Statistical root cause analysis of novel faults based on digraph models

chemical engineering research and design 9 1 ( 2 0 1 3 ) 87–99

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Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd

Statistical root cause analysis of novel faults based on digraph models Yiming Wan a , Fan Yang a , Ning Lv a , Haipeng Xu a , Hao Ye a,∗ , Weichang Li b , Peng Xu b , Liming Song b , Adam K. Usadi b a

Tsinghua National Laboratory for Information Science and Technology (TNList), Department of Automation, Tsinghua University, Beijing 100084, China b Corporate Strategic Research, ExxonMobil Research and Engineering Co., 1545 Route 22 East, Annandale, NJ 08801, USA

a b s t r a c t This paper investigates the challenging problem of diagnosing novel faults whose fault mechanisms and relevant historical data are not available. Most existing fault diagnosis systems are incapable to explain root causes for unanticipated, novel faults, because they rely on either models or historical data of known faulty conditions. To address this issue we propose a new framework for novel fault diagnosis, which integrates causal reasoning on signed digraph models with multivariate statistical process monitoring. The prerequisites for our approach include historical data of normal process behavior and qualitative cause–effect relationships that can be derived from process flow diagrams. In this new approach, a set of candidate root nodes is identified first via qualitative reasoning on signed digraph; then quantitative local consistency tests are implemented for each candidate based on multivariate statistical process monitoring techniques; finally, using the resulting multiple local residuals, diagnosis is performed based on the exoneration principle. The cause–effect relationships in the digraph enable automatic variable selection and the local residual interpretations for statistical monitoring. The effectiveness of this new approach is demonstrated using numerical examples based on the Tennessee Eastman process data. © 2012 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: Fault diagnosis; Novel fault; Digraph model; Causal reasoning; Multivariate statistical process monitoring

1.

Introduction

Industrial process anomalies and faults may degrade product quality and can lead to significant safety and environmental risks. The two primary steps in effective abnormal situation management are fault detection and root cause diagnosis, both of which have received wide attention from academia and industry (Venkatasubramanian et al., 2003a,b,c). In practice, fault diagnosis systems often encounter unanticipated and unknown events. For novel faults, there is often little prior knowledge about the associated fault mechanism. Nor is there typically historical data from which one may extract such information. Many existing fault diagnosis methods, such as the quantitative model-based methods (Gertler, 1998; Venkatasubramanian et al., 2003a), rule-based expert systems (Venkatasubramanian et al., 2003b), fault trees (Venkatasubramanian et al., 2003b), and pattern classification

∗

approaches (Detroja et al., 2006; Venkatasubramanian et al., 2003c), rely either on known cause–effect relationships or historical data containing abnormal situations. Hence, they are not applicable to these novel faults. Although contribution plots in multivariate statistical process monitoring (MSPM) techniques can potentially identify the contributing variables for a novel fault (Qin, 2003), they do not explain the root cause for the identified contributing variables (Venkatasubramanian et al., 2003c). Hybrid diagnosis approaches (Leung and Romagnoli, 2002; Musulin et al., 2006; Ündey et al., 2003; Zumoffen and Basualdo, 2008), which integrate the techniques mentioned above suffer from similar problems when faced with novel faults. Among existing fault diagnosis methods, causal digraphbased reasoning (CDR) uses only cause–effect knowledge about normal situations and is suitable for diagnosing novel faults (Venkatasubramanian et al., 2003b). CDR relies

Corresponding author. Tel.: +86 10 62790497; fax: +86 10 62785911. E-mail address: [email protected] (H. Ye). Received 25 March 2012; Received in revised form 7 June 2012; Accepted 12 June 2012 0263-8762/$ – see front matter © 2012 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cherd.2012.06.010

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on a digraph model in which nodes and arcs capture cause–effect relationships between different process variables during normal situations. CDR consists of backward and forward reasoning steps (Venkatasubramanian et al., 2003b). Candidate root nodes responsible for the deviations detected are identified from backward reasoning. Spurious root nodes are then eliminated through subsequent forward reasoning. A purely qualitative CDR approach, based on signed digraph models, has been described in the literature (Venkatasubramanian et al., 2003b). Several hybrid approaches have used quantitative information from CDR to improve diagnosis performance. In (Lee et al., 2004; Lv and Wang, 2008; Vedam and Venkatasubramanian, 1999), quantitative information from MSPM is used to trigger CDR, but the inference mechanism remains qualitative. The dynamic causal digraph method in (Cheng et al., 2008; Fagarasan et al., 2004; Gentil et al., 2004; Montmain and Gentil, 2000; Vachhani et al., 2007) incorporates quantitative models in the inference mechanism; however, the quantitative models employed are often unavailable. In practice it is realistic to assume that qualitative cause–effect knowledge and historical data in normal conditions are available. Therefore fault diagnosis performance can be improved by incorporating quantitative information from normal historical data in the inference mechanism of CDR. In this paper, a hybrid approach is developed by integrating MSPM and CDR to diagnose novel faults. First, traditional backward reasoning is used to hypothesize all possible root nodes responsible for the detected deviations. Then, in a forward reasoning step, a local statistical model is built for each identified root node. Subsequently, MSPM techniques are used to generate local residuals. Finally, diagnosis is conducted based on the set of resulting local residuals according to the exoneration principle. The main feature of this work is automatic variable selection and local residual interpretations in MSPM, which is enabled by combining the digraph model. The rest of the paper is organized as follows. In Section 2, CDR and MSPM are briefly reviewed. In Section 3, the proposed hybrid diagnosis strategy is detailed. Simulation examples in the Tennessee Eastman process are shown in Section 4, followed by conclusions in Section 5.

2.

Preliminaries

This section describes CDR and MSPM as the basis of the hybrid approach.

