all 2004 ICE opis: Poess Contol by esign 0.492 Letue Notes 2: Simulating te Sowe Poess te sowe poess is simple enoug tat we an simulate its opeation Ove te past deade ou tools fo simulating poesses ave geatly impoved. In patiula, we an pedit ow te poess will beave unde dynami onditions: statup, distubane, and sutdown. ynami simulation an aid in eifying te ontol seme and may at potential opeating poblems. o te sowe, we do not need a omplex ompute ode we an deive and solve a deent equation set by and. We will do tis simulation to illustate a simple ontol algoitm ow te poess beavio was indiated by te RGA and C tools Simulation tus allows us to ek te effiay of ou seening tools. fist we need to talk some moe about poess ontol Reall ow we defined feedbak ontol: measuement of CV used to motivate a ange in MV to keep CV at set point. o put tis into patie, we must asset some ontol algoitm, tat is, a way to alulate ow mu to move MV. We begin by defining eo: ( t) = SP CV(t) (2-) Eo is te diffeene between te desied value of CV, alled te set point SP, and CV. Of ouse, CV migt wande aound wit passing time, and so eo would, as well. If CV is at te set point, te eo is zeo; sould CV be distubed, te eo migt be positive o negative. e eo is te input to te ontol algoitm. te simple Popotional algoitm fo a ontolle An intuitively appealing algoitm is to make te eonse popotional to te eo. MV(t) = B (t) (2-2) C C MV te manipulated vaiable, wi may vay in time B C te value of MV wen eo is zeo; known as te bias C adjustable ontolle gain ( o -) we apply te popotional ontolle to ou poess Stat wit flow ontol. Call te set point. en te eo in te flow is = = = ' ' (2-3) By intoduing ou efeene value into te definition, we see tat te eo is also te diffeene between two deviation vaiables. In many ases, of ouse, we would take te set point value to be te same as ou efeene value, so tat is identially zeo. Howeve, evised 2004 e 6. Bay S. Jonston, Copyigt 2004.
all 2004 ICE opis: Poess Contol by esign 0.492 Letue Notes 2: Simulating te Sowe Poess distinguising fom allows us to easily desibe set point anges tat is, moving te poess fom one ondition to anote unde te supevision of te ontolle. te eo an be saled Wen we divide te eo by te opeating ange fo flow, we obtain a dimensionless eo. = = (2-4) If we pai CV flow and MV ot wate, te ontolle algoitm (2-2) is now witten in tese dimensionless tems. = (2-5) C C te saled deviation ot wate flow dimensionless gain (, so tat an inease in will deease ) e magnitude of te gain C may be ineased to povide moe aggessive ontolle ation. o example, a gain of means tat a 0% eo will motivate a 0% ange in manipulated vaiable. Ineasing te gain to 2 podues a 20% ange in MV. Notie tat te bias in (2-2) as been absobed into te deviation vaiables. Wen eo is zeo, no ange is made to, so is zeo. we teat te tempeatue ontolle te same way We will ontol by manipulating te old wate flow. = = (2-6) = (2-7) C In ontast to te flow ontolle, C sould be negative, so tat te ontolle will eond to an inease in tempeatue ( < 0) by ineasing te old flow. wat is a ontolle anyway? e ontolle is some devie tat eeives te eo input and puts out a dietion to te manipulated vaiable. Often, it is a ompute tat uns a ontol pogam; te pogam omputes te MV value at fequent intevals, and te esulting numbe is tansmitted to a tansdue tat moves a valve stem. e ontolle an instead be a meanial devie a ommon example is te leve meanism in te flus toilet. Wateve te adwae, its job is to exeute te algoitm, and if te devie and te mat ae easonably lose, we an desibe its beavio matematially. evised 2004 e 6 2
all 2004 ICE opis: Poess Contol by esign 0.492 Letue Notes 2: Simulating te Sowe Poess te ontol algoitm beomes pat of te mat model of te system We ave a linea system model of te sowe, and we popose to install ontolles. us te ontolle equations (2-5) and (2-7) must be intodued into te M&EB equations. ey eliminate te manipulated vaiables, esulting in ( ) ( ) = C C (2-8) ( ) ( ) = C C (2-9) Now we ave some eaangement to do: we must expess tese two equations so tat te outputs ( ) ae funtions of inputs (distubanes and set points ). We an do tis by solving te MB (2-8) fo substituting into te EB (2-9) and solving tat fo substituting bak into te MB and solving again fo te esult isn t petty wee C CC ( ) C = (2-0) C C C C CC = (2-) C C ( ) ( ) = C C C C (2-2) tese equations epesent a diffeent system: te oiginal piping plus ontolles at is wy tey ae moe ompliated tan ou oiginal M&EB. e fist ting we notie, peaps, is tat tempeatue distubanes affet te flow ate! Of ouse, tis is due to te feedbak stutue, wi begins tuning valves wen eite te flow o tempeatue depats fom set point. We notie, as well, tat anging eite set point affets bot ontolled vaiables. evised 2004 e 6 3
