A New Connuous-Tme Schedulng Fomulaon fo Connuous Plans unde Vaable Eleccy Cos Pedo M. Caso * Io Hajunkosk and Ignaco E. Gossmann Depaameno de Modelação e Smulação de Pocessos Insuo Naconal de Engenhaa Tecnologa e Inovação 1649-038 Lsboa Pougal ABB Copoae eseach Cene Wallsade S. 59 68526 Ladenbug Gemany Depamen of Chemcal Engneeng Canege Mellon Unvesy Psbugh Pennsylvana 15213. Absac Ths wok addesses he schedulng of connuous plans subjec o enegy consans elaed o me-dependen eleccy pcng and avalably. Dscee and connuous-me fomulaons ae pesened ha can addess hese ssues ogehe wh mulple nemedae due daes. Boh fomulaons ely on he esouce-ask newok pocess epesenaon. The compuaonally pefomance s compaed fo he objecve of oal eleccy mnmzaon wh he esuls favong he dscee-me model due o he moe naual way of handlng such a wde vaey of dscee evens. In pacula can successfully handle poblems of ndusal sze. Neveheless he new connuous-me model s a majo beakhough snce s he fs model of s ype ha s able o effecvely ncopoae me-vaable uly pofles. When compaed o a smple manual schedulng pocedue he poposed schedulng appoaches can lead o poenal eleccy savngs aound 20% by swchng poducon fom peods of hgh o low eleccy cos. * To whom coespondence should be addessed. Tel.: 351-2109643. Fax: 351-2116016. E- mal: pedo.caso@ne.p. 1
1. Movaon Enepses ae cuenly unde pessue o poduce a he lowes possble cos whn connuously changng economc consans 1. To acheve hs goal hey mus acvely look a he bes opeang pacces and opmze hese boh globally and locally. Whn hs oveall goal schedulng plays an mpoan pa. Ths wok s movaed by a eal ndusal poblem ha we canno dsclose fo confdenaly easons. I nvolves he fnal sage of a mulpoduc plan wh eleccy nensve paallel equpmen uns whee schedulng nvolves decdng when each un has o poduce a cean poduc. Mos of he mes hs s made manually by he opeao accodng o heusc ules. The poducs ae hen sen o soage uns whee hey ae soed unl dspachng akes place. Meeng cusome demands on me s val and fo hs eason n some plans no ohe facos ae aken no accoun besdes yng o keep he soage uns full n ode o be able o fulfll he odes. Plan schedulng s dffcul due o he followng facos: lage combnaoal sze asng fom he numbe of equpmen uns poducs and soage uns; vaous opeang and conacual consans; lbealzed eleccy make wh nonanspaen bllng pacces. Due o he nheen complexy he opeao schedulng choces may be fa fom he opmal ones. The mos challengng aspec of plan schedulng s undoubedly he ncopoaon of enegy consans elaed o eleccy pcng and avalably. We consde n hs pape he case whee n he plannng sage conacs ae ageed beween he eleccy supple and he plan whch ofen specfes maxmum levels of powe usage. If eleccy consumpon exceeds hs heshold he plan ncus n sff penales wheeas undepoducon coss he same as planned poducon. Even fo nomal poducon eleccy cos vaes sgnfcanly houghou he day and hs mus be aken no consdeaon whn he modelng famewok. Anohe mpoan aspec concens meeng he sales foecass whch epesen he mnmum amoun of poducs ha has o be avalable n he soage uns dung he me hozon. These ae ypcally consdeed o occu a he end of each day. Accounng fo evens ha occu a pedeemned pons of me fo nsance a a specfc hou of a gven day beng ehe a change n eleccy cos level o he occuence of a demand pon may 2
be elavely easy o exemely complex dependng on he ype of me epesenaon employed. Ths s saghfowad wh a dscee-me appoach wheneve a suffcenly fne me gd can be used. In such cases he peseleced me of some of he gd me pons wll mach exacly hose of he pons of change and so he consans ae ease o enfoce. In conas wh connuous-me he absolue me of all even pons s deemned by he solve and hus s much hade o elae he evens wh he pons of change. Ths pape pesens a new connuous-me fomulaon ha effecvely handles me dependen cos paamees and dscee demand pons. Incopoaon of he fome aspec whn a connuousme fomulaon has no been epoed befoe o he bes of ou knowledge wheeas he consans used o model he lae aspec ae concepually smla o hose used by Maavelas and Gossmann 2. The poposed appoach bulds on he geneal mulpupose fomulaon of Caso e al. 3 whch can addess poblems nvolvng bach and connuous asks effcenly. Indeed fo bach plans a sudy 4 has found o be he bes sngle me gd fomulaon. Neveheless he consans ha ae gven ae only suable fo connuous plans meely because bach asks unlke connuous asks canno be dvded n as many mes as equed. In ohe wods whle he oupu maeal fom a bach ask s poduced enely a s end afe a specfed peod of me he one fom a connuous ask s connuously beng poduced. Thus he execuon of a sngle nsance of a connuous ask s equvalen o he execuon (a he same pocessng ae) of mulple nsances poducng he same oal amoun of maeal n sequence dung he same me peod. The new fomulaon s also moe dealed wh espec o he esouce balances. In ode o goously accoun fo nvenoy consans balances need o be consdeed boh a he sa and end of me slos so ha he effec of esouce consumpon and geneaon occung due o connuous asks s consdeed n full as explaned n Schllng & Paneldes 5. The new fomulaon s bul on a unfed famewok fo pocess epesenaon he esouce-task Newok (TN) of Paneldes 6. Ths means ha he model vaables and consans ae wen s ems of absac enes lke esouces asks and even pons so has a much wde scope ha he sngle sage ndusal case sudy used fo llusave puposes. 3
2. Poblem Defnon Dung he fnal sage of he pocess an nemedae maeal s ansfomed no one of dffeen fnal poducs chaacezed by chemcal composon and pacle sze dsbuon hough he use of eleccy. These ae hen sen o soage uns whee hey wa unl cusome dspach akes place. Ths pocess s llusaed n Fgue 1. I emphaszes ha a pacula soage un may be suable fo jus a subse of he poducs. In fac he poduc allocaed o a soage un nomally neve changes. Snce s saghfowad o ncopoae such consans n he upcomng models s pefeable o consde full connecvy beween uns o ncease he flexbly. In ohe wods we wll be assumng ha evey soage un can handle all poducs (shaed soage) bu only one a a me. Machne 1 Poduc 1 Soage 1 Cusome 1 Inemedae Machne 2 Poduc 2 Soage 2 Cusome 2 Machne M Poduc P Soage S Cusome 3 Fnal pocessng sage Fgue 1. Fnal pocessng sage of ndusal case sudy Typcal plan schedules ae esablshed ove one week so hs wll be he me hozon (H=168 h) assumed fo he emande of he pape. Naually s saghfowad o consde ohe values. Le M epesen he se of machnes P he se of poducs and S he soage uns. Machnes ae chaacezed by: () powe equemens fo each poduc pw pm [MW]; () pocessng aes ρ pm [on/h]. Le DY and H be ses whose elemens ae he days of he week and he hous of he day especvely. Each poduc may have mulple demands ove he week and can occu a any hou of he day d pdyh [on]. Fnally soage uns have known maxmum capaces cap s [on]. 4
