A Process Decomposition Strategy for Qualitative Fault Diagnosis of

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A Process Decomposition Strategy for Qualitative Fault Diagnosis of Large-Scale Processes Gibaek Lee* Department of Industrial Chemistry, Chung-Ju National University, Chung-Ju, Chungbuk 380-702, Korea

En Sup Yoon School of Chemical Engineering, Seoul National University, Seoul 151-742, Korea

Most chemical processes are very large or complex. Because of this size and complexity, it is very difficult to make a diagnostic system for an entire process. Therefore, a systematic approach is required to decompose a large-scale process into subprocesses and then diagnose them. This paper suggests a method for minimization of the knowledge base and flexible diagnosis to be used in qualitative fault diagnosis based on a fault-effect tree model. The system can be decomposed for flexible diagnosis, size reduction of the knowledge base, and consistent construction of a complex knowledge base. The new node, called a gate variable, is introduced to connect the cause-effect relationships of each subprocess. For on-line diagnosis, off-line analysis is performed to construct both the fault-effect trees and the activation conditions for the gate variables. The on-line diagnosis strategy is modified to yield the same diagnosis result without system decomposition. Also, this work establishes that a plus cycle of the gate variables makes the diagnosis fail and proposes a method for resolving the problem of the plus cycle by minimizing the number of fault-propagation paths evaluated. The proposed method is illustrated with a fault diagnosis system for a large-scale boiler plant. Introduction As there have been many serious accidents in chemical industries during recent years, the importance of having fault diagnosis systems for chemical plants has become evident. To assist operators in making the right judgments in critical situations, various approaches have been developed,1,2 such as rule-based expert systems,3,4 state estimation,5,6 signed digraphs,7,8 qualitative simulation,9 and neural networks.10,11 However, although much research has been executed in the academy and in industry, few resulting systems can be successfully implemented in actual chemical plants. Therefore, it is important to resolve practical matters in the application of fault diagnosis systems to the actual plants. The main difficulties in developing diagnostic system are due to the exclusive features of chemical processes such as nonlinearity, uncertainty, large size, and complexity. In particular, this paper concentrates on the problems related to the size and complexity of the processes. Most chemical processes are very large or complex. As the number of variables used is enormous, it can be difficult to collect consistent information for fault diagnosis. In particular, maintenance of the knowledge base can become increasingly difficult. In addition, the online calculation time increases linearly or exponentially with the number of system variables. Therefore, a systematic approach is required to decompose systems targeted for fault diagnosis into subsystems and diagnose them. Two methods of system decomposition for fault diagnosis have been developed. One is a two-step diagnostic * Author to whom correspondence should be addressed. Tel.: +82-43-841-5230. Fax: +82-43-841-5220. E-mail: glee@ gukwon.chungju.ac.kr.

Figure 1. Block diagram of a boiler plant.

strategy, which narrows diagnostic focus to a decomposed subsystem and performs diagnosis for the subsystem.1,9,12 The other is a diagnostic method that uses data for the whole system. In this method, a diagnosis knowledge base is developed separately for each subsystem, and the diagnostic results from the subsystems are collected for the final conclusion during on-line diagnosis.13 When the system is decomposed, the knowledge base of each decomposed subsystem can have information about the associated subsystem only. This provides the advantage of reducing the size of the knowledge base. It also facilitates the construction and maintenance of the diagnosis model, because each knowledge engineer can build the knowledge base of a single subsystem and clearly understand the knowledge bases built by others. Therefore, the consistency and

10.1021/ie0001366 CCC: $20.00 © 2001 American Chemical Society Published on Web 05/01/2001

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Figure 2. Process schematics of a water supply unit and boiler main unit.

reliability of the diagnostic knowledge base can be improved. Because of these advantages, this paper suggests the latter decomposition method for qualitative fault diagnosis based on signed digraphs, while focusing on the fault-effect tree.14 As an example of a large-scale process, this paper considers a boiler plant operating in a petrochemical plant (Figure 1). The target plant involves three water supply units to supply feedwater into boilers, five boilers, and two steam distribution units to distribute steam to each steam-consuming plant. Among the 10 unit processes, this paper shows examples for one water supply unit, one boiler, and one steam distribution unit. Each boiler consists of a main unit to generate steam, an air supply unit to supply air for fuel combustion, and a fuel supply unit to provide oil and waste gas to the burner (Figure 2). Fault-Effect Tree A fault-effect tree (abbreviated to FET) is based on a signed digraph. A signed digraph (abbreviated to SDG) offers a simple and graphical representation of the causal relationship between process variables and has been widely used for fault diagnosis.5 A SDG consists of nodes that represent process variables and arcs that show the causal relationships between the nodes. The direction of a causal relationship is indicated by the sign of the arc. A positive (+) sign is used when the source and target nodes move in the same direction, and a negative sign (-) is used when they move in opposite directions. Mohindra specifies the basic concepts for using SDGs.15 Some keywords follow: (1) Qualitative States: The qualitative states for a process variable correspond to the range of possible values for that variable. Usually, the values of normal (0), high (+1), and low (-1) are used. (2) Symptom: A symptom is an observed nodes that does not have the qualitative state of “normal” in the SDG.

