design o f a sc3nsolnetwork. Although this problem i s an irnportant one i n design of new plants, it can also be used to retrofit the nieasurement structure of existing plants by identifying new variables that need to by measured for improved monitoring and control of the process. The design of a sensor network is influenced by different considerations, such as controllability of the plant, safety, reliability, environmental regulations, and accurate estimation of all important variables. If the estimates of variables are used in control, then the accuracy of estimation also has an effect on control performance. Keeping the scope of this text in mind, in this chapter we only consider the design of sensor netwolks for maximizing accuracy of estirnatior: through data reconciliation, while giving due considerations to the cost of the design. Moreover, the treat~nentin this chapter is limited to linear (,flow) pmcesses only. It should be noted that the objective of maximizing estimation accuracy is only one of the impolttnt considerations and a compreher~sivedesi~1i should also take into account other requirements mentioned ahove. T11ix problern is receiving increasing attenticn from different resealihers in recrrlt years and new solution strategies are being developed. It may req~~ire se.r era1 vears of additional effort before these solutioris are irnplernzntrd in practice.
ESTIMATION ACCURACY OF DATA RECONCILlATfON Eefore .risediscuss rhe rna:bematical iormu1a;ion of t!lt; sensol nctc,o:h design problern. we first e x ~ ? m i n the e estiri:ztioi~ accuracq obtained through data reconciiiation and ihe effect that the cilcice of 111casure(! variables have cn it. The flow reco:~ci!iaticn exampie discussed in Exain111e 11 i l l Chapter 1 doe:; hi~hiightsome of these issuzs. \17e ~eexai~lii~c~ this problem in greater depth.
Example 101
.. .;concilcd c.,:;i,n:;., .
:;f :hc stream flows for the process shown iii Figure 1? were presented in Tables 1  1 and 12 for different c h v i c e ~0;' measured variables. Let us consider the results of Table 11 fcr which ail flows are measured. and Case 2 of Table !2 for which the only the i l o . \~ of streams 1 and 2 are measured. Thc difference bettileen the cstimatcd and true values of all streams (estimation errors) can be computed t r  o i ! ~ these results and are shown in Table [email protected]!, along with the surn of squarrs of the cstirnation errors. T I . A
From these results, we can observe that the error in the flow estimates of streams 1, 3, 5 , and 6 are much less for the case when all flows are measured as compared to the case when only the flows of streams 1 and 2 are measured. The estimation errors, however, for streams 2 and 4 are marginally more when all flows are measured. Although, from a purely intuitive viewpoint, we expect the estimation errors to be reduced if more measurements are available, it is clear from this example that the estirnalion errors for all variables are not reduced &hen more nzeusurenzents are nzade. This is more forcefully brought out by the example presented by Mah [4] where it was shown through simulation that, as more measurements are made, a larger fraction of the reconciled estimates have smaller errors. It is thus clear that it is not appropriate to focus on any perticular variable for the purpose of designing sensor networks to increase accuracy of estimation. Table 101 Estimation Errors in Reconciled Flows for Process in Figure 12
The overall measure for estimation accuracy was first proposed by Kretsovalis and Mah [5] and is defined by
It should be noted that as J decreases the estimation accuracy irzcreases and therefore we refer to it as the measure of the overall estimation accuracy. It is implicitly assumed in the above definition that all variables are observable. If there are unobservable variables, then the measure of overall estimation accuracy can be written as the expected sum of squares of estimation errors for observable variables only. We will ignore this modification and restrict our considerations throughout this chapter to the design of sensor networks which ensure L I I ~observability of all variables. It can be proved [ 5 ]that the overall estimation accuracy given by Equation 101 for a data reconciliation solution is given by
Estimation Errors
All Flows Measured
Flows 1, 2 Measured
!
2 3 4
5 6 Sun) Squares of E5tinlarion Esrors
As an overall measure of estimation accuracy, we can use thz s u n cf squares of the estimation errors of all varizbles (which represents the overall inaccuracy). Table 101 shows that the sum of square of estimation errors is less when more meacurements are available. We can therefore use this .measure in order to design sensor networks. This measure, however, also depends on the measured values which can be different each time due to their random characteristics. The appropriate measure that we can use for design purposes is the expected value of the sum of squares of the estimation errors which will be independent of the actual outconx of the measurements and will depend only on the sensor network design as well as the inherent process structure.
where S is the covariance matrix of esiinlaiion errors and the operator Tiis the trucr of the matrix. It should be noted t!lat the diagonal elements of S are rhe varia~lcesof the estimation eIrors and J is therefore the sum of the variances of estimation errors, which is eqtiivalenr to the expecced sum of squarcs of the estimation errors. Alti30;lgh it is possibk to cierive the estimatior? error covariance matrix from the data reconciliation soluticns for the measured and unmeasured variables ziveri in Chapter 3, we desclibe later an alternative sequential update procedure which is more usefa! in the context of the sensor ~ e t w o r kdesigri problem. Different approaches have been developed to soive the sensor network design problem. In the fclioviing section, we discuss these methods which consider objectives of estimation accuracy, observability and cost.
SENSOR NETWORK DESIGN Methods Based on Matrix Algebra
Let us consider a linear flow process for which the material balances are given by Equation 32:
Let us separate the flow variables into a set of nm irzdeperzdenr variables XIand a set of rn dependent variables x, and rzcast Equation 103 as where the variables x represent the strean? flows. In general, only some of thew flow variables are measured and the relationships between the measurements and stream flows can in general be represented uslng
where each row of matrix H is a unit vector with unit in the column corresponding to the flow variable which is measured, the number of rows being equal to the number of rneasurernents. We have chosen Equations 103 and 104 to represent a partially measured linear process rather thari the equivalent alternative model Equations 31 and 311, because it is Inore convenient for the purposes of designing sensor networks. Mininzum Observable Serzsor Networks We are interested in sensor networks which ensure the observability of all variables. We, therefore. first address the question of the minimum nilrnhe; of measurer?lents to he made in order to ensure that every flow variable is observable. LVe will for con\.e~iiencerefer tci silch a design .ts a !nitllmrlin c!O.setvni)!e seizsot. rzerlcoik. If there are 11 stream flows to be ebtinlated and :ye have nz flow ccnstrair~ts,then it i:, evident rh:tr at leaxt I:m flows must be specified. In other words, :tic n~inirnumr,umSer cf ~neasuremerltsis 11171. For the flow process cor~sideiedin Example 11. the ~niniaurnof meas~~rernenis io be made in order to ecsure that all stream fiou,s are observable ic 2, since there are 6 streams and 4 flow balanczs. Czse 3 of Ex:imple 12 is a specific instance of a minimum cbserl~ablesensor uet\vork for this process. AII additional poinl to be noted is that in a niinirnv111 obsergable sensor network none of the measured variables is reduldant and the rtconci!ed values of these variables are exactly eqttal to their respective lneasured values. No: every combination of n  ~ n measureinents will give rise to an observable systern. however. For cxaruple. in Case 3 of Example 12. although two rneasurernents are made, the flowis of streams 2 to 5 are unobservable. The condition that a sensor network must satisfy in orde1 to ensure observability of all variables in a linear process is discussed as follows:
The dependent variables are chosen in such a way that the matrix A, is nonsingular. If we ~neasureonly the independent variables, then we can use Equation 105 to compute unique estimates of the dependent variables as
where
It is thus clear that this sensor network is a minimum observable sensor network. Therefore, the condition to be satisfied by a minimum sensor network in order to give rise to an observable system is that the submatrix corresponding to the unmeastlred variables should be nonsingular. Note that this irnplies that the columns of the constraint matrix corresponding to unmeasured variables are lineariy indspendent (which is the oilservability condition in Exercise 33). If :nore measurenients zre made than the niiinirnurn 1equired to ensue cbservabiiity of all variables, then we obtain a rt.du:zcIatarl: seizscr r~erwork desigiz. Ever: if a rzdundant sensor netwoik is designed. it does not automatically imp!y !hat all flows are observable. There coltld be stlosets of variables v,.hich are unobservabIc while the rest are redundant. A redund it' we dant se:?sor network gives rise to an observable process if a ~ only can choose rzin independent t~ariablesfrom among the set of rneaa:wci vzriables such that the coi~straintsubrrlatrix correspcnding to the remaining variables is nonsingular. In this case, the dependent set contains oce or more measured variabies. We refer to such a design as a 1eclr~nd~znr obsen>ablrscrzsor network. We can always obtain a redundant observable sensor network starting from a miniinurn observable sensor network by choosing to additionally ~xeasureone or niore of the unmeasured variables in the minimum scnsoi network.
Estir~mtionAccuracy of Minimurn Observable Sensor Networks now consider a minimum observable sensor network design we and obtain the overall estimation accuracy for reconciled estimates. Based on the discussion above, we can choose th2 measured variables as the independent variables. For any observable sensor network (nonredundant or otherwise), the estimates obtained using data reconciliation must satisfy constraint Equation 103. Thus, Equation 106 can also be used to relate the reconciled estimates for an appropriate choice of the independent and dependent variables. Since there is no redundancy in a minimum observable sensor network, the estimates of the independent (measured) variables are equal to their respective measurernents. This implies that the covariacce matriy of estimation errors in the independent variables is equal to QI,which is the covariance matrix of meastlrenlent errors of the independent variables. Let us denote the covariance matrix of estimation enors corresponding to the independent and dependent variables by SIand S,,. respectively. Then \ve obtain from the preceding arguments that
matrix corresponding to the dependent variables is nonsingular). For each combination, the independent variables can be chosen as the measured variables and the measure J for each sensor network design can be computed using Equation 1010. The combination that gives the least J is the optimal sensor network design that we seek.
Example 102 We will illustrate the minimum observable sensor network design that maximizes estimation accuracy for the ammonia process shown in Figure 101. We will limit our consideration only to the overall mass flows of this process. For simplicity, let us consider the case when the flow sensors used for measuring any stream have an error with variance equal to I. Since there are 8 streams and 5 process units, we require a minimull1 of 3 sensors to observe all variables. The different feasible combinations of sensor locations along with the corresponding measures of estimation accuracy are shown in Table 109. We can observe that there are 6 optimal sensor network designs corresponding to sensor locations (1, 2. 6). (2, 5, 7), (1, 3, 6). (3, 5. 7). (1, 4, 6 ) . and (4, 5, 7) with a minimum expected sum square of e s t i ~ ~ a t i o n errors equal to ! 1 units. It can also be observed tllar, although there ale
Lising Equatians 188 and 106 :rJecan show that
Combining 5quatic)ns 102. 103. a i ~ d109. ihc measure tiir overall esti:natio:: accuracy for a aiinimurn obszrvahle ser?s(>rnetwork car1 be expressed as
A niininit~rnobservable sensor rietwork design that minimizes J defined by Equation 1010 is desired. 111order to solve this problen~.a nlixed integer optilnization problem can be used, which is desciibed later in this chapter. Hcre we \vill use a nai've approach and examine every ft.trsil2le combination to d e ~ c r ~ n i nthe e optimal solution. We can select every co111bin;ttion of n111 independent variables (such that the sub
Figure 103. Simplified ammonia process
308
[)(,r
56 combinations of choosing 3 sensors locations out of 8 sensors locations, only 32 of these combinations give rise to observable sensor network designs. Table 102 Minimum Observable Sensor Network Designs for Ammonia Process No.
Measured Variables
Overall Expected Estimation Error
Redundant observable sensor networks. The measure of estimation accuracy for redundant observable sensor networks, can be computed using simple update formulae developed by Kretsovalis and Mah [ 5 ] .Let us begin with a minimum observable sensor network corresponding to a set of nm measured independent variables, xl and the remaining unmeasured variables xD. The measure of estimation accuracy for this sensor network is given by Equation 1910, Let us consider the addition of a new . q be the variance sensor to measure one of the variables in the set x ~Let in the error of this new measurement. As in Eqazuation 104, the new measurement y can be related to the variables x by
where
and hT is a unit row vector with unity in the colun~nposition conesponding to the new @variablebeing measured. The expected es~imateerror covariance mr:trices of the independe~lf and dependent v~ariaSles after the ., addition of this new measurenient, S,, and S1, respectively are $\.en by
Iji
where
The change in the nteasule of ictirnntinn :Icc:vracy due t~ of this new measurement can be show11 to he
!hi_. additioit
Equations 1012, 1015, and 1016 can be directly used to compute the change in the measure of estimation accuracy due to the addition of a new measurement using the covariance matrix of estimate errors in the preceding sensor network design solution. Thus, starting from a minimum sensor network design solution the measure of estirnation accuracy for a redundant sensor network design can be obtained by successively adding the required measurements, and using the update equations after each addition. Similar equations can also be derived for deletion of a measurement from a redundant sensor network design solution. In this case, the change in the measure of estimation accuracy is given by
Choosing these measured variables as the independent variables, the covariance matrix of estimation errors in the independent variables is the identity matrix (of dimension 3). The matrix F is given by
If we choose, in addition, to measure the flow of stream 1 , then the vector which relates this measurement to the independent variables is given by
where
and the updated covariance matrix of estimate errors i n the independent va1iables is given by
S, = S, + ~ , s , K G ~ s ,
The value of kl from Equation 1015 is equal to 1!(1+2) = 113 and hence the decrease in estirnation error can be calculated from Equation 1016 a5 3.3333. The updated covariance matrix of estimate errors of independent variables is given by
(1020)
It should be noted that rhe sct of independent variables and d~.perldzni lsariablcs do not change as new ~nea:;iirzinei;ts arc added or dclztzd. so that after a series of additions 01 ciele:ic?ns each of these sets can contain a mixture of measured and unl~ieascredvariables. Care must be taken. hou~ever.when a measuremelit is deleted to mrure that an ~~nobszr\,aS!e dzsign is not ob;ai!lcd. In f x t , if a mea\urernerlt is deleted which can icad to :in l.:nobservable process. then the der,c;min'ltor :II Equatior? 10 19 will become zero and this can be used as an indicator to iivoiti 5uch choices.
Example 103 We will consider the arurnonia process example and cotiipute the decrease in the expected enoi in estirnatcs for the addition of a single measurement to a minimurn obselvable serlsor network design. The 1:;11iances of all sensor errors are taken as unity as before. For this purpose. we will start with the optimal minimum observable sensor design obtained in Example 101 in which the variables 2, 5. 7 are measured.
ir, order to design an optimal redunda~tobservable sensor netm.orh ffir a specified number of sensors, say i  ( I  > I Z  ~ ? : ; ive can start with any ~:lini m u ~ ngbsen~ablesensor network design and add iiz+rt: additiocal sensors, Gne a: a tirnz and update the 1neasu1eof estination r;cclilacy usi:iz FAuations 1012. iii 14, ail3 i0 15. \Ve can then iriocc2ir the sc11sa:sby. in tGm. add:r?g 3 new nieasurenlent and deleting an exis:ing rnzasal=meill m get a new redu~ldantobservable sensor ~ e t w o r kdesigil conristing of isensors. Equations 1017 through 1019 can be used ior upaatir~gthe measure of estimation accuracy when a measurement is deleted. In [his mar  > a 1 'l l I'" ' L ~ S ~ ~ Y L L ~ l l b i ~ l a ic~lr i ori isensor ~ iocations can be examined i r c\r'~,rtr, find the design which gives the maximum expected estimation accuracy. 'This will result, however, in an exponential number of solutions to be examined for a general problem. Kretsovalis and Mah 151 outlined two suboptimal design procedures for a redundant censor network desigfi for a specified number of sensors, as described o n the following page. L L I Y ~ ,
I_fe\ry~r4 .SL'I~ior NL'III.OIk \
Algorithm 1 Step 1. Determine the optimal minimum observable sensor network. Step 2. Add a new sensor in turn to each of the remaining unmeasured variables and co~nputethe reduction in estimation error using Equation 1016. Step 3. Based on the results of Step 2, select the best r17+177 sensor locations (the locations that give maximum reduction in estimation error) to obtain the redundant sensor network design.
Algorithm 2
313
addition of a single sensor as observed from colu~nn1 of Table 103. On the other hand, if we apply Algorithm 2, then we would first select variable 1 (or 6) to be measured since this gives the maximum estimate error reduction (column 1). In the next iteration, we select variable 6 (or 1) to be measured since this gives maximum estimate error reduction (column 2) among all remaining variables, and, finally, variable 8 is chosen to be measured due to the same reason. In this example, both algorithms give the opti~nunlsensor network design. although in general this may not be the case. Table 103 Expected Estimation Error of Redundant Sensor Network Design for Ammonia Process 
Step 1. Same as in Algorithm 1
Change in Expected Estimation Error (MeasurementsAdded)
Step 2. Same as in Algorithm 1. Stcp 3. Determine the sensor placement that gives the rnaximur~treduction in expected estimation error from the results of Stcp 2 and add it to measured set of I anables. Stop if the number of measurements inade so f a  is. I  : or e!se retcrn to Step 2. Both the abave algorithn:~3c not necessariij~give the s e x o r nztwork design that Inaxirnizes cstirnation accuracy. 5l;t reduces :he computationa1 burdsn significantly. hfirzilnrin~Cost Sensor iVefu~orkL)esig:zs
T.\le apply the above two algorithms for designing redundan: observabie senso! networks using six measurements for the ammonia process. Frorn Example 102. the optimal minifzlum observable sensor network design corresponding to measured variables 2, 5, 7 is chcsen. We have tc. select three addiiio~~ai villidbies TG be measureci with the objective of reducing the expected es:i,.;ation error as milch as possible. Table 103 shows the expected decrease in estimation error achieved by addin,0 one. t\vo, and thtee additional sensors for different coinbinations of the variables selected to be measured. The ~naximumesti~ilateerror reduction is achieved by choosing to measure additionally the variables 1 , 6, 8. If we apply Algorithm 1 above we would select the variables 1, 6, 8 to be measured since these give the maxi~numestimate error reduction for
Insea2 of maxifilizing esti~nationaccutacy. a n:inimun~ cost sefisor network may be designed that ensures observability of all variables. This ob.iectivc fur~ctionwas considered by Madron and Vevcrka [6j for sensor network design. Althougl~several other issues were considered in their work, we limit our consideration to the design of minimum observable sensor networks at minirnun? total cost. The design algorithm proposed by Madron and Ververka [6]essetitially attempts to obtain a set of dependent variables such that the measured independent variables wi!l have the least total cost. The colun~nsof the constraint matrix are first arranged in decreasing order of the cost of the sensor for rneasuri~lgthe corresponding variables. A Gaussian elimination procedure is applied, with the pivot element
being chosen from the next available column if possible, and reordering of the rows and columns is done if required. This procedure stops once the first m columns form an identity matrix. The least cost minirnum observable sensor network design is obtained by measuring the variables corresponding to the remaining rzm columns of the constraint matrix. We will illustrate this procedure by means of the following example.
Example 105 We consider the ammonia process with the sensor cost data for nieasuring different variables given by Table 104. Table 104 Flow Sensor Costs for Ammonia Process 
Stream
Sensor Cost
1 2 3 4
2.5 4.0 3.5 3.0
5
1.0 2.0
7 8 .. 


