Recent Developments in the Application of Mathematical Programming to Process Integration

Transcription

1 Recent Developments in the Application of Mathematical Programming to Process Integration Ignacio E. Grossmann Center for Advanced Process Decision-making Department of Chemical Engineering Carnegie Mellon University Pittsburgh, PA 15213, U.S.A. International Process Integration Jubilee Conference Gothenburg, Sweden March 18,

2 Major approaches to Process Integration/Process Synthesis Heuristics (Knowledge Base) Physical Insights (Pinch Analysis) Enumerative Search (Means-ends Analysis, Hierarchical Decomposition) Mathematical Programming (MINLP) After 30 years, religion war is over! Take best of each approach and combine Gundersen, T. and I.E. Grossmann, "Improved Optimization Strategies for Automated Heat Exchanger Networks through Physical Insights," Computers and Chemical Engineering 14, 925 (1990). Goal: Overview state-of-art and progress of mathematical programming techniques and their application in Process Integration a) What progress has ben made with mathematical programming tools (LP, MILP, NLP, MINLP, GDP)? b) What has been their impact on Process Integration Problems? 2

3 Math Programming Approach to Process Synthesis/Integration 1. Develop a superstructure of alternative designs 2. Develop an LP/NLP or (MILP/MINLP, GDP) model to select topology and parameters of design 3. Solve LP/NLP or (MILP/MINLP, GDP) model to extract optimum design embedded in superstructure LP = Linear Programming NLP = Nonlinear Programming MILP = Mixed-integer linear programming MINLP = Mixed-integer nonlinear programming GBD = Generalized Disjunctive Programming 3

4 4 Mathematical Programming MINLP: Mixed-integer nonlinear programming m n y R x y x g y x h s t y x f Z {0,1}, 0, 0 ), (.. ), ( min = = ) ( LP: f, h, g linear, only x q n m n n R R x g R R x h R R x f : ) (, ) : (, ) : ( 1 NLP: f, h, g nonlinear, only x MILP: f, h, g linear

5 Evolution of Mathematical Programming x 2 LP: Linear Programming Kantorovich (1939), Dantzig (1947) x 1 x 2 NLP: Nonlinear Programming Karush (1939); Kuhn, A.W.Tucker (1951) x 1 y 2 IP: Integer Programming R. E. Gomory (1958) y 1 5

6 Major developments in last 30 years - Interior Point Method for LP Karmarkar (1984) - Convexification of Mixed-Integer Linear Programs Lovacz & Schrijver (1989), Sherali & Adams (1990), Balas, Ceria, Cornuejols (1993) - Modeling Systems GAMS, AMPL, AIMMS - MILP codes: CPLEX, GUROBI, XPRESS - NLP codes: MINOS, CONOPT, SNOPT, IPOPT - MINLP Duran & Grossmann (1986) - Global Optimization Floudas(1990), Sahinidis (1996) - Logic-based optimization Hooker (1991), Raman & Grossmann (1994) - Hybrid-systems Barton & Pantelides (1994), Bemporad & Morari (1998) 6

7 Applications of Mathematical Programming in Chemical Engineering Process Design Process Synthesis/Integration Production Planning Process Scheduling Plant Material flow Information flow (Orders) Plant Warehouse Distr. Center Retailer End consumers Demand for A Supply Chain Management Making of A, B & C Demand for B Demands for C Process Control Parameter Estimation LP, MILP, NLP, MINLP, Optimal Control u(t) u(t+k t) w(t+k t) w(t) y(t+k t) y(t) N t-1 t t+1... t+k... t+n Major contribution: new problem representations and models

8 Progress in Mixed-Integer Linear Programming Bixby, Rothberg and Gu (2011) => 80,000 speed-up!! on same computer CPLEX 1.2 to Gurobi 3.0 Unit-Commitment Model: California 7-Day Model 2, vars, 22,899 cont vars, 48,939 constr CPLEX 7.0: 1 hr initial LP, unfinished after 8 hours 2010 Gurobi 3.0: ~3 min (195 secs) to optimality! 8