2.1.

Causal reasoning on a signed digraph

The purpose of diagnostic reasoning is to identify root causes for the observed malfunctions through analyzing cause–effect relationships in system behavior. Following Venkatasubramanian et al. (2003b), we represent the cause–effect relationships by a digraph G = (V, M) where V = {v1 , v2 , . . . , vn } is the set of nodes, and M ⊆ V × V is the set of directed arcs. In the digraph, node vi represents a measured process variable or a controller output, and directed arc arcij = (vi , vj ) symbolizes the relationship from the cause node vi to the effect node vj . The simplest digraph is the signed digraph (SDG) (Venkatasubramanian et al., 2003b), in which each directed arc has a positive or negative sign. The state of a node can be normal (0), higher than normal (+), or lower than normal (−). The positive arc sign “+” means that

Fig. 1 – Flowchart of the qualitative SDG reasoning.

the nodes of the arc evolve in the same direction, while the negative arc sign “−” means that the nodes of the arc evolve in the opposite direction. Without assuming first-principle models, in this paper we derive the qualitative SDG model from operators’ knowledge about the process flow diagram (Maurya et al., 2004; Palmer and Chung, 2000; Thambirajah et al., 2009). The SDG reasoning follows backward/forward reasoning approach to identify the minimum set of root nodes to explain the detected abnormalities (Gentil et al., 2004), as shown in Fig. 1. Univariate monitoring is used online to determine the state of each node as normal (0), higher than normal (+), or lower than normal (−). An arc is defined as consistent if the product of the signs of cause node and effect node is the same as the sign of the arc (Vedam and Venkatasubramanian, 1997). If any abnormality is detected, the backward reasoning is done recursively starting from each deviated node to the cause node via consistent arcs till the root nodes are identified. A root node has at least one consistent arc connecting it to an effect node and no consistent arc connecting it to a cause node (Vedam and Venkatasubramanian, 1997). In the forward reasoning, the hypothesized root nodes are further tested to check whether there exists a valid causal path, which consists of consistent arcs, from each root node to the detected abnormal nodes (Vedam and Venkatasubramanian, 1997). Only the validated root nodes are kept to explain the detected abnormalities. The employed backward/forward reasoning relies on consistency tests with normal behavior model, and requires no knowledge or data about faults. Hence the SDG reasoning can handle novel faults (Venkatasubramanian et al., 2003b). Since the SDG reasoning is qualitative in nature, several hybrid approaches were proposed to improve diagnosis performance by incorporating quantitative information (Cheng et al., 2008; Fagarasan et al., 2004; Gentil et al., 2004; Lee et al., 2004; Lv and Wang, 2008; Montmain and Gentil, 2000; Vachhani et al., 2007; Vedam and Venkatasubramanian, 1999). But the incorporated quantitative information is either insufficiently used in the inference mechanism (Lee et al., 2004; Lv and Wang, 2008; Vedam and Venkatasubramanian, 1999), or unavailable in most practical situations (Cheng et al., 2008; Fagarasan et al., 2004; Gentil et al., 2004; Montmain and Gentil, 2000; Vachhani

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which is only affected by process noises and measurement noises in the normal situation because, according to (2), (3) and (5), there is

et al., 2007). Considering the limited a priori knowledge about novel faults, this paper focuses on utilizing normal historical data in the inference mechanism.

 T



Hd,s ds,k + os,k

2.2. Consistent dynamic PCA based residual generation

rk = P˜

MSPM is based on normal historical data to build process models, which are used to generate residuals for fault detection and diagnosis. MSPM methods include principal component analysis (PCA), partial least squares analysis, canonical variate analysis, and their variants (Russell et al., 2000). In this paper, dynamic PCA (DPCA) (Gertler and Cao, 2004; Ku et al., 1995; Li and Qin, 2001) is employed, but similar principles also apply to other MSPM methods. Assume that the monitored process is represented by

To measure the deviation from the normal process behaviors captured by the DPCA model, the squared prediction error (SPE) index is defined as (Qin, 2003)

xk+1 = Axk + Bu (uk − wk ) + Bd dk

where ı2 denotes the upper control limit for SPE (Ku et al., 1995; Qin, 2003). To identify the linear relation satisfying (3), we adopt the consistent DPCA method in (Li and Qin, 2001) which perT forms singular value decomposition on (1/(N − 1))Zk Zk−s−1 . Then the estimated linear relation P˜ is the left singular eigenvectors corresponding to the smallest (ms − n) singuT , and the number of principal lar values of (1/(N − 1))Zk Zk−s−1 components is ls +n. To determine the number of lagged variables s and the order of system states n, Akaike Information Criterion can be used by following Wang and ˜ the priQin (2002). With the estimated linear relation P, mary residual r1,k in this paper is generated according to (5):

T

SPE ≡ ||rk ||2 = ||(I − P˜ P˜ )zs,k ||2 .

SPE ≤ ı2

where uk ∈ Rl , yk ∈ Rm , xk ∈ Rn are measured inputs, measured outputs, and state variables respectively; wk , ok , and dk ∈ Rp are input noises, output noises, and process disturbances respectively. By manipulating (1), we have (Li and Qin, 2001)

 zs,k = z∗s,k +

Hd,s ds,k + os,k

(7)

The monitored process is considered to be normal if

(1)

yk = Cxk + Du (uk − wk ) + ok

(6)

.

ws,k

 (2)

ws,k

where

T

r1,k = P˜ zs,k .

zs,k = [ yTs,k

T

(8)

T

uTs,k ] , z∗s,k = [(s xk−s + Hu,s u∗s,k ) u∗T s,k ] , T

u∗k = uk − wk ,

⎡

yk−s

⎤

⎡

⎢ ⎥ ⎢ yk−s+1 ⎥ ⎢ ⎢ ⎥ ms ⎢ ys,k = ⎢ ⎥ ∈ R , s = ⎢ ⎢ .. ⎥ ⎣ ⎣ . ⎦

⎡

⎢ ⎥ ⎢ ⎥ ms ×n ⎢ ∈ R , H = ⎥ d,s .. ⎢ ⎣ . ⎦

CA

CA

yk

⎤

C

s

0 CBd

0

.. .