I
4
3
2.0 1.5

The order in which the pivots were selected for Gaussian eliinination are (1, I), (2,2), (3,3), (4,4), and (5,6), where the elements within brackets indicate the row and column index of the pivot element. It should be noted that for selecting the pivot elemelits the columns had to be rearranged since a nonzero pivot element was not available in the next column. The least cost mininiuni observable sensor network design is obtained by measuring the variables 6, 8, and 5 corresponding to the last three colun~nsof modified matrix A. Madron and Veverka [ 6 ] also considered constraints on the sensor location problem such as specifications of which variables are unmeasureable and which variables were required to be estinlated. They also considered the problem of locating additional sensors in a given partially n~easuredprocess in crder to obtain an observable sensor network dcsign at mini~nlurnadditional cost. In order :o solve these problems. tire columns of the constraint matrix A have to be ordered appropr~atcly befcre applying Gaussian elimination. 'The details of the pracedure tnay be obtsined from the pubiication by Madron and Veverka i61.
The constraint matrix for this process is giver, by Methods Based on Graph Theory
i
where the colurnns are arranged in order o! decreasing sensor costs for measuring the correspo~ldingstrearn flows and the rows are the flow balances for nodes 1 to 5. After applying Gaussian elimination to obtain an identity matrix in the first t n columns. we get the following modified matrix
Sensor networks far linear tlow processes can be desi%nzd elegantly dsifig graphtheorecic techniques. Unlike other methods. powerful insights are obtained concernkg the :structure of the seasor neiwork which make it possible to develop efficient algorithms for solving :he design problem. We wili again consider the design of sensor networks for maximizing esti~nationaccuracy or for minimizing the total cost. The additional graphtheoretic concepts required for understanding r h r method discussed in this section can be found in Appendix B.
Maxit~zuazEstinzatiotz Accuracy Serzsor Network Design
B
*
Minimum observable sensor networks. In the preceding section we stated that a miniinunl observable sensor network can he designed by
choosing nm independent variables to be measured such that the constraint submatrix corresponding to dependent variables is nonsingular. In other words, our choice of independent variables should make it possible to express each of the dependent variables as a linear combination of independent variables only. In Chapter 3, we showed that all unmeasured variables are observable, if no cycle containing only unmeasured variables exists in the process graph. We also showed that in order to ensure observability of all unmeasured variables using a minimum number of measurements, the unmeasured variables should fonn a spanning tree of the process graph. In other words, a minimum observable sensor network can be designed by simply constructing any spanning tree of the process graph and choosing the flows of chords of the spanning tree as the measured variables. In this case. the chord stream flows are the independent variables and the branch stream flows are the dependent variables. Note that this is similar to the choice of independent and dependent valiable choice made in Simpson's method for solving bilinear data reconciliation problems efficiently that was discussed in Chapter 4. The relationship between dependent and independent variables can also be obtained easily using the fundamental cuisets of the spanning tree. As descri!>ed in Appendix R , a fundarnenta! cutset, with respect to a branch of the spanning 1:ze, contains ope or more chord:; acd the stream !low coiresponding to the branch can be written in ternis (jf these chord streams floivs as
where K: is the fundamental cutset with respect io branch i. The elr,rnent.; pij are O if chord j is not an element of K,': otherwise they are +1 or 1 +:2rzc!i7g cr. whether chord .j has the same or 3pposite orientatlnr. 15 branch i. If the variance in the measurement error of chord flow j is o'. J then fro111Equation 102 1, the expected variance of the estimate crror of branch flow i can be obtained as
For a minimum observable sensor network, the estimate of the measured stream is given by the measured values themselves. Thus, the expected variance in the estimate of a measured variable is equal to its measurement error variance. The overall expecled estimation error (rneasure of estimation accuracy) is the sum of all the expected variances in the estimate of all variables. Using Equation 1022, we get the overall measure of estimation accuracy as
where k, is the number of fundamental cutsets of the spannin,o tree in which chord j occurs. Equation 1023 is exactly equivalent to Equation al concepts 1010 except that it uses spanning tree and f u n ~ ~ m e n tcueset instead of their matrix equivalents.
Example 106 The process graph of the ammonia process considered in Example 102 is depicted in Figure 102. A spcnning iree of ths process graph is s h o w in Figure 103 which consists of branches 2, 3, 2. 5, and 8 and chords 1. 5. arid 7. ~Zorrsspcrndirigio this spannirig tree, in the minimurn obse:vabls sensor cetwork design ihc flclws c!f strear::s 1, 6. and '7 ac mcasu~26.Thifilndamental cutsets of this spanning tree are [2. 1, 71, 13.1, 71, [&, 1 . 71. 1, 6, 7j, and [8, 1, GI whsrc the branch in each fundamenral cutsit =irt: denoted by 2n underscore. Chord 1 occurs ir. five fcndamental cutssts. chord 6 in two. and chord 7 in four. If we :issunie all measurerrterlt enor variances as unity, then from F4uation 1013 we get the overall expected estimaiicn error for this sensor design as 14. This va!ue may be compared with the solution given in Table 101. A process graph can contain several spanning trees. The numbel of spanning trees which is equal to the number of fcasible minimum ohsenable sensor network designs can be as large as nu', where n is the number of nodes in the process graph [ 7 ] . There are several algorithms for constructing a spanning tree of a process graph and for finding the fundamenla1 cutsets of the spanning tree. Some of these algorithms alons with computer programs i11 FORTRAN language have been described in Deo (71. It should be kept in mind that, for the purposes of constructing a spanning tree the direction of ihe streams are ignored, that is, the process
is,
We can start with an arbitrary spanning tree and use the elementary tree transformation technique to successively obtain new spanning trees which gives improved overall estimation accuracy. The o~llyissue to be resolved is to select the chord and branch to be interchanged to improve estilnatiorl accuracy. An algorithm for this purpose is outlined below.
Algorithm 3 Step 1. Construct an arbitrary spanning tree of the process graph. Figure 102. Process graph of simplified ammonia plant
Step 2. Determine all fundamental cutsets of the current spannin,0 tree solution and compute its overall estimation error using Equation 1023. Step 3. Find the number of occurrences of each chord in the fundamental cutsets and compute the contribution ( k i + l ) o t of each chord i to the overall estirnation inaccuracy, and Iank the chords in decreasing order of their contribution.
Figure 103. Miqiniurn observabie sensor neiwork.
graph is treated as an undirected graph. Tile directions of thc sirzans arc csed only to obtain the c~cfficien!~,piJin Equation 1021 if reqnircd. Tile problem cf designing a minimum observable sensor actwork thai rnaxirnizes .,stirnation accuracy (cr e.qilivalently miilimizes J ) ca:] be restated as ille prob!cm of constructi~lga span!~ingtree of the process graph which gives the least value of J as defined by Equatior? 1923. Starting with a spanning tree. we call generate a new spanning tree by means of a chardbranch interchange described in Appendix B as an elementary tree transformation. In this technique, if we add 3 chord to the spanning tree, then we aflould delete a branch fro111 the fundaruental cycle forrned by tlie chord. in order to obtain a new spanning tree. This elementary tree transforma[ion implies that the sensor ~neasuringthe chord stream flow is removed and instead a sensor is used to measure the branch flow deleted from the initial spanning tsee solution.
Step 4. Select tlie next ranked chord. say c,, from the crdered set of chords arid find the fundamental cycle formed h \ chord c,. Stop if !liere are no more chords to be examir?ed: else ma!; :he branches in the fundaniental cyclc in increasing order of their ineasure~nenterror \ arinnces. Step 3. Select the next ranked branch, say bJ, from the fu!u!?darne~:talcycle and interchange cliord ci and t;ranch b, to <)btain a ilew spanning t~ec.i f there are no rr,ore brafiches to be examir~ed,return ro Step 4. Step 6. Obtain thc fundamental zutszts v.4t.b respect to T ~ new E spannjng tres and cornputz the overali estimation error '~si;tgEquation 1023 colresp~ndingio this new soluticn. If the overail estimatic;n error of the nex. solution is less than that of the old spanning tree. replace old solutio:~ v:ith new spanning tree and return to Step 2: or eise icstore old spanning tree solution and return to Step 5. In the above algorithm, at each stage an atle~i~pt is rnadc to obtain a new spanning tree solution having better estimation accuracy, by a chordbrancli i~lterchangein the c u ~ ~ e spannin~ nt tree. If this artetnpt is successful, then the current solution is replaced by the new one arid the procedule repeated. If after systematically exarilining all possible chordbranch intercha11;L> an improved solution is not obtained. then the algorithm stops.
The procedure adopted in the above algorithm is known as a loc.al n e i g h b ~ r l z ~search ~ d technique since only the neighboring spanning tree solutions which differ from the current one in respect of only one branch are examined for obtaining a better solution at every iteration. 'The algorithm, therefore, gives only a local optimum solution and not the global optimum (sensor network design with least estimation error).
the maximum (or minimum) weight spanning tree is one of the classical problems of graph theory that has been well studied. Several algorithms are available for determining the maximum weight spanning tree in a straightforward manner [7], and we choose to describe Kruskal's algorithm below 18):
Example 107
Step 1. Sort the streams (edges of the process graph) in decreasing order of their weights. Initialize the set of edges in the tree, T to be a null set.
The above algorithm is applied to the ammonia process using the initial spanning tree with branch set [2, 3, 4, 5, 81 considered in Example 106. From the fundamental cutsets of this spanning tree obtained in Example 106, the overall estimation error is computed as 14. Moreover; the chord set ranked in order of their individual contributions is [I, 7, 61. We select chord I and find the branch set which forms a furldamental cycle with this chord as [2. 3, 4, 5, 81. Since all nleasurernent error variances are equal, we can choose any branch for the interchange. If we arbitrarily select branch 2 and interchange with chord 1, we get a new spanning trse with branch set [ I , 3, 4. 5 , 81. The overall estimation enor of this solutio11 is 12 which is less than tlle current solution. Therefore, we accept this new solution. If we repeat this procedure with i h ~ new solution we find that none of ihe chordbranch interchanges results in a bettzr solution. Thus, rhe satsor n e t w ~ r kdesign ob:ained by this algorithm corresponds i G the measurelnents of streams 2, 6, and 7 with estimatioi? error of I?. This soIution is worse than the global optimilm desigr, which has an estimation error of 1 1 a!: observed from Table 10 1. P.lgorithms for redunc!ant sensor network desiyts for n~sxintizingesti:na:iolr Liccilrilcy using graph theoretic techniques are >let to he de\,eloped Minimum Cost Sensor Aretwork Design
The desipn of a n~inimumobservable sensor netviork which has the least cost amang all minimum observable sensor networks can be easily accomplished using graphtheoretic techniques. From the discrtssion in the preceding section. we note that every spanning tree corresponds to a minimum observable sensor network. If we assign a weight to each stream which is equal to the cost of the sensor required to measure the flow of the stream then the problem considered here is to determice tiis maximum weight spanning tree, where the weight of a spanning tree is the sun1 of the weights of its branch streams. The problem of deterrninirip
Step 2. Pick the next edge from the sorted list. Step 3. Check if the edge picked forms a cycle with edges of the partial tree constructed so far. If so, discard this edge or else add edge to set T. If the number of edges in T is less than n (number of process units), return to step 2, or else Stop. The method of checking if an edge forms a cycle with other edges in a set, and the other operations to be carried out when an edge is added to T, are explained in Deo 171. We illustrate this algorithm in the following exalnple.
Example 108 VJe repcat Exan~pie105 tc find the least cost minimum observable sensor network of the ammonia prccess, bur this time using Kruskal's algorjthm. The c~rderof the streams in decreasing order of weights (sensor costs) is 12, 3, 4, 1. 6 , 7 , 8. 51. We pick edge 2 and add it to the ;see being constr~ited.h'ext, we pick edge 3 from the sorted !is[ arrd add it co the tree (since it does not form zi cycle with edge 2). \Ve continue io pick and add edges 4, and 1 because they do nct fom cycles with ~ t h e edges r added so far to the tree. When we next pick edge 6, we find that it cannot be added to tree since it forms a cycle with edges 2, 3, 4, and 1 added to tree so far (can be visually verified from the ammonia process graph of Figure 102). So we discard it and pick the next edge in the sorted list and add it to tree since it does not form a cycle with other edges. \Ve stop because we now have added 5 edges to the tree which is equal to the number of process units. The resulting spanning tree is shown in Figure 104. Corresponding to the spanning tree the sensor network design measures chord streams 5, 6, 2nd 8. Comparing with the solution obtained in Example
105, we can observe that this is the minimum cost sensor network design among all rnini~nunlobservable networks. The order of choosing the edges of the spanning tree is identical to the order of picking the pivots from the columns of the constraint matrix in the linear algebraic method used in Example 105.
qi
={01
if x,is not measured if xi is measured
Let c, be the cost of sensor for measuring flow of stream i, and let ci; be the maximum allowable standard deviation of the error in the estimate of variable i. A minimum cost sensor network design which satisfies the constraints on estimation error can be formulated as
Min
zciqi
qi subject to
Gi I o,
i = I...n
Figure 104. Minimum cost observable sensor network
oi
Methods Based on Optimization Techniques
The p~oblemof ser:sor !letwork design can also be forrnulateti ss s 11iathema:ical cpti~ni/.aiionprob!eni ant1 solved usi:lg appropriate o ~ j t i rnization tecll~iiques.In the preceding sections, the design cf sensor networks with the objective of either it3inimizing cots or filaxiriz~ngestimation accuracy was considered. Optimization tcchciques offer the 1:ossihility of siinllltaneo:~s!y considering differecr objectives arid a l s i inlposin~otlier constraints. Fur!hermore, i: can be estended to consider more gel~eralprocesses illvolving flows. temperatuixs, pressures and concentration measurements. H3giijeivicz [9j u.as the first to formulate sensor nerwork design opti~nization131oblern.We describe here only the Sonnulation of the problem and refer the reader to standard texts on op~imizationfor the details of the optimization rechnique used to solve the probiern. In sensor network design. the important decision to be rnade with regard to each stream flow variable is whether to measure it or not. In order t o ~nathematicallyformulate these decisions, we can convenicntly make LISZ nf binary (01) integer decision variables qi, one for each stream i which have the following interpretation.
where is the standard deviation of the esror in the estimate of flow of stream i and is thc square root of the dlagonal e!ement of cokariancz matrix 3f estimation errors, S, which can be computed for ar?). ciloice of sensor locations (defined by the valuzc chosan Sol the binary variables o,,) using Equation 310, The above problsm is a mixed integer optimizatior, proble~nand can be sclved using techiliques such a5 branch and bound. Alternatively, commercial optimization packages such as GAMS, GINO or MINCS ~ h i c hhavc a suite of opti:nization iechniq~ebcan be efl'cctivelq used for solviny the atxve problern. In the aboie optimization problern for:~:ula:icn, miui;nizution of the sensor network cos: was used as the objective (Equarion 1025>,subject to a minirnum accuracy specification for the estirnaies. Alternativelyl we can choose to maximize the overall estilnation accuracy subject to a maximum limit on the cost of the sensor network. Ocher constraiilts, such as minimurn deslred reliability, can also be included in the problem S(.>i.mulation.
DEVELOPMENTS IN SENSOR NETWORK DESIGN The earliest statement of the sensor network design problem for maximizing estimation accuracy through data reconciliation was given by Vaclavek [I01 for linear (flow) piocesses. Later, Vaclavek and Loucka
324
Dura Recol~cili~~rio~l and G'rus~Errol Derecriorl
[I 11 proposed algorithms for sensor network design for ensuring observability of important variables in linear as well as multicomponent (bilinear) processes. Almost two decades later, Kretsovalis and Mah [ S ] proposed a systematic strategy for solving this problem, where a measure of estimation accuracy was formally defined and algorithms proposed for design of redundant sensor networks for maximizing estimation accuracy. Ali and Narasimhan [12, 13, 141 addressed the problem of sensor network design for maximizing reliability and developed graphtheoretic algorithms for this purpose. Matrix methods for sensor network design for maximizing reliability were independently developed by Turbatte et al. [15]. Observable sensor network designs for linear and bilinear processes using matrix methods were also addressed by their group [16]. More recently, the issue of sensor network design for improving diagnosability and isolability of faults have beck1 tackled by Maquin et al. [17] and Rao et al. [IS]. Independently, Madron and Veverka [6] tackled the problem of ~ninimum cost observable sensor network design using matrix methods. Bagajewicz [9] formulated the sensor network design as an optimization problem. The use of generic optimization algorithms fr3r sensor network design considering different objectives such as cost, estimation accuracy, and reliability was reported bj. Sen et al. [19]. Recent!y, Bagajewicz and Sanchez [20] presented a n~ethodolozyfor designing or upgrading a sen53r netork in a process plant with the goal of achieving a certair, degree of observability and redilndancy for a specific set of variables. Although significafit progress has Seen made in t l i ~design of sensor networks, a compreliensive strategy sirnul:aneously considering different objectives still has to be d=.ve;oped.
Design of Ser~sorNerworks
325
SUMMARY The location and accuracy of sensors determine the estimation accuracy of data reconciliation and performance of gross error detection methods. The unmeasured flows in a minimum observable sensor network design for a flow process forms a spanning tree structure. A minimum cost minimum observable sensor network design is equivalent to the minimum weight spanning tree. The general sensor network design can be formulated and solved as a mixed integer nonlinear optimization problem.
REFERENCES 1. Liptak, B. G. hzstrument Engineers' HandbockProcess Measureilzer!rciizd Aizalysis, 3rd ed. Oxford: EuttenvorthHeinemann. 1995. 2. Peters, M. S.. and K. D. Tinmerhaus. Plui~tDesigiz u ~ z dEcoiloiliic~,;OIC1ze~i:icaiEi7gitzet.r~.New York: McGrawHill. :980.
3. Coulson, J. M.: J. F. Richardson, and R. K. Sinnott. CI:er~zicalGrgiizeerill?P'ol. 6. Desigiz. Oxfard: Pergamcn. 1983. 4. Mah, R.S.H. C h ~ i ~ i c Process al S?iticturesa;ld fi:Jonmtioil Flotcs. Boston: Butte~xorths.1390. 5. Kretsovalis. A . , and X.S.H. Atah. "Eifect s f Redu~dattcyoil Esiimation Accuracy in Process Data Reconciliation." C11en1.E;iig. Sri. 42 ( 1 9 S 7 f
2! 152321. 6. Madfor?,F., an* V. Veverka. "Optimzl Selection of Measuri~lgPoinr in Complex Plants by Linear Models." AIChE Joumal38 (1992): 227236. 7. Ceo. N. Grap11 Thecln r+
8. Kruskzl, J. B.. Jr. "On the Shortest Spanning Subtree of a Graph and rhc Travelling Salesman Problem." Pioc. .4m. Marlz. Soc. 7 (1956): 4850.
326
Data Rrco~lciliatio~~ olld GIossEiIor Detecr~orr
9. Bagajewicz, M. "Design and Retrofit of Sensor Networks for Linear Processes." AIChE Journal 41 (1 997): 23682306. lo. Vaclavek, V. "St~~dies on System Engineering111. Optimal Choice of the Balance Measurements in Complicated Chemical Engineering Systems." Chem. Eng. Sci. 24 (1969): 947955. 11. Vadavek, V., and M. Loucka. "Selection of Measurements Necessary to Achieve Multicomponent Mass Balances in Chemical Plants." Chein. Eng. Sci. 31 (1976): 11991205. 12. Ali, Y., and S. Narasimhan. "Sensor Network Design for Maximizing Reliability of Linear Processes." AlChE Journal 39 (1993): 820828. 13. Ali, Y., and S. Narasimhan. "Redundant Sensor Network Design for Linear Processes." AIChE Joccrnal4 1 ( 1 995): 22372306.
e?
c)
Industrial Applications of Data Reconciliation and Gross Error Detection Technologies
14. Ali, Y., and S. Narasimhan. "Sensor Network Design for Maximizing Reliability of Bilinear Processes." AICIiE Journal 42 (1996): 25632575. 15. Turbatte, H. C., D. Maquin, B. Cordier. and C. T. Huynh. "Analytical Redundancy and Reliability of Measurement Systern," presented at IFAC/IMACS Symposium Safeprocess '9 1, BadenBaden. Germany, 199 1. 4954.
16. Ragot, J., D. Itlaquin, and G. Bloch. "Sensor Positioning for Processes Cescribcd by Bi!inear Equations." Diczyrlo.c.tic rr srtr?te c/e fntzc:ior~inrrlr 2 (1992): 115132. 17. Maquin. 3., M. Luong. and J. Ragot. "Fault Detection and Isolation and Sensor Network Design." R4IRGAPIIJESA 3 1 (1 997): 393406.
13. Rao, R., M. Ghushan, 2nd K. Rengaswa~ny."Locating Sensors in Complex Chemical P1a.n:~ Eased on Fault Diagnostic Ohstr\;a'oility Criteria." AI'C!IE J~!r7tzul15(1999): 3 !O322. 19. Sen, S., S . Narasi~n!ian. and K. Deb. "Sensor Network Design of Linear Processes Using Genetic Algorithms." Conzpuretr Clzen~.Etzg~lg.2'2 (1993): 385390. 20. Bagajcwicz, M. J., and M. C. Sanchez. "Design and Upgrade of NonredunAavt r!.l,Rcdn~dantLinear Sensor Networks." AICIzE J o ~ ~ r t ~ 45a l(!?93!: 19271938.
