Planning of production and utility systems under unit performance degradation and alternative resource-constrained cleaning policies

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1 Plannng of producton and utlty systems under unt performance degradaton and alternatve resource-constraned cleanng polces Nur I. Zulkafl and Georgos M. Kopanos * Cranfeld Unversty, School of Water, Energy and Envronment, Bedfordshre MK43 0AL, Unted Kngdom. * Correspondng author: (Georgos M. Kopanos) Abstract A general optmzaton framework for the smultaneous operatonal plannng of utlty and producton systems s presented wth the man purpose of reducng the energy needs and materal resources utlzaton of the overall system. The proposed mathematcal model focuses manly on the utlty system and consders for the utlty unts: () unt commtment constrants, () performance degradaton and recovery, () dfferent types of cleanng tasks (onlne or offlne, and fxed or flexble tme-wndow), (v) alternatve optons for cleanng tasks n terms of assocated duratons, cleanng resources requrements and costs, and (v) constraned avalablty of resources for cleanng operatons. The optmzaton functon ncludes the operatng costs for utlty and producton systems, cleanng costs for utlty systems, and energy consumpton costs. Several case studes are presented n order to hghlght the applcablty and the sgnfcant benefts of the proposed approach. In partcular, n comparson wth the tradtonal sequental plannng approach for producton and utlty systems, the proposed ntegrated approach can acheve consderable reductons n startup/shutdown and cleanng costs, and most mportantly n utltes purchases, as t s shown n one of the case studes. Keywords: energy plannng; producton plannng; performance degradaton; cleanng; optmzaton; utlty system; combned heat and power. 1. Introducton In the hghly dynamc and compettve global market wth strngent envronmental and safety regulatons, t has grown the sgnfcance of systematc operatonal and mantenance plannng for energy ntensve process plants n order to mze proft, mprove plant relablty and enhance the effcent management of assets, resources and energy. Major ndustral facltes consst of nterconnected producton and utlty systems. The producton system produces the desred fnal products from raw materals that can undergo dfferent producton processes, such as chemcal reactons or separatons. These processes requre sgnfcant amounts of several types of utltes, such as electrcty, steam, ndustral gases and water. In general, most ndustral process ndustres have bult onste utlty systems that are drectly connected va ppelnes to the man producton system so as to satsfy ts demands for utltes. Combned heat and power systems, bolers, gas and steam turbnes, compressor statons and ar separaton systems are some typcal examples of onste utlty systems. Combned heat and power systems cogenerate electrcty and heat usually from natural gas are among the most mportant types of utlty systems n process ndustry, because they generate effcently the man utltes needed for the operaton of major equpment of the producton system. In another example, for a cryogenc ar separaton system, the atmospherc ar s frst compressed and then undergoes a cryogenc process before beng separated nto ts prncpal components (ntrogen and oxygen) that consttute some of the key ndustral gases used broadly n process ndustres.

2 For nstance, ntrogen may be used for nertng process vessels for cleanng purposes and ppelne purgng, whle oxygen could be used for the oxdaton of chemcals compounds [1]. The nteracton of the producton system wth utlty systems n process ndustry s llustrated n Fgure 1. Utlty raw materals can be any type of fuel or other resource, such as natural gas or atmospherc ar. These raw materals then undergo a converson process n utlty unts and they generate the desred set of utltes. Compressors, bolers, turbnes, combuston chambers and combned heat and power systems are some representatve examples of utlty unts. Dependng on the type of utltes, dfferent types of converson processes may take place n a utlty unt, such as reacton, combuston or compresson. The generated utltes are then suppled to the producton system for ts own operaton and producton of the fnal products. Excessve amounts of utltes can be stored n buffer tanks (e.g., hot water), be recycled so as to undergo the same process (e.g., steam), or n some cases be released to the envronment (e.g., exhaust heat). It s mportant to notce that the demands for utltes are determned by the needs of the producton system, as a result of the producton plannng problem. raw materal utlty unt 1 utlty type e1 utlty type e2 producton system utlty type e1 utlty unt 2 raw materal utlty type e3 materals or energy forms converson tasks Fgure 1. A representatve layout of the nteracton of producton and utlty systems. In the open lterature, the prmary nterest of producton plannng s usually n advanced equpment, such as chemcal reactors, dstllaton systems, heat exchanger networks and compressor networks. An overvew of operatonal plannng and schedulng n process ndustres can be found n Kallrath [2]. Generally speakng, apart from safety the performance of a process plant s measured by the desred product qualty, mnmum operatng cost and reduced envronmental mpact. Modern process plants are hghly ntegrated and nvolve a set of complex operatng equpment unts that requre mantenance (e.g., cleanng or parts replacement) based on specalzed mantenance montorng technques n order to perform ts requred functon n a tmely manner to avod equpment damage and the neffcent use of t. Effectve mantenance polces can sustan the operatonal level, reduce operatng and fxed costs and restran the equpment unt and the overall system from enterng hazardous states [3]. It s clear from the above dscusson that a holstc systematc approach s needed for the optmzaton of the nterconnected utlty and producton systems under mantenance consderatons n process ndustres. For ths reason, n ths work a system-wde optmzaton framework s developed for the smultaneous operatonal and cleanng plannng of producton