(3) Consistent Arc: An arc x f y is consistent if the product of the qualitative state of its source node and its sign is the same as the qualitative state of its target node, i.e. y ) x‚sign(x f y). (4) Single Fault Assumption: All observed abnormalities are assumed to have resulted from a single fault. (5) Consistent Path Assumption: Each observed symptom is assumed to be connected to its fault by an unbroken path of consistent arcs. However, the assumptions of SDGs, such as the single fault assumption and the consistent path assumption, have limited their applications to real processes. To overcome the weakness of SDGs under uncertainty, FETs use modified basic concepts of SDGs as follows: (1) Fault: With FETs, the physically feasible faults for each piece of equipment are defined and added to the root node in order that only physically meaningful faults be addressed. (2) Variable Cluster: A variable cluster is a group of nodes that show similar dynamic behaviors when a disturbance propagates. Each variable cluster has more than one measured variable and is treated as a single measured variable. The qualitative states of a variable cluster involve the states of all measured variables in the variable cluster and are denoted as subpatterns of the variable cluster. In addition to fault nodes and variable clusters, constraint variables are added to the SDGs. They represent the quantitative governing equations, such as balance equations and valve relations, as variables. After the subpatterns are determined, the FET is constructed by searching the possible fault-propagation paths in the modified SDG. The on-line diagnosis based on the model quantitatively evaluates the qualitative relations of the faultpropagation path and determines how similar the response of process variables is to the predicted qualitative model. The evaluation begins with evaluation of the subpatterns. As a result of the subpattern evaluation,

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Figure 3. (a) Process flow diagram and (b) SDG of an example.

the subpattern likelihood (SL) is obtained by using the sigmoid-type belief functions of the measured variables. The fault candidates are ranked by their fault likelihoods (FL), which are calculated using the equation NESP(F)

FL )

∑i

〈〈SPi,F〉〉

NESP(F)

In the above equation, ESP(F) indicates the set of subpatterns that can be explained by the fault F

ESP(F) ) {(SP,〈〈SP,F〉〉)|SP ∈ SF, SLSP > 0.5} and 〈〈SP,F〉〉 is the likelihood of the path from F to subpattern SP. The path likelihood is obtained from a simple quantitative evaluation of the fault-propagation path NPSP F

〈〈SPi,F〉〉 )

∑i SLi NPSP F

where PSP F denotes the sequence of subpatterns existing on the path from F to SP. This quantification enables the method to avoid the consistent path assumption and to diagnose multiple faults using the diagnostic model constructed for single faults. Process Decomposition As each unit of the target process corresponds to each SDG describing the unit, process decomposition means decomposition of the process and the SDG. For instance, the boiler plant can be decomposed into five subprocesses of water supply units, boiler main units, air supply units, fuel supply units, and steam distribution units. This decomposition divides the SDG into five parts as well. For simplicity, consider another example of a small-scale process, as shown in Figure 3a. The process is decomposed into three subsystems, as designated by the dotted lines. Also, the corresponding SDG is decomposed by the process decomposition (Figure 3b).

Process decomposition in qualitative fault diagnosis can provide the following advantages, and the process can be decomposed to maximize these advantages. (1) Flexible Diagnosis during Operational Condition Changes: In most chemical plants, operators might shut down some units in the plant for an operational condition change such as scheduled maintenance or for an unscheduled event. For the same reasons, they might operate only part of the plant. For example, consider our target boiler plant. Even though all five boilers are connected to other units, one boiler is a stand-by unit for use during abnormal shutdown of other boilers. During on-line diagnosis, the causal relationships of the stand-by boiler with other units are broken. Therefore, in off-line analysis, the developer can decompose the processes that can be shut down independently. The unnecessary knowledge bases of the subprocesses can be easily ignored during on-line fault diagnosis. (2) Size Reduction of the Knowledge Base: As the target plant is so large, the knowledge base of the qualitative diagnosis model for the process might be too extensive to be maintained properly. In particular, in the fault diagnostic methodology based on SDGs, the computation time increases exponentially with the number of unmeasured nodes and linearly with the number of arcs.16 When a process is decomposed, every fault-propagation path includes the paths in the subsystem of that particular fault only, and the size of knowledge base is greatly reduced. Also, searches are limited to each subsystem, so the computation time can be reduced much more. Therefore, process decomposition could be helpful in enhancing system performance and maintenance. (3) Ease in Understanding of Complex Processes with Process Interactions: If there is a complex interaction or control logic, diagnostic knowledge might be complicated and difficult to understand. In fact, a simplified model that removes the unmeasured variables is even more difficult to understand. If, instead, the process is decomposed at the appropriate positions, the developer can independently consider the diagnostic knowledge for each subsystem. Therefore, the building and understanding of the knowledge base become simpler, and revision of the knowledge base can be more clearly executed by other developers. For instance, the fuel flow rate of the target boiler is controlled by a series of four cascade controllers, which measure over 10 variables.

