GAMS Handout 2. Utah State University. Ethan Yang

Transcription

1 Uah ae Universiy All ECAIC Maerials ECAIC Repsiry 2017 GAM Handu 2 Ehan Yang yey217@lehigh.edu Fllw his and addiinal wrs a: hps://digialcmmns.usu.edu/ecsaic_all Par f he Civil Engineering Cmmns Recmmended Ciain Yang, Ehan, "GAM Handu 2" (2017). All ECAIC Maerials. Paper 64. hps://digialcmmns.usu.edu/ecsaic_all/64 his Lecure Maerial is brugh yu fr free and pen access by he ECAIC Repsiry a DigialCmmns@UU. I has been acceped fr inclusin in All ECAIC Maerials by an auhrized adminisrar f DigialCmmns@UU. Fr mre infrmain, please cnac dylan.burns@usu.edu.

2 Using GAM fr Reservir Operain Y. C. Ehan Yang Deparmen f Civil and Envirnmenal Engineering Lehigh Universiy Behlehem, PA UA YOU WILL LEARN Quic review f GAM example ingle reservir perain hery ingle reservir perain GAM example ingle reservir perain Hmewr Nex sep River basin managemen QUICK REVIEW OF GAM EXAMPLE Example seing Le s say yu are a farmer ha wans decide wha crp yu wan grw in yur farm maximize yur prfi. Yu have w differen chses: whea and cn. hey require differen amun f resurces (waer and land) and give yu differen ne prfi. All daa yu need are given in he fllwing ables. whea cn Ne prfi UD whea 2 cn 3 Resurces need waer land whea 2 3 cn 4 2 al available resurces 1

3 waer 20 land 25 Review he basic cmpnens in GAM: e, Parameer, able, calar, Variable, Equain, Mdel and lver. his example is ne f he fundamenal quesins in waer resurce managemen bu nly represens a small scale issue. Nex, le s enlarge he spaial scale river newr and cnsider a reservir perain quesin ha ries pimize he waer needs frm dwnsream. INGLE REERVOIR OPERAION HEORY here are hree majr differen purpses fr reservir perain: waer supply, flding cnrl, hydrpwer generain Q is he inflw, R is he uflw, is he live srage and K is he capaciy. represen he ime perid used fr mdeling. Waer supply Waer supply pimizain mdels may be frmulaed maximize he waer ha can be supplied given a reservir vlume, giving an indicain f he size fr reservir needed fr a specified demand. Alernaively, he mdel may be frmulaed prescribe he perains ha wuld allw he waer be supplied a specified demand as reliably as pssible, which a given reservir vlume. he hird frmulain belw shws a mdel frmulaed maximize he ecnmic value f waer supply releases. In all cases he decisin variables are he releases. ype 1 maximize al waer supply 2

4 R W P K W WD 1 Maximize s.. W Q R L 1,..., 11 1,..., In 12 1,..., 1,..., W is he annual al waer supply, WD is he waer demand in mnh, β is fixed rai f mnh demand annual demand, P is he spill in ime perid ype 2 maximize reliabiliy R W P K W WD 1 Maximize s.. Q R L W WD W min 1,2..12 WD 1,..., 11 1,..., In 12 1,..., 1,..., WD is he annual waer demand and λ is he weigh he secnd reliabiliy erm ype 3 real ime perain W s: 1 Q R L R W P K WD Maximize W / WD minw / WD 1,..., 1,..., 1,..., 11 1,..., WD is he waer demand in ime perid, = 1, 2,... (his value is nly available fr a real ime perain) ype 4 Maximize waer use prfi exended bjecive funcin (Will see mre in he example) s: 1 Q R L R W P K Maximize P( W 1,..., 1,2,.. ) 1,..., 11 3

5 Develp a prfi funcin fr waer supply W fr ne r mre paricular uses such as irrigain, indusry, hydrpwer generain, ec. Fld cnrl Reservirs wih flding cnrl purpse are eiher single purpse r cupled wih her bjecives, in which case srage fr fld cnrl (essenially empy space in he reservir) is a hard cnsrain n srage levels in he pimizain mdel. he fllwing example shws he reservir used fr waer supply bu wih a cnsrain fr flding cnrl. Again, he decisin variables are he releases. Maximize W R W P K v W WD 1 s.. Q R L 1,..., 11 1,..., In 12 1,..., 1,..., ν is he srage reserved fr flding cnrl in mnh Hydrpwer generain his pimizain mdel will aemp release waer prduce energy ha maches a schedule f energy demand (ype 1). Alernaive frmulains fr hydrpwer prducin migh maximize energy prducin (regardless f demand) r maximize ecnmic prducin by aing in accun he price f energy and i is changes a differen imes f year (ype 2). In he hydrpwer generain case, bh he release and he head (srage) are decisin variables. Reservir wih pwer plan 4