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10 Nonlinear Programming NLP: Algorithms (variants of Newton's method) Sucessive quadratic programming (SQP) (Han 1976; Powell Reduced gradient Interior Point Methods Major codes: MINOS (Murtagh, Saunders, 1978, 1982) CONOPT (Drud, 1994) SQP: SNOPT (Murray, 1996) OPT (Biegler, 1998) KNITRO (Nocedal, 2000) IP: IPOPT (Wächter, Biegler, 2002) Typical sizes: 100,000 vars, 100,000 constr. (unstructured) 1,000,000 vars (few degrees freedom) Convergence: Good initial guess essential (Newton's) Nonconvexities: Local optima, non-convergence

11 Mixed-integer Nonlinear Programming Algorithms Branch and Bound (BB) Ravindran and Gupta (1985), Stubbs, Mehrotra (1999), Leyffer (2001) Generalized Benders Decomposition (GBD) Geoffrion (1972) Outer-Approximation (OA) Duran and Grossmann (1986),Fletcher and Leyffer (19 LP/NLP based Branch and Bound Quesada, Grossmann (1994) Extended Cutting Plane(ECP) Westerlund and Pettersson (1992) Availability of New Codes: SBB GAMS simple B&B MINLP-BB (AMPL)Fletcher and Leyffer (1999) Bonmin (COIN-OR) Bonami et al (2006) FilMINT Linderoth and Leyffer (2006) KNITRO Nocedal (2009) DICOPT (GAMS) Viswanathan and Grossman (1990) AOA (AIMSS) α ECP Westerlund and Peterssson (1996) MINOPT Schweiger and Floudas (1998) BARON Sahinidis et al. (1998) Couenne Belotti, Margot (2008) Typical sizes: up to variables, 10,000 cont. vars./constraints 11

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13 OR operator Generalized Disjunctive Programming (GDP) Raman and Grossmann (1994) (Extension Balas, 1979) Motivation: Facilitate modeling discrete/continuous problems j J min Z = k Y jk s.t. r ( x ) 0 Y jk g ( x ) 0 jk c γ = k jk Ω = true x ( Y ) R c + n k, c k k f ( x ) R { true, false } 1, k K Objective Function Common Constraints Disjunction Constraints Fixed Charges Logic Propositions Continuous Variables Boolean Variables Codes: LOGMIP, EMP 13

14 Global Optimization Algorithms Algorithms based on spatial branch and bound method and use of underestimators/convex envelopes (Horst & Tuy, 1996) Nonconvex NLP/MINLP αbb (Adjiman, Androulakis & Floudas, 1997; 2000) BARON (Branch and Reduce) (Ryoo & Sahinidis, 1995, Tawarmalani and Sahinidis, 2002) Branch and cut (Kesavan, Allgor, Gatzke and Barton, 2004) Branch and Contract (Zamora & Grossmann, 1999) Transformation signomials (Bjoerk, Lindberg, Westerlund, 2002) Couenne (COIN-OR) (Belotti & Margot, 2008) Nonconvex GDP Two-level Branch and Bound (Lee & Grossmann, 2001) Bound strengthening (Ruiz, Grossmann, 2010) Typical sizes: up to variables, 1,000 cont. vars./constraints 14

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16 Synthesis of Heat Exchanger Networks Yee and Grossmann (1990) Stage k=1 Stage k=2 S1 H1 t H1,1 t C1,1 H1-C1 H1-C2 t C1,2 t H1,2 H1-C1 H1-C2 t H1,3 t C1,3 C1 CW S1 H2 t H2,1 H2-C1 t C2,1 t C2,2 H2-C2 t H2,2 H2-C1 H2-C2 t H2,3 t C2,3 C2 CW temperature location k=1 temperature location k=2 temperature location k=3 Multiple stages with potential heat exchangers z ijk = 0,1 3 hot, 4 cold streams MINLP , 270 cont vars, 258 constr. 3s CPU-time (DICOPT-2013) 16

