Ind. Eng. Chem. Res. 1994,33, 1943-1954
1943
Quantifying Signed Directed Graphs with the Fuzzy Set for Fault Diagnosis Resolution Improvement Chung-cheng Han, Ruey-fu Shih, a n d Liang-sun Lee' Department of Chemical Engineering, National Central University, Chung-li, Taiwan, 3254 ROC
The fault diagnosis method of signed directed graphs (SDGs) has difficulties verifying the seriousness of an abnormal process variable and providing an accurate resolution of fault origin, when the abnormal variable is in the neighborhood of the designed threshold. In this paper, the fuzzy set theory was incorporated into SDGs to overcome these obstacles. This proposed algorithm begins with an SDG representing the qualitative cause-effect relations among the process variables. The primarily formulated SDG is then simplified by edge consistency and the combination of unmeasured variables, an acyclic graph consisting of strongly connected components was formed, and then the technique of the depth-first search determined the specific strong component where the possible fault origin was located. Finally, the quantitative fuzzy set manipulation was introduced, the degree of membership of the process variables was determined, and then the variables were sequentially arranged by their degree of membership to determine the possible fault origins. A process consisting of a jacketed CSTR and a vaporizer with a designated plugged product valve was diagnosed by the proposed approach. The result shows that the proposed approach improves the accuracy of resolution and provides more valuable information on the arrangement of fault origin candidates by the seriousness of the abnormality of the variables. Introduction The product quality of a chemical plant is highly dependent on the process variables which are tactically controlled in an allowable range. The situation in which the process variable, such as temperature, pressure, or mass flow rate, deviates from the normal value but remains within the allowable deviation is called "process fault". A process fault may be the incipient cause of process failure; for example, the leakage of a cooling water pipe causes the increase of the reactor temperature, then the decrease of feed supply, and worse than that, the interruption of plant operation. Sometimes, a process fault causes hazardous gas leakage, fire explosion, and even more serious catastrophes. In a chemical plant, a process fault may alert the operators by the alarms or by the indicators on the control panel. The plant operators will judge and react immediately to find out the cause and location of the process fault by his experience and knowledge. The location of the fault is called the "fault origin". In the modern chemical industries, the chemical plants are always designed to be larger and more complex than before. The engineers' knowledge, experience, and training alertness are still not capable enough to handle the emergent situations due to the complexity of the plant; thus, an automatic diagnosis of fault is worth studying for assisting plant engineers in handling the emergency. Fault diagnosis has been studied for more than a decade, (Himmelblau, 1978;Pau, 1981); since then, many different approaches have been proposed. By theoretical aspect, they can be fitted into two categories: qualitative and quantitative (Kramer, 1987; Yu, 1991). The quantitative approach includes the Kalman fdter methods of parameter estimation (Willsky, 1976; Himmelblau,l978; Isermann, 1984). These methods require a rigorous process model and countless measurements to collect extensive process data for parameter estimation. The qualitative approaches include fault tree (Lees, 19831, event tree, and signed directed graph (Iri et al., 1979; Lees, 1983; Tsuge et al.,
* To whom all correspondence should be addressed. oaaa-~aa5194/ 2633-1943$04.50/0
1985; Shiozaki et al., 1985). The fault tree analysis is a composite of deep knowledge and shallow knowledge in a tree-shaped structure. The shallow knowledge representing the relation between system behavior and system fault is obtainedempirically by experience; while the deep knowledge is rigorous, as represented by the mathematical model of the process. Since the chemical processes always involve quite a few control loops and recycle streams, the fault tree analysis is not suitable for on-line fault diagnosis of transient process behavior, but it is always used for hazard evaluation attributed to control system failure (Radoux,1986). Another important qualitative approach based on the usage of process information is signed directed graph (SDG).The SDG is constructed with a vertex (or node) representing the process variable and an edge (or link) representing the qualitative cause-effect from one vertex to another with the sign of gain. The SDG approach usually utilizes the qualitative influence between process variables and depth-first search (Tarjan, 1972) to determine the fault propagation path and possible fault origin candidates (Iri et al., 1979; Shiozaki et al., 1985). Unfortunately, the resolution is usually not accurate enough that the advantages, fast manipulation and a nonheuristic basis, of the SDG approach are not so well recognized in practical use. In order to improve the resolution of this approach more information such as time delay collected from transient behavior is used. (Umedaet al., 1980, Tsuge et al., 1985; Hwang and Lee, 1991). The expert system, also belonging to the qualitative approach has been studied (Kumamotoet al., 1984;Andow, 1985; Shum et al., 1988; Ramesh et al., 1988; Lapointe et al., 1989; Calandranis et al., 1990; Macdowell et al., 1991). The expert systems provide rapid manipulation and accurate resolution, especially for the experienced process faults since the events were compiled in the knowledge bank. However, it has the shortcomings of a lack of selflearning ability and of an incomplete knowledge base since a complete experience is not possible. The construction of an expert system is essentially based on past experience, so that an expert system for a complete new plant will be developed from scratch. A modular-wise expert system similar to a process simulator for the chemical plants is 0 1994 American Chemical Society
1944 Ind. Eng. Chem. Res., Vol. 33, No. 8,1994