Data reconciliation technology is widely applied nowadays in various chemical, petrochemical. and other material processing industries. It is applied offline o r in connection with online applications, such a s process optimization and advanced process control. This chapter presents a review of major industria! applications of data reconciliatiori and gross elsc;r detecticr, reported in the 1itera:ure. Based or. this published infamation, w e only describe the brozd features of the applications without going inta the details about the particular solution technique o r the software ~ l s e d .However, we describe in greater detail two applications (with which the authors were personally associated) in order to highlight some practical prob:en~sand their resclution. Although there are many other industrial impleinentations and software for daia reconciliation applications, they are either proprietary (and n o detailed information is publicly disclosed) o r the p~lblishedsource of information is not easily accessible. . . Th,. ,,,, znaiysis In :his chapter is organized according ro the rnajor industrial types of app!ications for data reconciliation technology. From the multitude of industrial data recorlciliation app!ications, w e can distinguish three major types of applications:
1. Process unit balance reconciliatioil and gross error detection 2. Parameter estimation and data reconciliation 3. Plantwide material and utilities reconciliation
330
D:,to Hecr~~iriliurion arid Gross ErrorDetecrior~
Instrument standard deviations are very important in data reconciliation, and a syste~naticestimation procedure for the standard deviations should be employed. Redundancy is another important factor in data reconciliation solution and capability of detecting gross errors. Additional instrumentation may often be necessary to achieve a satisfactory level of redundancy. An optimal sensor location design software is ideal for data reconciliation applications. Gross error detection should be always followed by instrument checking and correction. Uncorrected instrument problems deteriorate the quality of the data reconciliation solution. Data reconciliation/validation is a complex problem that might require more than one solutlon technique. Since important flows and temperatures in tne pant may be nonredundant, data filtering and validation can be used to provide a quality solution overall. A satisfactory data reconciliation system should have enough flexibility to handle process configuration changes, variable standard deviations, missing nleasure~nentrand to accept various kinds of equations, including inequality process co~straints. More guidelints and challenges stiil facing data reconciliation technc?logy have bee^ poinied out by Ragajewicz and Mullick 131: For rerinery applicatioris wid, a large vzrie~yof stream cnrnposiriocs, proper assay cha;actenzation is the key to a successful data reconciliation. With inaccura~ecompositions, results may not saiisfy material balances and good measurements may 'Je ide~ltifiedas gross errors. For more acccrate darz reconcilia:ion, marerial and energy balance recoilciliatior: is necessary. Heat balances enhance the redundancy in the flow measuremenis and aIi in~proveciaccuracy in the reco~lciledflow rates is obtained. *The steadystate assumptior, and the data averaging m i ~ h tnot be satisfactory for some processes and material balances cannot be accurately closed. Dynanuc data reconciliation software needs to be developed for such processes and especially for advanced process control applications. Rigorous models do not necessarily increase the accuracy in the data reconciliation solution, but they enable merging data reconciliation and the associated parameter estimation problem in one run. Increased accuracy in gross error detection is still a current need, since none of the existent methods and strategies provide effective gross error detection for all types of errors and error locations simultaneously.
e
Typical software for process units material and energy data reconciliation and gross error detection are: DATACONT" (Simulation Sciences Inc.) [3, 5 , 61, DATREC (Elf Central Research) [4], RECON (part of RECONSET of ChemPlant Technology s.r.o., Czech Republic) 1211, VAL1 (Belsim s.a.) 1141, and RAGE (Engineers India Limited) [I, 161. Many NLPbased optimization packages designed for online applications have data reconciliation capabilities. They mostly use rigorous models, making the gross error detection more challenging. For that reason. only few have some sort of gross error detection. R o ~ e o ' " , a new product of Simulation Sciences Inc. designed for closed loop online optimization has data reconciliation and gross error detection capability [22].
PARAMETER ESTIMATION A N D DATA RECONCILIATION A problem associated with data reconciliation is the estimation of various model parameters. Data reconciliation and gross error detection algorithms make use of plant models, which might have totally unknown parameters, or parameters that are changing during the plant operation. Most of these paranezerssuch as heat transfcr coefficients, fouling factors, dis:illation column tray efficicncies, compressor efficiencies, etc.are fixed va!ues for the process optiinizziion; therefore, a high accurzcy in thcir es~imatedvalues is required. Cine zpproach to parameter es~imationproblen~is to solve it sirnullatzeously with the reconciliation problem. Thc model parameters can be treated as regular ucmeasured variables, or as tuning psrarneters that are adjusted in NLPtype algorithms to match the pla:?; measuremects. The major proble~nwith this irpproac'n is that in the presence of gross err015, the paaneters may be adjusted to wrong values or some measurements can wrongiy be declared in gross error because of errors i r ~model parameters. To obtain an accurate solution for both measured variables and which is time model parameters, ail iterative process is usually Iequ~red, consuming. especially with rigorous 111ddeib. An alternative approach is to separute and secjuc~r~tinll)~ srrlvc. the two problems. First, data reconciliation and gross error detection is performed using only overall material and energy balances. The model parameters are then estimated using the reconciled values. This is similar to projecting out the unmeasured model parameters from the reconciliation problem along with their associated model equations. The parameter estimates obtained using the sequential approach are identical to those
332
Dora Kfcotrriliizlior~ and G r o s Error 1)errction
obtained using the simultaneous approach if there are no a priori estimates of the paranzeters available. Moreover, the parameter estimates obtained using the sequential approach may not a1:vays satisfy bounds on parameter values. An iterative procedure may be used to eliminate such problems. This approach was applied to parameter estimation problems in connection with advanced process control applications. The computational time is a serious constraint for such applications, and usually only one iteration is applied 121.
PLANTWIDE MATERIAL A N D UTILITIES RECONCILIATION Plantwide reconciliation is a very important tool for material and utilities accounting, yield accounting, or monitoring of energy consumed by the process. Many refineries are already saving a significant amoxnt of money by using a production accounting and reconciliation system. The usual term for a plantwide reconciliation system is production nccountins;therefore, we will adopt this term for the description in this section. Keld arcol~nringis another frequently used tenn 123, 241. A production accounting system interacts with various groups iil the plant. The operations department provides the input infarmation and collects the reconciled results. The ins?nmenta:ion group obtains ins:^umenc status ar,d perfmlls instrumenr recalibration and correction if mcessary. Process engineers, accou~iinganc! financial personnel, and planning and scheduling management retrieve periodic reports for their v a r i o ~ sneeds. Daily, week!y, or msnthly reports are standard requirements for a production accounting system. ?arious types of data are required for piant reconci!ia:i~n as indicated in Table l i 1. For beacr data quality and timely infomation, a producriori accounting system is usually integrated with the plant historian and the entire process information/management system. Some data is retrieved automaticaily from a historian, but other data is entered manually. Human elrol1s a factor affecting the data accuracy (and the reconciled results). For that season, some sort of data and model validation is very important. Veverka and Madron 1251 describe an empirical procedure for detecting topology errors, such as a missing stream or a wrong stream onenlation. They used the balance residual (or deficit), defined as: r = inputs  outputs  accumulation
(1 1  1 )
Table 111 Data Types for a Production Accounting System Data Type
Plant Topology Process Data Tanks Inventories Movements (from unit to unitltank) and Transfers (from tank to tank) Blends
Receipts Shipments Meters/Sensors Accuracy Factors Idat:oratorq. Additioi~aiDxta Entry
Description
Process units, tanks, and their conn~ctingstreams PIocessdata (e.g.. comperisated flows) from the process and utility units Inventory from each tank Movement datastdstop tirne, source node, destination node, quantitjr transferred
Blend datastardstop time, source node, destination node, tank volumes and S!ending quantities Receipt datastartlstop time. source node, destination node, quantity receivei! Receipt datastartlstop time. source node, destination node, quantities measured Instrument acct~racyfacton (tolerance, reliability, etc.) for each of ihe measurement deilices Lab t a t results (density. %H2U. cctnpositioi!~, ctc.) Unspecified or adjusted data to hz nsc:i by the modzl pxior to its balarxs calculation. Includes missing valuzs. specified valueb, adjustments, etc.
Thz balance deficit for each node is compared with a critical vtilue, say r,,,. If for a paiticula node r > rCTi,, then the balance around that node is declared inconsistent. The major proble~nis how to set the rCritvalue f9r each node. A good portion of the node imba!ance can be attributed t o errors in data, and is therefore zdmicsible. The remaining of {hedifference is conside~c'x:d=!izg e n s . The vz!ze r,,, can be obtained f r o x the statistical analysis of the balance residual r, which is a racdoxn variable (sirnilar to the nodal test for gross error detection described in Chapter 7). Serious imbalance problems occur due to frequent (daiiy) changes in some movements [26].The p!ant topology for many refinery processes is rather dynamic. There are many "te~nporaryflows" that one day have a nonzero value, and in another day becomes zero (closed valve) or the flow is redirected to a different tank or unit. Mistakes in the reconciliation topology input can very easily be produced. Kelly 1261 proposed a
inore complex strategy for finding the wrong material constraints followed by detection of gross errors in measured data. His strategy is based on a previous algorithm for deleting different combinations of measurements in order to assess the ~eductionin the objective function, developed by Crowe [27]. Since the deletion process gives rise to a large combinatorial problem, Kelly designed an algorithm to narrow down the number of possible combinations to delete. A good method for gross error detection is crucial for a productiorl accounting system, which is exposed to many sources of errors. In the presence of significant topology and measurement errors, the reconciled result may become meaningless. The gross error detection task, however, is very challenging for this type of problem because it is very difficult to distinguish between the true measurement errors and topology errors. Leaks and losses and existence of unmeasured flows create an even higher level of complexity. A lot of research effort in data reconciliation is done to resolve these issues and provide more accurate production accou~itingalgorithms. Some unmeasured flows are observable and can be estimated based on their relationship with measured flows. But the precision of such estimaricn is often unsatisfactory. due to propagation of errors 1251. An alternative way to get an estimate for unmeasured flovis and other variables i n v o l v ~ din !he p ! a ~ ~reconciiiation t is by using appropriate chemical enzineering cr,lculations. which is best accomplished with a process simulator 14, 251. Process unit data reconciliation performed before plzntwide reconci!iation is another possible approach [28]. A more compliczted problem for plant reconcilia~ionproblems is the estimatior? of material and energy iosscs [26' 291.There are many sources f ~ r mzterial and energy losses in a refinery or chemical plant such a,; flares, fugitive emissions (from volatile organic compounds), leaking valves, fittings, pumps, or heat exchangers, and tank losses (by evapcration or liquid leaks). In addition to the real losses nieiltioned above, there are apparent losses caused by rneasc~mer.:errors, lab density errors, line fills, or timing errors due to unsynchronizetl readings in tank gauges and meters. Many loss models and loss estimation fonnulas arc available. Governmental agencies in the United States such as the American Petroieum Institute (API) and the U.S. Environmental Protection Agency (EPA), provide publications with procedures for predicting fugitive emissions, tank losses, and various leaks. For plantwide data reconciliation, i t is important to clarify how to use these estimates. There are three major ways of including the loss estimates in the data reconciliation model:
1. Treating the losses as un~neasuredflows, and reestimating them based on the measured data. This approach does not require a "good" estimate of the loss flows, but requires observzbility of all loss flows, which is very unlikely to be obtained in a real plant. 2. Modeling leaks separately as explained in Chapter 7. A GLR test procedure can be used to detect leaks and estimate the order of magnitudes of the leaks (or losses). The method might have practical limitations if too many leaks are included in the model (it becomes a large combinatorial problem; also there might not be enough redundancy to accurately estimate the magnitudes of all leaks and losses). 3. Treating the losses and leaks as "pseudomeasured" flows. The estimated loss or leak value is used as a "measured" value, and a relatively higher standard deviation than that for the real measured flows is given to the loss flow. The values of the flow rates for the loss flows are reconciled together with the other measured flows. Typical software for plantwide reconciliation and yield (productiort) accounting is [21]: OpenYieid (Simulation Scie~lcesInc.) and SIGMAfine (KBC Advanced Technologies).
CASE STUDIES Reconciliaiion of Refinery Crude Preheat Train Data 11 63
A crude preheat train is an impcfiant subsystem of a refirieiy used to rni~iimizethe external energy ronsitmption reqsired for heaticg crude oil. It consists o i a network of heat exchar?gers which 31.5 used to preheat crude oil before the crude is sent to a furnace for further heating prior to distiliation. The hot streams used for preheating the crude are tile distilS. 1 !I shows the crude late streams from the downstream C O ~ U I I I ~ ~Figure preheat train of a refinery consisting of 2 1 exchangers in which the crude ;.: h w t d by 1I distillate streams. The flow of the crude before the splitter. as well as the two split f l o w qf the crude, are measured. The inlet flows of all distillate streams are measured as are also the inlet and outlet temperatures of both hot and cold stream to every exchanger. Table 112 shows a typical set of measured flows and Table 113 shows the measured temperatures. Tile motivation for reconciling these measurements arises from the need to optimize the crude split flows every few hours. In Chapter 1. we have already described this application and here we will focus only on the ~econciliationproblem.
lndrrstrial Ap;,dicutioii.sof Dara Rrcorzr:iliarior7 and Grwsv Error DelectiOll Tcchr,olr,sies 337
Dur(~K~~coi~ci!iuriori and Gross Error Drraclion
Table 1 13 Measured and Reconciled Temperatures of Crude Preheat Train Crude
Stream
Tcble 1 12 Measured and Reconciled Flows of Crude Preheat Train
CR 1 CR2 CR3 HN KE;I
TPl DS I MPl HV 1 BPI VR 1 HV2 VR2
Reconciled Flows (tons/hr) B e i ~ r eGED Afier GED
399.2352 153.1427 246.0925 8.3865 58.4888 267.0283 86.5460 454.8024 54.5657 209.2158 106.7616 106.6069 170.3060
Reconciled Temperature (C] After GED Before GED
CR I CRl A CRIB CRlC CRlD CRlE CRlG CR2 CR2A Ck23 CR2C CR2D CR2E CR2F CR2G CK2H CR3 CR3A CR3B CK3C CR3D CR3E CR3F CR3G CP.3H CR4 HN HNA KE 1 KEiA KElB TP 1 TP l A DS 1 DS 1A DSlB DSlC MP I MPl A MPlB
Figure 1 11. Crude preheat train of a refinery.
Measured Fiow (tonslhr]
Measured Temperature (C]
409.5705 151 8511 257.7!94 8.3885 58.6845 267.7579 8 1SO42 455.0960 54.5277 209.4634 109.9967 106.7458 171.2554 0
*
(table contit~uedon tieir page)
Table 1 14 Constants for Specific Heat Capacity Correlation
Table 1 13 (continued] Measuredand Reconciled Temperatures of Crude preheat Train Stream
Measured Temperature (C)
HV 1 HVlA HVlB HVlC BPI BPlA BPlB VR 1 VRlA VRlB LV 1 LVl A HV2 HV2A HV2B VR2 VR2A VR2i3
302.300 299.800 263.800 229.492 309.458 253.525 230.667 283.885 258.167 152.550 212.142 190.267 302.300 273.617 233.075 345.925 345.925 345.925
Recor,ciled Temperature (C) After GED Before GED
301.0654 301.0387 266.6501 227.4435 304.7259 253.0694 233.4092 277.3424 242.7938 158.8106 206.1244 195.0234 295.0904 268.48 12 240.6166 345.4640 355 4640 345.4640
301.4735 300.7566 265.2976 228.3426 306.4544 253.1749 232.4505 283.3504 260.7749 198.7786 2 1 1.7749 190.5526 30U.22 11 275 4542 25 1.9620 347.0395 347.0395 317.0395
The problem iri this case is t c reconci!e all t!le f! o u ~ sand ccmperatures so as to sa~isfymaterial arid energy baiances of each process unit ef this subsystem 1x1 addition, ir is required to estimate the overall heat transfer coefficient of ezch exchanger given the area and number of tube and shell passes. It is assuned that ail the streams zre single phase Auitls and ihe specific heat capacity of ~ a i strsam h is giver. by
where ai and bi are constants and Ti is the temperature of the strean? in Jclgrees C. T i l ~~ulibcanrsai anci bi ful rne different streams are given in T~!J:E114. The are2 and the iiumber tube passes for each exchanger are given in Table 115, while all exchangers have a single shell pass.
Stream
a
b x 100
CRUDE HN KEl TP 1 DS 1 MP I HV I BPI VR 1 LV I HV2 VR2
0.4442 0.458 1 0.4455 0.48 19 0.4263 0.4455 0.4143 0.4263 0.4092 0.4285 0.4143 0.4062
0.101 1 0.1036 0.1011 0.1081 0.0975 0.1011 0.0959 0.0975 0.0962 0.0986 0.0959 0.0957
Table 1 15 Heat Exchanger Areas and Number of Tube Passes for a Crude Pr9heat Train Exchanger
C
*
Areo (m2)
Tube Passes
lrzdu~trialApplicarionr of Dura Keconciliariorr arrd G r m Err<,r Detertiort TPchJloiogies341
j
i
Using a standard deviation of 1% of the measured values for all stream flows and temperatures, the reconciled estimates are obtained assuming that no gross errors are present in the data. In the third colunln of Tables 112 and 1 13, the reconciled flows and temperaiures, respectively, are shown. (All the results of this case study were obtained using the software package RAGE.) The constraints that are used for each exchanger are the flow balances for the hot and cold streams, as well as the enthalpy balance. For the mixer, flow and enthalpy balances are imposed while, for the splitter, a flow balance and equality of temperatures across the splitter are imposed. No other feasibility constraints or bounds on the variables are imposed. In order to remove gross errors from the data, the GLR test along with serial compensation strategy is applied for multiple gross error detection after linearization of the constraints around the reconciled estimates. The final reconciled estimates after all gross errors are identified and compensated are also shown in the last column of Tables 112 and 113. Ure focus on some interesting problemsffeatures of the measured and reconciled dara. I i we consider the measured temperatures of streams incident on exchanger E38A (the streams CR2C, CR2D, D S l A , and DSI B), we note :hat the crude stream is getting cooled fro111 217.608 to 216.992 degrees C. whi!e the intended "hot" distillate strearn is getting heated froin 224.333 io 254.908 dcgrees C . A1:hough it may bc possib!e for the roles of ho: and cold streams ti: bz reversed dcpznding on the prevailing flows and temperatures. what is urlacceptable here is :bat heat is being transferred fiom the !owel temperature ciudc to the higher ternperature diitilla~cstream which is thermodynamically infeasible. 11 can be verified thai reconciliation before cr after gross errpoldetecfion (GED) does riot correct this probleni 2nd thz estiniates for exchangcr E3YA still \&)late thermody~ialnicfea;ibi!ity. If wc use these estimates to obtain an estimate of the overall heat iransf'er coefficient for this exchanger. then we obtain a ne9a:ive value for it which ic absilrd. In order to obtain thermodynamically feasible estimates, several possibilities were examined. One general z p ~ r c n c his $5 inc!i,de feasihi!i?y constraints at the ho; and cold end of each exchpqger of the form
where the subscript i is the inlet and subscript o is the outlet end of the exchanger. This would, however, increase the number of constraints significantly. Moreover, this presupposes knowledge of the cold and the hot streams for each exchanger and does not allow any role reversal. A simpler technique is to include the relation between overall heat transfer coefficient (U) and heat load for every exchanger and impose bounds on the overall heat transfer coefficient. If we impose a nonnegativity restriction on U, then we can ensure that thermodynamic feasibility is maintained regardless of which of the streams plays the role of the hot stream and which plays the role of the cold stream. Using this approach, the reconciliation problem was solved again. (Note that, as explained in Chapter 5, in order to solve this problem a constrained nonlinear optimization program has to be used and the unmeasured heat transfer coefficient parameters cannot be eliminated using a projection matrix.) The reconciled temperature estimates of the four streams incident on exchanger E38A alone before GED and after GED are shown in the second column of Tables 116 and 117. respectively. For comparison, we also reconcile the problem by deleting each of the four suspect temperature measurements in turn and also after deleting all the four temperature measilrenients which violate thermodynamic feasibility. The reconciled temperatcre esiimrrtes for these four streams before and zfter GEU are also shown in Tables 116and I 17. Table 1 16 Reconciled Ternperstvres 6efore GED Around Exchanger E38A for Different Cases


