Comprehensive Design of a Sensor Network for Chemical Plants Based on Various Diagnosability and...

Comprehensive Design of a Sensor Network for Chemical PlantsBased on Various Diagnosability and Reliability Criteria. 2.Applications

Mani Bhushan and Raghunathan Rengaswamy*

Department of Chemical Engineering, Indian Institute of Technology, Bombay Powai, Mumbai 400 076, India

In part 1 of this series of papers, a framework for finding a reliable sensor network from a faultdiagnosis perspective was presented. To illustrate the issues involved and underscore the utilityof the discussed approach, a case study is presented in this part. A large flowsheet, theTennessee-Eastman flowsheet, is considered for the case study. For various cases, sensor locationis performed, and the results are compared.

1. Introduction

A framework for reliable sensor location for faultdiagnosis for chemical processes was presented in part1 of this series of papers. In this paper , sensor locationis performed for the Tennessee-Eastman (TE) flow-sheet. The TE test problem was presented by Downsand Vogel.1 The idea behind the test problem was topresent a case study which could be used by researchersto try out a wide variety of ideas1 in diverse areas, suchas plantwide control, multivariable control, optimiza-tion, nonlinear control, estimation, and diagnostics.Various researchers have since then used this problemto test their ideas.

In the area of process control, various researchershave given control structures for the TE process. Mc-Avoy and Ye2 gave single-input single-output controlloops to satisfy the objectives listed by Downs andVogel.1 Banerjee and Arkun3 have also given single-input single-output loops for the TE process. Kanadib-hotla and Riggs4 used a combination of nonlinear modelbased and proportional-integral (PI) controllers for theprocess. Ricker5 used a decentralized strategy to designcontrollers for the TE process. His approach is similarto that of Price et al.6 and Lyman and Georgakis,7 whoused throughput manipulation and inventory control todesign control structures. Among model-based tech-niques, Sriniwas and Arkun8 used an input-outputmodel identified from plant data as the basis of a modelpredictive control (MPC) scheme. McAvoy9 gave anoptimization-based approach for the synthesis of aplantwide control system. Ricker10 has given optimalsteady-state operating points for all of the operatingmodes of the TE process.

Other applications tested on the TE process includediagnosis of process disturbances using statistical11 andspeech recognition12 methods, predictive online monitor-ing,13 and online quality improvement using a multi-variate statistical controller.14

Downs and Vogel1 gave the process flowsheet, steady-state material and energy balances, some physicalproperty data, and some qualitative information onreaction kinetics. However, no process model was given

by them. Instead the model has been coded into acomplex FORTRAN code. While the code containsinstructions for its usage, the calculations have not beendocumented and the variable names are cryptic.1 Rickerand Lee15 have derived a nonlinear mechanistic modelof the TE process. Their aim was to reproduce theoriginal TE process characteristics without introducingunnecessary detail. They used certain tuning param-eters to match their model with the original TE process.A FORTRAN code which simulates their TE model hasalso been given.15

In the sensor location framework presented by us inpart 1 of this series, a qualitative, cause-effect repre-sentation is used. The cause-effect representation isgenerated from a process directed graph (DG) or processsigned directed graph (SDG). The process DG and SDGare generated from the model equations. For thispurpose, we chose to work with the TE model given byRicker and Lee15 rather than with the original TEproblem of Downs and Vogel.1 As was already men-tioned, the essential characteristics have not beenchanged in Ricker’s model; only some unnecessarydetails have been omitted. Ricker and Lee16 have usedthe modified TE model as the basis for nonlinear MPCof the original TE process. The use of process DG andSDG to represent chemical processes is presented next.

2. DG and SDG Representations of ChemicalProcesses

Over the years, digraph representation of chemicalprocesses has become quite popular with researchersworking in the area of fault diagnosis. The SDGrepresentation was first developed by Iri et al.,17 whoused it to identify a failure origin for a given set ofmeasurements. Mylaraswamy et al.18,19 have given asystematic procedure for the development of SDG forchemical processes. The SDG is a collection of nodes andbranches which represent cause-effect relationshipsbetween process variables. The nodes represent processvariables and fault origins, while branches representthe causal influence between the nodes. The directionof deviation in the nodes is represented by signs on thebranches, with + (-) indicating that the cause and effectvariables tend to change in the same (opposite) direc-tion.20 The values of the nodes (variables) are specifiedas being either 0 (normal), + (high), and - (low). If thebranch signs are not considered, it becomes a DGrepresentation.

* To whom correspondence should be addressed. Currentaddress: Department of Chemical Engineering, ClarksonUniversity, Potsdam, NY 13699-5705. E-mail: [email protected].

1840 Ind. Eng. Chem. Res. 2002, 41, 1840-1860

10.1021/ie010437v CCC: $22.00 © 2002 American Chemical SocietyPublished on Web 03/01/2002

The SDG representation has been used by variousresearchers for fault diagnosis. These include Kramerand Palowitch,20 Mohindra and Clark,21 Chang andYu,22 Gujima et al.,23 Vedam and Venkatasubrama-nian,24 and Pramanik and Venkataram.25 To enhancethe resolution of the diagnostic algorithm, variations inSDG have also been used. Umeda et al.26 incorporatedmultiple time stages and time delays in the SDG.Kokawa et al.27 used delays and failure propagationprobabilities to improve the performance of their diag-nostic algorithm. Other researchers who have modifiedthe SDG include Wilcox and Himmelblau,28,29 Nam etal.,30 and Wang et al.31 Raghuraj et al.32 and Bhushanand Rengaswamy33 have used the DG and SDG repre-sentations of the process to design sensor networksbased on fault observability and resolution criteria. Itis important to note here that the SDG can be con-structed either from a mathematical model of theprocess or from operation data and experienced opera-tors.17 A concise description of the TE model as givenby Ricker and Lee15 is presented next, after which theDG/SDG representation of the TE model is discussed.

3. TE Process Description

The flowsheet of the TE process is presented in Figure1. The process consists of five major unit operations: an

exothermic two-phase reactor, a product condenser, aflash separator, a reboiled stripper, and a recyclecompressor. The process produces two products (G andH) from four reactants (A, C, D, and E), with an inertcomponent (B) and a byproduct (F).

The gaseous reactants are fed to the reactor wherethey react to form liquid products. The gas-phasereactions are catalyzed by a nonvolatile catalyst dis-solved in the liquid phase. The reactor has an internalcooling bundle for removing the heat of the reaction. Thevapor products, along with the unreacted reactants,after leaving the reactor pass through the productcondenser before entering the separator. The productcondenser condenses the products. Noncondensed com-ponents are recycled back to the reactor through thecompressor. Condensed components move to the strip-per to be stripped with one of the feed streams to removethe unreacted reactants. Products G and H exit throughthe stripper liquid outlet. The inert and byproduct aremainly purged from the system as vapor from thevapor-liquid separator.

Downs and Vogel1 have given the salient features ofthe original TE model. The salient features of the modelpresented by Ricker and Lee15 are presented as follows:

(i) A key simplification is the elimination of energybalances. This is accomplished by adding appropriate

Figure 1. TE flowsheet.

Ind. Eng. Chem. Res., Vol. 41, No. 7, 2002 1841

PI temperature-control loops that convert the reactorand separator temperature from dependent variablesto independent variables. This removes the need ofwriting energy balances for the reactor and separatorand, hence, avoids modeling heat-transfer rates andheat effects in chemical reactions. Also, this makes theprocess more tractable, because instead of specifying theposition of the valve controlling the coolant rate, onemay specify the operating temperature.

(ii) Recycle flow is defined as an independent variable,eliminating the need to model the compressor.

(iii) Detailed modeling of the stripper is not donebecause it provides adequate removal of volatiles re-gardless of the steam rate.

When the above modifications are incorporated, thenumber of states to be modeled is 26, which is abouthalf the number of states (50) present in the originalTE model. Other assumptions involved in modeling theprocess may be referred to in work by Ricker and Lee.15

The strategy used by them was to write a first principlesmodel with a sufficient number of free parameters(disturbances), which can be tuned so that most of theoutputs predicted by the model agree with those givenby the Downs and Vogel1 TE code. The model is writtenin nonlinear state variable form:

where x is the state vector, u is a vector of knowninputs, d is a vector of unmeasured inputs (disturbancesand time-varying parameters), and y is the outputvector. The state variable vector is

where Ni,r is the molar holdup of chemical i in thereactor [kmol] and Ni,s, Ni,m, and Ni,p are those in theseparator, feed mixing zone, and product reservoir(stripper base), respectively. The u vector contains 10variables:

where Fi is the molar flow rate of stream i (kmol/h)(stream numbers are as given in Figure 1) and Tcr andTcs are the reactor and separator temperatures (°C). Themodel equations are given next.

where

where

Other equations are

ddt

x ) x3 ) f(x,u,d) (1)

y ) g(x,u,d) (2)

xT ) [NA,r, NB,r, ..., NH,r, NA,s, NB,s, ..., NH,s, NA,m,NB,m, ..., NH,m, NG,p, NH,p] (3)

uT ) [F1, F2, F3, F4, F8, F9, F10p , F11, Tcr, Tcs] (4)

State equations:

dNi,r

dt) yi,6F6 - yi,7F7 + ∑

j)1

3

νijRj i ) A, B, ..., H (5)

dNi,s

dt) yi,7F7 - yi,8(F8 + F9) - xi,10F10

i ) A, B, ..., H (6)

dNi,m

dt) zi,1F1 + zi,2F2 + zi,3F3 + Fi,5 + yi,8F8 + Fi* -

yi,6F6 i ) A, B, ..., H (7)

dNi,p

dt) (1 - φi)xi,10F10 - xi,11F11 i ) G and H (8)

Output equations:

Pr ) ∑i)A

H

Pi,r (9)

Pi,r )Ni,rRTr

VVri ) A, B, C (10)

Pi,r ) γi,rxi,rP isat(Tr) i ) D, E, ..., H (11)

VLr ) ∑i)D

H Ni,r

Fi

(12)

Ps ) ∑i)A

H

Pi,s (13)

Pi,s )Ni,sRTs

VVsi ) A, B, C (14)

Pi,s ) γi,sxi,10P isat(Ts) i ) D, E, ..., H (15)

VLs ) ∑i)D

H Ni,s

Fi

(16)

VLp )NG,p

FG+

NH,p

FH(17)

Pm ) ∑i)A

H

Ni,m

RTm

Vm

(18)

F6 ) (â62413.7

M6x|Pm - Pr| (19)

yi,6 )Ni,m

∑i)A

H

Ni,m

i ) A, B, ..., H (20)

yi,8 ) yi,9 )Pi,s

Psi ) A, B, ..., H (21)

xi,11 ) øGH

Ni,p

NG,p + NH,pi ) G and H (22)

R1 ) R1VVr exp[44.06 - 42600RTr ]PA,r

1.08PC,r0.311PD,r

0.874

(23)

R2 ) R2VVr exp[10.27 - 19500RTr ]PA,r

1.15PC,r0.370PE,r

1.00

(24)

R3 ) R3VVr exp[59.50 - 59500RTr ]PA,r(0.77PD,r + PE,r)

(25)

VVr ) Vr - VLr (26)

1842 Ind. Eng. Chem. Res., Vol. 41, No. 7, 2002

Ricker and Lee15 tuned some model parameters sothat some critical outputs match the correspondingvalues predicted by the Downs and Vogel1 code. Theoutputs which were matched are given in Table 1. Theparameters which were adjusted online to match theoutputs are also given in Table 1. This has been donefor various modes of operation, including the base casespecified by Downs and Vogel.1

For the purpose of sensor location, we have assumedthe parameters to be constant, with their values beingthe base case values. The faults considered are theelements of the u vector (eq 4). We introduce anotherfault corresponding to catalyst deactivation (Cd

-) andmodify the expressions (eqs 23-25) for reaction ratesto account for the catalyst activity:

The normal value of catalyst activity Cd ) 1. Unlikeother faults (elements of vector u) which are flow rate

and temperature changes and hence can both increaseor decrease, the catalyst activity can only decrease. Thisis indicated by a “-” superscript to Cd. Hence, for theprocess SDG, for the 10 faults corresponding to vectoru, both positive and negative deviations will be consid-ered. For catalyst activity, only a negative deviation willbe considered as a fault. The process SDG will thenconsist of 10 × 2 + 1 ) 21 faults, whereas the processDG will have only 10 + 1 ) 11 faults.