The enegy conac sgned by he plan and eleccy povde esablshes a cean pcng polcy. Eleccy cos s ypcally lowe dung he ngh and hghe dung he day. Smlaly o he poduc demands s geneally assumed ha he cos can change evey hou of evey day 19. The maxmum level of oal powe consumpon s n un specfed hough he paamee pwx hdy [MW]. Fo llusaon puposes he enegy polcy gven n Duae e al. s used. I consss of hee enegy levels E wh pces c e of 0.0481 0.0945 and 0.2162 [ /kwh]. The weekly dsbuon s gven n Fgue 2. MO F SA SU 0.25 Eleccy cos ( /kwh) 0.2 0.15 0.1 0.05 0 Hou Fgue 2. Eleccy cos polcy whn a wokng week. The objecve wll be o mnmze he oal enegy cos subjec o consans on esouce avalably ha ncludes pocessng and soage uns and ules (eleccy). 3. esouce-task Newok epesenaon of he Pocess The models poposed n secons 5 and 6 ae bul on he esouce-task Newok o make hem as geneal as possble. The nex sep s hus o denfy he se of pocess esouces and asks whch wll be denfed as a membe of one o moe subypes n ode o bee undesand he sucual paamees geneaon pocess and he dffeen ems n he model consans. Fo ha pupose we use he gudelnes gven n Caso e al. 8 o defne new subypes such ha he specfcs of he ndusal poblem a hand can be ncopoaed. The fnal esul s he TN gven n Fgue 3. Noe ha s helpful o geneae he complee dawng even hough s no absoluely necessay. 5
EL S 1 M 1 Soe_P 1 _S 1 Pocess_P 1 _M 1 ae P 1M1... pw P 1M1 pw P 1MM Pocess_P 1 _M M ae P 1MM P 1 _M Send_o_soage ae=gndng ae Hold_n_soage Duaon=1 me neval Soe_P 1 _S S Send_o_soage ae=gndng ae Hold_n_soage Duaon=1 me neval P 1 _S 1 P 1 _S S emove_p 1 _S 1 Insananeous emove_p 1 _S S Insananeous P 1 M M M... S S... Soe_P P _S S Pocess_P P _M M ae P PMM Send_o_soage ae=gndng ae... pw P PMM pw P PM1 P P _M Hold_n_soage Duaon=1 me neval P P _S S emove_p P _S S Insananeous Soe_P P _S 1 Pocess_P P _M 1 ae P PM1 Send_o_soage ae=gndng ae P P Hold_n_soage Duaon=1 me neval P P _S 1 emove_p P _S 1 Insananeous Equpmen esouces nvolved n mng consans ( TC ) Equpmen esouces no nvolved n mng consans esouces poduced o consumed connuously ( CT ) Fnal poduc esoces ( FP ) Uly esoces ( UT ) Connuous asks (I c ) Soage asks (I s ) Tansfe asks (I ) Connuous neacon Dscee neacon Fgue 3. esouce-task Newok epesenaon of fnal pocessng sage. 3.1. esouces Idenfcaon In an TN esouces ae epesened as ccles. Gong fom lef o gh n Fgue 3 we have he maeal npu o he schedulng poblem ha wll be called aw-maeal and named M (he sole membe of se M ) despe he fac ha s n ealy an nemedae pocess maeal. I s connuously consumed whle a poduc s beng poduced. Whle aleady n he sae of fnal poduc sll has o go hough soage and wa unl dspachng. I s hus equed o denfy poduc locaon whch can be n hee dffeen places: () mmedaely afe he machnes LM ; () conaned n soage LS ; o () nsde he fnal anspoaon vessel FP. In he fs suaon hee s no need o dffeenae he machne whee he poduc s poduced so a oal of P esouces ae 6
geneaed fom P 1 _M o P P _M whee he M sands fo afe he machnes. In conas n he second possble place we have o dsngush n whch soage un (o goup of uns wokng ogehe) he poduc s locaed snce hey can have dffeen capaces. The esul s he geneaon of addonal P S esouces. esouces connuously poduced and/o consumed ae he elemens of se CT = M LM LS. These ae coloed n dak gey o faclae denfcaon. The fnal locaon gves se o he fnal poduc esouces P 1 o P P whch ae flled n black. And hs ends he maeal esouces. Movng on o he equpmen esouces se EQ we have pocessng equpmen soage equpmen and evenually anspoaon equpmen. They ae howeve handled dffeenly so subse TC s needed. Ths ncludes solely he fome ype whch ae geneally he mos mpoan esouces n a schedulng poblem. They wll have a song mpac on he fnal fom of he connuous-me gd due o he pacpaon n he mng consans. In hs specfc case he machnes ae he membes of TC and ae flled n shadowed whe unlke soage ubs whch have a lgh gey backgound. Fnally hee ae he ules UT he subjec of hs pape. Connuous-me fomulaons 9-12 have so fa consdeed a maxmum uly avalably ha emans consan houghou he me hozon. Fo seam and coolng wae we ae dealng wh flowae values ypcally gven n [on/h] whle fo eleccy we deal wh powe [MW]. Howeve as dscussed eale (secon 2) we wll geneally assume a uly avalably pofle whee n he lm he specfed maxmum avalably can change evey hou of he day dung he ene week. 3.2. Tasks Idenfcaon Tasks ae epesened as ecangles n an TN and mus be chaacezed fully n ems of s esouces unlke n a Sae-Task Newok (STN) epesenaon 91013. Moe specfcally he STN epesenaon of hs pocess would show jus P pocessng asks snce hee s no need o explcly say n whch un he poduc s gong o be execued. In conas n he TN of Fgue 3 he pocessng asks mus also be dsaggegaed no all he possble machnes.
In Fgue 3 hee ae hee ypes of asks. The pocessng asks flled n whe ae pefomed connuously a a known ae se I c. The execuon of a pocessng ask nvolves fou esouces he ones lnked by aows o he asks. In ode o dffeenae beween connuous and dscee neacon sold and dashed lnes ae especvely used. Some lnes wll have aows on boh decons denong consumpon of he esouce a he sa of he ask and poducon a s end. As an example Pocess_P 1 _M 1 connuously consumes M whle poducng P 1 _M. I also consumes powe fom eleccy EL and machne M 1 a s sa only o egeneae M 1 a s end. Soage asks come nex (I s n lgh gey). They ae moe complex han hose descbed n Caso e al. 8 snce now he soage uns ae shaed ahe han dedcaed o a sngle maeal esouce. These soage asks ae hybd conssng of wo pas: (a) Send_o_soage connuous; (b) Hold_n_soage bach. Any pa wll gge he consumpon of he coespondng desnaon soage un (e.g. S 1 ) such ha s made unavalable fo ohe poducs dung ha me. Boh can be acve smulaneously n ode fo a poduc o ene a paally flled soage un wh he same qualy. The connuous pa consumes a poduc-afe-machnes esouce (e.g. P 1 M) whle s beng poduced by Pocess_P 1 _M 1 Pocess_P 1 _M M and poduces a poduc-a-soage esouce (e.g. P 1 _S 1 ). The bach pa can be vewed as empoaly hdng he maeal esouce (e.g. P 1 _S 1 ) fo one me neval. The las subse of asks ansfe he poducs fom soage o cusomes I (dak gey). They ae assumed nsananeous meanng ha hey las much shoe when compaed o he pocessng asks. They could have also been easly defned as bach o connuous. We ae also assumng ha no equpmen esouce s nvolved whch s agan no n any way lmng. 3.2.1. Fuhe equemens fo Connuous-me Model Ideally he TN epesenaon of he pocess should be ndependen on he model ha s employed o solve he schedulng poblem. Howeve hs s no always he case. In he poblem a hand he eleccy cos paamee s me dependen and we need o make sue ha he coec paamee s consdeed. Whle n dscee-me hs s no paculaly dffcul snce he enegy level can be decly lnked o a subse of he me nevals n connuous-me he lnk s a he ask level. 8