all 2004 ICE opis: Poess Contol by esign 0.492 Letue Notes 2: Simulating te Sowe Poess ese inteations may not in temselves be bad. Howeve, looking at te equations in moe detail will sow tat a pesistent ange in eite inlet tempeatue su tat o emains non-zeo will ause bot and to emain non-zeo. us ou ontolles ae not satisfying ou oiginal opeating objetives of keeping te sowe at te set point! is inability to etun to set point is a popety of popotional ontol; it is known as offset. Offset also ous fo a ange in set point: te ontolled vaiables annot attain te new desied values. Offset an be edued by ineasing te ontolle gain. It an be eliminated by a moe sopistiated ontolle algoitm, but tat is te subjet of 0.450. is is a good time to eall tat ou model egads all input anges as being immediately tansmitted to te output. Hene, equations (2-0) toug (2-2) an desibe a steady state, but tey also desibe time-vaying beavio, even toug time does not appea expliitly. always ty out a ompliated model on simple ases We an lean moe about ou system by examining ow te equations simplify fo limiting onditions. no tempeatue ontol tun C down to zeo (open te tempeatue loop) e mateial balane (2-0) beomes C = (2-3) C Witout te tempeatue ontolle, tee is no meanism fo tempeatue distubanes to affet te flow. A set point ange will neve be fully aieved (offset), but ineasing te gain C makes te appoa lose. e enegy balane (2-) beomes C = (2-4) C empeatue distubanes affet te sowe tempeatue witout intefeene fom te flow ontolle. A set point ange in sowe flow will affet te tempeatue by an amount tat ineases wit te ontolle gain. Hene, attempting to edue flow offset will ineasingly distub tempeatue. no flow ontol tun C down to zeo (open te flow loop) e balanes beome evised 2004 e 6 4
all 2004 ICE opis: Poess Contol by esign 0.492 Letue Notes 2: Simulating te Sowe Poess evised 2004 e 6 5 C C C = (2-5) C = (2-6) wee ( ) C = (2-7) As ontolle gain is ineased, te outlet tempeatue is less affeted by inlet tempeatue distubanes. Coeondingly, tempeatue set point anges ae moe faitfully followed. One again, oweve, ineased gain means tat te flow is ineasingly affeted by bot distubanes and tempeatue set point anges. does te simulation bea out ou RGA peditions? e RGA said tat ou ot-flow-to-flow and old-flow-to-tempeatue paiings would be good if te flow wee mostly ot, and bad if te flow wee mostly old. Let s take te best ase: te outlet flow at te efeene ondition is entiely omposed of ot inlet flow. us 0 = = (2-8) and = (2-9) Making tese substitutions, we find C C C C C C C C = (2-20)
all 2004 ICE opis: Poess Contol by esign 0.492 Letue Notes 2: Simulating te Sowe Poess C = (2-2) C C e flow (2-20) an be made to appoa set point by ineasing C. Ineasing C also elps to suppess te effets of tempeatue set point and distubane anges on flow. e flow ontolle gain as no effet on te outlet tempeatue (2-2). Ineasing C impoves tempeatue set point eonse and suppesses te effet of distubanes. Reduing loop inteation, as dieted by te RGA esults, etainly impoves te ontol eonse! In tis limiting ase, we seem to be able to inease ontolle gains abitaily witout ill effet. If we ae suffiiently uious, we an ek out te opposite limiting ase. does te simulation bea out ou C peditions? C said tat te most diffiult distubane ombination to oveome was a ange by bot inlet tempeatues in te same dietion. Of ouse, by using popotional ontol in ou simulation, we ave allowed offset to ou in te ontolled vaiables. By aving offset, we violate te assumption of pefet ontol tat we used to deive C. Even so, we sould be able to examine wat te simulation pedits fo distubanes. We assume tat set points ae unanged, so tat te set point deviation vaiables ae zeo. om (2-0) toug (2-2) te distubane eonses ae wee = C C (2-22) = C C (2-23) ( ) ( ) = C C C C (2-24) If = (wete positive o negative) C = [ ] (2-25) = C [ ] (2-26) evised 2004 e 6 6
