Exptng vectr space prpertes fr the gbal ptmzatn f prcess netwrks Juan ab Ruz Ignac Grssmann Enterprse Wde Optmzatn Meetng March 00
Mtvatn - The ptmzatn f prcess netwrks s ne f the mst frequent prblems that s addressed n prcess systems engneerng (e.g. ptmzatn f plng netwrks, heat exchanger netwrks and water treatment netwrks - Mass and energy balances are the cmmn denmnatr f these systems and are ften represented thrugh equatns wth blnear terms. - Blneartes lead t nncnvex prblems hence, gbal ptmzatn technques are requred. - Varatns f the spatal branch and bund framewrk are used t slve the prblems. They heavly rely n tght relaxatns. Gal: rpse a methdgy t fnd strnger relaxatns fr the gbal ptmzatn f prcess netwrks.
Intrductn j n n j n n Buldng bck f prcess netwrks General mass and energy balance frmulatn I n ( n j n I n n n j n n 0 0 n N, j J n N (MB Sets: N : Ndes n the netwrk I n : Streams enterng nde n and J : rperty type
Vectral Representatn r a gven nde n and prperty j we defne the vectrs: v,,...,, v (,,...,, v,,...,, ( I (MB can be represented as: E v. E 0 v. v E 0 Or equvalently, n vectral frm: ( I 3-Vectr Representatn v v E v v E (VMB The nteractn between the vectr spaces v, v and the vectr v E s clearly expsed n the, 3-Vectr Representatn.
Mnmal Set We defne a mnmal set, the set cmpsed by three elements (.e. I + = 3 Lemma : Any system f the frm (VMB can be decmpsed as the ntersectn f I - 3-Vectr Representatn f mnmal sets Illustratn ( I = 4 : equvalent 5 5 3 4 3 4 4 mnmal sets
rpertes f the mnmal set Lemma : The prperty vectrs (v and fw vectrs (v n a mnmal set are related as flws: Or equvalently v v v v v v v. v E E 0, v. v 0 v v E E v v where v 3 sn, 0 v Illustratn: v The crss prduct between v and v E s parallel t v v v E
rpertes f the mnmal set (cnt. Lemma 3: The space defned by the mnmal set s nncnvex Illustratn: Gven tw pnts n the set v = {,,3} v = {,,-} v E = {,,-}, v = {,,} v = {3,,-} mnmal set v E = {,,-} The flwng pnt, whch s a cnvex cmbnatn, s nt n the set 0.5v + 0.5v = v = {.5,,.5} mnmal set 0.5v + 0.5v = v = {,,-.5} 0.5v E + 0.5v E = v E = {,,-} v. v 0
Cnvex relaxatn f mnmal set (Tradtnal Apprach A tradtnal relaxatn f (MB s gven by replacng the blnear terms wth the McCrmck cnvex envepes. {,} {,} ( 0 0 The rthganalty between v and v s st! v v v v v v E v v v mplctly defnes the rthgnalty between v and v
Vald cuts frm crss prduct Based n Lemma the flwng s a vald cut Whch n algebrac frm reads v v v Nncnvex! (CC rm where the flwng lnear cuts are derved: =,, where, and
Bunds fr rm the defntn f crss prduct: v max 3 sn v v v max mn 3 v v 3 Tghter wer and per bunds can be btaned by usng (CC: mn(max,max,max max(mn,mn,mn
rpsed vs Tradtnal Apprach rpstn The prpsed cuts are nt dmnated by the McCrmck cnvex envepes Illustratn Gven the mnmal set: 3 3 3 where:.5,.5.5, 4. 5 0 3 0.5.5, 0, 0 3 the regn n the space wth fxed =0.5, =.3, 3 =.8, =., =0. s 3 = [0.9-0.43] usng the McCrmck envepes and 3 = [0.3-0.36] usng the prpsed cuts
Case Study (Data Recnclatn rblem Statement: nd the set f values f fws and cmpstn that mnmze the squared errr when cmpared wth the measurements. System representatn (Instance -: rmulatn: I CI I CI I 3 CI 3 Mn Z w( w3( 3 w5( I 3 3 3 5 I I 3 5 w( w4( w6( I 4 6 I I Nncnvex set 4 6
Numercal Results System representatn (Instance 3-4: I 3 CI 3 I CI I CI I 3 CI 3 I 4 CI 4 I 5 CI 5 I 7 CI 7 I 8 CI 8 I 9 CI 9 I 6 CI 6
Numercal Results Tradtnal Apprach rpsed Apprach Instance GO LB Ndes Tme(s LB Ndes Tme(s 8.78 78.5 0 8.6 4 9 5.6 4.89 80 09 5.0 9 8 3 3.4 0.45 48 58.3 4 83 4 7.9 7.08 7 30 7.3 5 On average, the prpsed apprach led t 46% mprvement n wer bunds, /3 f ndes necessary t fnd the slutn and / the cmputatnal tme.
Cnclusns - rpsed a vectral representatn f the prcess netwrk mdels. - Expsed part f the nteractn between the vectr space defned by the fws and prpertes. - rpsed cuts that strengthen the relaxatn gven by tradtnal appraches. - The perfrmance f the methd tested n several nstances related t data recnclatn n prcess netwrks shws sgnfcant mprvements.