A Fault Isolation Method via Classification and Regression ... - MDPI

energies Article

A Fault Isolation Method via Classification and Regression Tree-Based Variable Ranking for Drum-Type Steam Boiler in Thermal Power Plant Jungwon Yu 1 , Jaeyel Jang 2 , Jaeyeong Yoo 3 , June Ho Park 1 and Sungshin Kim 1, * 1 2 3

*

ID

Department of Electrical and Computer Engineering, Busan National University, Busan 46241, South Korea; [email protected] (J.Y.); [email protected] (J.H.P.) Technology & Information Department, Technical Solution Center, Korea East-West Power Co., Ltd., Dangjin 31700, South Korea; [email protected] Chief Technology Officer (CTO), XEONET Co., Ltd., Seongnam 13216, South Korea; [email protected] Correspondence: [email protected]; Tel.: +82-51-510-2374

Received: 6 April 2018; Accepted: 2 May 2018; Published: 3 May 2018







Abstract: Accurate detection and isolation of possible faults are indispensable for operating complex industrial processes more safely, effectively, and economically. In this paper, we propose a fault isolation method for steam boilers in thermal power plants via classification and regression tree (CART)-based variable ranking. In the proposed method, binary classification trees are constructed by applying the CART algorithm to a training dataset which is composed of normal and faulty samples for classifier learning then, to perform faulty variable isolation, variable importance values for each input variable are extracted from the constructed trees. The importance values for non-faulty variables are not influenced by faulty variables, because the values are extracted from the trees with decision boundaries only in the original input space; the proposed method does not suffer from smearing effect. Furthermore, the proposed method, based on the nonparametric CART classifier, can be applicable to nonlinear processes. To confirm the effectiveness, the proposed and comparison methods are applied to two benchmark problems and 250 MW drum-type steam boiler. Experimental results show that the proposed method isolates faulty variables more clearly without the smearing effect than the comparison methods. Keywords: drum-type steam boiler; fault isolation; classification and regression tree; variable ranking; smearing effect

1. Introduction Modern industrial processes (e.g., power plants, and chemical and manufacturing processes) are becoming larger and more complex owing to efforts to fulfill safety and environmental regulations, to save costs, and to maximize profits. Therefore, process monitoring for accurate and timely detection and isolation of potential faults is more important than ever before. Faults are defined as abnormal events that occur during process operations. In thermal power plants (TPPs), main generating units (e.g., steam boilers and turbines) operate under very high pressure and temperature; if possible faults of the main units are not detected and isolated precisely, they may result in system failures, and eventually may cause significant losses of life and property. Properly designed monitoring systems can ensure safe, effective, and economical operations of target processes. Statistical process monitoring techniques fall into two categories: model-based and data-based methods. In the model-based methods, process monitoring is carried out by rigorous mathematical models derived from prior physical and/or chemical knowledge (e.g., mass or energy balances)

Energies 2018, 11, 1142; doi:10.3390/en11051142

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governing target processes; the monitoring performance may be degraded when the derived models are inaccurate. If target processes are heavily complex and large-scale, the derivation of the models is difficult, or may be impossible; even when possible, it is costly and takes much time. Despite their development efforts (e.g., mathematical derivations and proofs), they may be limited by boundary conditions for specific system behaviors or only applicable for particular system setups [1]. In the data-driven techniques, monitoring models are developed based on historical operation data. Recent advances in communications and measurement technologies facilitate the installation of huge numbers of sensors in target processes; various key variables related to process monitoring and control are measured by the installed sensors. Distributed control systems and/or supervisory control and data acquisition systems built into modern industrial processes enable efficient management and storage of the massive amounts of operational data. Without mathematical models based on the first-principles, the data-driven methods are capable of extracting useful health information from abundant operation data; they are receiving a great deal of attention from academia and industries. Process monitoring is carried out via the following four steps [2]: fault detection, fault isolation, fault diagnosis, and process recovery. Fault detection determines whether possible faults have occurred; early detection helps operators and maintenance personnel take corrective actions to prevent developing abnormal events from leading to serious process upsets. Fault isolation (also called fault identification) locates faulty variables closely related to detected faults; the isolation results assist field experts in diagnosing the faults precisely, i.e., it helps the system operators to determine which parts should be repaired or replaced [3]. Fault diagnosis investigates the root causes and/or sources of occurring faults. Process recovery corresponds to the last step in finishing the process monitoring loop; in this step, the effects of abnormal events are removed and target processes get back to normal operating conditions. So far, multivariate statistical techniques and machine learning have been widely used for fault detection and diagnosis of power plant equipment, such as principal component analysis (PCA) [4–7], independent component analysis (ICA) [8,9], auto-associative kernel regression (AAKR) [10,11], artificial neural networks [12,13], fuzzy models [14,15], support vector machine [15,16], neuro-fuzzy networks [17], and group method of data handling [18]. PCA and ICA can handle multivariate process data effectively via dimensionality reduction. AAKR is a nonparametric multivariate technique to obtain predicted vectors for new query vectors by updating local models in real time; it can perform fault detection tasks without any assumptions for target data properties (e.g., linear or nonlinear). In the case of machine learning techniques for black box modeling, considering nonlinearities of target processes, fault detection and diagnosis can be carried out without the need of physical or chemical knowledge. Although these methods can detect potential faults successfully, they may have several difficulties in isolating and diagnosing the faults. As explained above, after the fault detection tasks, fault isolation and diagnosis should be conducted to pinpoint the root causes of occurring faults. If there exist abundant historical faulty samples whose class labels are associated with all possible fault categories, multi-class classifiers [19–22] can be explicitly constructed. The designed classifiers assign the most similar fault categories to query samples; faults are diagnosed without the fault isolation procedure. When designing such classifiers, one might face the following difficulties. First, in general, the number of available faulty samples is much fewer than that of normal samples. Actual industrial processes are carefully designed to prevent abnormal events; the processes are equipped with closed-loop controllers compensating for the effects of unpermitted changes and disturbances. In addition, to build multi-class classification models, preparing complete training dataset encompassing all types of faults is nearly impossible; if all possible faults occurred several times in the past and the corresponding fault dataset was stored in a database, or if there are useful simulators that can emulate target processes as perfectly as possible, collecting the complete dataset may be possible. Second, collected training dataset for classifier learning is usually class-imbalanced [23]; the number of faulty samples may vary from class to class considerably. Furthermore, high levels of expert experience and knowledge are required to assign proper class labels

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to historical faulty samples; these labeling procedures may also be time-consuming. Lastly, classifiers based only on historical fault data do not function adequately when new types of faults develop. When the faulty data samples encompassing all possible fault types are not enough, after isolating faulty variables based on detection indices from fault detection models, the isolation results can be used for fault diagnosis. In the fault detection models, normal samples are only needed for training, and whether target processes fall outside normal operating regions is decided in real time; if detection indices for query samples deviate from predefined threshold values, fault occurrence is declared, and alarm signals are generated. After faults are detected, fault isolation should be performed to determine which variables are responsible for the detected faults. Contribution plots [24,25] are the most popular method for fault isolation; it is assumed that the contribution values of faulty variables to detection indices are higher than those of non-faulty ones. In this method, monitored variables are identified as faulty variables if their contribution values violate the confidence limits. Although the contribution analysis can identify faulty variables quickly and easily without a historical fault dataset and prior information on possible fault types, it has several limitations, as follows: first, the threshold values derived only from normal data may be unsuitable for isolating faulty variables, because contribution values of each variable within in-control regions may follow different distributions from those outside the regions. Second, it is well-known that isolation results from contribution plots confuse process operators and engineers due to the smearing effect [26–28]; smearing is a phenomenon that contribution values of non-faulty variables are enlarged by faulty variables. When complicated faults, rather than simple faults, have happened, fault isolation with contribution analysis may lead to an ambiguous diagnosis. In this paper, we propose a fault isolation method via classification and regression tree (CART)-based variable ranking for a drum-type steam boiler in TPP; it is assumed that some proper fault detection algorithms were already designed, and after possible faults are detected by the algorithms, the proposed fault isolation method is triggered. CART algorithm [29] with a divide-and-conquer mechanism, splits the entire input space repeatedly and builds binary trees. CART has been successfully applied to various fields [30–33] for which data mining techniques are necessary, since it can tackle multivariate data efficiently, and the constructed trees are transparent. In the proposed method, after designing classification trees by applying CART algorithm to training dataset composed of normal and faulty samples, variable ranking is performed by extracting importance values of each input variable from the trees. In the training dataset, Normal and Abnormal class labels are assigned to the normal and faulty samples, respectively; the faulty samples are relevant to the regions where the detection indices are larger than or equal to threshold values. In this paper, as will be explained in Section 3, we propose two approaches (see Figure 1) for fault isolation; the goal of the first approach is to locate faulty variables in entire fault regions, and the second approach (with time window sliding) aims to monitor how occurring faults propagate and evolve. The proposed method, based on the nonparametric CART algorithm, can be applicable to fault isolation of nonlinear processes; the method can locate faulty variables even when normal and faulty samples cannot be linearly separated from each other. Above all, the proposed isolation method does not suffer from the smearing effect, since it is based on CART algorithm defining decision boundaries only in the original input space. To verify the performance, the proposed method and comparison methods are applied to two benchmark problems and a 250 MW drum-type steam boiler. Experimental results show that the proposed method can effectively identify faulty variables without suffering from fault smearing, and can properly tackle nonlinearities of target processes. The remainder of this paper is organized as follows: Section 2 explains CART-based tree growing, pruning, and variable ranking. Section 3 describes the proposed isolation method via the variable ranking. Section 4 presents the experimental results and discussion, and finally, we give our conclusions in Section 5.