3 and utlty systems consderng mantenance aspects n terms of cleanng operaton, n order to obtaned solutons wth enhanced energy savngs and total cost reductons. The paper s lad out as follows. Secton 2 provdes a bref lterature revew on the plannng of producton and utlty systems and on cleanng operatons. In Secton 3, the problem statement of the subject study under queston s formally defned. The proposed optmzaton framework s then presented n Secton 4, followed by the descrpton and the dscusson of the results of all case studes n Sectons 5 to 7. Fnally, some concludng remarks wth ongong research drectons are provded n Secton Lterature Revew 2.1 Plannng of producton and utlty systems. Most process ndustres, and especally the most energy ntensve, have nstalled onste a utlty system for meetng the utlty requrements of the prncpal producton system. A sequental approach s typcally used for the plannng of utlty and producton systems, as s explaned below. Frst, the plannng of the producton system s performed consderng smply upper bounds on the avalablty of utltes. Once the producton plan s derved, the utlty needs of the producton are known. Ths nformaton s then used for obtanng the operatonal plannng of the utlty system. Ths sequental approach provdes suboptmal solutons (manly n terms of energy effcency and costs) because the two nterconnected systems are not optmzed at the same tme. For ths reason, ths work focuses on the smultaneous plannng of utlty and producton systems. A bref lterature revew on the subject follows. Most prevous studes have addressed ether the plannng of producton systems [4 7] or the plannng of utlty systems ndependently [8 10]. There are few works that dealt wth the smultaneous plannng of utlty and producton systems. For example, Agha et al. [11] presented a Resource-Task Network based mathematcal model for the smultaneous plannng of a manufacturng and a combned heat and power plant. The results of ther case studes demonstrated clearly that ths ntegrated approach reduces sgnfcantly the energy costs and the emssons of greenhouse gases compared to the tradtonal sequental approach. In another study, Zhang et al. [12] developed a mxed nteger nonlnear programmng model that ncludes the heat ntegraton of the process plant, the optmzaton of the utlty system and couplng equatons for the ste-scale steam ntegraton. Zhao et al. [13] presented mathematcal models for the smultaneous plannng of a refnery and ts onste utlty system. The results of all the above works showed that the ntegrated plannng of utlty and producton systems could result n sgnfcant energy savngs, emssons and overall costs reductons. 2.2 Plannng of cleanng operatons. The cleanng of specfc equpment that are characterzed by performance degradaton (e.g., due to foulng), such as compressors and heat exchangers, s one of the major mantenance actons n process ndustry [14 16]. The purpose of these cleanng operatons s to recover the performance (effcency) of equpment and that way decrease ther energy consumpton or ncrease the energy savngs over the operaton of the equpment. There are two man cleanng strateges to deal wth equpment performance degradaton, namely onlne and offlne cleanng. Onlne cleanng tasks take place wthout nterruptng the operatng status of the equpment and recover partally the performance of the equpment. An example of onlne cleanng task s the njecton of a cleanng soluton n the equpment whle t s stll under operaton. Offlne cleanng tasks can be performed only when the equpment s closed and t s generally assumed that they can recover the full performance of the equpment. The duraton of offlne cleanng

4 tasks can be consderably hgher than that of onlne tasks, because durng offlne cleanng other supplementary mantenance tasks, such as mechancal and electrcal nspectons, take place. The nterested reader could be referred to the works of Pattanayak et al. [17] and Tan et al. [18] for a more detaled dscusson on the cleanng of equpment. A few studes studed dfferent types of cleanng tasks, resource allocaton, cleanng duraton and costs. For example, Nguyen et al. [19] studed the trade-off between the number of workers, cleanng cost and economc losses. They show that for lmted avalable cleanng resources, the cleanng tasks dd not perform on tme and economc loss occurred. Whle for excessve avalable cleanng resources, the mantenance tasks can be done on tme but the total cleanng cost may become unnecessary hgh. Kopanos et al. [20] presented an optmzaton framework for the plannng of a network of compressors consderng lmtatons on the number of compressors that could be under mantenance smultaneously. Do et al. [21] studed a multcomponent system wth lmted mantenance team and they showed that the mnmum number of avalable resources can be obtaned by mnmzng the mantenance cost. Most of the works on the plannng of cleanng tasks dd not consder resources lmts for the cleanng operatons. To the best of our knowledge, ths s the frst work that addresses the smultaneous operatonal and cleanng plannng of producton and utlty systems consderng unt performance degradaton and recovery, unt commtment constrants and cleanng resources aspects (.e., selecton of alternatve cleanng optons, mum avalablty of cleanng resources). 3. Problem Statement The smultaneous plannng of producton and utlty systems consttutes the subject of ths 144 study. In partcular, the prncpal focus of ths work s on the detaled operatonal and cleanng 145 plannng of the utlty system consderng tradtonal and alternatve condton-based cleanng 146 operatons. Performance degradaton and recovery are consdered for the utlty unts that are 147 subject to condton-based cleanng. Alternatve optons for cleanng tasks wth respect to the 148 duraton, resource requrements and costs are also studed. The resultng problem s formally 149 defned n terms of the followng tems: 150 A gven plannng horzon dvded nto a number of equally-length tme perods t T. 151 A set of utlty types e E that are produced from a number of utlty unts I e wth 152 gven mum (mnmum) operatng levels κ (,t) (κ (,t) ). For every utlty unt 153 I, the mnmum (mum) runtme after ts startup ω (o ), the mnmum dle tme 154 after ts shutdown (ψ ) and the costs for startup (φ S ) and shutdown (φ F ) are defned. 155 A set of fnal products g G wth known demand profles ζ (g,t) that can be produced 156 from a number of processng unts n N g and mum (mnmum) producton levels 157 κ (n,g,t) (κ (n,g,t) ). For every processng unt n N and fnal product g G n, fxed 158 and varable requrements for utltes are gven (β (n,g,e) and α (n,g,e), respectvely). 159 var Fxed and varable operatng costs for the processng unts are also consdered (χ (n,g) fx and χ (n,g), respectvely). Each processng unt s lnked to a set of nventory tanks for 161 fnal product (n N l ) and to a set of nventory tanks for utltes (n N e ). 162 A set of utlty-dedcated nventory tanks z Z e that are connected to processng unts 163 n N z. These nventory tanks usually have a gven mum (mnmum) nlet total 164 flow ε (e,z,t) (ε (e,z,t) ) and mum (mnmum) storage capactes ξ (e,z) (ξ (e,z) ).