Construction of the DG and SDG for this process isdiscussed next.

4. DG and SDG Representations of the TEProcess

The nonlinear state variable model for the TE processwas discussed in the previous section. For a processmodel in the differential algebraic equations form,Mylaraswamy et al.18 have given a systematic procedurefor generating the SDG. The basic idea is to assign eachvariable as the output of an equation. For a givensystem of equations, this is similar to the bipartitematching34 problem. A matching is said to be perfect ifevery variable is matched to a different equation.Steward35 showed that, for a well-defined system ofequations, at least one perfect matching between thevariables and the equations has to exist. For ordinarydifferential equations, this matching is straightforward.The state variable for which the differential equationis written is matched to that equation. The causality ina differential equation is from right to left. In otherwords, the variables on the right-hand side (RHS) of adifferential equation have an effect on the correspondingstate variable. The sign of effect from a variable (onRHS) to the state node is given by the derivative of theRHS taken with respect to that variable. As an example,consider the following single differential equation:

During construction of the SDG, the above equation ismatched to variable x1, which will have arcs from nodesu1 and u2. The direction of the arc from u1 to x1 is thesign of ∂f/∂u1 evaluated at the steady state around whichthe SDG is developed. Similarly, the sign of the arc fromu2 to x1 is the sign of ∂f/∂u2.

xi,r ) 0 i ) A, B, C (27)

xi,r )Ni,r

∑i)D

H

Ni,r

i ) D, E, ..., H (28)

yi,7 )Pi,r

Pri ) A, B, ..., H (29)

VVs ) Vs - VLs (30)

xi,10 ) 0 i ) A, B, C (31)

xi,10 )Ni,s

∑i)D

H

Ni,s

i ) D, E, ..., H (32)

Fi,5 ) zi,4F4 i ) A, B, C (33)

Fi,5 ) xi,10F10 i ) D, E, F (34)

FG,5 ) 0.07xG,10F10 (35)

FH,5 ) 0.04xH,10F10 (36)

F7 ) (â75722.0

M7x|Pr - Ps| (37)

F10 ) F10p - F10

/ (38)

R1 )

Cd-R1VVr exp[44.06 - 42600

RTr ]PA,r1.08PC,r

0.311PD,r0.874

(39)

R2 )

Cd-R2VVr exp[10.27 - 19500

RTr ]PA,r1.15PC,r

0.370PE,r1.00

(40)

R3 )

Cd-R3VVr exp[59.50 - 59500

RTr ]PA,r(0.77PD,r + PE,r)

(41)

Table 1. Matched Output and Disturbance Variables forthe TE Model

outputs parameters

variable description variable

Pr reactor pressure zA,4VLr reactor liquid level zB,4Ps separator pressure R1VLs separator liquid level R2VLp stripper bottoms level F 10

/

Pm stripper pressure â7yA,9 A in purge (stream 9) â6yB,9 B in purge øGHyC,9 C in purge γG,syD,9 D in purge γH,syE,9 E in purge F C

/

yF,9 F in purge F D/

yG,9 G in purge F E/

yH,9 H in purge F F/

xG,11 G in product γrxH,11 H in product

dx1

dt) f(u1,u2) (42)


For algebraic equations, a bipartite matching has tobe performed. Various algorithms are available for doingthis.34 Once a perfect match is obtained, the SDG forthe system of algebraic equations can be easily drawn,as explained by the following example. Consider thefollowing algebraic equation:

If the above equation is matched to variable e1 in theperfect match, then it may be rewritten as

Arcs from nodes e2 and e3 are then drawn to node e1.The sign of the arc from e2 is the sign of ∂g′/∂e2 evaluatedat the steady state around which the SDG is developed.Similarly, the sign of the arc from e3 to e1 is the sign ofthe partial derivative of g′ with respect to e3, with thepartial derivative evaluated at the steady state. It isimportant to note that getting an explicit form for e1(eq 44) may require linearization of eq 43. Anotherimportant point is that independent variables (faults,inputs, etc.) are not considered for matching whileperforming the equation-variable matching. Also, statevariables which have already been matched to dif-ferential equations are not considered while matchingthe algebraic equations.

The above procedure is used to generate the SDG forthe TE process discussed in the previous section. As anillustration, consider the differential equation corre-sponding to state NA,r (eq 5)

This implies that variable NA,r will have positive arcsfrom nodes yA,6 and F6 because the derivative of the RHSof the above equation with respect to yA,6 and F6 ispositive (when the derivative is evaluated at the basecase values). Similarly, NA,r will have negative arcs fromnodes yA,7, F7, R1, R2, and R3. To generate the SDGcorresponding to the algebraic equations, equation-variable matching needs to be performed. For the TEprocess equations as given in the last section, this isnot required because the equations are already givenin a matched form. Each algebraic equation correspondsto a balance/definition for a variable and can directlybe used to calculate the effect of other variables on thatvariable. Moreover, it turns out that for the algebraicequations this is the only complete matching. Theimplication of this is that, in the TE SDG, there is noloop involving arcs generated only from algebraic equa-tions. As discussed by Mylaraswamy et al.,18 variablesinvolved in an algebraic loop (loop involving onlyalgebraic arcs) represent events which occur simulta-neously and causal ordering between these eventscannot be established. These variables should then becollapsed to a supernode. For the TE case study, becauseof the absence of algebraic loops, this situation does notarise. As an example of an algebraic equation, considereq 10 for variable PA,r:

This equation is used to calculate the effect on PA,r.From the equation, it can be derived that nodes NA,r

and Tr will have positive arcs to PA,r while variable VVrwill have a negative arc to PA,r.

Another feature which will be used to decrease thespurious solutions generated by SDG is the addition ofgains. With each arc, along with the sign of the effect,the magnitude of the effect is also considered. This isjust the value of the derivative which is used to calculatethe sign of the effect. For example, from eq 46, the gainof the effect of variable NA,r on variable PA,r is

where the derivative is calculated at the base casevalues. For effects generated from differential equations,the gain calculation involves multiplication by an extra∆t term, where ∆t is the step size used for integrationof the differential equations. For example, from eq 45,the gain of the arc from variable yA,6 to NA,r is

where the derivative is calculated at the base casevalues. For arcs generated from algebraic relationships,the extra term ∆t for gain calculation was not requiredbecause algebraic equations represent instantaneousdynamics. A change in any one variable causes instan-taneous changes in other variables such that therelationship specified by the algebraic equation ismaintained. For differential equations, on the otherhand, the amount of change is related to the timeelapsed since the change first occurred. For a lineardifferential equation, x ) ay, where a is a constant;the change in x is simply a × (the product of change iny and the time elapsed ∆t). Because the SDG generationis based on linearization around a steady state, the gainis calculated as discussed in eq 48. For gain calculationfor the TE case, a ∆t value of 0.0001 is used. The ideain choosing a small ∆t is to characterize the initialresponse of the system as it is perturbed around thesteady-state operation. If the system behavior was tobe free from inverse response, the ultimate effect of avariable on the RHS on the left-hand-side (LHS) vari-able will be of the same sign as the initial effect. Use ofthe arc gains based on order of magnitude argumentsto increase fault resolution will be discussed later.

Gain calculation for arcs involved in the control loophas to be done differently. For the purpose of gaincalculation, we will be considering the case where thecontrol loop is functioning perfectly and is able to keepthe controlled variable at its normal value. The impor-tant implication of considering perfect control is thatthe gain of the control loop has to be -1. The gain ofthe control loop is the product of the gains of all of thearcs involved in the control loop. Only when the loopgain is -1 will the control loop be able to rejectdisturbances entering the loop and ensure that thecontrolled variable is normal (0) at steady state. As anexample, consider the control loop shown in Figure 2.In the figure, variable CV is the controlled variable, Sis the controlled variable sensor, C is the controller, Vis the control valve, and MV is the manipulated vari-able. The disturbance originating at variable F entersthe control loop through the controlled variable CV. Fora change ∆F in variable F, initially, the effect on CV is

g(e1,e2,e3) ) 0 (43)

e1 ) g′(e2,e3) (44)

dNA,r

dt) yA,6F6 - yA,7F7 - R1 - R2 - R3/3 (45)

PA,r )NA,rRTr

VVr(46)

∂

∂NA,r(NA,rRTr

VVr) )

RTr

VVr(47)

{ ∂

∂yA,6(yA,6F6 - yA,7F7 - R1 - R2 - R3/3)}∆t ) F6∆t

(48)


∆F × k1. This activates the controller which, in turn,changes the manipulated variable through the controlvalve, such that, finally, the effect of the manipulatedvariable on the controlled variable is -∆F × k1. Onlythen will the net effect on the controlled variable be ∆F× k1 - ∆F × k1 ) 0, and at steady state the controlledvariable will be at its normal (0) value. The gains of arcsderived from model equations, such as arcs from MV toCV, can be calculated from the corresponding equations.The gain of the arc from the controller to the valvecannot be directly calculated because the valve openingis related to the error signal by an integral equation(for a PI controller). Hence, the gain of the arc from Cto V will be chosen such that the gain of the control loopis -1. Also, for the TE case, without any loss ofgenerality, we will assume the gains of arcs from CV toS, and from S to C, to be 1. The assumption here is thatthe controlled variable is measured as it is and theactual variable value is passed to the controller. In thecase where the value is transformed to some other form(for, e.g., an electrical signal) before being passed to thecontroller, the gains of the arcs from CV to S and/or fromS to C can be appropriately modified. In either case, thegain of the arc from C to V will be chosen such that thenet gain of the control loop is -1.