.. .

s−1

s−2

CA

and vectors u∗s,k ∈ Rls , us,k ∈ Rls , ws,k ∈ Rls , ds,k ∈ Rps , and os,k ∈ Rms are defined similarly to ys,k ; and matrix Hu,s are defined similarly to Hd,s . The task of DPCA modeling is to identify for the noise-free process variables z∗s,k the following linear relation T P˜ z∗s,k

=0

0

Bd

CA

..

.

···

Bd

⎤ ⎥ ⎥ m ×l ⎥ ∈ R s s, ⎥ ⎦

0

Suppose that there is a fault on the ith sensor, we have

 r1,k =

T

P˜ zs,k = P˜



(3)

= P˜

T

Hd,s ds,k + os,k

T

ws,k

Hd,s ds,k + os,k ws,k



 T

+ P˜

⎡



⎢ ⎣

zs,k−N+2

···

zs,k ]

+ F˜ s,i ⎢

.. .

⎤ (9)

⎥ ⎥ ⎦

f,i

yk (4)

f,i

where yk is the fault signal on the ith sensor at time k, and f,i

T Given (2) and the identified linear relation P˜ in (3), the residual is generated as (Qin, 2003)

T rk = P˜ zs,k



0 f,i yk−s+1

from the available noisy measurements Zk = [ zs,k−N+1

f,i

ys,k

ys,k = [ 0 · · ·

F˜ s,i

(5)

0

f,i

yk−s+1

···

0

f,i

yk−s+j

···

0

f,i

yk

0

···

T

0] .

It can be seen from (9) that the fault direction matrix T consists of the columns of P˜ corresponding to

f,i

f,i

f,i

yk−s+1 , yk−s+2 , . . . , yk



. Similarly, we can derive the fault

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T direction matrix for any actuator faults from P˜ . But the fault direction matrix for a novel process fault is unknown.

Remark 1. Special attention should be paid to sensor or actuator faults in a control loop. Due to fault propagation through feedback control, the fault direction matrix of a sensor fault or T an actuator fault cannot be obtained from P˜ if two conditions are satisfied: (1) the sensor or actuator is included in a control loop and (2) the control law related to the sensor or actuator is captured by the linear relation in the DPCA model (Gertler and Cao, 2004). ˜ i is available, the If the associated fault direction matrix  effect of fault fi,k can be removed from r1,k in (8) by using the reconstruction-based method in (Alcala and Qin, 2009). Define the reconstructed residual r2,k as (Alcala and Qin, 2009) ˜ i fi,k . r2,k = r1,k − 

(10)

What remains is then to find an estimate ˆfi,k such that rT2,k r2,k is minimized, which yields (Alcala and Qin, 2009; Qin, 2003) † T ˆfi,k = ( ˜ T ˜ i)  ˜ r1,k . i i

(11)

˜ T ˜ i is where “†” is the Moore-Penrose inverse for the case that  i singular. By substituting (16) back into (10), the reconstructed residual r2,k becomes †

˜ i ˆfi,k = (I −  ˜ i ( ˜ T ˜ i)  ˜ T )r1,k . r2,k = r1,k −  i i

(12)

Although the presence of other faults different from fi,k may have contributions to the estimated ˆfi,k , r2,k in (12) removes from r1,k the fault effect associated with fault direction matrix ˜ i. 

3.

Hybrid diagnosis strategy

Considering the lack of fault mechanism knowledge and historical data for novel faults, the proposed hybrid approach integrates SDG reasoning and MSPM, by replacing the qualitative forward reasoning in Fig. 1 with local MSPM-based residual generation and the diagnosis logic on multiple local residuals, as shown in Fig. 2. The proposed hybrid approach relies on relating graph elements to process components, interpreting local residuals in the digraph, and diagnosis logic on multiple local residuals, which will be discussed in turn before explaining the steps of the hybrid diagnosis strategy.

3.1. Description of graph elements and interpretation of root nodes

Fig. 2 – Flowchart of the proposed hybrid approach.

With the concept of “support”, we can translate the abstract root node in an SDG into practical root causes, i.e., possible faulty process components in the corresponding monitored process. In CDR, since a root node is one node that “has at least one consistent arc connecting it to an effect node and no consistent arc connecting it to a cause node” (Vedam and Venkatasubramanian, 1997), explanations of a root node for possible root causes rely on explanations of inconsistent incident arcs on the root node, which are listed in Table 2. Therefore, the subset of possible root causes corresponds to a root node rvi can be written as Rc(rvi ) =

∪

vj ∈ Par(rvi )

where Par(vi ) represents the set of parent nodes of vi . Thus the set of possible root causes for all root nodes is RC =

∪

rvi ∈ RN

Rc(rvi ),

(14)

with RN denoting the set of root nodes.

Table 1 – Description of support of graph elements. Graph element

The introduced qualitative SDG reasoning in Section 2.1 simply locates possible root causes on the root nodes. To facilitate automated interpretation of root causes in terms of possible faulty process components, each node and arc is labeled by its related process components in the digraph model. There are two kinds of nodes in a digraph model:one is Vy representing the set of measured process variables, the other is Vu representing the set of controller outputs. The concept of “support” is introduced to relate graph elements to process components (Fagarasan et al., 2004; Gentil et al., 2004), as shown in Table 1.

