ExxonMobil. Juan Pablo Ruiz Ignacio E. Grossmann. Department of Chemical Engineering Center for Advanced Process Decision-making. Pittsburgh, PA 15213

Transcription

1 ExxonMobl Multperod Blend Schedulng Problem Juan Pablo Ruz Ignaco E. Grossmann Department of Chemcal Engneerng Center for Advanced Process Decson-makng Unversty Pttsburgh, PA

2 Motvaton - Large cost savngs can be acheved f the correct blendng decsons are taken. - Models hghly nonconvex global optmzaton technques requred. - Effcent soluton methods for large scale systems remans as a challenge C D U FCC Blendng Upstream the CDU Blendng Downstream the CDU Goal: Develop tools and strateges amng at mprovng the effcency of the soluton methods for the global optmzaton of the multperod blend schedulng problem 2

3 General Problem Topology The general case of a blendng problem can be represented schematcally as follows Remarks: Examples of supply nodes: - tanks loaded by shps - feedstocks downstream the CDU Supply Intermedate Delvery Examples of delvery nodes: - tanks feedng the CDU - tanks delverng to fnal customers Man Model Assumptons -The qualty of each stream/nventory s constant for a gven perod. - A tank can receve or delver n a gven perod of tme but not both. - Supply tanks keep a constant qualty. - Delvery tanks keep the qualty wthn a gven range. - Streams enterng delvery tanks should satsfy a qualty condton. 3

4 Alternatve Formulatons Work lnes - Summary Proposed formulatons gven n the space of propertes and total flows and n the space of ndvdual property flows Reduced the number of blnear terms by usng GDP formulatons Explored the use of redundant constrants to mprove the relaxatons Soluton Methods Proposed a Logc Based Outer Approxmaton method to fnd local solutons Proposed a Lagrangan Decomposton method to fnd global solutons Novel Relaxaton Strateges Man Focus Proposed the use of new relaxatons based on vector space propertes 4

5 Alternatve Formulatons 5

6 Alternatve Formulatons Dfferent space formulatons Formulaton I Specfc Property Total Flow Mxer Formulaton II Total Property Flow Nonconvex F j' jt S qj' t Tf j j ' ( ' j)e n qjt f q j' jt Tf j' ( j' j)e n qjt Formulaton I Specfc Property Total Flow Spltter Formulaton II Total Property Flow Nonconvex jj j ' ( ') E F, S out jj ' t TF jt, qj f q Tf q j' jt ( jj') E jj' t jj' t 1 out qjt Formulaton I and II are equvalent but wth dfferent relaxatons! 6

7 Alternatve Formulatons Redundant Constrants* The total property flow to the delvery ste s constraned by an upper B B B and lower bound. Ths nformaton s lost when CFqjj' t1 Cqjt 1Fqjj' t1 s relaxed. L B B D C qj' F jj' t M ( 1 x jj' t ) CF qjj' t 1 q Q, j J, j' J,( jj') E, t T CF C F M 1 x q Q, j J, j' J,( jj') E t T B U B D qjj' t 1 qj' jj' t ( jj' t ), For any two propertes q and q (n any stream or nventory) the rato between the total property flow of q to q s the same as the rato between specfc property value. Ths s lost when the problem s relaxed. CF B qjj' t B B B B D B 1Cq ' jt1 CFq ' jj' t1c qjt1 q, q' Q, j J, j' ( J J ),( jj') E, t T CI B qjt B B B B 1Cq ' jt1 CIq' jt1c qjt1 q, q' Q, j J, t T The property balance around each spltter should be held. Ths s lost when the blnear terms are relaxed. f q j' jt j' ( j' j) E Tf out qjt q Q, j J B, t T * Ruz and Grossmann, 2010 Usng redundancy to strengthen the relaxaton of nonconvex MINLPs To appear n Computers and Chemcal Engneerng Journal 7

8 Alternatve Formulatons Generalzed Dsjunctve Programmng Tradtonal MINLP Formulaton One general state Proposed GDP Formulaton Two states By explotng the underlyng logc structure of the problem, a reducton of the number of blnear terms can be acheved 8

9 Soluton Methods 9

10 Soluton Methods Logc Based Outer-Approxmaton Outlne of the Logc Based OA Master Problem (Lnear GDP) NLP Subproblem LGDP Master: - Relax blnear terms usng McCormck envelopes. - Solve MIP usng the Hull Reformulaton. NLP Subproblem: - Fx boolean varables from master problem. - Elmnate not actve dsjuncts - Solve small NLP formulaton No guaranty a global soluton! Iteraton Step: - Generate lnear cuts on soluton pont of NLP subproblem. 10