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Figure 5. FETs of gate variables in the example. Figure 4. Subprocesses and gate variables in the decomposed example.

This series of cascade controllers can be decomposed for developers to make an exact diagnostic model of the complex control logic. To connect the cause-effect relationships between the subsystems, the gate variable is introduced as a new node in the SDG. A gate variable is included in two subsystems that are connected by the gate variable. For on-line diagnosis, this node requires a sign, just like other nodes in the SDG. Consider the example process of Figure 3. Three gate variables are introduced between F1 and P, P and F1, and F2 and L2 as shown in Figure 4. Also, a condition that represents activation of the subsystem is introduced on the arc from the gate variable to represent the disconnection of relationships, as explained earlier. In the example of Figure 4, there is a condition, SA(SS1), representing the activation of subsystem SS1 on the arc from GV21. In on-line diagnosis, the shutdown of SS1 can be handled simply through this condition. Although system decomposition can be helpful to build and maintain a fault diagnosis system, unnecessary decomposition can result in a complicated system. Therefore, it is noted that a system should be decomposed only to obtain the advantages stated previously. In addition, the decomposition strategy is conceptually simpler than the methods that consider structural or functional relationships, as it does not use strict criteria and focuses on the partition of the target processes into smaller parts. Therefore, we note that the above concepts can be applied to other SDG-based methods. Diagnosis Strategy Off-Line Analysis. For the diagnosis based on FETs and process decomposition, the following items should be added to the off-line analysis. (1) Fault-Propagation Path from the Gate Variable: When a FET is being built, the gate variable is regarded as a subpattern in the subsystem connected with an arc to the gate variable. On the other hand, it is regarded as a fault in the subsystem connected with an arc from the gate variable. As a gate variable can have two signs (+ or -), two FETs are required for a gate variable. As for a fault, the FET for a gate variable is constructed by searching the fault-propagation paths from the gate

Table 1. Subpatterns for the Example variable cluster VC1

VC2

VC3

FOS

subpattern

qualitative state of measured variables

VC1-1 VC1-2 VC1-3 VC1-4 VC2-1 VC2-2 VC2-3 VC2-4 VC3-1 VC3-2 VC3-3 VC3-4 FOS(+) FOS(-)

(L1S, +), (CL1, +) (L1S, +), (CL1, -) (L1S, -), (CL1, -) (L1S, -), (CL1, +) (PS, +), (CP, +) (PS, +), (CP, -) (PS, -), (CP, -) (PS, -), (CP, +) (L2S, +), (CL2, +) (L2S, +), (CL2, -) (L2S, -), (CL2, -) (L2S, -), (CL2, +) (FOS, +) (FOS, -)

variable. If the process is decomposed, the search space for a given gate variable is limited to the subsystem of that gate variable. In the case of fault, the search space is likewise limited. For instance, consider the example of Figure 3. In this process, there are three variable clusters with two measured variables and one variable cluster with one measured variable, and the subpatterns for the variable clusters are presented in Table 1. The FET of the gate variable GV12(+) includes the gate variable and subpattern in SS2 only (Figure 5). If LVBH denotes the biased-high level of the control valve in SS1, the FET of LV-BH includes the gate variable and subpatterns in SS1 only (Figure 6a). In the result, process decomposition divides the entire fault-propagation path into a number of pieces and stores the pieces as FETs. Therefore, it is evident that there is no redundancy or loss of diagnostic knowledge. For example, the FETs of LV-BH and L1S-BL without decomposition look like Figure 6b. With process decomposition, the FETs of LV-BH and L1S-BL become Figure 6a. The subpaths of LV-BH to VC2-1 (L1S-BL to VC21) and VC2-1 to VC3-1 are stored in the FETs of GV12(+) and GV23(+), respectively (Figure 5). Therefore, the efficiency in knowledge storing can be greatly improved. (2) Basic Symptoms of Gate Variables: If the basic symptom of a gate variable is decided by the same method as that of a fault, small symptom variations might prevent gate variable from activating. Therefore, this study uses a new method. The OR activation condition of all subpatterns in the FET of a gate variable can be used. However, if the sensors are highly reliable or the fault-propagation path of the gate variable is too long, the system might activate too many insignificant