5 A set of product-dedcated nventory tanks l L g wth mum (mnmum) storage capactes ξ (g,l) (ξ mn (g,l) ). A number of cleanng polces for the utlty unts are consdered. More specfcally, a utlty unt could be subject to: () flexble tme-wndow offlne cleanng ( FM ) wth gven earlest (τ es ) and latest (τ ls ) startng tmes, () n-progress offlne cleanng carred over from the prevous plannng horzon ( DM ), or () condton-based cleanng ( CB ) wth known degradaton rates (ρ ) for the utlty unts. Furthermore, two types of condton-based cleanng tasks are consdered, namely: onlne cleanng tasks (CB on ) wth gven recovery factors (ρ rec ), and offlne cleanng tasks (CB off ). A set of alternatve cleanng tasks optons q Q. For each utlty unt (CB off FM ), there s a set of alternatve cleanng tasks optons Q that are characterzed by dfferent duratons (ν (,q) ), cleanng resource requrements (θ off (,q) ), and costs (φ off (,q) ). Gven purchase prces (or penalty costs) for acqurng utltes or fnal products from external sources, φ UT,ex (e,n,t) and χ ex (g,t) respectvely. A gven tme-varyng electrcty prce profle φ t pw. Some addtonal consderatons of the problem under study follow. All parameters are assumed to be determnstc. Also, the demands for the fnal products should be fully satsfed. And, the nventory tanks for utltes could be connected to multple processng unts. In addton, there s a lmted amount of avalable resources for cleanng tasks per tme perod. For every tme perod, the key decsons to be made by the optmzaton model are: The operatng status and the producton level of fnal products for all processng unts. The nventory levels for fnal products and utltes at each nventory tank. The utlty requrements of the processng unts. The operatng status for each utlty unt (.e., startup, n operaton, shutdown, under cleanng). The producton level for the utlty systems. The selecton of the types and the tmngs for the cleanng tasks to be performed n the utlty unts. And all the above wth the objectve to mnmze the total cost of the overall system, whch encompasses: fxed and varable operatng costs for processng unts, startup and shutdown costs for utlty unts, power consumpton costs for utlty unts, total cleanng costs for utlty unts, and costs for acqurng utltes or fnal products from external sources. 4. Optmzaton Framework

6 A lnear mxed nteger programmng model s presented for the operatonal and cleanng plannng of utlty system and the operatonal plannng of the producton system of a process plant. The man part of the presentaton of the optmzaton framework s dvded nto the followng parts: () the utlty system, () the producton system, and () the objectve functon. 4.1 The Utlty System Constrants related to startup and shutdown actons. In order to model the man operatonal aspects of the utlty unts, the followng bnary varables are frst ntroduced: X ( t, ) 1 f utlty unt s operatng durng tme perod t, 0 otherwse. 214 S ( t, ) 1 f utlty unt starts up at the begnnng of tme perod t, 0 otherwse F ( t, ) 1 f utlty unt shuts downs at the begnnng of tme perod t, 0 otherwse. Constrants (1) and (2) model startup and shutdown actons through the operatng status of the utlty unt. S ~ (, t) F(, t) X (, t) I, t T : t 1 (1) S F X X I, t T : t 1 (, t) (, t) (, t) (, t 1) S( F 1 I, t T (2), t) (, t) For nstance, accordng to constrants (1), f a utlty unt has not been operatng n the prevous tme perod but operates n the current tme perod, then a startup takes place (.e., S (,t) = 1 and F (,t) = 0). Parameter χ denotes the operatng status of utlty unt just before the start of the plannng horzon. If the utlty unt has been operatng just before the start of the plannng horzon, then χ = 1, otherwse t s zero. Constrants (2) excludes the smultaneous realzaton of a startup and a shutdown acton. If startup and shutdown costs are ncluded n the objectve functon, constrants (2) could be excluded from the optmzaton model, snce ther correspondng values wll tend to zero. Constrants (3) model the mnmum runtme (ω ) for a utlty unt after ts startup. These constrants ensure that f a utlty unt startups at a gven tme perod t, t wll operate for at least ω tme perods. X S, t T : 1 (, t) ( t, ) t {1, t 1} t X 1 I, t 1,...,( ) : 0 (, t) Parameter ω descrbes the ntal state of each utlty unt wth respect to ts mnmum runtme. More specfcally, ths parameter corresponds to the total number of tme perods at the begnnng of the plannng horzon that utlty unt has been contnuously operatng snce ts last startup. For example, f ω = 4 and ω = 2, the second part of constrants (3) gves X (,1) = X (,2) = 1. (3)

7 Smlarly, constrants (4) model the mnmum shutdown tme (ψ ) for a utlty unt after ts shutdown. These constrants ensure that f a utlty unt shuts down at a gven tme perod t, t wll not operate for at least ψ tme perods X F, t T : 1 (, t) ( t, ) t {1, t 1} t X 0 I, t 1,...,( ) :0 (, t) (4) Parameter ψ descrbes the ntal state of each utlty unt wth respect to ts mnmum shutdown tme, and corresponds to the total number of tme perods at the begnnng of the plannng horzon that utlty unt has been contnuously not operatng snce ts last shutdown. Obvously, constrants (3) and (4) are needed only f the duratons of the mnmum runtme and mnmum shutdown tme are greater than a sngle tme perod, respectvely. In addton, there may be a mum duraton of a contnuous operaton of a utlty unt ( MR ), called here a mum runtme (o ). Usually ths reflects n a way the performance deteroraton of a unt durng ts operaton and s used to prevent major mechancal damages and reduce the energy-neffcent use of the unt; when no more sophstcated methods for the performance degradaton are consdered. The mum runtme for a utlty unt s gven by: t t {1, t } t ( t, ) t {1, t ( )} X MR, t T X ( ) MR, t ( 1) : 1 ( t, ) Smlar constrants can be formulated for the mum dle tme of a utlty system, f needed. Here, the mum dle tme s defned as the mum duraton that a utlty system remans swtched off after ts last shutdown Constrants related to cleanng actons. Alternatve types of cleanng operatons for the utlty unts are consdered n ths study. More specfcally, the utlty unts could be subject to: In-progress offlne cleanng, carred over from the prevous plannng horzon ( DM ). Flexble tme-wndow offlne cleanng tasks ( FM ). Condton-based cleanng tasks ( CB ), where a performance degradaton and recovery model s used. There are two types of condton-based cleanng tasks: onlne cleanng tasks (CB on ) and offlne cleanng tasks (CB off ). Both condton-based cleanng task types could be avalable for a unt, or just one of them (e.g., a unt could undergo offlne cleanng but no onlne cleanng). For a utlty unt that could undergo condton-based offlne cleanng (CB off ) or flexble tmewndow offlne cleanng (FM ), alternatve optons of cleanng tasks (q Q ) are also consdered and modeled. These optons dffer n the: duraton, cost, and cleanng resources requrements. In order to model the aforementoned cleanng tasks of the utlty system, the followng bnary varables are frst defned: (5)