{Supp(vj ), Supp(arc(vj , rvi ))} ∪ Supp(rvi ), (13)

Arc arcij = (vi ,vj )

Node vi ∈ Vy

Node vi ∈ Vu

Support Supp(arcij ): a set of process components related to the cause–effect relationship represented by arcij Supp(vi ): the sensor transforming the physical variable into the measured variable represented by vi , and the unknown upstream disturbances directly affecting this physical variable Supp(vi ): the controller output that is related to vi ; all the controllers are assumed to be fault-free in this paper.

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Table 2 – Explanation of an inconsistent arc arcji . Suspected components Supp(vi )

Supp(vj ) Supp(arcji )

a b c

Scenarios

Possible root causes

vi ∈ Vy vi ∈ Vu vj ∈ Vy vj ∈ Vu vi ∈ Vy , vj ∈ Vy vi ∈ Vy , vj ∈ Vu vi ∈ Vu , vj ∈ Vy

Sensor fault or unknown external disturbances on the process variable represented by vi Fault-free a Sensor fault on the process variable represented by vj Fault-freeb Process fault on Supp(arcji ) Process fault or actuator fault on Supp(arcji ) Fault-freec

Controllers are assumed to be fault-free in this paper. Controllers are assumed to be fault-free in this paper. Controllers are assumed to be fault-free in this paper.

3.2.

Interpreting local residuals in the digraph

As explained in Section 3.1, from the qualitative SDG reasoning, we obtain the set of root nodes RN and the set of corresponding root causes RC in (14). For each root node rvi , the set of selected process variables, represented by V(rvi ), is used in the local consistent DPCA modeling. The set V(rvi ) consists of the process variables corresponding to rvi and its parent nodes Par(rvi ), i.e., V(rvi ) = rvi ∪ Par(rvi ).

(15)

And the local primary residual defined in (8) for root node rvi is denoted as res(rvi ). Each local residual res(rvi ) is sensitive to a subset of the hypothesized root causes represented by Lrc(res(rvi )). Thus there is Lrc(res(rvi )) ⊆ RC, and Lrc(res(rvi )) will be referred to as the set of locally hypothesized root causes in the following. Each hypothesized root cause in the set RC is examined to see whether it affects the local residual res(rvi ) according to cause–effect relationships in the digraph model; and if so, it should be included in the set Lrc(res(rvi )). The rules to determine Lrc(res(rvi )) for each local residual res(rvi ) are given in this subsection. A simple digraph example is given in Fig. 3 and will be frequently used to explain related concepts later. Some definitions are first introduced: • A causal path, path(vj ,vi ), is defined as a sequence of nodes starting with vj and ending with vi such that from each of its nodes there is an arc to the next node in the sequence. For example, in Fig. 3, v1 v2 v4 and v1 v3 v4 are two causal paths, but v2 v3 v4 is not a causal path. • A node vj is called an output node in the local model of the / ∅. An output node set V(rvi ) if vj ∈ V(rvi ) and Par(vj ) ∩ V(rvi ) = node corresponds to an output process variable in (1).

Fig. 3 – A digraph example (the solid arcs denote positive effects, and the dashed arcs denote negative effects).

• A node vj is called an input node in the local model of the node set V(rvi ) if vj ∈ V(rvi ) and Par(vj ) ∩ V(rvi ) = ∅. An input node corresponds to an input process variable in (1). For example, nodes v2 , v3 and v4 are selected for root node v4 , then nodes v2 and v4 are output nodes, and v3 is an input node. • A node vj is called a disturbance node for the local / V(rvi ), and there exist model of the node set V(rvi ) if vj ∈ a causal path path(vj ,vm ) from vj to a node vm ∈ V(rvi ), and path(vj ,vm ) does not include any input node in the node set V(rvi ). A disturbance node affects the local DPCA model related to V(rvi ) like external process disturbance in (1).

As mentioned earlier, determining whether a hypothesized root cause should be included in Lrc(res(rvi )) is equivalent to determining whether it affects res(rvi ), and the rules are as follows: (1) Supp(arckj ) ⊆ RC should be included in Lrc(res(rvi )) in either one of the following two cases: (a) vj ∈ V(rvi ). In this case, Supp(arckj ) corresponds to local process model captured in (A, Bu , C, Du ) in (1) if vk ∈ V(rvi ); otherwise, abnormalities in Supp(arckj ) affects the local primary residual res(rvi ) through external disturbances of the local process model, as shown in (6). (b) vj ∈ / V(rvi ), and vj is a disturbance node for the node set V(rvi ), i.e., abnormalities in Supp(arckj ) affect the local primary residual res(rvi ) through external disturbances, as shown in (6). (2) Supp(vk ) ⊆ RC hypothesizes a sensor fault or unknown external disturbances on the process variable vk , as stated in Table 2. The sensor fault hypothesized in Supp(vk ) should be included in Lrc(res(rvi )) in either one of the following two cases: (a) vk ∈ V(rvi ), and vk is an output node; (b) vk ∈ / V(rvi ), vk is used in a control loop by a controller output node vn , and vn ∈ V(rvi ), or vn is a disturbance node for the node set V(rvi ). In this case, the sensor fault propagates through a control loop and affects the local residual as external disturbances. (3) Unknown external disturbances hypothesized in Supp(vk ) ∈ RC should be included in Lrc(res(rvi )) in either one of the following two cases: (a) vk ∈ V(rvi ), and vk is an output node; (b) vk ∈ / V(rvi ), and vk is a disturbance node for the node set V(rvi ).