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Figure 6. FETs of LV-BH and L1S-BL (a) with decomposition and (b) without decomposition.

gate variables. Therefore, it is better to evaluate FL for a gate variable and compare it with the minimum fault likelihood for the primary fault candidate.14 If the FET of a gate variable includes other gate variables and the subpath between two gate variables is short enough, the developer might decide for the basic symptom of the gate variable to include the subpatterns of the other gate variables. On-Line Diagnosis. When the subpattern likelihood of each subpattern is evaluated, faults and gate variables are invoked by evaluating the basic symptoms. The next step is the evaluation of fault-propagation paths. In this step, the previous method is modified as follows in order to obtain the same results as those without decomposition. (1) The path likelihood in the set of explained subpatterns of a gate variable is not calculated, unlike that of fault. Once the path likelihood from a gate variable to a subpattern is calculated by the same method as for the path from a fault to a subpattern, the path likelihood from the gate variable to a fault cannot be calculated because the required data are removed from storage. Therefore, the denominator and numerator of equation used to evaluate the path likelihood should be separately stored as follows. D(〈〈SP,G〉〉) and N(〈〈SP,G〉〉) denote the denominator and numerator of the equation used to calculate the path likelihood from G to SP.14

ESPG ) {(SP,(D(〈〈SP,G〉〉),N(〈〈SP,G〉〉)))|SP is a subpattern that can be explained by G} NPSP G

〈〈SP,G〉〉 )

∑i

SLi

NPSP G

(2) In fault diagnosis based on FETs and system decomposition, the fault-propagation path from faults to subpatterns can be split into four types of subpaths: (i) fault to gate variable, (ii) fault to subpattern, (iii) gate variable to gate variable, and (iv) gate variable to subpattern. Therefore, the fault-propagation path from

Figure 7. Example of the plus clycle of gate variables.

faults to subpatterns can have the following types of links between subpaths: (i) fault to subpattern; (ii) fault to gate variable and gate variable to subpattern; and (iii) fault to gate variable, gate variable to gate variable, and gate variable to subpattern. To obtain the same results as found without decomposition, the links of the subpaths must be evaluated in the direction opposite to that in which the faultpropagation path from fault to subpattern was evaluated. For example, in the case of link-type iii, the subpath of fault to gate variable should not be evaluated before the subpaths of gate variable to gate variable or gate variable to subpattern. That is, the evaluation of the subpaths from gate variable to subpattern must be done prior to the evaluation of the subpaths of gate variable to gate variable. Also, the evaluation of the subpaths from gate variable to gate variable must be done prior to the evaluation of the subpaths of fault to gate variable. Therefore, the path evaluation order should be (i) fault to subpattern, (ii) gate variable to subpattern, (iii) gate variable to gate variable, and (iv) fault to gate variable. This ordering is achieved by a new attribute, the number of connected gate variables (NCG), introduced on faults and gate variables. Definition: A connected gate variable is defined as an activated variables among the gate variables included in the propagation path of a fault or a gate variable. If a gate variable is activated, its faults or gate variables, including the gate variable itself, is searched, and the NCG values of those faults and gate variables are increased by one. For example, if GV12(+) is activated in Figure 6a, then the NCG of LV-BH is 1. The evaluation order is determined by the NCG values. If the NCG of gate variable A is 0, then the subpath propagating from gate variable B to gate variable A is searched and evaluated, the evaluated

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subpath is inserted into the ESP of B, and the NCG of B is decreased by 1. This step is same for every subpath from faults to gate variables. If the NCG of a fault is 0, then the FL of the fault is evaluated. This step will be continued until the NCG of every fault is 0. For instance, consider the example of Figure 3. Assume that LV-BH, GV12(+), and GV23(+) are invoked. The following equations give the results of subpath evaluation from the fault, LV-BH, and gate variables to the subpatterns.

ESP(LV-BH) ) {(FOS(+), FOS, 0.7), (VC1-2, VC1, 0.85)} ESP(GV12(+)) ) {(VC2-1,VC2, (1,0.9))} ESP(GV23(+)) ) {(VC3-1,VC3, (1,0.75))} As GV12(+) and GV23(+) are invoked, the NCG of F1 is 1, that of GV12(+) is 1, and that of GV23(+) is 0. Therefore, the subpath of GV12(+) to GV23(+) is evaluated. Then, the evaluated subpath is inserted into the ESP of GV12(+), and the NCG of GV12(+) becomes 0.