In ohe wods we need o dsngush whehe he ask s execued a e 1 e 2 o e E wh subse I e c ndcang he pocessng asks pocessed a enegy level e. Fuhe dsaggegaon of he pocessng asks s hus equed (Fgue 4) leadng o a oal P M E of such asks. Neveheless he exac same esouces ae nvolved so Fgue 3 s sll vey much useful. Pocess_P 1 _M 1 _E 1 ae P 1M1 Pocess_P 1 _M 1 ae P 1M1... Pocess_P 1 _M 1 _E E ae P 1M1 Fgue 4. Due o me dependen eleccy coss pocessng asks need o be fuhe dsaggegaed fo he connuous-me model. 4. TN-based Model Enes Ths secon focus on he model enes ha ae used egadless of he me epesenaon employed by he esouce-task Newok based model. In he case of he paamees hs does no necessaly mean ha hey have he same doman o ake he same values. Addonal vaables fo he connuous-me model ae explaned followng he descpon of he undelyng me gd n secon 5.1. 4.1. Vaables Tasks ae geneally chaacezed by exen vaables one bnay N and one connuous se ξ. The fome denfy he sa of ask a me pon whle he lae nomally gve he amoun handled by he ask. We ae mplcly assumng ha he asks ae ehe nsananeous (sa and end a ) o connuous whch can be made o las a sngle me neval whou loss of genealy 3. In pacce hs means ha a few consecuve nsances of he ask wll have o be pefomed o mee lage demands. Bach asks ha las one me neval ndependenly of s duaon ae also ncluded. In fac he soage asks descbed n secon 3.2 ae hybd conssng of a connuous and a bach * pa ha can occu smulaneously so anohe se of connuous exen vaables ξ s equed fo 9
he full chaacezaon. I s also mpoan o hghlgh ha no equpmen esouce s nvolved n he ansfe asks. Thus bnay vaables ae no necessay. The beakhough of he TN 6 comes fom he unfed eamen of esouces. The mahemacal fomulaons keep ack of esouce avalably ove me hough he excess esouce vaables. The wod excess s cucal wh non-zeo values ndcang ha hee s sll some amoun of esouce avalable a even pon besde he amoun ha s allocaed o bach asks ha wll sa o nsananeous asks ha wll be execued a. Equpmen esouces ae mosly eaed as ndvduals so havng =0 ndcaes ha hee s one ask sang a ha uses un. If one wans o foce such equpmen o be dle dung hen mus be avalable n excess and he consan o use s =1. In he pesence of connuous asks anohe se of excess esouce vaables 5 end s somemes equed. These gve he excess amoun of esouce mmedaely befoe he end of neval. In hs pacula poblem hee wee wo easons fo he use: () he need o occupy a soage un gh fom he sa of a pocessng ask n ode o ensue ha hee s one avalable; () he need o know he amoun n soage o guaanee ha he maxmum capacy s no exceeded. Evens ggeed by he sa o end of asks wll affec he wo ypes of excess esouce vaables dffeenly (see secon 5.3). esouce avalably a he begnnng of he me hozon 0 s ofen known fo all esouces and so s nomally defned as a paamee. The value fo equpmen uns s gven n Eq. (1). Neveheless he scope of he mahemacal fomulaon s wde f such an eny s a vaable. Ths s paculaly useful f one wans o solve he smulaneous desgn and schedulng poblem 8 whee one of he goals s o selec he mos appopae uns fo he plan. In hs pacula case and even hough he aw-maeal equemens can easly be deemned fom he poduc demands ha ae fxed he nal avalably of he aw-maeal esouce M wll be a vaable n he model. I wll be he only npu of aw-maeal o he sysem ( s saghfowad o consde ohewse) and so wll be equal o he oal amoun equed ove he full me hozon. Fo all ohe esouces wll be se o zeo (Eq. 2). 10
EQ 0 = 1 (1) EQ M 0 = 0 (2) 4.2. Sucual Paamees The TN epesenaon of he pocess s bough no he model by he sucual paamees. These can be slghly dffeen dependng on whehe we ae usng a dscee o connuous-me fomulaon 14. Howeve n poblems wh asks lasng a mos a sngle me neval he dsceeme fomulaon can use he exac same paamees as fo he connuous-me 1 model. Sucual paamees gve ehe he oal esouce consumpon/poducon o he popoon elave o he amoun handled by he ask. Thee ae fve ses of sucual paamees coespondng o dffeen ypes of neacon wh he exen vaables. Dscee neacons occu ehe a he sa o end of he ask whle a connuous neacon akes place houghou he execuon of he ask. Paamees µ and μ ae he dscee neacons assocaed wh vaables N especvely fo he sa and end of he ask. They ae used wheneve he amoun of esouce consumed/poduced s ndependen on he amoun handled by he ask as fo equpmen uns. Fo maeal esouces one nomally eles on he dscee neacons assocaed wh he connuous exen vaables ξ.e. paamees ν and ν. Fnally paamees * λ hold he connuous neacons assocaed wh he connuous exen vaables ehe ξ o ξ. Whle hee can be many paamees n oal he lage majoy wll be equal o zeo and hose ha ae no wll mosly have a value of ehe 1 o -1. Afe some pacce devng hem fom he TN becomes a elavely easy ask. As an example fo he case sudy a hand we have: µ M1 =-1; μ 1; µ EL =-pw P1M1 ; λ M =-1; λ P1_M =1 - =Pocess_P 1 _M 1 (3) M1 = µ S1 =-1; μ 1; ν P1_S1 =-1; ν 1; λ P1_M =-1; λ P1_S1 =1 =Soe_P 1 _S 1 (4) S1 = P1 _ S1 = ν P1_S1 =-1; ν 1 =emove_p 1 _S 1 (5) P1 = 11
4.3. Ohe Paamees Whle sucual paamees lnk asks wh esouces hee ae also paamees ha ae ehe elaed o asks o o esouces. These have aleady been gven fo he pocess enes such as machnes and poducs so now we jus need o make he coespondence o he absac enes of he TN befoe assgnng he appopae values. An algohm was devsed fo hs pupose ha smply geneaes he asks and esouces fom he se of poducs soage uns machnes and eleccy levels whle defnng funcons (f x ) o keep ack of he coespondence. These may also depend on whehe he poblem s o be solved wh a connuous o dscee-me fomulaon (c o d supescp especvely). I c = ; I e c = =1; p=1 m=1 e=1 I c =I c { }; I ec =I ec { } ρ max =ρ pm =1 e=e1 e=e? YES m=m? YES p=p? YES END NO NO NO m=m1 p=p1 Fgue 5. Pa of he algohm esponsble fo geneaon of ses I c c and I e. Fgue 5 llusaes he pa of he algohm esponsble fo geneang he connuous asks (I c ) and he elemens of se I e c. As an example fo P=2 M=1 and E=3 we ge I c ={ 1 6 } and 12