8 V ( t, ) W ( t, ) on 1 f an onlne cleanng task for CB takes place n tme perod t, 0 otherwse. off 1 f an offlne cleanng task for ( CB FM ) begns at the start of tme perod t, 0 otherwse H (, q, t) off 1 f a cleanng task opton q for ( CB FM ) begns at the start of tme perod t, 0 otherwse. For the sake of clarty, an llustratve example of the major optmzaton varables s dsplayed n Fgure 2. F (1, ) = 1 (1 ) = 1, (1,q2, ) = 1 S (1, ) = X (1, ) = 1 R (1, ) = 0 1 offlne cleanng X (1, ) = X (1, ) = X (1, ) =X (1, ) = onlne cleanng (2,2) = 1 R (2,2) 0 Fgure 2. Illustratve example for the man optmzaton varables In-progress offlne cleanng tasks carred over from the prevous plannng horzon. At the begnnng of the plannng horzon of nterest, there may be some n-progress unfnshed offlne cleanng tasks carred over from the prevous plannng horzon (.e., started but not completed n the prevous plannng horzon). For the current plannng horzon, these are predetermned cleanng tasks wth known completon tmes and resource requrements per tme perod. Constrants (6) model ths type of cleanng tasks by keepng closed the unts DM for the all these tme perods that there s a known cleanng resource requrement (η (,t) ). X 0 DM, t T : 0 (6) (, t) (, t) (1, ) = 1, (1,q1, ) = t Flexble tme-wndow offlne cleanng tasks. offlne cleanng X (2, ) = X (2, ) = X (2, ) = 0

9 For utlty unts that are not subject to a condton-based cleanng polcy, a flexble tmewndow offlne cleanng polcy ( FM ) s usually employed. In general, these types of cleanng tasks have known duratons and they should start wthn a gven tme-wndow t = [τ es, τ ls ]. Constrants (7) ensure that the flexble tme-wndow cleanng task for each utlty unt FM starts wthn ts correspondng tme-wndow. q Q t ls es H( 1 FM (7), q, t) Notce that fxed offlne cleanng tasks can also be modeled through the above constrants smply by settng equal the earlest and startng tmes (.e., τ es = τ ls ) Extra power consumpton for utlty unts (devaton from clean condton). In ths study, the condton-based cleanng of a utlty unt s modeled through the extra power consumpton of the unt (U (,t) ) due to the devaton from ts full performance (.e., when the unt s completely clean). There can be an extra power consumpton only n perods that the utlty unt s under operaton and ths extra power consumpton cannot exceed an assocated upper lmt (υ ), as gven below: U X CB, t T (8) (,t) (,t) It s consdered that the condton of a utlty unt, whch here s expressed by the extra power consumpton, s related to ts cumulatve tme of operaton (R ) from ts fully clean condton and the correspondng degradaton rate (δ ) accordng to the followng set of constrants: U R (1 X ) CB, t T (,t) (,t) (,t) U R + (1 X ) CB, t T (,t) (,t) (,t) Accordng to these constrants, f a utlty unt s operatng (.e., X (,t) = 1), ts extra power consumpton s equal to δ R (,t), otherwse t becomes zero from constrants (8) Performance degradaton and recovery model for unts under condton-based cleanng. The performance degradaton and recovery of the utlty unts s expressed through ther cumulatve tme of operaton. It s assumed here that a utlty unt can retreve ts full performance after the occurrence of a condton-based offlne cleanng task. Ths s expressed by a zero cumulatve tme of operaton, as gven by: R μ (1 W ) CB off, t T (10) (,t) (,t) Parameter μ s a suffcent bg number. The evoluton of the cumulatve tme of operaton for any utlty unt that s subject to condton-based offlne or onlne cleanng s gven by constrants (11) and (12), respectvely. R ( + X ) ( W V ) CB, t T : t 1 (,t) (,t) (,t) (,t) R ( R + X ) ( W V ) CB, t T : t 1 (,t) (,t 1) (,t) (,t) (,t) (9) (11) 320 rec on R ( + 1)(1 ) (1 V ) CB, t T : t 1 (,t) (,t) R ( R + 1)(1 ) (1 V ) CB, t T : t 1 rec on (,t) (, t 1) (,t) (12)