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(a)

…

Normal region

Fault region

time normal tstart

normal tend

Step 1. Select normal and fault samples normal normal X normal =  x(tstart )  x(tend ) 

test tstart

fault tstart

fault tend

Step 2. Prepare training data matrix for CART Input vectors

T

Normal data matrix

X

fault

=  x(t

fault start

)  x(t

fault end

) 

Training matrix (for CART)

T

Fault data matrix

normal T  x(tstart )     x(t normal )T X =  endfault T  x(tstart )    fault T  x(tend )

Class

Normal     Normal   Abnormal     Abnormal 

Step 3. Construct classification tree and obtain variable importance values

(b)

…

Window sliding

time

w

t

test start

t

fault start

t

fault start

+ w −1

t

fault end

Figure 1. Proposed variable ranking: ranking: (a) isolation Figure 1. Proposed fault fault isolation isolation method method based based on on CART-based CART-based variable (a) fault fault isolation for the entire fault region (denoted as Method 1); (b) fault isolation using time window sliding for the entire fault region (denoted as Method 1); (b) fault isolation using time window sliding (denoted (denoted as Method 2). as Method 2).

2. and Regression Regression Tree Tree 2. Classification Classification and The algorithm [29] builds binary binary decision decision trees by recursively recursively dividing dividing aa given given training training The CART CART algorithm [29] builds trees by dataset dataset into into several several subsets; subsets; the the entire entire input input space space is is partitioned partitioned into into mutually mutually exclusive exclusive rectangular rectangular regions. In the constructed trees, nodes without any child nodes are named terminal nodesnodes (i.e., regions. In the constructed trees, nodes without any child nodes are named terminal external or leaf nodes); they form a partition of all the training samples. The remaining nodes except (i.e., external or leaf nodes); they form a partition of all the training samples. The remaining nodes for the for terminal nodes nodes are called internal nodes,nodes, whichwhich have have only only two child nodes; especially, the except the terminal are called internal two child nodes; especially, nodes located at the top of the trees are defined as root nodes. In classification trees, out of several the nodes located at the top of the trees are defined as root nodes. In classification trees, out of several class label is is assigned assigned to to each each terminal terminal node. node. class labels, labels, only only one one label The advantages of CART algorithm can be summarized as follows [29,34]: without any The advantages of CART algorithm can be summarized as follows [29,34]: first,first, without any prior prior domain knowledge, classifiers can be constructed only from input-output training dataset; domain knowledge, classifiers can be constructed only from input-output training dataset; multivariate multivariate data collected from complex target systems can be efficiently handled by its divide-anddata collected from complex target systems can be efficiently handled by its divide-and-conquer conquer approach. Second, one does not need to be concerned about target data properties in advance advance approach. Second, one does not need to be concerned about target data properties in (e.g., linear or nonlinear) because of its nonparametric nature. Third, the algorithm is robust (e.g., linear or nonlinear) because of its nonparametric nature. Third, the algorithm is robust against against statistical statistical outliers outliers and and mislabeled mislabeled training training samples. samples. Lastly, Lastly, final final tree tree structures structures and and if-then if-then rules rules taken taken from from the the trees trees explicitly explicitly describe describe predictive predictive principles principles of of target target problems; problems; one one can can clearly clearly understand understand how theclassification classificationprocedures procedures conducted bytrees. the trees. The following CART-based tree how the areare conducted by the The following CART-based tree growing, growing, pruning, and variable ranking steps explained in Sections 2.1–2.3 are based on [29]. pruning, and variable ranking steps explained in Sections 2.1–2.3 are based on [29]. 2.1. Growing 2.1. Tree Tree Growing In the CART CARTalgorithm, algorithm,splitting splittingaanote notet into t into two child nodes, tL and R, is achieved selecting In the two child nodes, tL and tR ,tis achieved byby selecting an an optimum binary that makes thenodes child purer nodesthan purer thenode; parent node; here, the optimum binary splitsplit pointpoint that makes the child thethan parent here, the subscripts subscripts L andLeft R and denote Leftrespectively. and Right, respectively. Therecriteria are several criteria (i.e., impurity L and R denote Right, There are several (i.e., impurity function i(t)), function i(t)), such as Gini diversity index, cross entropy, and twoing rule, used for impurity such as Gini diversity index, cross entropy, and twoing rule, used for impurity measures; in [29], it was measures; in [29], it was reported that the properties of constructed trees do not depend highly on the types of criteria. In this study, to measure node impurities, we use Gini diversity index defined as:

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reported that the properties of constructed trees do not depend highly on the types of criteria. In this study, to measure node impurities, we use Gini diversity index defined as: i (t) =

∑ p(l |t) p(k|t)

l 6=k

=



∑l p(l |t)

2

− ∑l p2 (l t) = 1 − ∑l p2 (l t) , l, k = 1, . . . , C,

(1)

where C is the number of assignable class labels, and p(l|t) is the ratio of samples with lth class label among all samples at node t, i.e., ΣC l =1 p ( l t ) = 1 . In two-class problems (i.e., C = 2), Equation (1) can be rewritten as i(t) = 2p(1|t)p(2|t). In (1), the cost, c(k|l), k 6= l, of misclassifying a sample with lth label into kth label is set to 1; the cost may vary from label to label. Gini diversity index with c(k|l) 6= 1, k 6= l, is calculated as: i (t) = ∑ c(k|l ) p(k|t) p(l |t) , (2) l,k

where c(k|l) = 0 for k = l. In two-class problems, Equation (2) can be written as i(t) = (c(2|1) + c(1|2))p(1|t)p(2|t). Let s denote a candidate split point dividing a node t into tL and tR , and let pL and pR be the ratios of training samples entering tL and tR , respectively, by the split point among all samples at node t. The candidate split points of jth continuous input variable xj are obtained from averaging all possible adjacent pairs of training samples with respect to xj . The goodness of split point s at node t, ∆i(s, t), is defined as a decrease in impurity: ∆i (s, t) = i (t) − p L i (t L ) − p R i (t R ).

(3)

Let S be a set that consists of all possible candidate split points for all input variables. The best split point s* that gives the largest decrease in impurity is obtained by: s∗ = argmax∆i (s, t).

(4)

s∈S

For example, when the split point s* divides a node t into tL and tR on the j* th input variable x j∗ , training samples belonging to tL and tR are satisfied with the conditions x j∗ < s∗ and x j∗ ≥ s∗ , respectively. By the above principles, root node t1 is split into descendant nodes t2 and t3 , and then, partitioning is continuously applied at each node separately. If ∆i (s∗ , t) is smaller than a predefined value, then node splitting is stopped and the node t is regarded as a terminal node; the class label of the terminal node can be determined by plurality rule [29]. 2.2. Tree Pruning In tree growing, it may be unhelpful for the performance of final trees to consider complicated rules for stopping node division. Instead of stopping node splits when reaching a tree with right-sized terminal nodes, after continuing the splits until the number of training samples at each terminal node is very small (i.e., constructing a very large tree with many terminal nodes), it is much better to decide the final tree structure through selective upward pruning. A decreasing sequence of subtrees can be obtained by applying sequential tree pruning to the large tree in the direction of the root node; the tree with the minimum misclassification rate can be finally selected from the subtrees. In this paper, after obtaining the sequence of subtrees through minimal cost-complexity pruning, the final trees with optimal number of terminal nodes are selected using cross-validation. This recursive pruning process is computationally efficient; this process requires only a small fraction of the total tree construction time [29]. 2.3. Variable Ranking Based on Surrogate Splits Assume that node t is divided into tL and tR by best split point s* on best split variable x j∗ . Surrogate splits are defined as split points that behave most similarly to s* on the remaining input

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variables, except for x j∗ . Suppose that node t can also be partitioned into t0 L and t0 R by candidate splits sj on input variables xj , j ∈ {1, . . . , m}\j* . Let NLL (t) and NRR (t) be the number of training samples going to the left and right nodes by both s* and sj , respectively; NLL (t) and NRR (t) are the number of samples that meet the conditions x j∗ < s* and xj < sj , and the conditions x j∗ ≥ s* and xj ≥ sj , respectively. The probability that sj correctly predicts the action of s* is defined as: p(s j , s∗ ) =

NLL (t) + NRR (t) , N (t)

(5)

where N(t) is the number of training samples belonging to node t. If the probability values are larger, sj closely mimics s* ; if s* sends a sample to tL , then the likelihood of sj sending the sample to t0 L is high. The best surrogate split e s j on xj can be obtained through: e s j = argmax p(s j , s∗ ).