Calculation of gains for normal arcs is based oncharacterization of the initial response of the variablesas the system is perturbed around the steady state. Forcontrol loops, on the other hand, a steady-state responseis modeled and not the initial response. This is donebecause, for control loops working normally, the steady-state behavior is known irrespective of the initialtransients. At steady state, the controlled variable isnormal and other variables in the control loop takevalues to cancel the effect of disturbance affecting thecontrolled variable. However, for other variables, thesteady state cannot be exactly characterized. So, theinitial response is used as an estimate of the steady-state response. Hence, gain calculations for control loopsbased on steady-state response are consistent with gaincalculations for other arcs based on the initial response.

Applying the methodology discussed above, the SDGwith gains, for the TE process, is generated. Because ofthe large number of variables present in the flowsheet,it is not possible to graphically represent process SDG.Fault modeling based on the generated SDG is discussedin the next section.

5. Fault Modeling Based on Process SDG

The generation of SDG from process model equationswas discussed in the previous sections. The faults forthe TE process were also listed. Given a process SDG,with a set of defined faults, the sensor location strategyconsists of two main steps:33 (i) fault modeling involvingthe prediction of the cause-effect behavior of thesystem, generating a set of variables that are affectedwhenever a fault occurs, and (ii) use of the generated

sets to identify sensor location based on various designcriteria, such as observability, resolution, reliability, andso on. The fault propagation behavior may be derivedfrom the qualitative models of the process, such as theDG and SDG representations. The use of generated setsto locate sensors for observability and resolution waspresented by Raghuraj et al.32 for the DG case and byBhushan and Rengaswamy33 for the SDG case. Aframework for sensor location based on reliabilitycriteria was discussed in part 1 of this series. Itsapplication to the TE problem will be discussed later inthis paper. In this section, generation of fault propaga-tion behavior is discussed for three cases: (i) processDG, (ii) process SDG, and (iii) SDG with gains.

5.1. Fault Propagation for the DG Representa-tion. For the DG representation of the process, faultsimulation is a straightforward task and may be ac-complished by a suitable graph theoretic algorithm suchas depth first search.36 For example, for the DG givenin Figure 3, to get the nodes affected by fault F1, onemay start a depth first procedure for labeling all of thenodes, starting from F1. The nodes which get labeledare N1, N2, and N3. This is the set of nodes which areaffected when F1 occurs. The other nodes which do notget labeled will not be affected by F1. Similarly, for faultF2, the affected nodes are N2, N3, and N4.

5.2. Fault Propagation for the SDG Representa-tion. When the process is represented in the form of aSDG, fault modeling is more involved than that for theDG. Apart from knowing whether a fault affects aparticular node, one is also interested in knowing aboutthe direction (sign) of the effect. Deducing this is not astraightforward task because of the qualitative ambi-guities that are inherent in the SDG analysis. Bhushanand Rengaswamy33 have discussed fault modeling forSDG in detail. A detailed discussion on steady-stateanalysis of SDG can be found in Oyeleye and Kramer.37

Here we summarize the main issues involved in analyz-ing the SDG for the purpose of sensor location.

(i) Presence of multiple paths: A fault may haveopposing effects on a node through different paths.Without the use of any quantitative information, it isnot possible to fix the sign of the node in such a case.While considering the SDG, we will consider the signon such a node to be indeterminate. Use of gains toresolve such cases based on order of magnitude argu-ments will be discussed later.

(ii) Occurrence of multiple faults: It is possible thattwo faults with opposite effects on a node occur together.Again, without the use of order of magnitude argu-ments, it may not be possible to determine the sign onthe node. Also, this scenario is relevant only for themultiple-fault case (and not for single-fault case).

(iii) Feedback loops: For both positive and negativefeedback loops (other than control loops), the feedbackpath is ignored. For a negative feedback loop (loop whoseproduct of branch signs is negative), the presence of oneinput to the loop eliminates the feedback path from allvalid interpretations.20 For a positive feedback loop also,

Figure 2. Control loop.

Figure 3. Example DG.


the feedback path is ignored. This is reasonable becausethe SDG represents only the initial response of thesystem.20 The control loops have to be treated differ-ently.

(iv) Negative feedback control loops: The purpose ofa control loop is to keep the controlled variable at thespecified value. This controller action implies that thecontrolled variable can pass disturbances without beingsignificantly altered. A disturbance entering a controlloop gives rise to one of two situations:20 (a) Perfectcontrol is achieved: if the disturbance is not large, thecontrol loop is able to bring back the controlled variableto its normal value. Thus, at steady state, the controlledvariable has a normal value, while other variables inthe control loop have normal/deviated states dependingon their position in the control loop, vis-a-vis thecontroller position and the entry point of the distur-bance. (b) The loop saturates: this happens when thedisturbance is large enough, so that the deviation in thecontrolled variable is too large to be compensated by thecontrol loop. Thus, at steady state, both the controlledand other variables in the loop have deviated values.Another situation which can occur is the failure of thecontrol loop. This will imply that the control loop willnot take any action even if there is any deviation in thecontrolled variable. As mentioned by Bhushan andRengaswamy,33 the control loop failure fault manifestsitself only for the multiple-fault case.

The three possibilities (perfect control, loop satura-tion, and loop failure) can be modeled to get the nodesaffected when a given fault occurs. The case of completecontrol will lead to a conservative design as comparedto the case of loop saturation for the observabilityproblem,33 and hence only the case of complete controlneeds to be considered in the sensor network design.

Chemical process flowsheets are often strongly coupled.This invariably leads to the presence of multiple pathswith opposing effects, from a fault to a node. Asmentioned above, for such cases, without the use ofquantitative information, it is not possible to determinethe sign of the effect. Use of gains to resolve such casesis discussed next.

5.3. Fault Propagation for the SDG Representa-tion Using Arc Gains. The procedure for calculationof arc gains was discussed in the previous section. Theinformation about arc gains may be used to decreasethe spurious solutions during fault modeling and therebyenhance fault resolution compared to that obtainedbased on process SDG. Specifically, this informationmay be used to resolve the cases when there are bothpositive and negative paths from a fault to a node. Thebasic idea is to find the magnitudes of the positive andnegative effects. If one of the effects is much greaterthan the other, then the smaller effect may be neglected.The magnitude of the positive effect is the product ofgains of the arcs constituting the positive path. Simi-larly, the negative effect is the product of gains of arcsencountered in the negative path. As an example,consider the SDG with gains shown in Figure 4. Fault

F in this SDG has both a positive as well as a negativepath to node N3. The positive effect on node N3 is theproduct of deviation in F and the gains of the arcs inthe positive path. The arcs involved in the positive pathare from F to N1 and from N1 to N3. Hence, the positiveeffect on N3 is equal to ∆F × 0.1 × 0.4 ) 0.04∆F, where∆F is the fault magnitude (deviation in F). Similarly,the negative path consists of arcs from F to N2 and fromN2 to N3. Hence, the negative effect of F on node N3 is∆F × 0.1 × 0.02 ) 0.002∆F. The effect on N3 due tochange in F can then be given a positive (+) sign becausethe positive effect is much greater than the negativeeffect. For the TE case, we will use such order ofmagnitude arguments to fix the sign. Let the magni-tudes of larger and smaller effects on a node be a andb, respectively. We will fix the sign of the node to bethe sign of a if the following inequality holds:

The above criterion is used to fix the signs on nodes,where both positive and negative effects are present. Ifthe above inequality is not satisfied, then the effect onthe node is considered to be indeterminate (() as before.

There might be cases when more than one positivepath from a fault to a node is present. In that case, thenet positive effect will be the addition of effects of all ofthe positive paths from the fault to that node. Similarly,for the case when more than one negative path ispresent, all of the negative effects will be added to getthe net negative effect. The net positive and the netnegative effects may then be compared to fix the signof the effect.

In literature, some work has been done on formalizingorder of magnitude reasoning for process engineering.Purely qualitative modeling approaches, though widelyapplicable, do not incorporate any quantitative informa-tion. Purely quantitative approaches, on the other hand,utilize the most detailed information which may notalways be available. Order of magnitude approachesutilize some quantitative information such as the rela-tive magnitude of variables to supplement qualitativereasoning. Their aim is to mimic the reasoning of humanexperts in process systems. Mavrovouniotis and Stepha-nopoulos38 have presented a formal framework for orderof magnitude reasoning in process engineering. Theyimplemented their system, labeled O(M), using Sym-bolics Common LISP language. It is based on sevenprimitive relations (such as “much smaller than”, “slightlysmaller than”, etc.) among quantities and some com-pound relations formed from valid combinations of thebasic primitives. The tolerance parameter, which de-cides under what conditions a variable is much smallerthan another variable, is kept unspecified because itdepends on the particular problem. The default valueof the parameter, though, is taken to be 0.1, whichcorresponds to the common idea38 that 1 order ofmagnitude denotes roughly a factor of 10. In our workalso, we have used this value to differentiate betweentwo opposing (+ and -) effects (eq 49).

The use of order of magnitude arguments as pre-sented here may lead to an increased fault resolutionas shown in Figure 5. In the figure, fault F1 has bothpositive and negative (() effects on node N2. Fault F2has a positive effect on N2. In the absence of any otherinformation, it is not possible to resolve faults F1 andF2 based on their effects on N2 (or N1). If order of

Figure 4. SDG with gains.

a - bb

g 10 (49)


magnitude arguments were to be used, then the sign ofF1 on N2 can be fixed to be negative. Now, faults F1and F2 can be resolved based on their effects on N2.

Use of such order of magnitude arguments leads toincreased resolution, but at the risk of making wrongpredictions. The gain calculation is done using thesteady-state values. As the process moves away fromthe current steady state, the arc gains may changesignificantly and the effect predicted using order ofmagnitude may no longer be valid. The chances ofmaking wrong predictions can be reduced by using afactor greater than 10 in inequality (eq 49) to resolveconflicting paths, but this may result in a loss ofresolution because it may no longer be possible touniquely fix the sign of some variables. For example, ifinstead of 10 the factor 20 were to be used, then it willnot be possible to fix the effect of fault F1 on node N2in Figure 5. For this case, it will no longer be possibleto resolve between faults F1 and F2. Thus, there is atradeoff between increased resolution and risk of ob-taining wrong predictions. An important observationhere is that, even though wrong predictions for somecases are obtained, the sensor location based on thesepredictions is still valid from the fault diagnosis per-spective. This point will be explained while performingsensor location for the TE case study later in this paperbased on results obtained, and further validation willbe provided based on sample simulation runs. Anotherimportant point to be noted here is that if the processwere to be free of inverse responses, then the cause-effect modeling based on the framework discussed herewill, in general, give correct predictions.

The methodology as discussed here may be used toreduce the ambiguities associated with fault propaga-tion behavior in the SDG. In the next section, sensorlocation is performed for the TE case. Results aregenerated for all three representations discussed sofar: (i) process DG, (ii) SDG, and (iii) SDG with gains.

6. Sensor Location for the TE Process

The TE process and its cause-effect representationwere discussed in the previous sections. In this section,sensor location design is carried out for the TE case.Results are generated for the three representationsdiscussed earlier, namely, (i) process DG, (ii) SDG, and(iii) SDG with gains. The sensor failure probabilities andtheir costs are given in Tables 2 and 3, respectively. Thesensor cost data are taken from Bagajewicz andSanchez.39 The faults and their occurrence probabilitiesare listed in Table 4.