N(〈〈VC3-1, GV12(+)〉〉) ) N(〈〈GV23(+), GV12(+)〉〉) + N(〈〈VC3-1, GV23(+)〉〉) ) 0.9 + 0.75 ) 1.65 D(〈〈VC3-1, GV12(+)〉〉) ) D(〈〈GV23(+), GV12(+)〉〉) + D(〈〈VC3-1, GV23(+)〉〉) )1+1)2 ESP(GV12(+)) ) {(VC2-1,VC2, (1,0.9)), (VC3-1, VC3, (2,1.65))} Next, the subpath from LV-BH to GV12(+) is evaluated, and the NCG of LV-BH becomes 0.

〈〈VC2-1,LV-BH〉〉 ) )

N(〈〈VC2-1,GV12(+)〉〉) D(〈〈VC2-1,GV12(+)〉〉) 0.9 ) 0.9 1

By the same procedure

〈〈VC3-1,LV-BH〉〉 )

1.65 ) 0.825 2

Therefore, the following result is obtained:

ESP(LV-BH) ) {(FOS(+), FOS, 0.7), (VC1-2, VC1, 0.85), (VC2-1, VC2, 0.9), (VC3-1, VC3, 0.825)} Because the NCG of LV-BH is 0, the fault likelihood of LV-BH can be calculated as follows:

FLLV-BH )

0.7 + 0.85 + 0.9 + 825 ) 0.819 4

The FET of LV-BH determined without process decomposition is shown in Figure 6b, and the set of explained subpatterns of LV-BH is the same as the result obtained above. Thus, the result with decomposition is identical to the result without decomposition.

Plus Cycle. When the proposed method is applied, sometimes, the system cannot terminate the diagnostic steps. This is due to the plus cycle of gate variables. A plus cycle of activated gate variables is a path in which the first gate variable is the same as the last and the first is connected to the last with consistent arcs. If there are plus cycles (henceforth, cycles) in the system, the NCG values of the gate variables in the cycles cannot be 0, so the diagnosis will fail. A modification of diagnostic strategy is required to evaluate fault-propagation path of each gate variable. Consider GV23(+), shown in Figure 7b, which is an artificial plus cycle that was constructed as an example. To obtain the correct set of explained subpatterns for GV23(+), evaluation in the order of GV45(+) to GV52(+), GV45(+) to GV57(+), GV34(+) to GV45(+), and GV23(+) to GV34(+) is required. Also, the same steps should be applied to all gate variables in the cycle. However, implementation on a fault diagnosis system is very difficult because the fault-propagation path of each gate variable overlaps in a cycle. This study suggests a method for finding the minimum number of paths that must be evaluated. In the off-line analysis, all possible cycles in the knowledge base must be found, and the evaluation order of the subpaths in the cycles must be determined. To judge cycle development in on-line diagnosis, one determines whether every gate variable in the cycle is activated and all subpaths from every gate variable in the cycle to other gate variables are evaluated. If this condition is satisfied, then the evaluation of the subpaths is forced, except for the subpaths capable of being evaluated by the previous strategy. The off-line analysis method for determining the evaluation order of the subpaths is as follows: (1) List the gate variable to gate variable subpaths of each gate variable in the cycle. (2) Choose a subpath from among the gate variable to gate variable subpaths. It is preferable to choose the subpath that has the most subpaths at the end of the path list of from step 1. (3) If the chosen subpath is at the end of the path list of each gate variable, then delete the subpath from the list and insert the chosen subpath at the end of the subpath evaluation order. (4) Choose the backward subpath of the chosen subpath. If more than two backward subpaths can be selected, choose the one that is not included in the subpath evaluation order. (5) Repeat steps 3 and 4 until all subpaths in the list are removed. (6) From the obtained subpath evaluation order, remove the subpaths that can be evaluated using the NCG. In on-line diagnosis, one determines whether a cycle is invoked. If the cycle is activated, the subpaths in the subpath evaluation order are sequentially forced to be evaluated. For continued evaluation, the NCG of the variable is decreased by 1, which is the terminal gate variable of the last subpath in the subpath evaluation order (see the example). Figure 7 displays the gate variable to gate variable subpaths only for the decomposed process. In this example, a cycle of GV23(+)sGV34(+)sGV45(+)sGV52(+) is possible. Off-line analysis determines the evaluation order of paths as follows. (1) Gate variable to gate variable subpaths of each

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gate variable in the cycle are listed as follows.