Stream
Rec~ncild Tempcrob~res(C) Bounds on U
T of DSl B T of CR?C T of CR2D At! Four T's unmecsured unmeasured unmeasurd unmeasbrd u n m ~ s u r e d
T of GSlA

DS iA US1 9
CR2C CR2D
235.6707 235.6655 217.2356 217.2356 
227.4071 248.6476 223.1970 212.1257 
222.3988 222.7002 232.2426 250.8886 2 19.9628 229.2839 214.8731 214.7073
223.2842 250.1673 ?21 1 179 207,0257
:95.7'.178 23 1.7550 227.4480 210.7080
It~da.\rricil Applicarions r,fData R~cc~ncriliarion nr~dGross Error Defection Techndogies 343
D~iroR,~ronciliarinr~ ortd Gross Ermr Drrc+criorz
342
Table 1 17 ~econciledTemperatures After GED Around Exchanger E38A for Different Cases
Stream
Bounds on U
DS l A DS l B CR2C CR2D
Reconciled Temperatures (C) T of DSlA T of DSl B T of CR2C T of CR2D NI Focr T's unmeasured unmeasured unmeasured unmeasured unmeasured
225.7176 221.1880 225.7 175 255.417 1 216.8586 235.1001 216.8586 217.0614
224.43 15 217.9601 217.0624 220.4571
224.4547 255.2667 236.3756 220.0339
223.9519 255.5525 233.8225 217.1436
ak
215.7303 223.5384 23 1.2648 227.2575
From the reconciled estimates, it can be observed that by imposing nonnegativity bounds on U, it is possible to obtain feasible estimates before and after GED. In fact, the results show that the heat transfer coefficient for exchanger E38A is at its lower bound of zero, which implies that this exchanger is being bypassed completely by one of the streams, resulting in the temperatures of both hot and cold streams being unchanged across this exchanger. T h e only other case when feasible temperature estimates were obtained was after deleting the temperature of stream DSlB and application of GED (refer to column 4 of Table 117). Even in this case, the stream ten1peratitl.e~change marginally across exchanger E38B, indicating tha: this exchanger is largely being bypassed. ( T h ~ swas also later confirmed after i n s p e c t i ~ gthe manual valve positions on the crude r exchanger.) 'fhe results clearly demonstrate that bypass line f ~ this imposition of 'oo~indcoristlaints on the parameters can be used as a gener;lc nlethod to obtain feasible estin~atzs. This case stildy also brought out other issues that needed to be addressed in practice:
C
optimization, it may be necessary to include a heat loss term in the enthalpy balances. However, enough redundancy does not exist for treating the loss terms as unknowns. One possibility is to assume a specified fraction of the heat load of exchangers to be lost based on past experience or based on recommended loss estimation methods. A s pointed out in Chapter 1, the reconciliation of the crude preheat train data was performed every four hours using averaged measured data for the preceding two hours. Since the heat transfer coefficients of exchangers cannot be expected to change dramatically from one time period to the next, it is possible to use their estimates derived in one time period as "measurements" for the next time period with a larger standard deviation. Due to this extra redundancy, better estimates can be obtained. Moreover, the heat transfer coefficient estimates change smoothly from one time period to the next and will not fluctuate wildly. A trend of the heat transfer coefficients can be used to decide when cleaningl~naintenance procedures have to be initiated for the exchangers. Reconciliation of Ammonia Plant Data [30]
Ammonia is a chemical product with many industrial ap~lications such as refrigerants and fertilizers. Figurz 11 2 shows a simpiified process flowsheet diazram for the sy11:hesis section of an ammonia prGcess [30]. Ammonia is produced by ar, exoti~ermicreaction of nitrogen and hydrogen:
I.
It was assumed that ali the streams arein a single phase. A more ri,oorous method would r.x..:..LyU,,+ *I.,. ..++ "I C +LU.L .,,,eaiz +!G be determined and an appropriate correlation tn he used for determining the stream enthalpy. Some fraction of the crude flows was being bypassed in a few other exchangers also, but sufficient measurements (redundancy) were not available for treating the bypass fractions as unknown parameters and estimating them as pa11 of the reconciliation problem. Heat losses fro111 exchangers were not accounted for in the enthalpy balances of e s c h a n ~ e r s In . order to use the reconciled data for better UIC
C
The feed stream S l t3 the synthesis section already contain.: ammonia from ilpstrezrn processes. To separate it, stream S! is cooled 2nd sent to flash drum F1, where the ammoniarich liquid SS is separated from the remaining vapor S2. Before entering the reactor section, the vapor stream is preheated by a product stream. The reactor section consis:> of t:vo reaction stages and two internal heat exchangers. Stream S4 is split into three streams (SS, S6, and S7) and the split fractions are used to control the reactor feed temperatures. Stream S7 is used to quench the hot product stream from the firststage reactor and stream S5 is used to recover some of the heat from the product of the second stage reactor (S13). The three streams are then recombined and fed to the firststage reactor. Most of the cooled reactor prod
irrd~si,i~~l,4,7,7iir:ar;r)t1r ofno?u Kecorrci1;at;on und Gro.5.s Error Derecrior: Tec/utologies 345
uct (S1.5) is rccycled to another section of the plant (stream S 16), while the remainder (S17) is further cooled with refrigerant (stream S22) to condense most of the ammonia (stream S20). The two condensed streams (S3 and S20) are combined and further purified downstream.
Table 1 18 Stream and Unit Reconciliation Solution for Ammonia Example Case A. No Gross Errors Present in Measurements STRM VBL
STAT
UOM
TAG NAME
STANDARD MT MEASURED DEVIATION STAT VALUE
CNC VALUE
SI RATE TEMP PRES XI X2 X3 X4 X5
U M F U ti
RATE TEMP PRES X1 X2 x3 X? X5
M M I
KATE TEXT? FRES X! X2 x3 Xd X5
M M F U
U U ti
M3MR C ATM MOL% MOL% MOL% MOL% MOL%
TO 1
1.50
M3lHR C ATM hlOLLio MOL% MCLC'c ,MC)L% MOL7c
F10 TI0
3075 000 500
M31flK C ATM MOLT MOLi/;c MOL% MOL'ir, MOL%
F! 1 TI 1
2880.000 5.00
1.50
23.00
256 367 22 46 150.001) 32.7478 11.8345 7.1737 2.3594 45.8846
S10
u u LT 2
u
Sll
S!3
Figure 1 12. An crmmonia synthasis industrial process.
L'
u I! I:
S12
The atnrnonid synthesis plant contains instrumentation for measuring flow rates, temperatures and \lariol~sstrezrn compositions (mole fractions). The measured values, their associated standard deviations and the reconciled values are reported in Tables 118 and 1 19. Tables 1 18, I 19, and 1110 show the reconciliatiort results for a case where no gross errors were found (Case A). Table 118 reports all stream calculation results, for both measured and unmeasured data. Other calculated values such as rcaction extents. heat exchanger duties, U A values, and flash data, art: also reported at the bottom of Table 118.
R.47E TEMP PRES X1 X2 X 3 X4 X5
U
M F U ii
li
il ti (tobie corzritzued on next pnge)
Ir~dus(riolAppiicuiions
Table 1 18 (continued) Stream and Unit Reconciliation Solution for Ammonia Example Case A. No Gross Errors Present in Measurements STRM VBL
STAT
UOM
TAG NAME
STANDARD MT MEASURED D M A l l O N STAT VALUE
STRM V B L
RATE TEMP PRES X1 X2 X3 X4 X5
M3MR C ATM MOL% MOL% MOL% MOIL%
RATE TEhlP PRES X1 X2 X3 X4 X5
hl3/HR C ATM MOL4 .MGL% XtO1*5c MOL8 MOL%
RATE l'Eh4P PRES xI x2 X3 X4
b13XR C ATM MOLQ MOL% MOLS; MOLq
X5
M O i Yc
STAT
UOM
TAG NAME
STANDARD MT MEASURED DEVIATION STAT VALUE
CALC VALUE
S17
0
S13
Table 1 18 {cont~nued) Stream and Unit Reconciliation Solution for Ammonia Example Case A. No Gross Errors Present in Measurements
I
I
CALC VALUE
of llara K~.cor~crliutiun arld Gross EIrorDctr~ctlor~ T~ci:ri:noio,~i~~ 347
I
MOT,%
RATE TEMP PRES X1 X2 X3
M U F U U U
MYHR C ATM MOL% MOL% MOL%
X4 X5
U U
MOL% MOL%
F17
48 000
I 84
1600 000
TI8
1 00
1 50
15 00
1665 585 127 69 150 000 49 5273 18 3357 14 2649 46916 13 I805
S18
S14 C
RATE
'J
M?/HR
TEMP PRES X1 X2 X X4 X5
M F
C
U
U U U C
1665 585
ATM MOL% MOIL% MOL"& hiOL% MOL%
15 24 150 O(K) 49 5277 18 3757 14 2649 46910 l i 1805
S 19
515
RA'IE TEMP PRES X! X2 X3 X4 X5
4.
h1
hl?/HX
U F
C
hf 1
h4 21 M
A IM MOL% MOL% MOL% MOL% 0 '6
l19
45 000
84
l200K'U
H I 19
1 0000 1 COO8
1 14 32 1 4C 78 19
57 0000 21 0OOr) 15 0000 5 0000 :!OOO(1
NI 19
C1 13 1 9GQG 4RI9 I 0090 Xt33 19 1 0000
1472177 8 15 i 50 OtXI 56 0256 20 74:'; 16 1 557 5 1069 1 7915
S2
S 16
RATE TEMP PIII,S X1 X2 X3 X4 X5
RATE rkMP PEES X1 X2 X3 X4 X5
M3NR C ATM MOLQ hlOL% MOLO MOLO MOL%
rJ U hl U U U U 1
M3ff3R C ATM MOL% MOL% MO1Q Mold% MOL%
PO:!
500
1 50
i50 000
1 02E+Oi 2 1 82 149 998 58 9875 21 3170 129215 4 2499 2 5238
able cor1rrr:lred or1 rzc,xt p u ~ e i 6
Irrdt~striuLA~~~~licrrriur~.s of Ilata ReconciliaZion and Gro.ss Error D~recriorlTechrrolo~ies349
Data Recoriciliatiorz and Grocs Error Derrcrion
348
Table 1 18 {continued) Stream and Unit Reconciliation Solution for Ammonia Example Case A. No Gross Errors Present in Measurements
Table 1 18 (continued) Stream and Unit Reconciliation Solution for Ammonia Example Case A. No Gross Errors Present in Measurements STRM VBL
STAT
UOM
TAG NAME
STANDARD MT MEASURED DEVIATION STAT VALUE
CALC VALUE
STRM VBL
STAT
UOM
TAG NAME
STANDARD MT MEASURED DEVIATION STAT VALUE
CALC VALUE
S3
S20 RATE TEMP PRES XI X2 X3 X4 X5
M3MR C ATM MOL8 MOL% MOL% MOL% MOL%
KATE TEMP PRES XI X2 X3 X4 X5
M3MR C ATM MOL% MOL% MOL% kfSL% h.IOL9
KATE TEMP PR ES XI X2 X3 X4 x.5
KCWR
RATE TEMP PRES X1 X2 X3 X4 X5
M U F F F F F F
M3i'HR C ATM MOL% MOL% MOL% MOL% MOL%
F03
.500
RATE TEMP PKES X1 X2 X3 X3 XS
M M F
M3MR C ATM MOI,% MOL% MOL% MOL8 MOL%
F04 TO4
3075.000 2.00
1.50
!OO.OOO
100.000 21.82 150.000 .0000 ,0000 .0000 ,0000 200.0000
S4
S2 1
M M M M M
H24 N24 C14 AK4 NFI34
1.0(100 1.0000 1.0000 1.0000 1.0000
.30 .31 1.31 .36 .09 .28 .76
1.03E+05 1.02E+05 135.00 134.86 150.000 58.0000 58.9875 21.0000 21.3170 13.0000 12.9218 3.0000 4.2499 2.0000 2.5238
S5
S22 L P
ATM MOL% MOLR Mot% MOL% MOi%
RATE TEMP PRES XI X2 X3 X4 X5
M U F U
RATE TEMP PRES XI X2 X3 X3 X5
PO5
1650.00C
81
U U U
ELI3MR C ATM MOLQ MOLB MOL'X MOL% MOL%
5.50E+04 5.6OEi4~! 134.86 150.000 55.9575 21.3!70 i2.921S 4.2499 2.5238
M U F U IJ U U U
M31kiK C ATM MOL9b MOL% MOL% MOL% MOL%
F06
450.000
1.1 1
l.50E+04 1.51E+i)l 134.86 150.000 58.9875 21.3170 12.921s 4.2499 2.5238
t
S6
S23 RATE TEMP PRkS X1 X2 X3 X4 X5
KGnIR
C ATM MOL% MOL% MOL% MOL% MOL%
Irulusrriul Ap~~licufi~~rrs qfDnra Keconcilinrior~and Cross Error Detccrrorr T~~cl~,lologres 35 1
Durn Recor
350
T a b l e 1 18 (continued) S t r e a m and U n i t Reconciliation S o l u t i o n for A m m o n i a E x a m p l e C a s e A. No G r o s s E r r o r s P r e s e n t in M e a s u r e m e n t s
T a b l e 1 18 (continued) S t r e a m and U n i t Reconciliation S o l u t i o n for A m m o n i a E x a m p l e Case A. No G r o s s E r r o r s P r e s e n t in M e a s u r e m e n t s STRM VBL
STAT
UOM
TAG NAME
STANDARD MT MEASURED DEVIATION STAT VALUE
SPLITTERS: SPLIT FRACTIONS
CALC VALUE
UNIT
S7
CALCULATED VALUE
S16 S 17
U U
.98191 .O 1809
S5 S6 S7
U U U
.55063 .I4869 30068
SP2
REACTOR UNITS: EXTENT O F REACTION A N D D U N
S8 RATE TEMP PRES XI X2 X3 X4 Xi
STAT
SPI
M31HR C ATM MOL% Mold% MOL% lMOL% MOLCie
RATE TEMP PRES XI X2 X3 X4 X5
STRM
U M
U
M3/I3R C ATP.4 >fOL':c MOLQ KOL %
b
M0I.C
U
hlOLCc
5 U
u
UNIT VBL
TO8
UOM
EX 1 DUTY
IJ U
KGMOL/HR M'KCAL~R
1 22E+03
EX I DUTY
iT
KGMOLIH R M*KCALIHII
91 67165 03000
RI
F
FLASH UNITS
ii ii
UNli VBL
I; U
STAT UOM
TAGNAME
STANDARD DEVlAT!ON
MEASURED VALUE
CALC VALUE
F1
U U
u u
TEWP II PRES F DUTY F
C
TEMP U PRES M DUTY F
C ATM ~2.p MKCAI~IHR
21 g2 150 oon Go003
ATV
M'KCALIHR
F2
HEAT EXCHANGER D U N A N D U A VALUES HEAT EXNGR
I lit19
R2
S9 RATE TEMP PRES X! X2 X3 X4 X5
CALCULATED VALUE
STAT
DUTY UA (M'KCAL/HR)
(KCAL/HRC)
8 15 sry)
!5000n !50000
NOTA 7'1OI\' STRM :STREAM ID VI3L : K4RI/1RLE NAME S T A T : LrARIABLE .TTATLrS I N THE MODE/, (M=MEASLfRED. U=UNMEASURE2. F=FIXED) UOM :UNIT OF MEASURE ,WTSTAT :MEASUREMENT TEST STATISTIC NK :IV'ONREDUNDANTMEASUREMENT
11ld1rsr1id A[q~licafions of Durn Kecoi~ciliuiro~~ rnflrlGross Error 13rierrio11T e c l ~ n o l o ~ i355 e.~
Darn Kecorrciha~ionand Gross Error Derec~ion
a1 test (GT), the measurement test (MT), and the principal component measurement test (PCMT), properly indicated that there are no gross errors in measurements. In Case B, three gross errors were simulated in the following measurements:
Table 1110 Summary of Calculation Results for the Ammonia Example Case A. No Gross Errors Present in Measurements
NUMBER OF ITERATIONS MEASURED VARIABLES UNMEASURED VARIABLES FIXED VARIABLES NUMBER OF EQUATIONS DEGREE OF REDUNDANCY
= = = =
5 40 (1 NONREDUNDANT) 59 (0 UNOBSERVABLE) 4 1 (29 FIXED BY USER) = 82 = 20
F07 (magnitude = 10000 m3/hr, ratio 610 = 10); C14 (magnitude = 5 rnol%, ratio 610 = 5); T22 (magnitude = 3.6 Deg. C, ratio 610 = 3).
GLOBAL TEST (.950 CONFIDENCE LEVEL) GT STATISTIC = 10.96
CRITICAL VALUE = 3 1.40
*** MEASUREMENTS PASSED THE GLOBAL TEST *** MEASUREMENT TEST ( .950 CONFIDENCE LEVEL) CRITICAL VALUE = 3.16
**" ALL MEASUREMENTS PASSED THE MEASUREMENT TEST """ PRINCIPAL. CORlPONENT MEASUREhlENT (PCM) TEST (.950 JOIlTI, .997 INDIVIDUAL CONFlDENCE LEVELS) CRITICAL VhLljE = 3.02
*** A1.L PR~NCIPALCOMP9NENTS PASSED THE FCM E S T """
Tr?ble I i9 coniains the reconciliation results far all measurcd vzriables. Table 1116 indicates general run data jncmber of iterations w t i i convergence, number of equations, and :lumber of variables for each caiegcrymezsured, unmeasured, fixed, the number of nonredunciant and unobservable variables, the degrees of redundancy and a suinlnary of w e r e
They have different detectability factors, as indicated in Table 119. To increase the chance of detection and correct identification, a higher ratio of the gross error magnitude 6 to the corresponding standard deviation o was used for the measurements with lower detectability factor. The i o b i r the detectability, the higher the ratio 610. Table 1111 shows the reconciliation results for Case B for all measured variables. No error elimination was used in this run. Table l l  12 indicates various run datr? and the summary results from the GT and MT. The G T indicates the existence of Fross errors, while the M T declares 7 measurements in gross error. In addition to the three true gross errors, foils other gros: errors were found by the MT. Serial eliminatior? was used for better enor iderttificarion. Table 1113 shows the summarized results from a first 1z11 with serial elimination calculations. 111this rdn. the MT was used in zddition to GT. We notice that in the first elimination step the]e were two measured variables sharing the s a n e MT statistic (the larges! MT statistic value). This pziticulzr algorithm used the detectability F&cior as a tie brezker in the elimination pracess. Since FO7 has a higher deteciabiliry factor (0.5 135) than FC6 (0.2297), F07 wzs chosen tc be eliminated first. This turned out to be the right choice. S~lbsequel?tly.F06 was not foanld in gross enor anymore. Next. CI4 and T22 were also eliminated and no more gross r v n r s were found by both GT and iviT. The estimated values for the three eliminated measurements are very close to the values reported in Table 119 for the no gross error case. Table 1114 shows a similar run, this Time using the PCMT for gross error identification. Initially. both GT and PCMT indicate existence of gross errors. The elimination path and final results. however, are some
' & a 356
Itzduir~~uI Appliculion.~of l>a/u Rc,conci!iutron awd Gross Error L)efrcri(~..: T~clzlzulogies 357
Duru Rcc(iri
what different from the run with the MT. In the first elimination pass, F05 was found to be the major contributor to the largest inflated principal component. It is true that F05, F06, and F07 are modelrelated to each other (they are all outlet streams of the splitter SP2) and the smearing effect of the gross error in F07 is easier. In this case, the calculation of the contribution shares to the first lxgest principal component that failed the PCMT, indicates that F05 should be first eliminated. Subsequently, an associated temperature, T13. also ha? to be eliminated and properly adjusted in order to satisfy the heat balance for exchanger E3. The last two eliminated measurements, T22 and C14, are true gross errors. The MT and PCMT tests usually detect correctly gross errors in measured variables with relatively higher detectability factor. The outcome of the two types of tests for gross errors in measured variables with relatively lower detectability factors could be the same for both tests (no gross error detection or wrong gross error identification) or one test can perfom better than another. it is not clear which test performs conslstently better. More analj~sisand comparisoil of the two type of tests for the ammonia example can be found in Jordache and Tilton [31].
9
I
C
Darn Reconciliutio~~ orld GIossEtror L ) C I E ( . I ~ U I I
Table 1112 Summary of Calculation Results for the Ammonic Example Case B. Gross Errors Present in Three Measurements Measurement Test Used for GED
1
i
*
NUMBER OF ITERATIONS = 5 MEASURED VARIABLES = 40 (1 NONREDUNDANT) UNMEASURED VARIABLES = 59 (0 UNOBSERVABLE)
b
FIXED VARIABLES = 4 1 (29 FIXED BY USER) NUMBER OF EQUATIONS = 82 DEGREE OF REDUNDANCY = 20
GLOBAL TEST ( .950 CONFIDENCE LEVEL) GT STATISTIC = 80.86
**"
CRITICAL VALUE = 3 1.40 DID NOT PASS THE GLOBAL TI35 1 .:**
MEASUREMENT TEST ( ,950 CONFIDENCE LEVEL) CRITICAL VALUE = 3.16
*** 7 MEASUREMENTS FAILED THE MEASUREMENT TEST*** STRMIUNIT
TAG
MEASUREMENT VALUE
CALCULATED VALUE
MTSTAT
F07
20000.0000
23 125 7732
6.1226
M3lHK
F05
550Ci0.0000
59426.0C03
3.7092
TEMP
C
TO8
.230.0000
421.9775
3.4451
S 13 TEMP
C
T 13
450.0000
461.2256
3.3358
VBL
UOM
NAME
RATE
h,13KR
RATE
S7
303 '30 C
255 mm
N
l
I
S5
S8
Industrial Applicarions ofI~u10Reconcilialion and Grosr Error I ) E ~ ~ ~rEch,zolog~Es I;(,~ 361