almost impossible to accomplish. Usually, there are more than 10 thousand rules in an expert system for a moderate size chemical plant. However, we may say that a process that is difficult to be described mathematically may suitably be diagnosed by an expert system. The SDG approach has not been completely developed in the manipulation aspect. Due to the fast manipulation and the only requirement being qualitative information, it has been the focus of many researchers in fault diagnosis study. In order to improve the practicality of the original SDG approach Kramer and Palowitch (1987) bridged the SDG and the expert system by proposing the idea of transforming qualitative information into rule-based formulations, and in order to improve the resolution of diagnosis, the quantitative concept is also applied, but it was limited to the case of a single transition which means the sign of the deviated state of a process variable is unvarying during the diagnosis. Oyeleye and Kramer (1988) applied the stability criteria and the qualitative concept of De Kleer and Brown (1984) to the steady-state process to explain the compensatory and the inverse response of a dynamic feedback loop when multiple transition occurs. In practice, the quantitative information of the process variables shown either on the control panel or at the plant site are always very useful for diagnosis in addition to the qualitative knowledge. This has been tried by Shiozaki et al. (1985), Kokawa et al. (1983), and Kramer (1987). When a plant is under the situation of fault, there exists uncertainty such that the operators might hesitate to decide what would be the proper response. For instance, the measurement noises cause the processvariables deviate from the true values, or the seriousness of the fault is so crucial that emergent action should be taken. To manipulate such uncertainties, the fuzzy set theory is considered very suitable and convenient. Qian (1990)used the fuzzy set to describe fault propagation. Yu and Lee (1991), based on the SDG approach with rule-based interpretation, applied fuzzy numbers to process variables and gains to improve the resolution. In this paper, instead of rule-based consideration, the SDG is constructed to represent the process knowledge, edge consistency is considered to simplify the primary SDG ,and the SDG is partitioned into an acyclic diagram of strongly connected components. In addition, the fuzzy set is applied to represent the deviated states of the procesa variables quantitatively. The final diagnosis is obtained, and it is found that the resolution of the original SDG approach can be improved as discussed in the following sections. This direct application of the fuzzy set manipulation to SDGs is the first attempt in fault diagnosis research. Signed Directed Graph (SDG)
An SDG is constructed to represent the cause-effect relations among the chemical process variables and to exhibit the process behavior. The vertex (node) in a SDG represents the process variable, and the edge (link) represents the qualitative influence of a process variable on the related variable (see Figure 1). We may define a SDG as MG = (V, E, A, A), where V is the vertex set, E is called the signed directed edge set or link influence function, A: E {+,-I, represents the forward influence in the direction of an edge, e.g., A(ek) = + means edge ek is a positive influence in the edge direction, and A is the
-
S t a t u s of variable(vertex): O(normal), +(high), - ( l o w ) Sign of influence( edge): ~ ( e ~ , ,=d -), A ( e x ~ , x ~ ) + .
Figure 1. Signed directed graph.
fault manner of a vertex explained as follows
Ixui- Zuj < e,, then A(ui) = “0” xui- Qui > eUi then A(ui) = “+” xui- lui> eUi then A(uJ = u-n
(1)
where Zvi is the set point of the process variable ui , cui is the allowable deviation of variable i. The above representation of the fault manner is called a three-ranges fault pattern, and the search for the fault origin in this fault pattern is called three-value logic gate manipulation. The sign of an edge is usually determined by experience or by the mathematical relation between two relevant variables, e.g. dxi
dt = fi(q,xZ,*..,xn)
(2)
xi = fi(xl,xz,...,xn)
(3)
or
if
‘fi.
> 0, then A(ek) = +; if
= I
< 0, then A(ek) = -.
Since the accuracy of the measurements in a chemical process is affected by environmental noises and disturbances, it is obvious that there exists uncertainty in the measurements. In addition, the inaccurate mathematical model and the performance degradation of the chemical plant will cause difficulties in estimating the process variables and the parameters correctly. Thus, the fault diagnosis either quantitative or qualitative has this inherent uncertainty, if we merely considerthe discrepancy between the designed and the estimated values. Fortunately, the mathematical uncertainty can be described by the dynamic randomness and the fuzzy concept. The former is used to describe the randomly distributed deviation of the process variables from the normal values, while the latter is used to quantitatively express the variables in a range. A fuzzy value always provides both quantitative and qualitative information but cannot be treated statistically. For example, how does an operator adjust the cooling water during the emergent situation if he is not certain of the seriousness of the reactor temperature being in the neighborhood of the allowable limit, and how much confidence does he have in his response. To handle the problem of confidence, the fuzzy set theory is very helpful. In the following, we suggest the application of the degree of membership of fuzzy set theory to the confidence measurement for fault diagnosis. Fuzzy Set Theory and Fuzzy Measurement
The fuzzy set theory was developed by Zadeh (1965) for dealing with the inaccuracy of human expression. For
Ind. Eng. Chem. Res., Vol. 33, No. 8,1994 1945 instance, the sentence “Itis very hot today” is only a vague expression and is difficult to expressmathematically,since it has different physical feeling to different people. The fuzzy set theory, distinguished from the precise mathematical expression, is suitable to the event without distinct bounds. It defines an interval or assigns a unique value to a degree of membership for the mathematical description similar to a precise numerical value. For instance, the phrase “very hot” represents a temperature in the interval 130 “C, 40 “C] or that the possibility of a value over 34 “C is 0.90. Although this expression is more or less subjective, it is a very useful expression to balance the vague variables in system modeling, qualitative analysis, and simulation. Some definitions related to the fuzzy set theory to be used in this paper are given here for reference. The objects to be measured are defined as the elements of a population, and the measurement outcome of an element is defined as its characteristic value. Definition 1. Let Urepresent the universe of discourse, R 3 U, and let X represent the elements of a population with the characteristic value x , x f R, then the fuzzy set is defined as
where ph is the characteristic function of x , i@ so-called membership function or belongingness, in set A. It gives the characteristic value an exact numerical value. Relationship 4 implies that X is mapped to a membership space M through subregion x . If M E {0,1)then it is nonfuzzy and belongs to the conventional Boolean set. Zadeh (1978) defined M E [0,11; 0 and 1 represent nonbelongingness and complete belongingness, respectively. In this paper, the set of elements of the population X is the set of process variables or parameters such as temperature, pressure, time constant, etc. The set points of process variables are denoted as a. The membership function denoted as pi@) represents the “closeness of x to 5 or belongingness degree of x to 5”. In 1970, Sugeno defined the fuzzy measurement as follows (Pedrycz, 1989). Definition 2. Any set X , R 3 X , and a set B represent the Bor.el field of X . If a set function g satisfies the following conditions, then g is a fuzzy measurement. 1.