6.1. Sensor Location for the DG Representation.For the DG representation of the TE process, the 11faults listed in Table 4 are considered. No control loopsare considered for this case. For each of these faults,the sets of affected nodes are generated. For observ-ability, measuring node xG,11 is sufficient. This isexpected because of the coupled nature of the flowsheet.For resolving the faults under the single-fault assump-tion, sensors should be placed on nodes [Pr, xG,11]. Theresults for the DG case are summarized in Table 5.

6.2. Sensor Location for the SDG Representa-tion. For the SDG representation of the TE process,control loops are also considered. The original TEprocess as discussed by Downs and Vogel1 is unstableat the base case operating values. The simplified modelof Ricker and Lee15 that is used in this paper is alsounstable at the base case values. To stabilize thelinearized Ricker and Lee model, Ricker and Lee16 usedthree level control loops. These loops are given in Table6. Ricker and Lee16 appended these three level loops tothe linearized Ricker and Lee model. This was then used

Figure 5. Increased fault resolution by using gains.

Table 2. Sensor Data for the TE Process

nodeno. var.

prob. offailure(log sj)

nodeno. var.


nodeno. var.


s1 Pr -3 s22 yA,8 -3 s42 xH,10 -3s2 Ps -3 s23 yB,8 -3 s43 xG,11 -3s3 Pm -3 s24 yC,8 -3 s44 xH,11 -3s4 F6 -3 s25 yD,8 -3 s45 xD,r -3s5 yA,6 -3 s26 yE,8 -3 s46 xE,r -3s6 yB,6 -3 s27 yF,8 -3 s47 xF,r -3s7 yC,6 -3 s28 yG,8 -3 s48 xG,r -3s8 yD,6 -3 s29 yH,8 -3 s49 xH,r -3s9 yE,6 -3 s30 yA,9 -3 s50 VL,r

s -2s10 yF,6 -3 s31 yB,9 -3 s51 CL,r -4s11 yG,6 -3 s32 yC,9 -3 s52 CVLr -4s12 yH,6 -3 s33 yD,9 -3 s53 VL,s

s -2s13 F7 -3 s34 yE,9 -3 s54 CLs -4s14 yA,7 -3 s35 yF,9 -3 s55 CVLs -4s15 yB,7 -3 s36 yG,9 -3 s56 VL,p

s -2s16 yC,7 -3 s37 yH,9 -3 s57 CLp -4s17 yD,7 -3 s38 xD,10 -3 s58 CVLp -4s18 yE,7 -3 s39 xE,10 -3 s59 F10 -3s19 yF,7 -3 s40 xF,10 -3 s60 F11 -3s20 yG,7 -3 s41 xG,10 -3 s61 Ts -2s21 yH,7 -3

Table 3. Sensor Cost Data for the TE Process

cost sensor nodes

100 s1, s2, s3, s51, s52, s54, s55, s57, s58150 s50, s53, s56200 s59, s60300 s4, s13500 s61700 s38, s39, s40, s41, s42, s43, s44, s45, s46, s47, s48, s49800 s5, s6, s7, s8, s9, s10, s11, s12, s14, s15, s16, s17, s18,

s19, s20, s21, s22, s23, s24, s25, s26, s27, s28,s29, s30, s31, s32, s33, s34, s35, s36, s37

Table 4. Faults for the TE Process

faultoccurrence

probability (log fi) faultoccurrence

probability (log fi)

F1 -2 F10p -2

F2 -2 F11 -2F3 -2 Tcr -1F4 -2 Tcs -1F8 -2 Cd

- -2F9 -2

Table 5. DG-Based Sensor Location for the TE Process

casesensorsselected indistinguishable sets

observability xG,11single-fault resolution Pr, xG,11 [F11], [F1, F2, F3, F4, F8, F9,

F 10p , Tcr, Tcs, Cd

-]

Table 6. Control Loops for the TE Process

loop no. controlled variable manipulated variable

1 reactor liquid separator temp (Tcs)2 separator liquid separator underflow (F 10

p )3 stripper liquid stripper underflow (F11)


as a model for their nonlinear MPC strategy. For ourcase study, we also consider these three level loops.From Table 6, it is seen that the three manipulatedvariables are part of the input (fault) vector u (eq 4).Hence, they will not be considered as faults. Theremaining seven elements of u along with Cd

- (catalystdeactivation) will be considered as faults. Also, negativedeviations in the seven elements of u will be considered.Positive deviation in Cd is not considered becausecatalyst activity can only decrease and not increaseduring process operation. This leads to a SDG with atotal of 7 + 7 + 1 ) 15 faults. Sensor location resultsare presented for the set of these 15 faults.

To attain maximum resolution under the single-faultassumption, the sensor should be placed on node Pr. Allof the 15 faults are indistinguishable for this case.Compared to the DG-based results (listed in Table 5),it appears that lower resolution is attained based onprocess SDG than on process DG. This is not really thecase because the fault (F11) which could be resolvedbased on process DG has become a manipulated variablein the process SDG. Nevertheless, use of SDG does notoffer any increase in resolution compared to the DGresults.

6.3. Sensor Location for SDG with Gains. As seenabove, because of the coupled nature of the flowsheet,the sensor location based on process SDG does not offerany resolution. To enhance fault resolution, we considerSDG with arc gains. The generation of SDG with arcgains, for the TE process, was discussed earlier. For agiven fault, on each measurable variable, both thepositive and negative effects are calculated. If the largerof the two effects is much greater than the smaller one,then the effect on the node is considered to be the signof the larger one. For the TE case, eq 49 is used toresolve opposing effects.

Sensor location was performed for this case. Forsingle-fault resolution, sensors should be placed onnodes Ps, yD,6, yE,6, F7, and xE,10 to attain maximumresolution. With this set of sensors, the indistinguish-able sets of faults are (F1

+, F4+), (Cd

-, F8-), and (F1

-,F4

-). The effect of faults on these sensors based on SDGwith gains is presented in Table 7. When these arecompared with the results obtained for the SDG case,it is seen that the resolution increases significantly.These sensor location results are obtained by applyingthe greedy search heuristic presented in work by Bhus-han and Rengaswamy.33 It basically involves convertingthe single-fault resolution problem to an appropriateobservability problem which is a set cover problem.

6.3.1. Discussion on the Results of Sensor Loca-tion Using SDG with Gains. The main motivation forusing SDG with gains was to improve resolution.Though increased resolution was obtained by resolvingindeterminate effects using arc gains, the use of suchorder of magnitude arguments has the risk of leadingto wrong predictions. In this section, we will validatethe sensor location results that we obtained for the TEcase study through numerical simulation studies on theTE problem. The simulation results are reported here

Table 7. Effect of Faults on Variables Selected forSingle-Fault Resolution (Prediction Based on SDG withGains)

fault Ps yD,6 yE,6 F7 xE,10 fault Ps yD,6 yE,6 F7 xE,10

F1+ (1 -1 -1 1 1 F1

- (1 1 1 -1 -1F2

+ (1 1 -1 1 1 F2- (1 -1 1 -1 -1

F3+ -1 -1 1 1 1 F3

- 1 1 -1 -1 -1F4

+ (1 -1 -1 1 1 F4- (1 1 1 -1 -1

F8+ -1 (1 (1 1 -1 F8

- 1 (1 (1 -1 1F9

+ 1 (1 (1 1 -1 F9- -1 (1 (1 -1 1

Tr+ 1 -1 -1 -1 -1 Tr- -1 1 1 1 1Cd

- 1 -1 -1 -1 1

Figure 6. Effect of faults F1+, F8

+, F9+, and Tr+ on variable xE,10.


to demonstrate that this approach will provide goodsensor network design for general cases too. Quantita-tive simulation of the TE flowsheet was carried out toobtain the effect of faults on the sensors selected forsingle-fault resolution. To illustrate, consider fault F1

+.Table 7 lists the effects of this fault on the selectedsensors. From actual simulation, it was found that whilethe effects of F1

+ were correctly predicted for variablesyD,6, yE,6, and F7, the prediction was wrong for variablexE,10. In the presence of fault F1

+, the profile of variablexE,10 is plotted in Figure 6a. It is seen that the final effectof F1

+ on xE,10 is negative, whereas according to Table7, it is positive. From Table 7, it is seen that the effecton variable xE,10 can be used to resolve fault F1

+ fromfaults F8

+, F9+, Tr+, F1

-, F2-, F3

-, and F4-. Now, the

sensor location results would still be meaningful if,despite wrong prediction of the effect on xE,10, it couldbe used to differentiate fault F1

+ from other specifiedfaults. The plots of effect of these faults on xE,10 arepresented in Figures 6 and 7. From these figures it canbe seen that the pattern of the effect of F1

+ on xE,10 isquite different from those of other faults. Hence, apattern recognition based fault diagnosis techniquewould be able to resolve F1

+ from faults F8+, F9

+, Tr+,F1

-, F2-, F3

-, and F4-. The sensor location obtained

using SDG with gains, therefore, is still meaningfuldespite wrong predictions of the effects on some vari-ables.

To further illustrate this idea, two other cases areconsidered. First is the wrong prediction of the effect offault F2

+ on variable yE,6, and the second case is thewrong prediction of the effect of fault F3

+ on variableyD,6. These two cases are presented in Figures 8 and 9,respectively. Once again, it can be seen from thesefigures that a pattern recognition based fault diagnosisapproach would be able to achieve the desired resolutionas expected based in Table 7. To further verify that the

set of selected sensors is indeed able to resolve betweenfaults, the effects of faults F1

+ and F2+ on the selected

sensors are plotted in Figures 10-14. From thesefigures, it is clearly seen that measuring the set ofselected sensors is sufficient to resolve between faultsF1

+ and F2+.

A complete comparison between the SDG-based cause-effect analysis and the actual simulation results isshown in Table 8. The entries in parentheses are theresponses predicted by SDG. Only where the numericalresponse does not match the SDG response the entriesare listed in brackets. The responses that are predictedas indeterminate are not listed in the table because theyare handled, per se, and hence do not generate wrongsensor location results. In all other cases in the table,the simulation results and SDG results match. It canbe seen from this table that to a large extent the SDG-based cause-effect analysis does a good job of predictingthe response of the variables to different faults.

The sensor location obtained by the use of SDG withgains representation, therefore, serves as a very goodinitial solution which may be further refined based onthe experiences of process engineer and operator. Animportant point worth mentioning here is that whilesensor location is performed in the optimization frame-work, various constraints may be imposed to decreasethe chances of obtaining an “incomplete” sensor location.A typical constraint might be to require that more thanone sensor is used to resolve a given pair of faults.Hence, even if one sensor is not able to resolve a givenpair, the other sensor might be able to ensure thedesired resolution.

Sensor location for reliability maximization is alsoperformed for this case. The results for various casesare presented next.

Sensor Location for Reliability Maximization. Inpart 1 of this series, various optimization-based formu-

Figure 7. Effect of faults F1-, F2

-, F3-, and F4

- on variable xE,10.


lations for sensor location were presented. Maximum-reliability and minimum-cost models were proposed. Aone-step optimization model was also proposed, whichmaximizes reliability and, among candidates giving themaximum reliability, selects the one having minimumcost. Results for the maximum-reliability and one-stepmodels are presented here. The advantage of using theone-step optimization model over the maximum-reli-ability model will be discussed.