c I e 1 ={ 1 4 } c c I e2 ={ 2 5 } and I e1 ={ 3 6 }. Along he way he asks maxmum pocessng aes ae made equal o he pocessng aes of he coespondng poduc-machne pa ρ pm. Ths can be genecally epesened by eq 6. max ρ c = f ρ p m) I (6) 1( max ρ The algohm also geneaes he dffeen demand pons dtd and he me peods ptp e fo eleccy levels e ogehe wh he fxed fx d and sang and endng mes lb ep and ub ep [h]. Boh he pofle of cos and maxmum powe avalably nfluence he numbe of me peods. Takng as an example he pofle of Fgue 2 and assumng consan maxmum powe avalably hee ae a oal of 8 1 and 10 peods fo he cheap medum and expensve cos levels especvely. Fo he dscee-me fomulaon he eleccy cos levels c e can be easly assocaed o specfc me nevals. I was assumed ha he cos can change evey hou so hee s no poblem f 1 s an nege mulple of he duaon of evey me neval δ. Fo he ohe cases a weghed mean can be used even hough he oal cos wll no be exac wheneve hee ae paally execued asks (see lae on secon 6.1). Equaon () emphaszes ha he cos fo neval ce s a funcon of δ. ce d = f ( c δ ) T () 2 e T end max mn Excess esouce vaables may be subjec o gven uppe ( ) and lowe bounds ( end mn ). The lae ae se o zeo whle one should also do he same fo he fome wheneve possble. In hs way soluon degeneacy wll be educed by ensung ha asks ae execued sequenally whou wang peods n beween. Fo nsance dong hs fo poducs locaed mmedaely afe he machnes wll foce sendng o soage he exac same amoun ha s pocessed. On he ohe hand poducs should be allowed o exs n soage vessels ( LS ) up o an amoun equal o he maxmum capacy Eq. (8). Neveheless an excess of such esouces s only empoay ehe because hey wll be ansfeed o he clen o because hey need o be hdden empoaly so ha ohe poducs do no occupy he same soage vessel. Ths s accomplshed hough Eq. (9). We ensue ha he sysem eceves he mnmum amoun of aw-maeal f hee s max none avalable a he las even pon (Eq. 10). 13
end max CT M 0 f3( cap ) \ T T = s LS (8) max CT FP M = 0 \ T (9) = 0 max M T (10) The sysem can exchange maeal wh s suoundngs a any me. We know exacly when a cean quany of a esouce becomes avalable o o s emoved fom he sysem. In a dsceeme model hee s a dec lnk beween eal me and he me pons of he gd so paamees ou Π and Π n can easly be deved see Eqs. (11-12). In hs case hey wll be used o povde he machnes wh he maxmum level of powe consumpon and o mee he poduc demands. When usng connuous-me fomulaons howeve he coespondence beween he dscee npus/oupus and he even pons of he me gd wll be accomplshed hough he use of wo addonal ses of bnay vaables. Thus he me ndex of he npu paamees needs o be changed o me peod p of enegy level e whle ha of he oupu paamees s changed o demand pon d (Eqs. 13-14). d UT Π = f pwx ) T (11) n 4 ( h dy T ou d FP Π = f d p dy h ) T 1 (12) 5 ( n c UT Π p e = f4 ( pwxh dy) ptp e E (13) ou c FP Π d = f5 ( d p dy h ) d TD (14) 5. New Connuous-Tme Fomulaon (CT) The new connuous-me fomulaon fo sho-em schedulng of connuous mulpoduc plans unde vaable uly avalably coss/pofles and mulple nemedae due daes s gven below. The man focus wll be on ssues elaed o me modelng and he esouce balances ove me. The model enes ae summazed n he Nomenclaue secon wh all connuous vaables beng nonnegave. 14
5.1. Tme epesenaon The poposed connuous-me fomulaon uses a sngle me gd o keep ack of evens akng place see Fgue 6. I uses T even pons also named global me pons 15 ha can be placed anywhee beween he ogn and end (H) of he me hozon. Tadonally 38-11 asks sang a even pon have been assumed o sa a he absolue me deemned fo ha even pon T. Fo bach asks wh duaon shoe han T 1 -T s assumed ha he oupu esouces say n he coespondng equpmen un fo he emanng me. Wh connuous asks s assumed ha hey las exacly he duaon of he me neval whle beng pocessed a a ae lowe han he pedefned maxmum ae. In ealy howeve he asks have oal feedom o sa anywhee povded ha hey end befoe T 1. Gménez e al. 16 hghlgh ha he numbe of even pons needed o epesen a soluon can be educed f asks ae no equed o sa (end) exacly a a me pon whle pesenng a new connuous-me fomulaon fo he sho-em schedulng of mulpupose bach plans. The cuen case sudy s a pefec example fo he advanages of such an appoach as wll be explaned nex. Slo 1 Slo 2 Slo T-2 Slo T-1 2 3 T-2 T-1 1 T 0 H Fgue 6. Connuous-me epesenaon. Consde a smple example nvolvng he execuon of a sngle pocessng ask a slo/neval see Fgue. The neval boundaes T and T 1 can esul fom wo consecuve demand pons n cases whee he duaon of he ask s lowe han he duaon of he lowes cos peod locaed nsde he neval (shown n geen). Vaable Ts wll gve he sang me of he ask execued dung neval (o he eales sang me amongs all asks execued). Noe ha no fuhe even pons ae equed. Hgh cos Low cos Pocessngask Medum cos T ub e2p-1 ub e3p Ts ub e1p T 1 lb e 3p lb e 1p lb e 2p Fgue. Connuous asks execued a neval do no necessaly sa a even pon. 15
In he moe geneal case he amoun o poduce wll eque an exenson lage han he duaon of he cheapes me peod. In such case he pocessng ask wll be dvded by he solve no as many mes as he numbe of dffeen enegy cos levels whch may nclude consecuve me peods of he same enegy level. Each such dvson wll eque an addonal even pon as can be seen n Fgue 8. Medum Hgh cos Low cos Medum cos P. Task Pocessngask Pocessngask T ub e2p-1 T 1 ub e3p T 2 ub e1p T 3 Ts lb e 3p Ts 1 lb e 1p Ts 2 lb e 2p Fgue 8. Mulple ask nsances mgh be equed o mee he demands one fo evey acve enegy cos level. 5.2. Tmng Consans The fundamenal mng consan fo a sngle me gd fomulaon 3 saes ha he dffeence n me beween wo consecuve even pons mus be geae han he duaon of he ask akng place. Equaon (15) s wen fo all equpmen esouces nvolved n he mng consans. The summaon s used o educe he negaly gap 3 and mplcly assumes ha hee can only be one ask execued n such equpmen un a a cean me. Ths s ensued by he nal esouce avalably and he excess esouce balances (see Eqs. (1) and (2)). μ ξ TC T 1 T T T max c ρ I (15) A smla consan can be used o guaanee ha he asks ae fully execued whn me neval. Equaon (16) ogehe wh Eq. (1) also sasfy Eq. (15). Howeve compuaonal sudes have shown ha s bee o keep Eq. (15). μ ξ TC T 1 Ts T T max c ρ I (16) Ts T T T (1) We now defne a new bnay vaable Y pe whch akes he value of one f dung neval asks ae pocessed whn me peod p of enegy level e. Smlaly we defne bnay vaable ou Y d o denfy whehe o no even pon coesponds o demand pon d. I can be assumed whou loss 16