(6)

sj

If there are no split points on some input variables in final trees, we may think that these variables are useless for classification; however, this is a misinterpretation of the final trees when the effects of the variables are masked by other variables. In other words, although node splits on some variables do not appear in the final trees, the importance of those variables may be high; this may be the evidence of the masking. For example, for convenience of explanation, let us assume that two input variables x1 and x2 have greater effects on classification than other variables; here, the effects of x1 and x2 are similar but the effect of x1 is a little larger than that of x2 . In this case, node splits may be performed only on the masking variable x1 (i.e., x2 may be masked by x1 ); if x1 is removed, x2 may be prominently used as split variable. In summary, under the above situations, instead of determining variable importance only by examining split variables in final tree structures, we should decide variable importance through the variable ranking based on the surrogate splits [29]. The masking may occur when input variables are highly correlated to each other. For CART users, it is essential to confirm which variables are the most significant for classification and to rank variables according to their importance. Based on the surrogate splits explained above, one can calculate the importance values of each input variable. s j relevant to all input After calculating the goodness of split ∆i(s, t) for best split s* and surrogate splits e variables at all internal nodes, the importance of the jth variable can be measured by: M( x j ) =

∑

∆i (s, t) p(t),

(7)

t∈ T 0

where T 0 is a set composed of all internal nodes in the final trees, and p(t) is the ratio of samples at node t to entire training samples. If xj corresponds to the best split variable at node t, then the best split s* is substituted into Equation (7) (i.e., in Equation (7), ∆i(s, t) ≡ ∆i(s* , t)); otherwise, the value of s takes surrogate splits e s j of xj instead of s* (i.e., in Equation (7), ∆i(s, t) ≡ ∆i (e s j , t)). Since we are usually interested in the relative importance of input variables, variable ranking is performed through the normalized importance values as follows: M( x j ) = 100M( x j )/max M ( x j ). j

(8)

After extracting variable importance values via Equation (8) from constructed trees, faulty variables can be isolated by comparing these values. In the fault region, faulty variables have great explanatory powers to discriminate normal and faulty samples; their importance values are nearly 100 (i.e., M ( x j ) ≈ 100). On the other hand, the values of non-faulty variables are much smaller than those of faulty. In this paper, we decide monitoring variables whose importance values are larger than 80 (i.e., M( x j ) ≥ 80) as faulty variables. After sorting monitoring variables according to their importance values, we can also confirm the priority of variables responsible for occurring faults.

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3. CART-Based Fault Isolation This section provides the proposed fault isolation method via CART-based variable ranking as explained in Section 2.3. The purpose of fault isolation is to identify which monitored variables are closely related to occurring faults, and then to support process operators and engineers in deciding on correct fault types. In this paper, two kinds of approaches are employed for fault isolation. In the first approach, we check how much each variable contributes to the occurring faults in the entire fault region. In the second approach, time window sliding is used to track how the faults evolve and propagate over time. The first approach is suited to offline analysis of process anomalies; the second approach aids in understanding fault evolution and propagation mechanisms online. Figure 1a,b describe the proposed first and second fault isolation approaches, respectively. From now on, let us take a close look at the first approach (see Figure 1a). In Step 1, we prepare normal ), and fault data normal data matrix Xnormal composed of normal samples x(tnormal start ), . . . , x( tend fault fault fault matrix X composed of fault samples x(tstart ), . . . , x(tend ). The normal samples may correspond to training samples used to build fault detection models such as PCA and ICA; the fault samples are brought from a fault region where alarm signals are consistently generated. And then, training matrix X for tree construction is organized by augmenting Xnormal and Xfault ; class labels (Normal or Abnormal) are assigned to the last column in X, based on whether input vectors are normal or faulty. Finally, after building a classification tree based on X, the importance values of each variable are calculated, as explained in Section 2.3. In Step 3, after constructing a large tree with many terminal nodes, a sequence of subtrees is obtained through the minimal cost-complexity pruning; then, the optimal tree is selected by cross-validation, and the importance values are computed by Equation (8). The input variables with high importance values are dominant to separate normal and faulty samples in the original input space; they have great explanatory powers when classifying normal and faulty samples. The importance values of non-faulty variables are small, because their behaviors are similar in both normal and fault regions. On the other hand, faulty variables have high importance values, since their patterns are significantly different in the two regions. In summary, the first approach identifies the input variables with high importance values as faulty variables in the entire fault region. This approach cannot confirm fault how faults evolve and propagate from start time tfault start to end time tend . In the second approach in Figure 1b, calculations of the importance values are repeated by sliding fault fault , composed of w time window with size w along the time axis, i.e., from tfault start to tend . The matrix X fault samples, changes every moment; except for this, the remaining procedures are same as those of fault consists of faulty samples x( tfault ), . . . , the first approach. For example, at time t = tfault start + w − 1, X start fault fault consists of faulty samples x( tfault + 1), . . . , x( tfault + w ), x(tfault start end + w − 1), and at time t = tstart + w, X end and so on. Whenever Xfault changes, a new tree should be constructed and variable importance values are extracted from the newly constructed tree; computation time in this approach is longer than the first approach because many final trees are constructed. In this case, to save time, the pruning process may be skipped; the importance values before and after the process may not be considerably different. The importance values obtained through the window sliding from time t = tfault start + w − 1 fault to t = tend enable proper monitoring of fault evolution and propagation effects. From now on, for convenience of explanation, the two approaches presented in Figure 1a,b are referred to as Method 1 and Method 2, respectively. 4. Experimental Results and Discussion In this section, to verify the performance, the proposed method (i.e., Method 1 and Method 2) is applied to two benchmark problems (i.e., a simple linear process and a three variable nonlinear process), and 250 MW drum-type steam boiler. In the first and third problems, contribution plots based on PCA and AAKR are employed as comparison methods; in the nonlinear process, since PCA cannot detect bias and drift faults accurately, it is excluded. Before applying fault detection and isolation methods, each monitored variable is normalized to zero mean and unit variance. In PCA, to reduce dimensions in principal component (PC) subspace, PCs capturing more than 85 percent

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of variations are only retained; contribution analysis for detection indices (i.e., T2 and Q statistics) is performed for fault isolation; kernel density estimation (KDE) [35] is employed to define threshold values for fault detection and contribution analysis. In the case of AAKR [11], squared prediction error (SPE) is used as a detection index, and contribution values of monitored variables for SPE are also calculated; in common with PCA, thresholds for fault detection and contribution analysis are determined by KDE. In both PCA and AAKR, level of significance α is set to 0.01. To construct CART classifiers and to extract variable importance values from the classifiers, we employee the ‘fitctree’ and ‘predictorImportance’ MATLAB functions built into the Statistics and Machine Learning Toolbox; the ‘ksdensity’ function in the same toolbox is used to estimate cumulative distribution functions of detection indices and contribution values via KDE. Method 1 uses the impurity function in Equation (1) to measure node impurities. To take into account imbalance between normal and abnormal classes, Method 2 employs Equation (2) instead of Equation (1). In the case of Method 2, there are much more samples with Normal class labels than those with Abnormal in the training matrix X (see Figure 1) for CART classifiers. In Equation (2), c(1|2) and c(2|1) denote misclassification costs associated with classifying an abnormal sample into a normal sample and vice versa; for example, c(1|2) and c(2|1) can be set to 10 and 1, respectively. In Method 2, the size of time window w is set to 10. 4.1. Simple Linear Process First, let us describe the results of applying the proposed and comparison methods to the simple linear process, also used in [28]. Simulation data is obtained from the following multivariate linear system:     x1 −0.1681 0.2870 −0.2835  x   0.4354   0.3812 0.1455   2     x   0.0247 −0.0235 0.4096  s1  3     (9)  =   s2  + e  x4   −0.1173 −0.1763 0.4382      s3  x5   0.0825 0.1398 0.3204  x6