Table 1 1 1 3 (continued) Summary of Calculation Results for the Ammonia Example Case 6. Gross Errors Present in Three Measurements Serial Elimination of Gross Errors Applied Measurement Test Used for GED
Table 1113 Summary of Calculation Results for the Ammonia Example Case 6. Gross Errors Present in Three Measurements Serial Elimination of Gross Errors Applied Measurement Test Used for GED
MEASUREMENT TEST ( .950 CONFIDENCE LEVEL)
INITIAL DATA RECONCILIATION
CRITICAL VALUE = 3.14
*** 7 MEASUREMENTS FAILED THE MEASUREMENT TEST (SEE TABLE I 1 1 2) *** TAGNAME
*** 2 IMEASUREMENTSFAILED THE MEASUREMENT TEST *"*
DETECTABILIN FACTOR
STRM/UNlT VBL
UOM
TAGNAME
MOL%
C14
C
T22
MEASUREMENT VALUE
CALCULATED VALUE
MTSTAT
S4 X3 S22 TEMP
*** MEASUREMENT F07 WJLL BE DELETED IN THE NEXT PASS ***
= 39 (2 NONREDUNDANT) = 60 (0 UNOBSERVABLE)
TAGNAME
= 41 (29 FIXED BY [JSER) = 82 = 19
MEASURED VARL4BLES UNMEASURED VARIABLES FIXED VARIABLES NUMBER OF EQUATIONS DEGREE OF PADUNDANCY
STANDARD MEASUREMENT DEV~ATION \!ALUE
CAiCULATED VALUE
29.1232
3.7306
GLOBAL, T'dSTT ( .95G CONFIDENCE LEVEL) CRITICAL VALUE = 30.10
*** DID NOT PASS THE GLOBAL TEST "**
= 38 (2 NONREDUNDANT) = 61 (0 UNOBSERVABLE)
= 4i (29 FIXEC BY USER) = 82 = 18
SilMbMRY OF ELiMlNATED MEASUREMENTS .*
GT STATISTIC = 43.42
4.5065
PASS 2 OF SERIAL ERROR ELIMINATION
SYECNARY OF ELIMINATED MEASUREMENTS PASS STRMiUNlT VARIAGLE iJOh5
33.6000
14.0417
*** MEASUREMENT C14 WILL BE DELETED IN THE NEXT PASS ***
PASS 1 OF SERIAL ERROR ELIMINATION MEASUPSD VARIABLES UNMEASURED VARIABLES FIXED VARIABLES NUMBER OF EQUATIONS DEGREE OF REDUNDANCY
18.0000
PASS STZM/UNIT VARIABLE UOM
iAGP4AME

STANCARD DEVlATiON
MEASUREMENT VALUE 
CAlCtiLATEG VALUE
i
s7 RATE
MiMR
F07
IOOO.L7(iOC
19999.9992
32202.97 16
2
S4 X3
MOL%
Cl4
1.0000
18.0000
12.8745
S1.3BAL TEST (.S3G COiu'FiDENCE LEVEL) G?' STATISTIC = 23.10
CRITICAL VALUE = 28.90
*** MEASUREMENTS PASSED THE GLOBAL TEST ***
fndustr;ul A,vpliccitinrr$ of Dafa Recorlciliatiorr and Gross ElrorDerecrior~T e c h / o g j u s
Table 1 1 13 (continued) Summary of Calculation Results for the Ammonia Example Case B. Gross Errors Present in Three Measurements Serial Elimination of Gross Errors Applied Measurement Test Used for GED
Table 1114 Sr~mmary of Calculation Results for the Ammonia Example Case B. Gross Errors Present in Three Measurements Serial Elimination of Gross Errors Applied Principal Component Measurement Test Used for GED
MEASUREMENT TEST ( ,950 CONFIDENCE LEVEL)
INITIAL DATA RECONCILIATION
CRITICAI, VALUE = 3.13
PRINCIPAL COMPONENT MEASUREMENT TEST
***
1 MEASUREMENT FAILED THE MEASUREMENT TEST
STRM/UNIT VBL
S22 TEMP
UOM
TAGNAME
MEASUREMENT VALUE
C
T22
33.6000
**"
( .950 JOINT, .997 INDIVIDUAL CONFTDENCE LEVELS)
CRITICAL VALUE = 3.02
CALCULATED VALUE
29.1232
MTSTAT
*** 3 PRINCIPAL COMPONENTS FAILED THE: PCM TEST ***
3.7306
MAJOR CONTRIBUTING MEASUREMENTS TO TKE FAILED PRINCIPAL COMPONENTS AND SHARES
*** MEASUREMENT T22 WILL BE DELETED IN THE NEXT PASS "*" PASS 3 OF SERIAL ERROR ELIMINATION  

MEASURED VARIABLES UNMEASURED VARIABLES FIXED VARIABLES NUMBER OF EQUATIONS DEGREE OF REDUNfiAh'CY
= = = = =
37 (2 NONREDUNDANT) 62 (0 UNOBSERVABLE) 41 139 FIXED BY USER) 82 17
OUTLIER PC#
PC SCORE
10
4513
SUMMARY OF ELIMINATED NLEASUREMENTS ?ASS STRMjUNlT VARIABLE UOM
TAGNAME
STANDARD DEVIPT!ON
MEASUREMENT VUUE
CALClJLAED VALUE

STRM/UNU VARIABLE, SHARE %
STGM/UNIT VARIABLE, SHARE %
STRM/UNIT VARIABLE, SHARE %
STRM/UNIT VARIABLE, SHARE %
S5 RATE 28
S8 TEMP 20
S 16 RATE 17
S13 RATE i4
Slr) KATE 12
S4 RATE 12
S!4 RATE II
Sf2 TEMP 8
S7 RATE
s11 TEMP 7
S10 TEMP 7
Sll RATE 6
3l.r XS
S 14 X1 7
7
I 2
3
S7 RATE M3!HR S4 MCLQ X3 S22 TbMP C
I
F07
1030 @600
!9a99 9992
22203 3128
C14
I 0009
18 0000
!2 8741
TS2
1 2000
11 6000
29 1212
S14 TEMP 5 4
3 731
S22 TEMP !00
I3
3 049
S4 X3
GLOBAL TEST (.950 CONFIDENCE LEVtL)
GT STATISTIC = 9.1 6
3453
CRITICAL VALUE = 27 60
*** MEASUREMENTS PASSED THE GLOBAL, TEST ""*
Sf4 X4 74
S 19 X2
RtEASUKEMENT TEST (.950 CONFIDENCE LEVEL) CRITICAL VALUE = 3.12 ** ALL MEASUREMENTS PASSED THE MEASUREMENT TLST ^'^
S13 TEMP 4
d
8
S19 X! 5
J
10
4
""" MEASUREMENT F05 WILL BE DELETED IN THE NEXT PASS ' "' (tciblr ~o~i~irrrrcd o n tletr j~aqc!
Dnro Rer on( ~lratronand Gloss Error Deii,ctron
364

Table 11 14 (continued) Summary of Calculation Results for the Ammonia Example Case B. Gross Errors Present in Three Measurements Serial Elimination of Gross Errors Applied Principal Component Measurement Test Used for GED
Table 1114 {continuedj Summary of Calculation Results for the Ammonia Example Case B. Gross Errors Present in Three Measurements Serial Elimination of Gross Errors Applied principal Component Measurement Test Used for GED
PASS 1 OF SERIAL ERROR ELIMINATION MEASURED VARIABLES UNMEASURED VARIABLES FIXED VARIABLES NUMBER OF EQUATIONS DEGREE OF REDUNDANCY
MAJOR CONTRIBUTING MEASUREMENTS TO THE LARGEST PRINCIPAL COMPONENT AND SHARES
= 39 (1 N3NREDUNDANT) = 60 (0 UNOBSERVABLE) = 41 (29 FIXED BY USER) = 82 = 19
OUTLIER PC#
2
PC SCORE
4.348
SUMMARY OF ELIMINATED MEASUREMENTS PASS STRMIUNIT VARIABLE UOM
TAGNAME
STANDARD DEVIATION
MEASUREMENT VALUE
CALCULATED VALUE
* 
RATE
MYHR
F05
1650.0000
54999.9996
6368 1.9602
STRM/UNIT, VARIABLE, SHARE %
STRMIUNIT, VARIABLE, SHARE %
STRM/UNIT, VARIABLE, SHARE %
S13 TEMP 38
58 TEMP 30
S 14 TEMP
SI 1 TEMP 6
S 10 TEMP
S12 TEMP 5
6
C
4
GLOBAL TEST (950 CONFIDENCE LEVEL) S T STATISTIC = 66.85
STRMIUNIT, VARIABLE, SHARE %
3.731
16
S22 TEMP 100
CRITICAL VALUE = 30. I(!
*** DIE NGT PASS THC GLOEAL TEST **" PKZKCfl'AZ COMPONENT MEASUREMENT TEST ;.SjCi JOIN'T, .997 INDIVIDUAL CONFIDENCE LEVELS)
CRITICAL \IALUE = 3.00
*** 3 PR INC!PAL COMPO3ENTS FAILED TIiE BCM TEST **?' 

**" MEASUREMENT TI 3 W!Lt BE DELETED IN THE NEXT FASS * * & 
PASS 2 GF AUTOMATIC ERRQR ELIMINATION 

MEASURED VARIAi31AbS= UNME4SURFD VAKlABLE'5 = FIXEL: ',.\RIABLLS = NUMBER OF EQIJATTOhS = TrLZGREEOI REDUNDANCY =
.
38 (1 SONREDUNDANT) 61 (0 klhOBSERVABLE) 41 (?9 FIXLD BY USER) 82 18
litdustrial A/~~~licutiuns of Dara Rrconci1iu:;orz nizd GIus5Elror L)e:ecrio~~ T r c h n ~ l o ~ i e357 s


Table 11 14 (continued) Summary of Calculation Results for the Ammonia Example. Case 6. Gross Errors Present in Three Measurements. Serial Elimination of Gross Errors Applied. Principal Component Measurement Test Used for GED.
Table 11 14 [continued) Summary of Calculation Results for the Ammonia Example Case 6. Gross Errors Present in Three Measurements Serial Elimination of Gross Errors Applied Principal Component Measurement Test Used for GED 
PASS STRM/UNIT VARIABLE UOM
1
*+
MEASUREMENT VALUE
F05
1650.0000
54999.9996
65494.10 18
TI3
5.0000
450.0000
48 1.3397
MEASURED VARIABLES UNMEASURED VARIABLES FIXED VARIABLES NUMBER OF EQUATIONS DEGREE OF REDUNDANCY
CALCULATED VALUE
S5
RATE M3mR
2
STANDARD DEVIATION
TAGNAME
S13 TEMP C
GI,OBAI, TEST ( .950 CONFIDENCE LEVEL) CRITlCAL VALUE = 28.90
GT STATISTIC = 49.74
PASS STRMIUNIT VARIABLE UOM
1
*:** DID NOT PASS THE GLOBAL TEST *""
2
1.950 JOIKT, .a97 INDIVIDUAL CONFIDENCE LEVELS) 3
CRITICAL VALUE = 2.98 2 I'XJNCIPAL COMPONENTS FAILED THE PCM TEST *<:":
MAJOR CONTRIBUTING MEASUREMENTS TO THE LARGEST PR!NCIPAL COMPONENT AND SHARES
>
PC SCORE
3.731
STRM/UNli, VARIAGE, SHARE %
STR!d/UNIT, VARIABLE, SHARE %
STPA/UNIT, VAR!ABlE, SHARE %
STRM/UN:T, VARIABLE, SHARE %
S22 TEMP 100
S5 RATE Sl3 TEhIP S27TEMP
TAGNAMi
STANDARD DEVIATION
MEASUREMENT VALUE
CALCULATED VAlUE
M3/HR
F05
1650 0000
54999 9996
65494 1276
C
Ti3
iO(i00
liO GO00
4s 1 3396
C
T22
1 2000
33 6000
39 3 232
GLOBAL TEST ( 950 CONFIDENCE LGVEL) GT STL4TISTJC= 35 82
"*"
CRITICAL VAL'JE = 27 fv0
33113 NOT P4SS THE GLOBAL TEST '
PRINCIPAI, CORIPONBNT MEASUREMENT TEST (.95C JOINT, .997 INDIVIDUAL, CONTlDENCE I E V E L S: CIIITICAL VALUE = 2.97
"** 1 PRINCIPAL COMPONENTS FAILED THE PC3f TEST "':* . 
.
*** MEASUREMENT T22 WILL BE DELETED IN THE NEXT PASS "**
= 37 (1 NONREDUNDANT) 62 (0 UNOBSERVABLE) 41 (29 FIXED BY USER) 82 17
= = = =
SUMMARY OF ELIMINATED MEASUREMENTS
PRINCIPAL COMPONENT RlEASUREMENT TEST
OUTLIER FC#

PASS 3 OF SERIAL ERROR ELIMINATION
SUMMARY OF ELIMINATED MEASUREMENTS


Table 1 1 14 (cont~nvedj Summary of Calculation Results for the Ammonia Example Case B. Gross Errors Present in Three Measurements Seriai Eiimination ofGross Errors Applied Principal Component Measurement Test Used for GED
lndirsrriai Applicurions of L)ata Reronciliation and Gross Error Def. riior~Techno!nRie.y
MAJOR CONTRIBUTING MEASUREMENTS TO THE LARGEST PRINCIPAL COMPONENT AND SHARES
Steadystate process unit data reconciliation and gross error detection technology is widely used in chemical, petrochemical and other related industrial processes. Online data reconciliation is important for enhancing the accuracy in process optimization and advanced process control. Steadystate detection is necessary in order to increase the accuracy in the reconciled values and to provide meaningful gross error detection. If the process is not operated at steadystate for a longer period of time, dynamic data reconciliation should be applied. Proper component and thermodynamic characterization and accurate compositions are very important for a successful data reconciliation and gross error detection. Rigorous rnodel enables merging the data reconciliation and parzmeter estimation into one problem which can be solved simultaneously. Plantwide material and utilities reconciliation is an important tool for production (or yie!d) accounting. 'The most challenging prcblem in produciion acccunting is the estimatior? of various leaks and losses. With enough redundancy, data reconciliation can provide reasonsble estimates :or the magni:udes of materials that are not accounted for. Irnposing bounds on vaiiables is often necessary to eilsure a feasible solutior, for the daia reconciliation problem. 7'0 so!ve a botinded problem, an NLPbased software is n~eded. The existen: gross error detection meThods do not accurately detect gross errors ail the time. Some methods are better than 0thers, but their overal! performancz depends upon the model accuracy and the level of data redundancy.
OUTLIER PC#
9
PC SCORE
3 048
SUMMARY
STRMIUNIT, VARIABLE, SHARE%
STRM/UNIT, VARIABLE, SHARE %
STRMIUNIT, VARIABLE, SHARE %
STRM/UNIT, VARIABLE, SHARE %
S4 x3
S19 X3
S4 X1
14
S19 X1 11
S14 X3
4
S 14 X4 3
58 S4 X5
S14 X5 5
8 *
=** MEASURZMENT T22 WILL BE DELETED IN THE NEXT PASS ""'
PASS 4 OF AUTOMATIC ERROR ELIMINATION bIEASURED VARIABLES UNMEASURED V4RIABLES FIXED VARIABLES NUMBER 01 EOUATIONS DEGREE OF KtDIJNGAWCY
= 36 ( 1 NONREDLINDANT) = 63 (0 UNOBSERVABLE)
= 41 (29 FIXED BY USER) = li2 = 16
SUMMARY GF EL!MINATED MEASUREMENTS PASS SIRMIUNIT VARIABLE UOM
1 2
3 4
S5 RATE S13 TEMP S2L TEMP S4 X3
TAGNAME
STANDARD DEVIATION
MEASUREMENT VALUE
CALCUL4TED VALUE
M3mR
FOS
1650.0000
54999.9996
65795.7594
C
TI 3
5 OC00
450 0000
181 4463
C
T22
1 2000
37 6000
29 1232
MOLLrc
CI4
1 0000
18 0000