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18 LP Transshipment Model for Min Utility Cost Papoulias and Grossmann (1983) Q Fuel Unknowns: 5600 H1 R 1 a) Utility loads Q Fuel, Q HP, Q LP, Q CW H Q HP R 2 C b) Heat residuals: R 1, R H3 R C H4 Q LP R R Q CW 18

19 Transshipment Model Chen, Miller, Grossmann (2013) Case Study Results for Similar Fcps (Academic) Problem Size LP Transshipment Model MILP Transshipment Model # of Hot Streams * # of Cold Stream # of Variables # of Constraints CPU Time (seconds) # of Continuous Variables # of Binary Variables # of Constraints Optimum Solution / Best Bound LP Relaxation CPU Time (seconds) 2 * * * * * * 20 * * * Time limit 19

20 Transshipment Model Case Study Results for Dissimilar Fcps (Industrial) Problem Size LP Transshipment Model MILP Transshipment Model # of Hot Streams # of Cold Stream # of Variables # of Constraints CPU Time (seconds) # of Continuous Variables # of Binary Variables # of Constraints Optimum Solution / Best Bound LP Relaxation CPU Time (seconds) 2 * * * * * * * MILP Industrial Problems Easier to Solve! * Time limit 20

21 Superstructure of the integrated water network Carnegie Mellon MINLP: vars, 233 cont var, 251 constr optcr= CPUsec (BARON) 21

22 Optimal design of the simplified water network with 13 removable connections Optimal Freshwater Consumption 40 t/h vs 300 t/h conventional Carnegie Mellon 22

23 Strategy for simultaneous optimization Duran, Grossmann (1986) Yang, Grossmann (2011) PROCESS STRUCTURE PROCESS FLOWSHEET Fresh water Wastewater Water streams WATER TARGETING MUC* Utility networks Cold streams Hot streams MUC* HEAT TARGETING Cold utility Hot utility WN STRUCTURE HEN STRUCTURE PROCESS FLOWSHEET WITH HEN AND WN *MUC minimum utility consumption 23

24 LP Targeting model for minimum freshwater consumption Assumes only process units (reuse, recycle) Yang, Grossmann (2012) min s.t. F F C F Z = F fw k k F C k k j k = i m ku j = = i s i C j = P in out p in p out F i i m i F i F = P i ( F Loss p in FC i i j iu j j, s SU, p PU, k p p j j, p PU, i p m MU, k m j, s SU, k s in in out, k p out out i s p PU, i pin k ) C + L = ( F Gain m MU, k m p out, ) C k j k s out in Mixer mass balances Splitters mass balances Process unit mass balances Carnegie Mellon 24

25 Sequential vs. simultaneous result comparison Methanol Process from Syngas Carnegie Mellon SEQUENTIAL SIMULTANEOUS Profit (1000 $/yr) 62,695 73,416 Investment cost (1000 $) 1,891 1,174 Operating parameters Material flowrate (10 6 kmol/yr) electricity (KW) freshwater (kg/s) heating utility (10 9 KJ/yr) cooling utility (10 9 KJ/yr) Steam generated (10 9 kj/yr) overall conversion feedstock product % improvement LP exact target! 25

26 Optimization of Number of Trays Viswanathan & Grossmann (1993) Non-existing tray zr i = 0,1 zr m =1 No liquid on tray Vapor Flow MINLP => Number trays Existing trays zb n =1 Liquid Flow zb i = 0,1 Non-existing tray No vapor on tray Discrete variables: Number of trays, feed tray location. Continuous variables: reflux ratio, heat loads, exchanger areas, column diameter. Acetone-acetonitrile-water, max 25 trays, Virial-UNIQUAC MINLP: vars, 891 cont, vars, 957 const. 40 minutes in 1992, 10 secs in 2013!! 26