−0.3825

0.1250

0.4836

where x1 , . . . , x6 are directly measurable variables, s1 , . . . , s3 are source variables that follow normal distributions with mean of 0 and variances of 1, 0.8, and 0.6, respectively, and e is white noise with mean of 0 and variance of 0.04. From Equation (9), the six monitored variables are calculated through the linear combination of three source variables. To train PCA and AAKR, 3000 samples are generated; these samples are also used to form the normal data matrix Xnormal in Figure 1. Faulty samples are obtained through the addition of the term ξf (t), reflecting the effects of anomalies, into normal samples x* (t) as follows: x ( t ) = x∗ ( t ) + ξ f ( t ), (10) where the normal sample x* (t) at time t is corrupted by ξf (t), and ξ and f (t) denote the fault direction vector and fault magnitude, respectively. In this paper, in common with [28], we consider two types of fault (i.e., simple and multiple faults) to verify the fault isolation performance. In the simple fault, the fault direction vector and its magnitude are set to ξ = [0 0 0 1 0 0]T and f (t) = 10−5 t2 , respectively, and 1000 faulty samples are generated. The multiple fault is represented with ξ = [1 0 1 0 0 0]T and f (t) = 6 × 10−6 t2 , and the corresponding 1200 faulty samples are also generated. Figure 2 shows the fault isolation results of the comparison methods and Method 2 (see Figure 1b) for the simple and multiple faults. In the figures related to the comparison methods, if variable contribution values at a certain time are larger than or equal to thresholds, the relevant entries display as black; otherwise, the entries are marked with white. In the figures associated with Method 2, variable importance values at a certain time are indicated through different colors; as the values are close to 100 (or 0), the colors for relevant entries are close to black (or white). The simple and multiple faults start to be detected by PCA and AAKR at about time t = 300; the monitoring charts of PCA and

variable importance values at a certain time are indicated through different colors; as the values are close to 100 (or 0), the colors for relevant entries are close to black (or white). The simple and multiple Energies 2018, 11, x 9 of 20 faults start to be detected by PCA and AAKR at about time t = 300; the monitoring charts of PCA and AAKR for fault detection are omitted due to are space constraints. Asdifferent presented in Figure 2, PCA-based variable importance values at a certain time indicated through colors; as the values are Energies 2018, 11, 1142 9 of 19 contribution plots suffer terribly from smearing effect. In the simple fault, in addition to faulty close to 100 (or 0), the colors for relevant entries are close to black (or white). The simple and multiple faultsx4start to be detected by PCA and AAKR at about 300; xthe monitoring of PCAlimits. and In variable , contribution values of normal variables x1,time x2, xt3,=and 5 deviate fromcharts confidence AAKR for fault detection are omitted due to space constraints. As presented in Figure 2, PCA-based the multiple can confirm severe over non-faulty variables x2, x5, and x6. In AAKRAAKR forfault, fault we detection are omitted duesmearing to space constraints. As presented in Figure 2, PCA-based contribution plotsanalysis, sufferterribly terribly from smearing effect. Insimple the simple fault, in addition faulty its contribution plots suffer from smearing effect. Invariables the fault, in addition to faultyto variable based contribution smearing over normal appears consistently, although variable x 4 , contribution values of normal variables x 1 , x 2 , x 3 , and x 5 deviate from confidence limits. In not x4 , contribution of normal x1 , x2, x3 , and x5 deviate from confidence limits. In the severity is less thanvalues PCA. In the casevariables of Method values of non-faulty variables are 2 , importance the multiple fault, we can confirm severe smearing over non-faulty variables x 2 , x 5 , and x 6 . In AAKRfault, wein canthe confirm smearing non-faulty variables x2 , variables x5 , and x6 . In equalmultiple to zero only earlysevere stages of the over faults. However, faulty of AAKR-based the simple and based contribution analysis, smearing over variables normal variables appears consistently, although its contribution analysis, smearing over normal appears consistently, although its severity is multiple faults are clearly identified after approximately time t = 370. Contrary to comparison severity is less than PCA. In the case of Method 2, importance values of non-faulty variables are not less than PCA. In the case of Method 2, importance values of non-faulty variables are not equal to zero methods, as zero the magnitude of thestages faultsof becomes larger over time, variables importancethe values of normal equal only in the early the faults. However,offaulty simple only intothe early stages of the faults. However, faulty variables the simple and of multiple faultsand are variables obtained 2 becomeafter closer to zero. multiple faults by are Method clearly identified approximately time t = 370. Contrary to comparison

Simple fault

clearly identified after approximately time t = 370. Contrary to comparison methods, as the magnitude methods, as becomes the magnitude of the faults becomesvalues larger of over time,variables importance values normal2 of the faults larger over time, importance normal obtained byof Method (c) (a) (b) variables obtained by Method 2 become closer to zero. become closer to zero.

(f)

(e)

Multiple fault

(d)

Figure 2. (Simple linear process) Fault isolation results of comparison methods (PCA- and AAKRFigure 2. (Simple linearand process) Fault2:isolation results of comparison methods (PCA- plot; and AAKRbased contribution Method (a) (simple fault) PCA-based contribution (b) (simple Figure 2. (Simpleplots) linear process) Fault isolation results of comparison methods (PCA- and AAKR-based based contribution plots) and Method 2: (a) (simple fault) PCA-based contribution plot; (b) (simple fault)contribution AAKR-based contribution plot; importance values with plots) and Method 2: (c) (a)(simple (simplefault) fault)variable PCA-based contribution plot; (b)Method (simple2; (d) fault) contribution plot; (simple fault) variable importance values with Method 2; (d) fault) AAKR-based AAKR-based contribution plot;(c)(c)plot; (simple fault) variable importance values with Method 2; (f) (multiple fault) PCA-based contribution (e) (multiple fault) AAKR-based contribution plot; (multiple fault) PCA-based contribution plot; (e) (multiple fault) AAKR-based contribution plot; (f) (d) (multiple fault) PCA-based contribution plot; (e) (multiple fault) AAKR-based contribution plot; (multiple fault) variable importance values with Method 2. (multiple fault) variable importance values with Method 2. 2. (f) (multiple fault) variable importance values with Method

Figure 3 shows variable importance relatedtotothe thesimple simple and multiple faults, which Figure 3 shows variable importancevalues values related and multiple faults, which are are Figure 3 shows1variable importance values related to the simple and multiple faults, which are calculated by Method (see Figure 1a). For the simple fault, faulty samples from time t = 301 calculated by Method 1 (see Figure 1a). For the simple fault, faulty samples from time t = 301 to t =to t = calculated by Method 1 (see Figure 1a). For the simple fault, faulty samples from time t = 301 fault;; for 10001000 constitute thethe fault data multiplefault, fault,thethe matrix Xfault consists of to faulty constitute fault datamatrix matrixXXfault for the the multiple matrix Xfault consists of faulty t = 1000 constitute the fault data matrix Xfault ; for the multiple fault, the matrix Xfault consists of faulty samples from time = 301 1200.As As shown shown in importance values of faulty samples from time t =t 301 totot =t =1200. in the thefigure, figure,variable variable importance values of faulty samples from time t = 301 to t = 1200. As shown in the figure, variable importance values of faulty variables are much larger than those of normal (i.e., close to 100); the values of non-faulty variables variables are much larger than those of normal (i.e., close to 100); the values of non-faulty variables variables are much larger than those of normal (i.e., close to 100); the values of non-faulty variables are close to zero. other words,ininthis this benchmark benchmark problem, 1 can locate faulty variables are close to zero. In In other words, problem,Method Method 1 can locate faulty variables are close to zero. In other words, in this benchmark problem, Method 1 can locate faulty variables accurately in the entire fault regions. accurately in the entire fault regions. accurately in the entire fault regions. (a)

(a)

(b)

(b)

Figure 3. (Simple linear process) Variable importance values calculated by Method 1: (a) simple fault fault fault fault fault ( tstart = 301 and tend = 1000); (b) multiple fault ( tstart = 301 and tend = 1200). Figure 3. (Simple linear process) valuescalculated calculated Method 1:simple (a) simple Figure 3. (Simple linear process)Variable Variableimportance importance values byby Method 1: (a) fault fault

fault

fault

fault

fault

= 301 1000);(b) (b) multiple multiple fault andand tfault = 1200). tendtfault tstart= 301 ( tstart(tfault = 301 andand fault(t(fault = 301 = 1200). start start end ==1000); end tend

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4.2. Three Variable Nonlinear Process Energies 2018, 11, x

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Second, let us look at the results of fault isolation for the nonlinear process (also used in [36,37]) Three Variable Nonlinear with 4.2. three monitored variablesProcess and one latent variable described as follows: Second, let us look at the results of fault isolation for the nonlinear process (also used in [36,37]) + e1 described as follows: 1 = s variable with three monitored variables and onexlatent

x2 = s2 − 3s + e2

(11)

x1 = s + e 1

−2s3

+ 3s2

x3 x== s − 3 s + e + ,e3 2 2

(11)

x3 = − s 3 + 3s 2 + e 3

where x1 , . . . , x3 are the monitored variables, s ∈ [0.01, 2] is the source variable, and e1 , . . . , e3 follow ..., x3 are the monitored variables, s ∈ [0.01, 2] is the source variable, and e1, ..., e3 follow where x1, Gaussian independent distributions with mean and variance of 0 and 0.01, respectively. We obtain independent Gaussian distributions with mean and variance of 0 and 0.01, respectively. We obtain 100 normal samples from Equation (11), used to build fault detection models and to form the normal normal samples Equation (11), used to build fault detectiontwo models to form the normal data 100 matrix Xnormal . To from validate the performance of fault isolation, faultand cases, where each case is data matrix Xnormal. To validate the performance of fault isolation, two fault cases, where each case is composed of 300 test samples, are generated as follows: composed of 300 test samples, are generated as follows:

• •

Case 1: A step change of x2 by −0.4 was introduced starting from sample 101. • Case 1: A step change of x2 by −0.4 was introduced starting from sample 101. Case 2: x12:was linearly increased to270 270by byadding adding 0.01(t − 100) to xthe x1 value • Case x1 was linearly increasedfrom fromsample sample 101 101 to 0.01(t − 100) to the 1 value of each sample in this range, where t is the sample number. of each sample in this range, where t is the sample number.