12 8642 
GLOBAL TEST ( 950 CONFIDENCE LEVEL) GT STATISTIC = 15.42
369
CRITICAL VALUE = 26.30
**" MEASlJREMENTS PASSED TIIE GLOBAL TEST """ '** ALL PRlNCIPAL COhll'ONENTS PASSED THE PCM TEST 'I"* *'* ALL VARIABLES ARE L1'17'HIN i3O{JNI)S ""*
Uuf
12. Christiansen, L. J., N. BrunicheOlsen, J. M. Carstensen, and M. Schroeder. "Performance Evaluation of Catalytic Processes." Co17zputersClzern. Engng. 21 (Suppl., 1997): S 1179S1184.
REFERENCES 1. Ravikumar, V. R., S. Narasimhan, S. R. Singh, and M. 0. Garg. "RAGEThe State of the Art Package for Plant Data Reconciliation and Gross Error Detection," presented at the International Symposium on Automation and Control Systems, New Dehli, India, 1992.
0
*
2. Chi, Y. S., T. A. Clinkscales, K. A. Fenech, A. V. Gokhale, C. Jordache, and V. L. Rice. "Online, Closed Loop Control and Optimization of an FCCU Using a Self Adapting Dynamic Model," presented at the AIChE Spring National Meeting, Houston, Tex., 1993.
5. Leung, G.. and K. H. Pang. "A Data Reconciliation Strategy: From OnLine Implementation to OffLine Applications," prese~ltedat the AIChE Spring National Meeting. Orlando, Fla., 1993.
16. Ravikumar, V., S. Narasimhan, M. 0. Garg, and S. R. Singh. "RAGEA Software Tool for Data Reconciliation and Gross Error Detection," in Foul*darions of ComputerAided Process Operations (edited by D.W.T. Rippin, J. C. Hale, and J. F. Davis). Amsterdam: CACHE/Elsevier, 1994, 329L.436.
1
17. Stephenson, G. R., and C. F. Shewchuck. "Reconciliation of Process Data with Process Simulation." AICIzE Jo~cl1lal32 (1 986): 247254.
i.
18. Meyer, M.. B. Koehrct, and h4. Enjalbert. "Data Reconciliation on Multicomponent Network Frocess." Conipzito..~Clzenz. E!zgng. 17 (no. 8. 1993): 8078 17.
6. Scott, M. D., J. M. Tiessen. and S. L. Muilick. "Reactor Integ~atedRigorous online rnodel (ROMTM) for a MuitiUnit HydrctrearerCatalytic Refonner Complex Optimization," prescnted at the NPRA Computer Conference, Anaheim, Calif., 1994. 7. Chiari. M., G. Bussari, M. G. Grotto!;, and S. Pierucci. "Online Data Reconciiiatioil and Optitnization: Refinzry P,pplizat~ons." Co:nputer.s C!l?n~. Elzgng. 21 ([email protected],1997): S 1!85S 1 190.
8. Nair, P., and C. Jordache. "Rigorctus Data Reconciiir?tioi~is Key to Opti~nal Operations." Contro! (Oct. 1991): 118123. 9. Tan~ura,K. I, T. Sumioshi. G. D. Fisher, and C. E. Fontenot. "Octimization of Ethylene Plant Operatioils Using Rigorous Models," presented at the AIChE Spring National Meeting, Houston. Tex.. 1991.
10. Sanchez, M. A., A. Bandoni. and J. Romagnoli. "PLAPATA Package fur Process Variable Classification and Plant Data Reconciliation." Computers Chern. Erzgng. (Suppl. 1992): S499S506. 11. Natori, Y., M. Ogawa. and V. S. Vemeuil. "Application of Data Rcconciliation and Sirnulatior1 to a Large Chemical Plant." Proceedings of Large Chemical Plants 8th International Symposium, Antwerp, Belgium, 1992, pp. 1011 13.
14. Dempf, D., and T. List. "Online Data Reconciliation in Chemical Plant." Cornj~utersChenz. Engng. 22 (Suppl., 1998): S 1023S 1025. 15. Placido, I., and L. V. Loureiro. "Industrial Application of Data Reconciliation." Conllputers Chem. Engilg. 22 (Suppl., 1998): S 1035S 1038.
3. Bagajewicz, M., and S. L. Mullick. "Reconciliation of Plant Data. Applications and Future Trends," presented at the AIChE Spring National Meeting, Hcuston. Tex.. 1995. 4. Charpentier, V., L. J. Chang, G. M. Schwenzer, and M. C. Bardin. "An OnLine Data Reconciliation System for Crude and Vacuum Units," presented at thc NPRA Computer Conference, Houston, Tex., 1991.
13. Holly, W.. R. Cook, and C. M. Crowe. "Reconciliation of Mass Flow Rate Measurements in a Chemical Extraction Plant." The Canadian J1. of Chenz. Erlgr~g.67 (1989): 595401.
13. Nar:isirnt;an. S., R.S.H. Iviah, atid A. C.Tamhaiie. "A Composite Statistical l'cst foi Detecting Changes in Steady State." AICIIE Joii:~zal32 (1386): i4i19:118.
i
20. Narasimhan. S., C. S. Kao, aitd R.S.H. Mah. "Detecting Changes in Stzady Stzte Using the Mathematical Theory of Evidence." AIChE J o ~ c r ~ l a l 33(1990): 19301 931.
ii j l!
I
d
21. CEP Software Cirectory, a Sup;~!ement to C;le~ti.C:,qng. P~.o,oress.p ~ i b lished by thz American I ~ s t i t u ~ofe Chcrnical Er;gineers. 1998.
22. Tong. Fi., aiid D. Bluck. "An Indurtrir?I kpplicaticn of Principa! Ccinponer:t lest ro Fault Cetec~ionand Ideniifisatiott." presented at tile IFAC Confcrence, 1998.

23. Reagan E.. B. Tilton, and S. Sa!nmcx!.. “Yield A ~ ~ ~ i i i i tai ndg Data intezraiion." presented at the NPKA Compi1ti.r Conference, Atlanta. Ga., 1996. 24. Grosdidier. P. "Understand Operation Information Systems." H~drorar1,on Pmcessii~y(Sept.1998): 6778, 25. Veverka. V. V., and F. hladron. Marc.)in1 cr~iclG~arg?Ralancirig ill PIocess 1rzdu.srries: Fronz Microscopic. Ra1~11ce.s to L A I I X ~ Plants. Amsterdam: Elsevier. 1997.
By suitably adding vectors which have the same number of elements, O ln(71evectors other vectors can be formed. A linear coinhii~afionof ~ L L ' oris a vector which is formed by niultiplying each vector by a real number (scalar) and adding them. For example, consider a linear combination of the above vectors a and b represented by the vector d = a l a + a 2 b . If we choose the scalars a, = 1.5, and a2= 2.0, then the vector d is given by
ber of elements in any vector in a basis set (components of a vector) need not be equal to the dimension of the vector space spanned by the basis. This is illustrated clearly by the vector space spanned by the basis set a and b, where each of these vectors has 3 components but the dimension of the vector space spanned by them is only 2. Note that we often speak of a vector having n elements as an ndimensional vector. This only implies that the vector having n elements is a member of the ndirnensional space of vectors. We use the notation 157," for the ndimensional real vector space.
Given a set of 11 vectors, S = { a , , a?; . ., a n } ,we can generate all possible linear combinations of vectors in this set, a l a l + a2a, + . . . a n a n ,by choosing all possible values for the scalars a i . We refer to the col!rrt;qn of vectors thus generated as a vector space sl~arz~zed by the vectors in set S and denote it as V(S). It should be noted that the zelo vector is a laernbe1 of this space. A set of vectors S is said to be linearly independent, if a linear combinstion of the \iectors in this set equals 0 only f ~ the r case when all the scalars a, are equal to 0 and not for any other choice of the scalar values. if a set of vectors is lineally dependc~it,then t!?eic is a vector in thi:; set hose scalar multiplyin2 factor (a,is i l o n ~ ~ r \vhich o) can Sc expressed a.; a li!lear con:birlation of the other vec!ors in this st!. We can delere this vector and again check if the Ienlaining vccto:s are lineariy independent If' nct, we can repeat this procedure until we are left with a sct of veciors that arc lineal!y ir~dependent. The sr,t of vectors \vhich remain f ~ r ma ~:zir~irnnl set ~flineal11ii7d~pe17ilclzt ;Jcc.to~..i which span the vectar space V ( S ) .This mijtimal set of x c t o r s is said to for111a hcrsis sct for V(C). For example, if we consider the set S consisting of vectors a. b. anti d defined above, then this forms a linearly depertcicnt set because an! vector in this set can be expressed as a lineas combination of the other two vectors in this set. We can choose to delete \,ector d tiom this set. in which case we are left with the two vectors a afid r j which can be veitfied to be linearly independen!. Thils. the vectors a and b form a basis set for the vector space spanned the three vectors a. b, and d. Another basis set for the same vector space is a and d. There can be many different choice of a basis set for a vector space. but the number of vectors in every basis set is the same, and is denoted as the di171~17~iolr of the \.ector space. It must be borne in mind that the num
MATRICES A N D THEIR PROPERTIES A real matrix of order nz x n is an ordered set of eltmentc consisting of rows of i z elements each. Each row of a matrix can be regarded as an ndimensional row vector and each column of the matrix can be regarded as an trzdimensional column vector. Thus, an nz x rr matrix can either be ccnsidered as an ordered set of ~nrow vectors each of dimension n. or as a set of 11 column vectors each of dimension 111. Two specla1 r ~ a t r i c e sare the zero matrix dent~tedby 0. whose elements are 0, and the identity matrix of orcier I? s 17,denoted by the sylnhol 1, whose column i is the unit vector ei. it shou!d be noted tha: we do not explicitl!. denote the dimensions of the matrix in the notation because it is usua!ly c!ear from the context. r vectcr spaces associated with eveiy matrix as There 2re f ~ u iinpostant definzd below: 171
(i) Row Space: the space spanned by the roklJs. (2) Column Space: the space spanned by the co!unins. This space is 2 1 kno~v11as the Iunge sj~oceof a mat:ix. (3) Null Space: the space spanned by all veclors x which satisfy A x = G. where x is a vector belonging to 9i". (4) Left Yiuli Space: the null space of the transpose of a matrix.
There are some important properties that link these vector spaces. The la~zkof a matrix is equal to the dimension of its row space, u~hichis also eqiial to the dimension of its column space. This immediately implies that the rank of a matrix r I ri~in(nz,11). A matrix of order 11 x iz is know11 as a square matrix of order n. If the rank of such a matrix is n, then the matrix is known as a nonsingular matrix and its inverse exists.
~
~
376
Appendix ABasic
liurir Kci ori<ilinrionar~dGross Etror Deteciion
The nonzero vectors xiof size n that satisfy the equation
T h e following equality can also be proved:
where N(A) is the null space of matrix A. From the above equation, it follows that if the rank of a matrix is equal to n (which implies that the columns of a matrix are linearly independent), then the dimension of the null space of the matrix is 0. The only vector which satisfies Ax = 0 in thls case is the 0 vector. In general, we are interested in obtaining a vector x which is the solution of the linear set of equations Ax = b. In other words, we wish to express the vector b as a linear combination of the colurnns of A.This is possible only if b is a member of the column space of A. Furthermore, the solution is unique if the null space of A has d i w  n ~ i o n 0. This property is used in obtaining the solution of the unmeasured variables in data reconciliation discussed in Chapter 3. In general, we can express the solution vector x as
x = X r + X,,
(A  2)
where x, belongs to the column space of .4 and x, belongs to the null space of A. This is known as the ralrgr and rzull s p c e dc~co)npo.~itio17 (RND). which is used in the RNDSQP nonlinear constrained opti~nizatic11algorithm discussed i i l Chi!pter 5. 'The el;.erzlv!urr of a square matrix A of clrder n are the 12 roots of its characteristic equation:
={hl,jL2, ) . . .. A,,}. !f we The set of these roct\ is denoted by h < ~ define the t1at.r of matrix A by
then tr(A) = h, + h2+ . . . + &,
Coricrpfsin Linear Algebra
are referred to as eigenvectors. The eigenvalues and eigenvectors of certain matrices are used to build the principal component tests in Chapter 7.
REFERENCES 1. Noble, B., and J. Daniel. Applied Linear Algebra. Englewood Cliffs, N.J.:
PrenticeHall, 1977. 2. Strang, G. Linear Algebra and its Applications, 3rd ed. Orlando, Fla.: Harcourt Brace Jovanovich, 1988. 3. Golub, G. H., and C. F. Van Loan. Matrix (3omnputations. BaltimorcJohns Hopkins University Press, 1996.
Appendix B
Graph Theory
I
b
0
Graph theory deals with problems related to topological properties of figures. It is also useful for analyzing problems concerning discrete objects and their interrelationships. In this appendix, we define some of the important concepis of graph theory used ir? the book and illustrate them usins era~npies.Son?e facts are simp!;' stated withou: proofs and we direct the interested render to the book by Deo [ I j for thest: proofs 2nd add~tionalconcepts and theorems.
ir'
8
.
Figure B1 A graph.
GRAPHS, PROCESS GRAPHS, AND SUBGRAPHS A grcrplz consists of a set of /?odes,V, and a set ~f edges, E. Each edge is associatei with a pair of nodes, which it joins. An exampie of a graph is shown in Figure £31, which has six nodes draw2 as circles and eight edges shown as lines. Each edge is said to be incident on the nodes with which it is associated. The degree of a node is the number of edges incident on it. Aprocess graph is a graph which is simply obtalned trom a process flowsheet by adding an additional node called the envi~c)nrnent node to which all process feeds and products are connected. For example, the graph in Pisure H I is the process graph of a simplified ammonia process whose flowsheet is shown in Figure 101. The nodes of a process graph cotrespond to process units and the edges of the process graph correspond to streams that interconnect the units.
Thus, if the process contains n units and e streams, then the corresponding process giaph contains n+l nodes and e edges. For ease of reference, we nurnbei or label the edges and nodes of the graph using the same numbers or labels as used in the process flowsheet, except for the environment node which is labelled as E. If the directions of the edges are ignored, as in Figure RI, then an undirected graph is obtained; otherwise, the graph is directed. In this text, we are only concerned with undirected graphs.
C
d
A suligtaph of a graph consists of a subset of nodzs ar.d edges of the graph. Each edge of the subgraph jsins the s a n e two nodes as it does in the graph. In other words, if an edge is part of a subgraph. then the end nodes with which it is associated in the graph should also be pait of the r z is a subgraph of thz graph shcwn in subgraph. The graph in F l g ~ ~ B2 Figure B 1.
Figure 62. A svbgraph of the graph in Figure 61.
L)UIURerr,ncili:ztion and Gross Elror 1)efecfion
PATHS, CYCLES, A N D CONNECTIVITY A pat11 between two nodes (denoted as the initial and terminal nodes of the path) is a finite alternating sequence of edges and nodes such that each edge in the sequence is incident on the two nodes preceding and succeeding it, and no node appears more than once in this sequence. A path is called a cycle if the initial and terminal nodes are the same. For example, in Figure B 1, the alternating sequence of nodes and edges, E 1M2H3R, is a path between initial node E and terminal node R, while the sequence E6S5SP8E is a cycle. A graph is connected if there exists a path between every pair of nodes of the graph. The graph in Figure B1 is a connected graph as is generally the case for all process graphs.
Figure B3. Subgraph formed by deleting node E from graph in Figure B1
S P A N N I N G TREES, BRANCHES, A N D CHORDS A connected subgraph of the graph which does not contain any cycles and which includes all nodes of the graph is called a spanning tree of the graph. An edge of the graph that is part of the spanning tree is called a hlmlclz, while edges of the graph not part of the spanning tree are called c1zords. Figure B2 is a spanning tree of the graph of Figure bi. Corres ~ o n d i n gto this spailning tree, edges 2, 4, 5 , 7, ar,u 8 are branches while the remaining edges 1, 3, and 6 arc chords I? should be noted that brafiches and chords are defined with respect to the specified s p a ~ n i n gtree of a graph. If a different spafining tree of the graph is chosen then, accordingly. different edges of the graph are c1sssifie6 as brancl~esc r chards. It can be proved tha: n spanning tree contains n branches and e1: chords where n is the number of upits in the process flowsheet (or one less than the number of nodes o l the process grzph). and e is the number of streams or edges of the graph.
Figure 64.Graph fcrmed by m~rgingnodes E and M of graph in F i ~ u r e51.
*
%
GRAPH OPERATIONS A graph can be modified by operations such as deletion of edges or nodes and by merging of nodes. The deletion of an edge from a graph results in a subgraph which contains all nodes and all edges except the deleted edge. For example, the spanning tree shown in F i g ~ r eB2 can be obtained from the graph in Figure Bl by deleting edges 1, 3, and 6. The deletiorz of a node from a graph results in a subgraph which contains all the nodes of the graph except the deleted node, and contains all edges of the graph except the edges which are incident or1 the deleted node.
The subgraph shown in Figure B3 can be cbrained from :he graph of F i g ~ r zB1 by deleting the node E. The t t 7 e t ~ i n gcf nvo izodes cT a graph resul:s in a modified graph obtained by replacing the two merged nodes by a new node and dcieting the edge,; ir~cidcnt(111 both these :lodes. E d p which are incident 011 only one of the two merged nodes in the orig~nal graph arc now incident on the new node of the modified graph. The graph in Figure B4 is obtai~iedCrom erzph in Figure Bl by n~ergingnode5 E and M. The new merged node in Figure B4 is denoted as EM.
CUTSETS, FUNDAMENTAL CUTSETS, A N D FUNDAMENTAL CYCLES
!
5
'3
A crttret of a graph is a set of edges of a graph whose deletion disconnects the graph, but the deletion of a proper subset of the edges of a cutset does not disconnect the graph. The set of edges 12. 5. 61 is a cutset of the zraph in Figure Bl since the de!etion of this set of edges disconnects the graph into two node sets one cont;:irmin~ ikf. E, and SP, and :he other
382
Doru Keconciliulion and Gmss Erro~Dprecriotz
containing H, R, and S. On the other hand, the set of edges [l, 2, 5, 61 is not a cutset although the removal of this set of edges disconnects the graph si~iceits proper set of edges [2,5,6] is a cutset. There is a correspondence between cutsets and flow balances that can be written for a process. A flow balance can be written around every unit of a process, which will involve the flows of streams that enter or exit this unit. It can be verified that the edges corresponding to these streams form a cutset of the process graph. Thus, corresponding to every cutset consisting of all edges incident on a node, a flow balance can be written. Flow balances can also be written corresponding to other cutsets which are essentially linear combinations of flow balances around individual process units. Thus, corresponding to the cutset [2, 5, 61 of the graph in Figure B1, the flow balance equation involving the flows of streams 2, 5 , and 6 is a linear combination of the flow balances around process units H, R, and S of the process. It should be noted that the direction of the streams should be taken into account when writing the flow balances corresponding to cutsets of the process graph. A cutset of the graph which contains only one branch of a spanning tree of the graph and zero or more chords is called a fu1dui~let7talcutset corresponding to the spanning tree. For exa~nple,edge set [I, 3, 71 is a filndarnenta! cutset of the graph ill Figure EZ1, corresponding to the spanning tree shown in Figure B2. However. although the set of edges [2. 5, 61 is aiso a cutse! of the gra.ph in Figure B1, it is no: a fundamental cutset with respect to the spanning uee of Figure 3  2 because it c o n ~ ~ i two n s branches, 2 and 5 of the spalning tree. With respect to every branch of a spanning tree of a graph, 2 fundamental cutset can be identified. The fiindarnental cu:sets corresponding to t!e spznning tree, Figure B2, sf rhe graph in Figure E1, are @, 31, @, 31, [5,3 , 6 ] , [Z, I , 31, and [& 1, 61, where the branch in each fundamenral cutset is indicated by an undersccre. A concept, which is complementary to a fundamental cutset, is that cf a fundamental cycle with rzspect to a spanning tree of a graph. A funriclr?ze:7tul cycle with rcspect tc a s p n x i n g :Tee cf ;&;'$I is r; c j x k ;f ;:.: graph farmed by exactly one c h ~ r "2nd 9ne or more brq.r?ches. The cycle E1M7Sf8E, which consists of edges [I, 7, 81, is a fundamental cycle of the graph of Figure B1 with respect to the spanning tree B2, which consists of chord 1 and branches 7 and 8. For each chord of a spanning tree of a graph a fu~idameatalcycle can be identified. The fundamental cycles with respect to the spanning tree, Figure B2, of the 4, 5, 7, 21, where the graph in Figure BI are [I,7, 81, [G, 5, 81, and [3, chords are indicated by an underscore.
Fundamental cutsets are complementary to fundamental cycles in the sense that if a chord cj occurs in the fundamental cutset of a branch b;, then branch bi occurs in the fundamental cycle of chord cj. This may be verified from the fundamental cycles and funda~nentalcutsets with respect to the spanning tree of Figure B2 listed in the preceding paragraph. This property can be used to identify the fundamental cycles with respect to a spanning tree given the fundamental cutsets with respect to the same spanning tree. Fundamental cycles (or fundamental cutsets) can be used to generate new spanning trees of a graph starting from a given spanning tree. The technique known as an elementary tree ttunsfottnatioir(ETT) involves the interchange of a chord with a branch. In this technique, we add to the spanning tree a chord and delete a branch belonging to the fundamental cycle with respect to the original spanning tree formed by the chord which has been added. For example, the spanning tree shown i n Figure B5 is a new spanning tree of the graph in Figure B1 obtained from the spanning tree in Figure B2 by adding chord 1, and deleting branch 7, which belongs to the fundamental cycle forrned by chord I. The new spanning tree differs from the initial spanning tree in respect of one cizoid and one branch, and is also referred 1 0 as the neighbor of !he initial spanning tlee.
Fioure 65. Spannina tree formed by ETT of spannir.9 tree in Figure 82
REFERENCE 1. Deo, N.Gruplz 777eor~~ , i t kA/~j)Iicafiorr.~ to E I I ~ ~ ~ I C( 'zC ~ d~(:Ott~j~i~ietY I I ~ Science. E~~glewood Cliffs, N.J.: PrenticeHall. 1974.
Appendix C
Fundamentals of Probability and Statistics
R A N D O M VARIABLES A N D PROBABILITY DENSITY FUNCTIONS Probability is a mathematical theory dealin: with the laws of rundonr example, the rzsu!t of a phy~icdlor cheinical experiment is a random event. The measti!ed ol inferred va!m obtained at the end o i experiment is a ru?zdoin variirh!e which lies within a specified interval with a certain prcbability. it is easier to understand the behavior of random variables if we analyze a discrzte event. The rolling of a pair of dice provides a good exam~ l aei a random variable. It is impossible to PIcdict the outcome of an individ~lalroll; however, i t is nmre likely that the summation ngmher lor the pair of dice i~ a 7 rather [hail a 12. This is because, of the 35 possible rolls, there is oaiy one way :o roll a 12naniely (6.6). while thcre are six ways to roll a 7namely (6,l). (5,2).(4.3), (3,4); (2.5). (1,6). Let us assume that we roll the dice thousands ui' i i ~ ~ dlrd ~ c arecord how many times each ro!l occurred. Then. the probability fur,c:ion for each roll R. i.e., P(K), or socalled prohahilip derl.si~~.fir17crior? (PDF, or p.d.f.1 is: e1.e77t~.For
P(K) =
Number of tiriles roll K occuned Total number of rolls
Figure C1 shows the graph obtained by plotting P(R) for all possible rolls R. The probability of rolling a given valne R in a single throw of the dice is the area under its rectangle. For example, you have a 4 in 36 chance of rolling a 9. The probability of rolling 3 or 12 in a single throw of the dice is the total area under their rectangles, 2/36+1/36=3/36. The PDF graph in Figure C1 is discontinuous, because rolling a pair of dice produces only discrete values, i.e., the resulting value must be an integer between 2 and 12 inclusively. Integrals of the PDF are quite useful because they determine the probability of occurrence of a group of events. For example, we can obtain P (5
i I
6136
Probability o f rolling 9 is 4/36
5/36 PIobability of vaiue rolled
2 3
4
5
6
7 8
9
101112
Value rolled with a pair of dice, R Figure C1. Probabiiiiy density function for r o l l i a ~ poir ~ of dice.
S
386
,
@
Daru R~rcoi~cil~o;ii~ri and GIossError ijc.rec1ir.1
probabilities are of special practical interest. For example, we can make statements such as "there is a 95% chsnce that the true flow rate for stream S10 lies between 5,000 and 6,000 BPD." Or, "there is a 2.5% probability that the true flow rate in stream S10 exceeds 6,100 BPD." In order to calculate such probabilities, a continuous probability density function is required. For reasons specified in Chapter 2, the most widely used density function for continuous random variables in physical and chemical sciences is the normal dis~ributiorzfir7ctiorz. The normal distribution is also known as the Gaussiarz distribution, and its PDF IS described by the formula:
The most important properties of the normal distribution PDF are as follows:
'4
B
1. The maximum value of F(X) occurs at the mean. p. 2. The standard deviation o determines the width (or the skewizess) of the curve. For a very accurate instrument (small o),the density function will look like a sharp peak centered at zero. On the contrary, for an inaccurate instrument (large o),the PDF will look rather flat.
3. The
yoG
factor normalizes the density so that
4. It is symmetric about the mean. 5. The probability of a measurement error lying between XI and X2 is:
where p is the mean value of the random variable X,and o is its standard deviation. Since in practice we expect the errors to be zero on the average, p = 0 for the measurement enor density filnction. Figure C2 shows a Gaussian PDF with a mean of zero and a standard deviation of 1. This funct~onis a conti!luous analog of the dice rolling density function. The ~ o r n t a ldistiibution with Lero mean and standard deviarion of I is also know11 as .srcrrzilcirdIZOIIIK~Idi.~trib~4tto1z.
e
3
This probability is equivalent to :he area under the curve between XI and X7. 6. Similarly, the probability that a nieasurement error (it\ ab~olutevalue) is greater than a panicuiar value X*is
Tnis probability is equivalent to the area u:ldcr the curve ou:side the interva! (X* and X*). 4 s iilcstrated in Figcre C2, 95% of !iie randorn errors should lie within 1.96 standard deviztions. Anaiytlcaily, this means:
Figure C2. Normal distribution density function
This is often ca!led the 95% coi~fiderlceirztet::nl.The 99410 c o n f dence interval occurs within 2.58 standard deviations of the mean. Note that these figures are true when there is only one measured variable. For ~nultiplemeasured variables, the threshold is recalculated based on the rules given in Chapter 7 (see Sidak's rule). Another distribution of interest for the statistical applications in this book is the chisquaw (x" disltihution. If' Kl, K?, . . . R,, are indepen
dent random variables described by a standard normal distribution. then a chisquare random variable is defined by:
2.As v approaches infinity, the chisquare density approaches the normal distribution. The x2(8) curve in Figure C3 is starting to illustrate this behavior.
r=l
The integer v is usually known as the tzumher of degrees of fieedot~z. The probability density function for a for different degrees of freedom v is illustrated in Figure C3. The probability distribution function for the chisquare distribution l i described analytically by the following fomlula:
x'
where r is the gamma function. The most important propmies of the chisquare distribution are:
1. The inean m!ue of %'(\I) is v.
a'
.a
0.18 0.16 0.14 0.12 F(X) 0.1 0.08 0.06 0.04 0.02
P ( X <~ 9.49) = 0.95
W2>9.49) = 0.05 95% o f total area
I iI
I1
I
n
Figure C4. Ccnfidence intervals for chisauure distribution.
Confidence intervals can also be constructed far the X' distributioi;. For e?tar;lple. Fignre C4 silows a 95% icrifidence region for a 2 ' distn1;ution f ~ n c t i o nwith 4 degrees nf freedom. in this particular cxce, 95%of the random variab!es siiould lic between 0 and 9.49.
x2
STATlSTICAL PROPERTIES OF RANDOM VARIABLES The statistical properties for random variables were indirectiy mentioned as part of the ar?alytical description of :he probability density tunctions above. Now we are going to define thein i n ;1 more gencia1 fratncwork. There are two basic properties for the random variables (also known in statistics as ~uot?zer~!s): the mean and the variurlce of the random variable. The mean vaiue of a random variable X.px. is defined as the e.rpected v~ilueof X . For a continuous variable. it can he expressed analytically as:
Figure C3. Chisquare distribution dens+ function
The expected value defined above can also be defined as the first moment about zero. In general, the first moment about a constant value 6 is given by E[X6). If 6 is equal to the expected value of X, the central first moment is obtained, and the corresponding distribution is called the centiul distribution. Otherwise, a noncentia1 distrib~~rior~ is obtained. The mean (expected value) of a random variable Z whose distribution is a joint distribution of other random variables, X I , X2 . . . X, (i.e., a tnultivariate distribution) is defined as
!
C.
'
a
since the variance of a constant is zero. A practical application of the above definitions is the derivation of thc mean vector and the covariance matrix of a vector of randorn variables which is a linear function of other vector of random variables. For exaniple, let us assume a iinear equation in vector form such as:
where Z = f(X1. X2 . . . XI,) and @(XI,X2 . . . X,) is the joiilf piobabilitl), density$i17ctioil of the random variables X I , X2. . . X,,. If f(X1,X2. . . XI,,) is a linear function, i.e.,
then the mean
:,slue of %
1s
If f(X,, X2 . . . X,) is a linear function and the errors of the primary random variables X I , . . . X, are mutually independent random variables, (C 13) reduces to:
Let E(x)=O arid Cov(x) = Q, ~ v h e r zCov(x) means tha covarisnce matrix of vector of laildo111 valiables x. Then, tile expccted \~alueof y is:
a 1 ~ Ilnear. 1
l!ic variance of a random variable X. var(Xj, i s defir~eda< the reccnd
P
z
~uoinerztabout Lero, 1.e.. These results for linear transformations of vector :)f randon, variables arc used in Inany derivations throu,?hoot this book. 'I'he relationship between the variance and the standard deviation of a random variable is given by: Var(>o = o i
(C 13)
The variance of the multivariate random variable Z = f(X,. X ? . . . XI,) is defined as
HYPOTHESIS TESTING Hypothesis testins is a very inipoltant statistical tool for making decisions about random variables. The procedure uses information from a random sa~npleof data to test the truth or falsity of a statement. 7'he basic staternent about a random variable is usually called the rzuil Izy7othe.si.s,
denoted by Ho. The opposite hypothesis about same random variable is ca!!ed :he alternative hypothesis, here denoted b y H I . The decision to accept or reject the null hypothesis is based on a statistical test. The test statistic (the value of the statistical test for given data) is first calculated with the data in the random sample. A decision criterion (a threshold of the statistical test) is used to make the decision about the hypothesis Ho. Two kinds of errors may be made at this point. If the null hypothesis is rejected when it is actually true. a Type I error is made. Alternatively, when the null hypothesis is accepted when it is actually false. then a Type II error is made. The probabilities of occurrence of Type I and Type I1 errors are as follows: N
= P(Type I error) =
y = p(Type I1 error) =
eject Ho I I&
= true)
(C  19)
eject H , I H , = true)
(c 20)
Power = ~ ( ~ c c e N, p t 1 H I = true) = 1  y
(c21) tc: tetit
REFERENCES