g(4) = 0
3.
if A, B E R, B C A, then g(A) 1 g(B)
as
and with property of *(
n--
n-m
Conditions 1 and 2 mean that the set function g is nonnegative and bounded; condition 3 implies that the fuzzy measurement must be continuous and monotonous. If X is a finite set, then condition 4 may be neglected. Zadeh (1978)suggested that the possibility distribution function, the possibility measurement, and the probability measurement are useful to explain systems with inadequate of fuzzy behavior and_dynamic randomness. Definition 3. If X 2 B, A is a f p z y %et,and x f B, then the possibility measurementof A, *(A),is expressed
(7)
here p(AJ represents p ~ , ( x )I, is an integer set
X >A,, i E I Equation 6 means that the element x in X belongs to the possibility measurement of fuzzy set A. ?rz(x) is the possibility distribution function of x defiied in the interval [0,11. Equation 7 is an important property for possibility measurement; it gives the subset to be measured. In this paper, the possibility measurement of the fault origin will be the subset of the largest possibility. The eligible fault origin candidates, which should remain in the candidate set during the diagnosis process, can be determined by this property. It is worth noting that a possibility measurement is not necessarily a fuzzy one. But they are viewed to be identical if X is a finite set and its distribution is normal (Puri et al., 1982) and the corresponding region is the interval [0, 11. In this paper, the possibility measurement is applied to represent the fuzzy measurement of the confidence measurement of either normal or abnormal process variables. And the belongingnessfunction of a fuzzy set, as defined in the following, is used to represent the possibility distribution function of the process variables. Definition 4. Let the residual be R = x - xsp Assume R is a normal distribution iV(li.,s),and the conventionally used deviation of a process variable, Th, is represented by the normal distribution tolerance, then xh
= xspf K8
(8)
where R is the residual and xspis the set point of a process variable. s is the standard deviation of that variable, K is a restriction parameter. We use a fuzzy set which has three elements, N (negative), Z (zero), and P (positive), to describe the corresponding deviated states, “too low”, “normal”, and “too high”, of the process variables, respectively. The membership degree of each element is used to indicate the quantitative value of the expression. These elements are the subsets of the fuzzy set so that the membership functions of the subsets are defined as 0
4. if A , E B, n E N, n is a monotonous sequence, then
lim g(An)= g(1im A,)
UAi)= sup(pL(Ai),i E 4
P&)
=
- x-2h)/(”sp - x-2h))’ ((x2h - x)/(x2h - xsp))’ ((x
0
when x I X-Zh when ’-Zh when ’Sp when x 1 23,
’2h
1946 Ind. Eng. Chem. Res., Vol. 33, No. 8, 1994 Table 1. Truth Table
TLN,
MN I SN
SP I
MP
1
LP
ek'
N
c
Iz
I
P
Iz
I
P
ek-
0 C M
n
Observed v a r i a b l e v a l u e Figure 2. Membership function. pm(x)
= 1- & ( X ) p&)
when Zap 2 X
> X-Zh
= 0 when x L xBp
(14) (15)
ek'
N
where x-h
= zap- KS
(16)
X2h
= XBP 2KS
+
(17)
x-ph
= xap- 2KS
(18)
And where p&), p&), and pp(x) are the degree of membership of x in the fuzzy subsets, N, Z, and P, respectively. The variable's deviation is greater than the upper limit, X2h, when pp(x) = 1,and if it is less than the lower limit, x - a , then p&) = 1. Variable deviation beyond these limits is considered in fault situation. The integral power, j3, is used to define the curve shape of the membership function that is thin or wide. Here, we select unity for the reason of simplicity since we are mainly demonstrating the concept of fuzzy set application. One may use a different value for his own purpose. The membership functions of the fuzzy subsets, N, Z, and P, indicating the seriousness of the deviation, defined in this paper, are shown in Figure 2. We can see that the deviated state of each measured variable is mapped into three different membership functions; the deviated state shall no longer be hard-partitioned as too low, normal, and too high, but it is indicated with all possible directions of deviation, N, Z, and P, simultaneously by the fuzzy subsets with grade of membership. Such mapping is purposely designed to handle the deviation of a variable, it gives the faulty information to the engineer so that the seriousness of the faulty variable may be judged. In order to judge the fault seriousness of each measured variable conveniently,the deviated state is divided to seven grades, LN, MN, SN, Z, SP, MP, and LP, though a fuzzy set which has N (negative), Z (zero), and P (positive) has been used to represent the states of variables in this work. The grades of LN, MN, and SN indicate the negative deviation of variables, the grades of LP, MP, and LP indicate the positive deviation of variables. SP and SN are the grades of possible fault; MP and MN are the grades of warning; LP and LN are the grades of faulty variables; and Z is that of variables at normal condition. It is obvious that a vertex has three different memberships of fuzzy subsets at the sampling time by fuzzy set manipulation. The direction of the deviated variable is taken from one of the subsets which is the largest degree of membership, even these variables are still in the tolerance limit.