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The reconstructed residual in (12) can be generated only when the sensor or actuator fault direction matrix is available. Let cf represent the faulty component with known fault direction matrix, we denote the reconstructed residual by res(rvi ,cf ), and we have Lrc(res(rvi , cf )) = Lrc(res(rvi )) − {cf }

(16)

which means that cf is removed from Lrc(res(rvi )) because the reconstructed residual is not sensitive to cf . Example 1. Suppose arc(v1 ,v2 ), arc(v2 ,v4 ), and arc(v2 ,v4 ) are inconsistent arcs in Fig. 3. In addition, v2 and v4 are assumed to form a control loop in which v2 is a controller output node and v4 is a sensor node, and all the other nodes are sensor nodes. According to Table 2 the set of hypothesized root causes is RC = {Supp(arc(v1 , v2 ), Supp(arc(v2 , v4 ), Supp(arc(v3 , v4 ), Supp(v1 ), Supp(v3 ), Supp(v4 )}. For the root node v4 , nodes v2 , v3 , and v4 are selected for the local DPCA modeling. Then we have Lrc(res(v4 )) = {Supp(arc(v1 , v2 )), Supp(arc(v2 , v4 )), Supp(arc(v3 , v4 )), Supp(v1 ), Supp(v3 ), Supp(v4 )}. Note that the control law in Supp(arc(v4 ,v2 )) is captured by the linear relation in the local DPCA model, the fault direction matrix of sensor fault in v4 or actuator fault in Supp(arc(v2 ,v4 )) cannot be obtained from the local DPCA model according to Remark 1, and only the fault direction matrix of sensor fault in v3 can be obtained. Therefore we have Lrc(res(v4 , sensor fault in Supp(v3 ))) = Lrc(res(v4 )) − {sensor fault in Supp(v3 )} = {Supp(arc(v1 , v2 )), Supp(arc(v2 , v4 )), Supp(arc(v3 , v4 )), Supp(v1 ), Supp(v2 ), external disturbances in Supp(v3 ), Supp(v4 )}.

3.3.

Diagnosis logic on multiple local residuals

Let resi denote a local primary residual res(rvi ) or a local reconstructed residual res(rvi ,cf ), and let Lrc(resi ) represent the set of locally hypothesized root causes corresponding to the local residual resi . Each local residual resi is evaluated with the chosen threshold, where eval(resi ) = 0 means that the residual is below the threshold, and eval(resi ) = 1 means that the residual is above the threshold. It will be shown in the following that {Lrc(resi ),eval(resi )} enables deep root cause analysis. Exoneration principle (Cordier et al., 2004; Fagarasan et al., 2004): Suppose the monitored process consists of a set of components COMPS, and  is the set of root causes (faulty components),  ⊆ COMPS, then

unavoidable that we may have eval(resi ) = 0 in the presence of faults, the exoneration principle may lead to miss detections. Although the exoneration principle has its limitation, it is a fundamental concept widely used in diagnosis (Cordier et al., 2004; Fagarasan et al., 2004), and will be adopted in this paper. Various approaches can be proposed to reduce miss detections when using the exoneration principle, but this issue is not the major concern in the scope of this paper. Following the above exoneration principle, with the evaluation of multiple local residuals {eval(resi )}, the refined set of root causes is a set of components RC ⊆ COMPS given by RC =

3.4.

∪

Lrc(resi ) −

eval(resi )=1

∪

Lrc(resj )

eval(resj )=0

(18)

Steps in hybrid diagnosis strategy

Based on the previous discussions, the hybrid diagnosis strategy is proposed to follow three steps: Step 1. SDG backward reasoning. After detecting the nodes deviated from normal range, SDG backward reasoning is performed as explained in Section 2.1. According to the set of identified root nodes RN, the set of hypothesized root causes RC can be obtained by following (13), (14), and Table 2. Step 2. Diagnosis with local primary residuals. For each root node rvi ∈ RN obtained from Step 1, the process variables corresponding to the node set V(rvi ) in (15) are selected to generate residual res(rvi ) by following (8) and related procedures described in Section 2.2. And the set of hypothesized root causes to which res(rvi ) is sensitive, i.e., Lrc(resi ), is determined according to rules in Section 3.2. With the evaluation results of multiple local primary residuals {eval(res(rvi ))rv ∈ RN , the refined set of root causes RC1 (RC1 ⊆ i RC) is obtained by following (18). Step 3. Diagnosis with local reconstructed residuals. If the fault direction matrix of some hypothesized faulty components in RC1 obtained from Step 2 is known, it is utilized to generate the local reconstructed residual according to (12). After that, by following similar procedures in Step 2, we refine RC1 into RC2 (RC2 ⊆ RC1 ). If no fault direction matrix is available, Step 3 should be skipped. Example 2. Assume that all the nodes in Fig. 3 are sensor nodes except that v4 is a controller output node. In Step 1, suppose the detected deviations of the nodes are v1 (+), v2 (−), v3 (−), v4 (−), and v5 (+). From the SDG backward reasoning, the set of root nodes is {v1 ,v2 ,v5 }. Then according to Table 2, (13) and (14), we have Rc(v1 ) = Supp(v1 ),

(19)

Rc(v2 ) = {Supp(v1 ), Supp(v2 ), Supp(v3 ), Supp(arc(v1 , v2 )), Supp(arc(v3 , v2 )), Supp(arc(v4 , v2 ))}. Rc(v5 ) = {Supp(v5 ), Supp(arc(v4 , v5 ))},

(20) (21)

RC = Rc(v1 ) ∪ Rc(v2 ) ∪ Rc(v5 ) = {Supp(v1 ), Supp(v2 ), Supp(v3 ), Supp(v5 ), Supp(arc(v1 , v2 )), Supp(arc(v3 , v2 )), Supp(arc(v4 , v2 )), Supp(arc(v4 , v5 ))}.

eval(resi ) = 0

⇒

Lrc(resi ) ⊆ COMPS − 

(17)

According to (17), eval(resi ) = 0 implies that all the components in Lrc(resi ) are normal, i.e.,  ∩ Lrc(resi ) = ∅. Since it is

In Step 2, {v1 ,v2 ,v3 ,v4 } and {v4 ,v5 } are selected to generate primary residual res(v2 ) and res(v5 ), respectively. MSPM cannot be applied to the root node v1 because it has no parent node.