Figure 4 shows fault isolationresults results of of applying applying AAKR-based contribution analysis and and Figure 4 shows thethe fault isolation AAKR-based contribution analysis Method 2 to the two fault cases. Method 2 to the two fault cases.

Figure 4. (Nonlinear process) Fault isolation results of AAKR-based contribution analysis and Method

Figure 4. (Nonlinear process) Fault isolation results of AAKR-based contribution analysis and Method 2: 2: (a) (Case 1) AAKR-based contribution plot; (b) (Case 1) variable importance values with Method 2; (a) (Case 1) AAKR-based contribution plot; (b) (Case 1) variable importance values with Method 2; (c) (Case 2) AAKR-based contribution plot; (d) (Case 2) variable importance values with Method 2. (c) (Case 2) AAKR-based contribution plot; (d) (Case 2) variable importance values with Method 2.

AAKR-based contribution analysis starts to identify the faulty variable x2 in Case 1 at time t = 101, but its contribution values fluctuate constantly, and the alarms are variable repeatedly halted AAKR-based contribution analysis starts to identify faulty x2generated in Case 1and at time t = 101, over time; contribution values of x 2 do not steadily deviate from thresholds but depart from and but its contribution values fluctuate constantly, and alarms are repeatedly generated and halted over to normal regions and again. Also, smearing over normal variable x3 is frequently time;return contribution values of xagain 2 do not steadily deviate from thresholds but depart from and return observed. These phenomena may cause confusions to root cause analysis for uncovering the to normal regions again and again. Also, smearing over normal variable x3 is frequently observed. fundamental causes and/or sources of occurring faults. In Case 2, contribution values of faulty Thesevariable phenomena may cause confusions to root cause analysis for uncovering the fundamental causes x1 consistently violate thresholds from about time t = 160 to 270, but smearing over nonand/or sources of faults. larger In Case 2, contribution faulty variable x1 consistently faulty variablesoccurring x2 and x3 becomes as the magnitude of values the faultof increases. With Method 2, the violate thresholds from t = 160 to 270, but2smearing overtonon-faulty variables x2 In and x3 importance values of about faulty time variables in Cases 1 and are very close 100 in the fault regions. becomes as the magnitude of the fault increases. With Methodwidely 2, the over importance of faulty Caselarger 1, importance values of non-faulty variables x1 and x3 fluctuate time; thisvalues is not due to smearing effect but 2 is are duevery to theclose fact that the fault magnitude is not very large. the fault variables in Cases 1 and to 100 in the fault regions. In Case 1, Although importance values of occurred only in faulty variables, to determine boundaries for dividing normal andeffect faultybut is non-faulty variables x1 and x3 fluctuate widelydecision over time; this is not due to smearing samples in that the original input space, non-faulty variables well as faulty are needed due to the fact the fault magnitude is not very large.asAlthough thevariables fault occurred onlywhen in faulty the fault magnitude is small. If the magnitude is large enough, it is possible to establish the variables, to determine decision boundaries for dividing normal and faulty samples in the original boundaries using only faulty variables. In Case 2, the importance values of normal variables are

input space, non-faulty variables as well as faulty variables are needed when the fault magnitude is small. If the magnitude is large enough, it is possible to establish the boundaries using only faulty variables. In Case 2, the importance values of normal variables are slightly large when the fault

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Energies 2018, 11, 1142 Energies 2018, 11, x

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magnitude is not large enough; as the magnitude becomes larger, the importance values of non-faulty slightly large when the fault magnitude is not large enough; as the magnitude becomes larger, the variables decrease gradually. slightly large when the fault magnitude is not large enough; as the magnitude becomes larger, the importance values of non-faulty variables decrease gradually. Now, let usvalues look of more carefully into fault isolation results using Method 1. Figure 5 depicts importance non-faulty variables decrease gradually. Now, let us look more carefully into fault isolation results using Method 1. Figure 5 depicts the Now, let us look more carefully into fault isolation results usingconstruct Method 1.CART Figureclassifiers, 5 depicts the the results of applying Method 1 into Cases 1 and 2. In Case results of applying Method 1 into Cases 1 and 2. In Case 1, 1, to to construct CART classifiers, faultyfaulty fault in Figure fault is results of time applying Method 1 constitute into Cases the 1 and 2. In Case 1, to X construct CART classifiers, faulty samples from t = 201 to 300 fault data matrix 1a; in Case faultXis samples from time t = 201 to 300 constitute the fault data matrix Xfault in Figure 1a; in Case 2, X2, fault in Figure 1a; in Case 2, Xfault is samples from time t = 201 to 300 constitute the fault data matrix X composed of faulty samples from time Ascan canbebeseen seen from Figure 5, the importance composed of faulty samples from timet t==171 171 to to 270. 270. As from Figure 5, the importance composed of faulty samples from time t = 171 to 270. As can be seen from Figure 5, the importance values of faulty variables areare nearly 100; of non-faulty non-faultyvariables variables relatively smaller values of faulty variables nearly 100;the thevalues values of areare relatively smaller than than values of faulty variables are nearly 100; the values of non-faulty variables are relatively smaller than those of faulty. Examining the results for Methods 1 and 2 synthetically may greatly help field experts those of faulty. Examining the results for Methods and 2 synthetically may greatly help field experts those of faulty. Examining the results for Methods 1 and 2 synthetically may greatly help field experts diagnose developing faults target processes. diagnose developing faults in in target processes. diagnose developing faults in target processes. (a) (a)

(b) (b)

Figure 5. (Nonlinear process) Variable importance values obtained through Method 1: (a) Case 1 (

Figure 5. (Nonlinear process) valuesobtained obtained through Method (a) 1Case 1 Figure 5. (Nonlinear process)Variable Variableimportance importance values through Method 1: (a)1:Case ( fault fault fault fault t=start tend = and 201 and = 300); (b) Case 2 fault ( tstart 171and and tfault 270). faulttend fault fault fault= = fault == 270). (tfault 201 t = 300); (b) Case 2 (t 171 start t t = 201 and = 300); (b) Case 2 start (t = 171 and end = 270). end t start

end

start

end

Figure 6 shows the constructed trees from which the importance values in Figure 5 are extracted

Figure 6 shows the the constructed trees the importance importancevalues values in Figure are extracted Figure 6 shows constructed treesfrom fromwhich which the in Figure 5 are5 extracted and the scatter plots for normal and faulty samples used to build the trees. andscatter the scatter normal and faultysamples samples used used to and the plotsplots for for normal and faulty to build buildthe thetrees. trees. (a)

(b)

① ② ②

④

③

①

③

⑤

⑥

④

⑤

⑥

(c)

(d) ①

②

⑥ ⑤

①

② ③

③

④

⑤

④

⑥

Figure 6. (Nonlinear process) Constructed tree for variable ranking in Figure 5 and relevant scatter

Figure 6. (Nonlinear process) Constructedtree tree for for variable variable ranking inin Figure 5 and relevant scatter Figure 6. (Nonlinear process) ranking Figure 5 and scatter plots for normal and faulty Constructed samples: (a) (Case 1) final tree; (b) (Case 1) scatter plot for relevant 100 normal plots for normal and faulty samples: (a) (Case 1) final tree; (b) (Case 1) scatter plot for 100 normal plots for normal and faulty samples: (a) (Case 1) final tree; (b) (Case 1) scatter plot for 100 normal fault samples and faulty samples (from tfault start = 201 to tend = 300); (c) (Case 2) final tree; (d) (Case 2) scatter fault plot for 100 normal samples and faulty samples (from tfault start = 171 to tend = 270).

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12 of 20 fault

fault

samples and faulty samples (from tstart = 201 to tend = 300); (c) (Case 2) final tree; (d) (Case 2) scatter Energies 2018, 1142 plot for11, 100 normal samples and faulty samples (from

fault fault tstart = 171 to tend = 270).