Table 1 is a truth table of possible fault propagation. With this table the fault propagation path and the consistency test proceed easily. The integral power, j3,in the previous equations (9-16) of the membership function will affect the possibility of a process variable being included in the set of fault candidates or not. If j3 increases,the membership function becomes steeper. And when the process variables are close to the tolerance limits, the dilute effect (weight of inconfidence) of the membership is very obvious. If j3 = 0, the installation of a monitoring system is in vain due to the possibility that process variables in the fuzzy subsets of P, Z, and N are unity. Larger values of 6 will cause the analysis approach to use Boolean logic, such as three-values logic gate manipulation. In this paper 6 is chosen to be unity, representing a linear relation. In the following, we will address the confidence of fault origin diagnosis by SDG and fuzzy set manipulation on the degree of membership of the eligible fault origin candidates. Before, the process variables were represented by the fuzzy membership functions; let the relation of the fault origin candidate Vj and its descendent vertices Vi, i = 1, 2, ..., nj, be written as
where nj is the number of descendent vertices after vertex Vj, Conventionally, the above relation is a part of the mathematical equation of the energy balance or the material balance for quantitatively justifying the abnormality of vertex Vj. It is possible to consider this in terms
Ind. Eng. Chem. Res., Vol. 33, NO. 8, 1994 1947 of the max-min manipulation if the process variables are represented by fuzzy subsets, then the possibility of fault (fault grade) of a variable is represented by the membership function. For this purpose, we define a credible failure membership, CFj, of vertex Vj as
i = 1, ..., nj
CF, = Min(ppj(Vi))
simplified V.
v*
VI
-ckc-E A(e..) A(e,a)
A(e,,) .A(elr) .
= A(e,,) .A(etl).
. . . h(ek4.k)
...
A(ekb)
(20)
where pp(Vj) is the membership of vertex Vj. This definition gives the lowest degree of membership, CFj, of vertex Vj subject to whether it would stay in the set of fault origin candidates. If CFj 0 the vertex Vj should be deleted from the set of fault origin candidates even if pfi(Vj) 1,since the degree of membership of vertex Vj does not exist. Thus, in the case of fault origin candidates with identical degrees of membership, we may estimate credible failure memberships, indicating the possibility of it being a fault origin, and arrange the candidates by their credible failure memberships.
-
original
+
Algorithm of the Proposed Approach The fault diagnosis algorithm is aimed at locating the fault origin early enough to provide plant operators the necessary time to correct the abnormality before plant runaway. It is expected that the diagnosis be reliable and that the erroneous and the spurious resolutions not be included in the fault origin candidates. A chemical process always involves the following chemical reactions: mass transfer, heat transfer, and thermodynamic equilibrium. It is so complicated that diagnosis is not a easy task. However, a chemical process is characterized by the following features: (1) Process variables are nonlinearlyinterdependent and time variant. (2) Statesof process variables could be transient, and even the abnormality may be multitransition, so that the steadystateapproach usually considered by the researchersmight lead to an erroneousresolution. (3)Recycles in the process and feedback action in the control loops make the diagnosis more difficult by sequential or acyclic analysis (Kramer and Palowitch, 1987;Kokawa et al., 1983). (4)Time delay caused alarm actuated with lag, especially when the fault was near the threshold (Iserman, 1984). It is obvious that the alarm threshold and the measurement noises may affect the actuation of alarms. Usually, anarrow threshold causes the alarm trigger to be more sensitive. ( 5 ) Not all the process variables are measured. Thus, for many approaches,to verify the fault propagation path, the states of the unmeasured variables have to be guessed at the beginning of diagnosis. Before addressing our algorithm, we should mention some assumptions that were imposed (1)Only one fault origin exists in the process; but in case that multiple fault origins appear in the resolution, they should be carefully examined for the possibility of being a true fault origin by the magnitude of degree of membership of each candidate. (2) All sensors function properly. (3) The deviated states of the process variables are invariant at each sampling time. (4)The effect of time delay among process variables is not considered in this study. The proposed algorithm,generating a set of eligiblefault origin candidates with degrees of membership ranked for the operator's reference, is described step by step below. Step 0. Construct a complete SDG, called the primary SDG, for the process under consideration. Step 1. Collect information about process variables either from the computer simulation output or by direct measurement. Calculate the fuzzy degree of membership and the belonging area of each measured variable and
= A(e,). A ( e s ) . A ( e , , ) 'd(ee,,o) = A ( e , ) . A ( e , ) . A ( e , a )
Figure 3. Combining unmeasured vertices to simplify the SDG.