Sensor Location for Single-Fault Resolution. Forthe TE system, 15 faults have been considered. For thesingle-fault resolution case, this involves generation of15C2 ) 105 pseudo faults, so that the system now has15 + 105 ) 120 faults. Of the original 15 faults, threepairs of faults were indistinguishable. The set of affectednodes for the pseudofaults corresponding to these threepairs is null. These null faults are not considered forsensor location. That leaves 120 - 3 ) 117 net faults.These faults form the rows of the bipartite matrix, D.The columns of this matrix are the possible sensornodes. An entry dij of D is nonzero if fault i can beobserved by placing a sensor on node j. Each row of the

matrix D corresponds to a constraint in the sensorlocation optimization problem. Hence, the reliabilitymaximization based sensor location formulation consistsof 117 constraints. As discussed in section 4.2 in part1, it turns out that many of these are redundantconstraints and need not be considered in the optimiza-tion formulations. Row k of the matrix D can be removedif there exists another row l, such that Dk ⊇ Dl, and theprobability of fault k is less than the probability of faultl. In such a case, the unobservability of fault l willalways be greater than or equal to the unobservabilityof fault k. Because the sensor location framework isbased on minimization of the maximum unobservability,fault k need not be considered in the optimizationformulation. After such reductions are performed, outof the original 117 constraints, only 9 are left. Hence,the maximum-reliability optimization formulation con-sists of 9 constraints (plus the cost constraint) and 61decision variables (corresponding to measurable nodes).Sensor location for this problem was performed withdifferent available costs. Results for the maximum-reliability model (without any cost term in the objective


+, F1-, F2

-, F4-, and Tr- on variable yE,6.


function) are presented in Table 9. Correspondingresults for the one-step optimization model (with the Rterm in the objective function) are presented in Table10. The optimization was carried out using the optimi-zation package LINDO. For the simple maximum-

reliability model (Table 9), for all of the cases, all of theavailable resources are used. For the one-step lexico-graphic optimization (Table 10), on the other hand,considerably less cost is used for most of the cases, andyet the same system unobservability is obtained. This


+, F1-, F3

-, F4-, and Tr- on variable yD,6.

Figure 10. Effect of faults F1+ and F2

+ on Ps.


shows the utility of the one-step optimization formula-tion in choosing a minimum-cost solution among allsolutions which maximize reliability. Note that for allcases for one-step optimization, R ) 0.0001 is used. Forsensor location for the one-step lexicographic optimiza-tion problem, the greedy search-based heuristic pre-sented in part 1 of the series is also used. The resultsare listed in Table 11. When Tables 10 and 11 arecompared, it is seen that, for all of the cases, optimalresults are generated by the heuristic algorithm.

Sensor Location for the Double-Fault Scenario.Sensor location for the TE case is also performed forthe double-fault scenario. Besides the original faults,simultaneous occurrence of two faults is also considered.For the original set of 15 faults, this leads to 15 × (14/2) ) 105 faults corresponding to double faults. Of these,seven pairs are of the same fault occurring in oppositedirections. These pairs are not considered. This leads

to a system of 15 + 105 - 7 ) 113 faults. Double faultscorresponding to control loop failures are also consid-ered. For each control loop, the control valve gettingstuck and the controlled variable sensor failing areconsidered. For the three considered control loops, thisleads to six control loop failure faults. Cases where onenormal fault and one control loop fault occur simulta-neously are considered. This leads to 15 × 6 ) 90additional pairs of double faults. Simultaneous occur-rence of two control loop failure faults is not consideredbecause the affect is not manifested on any processvariable.33 So, we now have a total of 113 + 90 ) 203faults for the double-fault case. The probability ofoccurrence of a double fault is the product of theoccurrence probabilities of the two original faults.Single-fault resolution algorithm is applied to thesystem consisting of these faults, resulting in a total of203 + 203 × (202/2) ) 20706 faults. Each fault


+ on yD,6.


+ on yE,6.


+ on F7.


corresponds to a constraint in the optimization-basedsensor location. As for the single-fault assumption case,for this case also, it turns out that most of theseconstraints are redundant. The number of nonredun-dant constraints is only 27. So, the sensor locationproblem consists of solving for a system with 27constraints (plus the cost constraint) and 61 decisionvariables. The results for various total available costsfor the one-step optimization approach are presentedin Table 12. The value of R used is 0.0001 as before.

Once again, LINDO was used for solving the optimiza-tion problems. Similar to the single-fault resolution case,heuristic-based results are also generated for the double-fault resolution scenario, and the results are listed inTable 13. In comparison with Table 12, it is seen thatthe heuristic-based results are optimal for all of theconsidered cases. The ability of the heuristic to generateoptimal results for both the single- and double-fault


+ on xE,10.

Table 8. Comparison of SDG Prediction with ActualSimulation Results

fault Ps yD,6 yE,6 F7 xE,10

F1+ +1 -1 -1 +1 -1 (+)

F2+ -1 +1 +1 (-) +1 +1

F3+ -1 +1 (-) +1 +1 +1

F4+ +1 -1 -1 +1 -1 (+)

F8+ -1 -1 -1 +1 -1

F9+ -1 (+) +1 +1 +1 -1

Tr+ -1 (+) -1 -1 -1 -1Cd

- +1 +1 (-) +1 (-) +1 (-) +1F1

- -1 +1 +1 -1 +1 (-)F2

- +1 -1 -1 (+) -1 -1F3

- +1 -1 (+) -1 -1 -1F4

- -1 +1 +1 -1 +1 (-)F8

- +1 +1 +1 -1 +1F9

- +1 (-) -1 -1 -1 +1Tr- +1 (-) +1 +1 +1 +1

Table 9. Sensor Location for Reliability Maximizationfor the TE Case: Single-Fault Case (LINDO-BasedResults)

available cost(cost utilized)

unobservability(log values) sensors selected

500 -2 s611000 -2 s2(2), s82000 -2 s3(3), s8, s9, s582100 -5 s2, s8, s9, s13, s553000 -5 s2, s8(2), s9, s13, s604000 -5 s2, s8, s9, s13, s53(12), s54, s574200 -8 s2(2), s8(2), s9(2), s13(2), s55(2)5000 -8 s2(2), s8(3), s9(2), s13(2), s55(2)6000 -8 s1(9), s2(2), s7, s8(2), s9(2),

s13(2), s55(2),s576300 -11 s2(3), s5, s8(2), s9(2), s12,

s13(3), s55(3)7000 -11 s2(10), s8(3), s9(3), s13(3),

s55(2), s578000 -11 s2(3), s8(4), s9(4), s13(3),

s55(2), s57(2)8300 -14 s2(4), s8(4), s9(4), s13(4), s55(3)9000 -14 s2(4), s5(2), s7(2), s8(2), s9(2),

s13(4),s55(10)

Table 10. Sensor Location with One-Step Optimizationfor the TE Case: Single-Fault Case (LINDO-BasedResults)

availablecost

costutilized


500 100 -2 s21000 100 -2 s542000 100 -2 s572100 2100 -5 s2, s8, s9, s13, s543000 2100 -5 s2, s8, s9, s13, s584000 2100 -5 s2, s8, s9, s13, s544200 4200 -8 s2(2), s5, s8, s9, s11,

s13(2), s55(2)5000 4200 -8 s2(2), s5, s8, s9, s11,

s13(2), s58(2)6000 4200 -8 s2(2), s5, s8, s9, s12,

s13(2), s55, s586300 6300 -11 s2(3), s5, s8(2), s9(2), s12,

s13(3), s54(3)7000 6300 -11 s2(3), s5, s8(2), s9(2), s10,

s13(3), s54, s55(2)8000 6300 -11 s2(3), s8(3), s9(3),

s13(3), s58(3)8300 8300 -14 s2(4), s8(4), s9(4),

s13(4), s54(3)9000 8300 -14 s2(4), s5(2), s7(2), s8(2),

s9(2), s13(4), s58(3)

Table 11. Sensor Location with One-Step Optimizationfor the TE Case: Single-Fault Case (Heuristics-BasedResults)

availablecost

costutilized


500 100 -2 s541000 100 -2 s542000 100 -2 s542100 2100 -5 s2, s8, s9, s13, s543000 2100 -5 s2, s8, s9, s13, s544000 2100 -5 s2, s8, s9, s13, s544200 4200 -8 s2(2), s8(2), s9(2), s13(2), s54(2)5000 4200 -8 s2(2), s8(2), s9(2), s13(2), s54(2)6000 4200 -8 s2(2), s8(2), s9(2), s13(2), s54(2)6300 6300 -11 s2(3), s8(3), s9(3), s13(3), s54(3)7000 6300 -11 s2(3), s8(3), s9(3), s13(3), s54(3)8000 6300 -11 s2(3), s8(3), s9(3), s13(3), s54(3)8300 8300 -14 s2(4), s8(4), s9(4), s13(4), s54(3)9000 8300 -14 s2(4), s8(4), s9(4), s13(4), s54(3)


cases underlines its utility for designing sensor net-works for large problems.

“Best Sensor Location” for a Given AvailableCost. Comparison of the sensor location results for thesingle-fault (Table 10) and double-fault (Table 12) casesbrings out a powerful feature of the proposed approach.It gives a systematic procedure for obtaining the bestsensor location for a given total cost, irrespective of thesingle/multiple (double, triple, etc.) fault assumption.For the TE system considered here, the maximumprobability of simultaneous occurrence of two faults is0.001. Given a total cost, the procedure then is to firstperform sensor location for the single-fault case. If thesystem unobservability turns out to be greater than orequal to 0.001, then a globally optimum sensor location,irrespective of the single/multiple-fault assumption, hasbeen obtained. So, for that available cost, double faultsneed not be considered for sensor location. Even ifdouble faults are considered while sensor location isperformed, the system unobservability will not change.On the other hand, if the system unobservability for thesingle-fault case turns out to be less than 10-3, thensensor location for the double-fault case also needs tobe performed. From Table 10, for the first three cases,the system unobservability is 10-2, which is greaterthan the maximum double-fault probability (10-3).Hence, for these cases, even when the double faults areconsidered, the system unobservability does not change,as is seen from Table 12. For the next case, when thetotal available cost is 2100, the system unobservabilityfor the single-fault case is 10-5, which is less than themaximum double-fault probability. So, for this case,double faults need to be considered in the sensor location

design. The system unobservability for the double-faultcase may be different from that of the single-fault case.From Table 12, the system unobservability for anavailable cost of 2100 is 10-3, which is greater than theunobservability obtained for the single-fault case.