of genealy ha hee s: () one acve me peod of an enegy level dung ; () one even pon assocaed o due dae d eqs 18-19. p e ee ptpe Y = 1 T T (18) d T 1 ou Y = 1 d TD (19) I me neval s locaed whn me peod p of enegy level e hen he sang me of asks mus be geae han he me peod lowe bound Eq. (20). Lkewse hey mus end befoe s uppe bounday (see Fgue 8). Equaon (21) s a bg-m consan ha s only acve f hee s a connuous ask beng execued ha belongs o enegy level e. Ths consan s he one dffeenang beween enegy levels fo equvalen asks (efe o Fgue 4) so ha he pope eleccy cos s accouned fo n he objecve funcon see Eq. (33). Ts ee ptp lbe py p e T T e (20) Ts μ ξ ube py p e H ( 1 μ N ) max c ρ c I e ptpe I e TC T T e E (21) If even pon coesponds o demand pon d hen he me values mus mach as expessed n Eqs. (22)-(23). ou T fxdy d T 1 (22) dtd ou ou T fxdy d H 1 Y d ) T 1 (23) dtd ( dtd Fnally hee ae geneal bounds on he mng vaables. Accodng o Fgue 6 T [0H] so Eqs. (-25) apply. Equaon (25) enfoces ha no ask sas a he las even pon. Snce he model assumes ha asks can sa afe T he me of he fs even pon can be se o zeo whou loss of genealy (Eq. 26). T H T () Ts H T T (25) T 1 = 0 (26) 1
5.3. Excess esouce Balances esouce avalably ove he me gd s managed by he excess esouce balances. These ae mulpeod maeal balance expessons n whch he excess amoun a even pon s equal o ha a he pevous even pon (-1) adjused by he amouns dsceely o connuously poduced/consumed by all asks sang o endng a. Fgue 9 llusaes he conbuons o he values of he vaables and suppos he explanaon of Eqs. (2-28). In Eq. (2) he nal esouce avalably (fs em on he HS) s only o be consdeed a he fs even pon =1. Fom Eqs. (1-2) we know ha wll be dffeen fom zeo only fo a subse of he esouces 1 fo EQ and a value o be deemned by he solve 0 fo M. Fo he emanng pons he value fom he pevous even pon s used nsead. Fo connuous esouces CT he ones nvolved n Eq. (28) eceves he conbuon of end 1 whch gves he excess amoun mmedaely befoe he end of neval -1 (even pon ). Fo he ohe esouces bu he ules one should use he value a he begnnng of he pevous neval -1. Ths hd em on he HS s no used fo UT snce we wan o peven esouce avalably o popagae fom one even pon o he nex. Ths s because hee wll be npus fom he exeo a evey even pon (sxh em on he HS). EQ M 0 1... UT CT CT Π n -1 npu maeal -pw I 1-1 1 I s 1-1 1 I c -1 I c oupu oupu npu oupu npu EQ -1 I c I s 1 EQ CT -Π ou npu FP 1-1 -1 end Fgue 9. Evens affecng he value of he excess esouce vaables. 18
19 T Y Y v N v N TD d ou d ou d T E e TP p e p n e p I I T end FP UT e UT CT CT Π Π = = ) ( ) ( ) ( 1 1 ) ( 1 1 1 0 ξ μ ξ μ (2) ) ( * T T v CT I I end s c = ξ λ ξ ξ λ (28) In Eq. (2) he fouh and ffh ems deal wh mos of he dscee neacons of asks wh esouces. The excess amoun a even pon deceases by asks sang a and nceases by asks sang a -1 (endng a ) o (f hey ae nsananeous I ). The dscee neacon a he end of soage asks (I s ) affecs excess vaable end ahe han 1 see las em on he HS of Eq. (28). The pupose s o ge he amoun n soage fo evey u so ha he maxmum capacy consan can be enfoced. Vaable end also deals wh he connuous neacons of connuous (I c ) and soage asks. Fnally he las em n Eq. (2) emoves poduc esouces fom he sysem wheneve he me of even pon s equal o a demand pon d. The excess esouce vaables should le whn gven uppe and lowe bounds. The geneal consans ae gven n Eqs. (29-30) whle he values fo hs case sudy ae dscussed n secon 4.3 and gven n Eqs. (8-10). T max mn (29) max mn T T end end end (30) 5.4. Ohe Consans The emanng se of consans ensues ha he values of he connuous exen vaables ae se o zeo wheneve he ask s no execued.e. when he bnay exen vaable s equal o zeo. The uppe bound n Eq. (31) U can be gven by he poduc of he me hozon mes he sum of he maxmum pocessng ae on evey machne see Eq. (32). * T T I I N U s c I s ξ ξ (31) = M m m p P p H U max ρ (32)
5.5. Objecve Funcon The mahemacal fomulaon s compleed wh he objecve funcon. Equaon (33) maxmzes he oal eleccy cos [k ]. Ths s gven by he sum ove all eleccy levels e me nevals and asks of he poduc of eleccy cos c e [ /kwh] powe consumpon [MW] and duaon of he ask [h]. mn ee Ic UT e T T ξ ce ( μ ) (33) max ρ 6. Dscee-Tme fomulaon (DT) Exenal neacons wh he sysem a specfc pons n me such as dffeen eleccy coss and due daes ae handled much moe easly by a dscee-me fomulaon. Ths of couse povded ha he locaon of some of he gd s me pons s whn accepable accuacy of he exenal evens mng. Afe ecallng a few mpoan aspecs elaed o he epesenaon of me he model consans ae gven. These ae analogous o he non-mng consans of he connuousme fomulaon whch s no supsng gven he fac ha he dscee-me fomulaon s n fac a sub-model of he moe geneal connuous me fomulaon 18. 6.1. Tme epesenaon The sngle me gd employed by he dscee-me fomulaon has equal lengh me nevals of duaon δ see Fgue 10. As a consequence he absolue me of all even pons s known a po. Fxed lengh me nevals have mpoan consequences fo boh bach and connuous asks. Fo he fome he duaon mus ofen be ounded o a mulple of he neval lengh leadng o he consdeaon of an appoxmaed veson of he eal poblem. Fo connuous asks he exen can be made o vay whn [0 δ ρ max ] o ensue ha all possble demands can be me. Howeve hee may be nevals whee he ask s beng pefomed below s maxmum pocessng ae whch n vey consaned poblems may compomse opmaly o even feasbly snce no ohe ask can ake advanage of he emanng capacy of he un. Ths can be llusaed wh a smple example. 20
Slo1 Slo 2 Slo 3 Slo T-2 Slo T-1 δ 2 3 4 T-2 T-1 1 T 0 H Fgue 10. Dscee-me epesenaon. Demand 20 Demand 35 Demand 35 1 2 3 4 5 6 8 9 10 11 0 12 h 10 10 10 5 5 Idle 10 10 10 5 10 10 Subopmal 5 Idle 10 10 10 5 5 10 10 10 10 10 Opmal Fgue 11. Lmaon of dscee-me fomulaon when handlng connuous asks. In Fgue 11 can be seen ha fo δ=1 h we need o occupy he pocessng un fo he fs egh hous of he me hozon o mee he blue and ed poducs demands. The poblem s ha he las one s a medum-cos neval. Even hough he objecve funcon accouns only fo he half an hou ha he un s acve opmaly s compomsed snce he demands of boh poducs can be me enely n he low-cos peod (fs seven hous). Neveheless such soluon canno be obaned snce he excess esouce balances allow poducon of a mos one poduc on each me neval. Howeve can be acheved wh δ=0.5 h (o wh a connuous-me fomulaon) fo whch he nevals lengh wll mach exacly he me equed o poduce 5 of poduc. Thus he ed poduc can sa o be poduced mmedaely afe he blue ends n ode o fnsh befoe he medum-cos me peod sas. Moe mpoanly would sll be possble o poduc anohe 10 on befoe he due dae n case of hghe demands. 6.2. Model Consans The excess esouce balances ae gven by Eqs. (28) and (34). The lae dffes fom Eq. (2) n he ems nvolvng paamees Π n and Π ou. Now he doman s exacly equal o ha of he equaon (ecall he las paagaph of secon 4.3) so hee s no need o employ addonal bnay vaables. 21