construct the fault model by fuzzy three-value logic gate manipulation. Step 2. Simplify the SDG by (1)deleting the vertices which do not satisfy edge consistency, (2) combining the consecutive unmeasured vertices along the unbranched paths. This combination is accomplished by (a) replacing vertices u1, u2, ..., and Uk, along an unbranched path by a single vertex (b) The sign of the new edge is determined by n(elg)*h(e2,3). A ( e k - 1 ~ ) Three . cases of combination are illustrated in Figure 3. Step 3. Partition the simplified SDG into a graph consisting of strongly connected components (Iri et al., 1979; Swamy, 1981) by the method of depth-first search (Tarjan, 1972). Such a graph is a rooted tree with the strongly connected components as vertices and without any cycle between any two of them. Step 4. Arrange the variables sequentiallyby the degree of membership of each variable (each variable is mapped to three membership functions, and the largest one should be crucial) and, beginning from the strongly connected component containing the vertex of the largest degree of membership, search for the common ancestor of all strongly connected components. The vertex with the largest degree of membership in this common ancestor should be in the fault region P or N. Step 5. Construct and verify the fault propagation path for each fault origin candidate within the strongly connected component which was determined in the previous step via the followingconsiderations: (1)Starting from an arbitrary fault origin candidate, protrude the fault propagation path step by step by the depth-first search method. (2)For the unmeasured variables, assume a fuzzy subset of fault (P, Z, or N) and ensure the satisfaction of edge consistencyby the truth table. (3)If the fault propagation path includes all the annunciated variables, this fault pattern is recorded in a file; assign this starting vertex as an eligible candidate. If this starting vertex does not belong to any fuzzy subset, change the fuzzy subset of this candidate and return to step 5.2). (4) If the fault propagation path cannot proceed further before all the annunciated variables are included along the path, go back to the variable just ahead of the vertex where the path ends, and step forward again in a different direction. ( 5 ) If the unmeasured variables with the designated signs cannot satisfy the edge consistency, then change the sign of the previous unmeasured process variable and proceed from the beginning again. (6) If the protrusion is terminated at the chosen beginning vertex and all the
....
1948 Ind. Eng. Chem. Res., Vol. 33, No. 8, 1994
L
PB
PT
FH THO
Figure 4. CSTR and vaporizer example.
process variables are associated with a fuzzy membership subset, return to step 5.1) and proceed along another fault propagation path from another fault origin candidate. Step 6. Arrange a sequence of fault possibility memberships by the degree of membership of the eligible fault origin candidates. If there are more than one candidate with identical degrees of membership, then the sequence can be arranged again by the credible failure membership or by the credible fuzzy failure probability.
Illustrative Example In order to illustrate the application of the proposed algorithm employing the fuzzy theory on fault diagnosis, a process consistingof a jacketed reactor and an evaporator with three controllers, shown in Figure 4, is considered. The dynamic behavior of this process is simulated and executed on a personal computer. The dynamic values of the process variables are given in the computer output when a fault is imposed on this process purposely for illustration. The reactor temperature is controlled by cooling water taking away exothermic heat. About 10 96 of the product stream is recycled into the reactor, and the evaporator liquid level is controlled by the propane feed flow. The SDG for this process is shown in Figure 5. The sensors are represented by squares distinct from the process variables, so that the malfunction of sensors can be distinguished from the process variable faults for the reason of fault diagnosis. Every controller is divided into two parts; for example, the liquid level controller of the evaporator is separated into a liquid level recorder (LLR) and a level control output (MVL). This will avoid erroneous diagnosisattributed to incorrect measurements. When plant operation is abnormal, the measurements of the process variables deviate beyond the preset thresholds and the alarms may be triggered; the diagnosis system starts to work following the proposed procedure. Meanwhile, the sampling time and the measurements are recorded. The sampling time interval is denoted as T'. In this example, the fault origin is set purposely at the product outlet valve (VH) which is assumed to be stuck suddenly. The dynamic values of the measured variables corresponding to this fault are shown in Figures 6-14. The
Figure 5. SDG for the CSTR and vaporizer example. 1003
99'
t
997
Sampling time (second)
Figure 6. Downstream pressure (PHO)of vaporizer pressurecontrol valve.
diagnosis exhibits the abnormalities of the measured variables at different times as shown in Table 2. In order to show the difference, each diagnosis step is compared with the conventional diagnosis method of the nonfuzzy three-value logic gate approach.