10 Observe that the proposed modelng approach allows a utlty to be able to be subject to both offlne and onlne condton-based cleanng tasks, f needed Condton-based onlne cleanng tasks. Some addtonal constrants for the condton-based onlne cleanng of utlty unts are ncluded here. The duraton of an onlne cleanng task s equal to a sngle tme perod. Constrants (13) ensure that onlne cleanng could take place n a utlty unt at a gven tme perod only f the unt s under operaton. on V(,t) X (,t) CB, t T (13) In addton, there s usually a lmtaton on the frequency that onlne cleanng can take place n a utlty unt n order to avod potental damage or other negatve effects on the performance on the unt. Constrants (14) ensure that the necessary mnmum tme between two consecutve onlne cleanng tasks (γ on ) on a utlty unt s satsfed. 333 t on t {1, t 1} on V 1 CB, t T ( t, ) V 0 CB,t ( ) : < on on on on on (,t) (14) Parameter γ on provdes the ntal state of any utlty unt CB on wth respect to ts last onlne cleanng. Ths parameter represents the total number of tme perods that have passed snce the last onlne cleanng of a utlty unt at the begnnng of the current plannng horzon Operatonal constrants for offlne cleanng tasks. Constrants (15) ensure that f an offlne cleanng task takes place on a utlty unt, that unt remans closed (.e., X (,t) = 0) for the whole duraton of the selected offlne cleanng task opton. And, constrants (16) relate the two operatng bnary varables for offlne cleanng tasks. ls mn{, t} off es ls (, t) (, q, t ) (, q) es t {, t v(, q) 1} X H 1 ( FM CB ), q Q, t ( v 1) (15) off es ls W(, t) H(, q, t) ( FM CB ), t T : t (16) q Q For the condton-based offlne cleanng tasks, earlest and latest startng tmes should be set equal to the frst and the last perod of the plannng horzon, respectvely Resource constrants for cleanng tasks. In every tme perod, there s a lmted amount of avalable resources for cleanng operatons (η ) shared by all types of cleanng tasks consdered n ths study. CB on V on (, t) t off (, q) CB q Q t t v 1 off (, q) H (, q, t ) ls mn{, t} FM q Q t { off (, q) H(, q, t ) es, t v(, q) 1} (17) ~ t T t DM (, t)

11 on Parameters θ and θ off (,q) correspond to the resource requrements for onlne cleanng and dfferent offlne cleanng task optons, for every utlty unt. Parameter ν (,q) denotes the duraton of each offlne cleanng task opton Producton of utltes. The operatng producton level of any utlty unt (Q (,t) ) should be between ts correspondng lower and upper bounds (κ UT,mn (,t) and κ UT, (,t) ) when the utlty unts operates, as gven by: X Q X I t T (18) UT, mn UT, (, t) (, t) (, t) (, t) (, t), A utlty unt may produce at the same tme more than one utlty types (e.g., a combned heat and power unt). Then, constrants (19) specfy the amount of any utlty e produced by each utlty unt I e per tme perod. UT Q(,, ) (, ) Q(, ) I, e E, t T (19) e t e t Parameter ρ (,e) stands for the stochometry coeffcent that relates the operatng level of the utlty unt wth the produced amount of each utlty type that s coproduced by the utlty system (e.g., heat to power rato of a combned heat and power unt) Inventores for utltes. To contnue wth, the utlty system contans a number of utlty-dedcated nventory tanks. These nventory tanks can receve utltes (B UT,+ (e,z,t) ) from the utlty unts that are connected wth, accordng to: UT, UT B(,, ) Q(,, ) e E, z Z, t T (20) e z t e t e Ie Also, there are usually lower and upper bounds on the flows of utltes to nventory tanks: mn UT, ( e, z, t) B( e, z, t) ( e, z, t) e E, z Ze, t T (21) The utlty balances n the utlty-dedcated nventory tanks are gven by: B B B e E, z Z, t T : t 1 UT UT UT, UT, ( e, z, t) ( e, z) ( e, z, t) ( e, z, n, t) e n ( Ne Nz) B B B B e E, z Z, t T : t 1 UT UT UT, UT, ( e, z, t) ( e, z, t 1) ( e, z, t) ( e, z, n, t) e n ( Ne Nz) UT Parameter β (e,z) provdes the ntal nventory for each utlty nventory tank, varable B (e,z,n,t) gves the amount of utlty type e that leaves ts nventory tank so as to satsfy the correspondng demand for utlty of the connected processng unts at each tme perod. Mnmum and mum nventory levels for these nventory tanks are also set: B e E z Z t T (23) UT, mn UT UT, ( e, z) ( e, z, t) ( e, z), e, Demands for utltes The lnk between the utlty and the producton system. Constrants (24) consttute the lnkng constrants between the utlty and the producton system. More specfcally, the utltes demands of each processng unt consst of: () fxed utltes requrements dependng on the operatonal status of the processng unt, and () varable utltes needs dependng on the producton level of the processng unt. (22) UT,

12 NS B ( Q K ) e E, n N, t T (24) UT UT, FP ( e, n, t) ( e, z, n, t) ( n, g, e) ( n, g, t) ( n, g, e) ( n, g, t) e z ( Z Z ) g G e n n UT Notce that varables NS (e,n,t) gve the amount of unsatsfed demand for each utlty type per tme perod from the nternal utlty system. The acquston of utltes from external sources s allowed here but t s hghly undesrable and for ths reason a very hgh purchase or penalty cost s typcally ntroduced. 4.2 The Producton System Constrants related to the operatonal status and producton level of the processng unts. The producton system conssts of a number of processng unts that can produce the fnal products. The operaton of the processng unts along wth the product-to-unt allocaton are modeled through the followng bnary varables: K ( n, g, t) 1 f fnal product g s produced n processng unt n durng tme perod t, 0 otherwse. There are two man constrants for the processng unts. More specfcally, for every tme perod, there s a lmted number of products (λ n ) that a processng unt could produce, accordng to: g Gn K n N t T (25) ( n, g, t) n, Addtonally, the produced amount of a fnal product should be wthn the lower and upper producton level bounds (κ mn (n,g,t) and κ (n,g,t) ) of each operatng processng unt, as gven by: K Q K g G n N t T (26) mn FP ( n, g, t) ( n, g, t) ( n, g, t) ( n, g, t) ( n, g, t), g, Inventores for fnal products. In ths study, fnal product dedcated nventory tanks are consdered. These nventory tanks can receve fnal products (B + (g,l,t) ) from the processng unts that are connected wth, accordng to: B Q g G, l L, t T (27) ( g, l, t) FP ( n, g, t) g n ( Ng Nl) Hence, the materal balances n the product-dedcated nventory tanks are gven by: B B B g G, l L, t T : t 1 FP FP ( g, l, t) ( g, l) ( g, l, t) ( g, l, t) g B B B B g G, l L, t T : t 1 FP ( g, l, t) ( g, l, t 1) ( g, l, t) ( g, l, t) g (28) Parameter β (g,l) FP represents the ntal nventory level for each nventory tank, varable B (g,l,t) provdes the amount of fnal product g that leaves ts nventory tank n order to satsfy the correspondng product demand at each tme perod. Mnmum and mum nventory levels for each nventory tank are defned as shown below: B g G l L t T (29) mn FP ( g, l) ( g, l, t) ( g, l), g, Demands for fnal products.