22 T v N v N ou T n I I T end FP UT UT CT CT Π Π = = ) ( 1 1 ) ( 1 1 1 0 ξ μ ξ μ (34) Snce hee ae no mng consans he capacy consans need o ensue ha he maxmum amoun of maeal handled by connuous asks does no exceed he maxmum pocessng ae mes he duaon of he me neval. Fo soage asks Eq. (35) s dencal o Eq. (31). ) ( max * T T I I N U s c I I I c s s ρ δ ξ ξ (35) 6.3. Objecve Funcon The dffeences n compason o he objecve funcon of he connuous-me fomulaon gven n Eq. (33) esul fom he eleccy cos paamees whch have a dffeen doman. Because of hs Eq. (36) has one less summaon. c UT I T T ce max ) ( mn ρ ξ μ (36). Compuaonal Sudes The pefomance of he models s llusaed hough he soluon of 10 es cases wh daa geneaed andomly fom a eal ndusal poblem. In all of hem he poduc demands lead o plan opeaon below he maxmum capacy. Due o he dffeences n eleccy cos among he enegy levels assgnng poducon o he lowe levels wll have he bgges mpac on he oal cos. As we appoach full plan capacy all he enegy levels become acve and he solve swches focus o fndng he bes combnaon of poduc-machne assgnmen accodng o he pocessng aes and powe needs whch ae poduc dependen. Wh he excepon of EX5 he machnes ae occuped 41 o 56 % of he me. Decly elaed o he complexy of he poblem s he numbe of poducs machnes and soage uns so he poblems wll be manly chaacezed by (P M S). The mahemacal fomulaons gve se o mxed-nege lnea pogammng (MILP) poblems. They wee mplemened n GAMS 22.8 ogehe wh he algohm ha conves he poblem daa no he TN foma. All poblems wee solved by CPLEX 11.1 wh defaul opons up o a elave
opmaly oleance=10-6 unless ohewse saed. The hadwae conssed on a lapop wh an Inel Coe2 Duo T9300 pocesso unnng a 2.5 GHz 4 GB of AM and unnng Wndows Vsa Enepse..1. Illusave Example To llusae he capables of he new connuous-me fomulaon we sa wh a vey smple example nvolvng wo poducs one machne and one soage un (211). The machne powe equemens s n hs case poduc ndependen and equal o pw M1 = 5 [MW] and hee ae no consans on maxmum powe consumpon. The pocessng aes ae ρ P1M1 = 80 ρ P2M1 =0 [on/h] soage capacy s cap S1 =1200 [on] and he poduc demands can be found n Table 1. Table 1. Poduc Demands a he End of he Day fo Examples EX1-EX3 [on] MO ( h) TU (48 h) WE (2 h) TH (96 h) F (120 h) SA (144 h) SU (168 h) P1 1000-500 - - 1000 1200 P2-400 - - 1200 - - The opmal soluon fo EX1 coesponds o a oal eleccy cos of 18625. The esulng schedule and soage pofles ae shown n Fgue 12 whee he numbes nsde he boxes ndcae he ask lengh [h]. I eques a oal of T=11 even pons whch s he mnmum numbe ha ensues feasbly. No mpovemen n cos s obseved fo T=12. Such behavo s no common n connuous-me fomulaons 3411 whee ypcally he objecve funcon nceases wh T a leas a couple of mes. Theefoe s an ndcaon ha he poblem s hghly consaned whch s a dec consequence of: () lage vaably of he eleccy cos leadng o a oal of 35 me peods wh consan pce; () he asks no beng allowed o coss me peods. Thus some demands can only be me by execung mulple nsances of a pacula ask. Moe specfcally he Monday (MO) and Sauday (SA) demands of P1 need wo asks whle he Fday (F) demand of P2 eques hee. In Fgue 12 8 ou of 10 nsances ae execued n low-cos peods (geen egon) wh he emanng beng n medum-cos levels (yellow egon). Ideally all should be execued n he geen egon bu ha would only be possble fo opeaon well below he maxmum capacy of he plan. Powe shoages may foce asks o swch o a hghe cos level. In ode o show ha he model can 23
cope wh consans of hs ype EX2 was solved. I s uses he same daa of EX1 bu now he maxmum powe consumpon dung he fs seven hous of Tuesday and Thusday (pwx 0-6TU ; pwx 0-6TH ) s se o zeo. The opmal soluon fo EX2 s shown n Fgue 13 and coesponds o a oal cos of 2155. When compaed o EX1 anslaes no a 15.8% ncease n cos whch s vey sgnfcan consdeng such small changes n daa. Such a schedule s obvously a feasble soluon o EX1 whch s n many aspecs smla o he pevous one. Hence also seves o llusae he mpac ha advanced schedulng ools lke he models poposed n hs pape can have on plan pofably. We wll eun o hs dscusson n secon.3. M1 5.5 5.14 6.25 3.143 9 3.5 15 S1 1 1 9 15 1200 0 48 2 96 120 144 168 S1 Tme (h) 1000 800 600 400 P1 200 P2 0 (on) 0 48 2 96 120 144 168 Tme (h) Fgue 12. Opmal soluon fo EX1. M1 5.5 5.14 6.25.143 3 9 3.5 15 S1 1 13.143 5.85 9 15 1200 0 48 2 96 120 144 168 S1 Tme (h) 1000 800 600 400 P1 200 P2 0 (on) 0 48 2 96 120 144 168 Tme (h) Fgue 13. Opmal soluon fo EX2.
M1 6.60 1.85 5.143 6.5 6 15 S1 1.85 22.143 41 S2 1 1.85 22.143 6.5 1.5 1200 0 48 2 96 120 144 168 S1 P1 Tme (h) 1000 P2 800 600 400 200 (on) 0 1200 0 48 2 96 120 144 168 Tme (h) S2 1000 800 600 400 (on) 200 0 0 48 2 96 120 144 168 Tme (h) P1 P2 Fgue 14. Opmal soluon fo EX3. The amoun of avalable soage s ofen lmng. The pevous examples feaued a shaed soage un ha could only handle one poduc a a me. Ths made mpossble o mee demands of dffeen poducs n he same day whch explans he foma of such daa. Moe mpoanly poducon of p could only sa afe he oal amoun poduced of p had been dspached. In ode o allow fo moe flexbly EX3 consdes wo dencal soage uns and no maxmum powe consans. Wh he addonal soage un he oal cos can be educed o 18153 (2.5%). The opmal soluon s gven n Fgue 14. I bascally moves all he poducon of P2 (n ed) o low-cos peods whle moe P1 (blue) ges deced o medum-cos peods. Whle one mgh be emped o explan hs behavo based on he dffeen pocessng aes hee exss a degeneae soluon ha feaues P2 n boh low- and medum-cos peods. On Monday he poducon of P1 has nceased fom 1000 o 1089 on so slo S2 wll eman paally flled unl he end of Wednesday. S2 also holds P2 on Fday even hough could all be allocaed o S1. 25