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{Supp(v1 ), Supp(v2 ), Supp(v3 ), Supp(arc(v1 , v2 )), Supp(arc(v3 , v2 )), Supp(arc(v4 , v2 ))} {Supp(v5 ), Supp(arc(v4 , v5 ))} {Supp(arc(v4 , v5 ))} {Supp(v5 ), process fault in Supp(arc(v4 , v5 ))} res(v2 ) res(v5 ) res(v5 ,Supp(v5 )) res(v5 , actuator fault in Supp(arc(v4 , v5 )))

To implement Step 3, we first have to determine the fault direction matrix of the hypothesized faulty components in RC1 . As explained in Remark 1, although v2 is a sensor node, the fault direction matrix of its sensor fault cannot be obtained, because v2 forms a control loop with v4 , and the control law is captured in the DPCA model corresponding to {v1 ,v2 ,v3 ,v4 }. The fault direction matrix of suspected arcs in RC1 cannot be obtained either. Only the fault direction matrices of sensor fault related to v5 and actuator faults related to Supp(arc(v4 ,v5 )) can be obtained from the linear relation in the identified DPCA model. Therefore we have the local reconstructed residuals res(v5 ,Supp(v5 )) and res(v5 , actuator fault in Supp(arc(v4 , v5 ))). The set of locally hypothesized root causes and evaluation result for each local reconstructed residual are also listed in Table 3. Then RC1 in Step 2 can be refined into RC2 :

Locally hypothesized root causes

RC1 = RC.

Local residual

Residual evaluation

We have {Lrc(res(vi )), eval(res(vi ))}i=2,5 listed in Table 3. Then according to (18), we have the refined set of root causes

RC2 = RC1 − Lrc(res(v5 , Supp(v5 ))) = {Supp(v1 ), Supp(v2 ), Supp(v3 ), Supp(v5 ), Supp(arc(v1 , v2 )), Supp(arc(v3 , v2 )), Supp(arc(v4 , v2 ))}.

Simulation examples

In this section, some simulation examples on the Tennessee Eastman (TE) process are given to illustrate the effectiveness of the proposed hybrid approach in diagnosing novel faults. The TE process is a simulation program of a realistic industrial process that is widely used for evaluating process control and monitoring methods (Downs and Fogel, 1993; Russell et al., 2000). A brief description of the TE process is given in Appendix. Three fault scenarios are simulated, as shown in Table 4. The involved faulty components are C header pressure, separator temperature sensor of T11, and composition sensor of XE. The faulty component presented in one sce√ nario is marked with “ ” in the corresponding row. C header pressure loss is one of 20 faults simulated by the TE process FORTRAN code (Russell et al., 2000). The other two faults in the case studies are introduced by modifying the FORTRAN code. The simulation time is 1920 min, and the faults are injected at 961 min. All the faults in Table 4 are step change in the corresponding process variable. Based on the flow diagram of the TE process in Fig. A1 (Appendix A), a two-step procedure (Lv and Wang, 2008; Maurya et al., 2004; Palmer and Chung, 2000) is performed to build the SDG model in Fig. 4: first, the local SDGs for decomposed subsystems are built; then, the local SDGs are composed to derive the SDG for the entire process. In Fig. 4, each node is labeled with the names listed in Table A1 in Appendix. The whole process is decomposed into process components including: (1) 12 controller outputs correspond to Valves 1–12, respectively; (2) 41 sensors correspond to measured nodes from F1 to ZH, respectively; (3) mixing zone, reactor, condenser, separator, compressor, and stripper; and (4) the input process of streams 1–4, reactor cooling water, condenser cooling water, stripper input stream as the unmeasured input of the whole process. The above process components are related to graph elements in Fig. 4 by introducing the concept of “support”, as explained in Table 1. It should be pointed out

Table 3 – The sets of local root causes and evaluation results for local residuals.

4.

1 1 0 1

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Table 4 – Fault scenarios. Scenario

Faulty component C header pressure loss

1 2 3

Separator temperature sensor of T11 (positive deviation)

√ √ √

that more refined decomposition of the process into components can be incorporated if more process knowledge is available. As Step 1 of the proposed hybrid diagnosis strategy in Section 3.4, the result of SDG backward reasoning for fault scenario 1 at time 970 is shown in Table 5. Note that one process component can be in the support of two different arcs, revealing the dependence between these arcs. Intuitively, the dependence between two arcs comes from the fact that one process component includes multiple causal relations represented by different arcs, e.g., {Condenser, Separator} is appeared in the support of both arcs (P7, T11) and (T9, T11) in Table 5.

Composition sensor of XE (negative deviation)

√ √

According to the identified root nodes for fault scenario 1 in Table 5, local primary residuals are generated and diagnosed according to Step 2 of the proposed hybrid diagnosis strategy. The variable selection for each local model, number of lagged variables in the DPCA models, locally hypothesized root causes and the evaluation result of each local residual are shown in Table 6. Variable selection for each root node in column 2 follows (15). In the DPCA model, AIC is used to determine the number of lagged variables s (column 3 of Table 6) and the order of system states n by following Wang and Qin (2002), then the number of principal components is ls + n as explained in Section 2.2. Each local residual is normalized so that the threshold is unity, and is shown in Fig. 5.

Fig. 4 – SDG model of the TE process (solid arcs denote positive effects, and the dashed arcs denote negative effects).