6 Head-srage relainship H H ( 1 ) H( 2 ) y = x R 2 = Head (m) Head (m) Dead rage al rage 800 Dead rage al rage rage (millin m3) rage (millin m3) ype1 - Minimizing difference beween energy prduced and demanded Minimize subjec 1 K H H H 1 Q R L c H H H H q R / ime E D P (9.81) H q 2 q 2 1,..., max P 1 max E P * ime( ) /3600 E is energy generain, D is energy demand, H is head, q is release rae, P is pwer generain, ε is efficiency. ype2 Maximizing ecnmic prducin f hydrpwer generain Maximize subjec 1 K H H H 1 Q R L c H H H H q R / ime E price 2 q P (9.81) H q 1,..., max P 1 max E P * ime( ) /

7 price is he energy price a ime. INGLE REERVOIR OPERAION GAM EXAMPLE GAM example f reservir perain fr maximize waer use prfi (adaped and mdified frm example in Chaper 4, Lucs e al., 2005 and McKinney and avisy, 2006) Le s say his ime yu are he manager f he upsream reservir and yur jb is release waer every mnh fr dwnsream waer uses. here are hree firms dwnsream and each ne f hem has heir wn prfi funcin. All daa yu need are given in he fllwing ables. Mnhly inflw daa are given (implies perfec nwledge f inflws) Assuming he al waer uses frm all hree firms is equal he mnhly release and n cap n mnhly release. In his case, we emprarily ignre he upsream-dwnsream relainship beween firms. chemaic Firm 2 Firm 1 Firm 3 Prfi funcin fr firms NBn(Xn) = axn+bxn 2 Firm 1 Firm 2 Firm 3 a b

8 Waer use, X, in uni f cubic meer (m 3 ) f waer and Ne Benefi (NB) in unis f dllar Inflw m Q Reservir infrmain Capaciy: 9,500 m 3 dead srage: 5,500 m 3 iniial srage: 8,000 m 3 Le s see he GAM cde E i demand sies /i1, i2, i3/ ime / 1*12/ PARAMEER a(i) cefficien f benefi funcin (ax+bx^2)/ i1 280 i2 389 i3 482/ b(i) cefficien f benefi funcin (ax+bx^2)/ i1-0.8 i2-0.5 i3-0.3/ Q() inflw / / CALAR K reservir capaciy /9500/ _min reservir dead srage /5500/ 7

9 beg_ reservir iniial srage /8000/ POIIVE VARIABLE () reservir srage a ime, R() reservir release a ime, W(i,) waer use frm sie i a ime VARIABLE Obj bjecive value EQUAION Objecive Objecive funcin, Balance() Waer balance, Waercap() Waer use cap Objecive.. Obj =E= UM((i,),a(i)*W(i,)+b(i)*W(i,)**2) Balance().. () =E= beg_ $(ORD() EQ 1) + (-1) $ (ORD() G 1) + Q() - R() Waercap().. R() =E= UM(i,W(i,)).UP() = K.LO() = _min MODEL Reservir / ALL / OLVE Reservir UING NLP MAXIMIZING Obj INGLE REERVOIR OPERAION HOMEWORK Dry year inflw Assuming he weaher frecas indicaes nex year will be a dry year wih inflw as fllwing, rerun he mdel and cmpare he differences f 1) mnhly srage, 2) waer use a demand sies and 3) prfi a demand sies. Q() inflw fr dry year/

10 / Envirnmenal flw a dwnsream As we discuss in he very firs class, he waer uses are n limied human uses nly. Assuming a new envirnmenal requiremen saes ha he sreamflw dwnsream f he hree firms shuld be greaer han 150 m 3 per mnh, mdify he cde incrprae his new plicy. Use he nrmal year inflw and cmpare he difference f 1) mnhly srage, 2) waer use in demand sies and 3) prfi in demand sies wih he case wihu envirnmenal flw seing. In yur hmewr, yu need explain why and hw yu mdified he cde. NEX EP RIVER BAIN MANAGEMEN me issue hin abu befre we mve he nex sep Why desn he mdel use all waer in January? Wha if we wan cnsider he upsream-dwnsream relainship? Wha if we wan run he mdel fr muliple years? 9