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In Figure Figure 6b,d, 6b,d, normal normal and and faulty faulty samples samples are are denoted denoted by by black black circles circles and and red red asterisks, asterisks, In respectively; faulty faulty samples samples in inFigure Figure6b,d 6b,dcorrespond correspondto tosamples samplesatattime timett==201 201to to 300 300and andtt== 171 171 to to respectively; 270 in Cases 1 and 2, respectively. Index numbers are marked at the bottom of each terminal node in 270 in Cases 1 and 2, respectively. Index numbers are marked at the bottom of each terminal node in Figure 6a,c, 6a,c, and and the the corresponding corresponding rectangle rectangle regions regions in in Figure Figure 6b,d 6b,d are are also also indexed Figure indexed by by the the numbers. numbers. In the the constructed constructed trees, trees, not not only only faulty faulty variables variables but but also also non-faulty In non-faulty variables variables have have been been partitioned; partitioned; it is is important important to to note note that that identifying identifying faulty faulty variables variables via via variable variable importance importance values values in in (8) (8) is is more more it reasonable than doing this only through checking split variables in the final trees. In Figure 6b,d, it is reasonable than doing this only through checking split variables in the final trees. In Figure 6b,d, obvious thatthat non-faulty variables as as well as as faulty the decision decision it is obvious non-faulty variables well faultyvariables variablesare arerequired requiredto to define define the boundaries for classifying normal and faulty samples. In other words, if the magnitude of mean shifts boundaries for classifying normal and faulty samples. In other words, if the magnitude of mean shifts is small, small, both both faulty faulty and and non-faulty non-faulty variables variables are are needed needed to to derive derive the the boundaries. boundaries. As As the the magnitude magnitude is of the bias and drift faults becomes larger, the need for non-faulty variables shrinks. of the bias and drift faults becomes larger, the need for non-faulty variables shrinks. 4.3. Drum-Type Drum-Type Steam Steam Boiler Boiler in in Thermal Thermal Power Power Plant Plant 4.3. In this this subsection, subsection, we we present present the the results results of of applying applying the the proposed proposed fault fault isolation isolation method method to aa In 250 MW MWdrum-type drum-typesteam steam boiler. Figure 7 [11] describes the simplified schematic diagram the 250 boiler. Figure 7 [11] describes the simplified schematic diagram of the of target target drum-type drum-type boiler. boiler.

7. Simplified schematic of of the the target target drum-type drum-type boiler boiler [11]. [11]. Figure 7.

The boiler boiler raises raises steam steam by by boiling boiling feedwater feedwater using using thermal thermal energy energy converted converted from The from fossil fossil fuel. fuel. After being preheated by extraction steam from the turbines at feedwater heaters, the feedwater is After being preheated by extraction steam from the turbines at feedwater heaters, the feedwater is supplied to the economizer. The feedwater, heated again by flue gas at the economizer, flows into the supplied to the economizer. The feedwater, heated again by flue gas at the economizer, flows into drum. The The feedwater and and saturated water in the drum areare fedfed into thetheevaporator the drum. feedwater saturated water in the drum into evaporatorthrough througha downcomer. The evaporator raises saturated steam by absorbing radiant heat of the furnace. The a downcomer. The evaporator raises saturated steam by absorbing radiant heat of the furnace. saturated water and steam are separated at the drum. The superheater converts the main steam from The saturated water and steam are separated at the drum. The superheater converts the main steam the drum into high-purity superheated steam supplied to the pressure turbine. Working in the from the drum into high-purity superheated steam supplied to high the high pressure turbine. Working in high-pressure turbine, the main steam is reheated by reheater and provided to the intermediate the high-pressure turbine, the main steam is reheated by reheater and provided to the intermediate pressureturbine. turbine.The The main steam that from exits the from the low-pressure is condensed into pressure main steam that exits low-pressure turbine isturbine condensed into condensate condensate water. After being boosted by pumps and preheated by feedwater heaters, the water is water. After being boosted by pumps and preheated by feedwater heaters, the water is fed into the fed into the boiler again. For more details regarding TPPs, refer to the books of Sarkar [38], Basu and boiler again. For more details regarding TPPs, refer to the books of Sarkar [38], Basu and Debnath [39], Debnath [39], Kitto and Stultz [40], and Singer [41]. Kitto and Stultz [40], and Singer [41]. In the target boiler, key process variables such as main steam temperature, furnace pressure, In the target boiler, key process variables such as main steam temperature, furnace pressure, drum water level and condenser make-up flow need to be closely monitored to check whether they drum water level and condenser make-up flow need to be closely monitored to check whether they deviate from specified limits. For example, main steam temperature may show abnormal variations deviate from specified limits. For example, main steam temperature may show abnormal variations due to the following several factors: malfunction of attemperators, excess air ratio, fouling formed on due to the following several factors: malfunction of attemperators, excess air ratio, fouling formed outer surfaces of superheater, and slag attached to outer surfaces of waterwall. To avoid the failures on outer surfaces of superheater, and slag attached to outer surfaces of waterwall. To avoid the failures due to extremely high metal temperature, it is indispensable to control the steam temperature precisely [6]. When the steam temperature is lower than its rated value, moisture content in the main

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steam may increase; this may cause erosion of steam turbines. Also, if there are some problems with feedwater supply, drum water level may increase or decrease. The unusual rises of drum level may result in carryover of water into superheaters or steam turbines. On the other hand, if the level is too low, the amount of feedwater supplied into waterwall tubes declines. In this case, the tubes cannot readily absorb radiation energy generated in the furnace; this may reduce the life expectancy of the tubes due to overheating, and may lead to catastrophic disasters. Table 1 lists process variables for condition monitoring of the target boiler in Figure 7. Table 1. Summary of monitored variables for target drum-type steam boiler. Variable

Description

Unit

x1 x2 x3 x4 x5 x6 x7 x8 x9

Generator output Steam flow Main steam pressure Main steam temperature Reheater pressure Furnace pressure Drum level Condenser make-up flow Feedwater flow

MW t/h kg/cm2 ◦C kg/cm2 kg/cm2 m t/h t/h

4.3.1. Artificial Fault Cases From now on, we describe the results for fault isolation of artificially generated two fault cases in the target boiler. For the simulation study, 2500 historical normal samples from the target system are prepared; each sample is recoded in 5-min intervals. Among the normal samples, 2000 samples are used to train fault detection models (i.e., PCA and AAKR) and to form the normal data matrix Xnormal in Figure 1a; the remaining 500 samples are considered as test dataset. In the test dataset, we generate artificial drift and bias faults, respectively, starting at 201th samples as follows:

• •

Case 1: x4 was linearly increased from t = 201 to the end by adding 0.1(t − 200) to the x4 value of each sample in this range, where t is the sample number. Case 2: Step changes of x6 and x7 by 10 were introduced starting from t = 201 to the end.

Figure 8 shows the trajectories of faulty variables relevant with the Cases 1 and 2; blue and red solid lines describe normal and abnormal behaviors of the faulty variables without and with the artificial fault effects, respectively. As shown in Figure 8a, if the drift fault has occurred, main steam temperature (x4 ) starts to gradually increase from 201th sample by 0.1 ◦ C. In Figure 8b,c, furnace pressure and drum level have risen steeply at time t = 201, respectively. Figure 9 summarizes the fault isolation results obtained by applying comparison methods (PCAand AAKR-based contribution plots) and Method 2 into Cases 1 and 2. In Figure 9a,d, Q and T2 statistics-based contribution plots are used for fault isolation, respectively. First, let us look into the isolation results related with Case 1. As can be seen from Figure 9a, contribution plot for Q statistic suffers from smearing effect; it is observed that severe smearing arises at non-faulty variables x1 , x2 , x5 , x6 , x7 , and x9 . In Figure 9b, contribution values of x4 start to consistently depart from in-control region at about time t = 230; although the magnitude of the drift fault becomes larger, minor fault smearing in some normal variables (i.e., x2 , x5 , x8 , x9 ) is observed in Figure 9b. In the case of Method 2, normal variables as well as faulty variable x4 have relatively high importance values in the early stage of the drift fault (from about time t = 210 to 290); in the time period from t = 300 to the end, we can confirm that Method 2 can isolate faulty variable x4 more precisely than comparison methods.

Energies 2018, 11, Energies 2018, 2018, 11, 11,x1142 1142 Energies

14 of of 20 19 14 20 14

((ooC) C)

(a) (a)

(c) (c) (m) (m)

2 (kg/cm (kg/cm2))

(b) (b)

◦ C) Figure 8. (Artificial fault fault cases) cases) Behaviors Behaviors of of faulty faulty variables: variables: (a) (a) (Case (Case 1) 1) main main steam steam temperature temperature (°C) ((°C) Figure Figure 8. 8. (Artificial 2 (x ); (c) (Case 2) drum level (m) (x ). (x ); (b) (b) (Case (Case 2) 2) furnace furnace pressure pressure (kg/cm (kg/cm 6 (c) (x (x444);); (b) (Case 2) furnace pressure (kg/cm22)))(x (x66);); (c) (Case (Case 2) 2) drum drum level level (m) (m) (x (x77).).7

(b)

(c)

(d)

(e)

(f)

Case 2

Case 1

(a)

Figure Figure 9. 9. (Artificial (Artificial fault fault cases) cases) Fault Fault isolation isolation results results of of comparison comparison methods methods and and Method Method 2: 2: (a) (a) (Case (Case Figure 9. (Artificial fault cases) Fault isolation results of comparison methods and Method 2: (a) (Case 1) 1) contribution plot; (b) (Case AAKR-based contribution plot; (c) (Case 1) 1) Q Q statistic-based statistic-based contribution plot; (b) (Case1)1) 1)AAKR-based AAKR-basedcontribution contributionplot; plot;(c) (c)(Case (Case 1) 1) variable variable Q statistic-based contribution plot; (b) (Case variable 22 statistic-based contribution plot; (e) (Case 2) AAKRimportance values with Method 2; (d) (Case 2) T importance values with Method 2; (d) (Case 2) T statistic-based contribution plot; (e) (Case 2) AAKR2 importance values with Method 2; (d) (Case 2) T statistic-based contribution plot; (e) (Case 2) based contribution plot; (f) (Case 2) variable importance values with Method 2. based contribution plot; (f) (Case 2) variable importance values with Method 2. AAKR-based contribution plot; (f) (Case 2) variable importance values with Method 2.