all 2004 ICE opis: Poess Contol by esign 0.492 Letue Notes 2: Simulating te Sowe Poess If = - C = [ ] (2-27) = C [ ] (2-28) o te fist ase, in wi distubanes move in te same dietion, te tem in bakets is lage, wi implies a lage offset in te ontolled vaiables. us te ase identified by C as te most ostly to mitigate also sows te wost ontol eonse. is tee a best design to popose? e eal sowe will ave some degee of inteation between te loops, so tat avalie ineases in ontolle gain ae to be avoided. Pobably te easiest way to answe tis question (wen we ae not at a limiting ase) is to ompute numeial peditions fom te equations ove a domain of ealisti onditions. Peaps we an find gain settings tat allow two ontolles to maintain onditions at bette offsets tan would eite ontolle ating alone. e eadseet model sowe ontollability analysis.xls demonstates tese omputations. let s eview wat we ave done, as a guide to designing a poess We began wit a poess seme fo a sowe, wi we sketed out as a poess flow diagam. We made mateial and enegy balanes. is allowed us to ompute steady state efeene onditions. inking about tansients, we eified ou ontol objetives. We lassified te vaiables into useful input and output ategoies. Speifially, we identified ontolled, manipulated, and distubane vaiables. We simplified te mateial and enegy balanes by aylo seies lineaization. is allowed us to expess ou poess beavio using a standad linea system fom. y = Pm x m P d x d Using te linea system, we omputed te RGA to exploe paiing of CV and MV. We also omputed C to identify te most sevee distubane onditions. Upon oosing a ontol algoitm, we added ontolle equations to te system model and omputed te system eonse unde ontol. final message e next step would be te detailed design pipe sizes, and so fot wi will be beyond ou sope. Of ouse, we ould ave gone staigt to detailed design afte omputing te steady-state efeene onditions. Howeve, some tansient seening tools applied in te ealy stages may save you fom designing a fine-looking steady-state tat would be an opeating disaste. evised 2004 e 6 7
all 2004 ICE opis: Poess Contol by esign 0.492 Letue Notes 2: Simulating te Sowe Poess Atually, te RGA as sown us tat loop inteation will always be pesent fo ealisti opeating onditions. Hene a bette design would attak te inteation poblem dietly. o example, we ould edefine ou manipulated vaiables. We add a total flow ontol valve, dieted by te flow measuement. en we use a atio ontolle to set ot and old popotions. e ontolle algoitms ae standad and non-eifi to ou patiula poess. (Peaps you ave seen sowes in wi you set tempeatue by otating a valve andle and set flow by adjusting te sowe nozzle. ee is usually insignifiant inteation between te two manipulations.) R Altenatively, we ould wite a eial-pupose ontolle algoitm (it would suely inlude mateial and enegy balane equations!) tat would aept two measuement inputs and ompute two valve settings. Sometimes tis exta effot in ontol design pays off in bette poess opeation. C us we an do bette tan te simple two-loop ontol seme we poposed in te beginning. e RGA as allowed us to eognize tis in te peliminay design stage, befoe we ommitted too mu money on someting tat wouldn t wok. e sowe is a simple system, so we ould analyze te wole poblem wit mat. We will next go to a moe ompliated system fo wi analyti solutions ae impatial, even if possible. Howeve, we will follow te same design seme, wit te same intentions, even if we use diffeent tools to get te job done. nomenlatue B C bias; te value of MV wen te ontolled vaiable is at its set point evised 2004 e 6 8
all 2004 ICE opis: Poess Contol by esign 0.492 Letue Notes 2: Simulating te Sowe Poess C P d P m x d x m y a onstant denominato in seveal equations volumeti flow ate ontolle gain matix of gain oeffiients fo te distubane vaiables matix of gain oeffiients fo te manipulated vaiables tempeatue veto of input vaiables into te system, te distubane vaiables veto of input vaiables into te system, te manipulated vaiables veto of output vaiables fom te system, te ontolled vaiables te ange ove wi flow is expeted to opeate te ange ove wi tempeatue is expeted to opeate eo; te set point value minus te ontolled vaiable value at any time abbeviations CV ontolled vaiable, a system output tat we wis to maintain at a set point value C distubane ost V distubane vaiable, a system input tat we ave no influene ove MV manipulated vaiable, a system input tat we may adjust fo ou puposes RGA elative gain aay SP set point, te desied value of te ontolled vaiable subsipts old wate supply steam petaining to te flow ontolle ot wate supply steam a efeene opeating ondition aound wi we deive a linea appoximation set point value of ontolled vaiable o petaining to te tempeatue ontolle supesipts indiates a deviation vaiable; i.e., te pysial vaiable minus a efeene value indiates a saled vaiable; i.e., vaiable as been divided by its opeating ange evised 2004 e 6 9