11 Soluton Methods Lagrangan decomposton Orgnal Formulaton Fn 3S 3 I 3S 3 I 2S 2 Fout 3S 3 Fn 2S2 I 2S2 I1S1 Fout 2S2 Constrants are lnked together by the nventory and composton varables Fn 1S1 I1S1 I0S0 Fout 1S1 Decomposable Formulaton Fn3S3 I3 S3 I' 2 S' 2 Fout 3S3 I' 2 I2 ; S' 2 S 2 Dualze Fn 2S2 I 2 S2 I' 1 S' 1 Fout 2S2 I ' 1 I1 ; S' 1 Fn 1S 1 S I 1S 1 I 0S 0 Fout 1S 1 1 Duplcatng nventory and composton varables and dualzng the correspondent equaltes leads to a temporal decomposable structure 11

12 Soluton Methods Lagrangan decomposton (Algorthm) Outlne of the Lagrangan decomposton method Intalze uo, BestLB, BestUB, k uo represents the dual multplers; BestLB, the best lower bound; BestUB, the best upper bound and k, the ter counter Solve LR1 Solve LR2 Solve LRN LB = LB LR If BestLB < LB BestLB = LB Each subproblem (LR) from the decomposton s solved A lower bound for the orgnal nonconvex problem can be obtaned by addng up the soluton of each (LR) (.e. LB LR ) Obtan UB (Solve local MINLP) Update Multplers u t+1 =u t t *Error t =a/(b+k) If BestUB-BestLB < or k = maxter 1 STOP Any local optmzaton algorthm can be used to fnd an UB. (e.g. The logc based outer-approxmaton appled on the GDP formulaton) k=k+1 12

13 Soluton Methods Illustratve Example The mplementaton of the Lagrangan Decomposton method has been tested n the followng smple case Supply Intermedate Delvery Inlet Flows and dproperty Outlet Flows and dproperty Values are fxed Values are fxed Topology (allowed connectons) Network Descrpton: - Two Supply, Intermedate and Delvery nodes - Two propertes transported - Three tme perods 13

14 Lagrangan Decomposton Soluton Methods Numercal Results Representaton of the nonzero flow streams n the dfferent perods for the global optmal soluton Perod 1 Perod 2 Perod 3 Global l Soluton (Z = 14.22) (verfed wth BARON) Remarks - Forced dto stop after 20 teratons t (no mprovement observed). - Fnds the global soluton (Z = 14.22) - The exstence of the dualty gap s due to the nonconvex nature of the problem LB/UB Iter 14

15 Soluton Methods Generaton of Real-world Instances Lower Bound Upper Bound Structure Nodes Propertes Perods Edges J P, J B, J D 4 10 Q 4 10 T E 40% of all possble edges 100% of all possble edges P jt D jt S P qj 0 1 S B0 qj 0 1 S L qj, SU qj 0 1 I L j, IU I j, I0 I j F L jj, FU jj Bound ds on Varab bles Ths table was used to generate random nstances 15

16 Intalze uo, BestLB, BestUB, k Soluton Methods Observatons Numercal tests usng Lagrangan Relaxaton wth temporal decomposton Solve LR1 Solve LR2 Solve LRN LB = LB LR If BestLB < LB BestLB = LB 1- Hgh computatonal tme requred n subproblems (> 5mn) 2- Dffcultes to fnd local solutons Obtan UB (Solve local MINLP) Update Multplers u t+1 =u t t *Error t = a / (b + k) If BestUB-BestLB BestLB < or k = maxter 1 STOP How do we tackle these ssues? k=k+1 16

17 Soluton Methods Fndng Local Solutons Outer Approxmaton Method Lower Bound If MASTER nfeasble STOP MASTER Fx Integer (y j ) Add lnearzatons from soluton of NLP (y j ) The MASTER problem can be tghtened by addng McCormck Convex envelopes for the blnear terms Upper Bound NLP(y j ) If NLP(y j ) nfeasble remove y j If (Up Bound -Lo Bound) s less than STOP Bounds of varables Remarks - Reducng the number of blnear terms n NLP(y j ) leads to a more robust formulaton - Havng good bounds for the varables s of man mportance to fnd tght relaxatons 17

18 Soluton Methods Fndng Local Solutons Tghter bounds for varables (I) Observaton If two streams are mxed together, the concentraton of any gven component n the mxture s always hgher/lower than the mnmum/maxmum concentraton n the streams C qt F jt j Mathematcal Representaton C mn ( Cqjt ) q,, t qt 1 (, j) E C max ( Cqjt ) q,, t qt 1 (, j) E How can we use t to nfer bounds for the compostons? 18