27 Azeotropic Example Bartfeld, Aguirre, Grossmann (2004) Problem Specs Mixture: Methanol/ Ethanol/ Water Feed composition: 0.5/ 0.3/ 0.2 Feed: 10 moles/s Pressure: 1 atm Max no. trays: 20 (per section) Min purity: 95% Ideal/Wilson models GDP Model Superstructure Initialization 1.0 F methanol ehtanol ethanol Azeotrope Water Discrete Variables 210 Continuous Variables 9025 Constraints 8996 l o n a th e M n tio c r a F le o M Feed Liq. Col. 1 Liq. Col. 2 Liq. Col. 3 Liq. Col. 4 Liq. Col Mole Fraction Ethanol 27

28 Azeotropic Example Product Specifications 95% PP 1 = mole/sec 95% Methanol Optimal Configuration $318,400 /yr F kw PP 4 = mole/sec 95% Ethanol kw PP 5 = mole/sec Azeotrope Profiles Optimal Configuration kw 1.0 PP 6 = mole/sec 95% Water Mole Fraction Methanol Fedd Liq. Col 1 Liq. Col 2 Liq. Col 3 4 out of 10 sections deleted Optimal Solution Annual Cost ($/year) 318,400 Preprocessing (min) 6.05 Subproblems NLP (min) 36.3 Subproblems MILP (min) Iterations 3 Mole Fraction Ethanol Total Solution Time (min) MHz. Pentium III PC 28

29 Superstructure Separations Olefins Plant (Lee, Foral, Logsdon, Grossmann, 2003) 25 states 53 separation tasks BCDEFGH B/CDEFGH BCD/EFGH BCD/CDEFGH BCDEF/CDEFGH BCDEF/EFGH CDEFGH CD/EFGH CDEF/EFGH CDEF/GH CDEFG/H EFGH EF/GH EFG/H EFG EF/G GH G/H C5 H G A/BCDEFGH BCDEF/GH BCDEFG/H C4 C3H8 A- H 2 B- CH 4 C- C 2 H 4 D- C 2 H 6 E- C 3 H 6 F- C 3 H 8 G- C 4 H- C 5 Feed AB/CDEFGH ABCD/CDEFGH ABCDEF/CDEFGH ABCDEFGH ABCDEF/EFGH ABCD/EFGH ABCDEF/GH BCDEFG B/CDEFG BCD/CDEFG BCDEF/CDEFG BCDEF/EFG BCD/EFG BCDEF/G CDEFG BCDEF CD/EFG CDEF/EFG CDEF/G B/CDEF BCD/CDEF BCD/EF CDEF CD/EF EF CD F E/F E C3H6 C2H6 D C/D ABCDEFG/H ABCDEFG A/BCDEFG AB/CDEFG ABCD/CDEFG ABCDEF/CDEFG C C2H4 CH4 B STATES TASKS ABCDEF/EFG ABCD/EFG ABCDEF/G NON-SHARP ABCDEF A/BCDEF AB/CDEF ABCD/CDEF ABCD/EF ABCD BCD A/BCD AB/CD B/CD AB A/B A H2 GDP big-m MINLP: 5, vars, 24,500 cont. vars., 52,700 constraints ~3hrs CPU-time 29

30 MINLP optimal solution Dephlegmator first process 7 separation units Total cost: MM$/yr A H2 20M$/yr cost saving compressor AB -141F 900Psig Cold Box A/B 900Psig B CH4 heater compressor Dephlegmator ABCDEFGH 100F 480Psig valve AB CD 480Psig 480Psig valve CD -41F 160Psig C D -51F 140Psig Chemical Absorber -31F 140Psig 72F 140Psig C D C2H4 C2H6 cooler 1 dephlegmator 1 absorber 4 distillation columns 1 cold box 1 heat exchange CDEFGH 84F 190Psig D E 74F 170Psig Deethanizer 214F 170Psig EFGH 169F 160Psig F G 238F 140Psig 109F 140Psig Depropanizer pump pump 410 Mkwh/yr GH 226F 160Psig 236F 140Psig EF 83F 160Psig G H E F 123F 140Psig C3 Splitter 99F 140Psig Debutanizer E F G H C3H6 C3H8 C4 C5 30