GV23(+): sGV34(+)sGV45(+)sGV52(+) GV34(+): sGV45(+)sGV52(+)sGV23(+) GV45(+): sGV52(+)sGV23(+)sGV34(+) GV52(+): sGV23(+)sGV34(+)sGV45(+) (2) The GV45(+) to GV52(+) subpath is arbitrarily chosen and becomes the first evaluation subpath in the cycle. In the list of GV23(+) subpaths having GV45(+) to GV52(+) at the end, the GV45(+) to GV52(+) subpath is deleted.

GV23(+): sGV34(+)sGV45(+) The backward subpath of GV45(+) to GV52(+) is GV34(+) to GV45(+), which becomes the second evaluation subpath. This subpath is deleted from the subpath lists of GV23(+) and GV52(+).

GV23(+): sGV34(+) GV52(+): sGV23(+)sGV34(+) Through these steps, the following subpath evaluation order is obtained:

{GV45(+) to GV52(+),GV34(+) to GV45(+)} The next subpath is GV23(+) to GV34(+), which is deleted from the list of step 1. In this step, all subpaths in the list of GV23(+) are deleted.

{GV45(+) to GV52(+),GV34(+) to GV45(+), GV23(+) to GV34(+)} GV34(+): sGV45(+)sGV52(+)sGV23(+) GV45(+): sGV52(+)sGV23(+) GV52(+): sGV23(+) (3) Finally, the following result can be obtained by off-line analysis:

{GV45(+) to GV52(+),GV34(+) to GV45(+), GV23(+) to GV34(+),GV52(+) to GV23(+), GV45(+) to GV52(+), GV34(+) to GV45(+)} (4) If GV45(+) to GV52(+), GV34(+) to GV45, and GV23(+) to GV34(+) are sequentially evaluated, GV52(+) to GV23(+), GV45(+) to GV52(+), and GV34(+) to GV45(+) can be evaluated using the NCG and removed from the subpath evaluation order (refer to the example of on-line diagnosis). The final subpath evaluation order is as follows:

{GV45(+) to GV52(+),GV34(+) to GV45, GV23(+) to GV34(+)} Assume that all gate variables in Figure 7 are invoked. The procedure of on-line diagnosis is as follows: (1) The NCG of each gate variable is calculated as listed below, and it is judged that a cycle is developed because every gate variable in the cycle is invoked.

GV12(+): 1 GV23(+): 1 GV34(+): 1 GV45(+): 2 GV57(+): 0 GV52(+): 1 GV65(+): 1

(2) The subpaths of GV45(+) to GV52(+), GV34(+) to GV45, and GV23(+) to GV34(+) are forced to be evaluated, and the NCG of GV23(+) becomes 0. Therefore, the subpath from GV23(+) to the fault can be evaluated. (3) Because the NCG of GV23(+) is 0, the subpath of GV12(+) to GV23(+) is evaluated, and the NCG of GV12(+) becomes 0. GV52(+) to GV23(+), GV45(+) to GV52(+), GGV65(+) to GV52(+), and GV34(+) to GV45(+) are sequentially evaluated, and the subpaths from each gate variable to the faults are evaluated. Therefore, the evaluation order of all subpaths is

GV57(+) to GV45(+), GV45(+) to GV52(+), GV34(+) to GV45(+), GV23(+) to GV34(+), GV12(+) to GV23(+),GV52(+) to GV23(+), GV45(+) to GV52(+), GGV65(+) to GV52(+), GV34(+) to GV45(+) With the suggested method, a very complex cycle can be handled. Including plus cycle treatment, the diagnosis procedure is as shown in Figure 8. The proposed methodology was implemented on the expert system shell G2. The fault-propagation path and diagnosis procedure are described by the rule and procedure of G2, respectively. Example and Discussion Process and Diagnosis Description. The faults occurring in a boiler plant could cause a serious cutback of steam production, which could hinder the operation of the steam-consuming plants. Therefore, the proposed methodology is applied to the fault diagnosis system in order to achieve the safe operation of a boiler plant (Figure 1). In particular, the steam demand of each steam-consuming plant is very frequently changed, and changing steam demand is one of the external disturbances of the boiler plant. One fault occurring in steam demand change means multiple faults. Therefore, it is noted that the treatment of multiple faults in the diagnosis of a boiler plant is more important than it is for other plants. The scale of the target plant is shown in the following parameters. Number of measured variables: 107 Number of control-related measured variables: 51 Number of variable clusters with more than two measured variables: 33 Number of variable clusters with one measured variable: 18 Number of constraint variables: 12 Number of subpatterns: 150 Through an interview with experts, the faults of the equipment in the boiler plant were identified as shown in Table 2. Except for independent sensor faults, the number of target faults is 95. Among those 95 faults, the numbers of operator and external disturbances are 12 and 14, respectively. The equipment malfunctions are as follows: Number of control sensor faults: 36 Number of control valve faults: 18 Number of equipment leakages: 10 (6 pressure safety valves, economizer, riser, and primary and secondary superheaters) Number of other equipment malfunctions: 5