13 For every tme perod, the demands for fnal products (ζ (g,t) ) are gven and should be satsfed completely, accordng to the followng equaton: NS B g G t T (30) FP ( g, t) ( g, l, t) ( g, t), l Lg FP Varables NS (g,t) gve the amount of unsatsfed demand for every fnal product per tme perod from the nternal producton system. The purchases of fnal products s hghly undesrable and for ths reason a very hgh purchase or penalty cost s typcally used n the objectve functon. In the case that fnal products purchases are not allowed, NS FP (g,t) represent the lost sales of fnal products. 4.3 Objectve Functon The optmzaton goal s to mnmze the total cost of the producton and the utlty system. The total cost nvolves: () fxed and varable operatng costs for processng unts, () cost for purchasng fnal products and utltes from external sources, () startup and shutdown costs for utlty unts, (v) total power consumpton costs for utlty unts, and (v) cleanng costs for onlne and offlne cleanng tasks for utlty unts. The objectve functon s shown below: mn t T n N g G ( Q K ) var FP fx ( n, g) ( n, g, t) ( n, g) ( n, g, t) ( NS NS ) ex FP UT,ex UT ( g, t) ( g, t) ( e, n, t) ( e, n, t) t T g G e E n Ne ( S F ) ( Q U ) S F pw pw (, t) (, t) t (, t) t (, t) t T I t T I t T ( on off V (, t) (, q) H(, q, t) on off CB ( CB FM) q Q In the above expresson, all small-letter symbols multpled by the optmzaton varables correspond to cost coeffcents. A descrpton of them s provded n the Nomenclature. 4.4 Specal Case: No Storage of Utltes In fact, some types of utltes cannot be stored usually due to several factors, such as ther unstable nature, lack of good storage technology, and hgh storage energy needs. An example of such a type of utlty s compressed ar, whose storage s usually avoded due to hgh storage energy needs. Generally speakng, the absence of storage tanks for utltes n practce often results n a dfferent layout for the utlty system, where the utlty unts are connected drectly to the processng unts va connectng lnes (e.g., ppelnes). A representatve layout of such utlty systems can be seen n Fgure 3 of the frst case study consdered n the paper. From the operatonal pont of vew, n ths case the selecton of whch utlty unt s connected to whch connectng lne (and thus to whch processng unt) s an addtonal decson to be made for every tme perod. Typcally, multple utlty unts may serve a connectng lne and utlty property constrants should be consdered for the utlty unts that serve the same connectng lnes. For nstance, n the case of a network of compressors, whch s dsplayed n Fgure 3, the outlet pressures (.e., the property here) of the compressors that serve the same connectng lnes at any gven tme perod must be the same. The presentaton of the necessary set of constrants for ths type of utlty systems follows. ) (31)

14 Constrants related to the assgnment of utlty unts to connectng lnes. In order to model the actve connecton among utlty unts and connectng lnes (j), the followng bnary varables are ntroduced: Y (, j, t) 1 f utlty unt serves connectng lne j durng tme perod t, 0 otherwse. When a utlty s under operaton, t can serve at most one connectng lne at a tme, as stated by: j J Y X I t T (32) (, j, t) (, t), As already mentoned, property constrants should be consdered for the utlty unts that serve the same connectng lne. The type of the property of nterest and the related constrants depend on the utlty system. For ths reason, t s dffcult to provde a general constrant that would descrbe any such utlty system. However, n most of the cases the utlty property s defned for the connectng lnes and t could be expressed as a functon of: () the total amount that the lne receves, and () a fxed term related to the actve utlty unts connectons. In general, constrants smlar to those proposed by Kopanos et al. [20]) could be used (kndly refer to constrants (7) to (10) n Kopanos et al. [20]). Actually, these constrants have been used n Case Study Constrants related to assgnment changes of utlty unts to connectng lnes. In practce s sgnfcant to avod unnecessary assgnment changes of utlty unts to connectng lnes, snce ths would add unnecessary complexty n the plan mplementaton [20]. In order to model ths, the followng bnary varables are ntroduced: D ( t, ) 1 f utlty unt changes connectng lne at the begnnng of tme perod t, 0 otherwse. Then, the utlty unt to connectng lne assgnments can be modeled accordng to: D Y S I, t T : t 1 (, t) (, j, t) (, j) (, t) D Y Y S I, t T : t 1 (, t) (, j, t) (, j, t 1) (, t) Parameter φ (,j) represents the ntal state of the actve connecton between utlty unts and connectng lnes just before the begnnng of the current plannng horzon. A detaled explanaton of these constrants along wth an llustratve example can be found n Kopanos et al. [20]. Fnally, cost coeffcents should be defned for these assgnment changes and an assocated cost term (.e., total cost/penalty for assgnment changes among utlty unts and connectng lnes) must be added n the objectve functon of the problem Producton of utlty. UT The producton level (Q (,j,e,t) ) of utlty unt that serves connectng lne j J must be between the correspondng lower and upper bounds (κ (,j,t) UT,mn and κ (,j,t) UT, ), as gven by: ˆ Y Qˆ ˆ Y I, j J, e E, t T (34) UT, mn UT UT, (, j, t) (, j, t) (, j, e, t) (, j, t) (, j, t) (33)