In Case Case 2, 2, contribution contribution plot plot for for TT22 statistic statistic of of PCA PCA isolates isolates faulty faulty variables variables xx66 and and xx77 more more clearly clearly In In Case 2, contribution plot for T2 statistic of PCA isolates faulty variables x6 and x7 more than that that of of AAKR; AAKR; but but contribution contribution values values for for xx66 deviate deviate from from threshold threshold values values insignificantly. insignificantly. In In than clearly than that of AAKR; but contribution values for x6 deviate from threshold values insignificantly. Figure 9e, 9e, itit is is observed observed that that fault fault smearing smearing in in normal normal variables variables xx55,, xx88,, and and xx99 has has happened. happened. It It is is apparent apparent Figure In Figure 9e, it is observed that fault smearing in normal variables x5 , x8 , and x9 has happened. It is from from Figure Figure 9d,e,f 9d,e,f that that Method Method 22 achieves achieves clearer clearer isolation isolation results results than than comparison comparison methods. methods. apparent from Figure 9d,e,f that Method 2 achieves clearer isolation results than comparison methods. Next, let let us us take take aa look look at at the the isolation isolation results results based based on Method Method 1, 1, which are presented in Figure Next, Next, let us take a look at the isolation results based on Method 1, which are presented in Figure 10. fault fault fault fault 10. In In Cases Cases 11 and and 2, 2, the the values values of of ttend are set set as as 500 500 and and 400, 400, respectively, respectively, and and the the value value of ttstart is 10. are of is fault end set start In Cases 1 and 2, the values of tfault are as 500 and 400, respectively, and the value of t is set as start end set as as 301 incases; both cases; cases;2000 here,normal 2000 normal normal samples (also used to train PCA and AAKR) AAKR) constitute the set both here, 2000 samples (also train PCA and constitute the 301 in301 bothin here, samples (also used toused trainto PCA and AAKR) constitute the normal normal.. As normal normal normal data matrix X As presented in Figure 10a, importance value of main steam temperature normal data matrix X presented in Figure 10a, importance value of main steam temperature data matrix X . As presented in Figure 10a, importance value of main steam temperature (x4 ) is (x44)) is is nearly nearly 100, 100, and and the the values values of of the the others others are less than 30; as a result, we can decide xx44 as as major major (x

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nearly 100, and the values of the others are less than 30; as a result, we can decide x4 as major variable variable for discriminating discriminating normal andsamples faulty samples samples of1Case Case in the original original input space. In Figure Figure for discriminating normal normal and faulty of Caseof in the original inputinput space.space. In Figure 10b, variable for and faulty 11 in the In 10b, it can be observed that the variable importance values of furnace pressure (x 6 ) and drum level it canitbe observed that the variable importance valuesvalues of furnace pressure (x6 ) and(xdrum (x7level ) are 10b, can be observed that the variable importance of furnace pressure 6) andlevel drum (x77)) are are extremely extremely higher than those of others. others. extremely higher than those ofthose others. (x higher than of (a) (a)

(b) (b)

Figure 10. 10. (Artificial (Artificial fault cases) Variable importance values obtained byby Method 1: (a) (a) Case 1; (b) (b) Figure values obtained by Method 1: 1; (Artificialfault faultcases) cases)Variable Variableimportance importance values obtained Method 1: Case (a) Case 1; Case 2. (b) Case Case 2. 2.

Figure 11 11 shows shows the the constructed constructed tree from from which which the the importance importance values values in in Figure Figure 10b 10b are are Figure Figure 11 shows the constructed treetree from which the importance values in Figure 10b are extracted fault fault fault fault extracted and and the the corresponding corresponding scatter scatter plot plot of of normal normal and and faulty faulty samples fault (from ttstart == 301 301 to ttend fault to extracted (from and the corresponding scatter plot of normal and faulty samplessamples (from tstart = 301 start to tend = 400). end = 400). 400).faulty Only variables faulty variables variables and 7 are as used as variables split variables variables in Figure Figure 11a;other the other other variables do x6 andxxx667and arexxused split in Figure 11a; 11a; the variables do not 7 are used as split in the variables do =Only Only faulty not appear in the final tree. As shown in Figure 11b, the classification tree based on nonparametric CART appear in the final tree. AsAs shown in in Figure 11b, not appear in the final tree. shown Figure 11b,the theclassification classificationtree treebased basedon on nonparametric nonparametric CART algorithm can can successfully define decision boundaries forfor separating normal andand faulty samples. algorithm cansuccessfully successfullydefine definedecision decision boundaries separating normal faulty samples. algorithm boundaries for separating normal and faulty samples. (b)

(a)

②

③

① ①

②

③

Figure 11. Constructed Constructed tree for for variable ranking ranking in Figure 10b 10b and and relevant relevant scatter plot for normal and and Figure Figure 11. 11. Constructed tree tree for variable variable ranking in Figure Figure relevant scatter plot for normal normal faulty samples: (a) final tree; (b) scatter plot for normal and faulty samples. faulty samples: (a) final tree; (b) scatter plot for normal and faulty samples.

4.3.2. Failure Data Data Due to to Waterwall Tube Tube Leakage Leakage 4.3.2. 4.3.2. Failure Failure Data Due Due to Waterwall Waterwall Tube Leakage In this subsection, subsection, we provide provide the results results of applying applying Methods 11 and and 2 to failure failure data (also (also In In this this subsection, we we provide the the results of of applying Methods Methods 1 and 22 to to failure data data (also investigated in [5,11]) due to waterwall tube leakage, which is gathered from the target boiler in investigated investigated in in [5,11]) [5,11]) due due to to waterwall waterwall tube tube leakage, leakage, which which is is gathered gathered from from the the target target boiler boiler in in Figure 7. Failure from one or more tubes in the boiler can be detected by sound and either by an Figure Figure 7. 7. Failure Failure from from one one or or more more tubes tubes in in the the boiler boiler can can be be detected detected by by sound sound and and either either by by an an increase in the make-up water requirement (indicating a failure of the water-carrying tubes) or by an an increase of of thethe water-carrying tubes) or by increase in in the themake-up make-upwater waterrequirement requirement(indicating (indicatinga failure a failure water-carrying tubes) or by increased draft in the superheater or reheater areas (due to failure of the superheater or reheater increased draft in in thethe superheater or or reheater areas an increased draft superheater reheater areas(due (duetotofailure failureofofthe the superheater superheater or or reheater reheater tubes) [38]. [38]. The The boiler boiler tubes tubes can can be be influenced influenced by by several several damage damage processes processes such such as as inside inside scaling, scaling, tubes) tubes) [38]. The boiler tubes can be influenced by several damage processes such as inside scaling, waterside corrosion and and cracking, cracking, fireside fireside corrosion corrosion and/or and/or erosion, stress stress rupture due due to overheat overheat waterside waterside corrosion corrosion and cracking, fireside corrosion and/or erosion, erosion, stress rupture rupture due to to overheat and creep, vibration-induced and thermal fatigue cracking, and defective welds [42]. Tube leakage and vibration-induced and leakage and creep, creep, vibration-induced and thermal thermal fatigue fatigue cracking, cracking, and and defective defective welds welds [42]. [42]. Tube Tube leakage from a pin-hole might might be tolerated tolerated because of of an an adequate adequate margin margin of of feedwater and and the the leakage leakage can can from from aa pin-hole pin-hole mightbe be toleratedbecause because of an adequate margin feedwater of feedwater and the leakage be corrected after suitable scheduled maintenance [43]. However, if the boiler is continuously be corrected after suitable scheduled maintenance [43]. However, if the boiler is continuously operated with with the the leakage, leakage, much much of of the the pressurized pressurized fluid fluid will will eventually eventually leak leak and and cause cause severe severe operated

