Gross error management in data reconciliation

Transcription

1 Pepnts of the 9th Intenatonal Syposu on Advanced Contol of Checal Pocesses he Intenatonal edeaton of Autoatc Contol um3.4 Goss eo anageent n data econclaton M.J. uente*, G. Guteez*, E. Goez*, D. Saaba**, C. de Pada* Dpt. of Systes Engneeng and Autoatc Contol, EII, Unvesty of Valladold, c/ Real de Bugos, Span (e-al: aa@auto.uva.es, gloa@auto.uva.es, elena@auto.uva.es, pada@auto.uva.es, ) **Dpt of Electoechancal Eng., Unvesty of Bugos, Avda. Cantaba s/n, Bugos (dsaaba@ubu.es) Abstact: hs pape evews the pobles assocated to the pesence of goss eos n data econclaton pobles as well as the an appoaches to avod the undesable effects: detecton and eoval of the faulty easueents and nzaton of the effect on the econcled esults by usng obust estatos that odfy the cost functon. In the fst case, Pncpal Coponent Analyss s used to detect the goss eos whle, n the second, the a functon s used. A cobned appoach s also pesented that poves the PCA esults. he ethods ae evaluated n a ealstc lage-scale poble wth plant data coespondng to the hydogen netwok of a petol efney. Keywods: Data econclaton, Goss eo detecton, PCA, obust estatos, hydogen netwoks. 1. INRODUCION he use of elable nfoaton n the pocess ndusty s becong an potant ssue, as oe and oe systes ae based on odels and data fo akng o suppotng decsons nstead of elyng on the expeence of the pesonnel. hese decsons ae elated to dffeent applcaton felds, angng fo pocess ontong to unt evapng o opeaton optzaton. he elablty of any of these systes s dectly lnked to the qualty of the pocess easueents they eceve. It s well known that nstuents ae often affected by nose o sall bases that downgade the qualty of the easueents. In addton, dstubances of a dffeent natue act contnuously on the pocesses, so that what we easue does not always coespond to the condtons equed fo the specfc applcaton unde consdeaton. In patcula, egsteed data ay not be consstent wth the physcal laws that we know ust be satsfed, such as ass o enegy balances. In these cases, data econclaton s becong nceasngly popula fo obtanng a set of data consstent wth the subacent odels, and as close as possble to the easueents. Gven the N easueents y, coespondng to a subset of pocess vaables, and the odel f(y,x,p) = 0, wth p a set of paaetes, and x the pocess vaables, data econclaton look fo estatons of y and x (and possbly p) that nze the cost functon J, the su of squae eos between the easueents and the coespondng odel vaables, and satsfy the odel equatons as well as, possbly, a set of addtonal nequalty constants. he poble can be suazed as: n J {y,x,p} f(y, x, p) g(y, x, p) N 1 w 0 0 y y (1) whee w ae adequate weghtng factos povdng noalzaton and accountng fo nose levels and the elablty of the nstuents. ypcal applcatons of data econclaton ae, fo nstance, RO and KPI estaton. In the foe, t s used to update the pocess odel befoe ts use n anothe optzaton poble that looks fo the best (econoc) opeatng pont of the plant. In the second case, a set of poducton ndcatos s coputed fo the econcled data, avodng nconsstent nfoaton that could lead to wong ndcatos. Sensble solutons to poble (1) eque a cetan degee of edundancy n the easueents. If the eos n the easueents ae noally dstbuted aound the tue values, the data econclaton appoach s able to povde the best set of estatons coheent wth the odel. Nevetheless, fo te to te, cetan easueents ay be affected by sudden lage dstubances (outles) o systeatc sgnfcant bas that dstots the soluton, thus speadng the eo thoughout the est of the vaables, ceatng a seang effect. hese easueents ae called goss eos and the detecton and teatent s cucal fo obtanng good vaable and paaete estatons. If the dstbuton of the easueent eos s non-gaussan, as wll happen f goss eos ae pesent, the LS o WLS estatons ay gve ncoect esults, as they ae not obust aganst devatons fo the assued Gaussan dstbuton. hee ae seveal appoaches fo dealng wth goss eos n the lteatue, but only a few of the ae pactcal fo ndustal use. In ths pape, two an polces have been studed and copaed. he fst one s oented towads detectng the vaables that pesent goss eos, whch can be subsequently elnated fo the easueent set and the data econclaton poble (1), epeatng t late on wth the new goss-eo-fee set of easueents n a cyclc pocedue untl no oe goss eos ae detected, as n g.1. Copyght 015 IAC 64