31 Optimal Scheduling of Industrial Combined Heat and Power Plants under Time-sensitive Electricity Prices II Use the flexibility Demand of the CHP Side plant Management to adjust steam production for variability in electricity price Fuel X GT B 1 B 3 ~ B2 HRSG ST1 Electricity prices vary on an hourly basis ST2 Power Grid Electricity Steam at different pressure levels Mitra, Sun, Grossmann (2012) Raw materials I Account for variability in electricity prices in production planning 1 Products CHP plant Typicallymultiple boilers and turbines (steam, gas) Chemical plant The incentives given by utilities and grid operators to adjust power consumption/ production increase profitability, if the processes are able to cope with variability Mitra, S., I.E. Grossmann, J.M. Pinto and Nikhil Arora, "Optimal Production Planning under Time-sensitive Electricity Prices for Continuous Power-intensive Processes, Computers & Chem. Eng., 2012, 38, Samad, T.; Kiliccote, S., Smart Grid Technologies and Applications for the Industrial Sector, Computers & Chem. Eng., Wassick, J.M., Enterprise-wide optimization in an integrated chemical complex, Computers & Chem. Eng., 2009, 33,

32 A CHP plant has inherent flexibility in order to satisfy power and steam demand Fuel Air Fuel Water Exhaust stream (waste incineration plant) Water X GT B1 B3 ~ B2 Water HRSG Connection to power grid Exhaust Gas ST1 ST2 Power demand HP steam demand MP steam demand LP steam demand Condensate Given: Determine: - Boilers (B1, B2, B3) - Steam turbines (ST1, ST2) - Gas turbines (GT) w/ HRSG - Demand (HP, MP, LP, electricity) - Hourly electricity prices e h - Production levels and internal flows - Mode of operation for each equipment - Sales of electricity - Purchases of electricity for every hour for an entire week How should the plant be operated in order to maximize the operating profit 1, while satisfying power and steam demand of the chemical plant? 1. Operating profit = Electricity sales (internal and external) + Steam sales Fuel costs Start-up costs Electricity purchases 32

33 Disjunctive Programming (DP) is used as a modeling framework to represent the CHP plant Objective function: max Maximize operating profit over an entire week (7*24 hours = 168 discrete time periods) Electricity sales (internal and external) + Steam sales Fuel costs Start-up costs Electricity purchases 1 Disjunction over operating modes to describe feasible region of operation 2 -How to represent each component in terms of state graph and feasible region? Logic constraints to model restrictions of the state graph - How to address the previously mentioned model requirements? 2 for each CHP plant component Ramping constraints 3 Mass balances, demand constraints and additional constraints (e.g. energy exchange with the grid, restrictions on shutdowns) The model has 70,009 constraints, 49,535 variables (8,722 binaries) and can be solved within 2 minutes (CPLEX 12.4) 33

34 Weekly profiles for a case with 61% utilization 1 and 2 shutdowns allowed per component Electricity price profile Steam profiles for boilers and gas turbine Electricity profiles for steam and gas turbines HP steam input profiles for steam turbines MP, LP and cond. output profiles for steam turbines 1. Capacity utilization as percentage of maximum steam production. 34

35 Conclusions 1. Discrete/continuous optimization methods have had - tremendous progress in MILP - very significant progress in MINLP/GDP - good progress in global MINLP/GDP 2. Mathematical Programming has significantly impacted Process Integration - Heat Exchanger Networks - Water Networks - Power Systems - Separation Systems - Process Flowsheets 3. Math Programming models used increasingly by industry Math Programming can be successfully used in combination with heuristics, physical insights, enumerative methods 35