The powel of the test is often bsed to evaluate a particular statistical test. and ir is defined as:
In ihis book, the hypothesis testing is used
For multiple tests, as in the case of multiple measurements in tile piant, the probability of Type 1error is higher than a. An upper bound P can be designed, as explained in Chapter 7. Let zj be the test statistic for n?easurement j. If lzjl > Z I d 2 , then the null hypothesis Ho is rejected and hypothesis H I , is accepted. This means that z, is outside the +z1B/2confidence interval for a standard normal distribution. This is similar to value X being outside the interval (1.96. +1.96) for a = 0.05 in Figure C 1 . On the other hand, if a global test described in Chapter 7 is used to test the null hypothesis ifo against the global alternative hypothesis H,, thc threshold for the test is x;,, at a chosen level of significance. As in the null hypothesis Figure C4, if the test statistic is greater than x,f is rejected and a gross error is declared in the measurement set.
the nui! hypothesis:
HG : :here is uo gross error in proceys dera, \rcrsut; thz al:en~ativei\y?oti1r,sia:
HI : there is ar least a gioss error in process data. or. mori specifically: f3,, : there is a gross error in measurement j.
The choice of the test threshold depends on the statistical test that is used for hypothesis testicg. If the statistical test follows a standard nornlal distribution, such as some of the sta~isticaltests in Chapter 7, a thresli~ldZ , , , is used, at a chosen a level of significance. The value f( is used to control the probability of Type I error at value a.
1. Wadsworth, H. M. Hntzdbook of Statisrical Merizodstbr E12ginrer.r and Sc.ierzrisrs. New York: McGrawHill, 1990. 2. Hines. W. W., and D. C. Montgomery. PIuiiiihili~. nrd Srotirricr it; Ell,girierring atlci !Mnrlaget?~enr Scie;zre. New York: J o h n \Vile). & Sons. 1980.
3. CATACON Workbook. Brca, Calif.: Simulation Sciences. Inc.. 1996.
Accwacy of estimation, 301 of measurement, 36, 37 Adjustability, 21021 1 Ad.jastments. 1214, 16, 19 Amrnonia plant case study, 343368 synfnesis process, 219,307308, 3 14 Aritoine equation, 120 ,%verage error of estimatioi~(AEE), 258 nuicber cf Type I errors (AVTI), 258 Balance compone.nt flow, 88. 91,9395, 98 deficit, 332 elemental, 95 entlialpy, 115 oveia11low, 88,93, IUb, 107 residuals, 17: Ball mills, 1071 11 Bayes decision rule for identification, 266 formula, 267, 269 Bayesian algorithm, 27 1273,278 Bayesian test, 23, 264273, 278 sequential application of. 267 Bernoulli randorn variables, 265, 267
Bias in measurement, 32, 37, 176, 186.256,282,290,29 1,294 Bounded GLR method (BGLR). 249253 Bosnds cn variables. 22. 25, 61. 138. 23Y.241,246253,262.369 Branches of spanning trees. 1i0112. 3 16320,38Ct_:X? Broyden's r:l:it~:x ilpdste pmcedure. 127. 131 CauchySchwartz ineqnality, 192 Certainty Equivaleace Principle. 157 Chisquare distribution, 189, 225. 390351, 387,388 random variable, 358. 389 Cholesky Cactoiization, 159 Chords in a graph, I 101 12,316320. 380383 Circuits mineral beneficiation, 7 Coaptatioii subproblern, 77 Collective methods for bias and leak detection, 256 principal component tests. 200. 209 Combinatorial strategies, 253254
Confidence interval, 387 for nonlinear processes, 22,26, 138 Bonfmoni, 181 history of, 2 124 Connectivity, 382 in dynamic systems, 142173,282 Constant direction approach, 125 industrial applications of, 32737 j Constraint test, 23, 180182,201, 203, linear steadystate, 5984, 155 232,253255,259,278,334 (see material balance, 72 also Nodal test) nonlinear problems, 262 Constraints nonlinear dynamic (NDDR), 165, bilinear, 134 166, 168, 169, 170 equality, 122 nonlinear steadystate, 25, 119141 ' parameter estimation and, 33 1332 inequality, 128129: 166 nonlinear, 124, 131, 138 plantwide material and utilities, Continuous stirred tank reactor 332372 (CSTR), 1621 64, 1681 70,261 problem formulation, 7.910 Control law, 148 process unit, 328, 334 Coulers, 116 simple problems, 11 Correlation coefficients, 33 simulation techniques for Covariance matrix, 63, 121 evaluating, 8 182 of balance residua!^, 178 statistical basis of, 6163 of measurement adjustments, 183 steadystate, 4,5,6, 7, 10, 23, 25,27, Critical value of a statistical test, 177, 80, S1,85, 153, 154, 166, 329 successive linear (SL), 124128. 178, 180, Crowe's project matrix methcd, 135. 137 9'7104, 1131 14. 1 16, 126, DATACON'" software. 35 1, 354 1321 33, 138,219,333 DATREC s~ftware,33 1 Degrees Crude preheat tiair?,54, 86, 329, of frecdorn, 388 335339,340 of redundancy, 65 Delay split optirnrzatio!l, 5 , 10 CUSUM tests, 55 ir, instmment checking, 27; C11:sets in a giaptr, 1 i0! !2, 317, 319, in daia filtering, 39, 41 320,381383 Detectability factor, 21 1 Cycles in a greph. 380 Dirac delta function, 161 Distributed control system (GCS). 4G Distlibutions Data beta. 268 coaptation, 8, 15. 22 ceetral. 192 conditioning, 4, 56 multivariate, 392 filttlring. 27, 39, 5 1 noncen~ral,392 rectificaiion, 3 normal. 35,386387 smoothing. 39, 5 1 validation, 27, 56 standard normal, 386 Dyrlarnic measurement test (DMT). Data reconciliation (DR) benefits from, [email protected] 248149 bilinear, 25, 851 17, 119, 316 Edges, 321,378382 dynamic, 10,23,27, 142173,330 Eigenvalues, 196, 200, 209, 376 estimation accuracy of, 301303 Eigenvectors, 196, 377 flow reconciliation example, 1113
f
396
Uutrr Reco~~ciliufion mid Gross 6  r o l Derectior7
Elementary tree transformation, 383 Energy balances, 9 conservation constraints, 9 conservation laws, 8, 27 flows, 1I Enthalpy balance, 212,340,342 flows, treatment of, 114, 116 Equivalency classes, 215,2 16,258 Errorinvariables (EVM) estimation, 168 Error reduction methods, 38 Errors gross, 1 4 , 6 , 7 , 1 1, 1720, 21, 22, 23,24,26, 27,32,34, 35, 37, 60,XO81, 128, 174225, 226280,327372 normalized, !76 random, !A,7, 12,27,3237,41. 56.61, 81, 1 6 1 4 5 . 151, 154, 163, 168,175. 176,358 reduction methods. 3856 sqtiared prediction, 199 syr;temaric. 32 Type I. 177, 181, 184, !e8, l9c. 191, 198,223,22923 1,233. 234, 236.240,254,255, 257, 258,259,262,264, 27!, 273, 284 286,294,332,393 Type Ii, 177,223,234,236,254 255.284,286.294,392 E s t ~ m a t ~xcuracy o:~ of da:a reconc~l:at~on. 30 1 of mlnlmum obsen able scn\or net\\forka,306 Expected value of random errors. 3234.38939 1 of a furict~onof mndom varr~Sles, 3537 Extended measurement test (EMT), 248.249 Faults, 282, 295 additive. 289 diagnosis, 28 1,284,288, 295297 hard. 295
isolability, 295297 signature, 296 soft, 295 Filters analog, 38 digital, 38, 39, 54, 56 double exponential, 42,45 exponential, 4047,48, 54 exponentially weighted moving average (EWMA), 50 finite impulse response (FIR), 49,s 1 firstorder, 40,42,45, 53.54, 55 geometric moving average, 50 hybrid, 5456 infinite impulse response (IIR), 40 Kalman (see Kalrnan filterj leastsquares, 5 1 moving average, 48,49,54 nonlinear exponential. 4243, 45. 46.47, 50, 54 polynomial, 5 154 reverse nonlinear exponential, 44 seccndorder, 4 5 5 3 squareroot covariance, 158 Flow balznces, 12 energy, l I enthalpy, 114 estimated. 12 mass, 11 measured, 12 reconciliation. 13, 14. 16. I8 Fourthorder RungeKut:a method. 163, 169 Fundamental cutsets, 382383 cycles, 382383 CAMS, 323 GaussJordan elimination process, 124 Gaussian distribution, 101 1, 161, 284, 289, 290, 386 (see also Normal distribution) elimination, 31 3, 3 15
Generalized likelihood ratio (GLR) test, 23, 185194, 199, 201, 203205,214,223,227230, 234238,240,241244,252, 259, 261, 262, 266,269, 277, 288294,296,298,340 Generalized reduced gradient (GRG), 132133, 137, 138, 167 GINO, 323 Global test (GT). 23, 1781 80, 193, 194, 198, 199,201.203207,, 222. 230, 23 I, 236, 238, 240, 248,252,259,277,283288, 293,294,298,355,359362, 364,366,367,368 Graph, 378380 operations, 380 process, 378380 subgraphs, 379381.383384 theoretic methods, 22. 25, 72. 82, 1 10, I35,3 153 16.320,324 theory fundamentais. 378393 Grinding mills, I I3 Gross error detzction (GED). 1 4 . 23. 24.26, 174225,226280,330. 340541. 359 basic statistical tests for. 174195 benefits from, 202! forsteadystate processes. 226280 history of, 2124 i n linealdynaniic sysirms. 28 1299 in nonlinear processes. 260154 rrtedel. 185 $?rial strategies f ~ r236238 . signature models, 187 simultaneous strategies for, 227236,248 using principal componcni trsts, 195200 Gross errors. 1,6, 7, 1 I. 1720. 2 1 . 22. 23, 24, 26. 27. 32, 34. 35. 3738,60, 8081, 128, 174225. 226280,327375 equi\rale~~cy classes, 215. 216. 258 equivalent sets of, 2 14, 2 16
ident~fiabilityof, 21421 7 identification strategies, 256260 signature vectors, 20 1, 2 15, 216,230 HARWELL mathematic library, 82, 131 Heat balance equations, 25 exchangers, j4, 9, 10, 1 1, 2 1. 73, 74, 75, 1 15! 16, 162, 179, 212,233,252.274,276,29S. 336,339,340,342.343 transfer coefficients, 91 0. 2 1, 295,33 1 transfer fluid (HTF). 27G275 Heatcrs, 1 16 Hessian matrix. 1301 3 1 Hotelling T' test, 329 Hypotheses alternative, 176. 187. 228, 743. 241, 392 cornhinatoria!. 278 ~ l o b a alrema!ivc. l 395 null. 176. 2G3. 529. 231. 2%2%. 293. 391393 resting. 39 1393 irnpierner~tationof da::~recocziliarion yidz!ines. 339 oniine. stead) state. 329 IMSL niathen2a:ic library. S2 Independent equatic~~s. 88 rsr~cioinerrors. 3 ; I~~novations. 150. 2572813 inteeral o!'ahsolutc. eri.ors (IAEI, 3934.47,19 lritcgral dynamic rncasurerncnt test. 297 Itcr.:lti\.c rueasurcnient test (IMI'), 238240, 213.236. 248.277 Jacobian matrix, 1231 24. 125. 126. 127 Jo~ntprobability dzns~tyfunct~on.390
398
Doin Ilccor~crllarronand Gross Error Derrcrion
Kalnian filter, 148160, 163, 164, 165, 170, 171,283,285, 289, 293,294,296,298 extended, 23, 161, 163, 169 filtering methods, 26, 148160 gain matrix, 150, 158, 160 implementation, 158 steadystate Kalman gain, 151, 152 Kruskal's algorithm, 321322
left null space, 375 null space, 375,376 projection, 6 4 , 6 6 4 9 range space, 375 rank, 375 row space, 375 signature, 289 trace of, 303, 376 Maximum likelihood estimates (MLE), 122,230 Maximum power (MP) Lagrange multipliers, 60, 61, constraint tests, 181182, 199 122124, 131 measurement test, 184185, Leak detection, 185189,254256,335 1901 93,199,203206 Leaks, 37, 174, 185, 189. 190 Mean values, 392 Leastsquares Measurement formulation, 160 accuracy of, 37 minimization, 121 direct method, 78 op~imization,8, 13. 161 elimination, 23, 24 weighted objective fur~ction.8 error covariance matrix, 77,78, Le\ el of significance, 176. 392 79, 178 modified, 181 errors, 27,3238 Likelihood functi~n,52 indirect method, 78, 80 Line search. 127, 130 practically nonredundanr, 21 0 Idinear practically unobservable cc~nbinationtechniqw (LCT). variables, 210 254256.260 precision, 37 da:a rezonci!istior. pioblems. 9, test (MT). 20, 23, 1831 &5,201, 5982,155 222,227,255,355356,359, program (LP), 132 360,361 systcrns, 6377 test statistics, 183I85 L.ccal neigilhorhood search techniq~te. Mineral 323 beneficiatiotl circuits, 7, 104 120ss e5timatlon. 334 flotation process, 102 LU decomposition of matrix, process circuits, 23 134, 135 MINOS, 137,323 Mixers, 9 1, 94, 102, 113 Maznitude enthalpy balance, 114 of bias, 185, 262 twophase, 106107, I08 of gross error, 37,265.270. 273 Model MATLAB, 82.2 17.274 identification, 143 Matrices and their properties, 373, linear discrete dynamic system, 375377 143145 Matrix tuning, 2021 column space, 375 Modified iterative measurement test covariance, 63. 121 method (MIMT), 247249,25 1, decomposition methods, 7072 261,278
Modified serial compensation strategy (MSCS), 2 6 2 4 6 , 2 5 9 , 2 6 0 , 263264,277278 Moving window approach, 166167 NewtonRaphson iterative method, 123,132 Nodal test, 23, 180182, 208,232, 253255,259,278,334 (See also Constraint test) Nodes, 37838 1 Nonlinear data reconciliation, 9,26,85, 1641 70 GLR test, 263264 optimization strategies for data reconciliation, 136, 167, 171 programs (NLP), 23,25,26, 103, 104, 128129, 134, 137,261, 276,331,369 state estimations, 1601 64 Normal distribution, 101 1, 161, 284, 289, 290,386 (See also Gaussian distribution)
,
+
$
*'
Objective function (OF), 261263 for data :econciliation. 8, 60 difference. 26 i262 reduction in. 204205 Observzbiiiiy, 22, 6970, 71, 72. 74, 82, 135, 135,2lG definition of. '70 Oiliine data coilection and cocditioning, 5 implementation a:' data reconciliatiun. 329330 optimization, 10 Open Yield software, 335 Optimal !55 control and Kalmax fi1:::in;, state estimation, 148 Orthogonal collocation, 166167 Overall power (OP), 257 function (OPF), 257 function equivalency, (OPFE), 258 Parity equations, 296 Paths, 380
Performance measures for GE identification strategies average error of estimation (AEE), 258 sverage number of Type I errors (AVTI), 258 overall power (OP), 257 overall power function, 257 overall power function as equivalent sets, 258 Plantwide material and utilities reconciliation, 332372 Posterior probability, 266267 Power of statistical test, 177, 181, 190,392 Preheat train, 5 4 , 2 1 2 Principal component analysis (PCA), 296 scores, 196 measurement test (PCMT). 197, 355356,360 model, 197,297 of constraint residuals, 196 tests, 21, 176. 195200.207209, 223.232.259,364366 Prior distribution. 268 pr~babilitji,255 Probability density functions (PDF or p.d.f.). 35, 384339 Process control applications, 10 data conditioning methods. 1 4 unit balance reconciliatiotl, 32833 1 Produtti011accounting, 332, 335 Projection matrix, 22, 25,64,66,67, 70. 74, 8 1, 82, 101102, 132, 134,146,2U1, L ! Y , ZLI
Q statistic, 199 (See also Raostatistic error or squared prediction error) Quadratic objective function, 129 problem (QP), 131 QPSOL, 131 QR factorization, 22,667 1, 8 I, 127, 128
d Gross Elror I>etecrion
redundant observable, 305, RAGE software, 136,33 1,340 309313,320,324 Random errors, 1 4 , 7 , 10, 12, 27,3237, 38, Separation Theorem, 157 41,56,61, 82, 143145, 151, Separators, 94, 113 154, 163, 168, 175, 176,358 twophase, 105, 108 events, 384 Sequential variables, 384393 probability ratio test (SPRT), 284 Range and null space decomposition quadratic programming (SQP), (RND), 131, 136,376 129132, 135,136 Raostatistic error, 199 Serial Raoult's law, 120 compensation, 237, 241, 243, 260, Reactors, 9496, 113 277 Real correlation, 34 numbers, 376 elimination procedure, 24,204, 214, vectors, 376 RECON software, 33 1 Reconciliation of ammonia plant data. strategies, 236,246247, 259 343372 Shewhart test, 5455 RECONSET software. 33 1 SIGMAfine software, 335 Redundancy, 22,27,6973,82, 134, Signal 135,209,210, 21 1, 228, 300. aliasing, 38 330,342,343,354,369 processing, 25 classification: 71. 135 reconstruction. 45.55, 297 definition, 70 types, 55 degrees of, 65 Signature marrix, 289290 spatial. 4 Simpie seriai compensation strategy temporzl, 4, 149, 171, 282, 291 (SSCS:, 241244,245.259, Redundant sabproblem, 77 260,277 Rigorous online modeling. 2 1 Simpson's technique, 104105, 108. W4DSQP, 131. 133, 136,376 109,111,113, 114, 117, 133,3!6 ROh4e01"softwae. 331 Simultaneous strategies RungeKutta method, Courthorder. for multiple gross error 163, 169 identification, 227 using 2 Bayesian approaclr, Seiectivity, 258 264273 Sensor network using combinatorial hypothesis design, 70, 300326 testing, 22823 1 developments in des~gn,323 using simultaneous estimation of mar;im.;m es:imati~naccuracy GE's magnitudes, 232233 design, 306. 3 15 using single gross error test minimum cost designs, 3 133 15, statistics, 227 320,322,324 Smearing effects, IS, 228 minimum observable. 304306,307, SoaveRedlichKwong (SRK), 354 312,315,316,318,322,324 Spanning tree, 1101 12.316322, optimization techniques for, 322 325,380383
Spatial correlation, 33 redundancy, 4 Splitters, 9193, 96, 1131 14 enthalpy balance, 114 Squared prediction error, 199 SQPHP, 131 Standard deviation, 33, 3437,45,56, 2 18 nom~aldistribution, 386387 Statistical moments, 389, 390 process control, 38, 55 properties of innovations, 283284 properties of random variables, 389391 quality control tests (SQC). 3, 38 tests for general steadystate models, 200202 Steadystate linear reconciliarion, 25 processes. 4. 25, 282, 369 Subgraph, 379381 Subproblems coapta~ion.77 redundant. 77 Successive linear data reconciliation, 126128 Successive quadratic progran?minz (SQP), 2223, 13I. 135, 138. 167 Successively linearized hoiizon estiriiation (S1,IIE). 167 Systematic errors, 32 Systems bilinear, 25, 85 117 containing gross errors, 17 dynamic, 25, 27 linear. 63 linear dynamic, 28 1 noilli~~ear, 160 non1edundant, 16 observable, 17, 25 redundant, 25 unobservable, 17