2 IAC ADCHEM 015 g. 1. Scheatc of goss eo anageent wth the detect and supess polcy he second polcy as not to detect and eove vaables wth goss eos, but to tgate the effect by usng novel cost functons that educe the weght of vaables wth lage eos, called obust estatos. Repesentatve ethods of these appoaches have been selected and copaed n a ealstc ndustal case study coespondng to the data econclaton of the easueents of a lage scale hydogen dstbuton netwok of a petol efney, showng what can be acheved n these types of pobles. In addton, a new xed appoach s pesented that poves pevous esults obtaned wth PCA tools. he pape s oganzed as follows: afte the ntoducton, the ethods coespondng to the two polces and the xed ethod ae explaned n secton two. Secton thee descbes the hydogen netwok case study and the data econclaton poble. nally, secton fou pesents the esults and dscusses the. he pape ends wth soe conclusons and efeences.. GROSS ERROR MANAGEMEN As entoned above, two dffeent appoaches have been consdeed to deal wth the poble of goss eos n data econclaton, n addton to a new xed one. All of the wll be suazed befly n ths secton..1 PCA est fo Resduals of non-lnea Data Reconclaton he fst appoach uses the Pncpal Coponent Analyss (PCA) technque to detect goss eos and to dentfy the pocess vaables whch ae the oots of that eo. he typcal assupton ade n data econclaton s that the easueent eos ae ndependent, zeo ean and noally dstbuted, so the easueent odel s: y () (1),, whee s the tue value of the easued vaables, =1,,, and s the ando eo. In the absence of goss eos, the expected value of the esduals ( = y, y ),.e., the dffeence between the easued and the econcled values, s zeo. When goss eos such as senso bases and pocess leaks ext, the odel becoes: y b (3), and the esduals ae no longe zeo-ean. So, n ode to know f thee ae bases n the easueents, the esduals, also called the easueent adustents,, have to be analyzed. In ths pape, ths esduals vecto s analyzed by the PCA technque (ong and Cowe, 1995). o pleent the PCA odel, t s fst necessay to calculate the covaance atx of the esduals: H and pefo the sngula value decoposton on H : H U U (4) whee s a dagonal atx contanng the egenvalues of H n descendent ode ( 1, ), and the coluns of U ae the coespondng othonoalzed egenvectos. he pncpal coponents, t, of the esduals vecto can be calculated as: t W (5) wth W = U -1/. So, the set of coelated vaables,, s tansfoed nto a new set of uncoelated vaables, t. On the othe hand, the ognal vaables, the esduals (), can be bult fo the pncpal coponents when all the pncpal coponents ae etaned, but f only k < coponents ae etaned, coespondng to the k' bggest egenvalues of H, then the esduals ae calculated as: 1 / U t e e (6).e., the esduals can be decoposed nto the contbutons fo the pncpal coponents,, and the esduals of the pncpal coponents that take nto account the nfoaton contaned n the dscaded egenvalues: e = Pncpal coponent test he tadtonal ch-squae collectve test, k, s caed out wth the pncpal coponents etaned: t t (7) k whch can be tested aganst a theshold value to detect goss eos. Now, once an eo s detected, t s possble to dentfy the esduals,.e., the vaables that ost contbute to ths eo. he contbuton of each esdual,, to each pncpal coponent, t,, can be calculated as: g ( w, ), 1,..., (8) Defne g = (g 1, g,,g ), and let g be the sae as g, but soted n descendng ode of the absolute values. he contbutons ae donated by the fst few eleents, so the nube of vaables, K 1, whch ost contbute to the nceased value of the ch-squae test can be set as (ong and Cowe, 1995): K1 1 g ' t, t, 1 (9) whee 1 s a pescbed toleance. he nube of contbutos (K 1 ) wll decease when the value of 1 ncease. Also, anothe possblty to calculate the contbuton of the Copyght 015 IAC 65

