secondary loop in Unit-1 of Ningde NPP here considered). This also leads to a computational burden when performing the diagnosis.

To optimize the DUCG model, we have proposed to substitute it with a FDT and to optimize the symptom matrix of this FRB model by a single-objective GA. A wrapper search is carried out to look for the set of observations (coded into GA chromosomes) that best characterize the failure behavior of the system. This is expected to lead to twofold benefits: improvement of the classification rate and decrease of the complexity of the DUCG model.

Experimental results on a set data from a simulator of the secondary loop in Unit-1 of Ningde NPP demonstrate that the application of GA for optimizing a FDT-based DUCG is effective in optimizing the DUCG model.

However, some unique properties of DUCG, such as the effect of combination of several symptoms to a fault, the hidden relationships among different symptoms and the process of time sequence in some cases are not considered in this paper, which our future work focused on.