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16 of 20 suitable scheduled maintenance [43]. However, if the boiler is continuously operated with the leakage, much of the pressurized fluid will eventually leak and cause severe damage damage to to neighboring neighboring tubes. tubes. Tube Tube leakage leakage of of boiler, boiler, superheater superheater and and reheater reheater could could result result in in aa damage to neighboring tubes. Tube leakage of boiler, superheater and reheater could result in a serious serious efficiency decline. serious efficiency decline. efficiency decline. Target failure failure dataset dataset consists consists of of 4273 4273 training training and and 1054 1054 test test samples, samples, respectively; respectively; sampling sampling Target Target failure dataset consists of 4273 training and 1054 test samples, respectively; sampling interval isis equal equal to to 55 min, min, and and monitored monitored variables variables are are same same as as those those listed listed in in Table Table 1. 1. The The matrix matrix interval interval is equal to 5 min, and monitored variables are same as those listed in Table 1. The matrix training is composed of the training samples and they should be collected from the target boiler under training X X trainingis composed of the training samples and they should be collected from the target boiler under X is composed of the training samples and they should be collected from the target boiler under normal operating operating conditions. conditions. The The tube tube leakage leakage occurred occurred at at 911th 911th test test sample, sample, and and after after about about 12 12 normal normal operating conditions. The tube leakage occurred at 911th test sample, and after about 12 hours, hours,unplanned unplannedshutdown shutdownprocedures procedureswere wereinitiated. initiated.Figure Figure12 12shows showsthe thefault faultisolation isolationresults resultsfor for hours, unplanned shutdown procedures were initiated. Figure 12 shows the fault isolation results for the fault fault fault fault the failure failure dataset dataset based on on Methods Methods 11 and and 2; here, here, ttstart and ttend are set set as 951 951 and and 1050, 1050, the are fault and start end failure dataset basedbased on Methods 1 and 2; here,2;tfault and t are set as 951 and as 1050, respectively. start end respectively. As canFigure beseen seen from Figure12a, 12a, importance values ofcondenser condenser make-up flow(x (x88))flow and respectively. As can be from Figure importance values of make-up flow and As can be seen from 12a, importance values of condenser make-up flow (x 8 ) and feedwater feedwater flow (x 9 ) are larger than 80 in the time interval (from t = 951 to t = 1050); those of others in feedwater flowthan (x9) 80 areinlarger thaninterval 80 in the timet interval t = 951 to tof= others 1050); in those othersare in (x9 ) are larger the time (from = 951 to (from t = 1050); those the of interval the interval interval are smaller than 40. In Figure Figure 12b, such normal variables such as xx22,,high and exhibit high the than In 12b, normal as xx44,, and xx55 exhibit high smaller thanare 40. smaller In Figure 12b,40. normal variables as xvariables x5 exhibit importance values 2 , x4 , andsuch importance values temporarily at the beginning of the leakage; as the strength of leakage becomes importance temporarily at the beginning the leakage; as the strength of leakage temporarily values at the beginning of the leakage; as theofstrength of leakage becomes larger, faulty becomes variables faulty variables x 8and and x 9are are precisely isolated. Figure 13 describes the trajectories of the faulty larger, faulty variables x 8 x 9 precisely isolated. Figure 13 describes the trajectories of the faulty xlarger, and x are precisely isolated. Figure 13 describes the trajectories of the faulty variables (x x9 ) 8 9 8 and variables (x 8 and x 9 ) in the test period; it can be observed that condenser make-up flow and feedwater variables (x 8 and x 9 ) in the test period; it can be observed that condenser make-up flow and feedwater in the test period; it can be observed that condenser make-up flow and feedwater flow start to rise flowstart start to rise sharply atthe thetime time tt==911. 911. flow rise sharply at sharply atto the time t = 911.

(a) (a)

(b) (b)

(b) (b)

(a) (a) (t/h) (t/h)

(t/h) (t/h)

Figure12. 12.(Failure (Failuredata datadue dueto towaterwall waterwalltube tubeleakage) leakage)Results Resultsof offaulty faulty variable variableisolation isolation using usingthe the Figure Figure 12. (Failure data due to waterwall tube leakage) Results of faulty variable isolation using the proposed method: (a) Method 1; (b) Method 2. proposed proposed method: method: (a) (a) Method Method 1; 1; (b) (b) Method Method 2.

Figure 13. (Failure (Failure data due due to waterwall waterwall tube leakage) leakage) Trajectories of of faulty variables variables related to to Figure Figure 13. 13. (Failure data data due to to waterwall tube tube leakage) Trajectories Trajectories of faulty faulty variables related related to waterwall tube leakage: (a) condenser make-up flow (t/h) (x 8 ); (b) feedwater flow (t/h) (x 9 ). waterwall 9). waterwall tube tube leakage: leakage: (a) (a) condenser condenser make-up make-up flow flow (t/h) (t/h)(x(x8););(b) (b)feedwater feedwaterflow flow(t/h) (t/h)(x(x ). 8

9

4.4. Discussion 4.4. 4.4. Discussion Discussion In this subsection, subsection, we we summarize summarize the the strengths strengths of the the proposed proposed fault fault isolation isolation method, method, which which In In this this subsection, we summarize the strengths ofofthe proposed fault isolation method, which are are confirmed confirmed from from the the experimental experimental results results in in Sections Sections 4.1–4.3. 4.1–4.3. The The main main strength strength of of the the proposed proposed are confirmed from the experimental results in Sections 4.1–4.3. The main strength of the proposed method method isis that that itit does does not not suffer suffer from from smearing smearing effect. effect. As As shown shown in in Figures Figures 2, 2, 4, 4, and and 9, 9, PCAPCA- and and method is that it does not suffer from smearing effect. As shown in Figure 2, Figure 4, and Figure 9, PCA- and AAKR-based contribution plots are troubled with smearing effect; in contrast, the proposed isolation AAKR-based AAKR-based contribution contribution plots plots are are troubled troubled with with smearing smearing effect; effect; in in contrast, contrast, the the proposed proposed isolation isolation method(i.e., (i.e.,Method Method2) 2)can canclearly clearlylocate locatefaults faultswithout withoutthe thesmearing smearingas asthe thefault faultmagnitude magnitudebecomes becomes method larger. In PCA, the contribution values are obtained through projecting query samples in the original larger. In PCA, the contribution values are obtained through projecting query samples in the original space onto a reduced latent space; latent variables are linear combinations of all original variables. space onto a reduced latent space; latent variables are linear combinations of all original variables.

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method (i.e., Method 2) can clearly locate faults without the smearing as the fault magnitude becomes larger. In PCA, the contribution values are obtained through projecting query samples in the original space onto a reduced latent space; latent variables are linear combinations of all original variables. Therefore, contribution values of non-faulty variables are also influenced by faulty variables. In AAKR, each component of predicted vectors is a function of both non-faulty and faulty variables; contribution analysis based on the predicted vectors also suffers from fault smearing. The proposed method locates faults by selecting original input variables with higher explanatory powers to discriminate normal and faulty samples. In other words, the proposed method does not suffer from smearing effect because it is based on CART classifiers with decision boundaries only in the original input space. The second strength is that, as described in Figures 4 and 5, the proposed method can properly identify faulty variables of nonlinear process. PCA with the assumption of process linearity is unsuitable for fault detection and isolation of nonlinear processes. As shown in Figure 4, although fault isolation with nonparametric AAKR is troubled with fault smearing, the proposed method can accurately identify faulty variables of the nonlinear process. Except for the strengths mentioned above, the proposed method does not require historical faulty dataset and any prior knowledge and experience. 5. Conclusions In complex and nonlinear industrial processes, data-driven fault detection and isolation for process monitoring has become more important than ever before; compared with fault detection, fault isolation methods have not been studied very much. In this paper, we proposed a fault isolation method for a drum-type steam boiler via CART-based variable ranking. Method 1 is intended to identify faulty variables in entire fault regions, and it is proper for post analysis of occurring faults offline. In Method 2, fault isolation is carried out through time window sliding; Method 2 is suitable for monitoring fault evolution and propagation. To validate the fault isolation performance, the proposed method and comparison methods were applied to two benchmark problems and 250 MW drum-type steam boiler. The experimental results showed that the proposed method can locate faults more clearly than comparison methods without being affected by smearing effect. The proposed method, based on nonparametric CART, can properly deal with nonlinear processes, and can correctly identify faulty variables, even when normal and faulty samples are linearly inseparable from each other. In future research, we will consider the following three topics. The first topic is to verify the performance of proposed method for time-varying processes. For example, since the process equipment is gradually degraded over time, it usually shows time-varying behaviors. If components of the matrix Xnormal (see Figure 1) are updated with the latest normal samples at every moment, the proposed method may take the time-varying characteristics of target systems into consideration for fault isolation. Second, we will also perform comparative studies between the proposed method and recently developed methods in [27,28,44,45]; this will further ensure the validity of the proposed method. Lastly, to improve the isolation performance, we will develop variable ranking methods with consideration for structural properties (described in [46]) of binary decision trees. Author Contributions: Jaeyel Jang and Jaeyeong Yoo conceived and designed the simulations. Jungwon Yu analyzed the data and wrote the paper. The analysis results and the paper were supervised by June Ho Park and Sungshin Kim. Acknowledgments: This work was supported by the Energy Efficiency & Resources Core Technology Program of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) granted financial resource from the Ministry of Trade, Industry & Energy, Republic of Korea (No. 20151110200040). Conflicts of Interest: The authors declare no conflict of interest.

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