3 IAC ADCHEM 015 vaables to the goss eo detected s to cay out a contbuton analyss (Kouty and McGego, 1996). In ode to take nto account the esduals of the pncpal coponents,.e., the dscaded nfoaton, anothe statstcal test can be caed out, the Q statstc defned as: Q ) ( ) (10) ( hs statstc can also be copaed wth a theshold to detect eos. Both statstcs ae copleentay n so fa as the foe exanes the etaned and the latte the dscaded pncpal coponents, espectvely. Now, n ode to know the nube of esduals that ost contbute to the ncease of the Q statstc, the quantty f s defned as: f =, and f s the sae as f, but ts eleents ae soted n descendng ode, based on the absolute values. he nube of vaables K whch ost contbute to Q can be calculated as: K f ' 1 1 Q (11) whee, as befoe, s a pescbed toleance. he classcal contbuton analyss can also be caed out fo the Q statstcs to get anothe dea of the vaables that ost contbute to that the Q statstcs detects the eo.. Robust estatos An altenatve to the use of teatve goss eo detecton and elnaton ethods s the use of obust estatos that ae nsenstve to devatons fo deal Gaussan dstbutons, and whch tend to look at the bulk of easueents and lt the weght of vaables affected by goss eos (Aoa and Begle, 001). In contast wth WLS estatos, whch gve quadatc potance to the eos, the obust estatos lt the contbuton to the cost functon when the eo s too lage, as epesented n g., whee the Least Squaes functon s epesented n addton to the a and Welsch ones, used n obust estatos. In ths way, the effect of the goss eo s attenuated and the optze does not ty to dstbute the eo aong all the vaables n ode to avod the hgh penalty posed by a quadatc cost. In ths appoach, the goss eos ae not detected and elnated; nstead, they ean n the data set, but the effect on the soluton s educed, thus avodng the popagaton of the eo to othe nstuents. Notce that ths appoach, n addton to povng the data econclaton, also facltates the dentfcaton of faulty nstuents, as they tend to concentate the eo. he obustness of an ML-estato aganst devatons fo non-gaussanty s easued by the nfluence functon, whch s popotonal to the fst devatve of the estato. he estato s obust f the nfluence functon s bounded as the esduals go to nfnty. In patcula, the LS estato s not obust as ts nfluence functon s gven by: de e (1) de g.. Least squaes and a functons epesented fo a ange of values of the easueent eo e wth e epesentng easueent eos. hee ae any obust estatos poposed n the lteatue, e.g., the Redescendng, Welsch, etc. Aong the, the a functon has been chosen n ths study because t s a copose between the coplexty and dscontnuous natue of the Redescendng estato and the antenance of the shape of the WLS (Ncholson et al, 014). he a estato uses the expesson: e e c log 1 (13) c c as the cost functon of poble (1), whee c s a constant that can be tuned to the specfc poble. Its nfluence functon s gven by: d e e (1 ) (14) de c a LS e whch fulfls the obustness popety..3 Hybd detecton Welsch he ablty of the obust estatos to concentate the eos n the faulty vaables, avodng speadng the aong the whole set of vaables, can help the detecton task pefoed by the PCA based test. Because of ths, a hybd appoach s poposed that uses obust estatos nsde the econclaton steps of the PCA test, thus povng ts pefoance,.e., the esduals,, whch ae used to pefo the PCA ethod, ae calculated as the dffeence between the easued data and the econcled values usng the obust estato. 3. HYDROGEN NEWORK Hydogen has becoe a key utlty n petol efnees, beng used n any pocesses, anly n hydocabon desulphuzaton opeatons and n conveson unts. ypcally, hydogen netwoks nvolve thee types of plants: hydogen poduces, hydogen consues and platfoes, Copyght 015 IAC 66