Similarly, for the triple-fault case, the maximumprobability of occurrence of three faults simultaneouslyis 10-5. From Table 12, it is seen that the systemunobservability does not decrease beyond 10-5, for thecases when the available cost is <5500. So, for thesecases, even if triple faults were to be considered, thesystem unobservability will not change. For the caseswhen the total cost is g5500, performing sensor locationwith the triple-fault assumption might result in adifferent system unobservability and hence a differentsensor location. Therefore, for a given total cost between2100 and 5500, sensor location based on double-faultresolution is the optimal sensor location, independentof the multiple-fault assumption. This procedure can befollowed to obtain the best sensor network design for agiven total cost. An interesting observation for theresults presented here is that we get the same unob-servability with different sensor locations. For example,for the single-fault case (Table 10), for cases 1-3, thesame system unobservability (10-2) is obtained withdifferent sensors. This happens because the optimiza-tion problem has multiple optimal solutions, and theoptimizer (LINDO for our case) may converge to anyone of those solutions.

Sensor Location with Additional Constraints. Tofurther illustrate the utility of the sensor locationframework considered here, sensor location with someadditional requirements (constraints) is presented.

Table 12. Sensor Location with One-Step Optimization for the TE Problem: Double-Fault Case (LINDO-Based Results)

availablecost

costutilized


500 100 -2 s511000 100 -2 s522000 100 -2 s522100 2100 -3 s2, s8, s9, s13, s583000 2100 -3 s2, s8, s9, s13, s553100 3100 -4 s2, s7, s8, s9, s13, s52, s55, s584000 3100 -4 s2, s7, s8, s9, s13, s52, s55, s585000 4400 -5 s1, s2, s3, s5, s8, s9, s11, s13, s51, s52, s54, s55, s57, s585500 5500 -6 s1, s2(2), s3, s5, s8, s9, s10, s13(2), s49, s51, s52, s54, s55, s57, s586000 5500 -6 s1, s2(2), s3, s5, s7, s8, s9, s13(2), s49, s51, s52, s54, s55, s57, s587000 5500 -6 s1, s2(2), s3, s5, s7, s8, s9, s13(2), s49, s51, s52, s54, s55, s57, s588000 7900 -7 s1, s2(2), s3, s5, s8(2), s9(2), s10, s11, s13(2), s49, s51, s52, s54, s55, s57, s589000 7900 -7 s1, s2(2), s3, s5, s8(2), s9(2), s10, s11, s13(2), s49, s51, s52, s54, s55, s57, s589500 9200 -8 s1(2), s2(2), s3(2), s5(2), s8(2), s9(2), s10, s12, s13(2), s49, s51, s52(2), s54, s55(2), s57, s58(2)

Table 13. Sensor Location with One-Step Optimization for the TE Problem: Double-Fault Case (Heuristics-BasedResults)

availablecost

costutilized


500 100 -2 s551000 100 -2 s552000 100 -2 s552100 2100 -3 s2, s8, s9, s13, s553000 2100 -3 s2, s8, s9, s13, s553100 3100 -4 s2, s7, s8, s9, s13, s52, s55, s584000 3100 -4 s2, s7, s8, s9, s13, s52, s55, s585000 4400 -5 s1, s2, s3, s5, s7, s8, s9, s13, s51, s52, s54, s55, s57, s585500 5500 -6 s1, s2(2), s3, s5, s7, s8, s9, s13(2), s49, s51, s52, s54, s55, s57, s586000 5500 -6 s1, s2(2), s3, s5, s7, s8, s9, s13(2), s49, s51, s52, s54, s55, s57, s587000 5500 -6 s1, s2(2), s3, s5, s7, s8, s9, s13(2), s49, s51, s52, s54, s55, s57, s588000 7900 -7 s1, s2(2), s3, s5, s7(2), s8(2), s9(2), s13(2), s49, s51, s52, s54, s55, s57, s589000 7900 -7 s1, s2(2), s3, s5, s7(2), s8(2), s9(2), s13(2), s49, s51, s52, s54, s55, s57, s589500 9200 -8 s1(2), s2(2), s3(2), s5(2), s7(2), s8(2), s9(2), s13(2), s49, s51, s52(2), s54, s55(2), s57, s58(2)


(i) The case considered is the single-fault resolutionscenario, one-step optimization formulation with avail-able cost ) 2000. An additional constraint is addedwhich corresponds to the requirement that at least onesensor should be able to distinguish between faults F1

+

and Tr-. The sensors which can resolve faults F1+ and

Tr- are s7, s8, s9, s10, s11, and s12. Therefore, to ensurethat at least one sensor which can distinguish fault F1

+

from Tr- is selected, the following constraint is addedto the nine constraints considered for the single-fault,one-step reliability maximization formulation:

The problem is then solved in LINDO, which yields thesolution as x9 ) 1, with the system unobservability being10-2. The cost of the sensor s9 is 800. When it iscompared with the result presented in Table 10, it isseen that without constraint (50) sensor s57 is selectedfor available cost ) 2000, the system unobservabilityis 10-2, and the cost of the selected sensor is 100. Hence,sensor location with the additional constraint (50)results in the selection of a different sensor with ahigher cost.

Now, another constraint which corresponds to resolv-ing faults F1

+ and Cd- is added along with constraint

(50):

Performing sensor location for this case results inselection of sensors s4 and s8, with the system unob-servability being 10-2 and the cost of selected sensors1100. Hence, again the same system unobservability isobtained but at a higher cost. The advantage is that theselected sensors ensure that faults F1

+ and Tr- and F1+

and Cd- can be distinguished from each other.

(ii) The case considered is the single-fault, one-stepoptimization with available cost ) 4000. Two newconstraints are added for this case. They correspond tobeing able to resolve fault F1

+ from faults F8+ and Tr-

but using at least three sensors for both cases. Thesetwo constraints are as follows.

The constraint for resolving fault F1+ from F8

+ is

The constraint for resolving fault F1+ from Tr- is

The above two constraints ensure that there are atleast three sensors each to distinguish fault F1

+ fromfaults F8

+ and Tr-. Such constraints may be added forcases when the process consists of a critical fault whichshould definitely be distinguished from some otherfaults. These constraints are also useful when theunderlying fault models have some inherent uncertain-ties as the SDG with gains model used in this casestudy. When sensor location in LINDO is performedwith the above two constraints added as additionalconstraints, the selected sensors turn out to be s2, s8,s9, s10, s13, and s54(3), with system unobservability )10-5 and cost utilized ) 3100. Compared to the casewhen the above two constraints were not added (casecorresponding to available cost ) 4000 in Table 10), itis seen that the same system unobservability is ob-

tained, but the utilized cost increases from 2100 to 3100.Also, the selected sensor set has three sensors onvariable s54. Now, if the designer wants a distributedsensor network, a constraint restricting the values ofx54 e 2 may also be added.

After addition of this constraint along with constraints(52) and (53), the set of selected sensors turns out to bes2, s8, s9, s10, s13, s54(2), and s57 with the same systemunobservability (10-5) and utilized cost (3100).

Sensor location results for various cases where pro-cess specific requirements have to be satisfied by thefault monitoring system were considered in this section.The idea in presenting these cases was not to cover allpossible scenarios that the designer may wish to incor-porate in the selected sensor network but to demon-strate the utility of the approach developed in this paperfor incorporating such scenarios through appropriateconstraints in the relevant optimization problem.

6.4. Sensitivity Analysis. The sensor location for-mulations presented in this work use various quantita-tive information such as sensor failure and fault occur-rence probabilities. In this section, sensitivity analysiswith respect to some of these parameters is performed.The results for the single-fault resolution, one-stepoptimization case (Table 10) are used as the base casefor the sensitivity analysis.

Sensitivity to Sensor Failure Probabilities. Toperform sensitivity analysis with respect to sensorfailure probabilities, the failure probabilities of threerandomly chosen sensors, s9, s12, and s55, were simul-taneously increased by 10 times each. This is listed inTable 14. This is a sufficiently large increase in theprobabilities. The idea behind this large change is toconsider the sensitivity analysis as the worst casescenario analysis. The idea is to compare the earlieroptimal results in the presence of such large changesin the failure probabilities. Because the idea is toanalyze the sensor location in a worst case scenario, thecase where the actual failure probabilities are less thanwhat are used in the sensor location framework is notconsidered. For this case the estimation of systemreliability using the earlier values would be lower thanthat obtained by using the changed values.

With these changes, sensor location for single-fault,one-step optimization for various available costs wasperformed. The available costs were the same as thoselisted for the single-fault, one-step optimization resultsin Table 10. The new results are presented in Table 15.These results were generated in LINDO. For eachavailable cost, the new optimal results are listed in therow “optimal”. To compare the results, the unobserv-ability with the old solution is also listed in the row“old”. This is the system unobservability which isobtained in the changed scenario by the earlier, optimalsensor network (as listed in Table 10). Also, for caseswhere the earlier optimal sensor network is no longeroptimal, an optimal solution nearest to the earliersolution is listed in row “nearest optimal”. These resultswere obtained by adding explicit constraints in the

Table 14. Sensors Used for Sensitivity Analysis

sensor old failure prob. new failure prob.

s9 0.001 0.01s12 0.001 0.01s55 0.0001 0.001

x54 e 2 (54)x7 + x8 + x9 + x10 + x11 + x12 g 1 (50)

x4 + x13 g 1 (51)

x39 + x40 + x41 + x42 + x54 + x55 + x57 +x58 + x59 + x60 g 3 (52)

x7 + x8 + x9 + x10 + x11 + x12 g 3 (53)


optimization problem to ensure that all of the earlierselected sensors are again selected in the new solution.The optimal solution to this modified problem was thengenerated and is listed in the “nearest optimal” row.

From the results presented in Table 15, it is seen thatthe earlier solutions are still optimal for many cases.For cases where they are not optimal, the difference inunobservability from that obtained by the new optimalsolution is not very high, and in most cases by minoradditions to the earlier sensor location, an optimalsolution giving the minimum unobservability can bechosen. This indicates the robustness of the sensorlocation framework for selecting optimal or near-optimalsensor locations even in the presence of a large mis-match in the sensor failure probabilities. Anotherinteresting feature which comes out from these resultsis that this sensitivity analysis can be used to furtherscreen the optimal results generated by the one-stepoptimization framework. To understand this, considerthe unobservabilities for the old results for availablecosts of 5000 and 6000 as listed in Table 15. It is seenthat the old solution listed for available cost of 5000units gives a lower unobservability (10-7), as comparedto that obtained by the old solution listed for availablecost of 6000 units (10-6). From the earlier results (Table10), it is seen that both of these solutions had the sameunobservability (10-8) in the earlier scenario. Hence, theold solution listed for 5000 units is a better sensorlocation than the old solution listed for 6000 units. Thisis based on their objective function values in thechanged scenario and cannot be deduced from Table 10.Hence, if the designer can identify sensors whose failureprobabilities are not exactly known, then he can solvedifferent formulations in a lexicographic manner toidentify the optimal sensor location. As a first step,

failure probabilities which are most likely are used toobtain optimal sensor location. Among the multiplesolutions of this problem, the one which gives minimumunobservability in a formulation where the less likelyfailure probabilities are used is chosen. This idea is thesame as that of using cost as a criterion to identify anoptimal network from the multiple solutions obtainedby just solving a reliability maximization problem. Thisidea of performing sensor location with different prob-abilities can be integrated as another appropriatelyweighted objective function in the one-step optimizationframework developed in these papers.