15 In ths specal case of utlty system, a utlty unt produces a sngle type of utlty. A typcal example of such a utlty system s a compressor Demand for utlty. In ths specal case, constrants (24) are replaced by constrants (35). Now, these constrants are the lnkng constrants between the utlty and the producton systems. NS Qˆ ( Q K ) e E, n N, t T (35) UT UT FP ( e, n, t) (, j, e, t) ( n, g, e) ( n, g, t) ( n, g, e) ( n, g, t) e I j (J J ) g G e n n 4.5 Remarks The optmzaton frameworks presented n ths secton have been formulated n such a way that consders the complete set of parameters that defne the ntal state of the overall system. For ths reason, the proposed approach can be readly used n a rollng horzon framework to deal wth uncertanty aspects of the problem. Fnally, notce that one could solve the plannng problem of just the utlty system by replacng the rght hand sde of the constrants (24) or (35) by parameters that represent the gven demands for utltes per processng unt and tme perod. 5. Case Study 1: Plannng of a Utlty System - An Industral Network of Compressors. Ths case study s a modfed verson of the ndustral compressors network of the ar separaton plant of BASF SE studed by Kopanos et al. [20]. Compressed ar s the only utlty and product of nterest here. The purpose of ths example s to demonstrate the applcablty of the proposed optmzaton framework n an ndustral scenaro where condton-based onlne and offlne cleanng tasks for the utlty unts and operatonal tasks for the utlty and producton systems are consdered and optmzed smultaneously. In addton, dfferent optons for offlne cleanng tasks are consdered, thus ncreasng the complexty of the resultng decson-makng optmzaton problem. A smplfed verson of the layout of the network of compressors of ths example s dsplayed n Fgure j1 j j X 11 Y n n n Fgure 3. Case Study 1: Layout of the network of compressors.

16 Descrpton of Case Study 1. Ths case study consders a network of compressors that conssts of eleven compressors connected n parallel that supply compressed ar to three processng unts (n1, n2, n3) through three headers ( j1, j2, j3). There are fve small compressors (1, 2, 3, 4, 5) and sx large compressors (6, 7, 8, 9, 10, 11). Compressed ar cannot be stored here. A compressor can be connected to at most one header per tme perod. Compressors could be connected to any header, but each header serves wth compressed ar a dedcated processng unt. More specfcally, header j1 could supply compressed ar to processng unt n1, header j2 s connected to processng unt n2, and header j3 serves processng unt n3. Mnmum and mum levels of outlet mass flow rates and pressure ratos of compressors, can be found n Kopanos et al. [20]. A total plannng horzon of 30 days, dvded n day tme perods (.e., 30 tme perods), s consdered. Table 1 provdes the man operatonal parameters for ths case study. The values for mnmum and mum runtme, mnmum shutdown tme and mnmum tme between two successve onlne cleanngs are selected to reflect the typcal status of the process ndustry. A condton-based approach s used, therefore there s no need for ntroducng earlest and latest startng tmes for cleanng tasks. Cost related data for the operatonal and the cleanng tasks can be found n Table 2. Penalty costs for changng headers, startup and shutdown cost for compressors and onlne cleanng costs are taken from the hstorcal data of the compressors network by Kopanos et al. [20]. All parameters that descrbe the ntal state of the overall system under consderaton can be found n Table 3. Table 1. Case Study 1: Man parameters. Symbol Value Unt Descrpton t 1 day Duraton of each tme perod. T 30 days Total number of perods (plannng horzon). 6 days Mnmum runtme for compressors. 3 days Mnmum shutdown tme for compressors. 20 days Maxmum runtme for small compressors. 30 days Maxmum runtme for large compressors. 8 days Mnmum tme between two onlne cleanngs. on on 1 resource unt Necessary cleanng resources for onlne cleanng. 6 resource unts Avalable cleanng resources per tme perod. t rec Recovery factor after onlne cleanng. Table 2. Case Study 1: Costs for operatonal and cleanng tasks. Symbol Value Unt Descrpton m.u./change Penalty term for changng header. S 4,900 m.u./startup Cost of startup for small compressors. S 9,800 m.u./startup Cost of startup for large compressors. F 2,500 m.u./shutdown Cost of shutdown for small compressors. F 5,000 m.u./shutdown Cost of shutdown for large compressors.

17 on on off ( q, 1) off ( q, 1) off (, 2) m.u./ cleanng Cost of onlne cleanng for small compressors m.u./ cleanng Cost of onlne cleanng for large compressors m.u./cleanng Cost of offlne cleanng task q1 for small compressors m.u./cleanng Cost of offlne cleanng task q1 for large compressors. q m.u./cleanng Cost of offlne cleanng task q2 for small compressors m.u./cleanng Cost of offlne cleanng task q2 for large compressors. off ( q, 2) off ( q, 3) off ( q, 3) m.u./cleanng Cost of offlne cleanng task q3 for small compressors m.u./cleanng Cost of offlne cleanng task q3 for large compressors. Table 3. Case Study 1: Intal state of the network of compressors. ~ (, j) j3 - - j1 - j1 j2 j Fgure 4 dsplays the normalzed daly demand for compressed ar for each processng unt; havng as a reference the hghest demand observed throughout the plannng horzon. Demand s assumed to be determnstc. In addton, Fgure 5 shows the electrcty prce profle over the plannng horzon. 100% 90% 80% 70% 60% 50% 40% n1 n2 n3 Fgure 4. Case Study 1: Normalzed daly demand for compressed ar per processng unt.