4 IAC ADCHEM 015 whch consue hydogen but geneate oe as a by-poduct. hey ae nteconnected by a coplex netwok of gas ppes connectng geneatos and consues opeatng at dffeent pessues and putes. he pupose of the netwok s to supply the equed flows of hydogen to the consue plants. hs s potant, not only fo the pont of vew of the opeaton of the plants, but also n ode to potect the lfe of the expensve catalysts used n the eactos, whch need to guaantee nu hydogen to hydocabon atos. g. 3 epesents the an coponents of the one that was selected as case study, whch lnks sxteen plants. Colos dstngush poduces, consue and platfoe plants., out, n All flows ae easued as noal ones (N3/h), at noal condtons of tepeatue and pessue. Each stea s an deal xtue of hydogen ( H = g/ol) and putes wth a genec olecula weght I such that: 100 H H H X (100 X ) I (16) he data econclaton s then foulated as a non-lnea pogang poble that fnds the values of the odel vaables whch nze a cost functon that n tun penalzes devatons fo the easued values (Saaba et al, 01): w w /, ed X X, ed n n J p {, X, } 1 1 wth d 73 ( P 1) d ( ) (17) ( P 1) 73 d g. 3. Scheatc showng the an eleents nvolved n the hydogen netwok of a petol efney One of the an pobles concenng the pope anageent of the hydogen netwok s elated to the elablty of the ass flow easueents, whch suffe an nheent uncetanty due to the vaable olecula weght. Hydogen steas always have a cetan popoton of putes, lght gases that have a vey lage olecula weght copaed wth that of hydogen, whch s only two. As a esult, sall vaatons n the coposton of the putes ceate lage changes n the olecula weght of the stea, and hence n the ass flows. Hydogen s easued wth voluetc tansttes and then conveted to N 3 /h, accodng to the actual tepeatue, pessue and coposton of the cuent; but wthout pecse knowledge of ths coposton, the copensaton ade s only an appoxaton to ealty. In the sae way, hydogen puty s easued only n a few ponts, so that, f decsons have to be ade about the optal hydogen dstbuton, an estaton of the actual flows and putes ust be ade n advance. o ths pupose, a data econclaton syste has been pleented fo estatng consstent values of all elevant vaables n the netwok fo cuent easueents. he appoach assues that physcal laws, such as ass balances, etc., ust be satsfed exactly. All nodes n the coplete hydogen netwok ae odeled by ass balances n tes of puty, olecula weght and flow fo evey stea. Denotng fo flows, fo olecula weght and X fo puty of H, the equatons at a node wll be descbed by:, out X, n, out, n X (15) Hee, s the copensaton facto, efes to tepeatue and P to pessue, wth the sub-ndex d to denote desgn values, w s a weghtng facto, and s the vaance of the easued vaable used to noalze tes and to eflect the confdence n the easueent. he values of w can be adusted as a functon of the easued vaable s elablty. he ole of s to decopensate the odel vaables, so they can be copaed wth the easued values ed. Besdes the ass balances n the nodes (16), (17), addtonal balance equatons ae ncluded to copute the hydogen consued n the eactos and the lght gases poducton. In the sae way, anothe set of equatons takes nto account the elaton between the vaables of the sepaaton unts that ecupeate hydogen wth the pupose of ecyclng t. hese nclude flash sepaatons at dffeent pessues, whee lght gases ae eoved fo the hydocabon solutons, whee solublty equlbus ae foulated, and ebanes that sepaate hydogen fo the othe gases. In addton, a set of constants s ncluded that foce the odel vaables to be wthn a ange of the easued values calculated as a pecentage of the sensos span: p,n X,n,n p X X,ax,ax p,ax (18) as well as othe constants patcula to cetan types of equpent. In ode to guaantee a feasble soluton n spte of nstuentaton faults, a set of slack vaables ae added to the nequalty equatons. hese vaables squaed ae also added to the cost functon wth a lage weght. 4. GROSS ERROR DEECION IN HE H NEWORK A data econclaton syste has been pleented coposed of two odules: one connected to the pocess SCADA that pefos the tasks of pelnay data teatent and huan nteface, and anothe odule that solves the Copyght 015 IAC 67