Ind. Eng. Chem. Res., Vol. 33, No. 8,1994 1949 Table 2. Historical Deviation Data of the Measured Variables in Fuzzy Sets* time (8) L T LL PP F1 FR FH w1 1 7002 "0.99 n0.93 p0.97 ,0.71 n0.94 p91 p91 .0.82 i0.97 2 7007 .OB8 nO.86 n0.97 "0.92 J.00 ,0.79 ,0.99 .0.98 3 7012 ,0.93 J.00 nO.85 p0.96 .0.85 4 7017 ,0.73 p0.70 .0.69 ,0.78 p0.92 ,0.97 .0b7 5 7023 ,0.98 .0.80 ,,OB2 n0.77 .0.87 ,034 p0.76 6 7028 .0.98 .0.95 ,0.87 ,0.88 ,0.94 ,0.67 ,0.93 7 ,0.92 7033 ,0.95 .0.81 p0.55 .0.73 .0.62 ,0.92 8 ,0.89 7038 ,0.94 ,0.76 ,0.88 p90 ,,0.94 7043 ,0.65 9 p0.93 p0.94 p0.87 .0.82 ,0.98 7048 10 .0.78 ,O.Sl p0.92 p0.98 ,0.72 .0.99 11 ,035 7053 J.00 .0.94 ,0.79 .0.94 P73 ,0.90 12 ,0.92 7058 ,0.98 .0.82 nO.00 p0.86 .0.92 .0.91 13 ,0.69 7063 ,OB9 p0.94 *n0.32 .0.93 *.0.00 .0.91 14 .0.74 7068 .OB5 ,0.94 .0.00 ,0.78 .O.OO .0.91 7074 .0.57 15 ,0.79 n0.67 nO.00 ,0.93 nO.00 ,O.W 7079 16 00.87 ,031 .0.46 P78 nO.00 .0.00 ,0.90 17 7084 *n0.26 nO.84 ,0.87 .0.75 .0.00 n0.00 ,0.73 7089 18 p0.89 .0.94 no.09 p0.93 nO.00 no-00 ,0.92 7094 19 ,0.92 nO.OO ,0.68 nO.OO ,0.76 nO.00 n0.89 7099 20 .0.91 ,0.72 .0.00 .0.00 ,0.76 .0.00 n0.88 .0.80 21 7104 .0.00 .0.00 ,OS2 .0.00 ,0.88 p0.77 7109 22 ,0.64 ,0.00 ,0.00 ,0.88 p0.69 nO.00 p0.97 ,0.70 7114 23 .0.85 ,0.00 nO.OO ,0.69 .0.00 ,0.81 24 P98 7119 ,030 .0.00 .0.00 nO.OO ,0.76 .0.98 7125 25 *p0.49 .0.00 .0.00 ,0.91 .0.00 *p0.31 7130 26 ,0.59 .0.00 .0.80 .0.00 ,0.93 .0.00 ,0.71 27 ,0.57 7135 ,0.84 .0.00 nO.00 *p0.48 .0.00 .0.71 28 ,0.63 7140 ,0.50 .0.00 .0.00 ,0.84 .0.00 p0.78 a p, positive fuzzy subset; z, zero fuzzy subset; n, negative fuzzy subset. *, measured variables beyond tolerable limit.
PHO
no.
4.1
2
n0.79 p0.76 .0.83 J.00 .0.67 n0.94 ,0.75 ,068 .0.91 ,0.93 p0.88 p0.81 ,0.88 ,0.91 ,0.52 n0.81 ,090 ,0.76 ,0.73 ,0.49 ,0.76 P79 p0.79 ,0.73 *p0.46 ,0.68
0.011
0.010
h
0.008
E
v
2 3 9
0.006
L
0.004
0.001
3.6 3.5
t I
0.000
-0.002 4
7000
7100
Roo
7300
7400
7500
7600
Sampling time (second) Figure 7. Propane flow rata (Wl) response.
Sampling time (second) Figure 8. CSTR outlet flow rata (FH) response.
The very detailed diagnosis procedure is described as follows. Fault situation: At the 7060th second, the no. 7 alarm (product flow rate) sounds, the measured variable no. 5 (feed flow rate) exceeds the preset threshold. The fault diagnosis system starts to react. In the first diagnosis stage the algorithm will (1) Calculate the fuzzy subset membershipof the measured variables by the fuzzy threevalue logic gate manipulation and determine the fault belonging area. Contained in Tables 2 and 3 are the estimated degrees of membership of process variables at different sampling times and different stages respectively. (2) Delete the vertices with an inconsistent fault propagation edge, this can be done by checking with the fault propagation consistency true value table, and combine the consecutive unmeasured vertices along the unbranched path as shown in Figure 15. (3) Partition this simplified SDG into a graph of strongly connected components as in Figure 16. Notice that the vertices in a dashed-line block form a strongly connected component, and those vertices
with a single edge are also strongly connected components themselves. (4) Begin from the vertex FH, which has the largest degree of membership in fuzzy subset, and search backward to find the common ancestor component of all strongly connected componente. Since no such component preceding that containing the vertex FH has been annunciated (variables in the fault belonging area LP, MP, MN, or LN), and the strongly connected component containing FH has a complete belongingness fault subset (e.g., FH E fuzzy set N and l(FH) = l.O), this strongly connected component is the common ancestor of the rest of the other strongly connectedComponents. The vertices in this strongly connected component, {MP,VH, FH, PT, PPU, F2, PB, FR, L), are the fault origin candidates. (5) Select the eligible fault origin candidates by the consistency test of all fault origin candidates obtained in step 4, along the possible fault propagation path. Figure 17 shows the fault propagation path originating from the candidates obtained in the previous step. The eligible fault origin candidates are determined as MP, VH, FH, PPU, F2, and
1950 Ind. Eng. Chem. Res., Vol. 33, No. 8, 1994 3.200
0.7100 0.7750
3.1JO
0.7700 3.100
h
n
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v
v.050 ,0.7600
*2:* Q)
5 .e
e, 3.000 E
1w.75~0
2
3
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2950
0.7450 2900
' I
0.7400
2850 0.7350 0.7300 6900
2800 7000
7100
7200
7300
7400
7500
Sampling time (second)
Sampling time (second)
Figure 12. Vaporizer level (LL).