Note that, similar to the case presented above, it canbe deduced that the old solution listed for cost 7000units in Table 15 is better than the old solution listedfor 8000 units.

Sensitivity to Fault Occurrence Probabilities.Similar to the above case, to perform sensitivity analysiswith respect to fault occurrence probabilities, the oc-currence probabilities of two randomly chosen faults,F1

+ and Cd-, were increased by 10 times each. Fault

F1+ corresponds to an increase in the variable F1.

Because decrease in variable F1 has also been consid-ered as a fault (fault F1

-), the occurrence probability ofF1

- is also increased by 10 times. This is listed in Table16. As for sensitivity analysis with respect to sensorfailure probabilities, these are also sufficiently largechanges in the fault occurrence probabilities and are

Table 15. Sensitivity Analysis with Respect to Sensor Failure Probabilities (of s9, s12, and s55) for the Single-Fault,One-Step Optimization TE Case (Table 10)

soln. type avail. cost cost used unob (log) sensors selected

optimal 500 100 -2 s2old 500 100 -2 s2optimal 1000 100 -2 s54old 1000 100 -2 s54optimal 2000 100 -2 s52old 2000 100 -2 s57optimal 2100 2100 -4 s2, s8, s9, s13, s55old 2100 2100 -4 s2, s8, s9, s13, s54optimal 3000 2900 -5 s2, s8, s9(2), s13, s57old 3000 2100 -4 s2, s8, s9, s13, s58nearest opt. 3000 2900 -5 s2, s8, s9(2), s13, s58optimal 4000 2900 -5 s2, s8, s9(2), s13, s58old 4000 2100 -4 s2, s8, s9, s13, s54nearest opt. 4000 2900 -5 s2, s8, s9(2), s13, s54optimal 4200 4200 -7 s2(2), s5, s8, s9, s11, s13(2), s54, s58old 4200 4200 -7 s2(2), s5, s8, s9, s11, s13(2), s55(2)optimal 5000 5000 -8 s2(2), s8(2), s9(3), s13(2), s54, s58old 5000 4200 -7 s2(2), s5, s8, s9, s11, s13(2), s58(2)nearest opt. 5000 5000 -8 s2(2), s5, s8, s9(2), s11, s13(2), s58(2)optimal 6000 5000 -8 s2(2), s5, s8, s9(2), s10, s13(2), s54(2)old 6000 4200 -6 s2(2), s5, s8, s9, s12, s13(2), s55, s58nearest opt. 6000 5800 -8 s2(2), s5, s8, s9(2), s10, s12, s13(2), s55, s58optimal 6300 6200 -9 s2(3), s5, s8(2), s9(2), s10, s13(3), s54(2)old 6300 6300 -8 s2(3), s5, s8(2), s9(2), s12, s13(3), s54(3)optimal 7000 7000 -10 s2(3), s5(2), s7, s8(2), s9, s10, s13(3), s54(2)old 7000 6300 -9 s2(3), s5, s8(2), s9(2), s10, s13(3), s54, s55(2)nearest opt. not possible to generate an optimal solution after ensuring that all the earlier sensors are selectedoptimal 8000 7100 -11 s2(3), s5, s8(2), s9(3), s10, s13(3), s54(3)old 8000 6300 -8 s2(3), s8(3), s9(3), s13(3), s58(3)nearest opt. 8000 7900 -11 s2(3), s5, s7, s8(3), s9(3), s13(3), s58(3)optimal 8300 8300 -12 s2(4), s5(2), s7(2), s8(2), s9(2), s13(4), s54(2), s58old 8300 8300 -10 s2(4), s8(4), s9(4), s13(4), s54(3)optimal 9000 8300 -12 s2(4), s5(2), s7, s8(2), s9(2), s10, s13(4), s54(3)old 9000 8300 -12 s2(4), s5(2), s7(2), s8(2), s9(2), s13(4), s58(3)

Table 16. Faults Used for Sensitivity Analysis

fault old occurrence prob. new occurrence prob.

F1+ 0.01 0.1

Cd- 0.01 0.1

F1- 0.01 0.1


based on performing sensitivity analysis in the worstcase scenario. As before, the case where the actualprobabilities are less than what are used in the formu-lations is not considered. Hence, in the modified sce-nario, fault occurrence probabilities of three faults areincreased from their respective earlier values by 10times each.

With these changes, sensor location for single-fault,one-step optimization for various available costs wasperformed. The available costs were the same as thoselisted for the single-fault, one-step optimization resultsin Table 10. The new results are presented in Table 17.These results were generated in LINDO. Before theproblem was solved in LINDO, constraint reductionprocedures as discussed earlier in this paper wereapplied. This resulted in 11 constraints (apart from thecost constraint) in the formulation as compared to theearlier case of nine constraints. In Table 17, for eachavailable cost, the new optimal results are listed in therow “optimal”. As before, to compare the results, theunobservability with the old solution is also listed in

the row “old”. This is the system unobservability whichis obtained in the changed scenario by the earlier,optimal sensor network (as listed in Table 10). Also, forcases where the earlier optimal sensor network is nolonger optimal, an optimal solution nearest to the earliersolution is listed in the row “nearest optimal”. Theseresults were generated as explained while the sensorprobability sensitivity analysis was performed.

As for the sensor probability case, the earlier resultsare optimal or near optimal for most of the cases in thechanged scenario. This once again underscores therobustness of the sensor location framework proposedin these papers.

6.5. Effect of r on the Computational Effort. Theconstant R was used as a weight for the “cost saved”term in the objective function in the one-step optimiza-tion formulation. To ensure that the solution to the one-step optimization problem is optimal in a lexicographicsense, a valid range for R was given in eq 39 in part 1of this series:

where C* is the total available cost for sensor locationand the constant a depends on the probability data usedin the problem. For the TE case study considered here,the constant a ) 1. While any R within this range willensure that the solution obtained is optimal in alexicographic sense, selection of different values of Rmay result in different computational efforts requiredto solve the resulting optimization problems. This aspectis briefly discussed in this section.

As for sensitivity analysis for probability data, for thiscase also, results for various available costs as listedfor the single-fault, one-step optimization case of Table10, using various values of R, were generated. Someimportant results are listed in Table 18. These resultswere generated in the ILP solver CPLEX becauseCPLEX is one of the most popular solvers for solvingILP problems. In general, depending on its internalalgorithm, a solver may result in different behavior tovariations in values of R. Hence, to provide results basedon a well-known standard solver, CPLEX was used. Fora given total cost, the maximum allowed R was derivedbased on eq 55. From the results presented in Table 18,it is seen that while choosing different values of R withinthe maximum allowed limit does not seem to affect thenumber of nodes evaluated by the solver, use of a higherR (10-3) does result in a significant reduction in thenumber of evaluated nodes. This indicates that selectionof better problem-specific bounds for R and use of thehighest R within those bounds may result in a reductionin the computational effort. For the solutions listed in

Table 17. Sensitivity Analysis with Respect to FaultOccurrence Probabilities for the Single-Fault, One-StepOptimization TE Case (Table 10)

soln.type

avail.cost

costused

unob.(log) sensors selected

optimal 500 0 -1 noneold 500 100 -1 s2optimal 1000 0 -1 noneold 1000 100 -1 s54optimal 2000 1200 -2 s9, s13, s54old 2000 100 -1 s57nearest opt. 2000 1200 -2 s9, s13, s57optimal 2100 2100 -4 s2, s8, s9, s13, s54old 2100 2100 -4 s2, s8, s9, s13, s54optimal 3000 2400 -5 s2, s8, s9, s13(2), s54old 3000 2100 -4 s2, s8, s9, s13, s58nearest opt. 3000 2400 -5 s2, s8, s9, s13(2), s58optimal 4000 2400 -5 s2, s8, s9, s13(2), s58old 4000 2100 -4 s2, s8, s9, s13, s54nearest opt. 4000 2400 -5 s2, s8, s9, s13(2), s54optimal 4200 4200 -7 s2(2), s8(2), s9(2), s13(2), s54(2)old 4200 4200 -7 s2(2), s5, s8, s9, s11,

s13(2), s55(2)optimal 5000 4500 -8 s2(2), s4, s5, s8, s9, s11,

s13(2), s54(2)old 5000 4200 -7 s2(2), s5, s8, s9, s11,

s13(2), s58(2)nearest opt. 5000 4500 -8 s2(2), s5, s8, s9, s11,

s13(3), s58(2)optimal 6000 4500 -8 s2(2), s4, s5, s8, s9, s12,

s13(2), s55, s58old 6000 4200 -7 s2(2), s5, s8, s9, s12,

s13(2), s55, s58nearest opt. 6000 4500 -8 s2(2), s5, s8, s9, s12,

s13(3), s55, s58optimal 6300 6300 -10 s2(3), s5, s7, s8(2), s9(2),

s13(3), s54(2), s58old 6300 6300 -10 s2(3), s5, s8(2), s9(2), s12,

s13(3), s54(3)optimal 7000 6600 -11 s2(3), s5, s7, s8(2), s9(2),

s13(4), s54(3)old 7000 6300 -10 s2(3), s5, s8(2), s9(2), s10,

s13(3), s54, s55(2)nearest opt. 7000 6600 -11 s2(3), s4, s5, s8(2), s9(2), s10,

s13(3), s54, s55(2)optimal 8000 6600 -11 s2(3), s8(3), s9(3), s13(4), s54(3)old 8000 6300 -10 s2(3), s8(3), s9(3), s13(3), s58(3)nearest opt. 8000 6600 -11 s2(3), s8(3), s9(3), s13(4), s58(3)optimal 8300 8300 -13 s2(4), s5, s8(3), s9(3), s12,

s13(4), s55(3)old 8300 8300 -13 s2(4), s8(4), s9(4), s13(4), s54(3)optimal 9000 8700 -14 s2(4), s5(2), s7(2), s8(2), s9(2),

s13(5), s55(3), s57old 9000 8300 -13 s2(4), s5(2), s7(2), s8(2), s9(2),

s13(4), s58(3)nearest opt. 9000 8700 -14 s2(4), s5(2), s7(2), s8(2), s9(2),

s13(5), s58(4)

Table 18. Sensitivity of Computational Effort to r

avail.cost max R R

no. of nodesevaluated

costused

U(log)

10-5 23000 3.33 × 10-4 10-4 2 2100 -5

3 × 10-4 210-3 0

10-6 234000 2.5 × 10-4 10-4 23 2100 -5

10-3 2

10-8 226000 1.66 × 10-4 10-6 22 4200 -8

10-4 2210-3 2

0 < R < 1/(aC*) (55)


Table 18, even though R ) 10-3 was outside the rangespecified by eq 55, it still resulted in optimal solutionsfor the considered cases. This is in keeping with theobservation made in part 1 that the given range for Ris a sufficient condition and not a necessary one.Depending on the actual problem at hand, R valueswhich lie outside this range may also yield optimalsolutions. In fact, depending on the specific problemsbeing solved, the upper limit on R as given by eq 55 maybe increased. As an example, consider the case ofavailable cost ) 3000. From Table 10, it is seen that,for available cost ) 2100, the cost utilized is also 2100.Hence, while solving for the case when cost available )3000, we already know that the cost utilized will be atleast 2100 and the maximum cost saved (variable xs,which was the slack in cost constraint in part 1) can beonly 900 and not 3000. Hence, following the derivationfor the range of R as given in part 1, it can be seen thatthe upper limit of R is then 1/900, which is greater than10-3.