.2. Compuaonal Pefomance The compuaonal esuls of he 10 examples ae gven n Table 2. The example poblems can be dvded no wo levels of complexy. EX1-5 can be addessed by he connuous-me model (CT) whle fo he ohes one has o ely on he dscee-me fomulaon (DT). Ths s he fs majo esul: he connuous-me fomulaon s lmed o small poblems. Even fo EX5 whch feaues hee poducs wo machnes and wo soage uns we had o specfy low poduc demands (leadng o a 33% capacy) o ensue ha mos asks could f no a sngle peod of consan eleccy cos. In hs way a good soluon could sll be found wh a elavely small numbe of even pons. Howeve ook aleady moe han 1 hou o pove opmaly fo T=11 ( 26911) see Table 2 a value ha s sll 0.5% above he one fom DT. Fo lage poblems CT s nacable due o he followng facs: () he compuaonal effo s songly dependen on T wh expeence ellng us ha we ge ypcally a one ode of magnude ncease fo a sngle ncease n T; () s no saghfowad o fnd a numbe ha ensues feasbly; () may be even dffcul fo he solve o fnd ou ha he value of T s nsuffcen.e. fndng ha he poblem s nfeasble. The las examples have moe o less he complexy one expecs o fnd n a eal suaon. The encouagng esul s ha he dscee-me fomulaon wh 1-hou nevals can fnd vey good soluons o he poblem n sho compuaonal me. In fac fom he esuls n Table 2 one can see ha whn up o 5 mnues he opmal/bes soluons euned ae whn a elave opmaly gap of 0.8 %. Thus DT povdes an effcen decson-makng ool fo hs poblem. The gaps could be educed mos of he mes by up o one hou of compuaonal me essenally due o he fndng of bee soluons snce he value of he elaxed poblem emaned vually unchanged. Ths s an ndcaon of a hgh degee of degeneacy (moe on hs n secon.4). The esuls fom Table 2 also show ha he dscee-me fomulaon s sgnfcanly ghe han he connuous-me one. In EX1-4 and EX he negaly gap s even zeo and wh he excepon of EX8 dd no exceed 0.2%. The subsanally lage negaly gap fo he lae (3.3%) s pobably due o he use of hashe consans on maxmum powe avalably ha pevened he 26
machnes o opeae smulaneously n a sgnfcan pa of he week. Neveheless CT sll exhbed a easonable negaly gap (maxmum=23% fo EX2 mnmum=4.8% fo EX5 aveage=11.6%). Fnally was supsng o fnd ou ha he numbe of bnay vaables of DT was of he same ode of magnude as ha of CT n spe of he lage dffeence beween he T numbes used n he wo me gds. ecall ha T s one of he ndces of he bnay exen vaables N. Thee ae wo easons fo hs: (a) hee ae hee mes as many asks of he connuous ype (I c ) fo CT due o he enegy levels dsaggegaon see secon 3.2.1; bu moe mpoanly (b) CT employs wo ou addonal ses of bnay vaables Y and Y pe wh he lae feaung hee ndces and beng he d majo conbuo wheneve hee s a lage vaably on eleccy cos. Takng EX4 (T=12) as an example such vaables ae esponsble fo 69% of he oal numbe. Table 2. Compuaonal esuls (P Model T bnay sngle consans MIP MIP CPU nodes M S) vaables vaables ( ) ( ) s EX1 (211) DT 169 62 4552 3203 18625 18625 0.32 0 CT 11 490 9 34 16620 18625 8.53 4252 CT 12 539 85 380 16620 18625 9.8 5442 EX2 (211) DT 169 62 4552 3203 2155 2155 0.29 0 CT 11 490 9 34 16620 2155 26 2091 CT 12 539 85 380 16620 2155 21.6 1544 EX3 (212) DT 169 1008 643 4382 18153 18153 0.40 0 CT 11 510 934 420 16620 18153 23 69046 CT 12 561 1025 460 16620 18153 132 46962 EX4 (212) DT 169 1008 643 4382 2115 2115 0.52 10 CT 11 510 934 420 18896 21349 18 46512 CT 12 561 1025 460 18896 2115 140 23314 EX5 (322) DT 169 2016 1084 639 2638 2680 3600 a 1332 CT 9 528 1089 562 25625 2222.18 3989 CT 10 594 1221 629 25625 2008 369 138426 CT 11 660 1353 696 25625 26911 4131 1295540 EX6 (323) DT 169 2520 13986 8423 43250 43259 3600 b 85900 EX (334) DT 169 3528 18365 1080 68282 68282 18.4 61 EX8 (335) DT 169 4032 2156 164 101139 104622 3600 c 41252 EX9 (434) DT 169 404 23923 13810 881 8868 3600 d 400 EX10 (534) DT 169 5880 29481 16840 86505 86581 3600 e 500 a Inegaly gap a 300s (IG300) [%]=0.06; Inegaly gap a me of emnaon (IGT) [%]=0.04; b IG300=IGT=0.02; c IG300=0.61 IGT=0.18; d IG300=IGT=0.06; e IG300=0.8 IGT=0.09. 2
.3. Poenal Cos Savngs One mpoan queson s wha ae he poenal savngs ha one may ge by usng such a compehensve opmzaon schedulng ool? In ohe wods how mpoan s o accoun fo he eal eleccy cos pofle when devng he schedule? Gven: () he many consans n ems of soage avalably demand sasfacon and maxmum powe avalably; () he need o povde a schedule fas no only fo nomal opeaon bu moe mpoanly as a esponse o unpedced evens such as machne beakdown o ugen odes ha need o be sasfed; I s fa o assume ha a feasble soluon o he poblem povdes a vald bass fo compason. A so called blnd schedule can easly be deved by he dscee-me fomulaon unde he assumpon of consan eleccy pce houghou he me hozon. Unde such scenao and accodng o Eq. (36) he model wll smply y o poduce he poducs n he fase machnes. If n ems of pocessng ae we ge he same poduc ank on evey machne hen he schedule wll end o a sees poducon ahe han a paallel one whch as we can see n Fgue 15 s he pefeed choce unde a vaable cos pofle. Neveheless he nemedae poduc demands paally avod hs. Also he effec becomes less mpoan fo hghe plan capaces. Keepng hs n mnd we sck o he plan. Afe obanng he blnd schedule he eal cos paamees ae used o compue he ue oal cos. Table 3. Poenal Cos Savngs as a Funcon of he Plan Capacy. EX1 EX2 EX3 EX4 EX5 EX6 EX EX8 EX9 EX10 Capacy (%) 41 41 41 4 33 46 40 55 51 56 Savngs (%) 2 23 39 29 40 33 39 20 32 33 Table 3 lss he esuls obaned. Even afe consdeng ha an expeenced schedule by means of he heuscs n use a he plan could do a sgnfcanly bee job han he blnd schedule fo he easons menoned above hese ae vey sgnfcan cos savngs. The end s ha as he plan appoaches full capacy he poenal savngs dmnsh whch s no supsng gven ha hee ae fewe degees of feedom. I s elevan o hghlgh he esul fo EX3 when compaed o ha fo EX2. As dscussed n secon.1 hey have he same daa bu EX3 has one moe soage un whch consdeably nceases he plan flexbly. Ths makes much ease o geneae a feasble 28
schedule bu f one s no caeful one may make wong decsons concenng he mpac on cos (ncease n poenal savngs fom 23 o 39%). M1 9 6 26.25 1.143 M2 6.51 6.66 5.66 3.6 S1 9 13 2 1.5 S2 26.25 1.5 1200 0 48 2 96 120 144 168 S1 P1 Tme (h) 1000 P2 P3 800 600 400 200 (on) 0 1200 0 48 2 96 120 144 168 Tme (h) S2 P1 1000 P2 P3 800 600 400 200 (on) 0 0 48 2 96 120 144 168 Tme (h) Fgue 15. Bes soluon found by connuous-me fomulaon fo EX5..4. Fnal emaks To end he analyss we dscuss he ssue of soluon degeneacy and s mpac on he fom of he schedule. Fo a schedule o be mplemened n pacce should have as few evens as possble snce hee ae always facos ha ae no consdeed n he model. In hs case sudy he ssue of sequence dependen changeoves s no elevan. Howeve ha does no mean ha we ae allowed o change poducs on a gven machne evey hou! Theefoe we wsh o fs complee he poducon of a poduc and only hen swch o anohe one. The ealy s hus much close o he concep of evens used by me gd based connuous-me fomulaons o of pecedence used by sequencng based models 15. The eave pocedue used by CT of nceasng he numbe of even pons one by one mplcly ensues smplcy. In conas n DT a much lage numbe of evens ae allowed so hee wll ypcally be many moe ask nsances beng execued o acheve a pacula amoun of poduc. Thus hee wll be a lage numbe of soluons ha coespond o he same value of he objecve funcon wh he lage majoy of hem beng undesable due o many changeoves see 29