18 Fgure 5. Case Study 1: Electrcty prce per tme perod Results of Case Study 1 The resultng optmzaton problem has been modeled usng the general algebrac modelng language GAMS and solved by CPLEX 12 n an Intel(R) core(tm) 7 under standard confguratons and a zero optmalty gap. The optmal soluton was found n about half an hour. n1 offlne cleanng task optons n2 n3 compressor s off q1 q2 q3 onlne cleanng Fgure 6. Case Study 1: Optmal operatonal and cleanng plan for the utlty system. Fgure 6 presents the optmal plan for the operatonal and cleanng tasks for the network of compressors (.e., the utlty system). More specfcally, ths Gantt chart shows for each compressor: () ts actve header connecton at each tme perod, () the selected offlne cleanng tasks optons and ther correspondng tmng, and () the onlne cleanng plan over the 30-day plannng horzon. Compressors startups, shutdowns and header changes can be seen

19 n ths Gant chart as well. Accordng to Fgure 6, compressors 3 and 10 reman shutdown throughout the total plannng horzon. Accordng to the hstorcal data and the operators experence, ths s bascally due to the fact that these compressors are less-effcent compared to the other compressors. In addton, t s observed that exactly fve compressors are operatng at each tme perod n order to satsfy the total demand for compressed ar. More specfcally, three large compressors and two small compressors operate smultaneously from the begnnng of the plannng horzon untl day 13, two large compressors and three small compressors operate at the same tme from day 14 to day 22, and one large compressors and four small compressors operate smultaneously from day 23 to the end of the plannng horzon. Ths decrease n the number of operatng large compressors throughout the plannng horzon s partally due to the decrease of the total demand for compressed ar after day 14, as shown n Fgure 4. The hgher number of operatng large compressors durng the frst half of the plannng horzon s also due to the ntal state of the system where three large compressors were under operaton at the end of the prevous plannng horzon (see Table 3). Accordng to Table 3, compressors 2, 5, 7, 8 and 9 have been operatng just before the begnnng of the current plannng horzon. As t can be seen n Fgure 6, compressors 2 and 5 operate (except of a three-day offlne cleanng break each) throughout the plannng horzon, however compressors 7, 8 and 9 shutdown n day 14, day 3 and day 22 (and do not start agan untl the end of the plannng horzon), respectvely. Snce the ntal state of these three large compressors are qute smlar, ther observed shutdown sequence reveals ther energy consumpton performance. In other word, the more energy-neffcent compressors shutdown before the others (.e., 8 shuts down frst, followed by 7 and 9 s the last to shut down). Once compressor 8 shuts down, clean compressor 11 starts up and operates untl the end of the plannng horzon n order to meet the demand for compressed ar n processng unt n2. As expected, the ntal state of the system nfluences the optmal soluton. To contnue wth, t s observed n Fgure 6 that compressors 4 and 5 nterchange headers n day 30. In that day, there s a sgnfcant ncrease n the demand for compressed ar n processng unt n3 and an mportant decrease n the demand for compressed ar n processng unt n1 (see Fgure 4). These demand fluctuatons n tandem wth the output mass flow rates and the performance of these compressors n day 30 mght have trggered ths nterchange of headers. Accordng to the optmal plan of cleanng tasks dsplayed n Fgure 6, there are sx onlne and two offlne cleanng tasks. More specfcally, there are two onlne cleanng tasks for small compressors 1 and 2, and one onlne cleanng task for large compressors 9 and 11. Offlne cleanng tasks are observed for small compressors 5 and 2 n day 11 and 20, respectvely. In both cases the offlne cleanng tasks opton q1 has been selected. Ths cleanng task opton has the hghest cleanng cost but the shortest duraton n comparson wth the other cleanng task optons. Therefore, t seems that the optmal soluton tends to mze the total number of operatng perods for compressors 5 and 2 whch mght be an evdence of ther hgher energyeffcency per compressed ar unt produced n comparson the other compressors. The compressor wth the most cleanng tasks s compressor 2 that undergoes two onlne and one offlne cleanng tasks n order to restore ts performance and ncrease ts total operatng perod.

20 100% 80% 60% 40% 20% 0% Fgure 7. Case Study 1: Performance level profles for compressors per tme perod. Fgure 7 llustrates the profles of the performance level for all operatng compressors wthn the overall plannng horzon. The performance level of a compressor can be calculated as the devaton from mum performance level of the compressor (.e., 100%) mnus the fracton of the current cumulatve operatng tme and the mum cumulatve operatng tme. All onlne and offlne cleanng tasks can be seen n Fgure 7 as an ncrease n the performance level of the correspondng compressor. For nstance, observe the full recovery of the performance of compressors 5 and 2 n day 11 and 20 where ther assocated offlne cleanng tasks start. Onlne cleanng tasks recover a much smaller part of the compressors performance. Also, notce that because the performance recovery has been modeled as a proportonal functon of the cumulatve operatng tme, the lower the performance level of the compressor (.e., hgher cumulatve operatng tme), the hgher the performance recovery after an onlne cleanng task. For nstance, as t can be clearly seen n Fgure 7, the performance recovery of compressor 1 n day 23 s consderably hgher than that n day 15. In general, t s observed that most cleanng tasks take place n compressors performance levels lower that 50% and especally below 20%. For example, compressor 5 reaches a performance level below 20% n day 10 and ths nctes an offlne cleanng task to start n the next day. A smlar trend s observed for compressor 2. Before day 16, there are two onlne cleanng tasks to partally restore the performance level of compressor 2. In day 16, compressor 2 s at a crtcal low performance level and the opton of performng an addtonal onlne cleanng tasks n next day has been chosen aganst the opton to shut t down. Ths onlne cleanng task partally restores the performance level of ths compressor and allows t to operate for three addtonal tme perods before undergong an offlne cleanng task n day 20 so as to restore ts full performance. The performance level of compressor 2 s very low n day 19 and there are only two avalable opton for the next perod: () to shut t down, or () perform an offlne cleanng task. Notce that there s not avalable the opton of an onlne cleanng task because the mnmum tme between two consecutve onlne cleanng tasks n the same compressor s eght days, but there was an onlne cleanng took place n day 17. Some compressors, such as compressor 7 and 8, shutdown when ther performance levels reach a certan level and reman dle throughout the remanng plannng horzon. At ths pont, t should be emphaszed that havng n hand the compressors performance levels profles, the decson-maker may decde to perform offlne cleanng operatons to compressors 7 and 8 so as to restore ther full performance level, n case they need to operate them n the next plannng horzon. Offlne cleanng tasks could be performed on the dle compressors 3 and 10 as well.

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