Figure 9. CSTR recycle flow rate (FR) response. 0.012
1
-0.002
6900
7000
7100
Roo
-
I 7300
7400
7500
6900
7600
7000
7.3
Roo
7300
7400
7500
7600
Figure 13. CSTR temperature (T).
Figure 10. CSTR inlet flow rate (Fl)response. 7.4
7100
Sampling time (second)
Sampling time (second)
, 1
1.325
7.2 7.1
1.320
-
n
E
- 7 -
E * 6.9 v 2 6.1 -
W
E M
.-
Q)
1
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2 6.7 2 -
3
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& 6.6
6.5
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6.4
-
6.3
-
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el
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7300
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7500
7
1.300 6900
7000
7100
7200
7300
7400
7500
7600
Sampling time (second) Figure 14. CSTR level (L).
candidates have the same priority of being fault origin candidates except variable FH. This completes the first stage of diagnosis with the first information set when the fault appeared. We can see that by the conventional nonfuzzy approach the stateof variable L is still judged normally since ita deviation stays within the preset threshold. Since the edge consistency does not determine any eligible fault origincandidates,the diagnosis is a failure.
Ind. Eng. Chem. Res., Vol. 33, No. 8,1994 1951 Table 3. Fuzzy Manipulations of the Measured Variables at Different Stages of the Illustrative Example Fault Origin, VH stage variable
L T
LL PP F1 FR
FH
w1
PHO
1 0.93, 0.69, 0.89, 0.94, -0.68 0.93, -1.00 0.91, 0.93,
2 +0.78 0.84, 0.87, -0.74
3 +0.51 0.86,
4
0.84,
0.75.
0.91,
+0.52
0.73, 0.52,
0.69 0.79,
+0.55
a +, degree of membership belonging to subset P; -, degree of membershipbelongingto subset N;p, degreeof membershipbelonging to subset Z, fault grade SP;n, degree of membership belonging to subset Z, fault grade SN.
fault propagation path of F H PT + F2 L -+ L R -+ F1 p0.07 p1.0 n0.32 ”n1.0 p1.0 nl.O
+
FHb+
fault propagation path of PPU PPU -+ F2 -+ L --+ LR -+ F1 nl.O n1.0 p0.07 p1.0 n0.32 PPU -+ F 2 -+ PT -+ FH nl.O nl.O nl.O nl.O fault propagation path of F2 F2 -+ L -+ LR -+ F1 nl.O ~ 0 . 0 7 ~ 1 . 0 n0.32 F2 -+ PPU n1.0 ~ 1 . 0
F 2 -+ PT -+ FH nl.O nl.O nl.O fault propagation path of PB -+ F2 -+ F2 fault propagation path nl.O nl.O
PB
fault propagation path o f VH V H -+ F H -+ FH fault propagation path nl.O
nl.O
fault propagation path of MP VH fault propagation path nl.O
MP
-+
“:fuzzy subset,degree of membership I,:var iable/parameter Figure 17. Fault candidates and their propagation path Figure 15. Simplified SDG for the CSTR and vaporizer example in the first diagnosis stage.
Figure 18. Simplified SDG for the CSTR and vaporizer example in the second diagnosis stage. Figure 16. Strongly connected components of the simplified SDG in the first diagnosis stage.
The next stage of diagnosis will proceed with the second information set in which the process variables L and PP are beyond the threshold as given in Table 2. The simplified SDG after the inconsistent edges are deleted and the unmeasured variables along the unbranched path are combined is shown in Figurel8, and the strongly connected components of the current simplified SDG are shown in Figure 19. This diagnosis stage gives the same result as the previous one, the only difference is that the credible failure membership increases to 0.78 because of pp(L) = 0.78. A t the third stage of diagnosis (the sampling time is at
the 7125th second), the process variables T and W1 exceed their limits and the fault belonging areas of some measured variables have changed. The simplified SDG and graph of strongly connected components are shown in Figures 20 and 21, respectively. The fault origin candidate set and the eligible fault origin candidate set are the same (MP, VH, FH, FR, PT, PPU, F2, L, PB); the sequence arranged by the degree of membership is MP, VH, FH, PPU,F2, PB, FR, PT, and L. At the present stage the conventional three-values logic gate approach provided only six fault origin candidates, and this seemed superior to the proposed algorithm. However, the final diagnosis result has not been concluded so far. The process data at the fourth stage sampled at the 7135th second had the process variables FR and PHO
1952 Ind. Eng. Chem. Res., Vol. 33, No. 8, 1994 Table 4. Final Diagnosis Results of the CSTR and Vaporizer Example fuzzy three logic fault three logic gate value origin stage gate value (present method) nonea lb/60 VH 1 VH 2 6 116 VH 3 6 117 VH 4 none 115 Number of eligible fault origin candidates. Rank of the true fault origin in the eligible candidate set.
Figure 19. Strongly connected components of the simplified SDG in the second diagnosis stage.