Fgue 16. To ovecome such soluons a pos-pocessng pocedue s equed ha emoves he supefluous bu hs s beyond he scope of hs pape. 0 1 2 3 4 5 6 8 9 Unaccepable OK Fgue 16. Hgh soluon degeneacy may lead o an unaccepable numbe of changeoves n soluons fom he dscee-me fomulaon. 8. Conclusons Ths pape has focused on he modelng of dscee evens ha occu a pedeemned pons n me wh a connuous-me schedulng fomulaon. These ncluded mulple nemedae due daes uly avalably and vaable eleccy coss. A concepually new geneal model has been poposed ha eles on he esouce-task Newok fo pocess epesenaon. I s neveheless lmed o connuous asks. The valdy of he appoach has been demonsaed on a few es cases adaped fom a eal ndusal poblem. In such pocess soage uns ae used fo fnal poducs and may ac as shaed soage uns. To effcenly model hs ssue a novel hybd bach/connuous ask has been poposed. Despe he majo modelng beakhough he esuls have shown ha only poblems of small sze can be handled effecvely. Ths uned ou aenon o dscee-me models whee dscee evens can be handled n a moe naual and saghfowad way. The only doub was o whehe a suffcenly fne me gd could be used o epesen he poblem daa accuaely. The goal was o ensue a weekly schedule ha accouned fo houly changes n eleccy cos and hs has been successfully accomplshed. Poblems of ndusal sgnfcance wee ackled down o opmaly gaps below 1% wh paccal compuaonal mes of 5 mnues. Ths s mosly due o he chaacescally low negaly gap of TN dscee-me fomulaons whch wee found o be sgnfcanly lowe han he connuous-me counepas. The las pa of he pape has hghlghed he mpoance of akng vaable eleccy coss no consdeaon when devng he schedule. Sae-of-he-a schedulng fomulaons have he poenal o acheve majo savngs when compaed o pocedues ha ae mosly focused on 30
feasbly. Whle accuae values ae obvously dependen on poblem daa paculaly on he dffeen cos levels ageed wh he eleccy povde and on he schedulng pacce a he plan esuls have shown poenal cos savngs aound 20%. Clealy values of hs ode of magnude povde enough movaon fo he ncopoaon of he dscee-me mahemacal fomulaon pesened n hs pape as a cenal elemen of he decson makng ool used by ndusy. Acknowledgmens The auhos gaefully acknowledge fnancal suppo fom Fundação Luso-Amecana and he Cene fo Advanced Pocess Decson-makng a Canege Mellon Unvesy. Nomenclaue Ses/Indces DY/dy=days of he week E/e=Eleccy cos levels H/h=hous of he day I/=asks I c =connuous asks c I e =connuous asks execued dung enegy level e I s =soage asks I =nsananeous ansfe asks M/m=machnes P/p=poducs /=esouces CT =esouces connuously poduced/consumed EQ =equpmen esouces FP =fnal poduc esouces LM =poducs a a locaon afe he machnes LS =poducs a a locaon nsde he soage uns M =aw-maeal esouces TC =equpmen esouces nvolved n mng consans UT =uly esouces S/s=soage uns T/=even pons TD/d=demand pons TP e /p=me peods of eleccy level e Paamees c e =eleccy cos a level e [ /kwh] cap s =capacy of soage un s [on] ce =eleccy cos dung me neval [ /kwh] d pdyh =demand of poduc p a he end of hou h of day dy [on] H= me hozon [h] lb ep =sang me of me peod p of eleccy level e [h] 31
pw pm =powe equemen fo poduc p n machne m [MW] pwx hdy =maxmum powe consumpon dung hou h of day dy [MW] max =uppe bound on avalably of esouce a even pons end max = uppe bound on avalably of esouce dung nevals mn = lowe bound on avalably of esouce a even pons end mn = lowe bound on avalably of esouce dung nevals fx d =absolue me of demand pon d [h] ub ep =endng me of me peod p of eleccy level e [h] δ=duaon of evey neval on dscee-me gd [h] λ =connuous neacon of esouce dung execuon of ask µ =dscee neacon of esouce wh ask a s sa acng on bnay exen vaables μ =dscee neacon of esouce wh ask a s end acng on bnay exen vaables ν = dscee neacon of esouce wh ask a s sa acng on connuous exen vaables ν = dscee neacon of esouce wh ask a s end acng on connuous exen vaables Π Π n ou =amoun eceved no he sysem of esouce a even pon =amoun emoved fom he sysem of esouce a even pon max ρ =maxmum pocessng ae of ask [on/h] ρ pm =pocessng ae of poduc p n machne m [on/h] Vaables N =execuon of ask dung neval (bnay exen vaables) =excess amoun of esouce a even pon = nal avalably of esouce (can be a paamee fo some esouces) 0 end =excess amoun of esouce mmedaely befoe he end of neval T =absolue me of even pon [h] Ts =sang me of asks execued dung neval [h] Y pe =bnay vaable denfyng f dung neval asks ae execued whn peod p of level e ou Y d =bnay vaable denfyng f even pon coesponds o demand pon d ξ =amoun handled by ask a even pon/dung neval (connuous exen vaables) [on] ξ =amoun connuously sen o soage by ask dung neval [on] * 32
efeences (1) Casagnol D.; Gallesey E. Techncal epo CH-D 2003-12. ABB Copoae eseach Däwl Swzeland 2003. (2) Maavelas C.; Gossmann I. A Geneal Connuous Sae Task Newok Fomulaon fo Sho Tem Schedulng of Mulpupose Bach Plans wh Due Daes. Poceedngs ESCAPE-13 (Eds. A. Kaslawsk and I. Tuunen) 2003 pp. 215. (3) Caso P.; Babosa-Póvoa A.; Maos H.; Novas A. Smple Connuous-me Fomulaon fo Sho-Tem Schedulng of Bach and Connuous Pocesses. Ind. Eng. Chem. es. 2004 43 105. (4) Shak M.; Janak S.; Floudas C. Connuous-Tme Models fo Sho-Tem Schedulng of Mulpupose Bach Plans: A Compaave Sudy. Ind. Eng. Chem. es. 2006 45 6190. (5) Schllng G.; Paneldes C. Opmal Peodc Schedulng of Mulpupose Plans. Compu. Chem. Eng. 1999 23 635. (6) Paneldes C.C. Unfed Famewoks fo he Opmal Pocess Plannng and Schedulng. In Poceedngs of he Second Confeence on Foundaons of Compue Aded Opeaons; Cache Publcaons: New Yok 1994; pp 253. () Duae B.; Sanos L.; Maano J. Opmal szng schedulng and shf polcy of he gndng secon of a ceamc le plan. Compues Opeaons eseach. 2009 36 1825-1834. (8) Caso P.; Babosa-Póvoa A.; Novas A. Smulaneous Desgn and Schedulng of Mulpupose Plans Usng esouce Task Newok Based Connuous-Tme Fomulaons. Ind. Eng. Chem. es. 2005 44 343. (9) Maavelas C.T.; Gossmann I.E. New Geneal Connuous-Tme Sae-Task Newok Fomulaon fo Sho-Tem Schedulng of Mulpupose Bach Plans. Ind. Eng. Chem. es. 2003 42 3056. (10) Janak S.L.; Ln X.; Floudas C.A. Enhanced Connuous-Tme Un-Specfc Even-Based Fomulaon fo Sho-Tem Schedulng of Mulpupose Bach Pocesses: esouce Consans and Mxed Soage Polces. Ind. Eng. Chem. es. 2004 43 2516. (11) Shak M.; Floudas C. Novel Unfed Modelng Appoach fo Sho-Tem Schedulng. Submed o Ind. Eng. Chem. es. 33