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Figure 8. Diagnosis procedure. Table 2. Fault Classification According to Equipment in Boiler Plant equipment sensor

fault

bias high (including stuck) bias low (including stuck) valve bias high (including failure and stuck) bias low (including failure and stuck) pressure safety valve leak superheater leak riser leak economizer leak external disturbance flow rate change high flow rate change low temperature change high temperature change low pressure change high pressure change low operator disturbance setpoint change high setpoint change low

abbreviation BH BL BH BL LK LK LK LK FCH FCL TCH TCL PCH PCL SVCH SVCL

Process Decomposition and Off-Line Analysis. Because the boiler unit and the water supply unit can be shut down independently, they are decomposed. To simplify the model development, the boiler unit is decomposed into three parts: the boiler main unit, the air supply unit, and the fuel supply unit. Eleven gate variables are introduced to connect the cause-effect relationships of each decomposed subsystem. The names of the gate variables begin with GV, and the next two letters indicate which two units are connected by the gate variable. The final letter depends on the node connected to the gate variable (Table 3). For instance,

Table 3. Abbreviations Used in Gate Variable Names letter

abbreviation

meaning

third and fourth letters

M D A F W F T P Q S

boiler main unit steam distribution unit air supply unit fuel supply unit water supply unit flow temperature pressure heat duty control signal

fifth letter

GVMDT connects from the boiler main unit to the steam distribution unit, and the node connected to the gate variable is the temperature. In Figure 9, only the subpaths from gate variables to gate variables are drawn. Among the 11 gate variables, GVWMF has no connections with other gate variables. Through off-line analysis, 22 FETs for the gate variables are constructed. Let us compare the efficiency of the decomposition on the basis of the size of knowledge base. When the process is not decomposed, the number of rules representing the FET is 2337, and the number of items (nodes in the FET) in the rules is 19 413 in the knowledge base. With process decomposition, the numbers can be reduced to 658 and 1888, respectively. Thus, the knowledge base is reduced to one-tenth of its original size by system decomposition. In the system, a plus cycle of GVFMQ and GVMFS is possible. The evaluation order of subpaths is determined as follows. (1) The gate variable to gate variable subpaths of each

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Figure 11. Gate variables invoked in the boiler plant. Figure 9. Subpaths from gate variables to gate variables of the boiler plant.

Figure 10. Process schematics of a steam distribution unit.

gate variable in the cycle are listed as follows.

GVFMQ(+): sGVMFS(+) GVMFS(+): sGVFMQ(+) (2) The GVFMQ(+) to GVMFS(+) subpath is arbitrarily chosen and becomes the first evaluation subpath in the cycle. The next subpath is the subpath of GVMFS(+) to GVFMQ(+). Through the steps, the cycle evaluation list is obtained as

{GVFMQ(+) to GVMFS(+),GVMFS(+) to GVFMQ(+)} (3) If GVFMQ(+) to GVMFS(+) is evaluated, then GVMFS(+) to GVFMQ(+) can be evaluated using the NCG and is removed from the list. The same steps are used in the cycle where GVFMQ and GVMFS have a negative sign. Example: Steam Demand Increase (HP-FCH) and Pressure Sensor Bias High (PI2005-BH). The steam distribution unit of the target plant consists of four headers; each header has a different level of pressure (Figure 10). The steam from each boiler is sent to the main steam header (89 bar), produces electricity by passing through a turbine, and is sent to the highpressure header (41 bar), where steam is distributed to each steam-consuming plant. The steam sent to the middle-pressure header is also distributed to each plant. Three pressure sensors are used to measure the pressure of the high-pressure header because it is the most important controlled variable.