5 IAC ADCHEM 015 optzaton poble pleented n GAMS. In spte of good geneal pefoance, the soluton of the data econclaton poble s affected by faulty nstuents, ceatng bas n the vaables estaton. In ode to avod ths, the aboveentoned ethods fo goss eo anageent have been ncopoated to the data econclaton syste and a specfc test has been desgned fo evaluatng the. 4.1 Pncpal Coponent test Befoe applyng ths test, t s necessay to geneate a base case fo copason wth actual data,.e., t s necessay to get data fo the eal plant n noal opeaton condtons wthout any bas n the easueents, econcle ths data set, calculate the esduals as the dffeence between the easueent and the econcled values, and calculate the test statstcs, and Q, wth the espectve thesholds. hs base case can be acqued n two ways. he fst one s to develop a odel to descbe the physcal syste, addng Gaussan nose to the calculated easueents; the othe appoach s to get noal data fo the plant. In ths wok, as the H netwok s athe coplex, non-lnea, wth so any unts and vaables, the best soluton s to collect data fo the plant and suppose that ths stuaton s the noal one, and then ty to detect goss eo fo ths noal stuaton. Once a base case s obtaned, the actual data s collected fo the plant, econcled and the test statstcs calculated, and f one of the statstcs exceeds ts espectve theshold, a goss eo s detected. Now, n ode to know whee the easueent eo s, the contbuton analyss and equatons (9) and (11) ae caed out, whch gve the vaables that ost contbute to the goss eo detected. So, n ths exaple, a data set s fst collected fo the eal plant, on a day when the plant s wokng n good condtons. hen, a Gaussan nose s added to the collected data to geneate a oe coplete data set n ode to calculate the H atx. he nube of vaables collected fo the plant was 190, 171 flows and 19 putes of H n the ppes. hs data set s econcled, usng the non-lnea least squae poble pesented n secton 3, to obtan the esduals and the H atx used as base case. Soe exaples ae now caed out, atfcally addng a bas to the easueents, so the pefoance of the detecton test can be pecsely evaluated. In a fst step, the bas s added only n one senso, concetely, n senso 93. In a second expeent, a bas s added to senso 55, anothe n senso, and fnally, the thee bases n the thee sensos ae added sultaneously. All the bases added to the sensos ae aound 0% n agntude. he data sets of these fou expeents ae econcled by solvng two dffeent NLP, the non-lnea least squae poble and the obust poble wth the a functon, whch coesponds to the hybd appoach explaned n secton.3. he esults ae shown n ables 1 and fo the two econcled pocedues, whee the second and thd coluns ndcate whethe one of the two tests studed detected the goss eo, whle n the last two coluns, the vaables that ost contbute to the bas ae shown fo each of the statstcs consdeed,.e., the esults of applyng equatons (9) and (11), espectvely. As can be seen n tables 1 and, thee ae seveal vaables that ae esponsble fo the goss eo. In ode to decde whch ae the ost contbutng vaables, the contbuton analyss s caed out, and the esults fo the second exaple (.e., bas n senso 55) ae shown n gues 4 and 5 fo the two econcled pocedues, whee, n the left-hand pat, the contbuton fo the vaables s epesented and, n the ght-hand pat, fo the Q statstcs, espectvely. In both gaphcs, t s possble to see that the ost contbutng vaable s nube 55,.e., the goss eo s well dentfed. But at the sae te, t s possble to conclude that soethng wong happens wth vaable 18, as ts value was not odfed on pupose, but t appeas as a faulty one. he eason ay be that, n fact, ths nstuent pesented pobles n the eal plant, even f t was ncluded n the efeence set. able 1. Results fo the econcled data usng the nonlnea least squaes Bas Q Contbuton vaables: Contbuton vaables: Q 93 18, 4 18,93, , 18 55,18,138,18,18,4 93, 55, 55,,18,93 55,,93,18 able. Results fo the econcled data usng the obust NLP wth the a functon Bas Q Contbuton vaables: Contbuton vaables: Q 93 18,93,4 93,18, ,18 55,18,41,18,18,4 93, 55, 55,,18,93,4 55,,93,18 g. 4. he contbuton analyss fo the and Q statstcs usng the non-lnea least squae econclaton. g. 5. he contbuton analyss fo the and Q statstcs usng the obust econclaton ethod. Copyght 015 IAC 68