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Table 5 – Result of SDG backward reasoning for fault scenario 1. Root node F4 T11

XE

Incident arcs (MV4,F4) (P7,T11) (T9,T11) (F14,T11) (F3,XE) (F5,XE)

Hypothesized root causes for inconsistent arc

Hypothesized root causes for root node

{Sensor F4, Valve MV4, Stream4 input process} {Sensor P7, Sensor T11, condenser, separator} {Sensor T9, Sensor T11, reactor, condenser, separator} {Sensor F14, Sensor T11, Separator} {Sensor F3, Sensor XE, mixing zone} {Sensor F5, Sensor XE, mixing zone}

{Sensor F4, Valve MV4, Stream4 input process} {Sensor T11, Sensor P7, Sensor T9, Sensor F14, reactor, condenser, separator}

Fig. 5 – Local primary residuals for fault scenario 1 (solid line: residual; dashed line: normalized threshold; vertical dotted line: the time instant 970 min when doing diagnosis). (a) Local residual for node F4; (b) local residual for node T11; and (c) local residual for node XE.

{Sensor XE, Sensor F3, Sensor F5, mixing zone}

Locally hypothesized root causes in column 4 are obtained by following the rules in Section 3.2. And in column 5, the evaluation result of a local residual is “1” if it is above the chosen threshold for three consecutive instants; otherwise, the evaluation result of a local residual is “0”. Consider Step 3 of the hybrid diagnosis strategy. No fault direction matrix is available for Sensor F4 and Valve MV4 since they are involved in a control loop and the control law is captured in the local DPCA model corresponding to nodes {F4, MV4}. But in the local DPCA model related to nodes {T11, P7, T9, F14}, fault direction matrices are available for Sensor T11, P7, T9, F14. Thus according to (12) and (16), the local reconstructed residuals and their evaluation results are listed in column 3 of Table 7. With Tables 6 and 7, the refined set of root causes is {Sensor F4, Sensor P7, Valve MV4, Stream4 input process}. Note that C head pressure loss is the real fault located in Stream4 input process. In fault scenario 2, the identified root nodes from SDG backward reasoning are {Sensor F4, Sensor T11}. Then the local reconstructed residuals and their evaluation results for the local DPCA model related to the root node T11 are listed in column 4 of Table 7. The same procedures are implemented for fault scenario 3. The diagnosis results for three fault scenarios at time 970 are summarized in Table 8, where among the hypothesized root causes in column 3 the bold ones are included in column 4 while the others are eliminated according to the proposed hybrid diagnosis strategy. It can be seen that the refined set of hypothesized root causes in the proposed hybrid approach (column 4 of Table 8) includes the actual faulty components listed in column 5 of Table 8, but has fewer number of elements than that after SDG backward reasoning (column 3 of Table 8), which shows that the proposed hybrid approach improves diagnosis performance. In order to evaluate the proposed hybrid approach in the case of large measurement noises, the standard deviation of the measurement noise of each process variable is modified to be two times of that in the original TE process FORTRAN code (Russell et al., 2000). The obtained results for this case are given in Table 9. It can be seen that in all fault scenarios the Valve MV10 becomes an additional hypothesized root node from SDG reasoning, which shows that the step of SDG reasoning is sensitive to noise levels. The proposed hybrid approach is more robust by combining the MSPM step and the diagnosis logic: the false root node Valve MV10 is finally eliminated after performing MSPM, and the results for fault scenarios 1 and 2 in Table 9 are the same as those in Table 8. But as shown in Table 9, the actually faulty component Sensor XE is missed in the refined set of root causes, which can be explained as miss detection in the presence of large measurement noises. The selection of thresholds in MSPM is a trade-off between false alarms and miss detections. The above simulations show the effectiveness of the proposed hybrid approach in diagnosing novel faults, while

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Table 6 – Local primary residuals for fault scenario 1. Local primary residual

Selected variables

Number of lagged variables in DPCA

res(F4) res(T11)

{F4, MV4} {T11, P7, T9, F14}

4 3

res(XE)

{XE, F3, F5}

1

Locally hypothesized root causes

Residual evaluation

{Sensor F4, Valve MV4, Stream4 input process} {Sensor T11, Sensor P7, Sensor T9, Sensor F14, reactor, condenser, separator} {Sensor XE, Sensor F3, Sensor F5, mixing zone}

1 1 0

Table 7 – Local reconstructed residuals for fault scenarios 1 and 2. Local reconstructed residual

Locally hypothesized root causes

res(T11, T11)

{Sensor P7, Sensor T9, Sensor F14, reactor, condenser, separator} {Sensor T11, Sensor T9, Sensor F14, reactor, condenser, separator} {Sensor T11, Sensor P7, Sensor F14, reactor, condenser, separator} {Sensor T11, Sensor P7, Sensor T9, reactor, condenser, separator}

res(T11, P7) res(T11, T9) res(T11, F14)

Residual evaluation in fault scenario 1

Residual evaluation in fault scenario 2

0

0

0

1

1

1

1

1

Table 8 – Result of hybrid diagnosis for three fault scenarios. Fault scenarios

Identified root nodes

Set of hypothesized root causes after SDG backward reasoning

Refined set of hypothesized root causes in the hybrid approach

1

{F4, T11, XE}

{Sensor F4, Valve MV4, Stream4 input process}

C header pressure loss

2

{F4, T11}

{Sensor F3, Sensor F4a , Sensor F5, Sensor P7, Sensor T9, Sensor T11, Sensor F14, Sensor XE, Valve MV4, Stream4 input process, reactor, condenser, separator, mixing zone} {Sensor F4, Sensor P7, Sensor T9, Sensor T11, Sensor F14, Valve MV4, Stream4 input process, reactor, condenser, separator}

{Sensor F4, Sensor T11, Valve MV4, Stream4 input process}

C header pressure loss; Separator temperature sensor of T11 (positive deviation)

3

{F4, T11, XE}

{Sensor F3, Sensor F4, Sensor F5, Sensor P7, Sensor T9, Sensor T11, Sensor F14, Sensor XE, Valve MV4, Stream4 input process, reactor, condenser, separator, mixing zone}

{Sensor F4, Sensor XE, Valve MV4, Stream4 input process}

C header pressure loss; Composition sensor of XE (negative deviation)

a

Actual faulty components

The process components in bold in column 3 are the suspected root causes in column 4.