19 Soluton Methods Fndng Local Solutons Tghter bounds for varables (II) Lower Bounds Upper Bounds C LO C q,, t 1 C UP C q,, t 1 qt q0 qt q0 C LO qt LO mn ( Cqjt 1) (, j) E q,, t 1 C UP qt UP max( C 1) (, j) E qjt q,, t 1 Illustratve Example C 10 = C 20 =0.3 C 30 = C 40 =0.5 t=0 t=1 t=2 LO UP LO UP LO UP Node Node Node Node Lower and upper bound tghtenng can be acheved n the preprocessng step 19

20 Performance Analyss Remark Soluton Methods Fndng Local Solutons Tghter bounds for varables (III) Predcted lower bounds at frst MASTER problem Global Optmum Usng orgnal bounds Usng nferred bounds Instance Instance Improvements n the bounds predcton can be obtaned f lower/upper bounds of flows and nventory levels are consdered 20

21 Tradtonal MINLP Formulaton Soluton Methods Fndng Local Solutons Reduced number of blnear terms One general state Proposed GDP Formulaton Two states By explotng the underlyng logc structure of the problem, a reducton of the number of blnear terms can be acheved 21

22 Performance Analyss -11 random nstances Soluton Methods Fndng Local Solutons Numercal Results - Outer approxmaton solver DICOPT(GAMS) - Three dfferent formulatons (all usng McCormck envelopes): 1- Orgnal MINLP 2- Formulaton wth reduced number of blnear terms 3- Formulaton wth reduced number of blnear terms plus bound tghtenng - Forced to stop after 10 teratons or 30 mnutes Remarks - Formulaton (2) and (3) found feasble solutons n more than 70% of nstances - Formulaton (3) outperformed Formulaton (2) n 20% of the nstances - Formulaton (1) led to a large number of false nfeasble problems 22

23 Intalze uo, BestLB, BestUB, k Soluton Methods Soluton of LR sub-problems Spatal Decomposton Solve LR1 Solve LR2 Solve LRN Spatal Decomposton LB = LB LRn If BestLB < LB BestLB = LB Perod N Obtan UB (Solve local MINLP) Update Multplers u t+1 =u t t *Error t = a / (b + k) If BestUB-BestLB < or k = maxter 1 STOP How do we decompose t spatally? k=k+1 23

24 Soluton Methods Soluton of LR sub-problems Mnmal cut-edge wth fxed nodes Incdence Matrx mn s. t. Objectve: Mnmze the edges that cross the boundares of each subset jk A j ( y k z jk ) y k 1 Number of nodes k y k k k n dsjont subsets If y k = 1 then the node belongs to the subset k 1 y y z 0 y y k jk k z z jk jk jk jk jk jk jk z jk y k y jk j z jk 2 k k cut! 24

25 Soluton Methods Soluton of LR sub-problems Mnmal cut-edge wth fxed nodes example Sub-Set 1 Sub-Set 2 Sub-Set St3 Dualzed constrants necessary: 3(n+1) (n: number of propertes consdered) 25

26 Soluton Methods Soluton of LR sub-problems Numercal Results -Baron takes 347 seconds (~6mn) to solve the problem wth a soluton of The spatal decomposton solves the problem n 1 teraton: MIP separaton problem: 5 seconds Sub-problem 1: (sol: ) 1.6 seconds Sub-problem 2: (sol: ) 1.4 seconds Sub-problem 3: (sol: ) 0) 1.5 seconds TOTAL: (sol: ) 9.5 seconds Remarks: - Even though t s not expected for general problems to converge n one teraton, even wth 15 teratons, the tme necessary would be ~1 mn 26

27 Novel Relaxatons 27

28 Novel Relaxaton Strateges Vector space propertes to strengthen the relaxaton j F n P n Algebrac Representaton F P j on on Man buldng block of a process network Vectoral Representaton Explot nteracton to develop cuts (3-D Case) Cuts (for a gven j and n) 28

29 Novel Relaxaton Strateges Numercal Results* Table Comparson of the performance of proposed approach wth tradtonal relaxatons Tradtonal Approach Proposed Approach Instance GO LB Nodes Tme(s) LB Nodes Tme(s) Poolng problems! All problems were solved usng a Pentum(R) CPU 3.4 GHz and 1GB RAM *Ruz J.P. and Grossmann I.E. 2010, Explotng Vector Space Propertes for the Global Optmzaton of Process Networks, Optmzaton Letters 29

30 Remarks Proposed formulatons gven n the space of propertes and total flows and n the space of ndvdual property flows Reduced the number of blnear terms by usng GDP formulatons Explored the use of redundant constrants to mprove the relaxatons Proposed a Logc Based Outer Approxmaton method to fnd local solutons Proposed a Lagrangan Decomposton method to fnd global solutons Proposed the use of new relaxatons based on vector space propertesp Future Work - Implement spatal decomposton of the sub-problems wthn the global optmzaton framework. - Add cuts to strengthen relaxaton for LR (from Vector Space Analyss?) 30