6 IAC ADCHEM 015 Now, to know f ths pocedue s vald wth othe set of data, a new expeent s caed out whee soe plants wee n dffeent states (unnng/stopped). Anothe day s chosen, and data ae collected fo the eal plant on that day at dffeent hous. Each data set s econcled, and these data sets ae now consdeed as the base case. he fou exaples wth the bas ae then pocessed wth ths new base case. As the data ae so dffeent, the and the Q statstcs, calculated wth the data econcled as a non-lnea least squae poble, detect the fou eos. Yet the contbuton analyss does not wok at all, as the vaables esponsble fo the statstcs exceedng the thesholds ae vaables whose values, n the base case, ae zeo and, n the exaples wth bas, the value s dffeent fo zeo and have a vey bg value. hs occus because the plant s not wokng n the sae way on all days, and soe ppes, unts and valves n the plant ae wokng soe days and othe days ae stopped. Now, a geate effot has to be ade to pe-pocess the eal data befoe t can be econcled and the algoth to detect the goss eos can be appled Nevetheless, the obust econcled pocedue woks so uch bette n ths case, as can be seen n able 3. he statstc detects thee of the eos, but the vaables esponsble fo the eos ae the sae as those n the non-lnea least squae case; vaables wth value equal to zeo n the base case and vey dffeent fo zeo n the eo case, o vce vesa, whch asks the vaables wth bas. he Q statstc obtans bette esults, detectng thee eos, and the contbuton analyss gves the vaables esponsble fo the eo, togethe wth vaables that, as befoe, ae zeo n one data set and dffeent fo zeo n the second data set. So ths pocedue to detect goss eos s posng, especally wth the obust econclaton, but soe effot have to be ade to get a good base case and to pe-pocess the data befoe applyng ths PCA test. able 3. Results fo the econcled data usng the obust NLP wth the a functon Bas Q Contbuton vaables: Contbuton vaables: Q 93-93, ,108 4,,18 93, 55, 108,111 4,,93,55,18 4. Robust estato test he sae data set, wth the sae sulated faulty nstuents, has been used to test the data econclaton wth the a functon as obust estato wthout detecton and eoval of any vaable. It s then copaed t wth the esults obtaned wth the least squae functon, equaton (17), as cost functon. he statstcal suay of the test s shown n able 4. As entoned above, the data econclaton poble nvolves 190 total nstuents, of whch 3 have been gven a bas, the ones nubeed, 55 and 93. Usng the LS cost functon, the exta eo geneated by the speads aong the othes. In fact, the esults of the data econclaton gve 4 nstuents devatng between oe than 4 and 9 σ fo the easueent, devate between 9 and 14 σ, and one nstuent by 4 σ, whee σ s the coespondng standad devaton of the easueent. he thee faulty nstuents ae ncluded n ths lst. Usng the a estato wth the paaete c=15, the eos ae educed to the thee faulty nstuents, whch devate by 4 σ. Consdeng the eo wth espect to the span, the esults ae sla, but a false postve appeas. able 4. Copason between data econclaton esults wth the a functon and the LS functon Reconclaton ethod Nube of nstuents wth eos 4σ Nube of nstuents wth eos >15 %span LS a uncton 7 nstuents 3 nstuents 5 nstuents 4 nstuents 6. CONCLUSIONS he pape has shown the applcaton of two dffeent polces to deal wth the goss eo poble n data econclaton n a ealstc lage-scale poble. he esults show that the use of obust estatos poves the pefoance of the ethods, both as a stand-alone estato o as pat of the hybd polcy poposed fo the PCA detecto. he an poble assocated wth the PCA ethod s the need fo a sensble efeence test, fee of goss eos, as well as the pesence of false postves. In any case, both appoaches ae copleentay and can contbute to povng the pefoance of the data econclaton. ACKNOWLEGMEN hs wok was suppoted n pat by the Spansh MICINN unde poect DPI and EU P7 poect MORE unde Gant ageeent nube he authos wsh also expess the gattude to Repsol-Petono fo the facltes povded. REERENCES Aoa, N., Begle, L.., (001). Redescendng estatos fo Data Reconclaton and Paaete Estaton. Coput. Che. Eng. 5, 1585 Kouty,. and MacGego, J.. (1996), Multvaate SPC ethods fo pocess and poduct ontong, Jounal of Qualty echnology, 8, pp Ncholson B, Lopez-Negete, R. and Begle L.. (014), Onlne state estaton of nonlnea dynac systes wth goss eos, Coputes & Checal Engneeng, 70, pp Saaba, D., C. de Pada, E. Góez, G. Guteez, S. Cstea, J.M. Sola and R. González (011). Data econclaton and optal anageent of hydogen netwoks n a petol efney. Contol Engneeng Pactce, 0, pp ong, H. and Cowe, C.M. (1995), Detecton of goss eos n data econclaton by pncpal coponent analyss, AIChE Jounal, 41(7), pp Copyght 015 IAC 69