Further investigations on the effect of the value of Ron the computational effort are currently underway. Animportant point here is that we have just listed thenumber of nodes evaluated as an indication of thecomputational effort. Besides this parameter, there maybe several other important factors governing the overallcomputational effort.

6.6. Sensor Location with Fault Measurements.In the formulations considered so far in this paper,faults were not considered as potential measurements.In this section, some sensor location problems wheremeasurable faults are also considered as potentialmeasurements are presented. In general, placing sen-sors on all measurable faults may not always result ina cost optimal sensor network. Considering the measur-able faults as decisions variables in the sensor locationdesign problem gives an optimal solution which willindicate which measurable faults should be measured.Some such cases are presented in this section, and theresults are compared with the corresponding sensorlocation designs where faults were not considered asmeasurable variables. Some of the results and discus-sion presented here have been taken from Bhushan andRengaswamy.40 Once again, these results are based onthe use of SDG with gains to generate the requiredcause-effect information.

Sensor Location for Single- and Double-FaultResolution When Faults Are Not Considered asMeasurable Variables. The sensor location results forsingle-fault resolution were presented in section 6.3. Theresults presented there were obtained using the heu-ristic discussed in work by Bhushan and Rengaswamy.33

Results for single-fault and double-fault resolution,obtained by solving the appropriate set cover problemsin LINDO, are presented in Table 19. For the double-fault resolution case, the number of indistinguishablesets is quite large (4029). This is to be expected becauseof the tightly coupled nature of the process. Before the

sensor locations were performed, a redundant con-straints’ removal procedure, as discussed in Bhushanand Rengaswamy,40 was applied to reduce the numberof constraints in the set cover problem. For the single-fault case, the number of constraints decreased from 120to 8, and for the double-fault case, from 20706 (notconsidering simultaneous occurrence of two control loopfailure faults) to 18.

Sensor Location for Single- and Double-FaultResolution When Measurable Faults Are AlsoConsidered. Sensor location is also performed for thecase when faults are also considered as measurablenodes. For this case, faults which can be measured (suchas flow-rate changes) are considered as variables wheresensors can be placed. The corresponding cost data aregiven in Table 20. When the results in this table arecompared with the faults considered for this process, itis seen that only Cd

- is not a measurable fault. All otherfaults, being either flow-rate or temperature changes,are measurable. For this case, the LINDO-based single-fault and double-fault resolution results are presentedin Table 21. When these are compared with the resultspresented in Table 19, it is seen that, for the case whenmeasurable faults are considered, all faults are distin-guishable from each other for the single-fault resolutioncase. This is achieved by selection of some sensorsmeasuring the faults (such as s65). An interestingfeature is that not all measurable faults are selected assensor variables for the single-fault case in Table 21. Asolution where sensors are placed on all measurablefaults will not be optimal for the single-fault resolutioncase. For the double-fault case, the number of indistin-guishable pairs reduce from 4029 (in Table 19) to only28 (in Table 21), which represents a significant increasein the attained resolution. Once again, redundantconstraints were removed before solving these problemsin the LINDO framework. For the single-fault resolutioncase, this resulted in the reduction of constraints from120 to 33, and for the double-fault case, the number ofconstraints reduced from 20706 to only 21.

The results presented in Tables 19 and 21 are basedon solution of the set cover problem (of type problemIV in part 1 of the series). They minimize the numberof selected sensors to achieve the desired resolution butdo not take the cost of the sensors into consideration.We next present the results for sensor location for theabove cases with minimum cost as the objective function(problem V in part 1).

Minimum-Cost Sensor Network. (a) Single- anddouble-fault resolutions when faults are not consideredas measurable variables: The results for these two casesare presented in Table 22. Once again, these results

Table 19. Sensors for the TE Process withoutConsidering Measurable Faults (LINDO Solution)

case sensors selected indistin. setstotalcost

single-fault res. s2, s8, s9, s13, s59 [F1+, F4

+], [Cd-, F8

-],[F1

-, F4-]

2200

double-fault res. s1, s2, s3, s5, s8, s9,s12, s13, s49, s50,s51, s54, s55, s56,s60

5300

Table 20. Sensor Data for Measurable Faults

fault sensor no. sensor cost fault sensor no. sensor cost

F1 s62 300 F8 s66 300F2 s63 300 F9 s67 300F3 s64 300 Tcr s68 500F4 s65 300

Table 21. Sensors Selected for the TE Case afterConsidering Measurable Faults (LINDO Solution)

case sensors selectedindistin.

setstotalcost

single-fault res. s4, s8, s9, s65, s66, s67, s68 none 3300double-fault res. s2, s8, s9, s13, s49, s58, s59,

s61, s62, s63, s64, s65, s66,s67, s68

5800


were generated by solving the appropriate minimum-cost set cover problem (problem V in part 1 of the series)in LINDO. Before these solutions were generated,redundant constraints were removed, as was doneearlier. Hence, for the single-fault resolution case, only8 (instead of 120) constraints were considered, while forthe double-fault resolution case, 18 (instead of 20706)constraints were taken into account.

(b) Single- and double-fault resolutions when faultsare also considered as measurable variables: TheLINDO-generated results for this case are presented inTable 23. As before, redundant constraints were re-moved before obtaining these results.

When these results are compared with the cases whenfaults were not considered as measurable variables inthe sensor location framework (Table 22), it is seen that(a) for the single-fault resolution case, the cost of thesensor network is lower when faults are allowed to bemeasured and also better resolution is achieved for thiscase and (b) for the double-fault resolution case, eventhough the cost of the sensor network has gone up whenfault measurements are allowed, the resolution attainedincreases significantly (only 28 indistinguishable pairsas compared to 4029 earlier).

Hence, consideration of measurable faults in thesensor location framework allows a better sensor net-work design, which yields significantly more resolution.

Comparison of Table 22 with Table 19 and of Table23 with Table 21 clearly indicates that, instead of justthe simple set cover problem being solved without costconsideration (problem IV in part 1), the sensor networkdesigner should locate sensors based on the minimum-cost set cover formulation (problem V in part 1). Theminimum-cost set cover solutions give the same perfor-mance at a lower cost.

7. Conclusions

A framework for reliability-based sensor location forfault diagnosis was presented in part 1 of the series.The application of the proposed framework was pre-sented in this part to design a sensor network for theTE case study. Issues involved in generating the SDGwere discussed. Use of arc gains to reduce the ambiguityin fault modeling based on SDG was presented. Sensorlocation was performed for various cases, and resultswere compared. A systematic procedure to design the

best sensor network irrespective of the single/multiple-fault assumption was presented. The procedure pre-sented here enables the designer to obtain a good initialdesign without too much quantitative information aboutthe process.

Integration of sensor network design for fault diag-nosis, data reconciliation, and controls appears to be aninteresting area for further research.

Nomenclature for the TE Model

Cd- ) catalyst deactivation, with the nominal value being1

Fi ) molar flow rate of stream i (kmol/h)Fi* ) molar “pseudofeed” of component i (kmol/h) where i

) A, B, ..., HF 10

p ) apparent stream 10 flow rate (kmol/h)F 10

/ ) bias adjustment for stream 10 (kmol/h)Fi,5 ) molar flow of component i in stream 5Mi ) molecular weight of stream iNi,m ) total molar holdup of component i in the feed mixing

zone (kmol), where i ) A, B, ..., HNi,p ) total molar holdup of component i in the product

reservoir (stripper base) (kmol), where i ) G and HNi,r ) total molar holdup of component i in the reactor

(kmol), where i ) A, B, ..., HNi,s ) total molar holdup of component i in the separator

(kmol), where i ) A, B, ..., HP i

sat,(Tr) ) vapor pressure of pure i at the reactor temper-ature, where i ) D, E, ..., H

P isat,(Ts) ) vapor pressure of pure i at the separatortemperature, where i ) D, E, ..., H

Pi,r ) partial pressure of i in the reactor (kPa), where i )A, B, ..., H

Pi,s ) partial pressure of i in the separator (kPa), where i) A, B, ..., H

Pm ) total pressure in the feed mixing zone (and stripper),kPa

Pr ) total pressure in the reactor, kPaPs ) total pressure in the separator, kPaR ) gas constantRj ) molar rate of reaction j (kmol/h), positive if the

reaction goes to the products as writtenTm ) absolute temperature in the feed mixing zone (359.3

K), assumed constantTcr, Tr ) reactor temperature in °C and Kelvin, respectivelyTcs, Ts ) separator temperature in °C and Kelvin, respec-

tivelyVLp ) liquid volume in the product reservoir (stripper) base,

m3

VLr ) liquid volume in the reactor, m3

VLs ) liquid volume in the separator, m3

Vm ) total volume of feed mixing zone (150 m3)Vr ) total reactor volume (36.8 m3)Vs ) total separator volume (99.1 m3)VVr ) vapor volume in the reactor, m3

VVs ) vapor volume in the separator, m3

xi,j ) mole fraction of component i in liquid stream j, wherei ) A, B, ..., H

xi,r ) mole fraction of component i in the reactor liquid,where i ) A, B, ..., H

yi,j ) mole fraction of component i in vapor stream j, wherei ) A, B, ..., H

zi,j ) mole fraction of component i in feed stream j, wherei ) A, B, ..., H

Greek Symbols

Rj ) parameters used in the reaction rate equations,dimensionless with a nominal value of unity

âi ) used to adjust the flow/pressure drop relation

Table 22. Minimum-Cost Sensor Location When FaultsAre Not Considered as Measurable Variables (LINDOSolution)


setstotalcost

single-fault res. s2, s8, s9, s13, s57 [F1+, F4

+], [Cd-, F8

-],[F1

-, F4-]

2100

double-fault res. s1, s2, s3, s5, s8, s9,s11, s13, s49, s51,s52, s54, s55, s57,s58

5100

Table 23. Minimum-Cost Sensor Location When FaultsAre Also Considered as Measurable Variables (LINDOSolution)


setstotalcost

single-fault res. s1, s3, s51, s62, s63, s64,s65, s66

none 1800

double-fault res. s2, s8, s9, s13, s49, s52,s54, s58, s62, s63, s64,s65, s66, s67, s68

5300


γi,r ) activity coefficient of i in the reactor liquid phase,where i ) D, E, ..., H

γi,s ) activity coefficient of i in the separator liquid phase,where i ) D, E,..., H

φi ) stripping factor for component i, where i ) G and Hνij ) stoichiometric coefficient of component i in reaction j,

where i ) A, B,..., H, j ) 1, 2, 3 (the sign convention isthat νij < 0 if i is a reactant in reaction j)

Fi ) molar density of pure liquid i (mol/m3), where i ) A,B, ..., H

øGH ) purity of G + H in the product (as a fraction)

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Received for review May 15, 2001Revised manuscript received October 30, 2001

Accepted November 29, 2001

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