with all measured variables, 1114, 15, 16,22,26 with unmeasured variables, 1417,22 Taylor's series expansion, 36, 123, 129 Test statistics, 175203, Theory of evidence, 329 Truncated chisquare test, 199 Unbiased estimated techniques (UBET), 232233,236 Univariate tests, 180, 184. 199 VALI, 33 1 Variables basic, 132 classification methods, 77 dependent. 132 independent, 132 measured, 1114, 15, 16, 22,26. 32, 33,62, 6356.69, 70,72, 81, 101, 109. 121, 132, 135. 149, 177,212,238,249,257.26C, 305,314,3 16,358 nonbasic. 132 nonredundant, 16,210 observable, i7, 25 primary, 35 random, 3 5 5 7 , 119,256,257. 184.389 redundsqt, 17%136. 200, 2 I?. restricted, 250 secondary random, 35 splitfractior., 1 131 14 superbasic, 132 unmeasuied, 1417. 21, 27.6358. 81, 100, 103, 110, 121. 126, 132133, 135, 136, 177,200. 202, 204,2 12, 24 I, 248, 249. 305, 312, 315, 316,331. 358 unobservable. 17, 135, 136. 210 Variance. 392393 of the estimated error, 302303 of random variables, 33,38939 1
402
Dora Rccoitciliatiun aud Gross Error Urrecriol~
Vectors and their properties, 373375 column, 373 dimension of, 374 gross error signature, 187, 193, 215 of balance residuals, 178 of measurement adjustments, 183 real, 373
row, 373 space scanned, 377 Weighted leastsquares objective function. 8 Windows, 1661 68,285,293
Author
Yield accounting, 332, 335, 369
Abadie, J. i 32 Albcquerque, J. S., 134,297 Ali, Y., 324 Almasy, G. A.. 23,79.80
i'
Bagajewicz, M., 77, 155,214,260, 297,322,324,330 Bagchi, A., 158 Basseville, M., 295 Bellingharn. B., 145, 295 Bequette. B. W., 128, 137 Biegier, L. T., 22, 134, 136,297 Bodirrgtcn, C. E., 24 Borrie, J. A,. 158 Eritt, H. L., 12, 25,!26 Carpani, R. E., 23 Charpentier, V., 2 10, 329 Chen, J., 80 Clinkscale>,1. A., 5 1, 52, 54 Crowe, C. M., 22,24,66, 72.97,98, 101, 113, 135, 181, 190, 196, 198, 199,204,239,210,314 Daniel, J., 373 Darouach, M.. 155
Davidson, H., 22 Davis, J. F., 24, 181,232, 243,254, 257,259,260 Dee, N., 317, 321, 378 Devanathan, S., 155,2S4 Dunia, R., 37. 296, 297 Edgar, T. F., !39, 137, !62 Everell, M. D.. 23 Fisher, G., 22, 127. 133 Fisher, D. G., 151 Gelb, .4., 153 Gertler, J . J., 281,295. 296 Gill, P. E., i29 Gonnan, J. 'vSi ,22, 125
Har~humar,P., 256 neenan, W. A., 24,203,228,238, 239,247,255,261 Heraud, N., 23 H~mmelblau,D. M.. 129.295, 295 Hlavacek, V., 24 Hodouin, D., 23 Howat, C. S., 24, 120, 125, 126
..
n' i;rois EI ror Ilerectian
Ichiyen, N., 102 lordache, C.. 210,214,269,273 Isermann, R, 295 Jazwinski, A. H., 160 Jiang, Q., 155,214,233,260,297 Jones, H. L., 285,293 Jordache, C., 24, 37,53,55, 136,260, 360 Kalman, R. E., 149 Kao. C. S., 34 Keller, J. Y., 80,238. 244. 245, 259 Kelly, J., 333 Kim, Y. $1..39, 168. 261 Knepper, J. C., 22. 125 fieisovalis, A., 72. 135. 303, 309, 3 12,324 Kuehn, D R., 22 Lasdon. L. S.. 132. 13'7 Lee, J. M.. 39 Lees, F. P.. 145, 295 Lucclte,R . H.. 22. 125. 126 Liebman, M. J.. 23. 137. 165, 160, 169 Liprak, B. (3.. 57 Loucka, M., 72.323 !,uo, Q.. 296 MacDonald, R. J.. 24. 120. 125, 126 MacGrcgor, J. F.. 50 Mac!:on. F., 24. 32. 31. 35. 37. 123. 121.209,3 13.3; 5. 324, 332 h4ah. R.S.H., 22. 23.24, 34, 72, 74, 79,SC, 135.151.181.185,200, 206, 2 14. 237. 253, 255, 288, 303,309,3 1 1.321 Makn~.S.. 15 1 Maquin. D., 32 : Mehra. R. K., 285 Melsa. J. I.., 148. 150. 15 1 Meyer, M., 72.96 Montgomery, R. C.. 284 Mullick, S. L., 330 RQu~tagh, B. .A,, 137 Muthy, A.K.S.. 23 Muske, K. R., 162
Nair, P., 37, 136 Narasimhan, S., 23, 24, 185, 190, 200, 2 14,237,256,26 1,264,288,324 Nikiforov, I. V., 295 Noble, B., 373 Pai, C.C.D., 22, 127, 133 Parr, A,, 45, 55 Patton, R., 281, 295 Peschon, J., 285 Press, W. H., 159 Ragot, J., 72 Ramamurthi. Y., 128, 137, 165, 167, 169 Rao, R., 324 Ravikumar, V., 23, 24. 136 Reid, K. J., 23 Reilly, P. M., 23 Reklaitis. G. V., 8, 95 Kenganathan, T., 261,264 Rhinehart. R. R., 45 Rlpps, D. L., 23, !95, 204, 237 Rolllns, D. K 24, 155, 181, 232, 243. 257,259,260,284 Romagnol~,J. A., 22. 24.66. 77, 116,258 Rosenberg. J ,24,204,231.233,236. 248.249,260
.
Sage, A. P., 148, 150, 151 Sanchez, M., 22.66,77, 116.233,258 Saunders, M. A., 137 Seborg, D. E., 38,45 Sen, S., 324 Serth, R. W., 24, 123, 204, 228, 238, 239,247,255,261 Sheel. J. P., 24 Shewchuck, C. F.. 123 Shinskey, F. C. , 37 Sirripson, D. E., 23, 104, 108 Smith, H. W., 102 Sorenson, H. W., 150 Stanley, G. M., 23, 56,72, 151 Stephanopoulos, G., 24,77 Stephenson, G. K.123 Strang, G., 373
Swartz, C.L.E., 22. 66, 128, 1.34, 135 Sztano, 'r., 23
Wald, A., 284 Wang, N.S., 24 Warcn. A. D., 132. 137 Weher. R.. 45 Wiegel. R. I., 23 Williams, J. P., 284 Wi!lsky, A. S., 288, 293 Wisltner, R. P., 162
Tamhane. A. C.. 23.24, 1 8 I , 183, 264,273 Tham, M. T., 55 Tilton, B., 260, 356 Tjoa, I. B., 22, 136 Tong, H., 24, 196. 198, 199.209 Turbatte, H. C., 324
Yang. Y., 255
Vaclavek, V., 72, 323, 324 Veverka, V. V., 24, 3 13,315,324,332
Zalkind. C. S.. 37 Zasadzlnski, M., 155