In this example, the steam demand of a steamconsuming plant connected to the high-pressure header is increased at 120 s (HP-FCH). After 10 s, two of the three pressure sensors measuring the pressure of highpressure header are biased high (PI2005-BH). As the middle value of the three measured pressures is used for pressure control, one sensor malfunction is considered to be an independent sensor fault. The required data set is obtained by dynamic simulation of the plant.17 The diagnosis procedure will be explained with the data for 190 s. (1) Faults and gate variables are invoked by the evaluation of basic symptoms. The invoked gate variables are shown in Figure 11, and the NCGs of the invoked gate variables are

GVMDT(-): 0 GVFMQ(+): 3 GVMWP(-): 0 GVMDF(+): 0 GVMDF(-): 1 GVMFS(+): 3 GVDMF(+): 2 GVWMF(-): 0 GVDFS(+): 2 GVFAQ(+): 0 GVAFS(+): 0 (2) The subpaths from the invoked faults and gate variables to subpatterns are evaluated. (3) Because the NCGs of GVMWP(-), GVMDF(+), GVFAQ(+), GVAFS(+) are 0, the following subpaths can be evaluated:

GVFMQ(+) to GVMWP(-), GVDMF(+) to GVMWP(-), GVFMQ(+) to GVMDF(+), GVDFS(+) to GVFAQ(+), GVMFS(+) to GVFAQ(+), GVMFS(+) to GVAFS(+) Now, the following result is obtained.

GVFMQ(+): 1 GVMDF(-): 1 GVMFS(+): 1 GVDMF(+): 1 GVDFS(+): 1 (4) Because GVFMQ(+) and GVMFS(+) are invoked, it is judged that the cycle is developed. The GVFMQ(+) to GVMFS(+) subpath is forced to be evaluated, and the NCG of GVFMQ(+) becomes 0.

GVFMQ(+): 0 GVMDF(-): 1 GVMFS(+): 1 GVDMF(+): 1 GVDFS(+): 1 (5) Because the NCG of GVFMQ(+) is 0, the subpaths of GVDFS(+) to GVFMQ(+) and GVMFS(+) to GVFMQ(+) can be evaluated.

GVMDF(-): 1 GVMFS(+): 0 GVDMF(+): 1 GVDFS(+): 0

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Figure 12. FLs and ESP numbers of the fault candidates in the example. Table 4. Diagnosis Result of the Multiple-Fault Example fault candidate

ESP no.

FL

HP-FCH PI2005-BH PV2005-BL PC2006-BL HP-FCH and PI2005-BH

12 12 10 10 13

0.739 0.688 0.713 0.500 0.808

(6) After the subpaths of GVDMF(+) to GVMFS(+) and GVMDF(-) to GVDFS(+) are evaluated, all gate variables have NCGs of 0. Therefore, the subpaths from faults to gate variables can be evaluated. The numbers of explained subpatterns (ESP numbers) and FLs of the primary fault candidates at 190 s are shown in Table 4. True single faults show the largest values of (FL × ESP number) among the primary fault candidates. After the search for multiple faults, one multiple-fault candidate is obtained (Table 4), and other combinations of single faults do not show better results than single faults. Therefore, the diagnosis gives the true solution. Figure 12 gives the FL and the number of explained subpatterns for each single fault and multiple fault. The dotted line in Figure 12 is the number of variable clusters that have the detected subpatterns. HP-FCH is diagnosed after 5 s from the time when the first fault (HP-FCH) occurs, and the second fault (PI2005-BH) is detected at 130 s. Although the system does not miss two true single faults during the whole diagnosis time, the true solution of a multiple fault is missed during the range 155-175 s. This is because the first single fault can explain all of the symptoms from the second single fault for 25 s (masked multiple fault).14 Conclusion Our previous study resolved the problem of diagnosis failure under uncertain conditions and expanded the diagnosis area of signed digraphs to multiple faults.14

In this study, the method of system decomposition for qualitative fault diagnosis is discussed to facilitate the development of a diagnosis system for large-scale chemical processes. Through process decomposition, the following advantages can be obtained: flexible diagnosis throughout operational condition changes, size reduction of the knowledge base, ease in understanding of complex knowledge bases with process interactions, and consistent and reliable construction and maintenance of the knowledge base. The advantages are successfully shown by the fault diagnosis system of a boiler plant. However, the FETbased method cannot diagnose masked multiple faults as described in the previous paper.14 Presently, we are studying a hybrid diagnosis model using FETs and other models to diagnose masked multiple faults. Acknowledgment This work was partially supported by the BK 21 Program supported by the Ministry of Education and the National Research Lab Grant of the Ministry of Science & Technology. Notation C ) controller output D ) denominator ESP ) set of explained subpatterns F ) fault FL ) fault likelihood G ) gate variable I ) indicator L ) tank level LC ) level controller LT ) level transmitter N ) nominator P ) pressure P ) sequence of subpatterns in the equation used to calculate the path likelihood

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PC ) pressure controller PT ) pressure transmitter SA ) logical function of subsystem activation SL ) subpattern likelihood SP ) subpattern SS ) subsystem V ) control valve VC ) variable cluster Subscripts L ) level O ) output P ) pressure S ) sensor

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Received for review January 31, 2000 Accepted February 28, 2001 IE0001366