Table 9 – Result of hybrid diagnosis in the case of large measurement noises. Fault scenarios

Identified root nodes

1

{F4, T11, MV10}

2

{F4, T11, MV10}

3

{F4, T11, XE, MV10}

Refined set of hypothesized root causes in the hybrid approach {Sensor F4, Valve MV4, Stream4 input process} {Sensor F4, Sensor T11, Valve MV4, Stream4 input process} {Sensor F4, Valve MV4, Stream4 input process}

the existing hybrid approaches combining CDR and MSPM in (Lee et al., 2004; Lv and Wang, 2008; Vedam and Venkatasubramanian, 1999) required to describe anticipated faults in the digraph model and were not capable to handle novel faults.

5.

Actual faulty components

C header pressure loss C header pressure loss; separator temperature sensor of T11 (positive deviation) C header pressure loss; composition sensor of XE (negative deviation)

Conclusions

We have developed a new method for novel fault diagnosis which combines qualitative SDG reasoning and quantitative MSPM techniques. The new approach requires only normal

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process flow diagrams and historical data. The first step for this approach is qualitative backward reasoning, which identifies the root nodes accounting for the deviation. After that, a local residual is generated to quantitatively evaluate the consistency of suspected arcs implied by each root node. By relating graph elements to process components, the set of root nodes is translated into the set of hypothesized root causes in terms of possible faulty process components, and different local residuals are interpreted to be sensitive to different subsets of the hypothesized root causes. Next, the diagnosis logic based on the exoneration principle refines the set of root causes by evaluating multiple local residuals. Potential fault direction information from normal historical data is also incorporated to improve the diagnosis performance. Although quantitative information from MSPM has been previously utilized to trigger CDR (Lee et al., 2004; Lv and Wang, 2008; Vedam and Venkatasubramanian, 1999), the inference process is still qualitative. In comparison, the proposed approach utilizes MSPM in the inference mechanism. As a result, the quantitative information from MSPM not only eliminates spurious root nodes, but also facilitates deep root cause analysis by interpreting multiple local residuals in the digraph. Additional work, incorporating normal historical data in backward reasoning and handling uncertainties in the digraph model, is currently on-going.

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Acknowledgments The work has been supported by the 973 Program of China under Grant 2010CB731800, the National Natural Science Foundation of China under Grants 60974059, 60736026, 61021063, and 60904044, Tsinghua National Laboratory for Information Science and Technology (TNList) Cross-discipline Foundation, and research funding from ExxonMobil Research and Engineering.

Appendix A. Technical description of the TE process There are five major units in the TE process, including a reactor, a condenser, a recycle compressor, a vapor/liquid separator, and a stripper, as shown in Fig. A1. The TE process is fed with the gaseous reactants A, D, D, E, and the inert B, and produces the liquid products G and H and by-product F from the stripper (Russell et al., 2000). The whole process is open-loop unstable and controlled by PI-controllers in 9 loops and 21 set points, with 41 process variables and 12 manipulated variables (see Table A1) (Russell et al., 2000). More details of the TE process are referred to Downs and Fogel (1993).

Fig. A1 – Process flow diagram of the TE process (Russell et al., 2000).

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Table A1 – Manipulated and measured variables in the TE process (Russell et al., 2000). Variable

Description

Sampling interval (min)

Variable

Description

Sampling interval (min)

Stripper pressure Stripper underflow (stream 11) Stripper temperature Stripper steam flow Compressor work Reactor cooling water outlet temperature Condenser cooling water outlet temperature Composition of A (stream 6) Composition of B (stream 6) Composition of C (stream 6) Composition of D (stream 6) Composition of E (stream 6) Composition of F (stream 6) Composition of A (stream 6) Composition of B (stream 6) Composition of C (stream 6) Composition of D (stream 6) Composition of E (stream 6) Composition of F (stream 6) Composition of G (stream 6) Composition of H (stream 6) Composition of D (stream 6) Composition of E (stream 6) Composition of F (stream 6) Composition of G (stream 6) Composition of H (stream 6)

1 1 1 1 1 1

MV1 MV2 MV3 MV4 MV5 MV6

D feed flow (stream 2) E feed flow (stream 3) A feed flow (stream 1) Total feed flow (stream 4) Compressor recycle valve Purge valve (stream 9)

1 1 1 1 1 1

P16 F17 T18 F19 J20 T21

MV7

Separator pot liquid flow (stream 10)

1

T22

MV8 MV9 MV10 MV11 MV12 F1 F2 F3 F4 F5 F6 P7 L8 T9 F10 T11 L12 P13 F14 L15

Stripper liquid product flow Stripper steam valve Reactor cooling water flow Condenser cooling water flow Agitator speed A feed (stream 1) D feed (stream 2) E feed (stream 3) Total feed (stream 4) Recycle flow (stream 8) Reactor feed rate (stream 6) Reactor pressure Reactor level Reactor temperature Purge rate (stream 9) Separator temperature Separator level Separator pressure Separator underflow (stream 10) Stripper level

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

XA XB XC XD XE XF YA YB YC YD YE YF YG YH ZD ZE ZF ZG ZH

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