The diagnosis gives the fault origin candidates set (MP, VH, FH, PHO, PT, FR, L, F2, PPU, PBJ, and the eligible fault origin candidates set (MP, VH, FH, FR, PHOJ.The sequence arranged by the degree of membership is MP, VH, FH, PHO, and FR. The variable MP should not be considered as a fault origin candidate, since it is an unmeasured variable with the designated degree of membership of unity mentioned before. Thus, the final conclusion is that variable VH is the most possible fault origin with the largest degree of membership. It should be mentioned that by the conventional three-value logic gate approach, an erroneous fault origin PH is obtained, while the proposed approach is able to avoid such an erroneous diagnosis. This showed the superiority of the proposed approach to the conventional ones. Table 4 gives the final diagnosis result of these four stages.
Discussion
Figure 20. Simplified SDG for the CSTR and vaporizer example in the third diagnosis stage.
Figure 21. Strongly connected components of the simplified SDG in the third diagnosis stage.
exceed the thresholds. Although some variables have their belonging areaschanged, the simplified SDG and the graph of strongly connected components are still the same as those a t stage 3. The common ancestor of the strongly connected components is PHO, which belongs to region MP, with 0.52 of the degree of membership in the LP subset and 0.48 in the Z subset. We change the belonging subset of PHO into normal subset Z, with possibility of 0.48 followed by step 5.3). This value represents the confidence limit of the ancestor PHO. Then we search again for new fault origin candidates.
In this paper, the fuzzy set theory is directly applied to fault diagnosis for better resolution and to avoid the erroneous fault origin. The real time process variable information is used for the proposed methodology. We find by experience that the sampling time and the accuracy of measurement usually affect the fault possibility membership. In this paper, the sampling time is every 5 s. The smaller the time interval, the smaller the delay error and the better the resolution that can be obtained. But too frequent data acquisition will waste computation effort, thus, there should be a optimal sampling time decided by the transportation delay in the process. In addition, the measurement noises might cause erroneous resolution. In the present study the measurement noise distributes normally with a magnitude order between lo6 and le2. According to our simulation experience, a large process disturbance or measurement noise might lead to an erroneous resolution, while a small disturbance and large noise might induce a spurious resolution. Thus an accurate measuring device is not only the essential necessity of process control it also will avoid spurious resolution in fault diagnosis. Although many researchers still invested the effort into the modification of the SDG approach by maximizing the advantages mentioned earlier, unfortunately, the SDG approach still has some limitations for application to date. Several problems encountered in the real chemical processes needed to be overcome if the SDG approach is adopted asthe methodology to handle process fault. Those problems are, for instance, time delay in fault propagation, cyclic loops in the process flows and the control systems, multitransition of deviation signs of process variables, and merging of fault propagation paths.
Ind. Eng. Chem. Res., Vol. 33, No. 8, 1994 1953
Conclusion In this paper, the fuzzy set theory is applied directly along with the real time information to the diagnosis of process faults with seven fault belonging areas in stead of the conventionalthree. The membership function is linear in our study. It can be modified to other mathematical forms to study the consequence of the modification as we wish. For the measured variables near the thresholds that are the suspicious faults,a min-max manipulation of fuzzy set theory is employed to determine the possible fault membership sequence and the credible failure membership. The proposed method is able to identify faults with low possible fault membership and low credible failure membership which cannot be detected by the conventional fault diagnosis.
Nomenclature A = fuzzy set defined in eq 4 CA1 = initial concentration of reactant (kg mol/m3) CA = reactant concentration in CSTR (kg mol/m3) CF = credible failure membership E = signed directed edge set e = edge F1 = CSTR inlet flow rate (m3/s) F1S = CSTR inlet flow rate sensor (m3/s) F2 = CSTR inlet flow rate (m3/s) FC = cooling water flow rate (m3/s) FH = CSTR product flow rate (m3/s) FHS = CSTR product flow rate sensor (m3/s) g = set function as given in definition 2 FR = recycle flow rate (m3/s) FRS = recycle flow rate sensor (m3/s) h = designated threshold of a variable L = reactor liquid level (m) LL = vaporizer liquid level (m) LLR = vaporizer liquid level recorder (m) LR = reactor liquid level recorder (m) M = membership space ML = reactor liquid level controller output signal MP = vaporizer propane vapor pressure controller output signal MT = reactor temperature controller output signal MVL = vaporizer liquid level controller output signal PB = reactor outlet pressure (atm) PHO = propane vapor outlet pressure (atm) PHOS = propane outlet pressure sensor (atm) PP = propane pressure in vaporizer (atm) PPR = propane pressure in vaporizer recorder (atm) PPU = pump head (atm) PT = split point pressure (atm) PW1 = opening of liquid propane flow rate controller s = standard deviation T = reactor temperature ("C) T1 = liquid propane inlet temperature ("c) TI = CSTR inlet temperature ("C) TC = cooling water outlet temperature ("C) TCI= cooling water inlet temperature ("c) TR = reactor temperature recorder ("C) TT = vaporizer temperature ("C) U = total heat coefficientbetween jacket and reactor (J/ m2 OC)
V = vertex set VH = opening of CSTR product flow controller VL = opening of reactor inlet flow controller VT = opening of cooling water controller VW = opening of liquid propane flow controller W1 = liquid propane flow rate (m3/s) W1S = liquid propane flow rate sensor (m3/s) W2 = vapor propane flow rate (m3/s)
Greek Letters
A = fault manner of a vertex
= membership function = possibility measurement as defined by eq 5 K = restriction parameter as defined by eq 8 j3 = integral power in eq 9 p
?r
Superscripts
+ = positive deviation - = negative deviation Subscripts i = ith process variable or mathematical relation
sp = process variable set point
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