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- 1. . A Lindenmayer system for heat exchanger network design with stream splitting Eric S Fraga Centre for Process Systems Engineering Department of Chemical Engineering UCL (University College London) 29 April – 1 May 2008 c 2008, All rights reserved. Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 1 / 30
- 2. Heat exchanger networks. Outline Heat exchanger networks 1 An evolutionary strategy 2 Implementation 3 Results 4 Conclusions 5 Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 2 / 30
- 3. Heat exchanger networks. Heat exchanger networks Energy consumption is often the largest cost of a process. One means of reducing energy use is through process integration: Identify matches for transfer of excess heat in one part of the process to another part. Problem is highly combinatorial, discontinuous and non-convex. Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 3 / 30
- 4. Heat exchanger networks. Problem deﬁnition and costing Example problem† Exchangers are costed using ˙ Stream Tin Tout Q Ccapital = α + βAγ (K) (K) (kW) 2 typically with γ ≈ 3 and the area, H1 443 333 30 A, is a function of the log-mean H2 423 303 15 temperature diﬀerence (LMTD): C1 293 408 20 ∆Tin − ∆Tout LMTD = log ∆Tin − log ∆Tout C2 353 413 40 † Pariyani et al. (2006). Computers & Chemical Engineering 30:1046. Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 4 / 30
- 5. Heat exchanger networks. Combinatorial aspects H1 Many possible alternative matches ... H2 ... with alternative order of placement. C1 Streams may be split as well. Split C2 Mix Result is highly combinatorial. Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 5 / 30
- 6. Heat exchanger networks. Combinatorial aspects H1 Many possible alternative matches ... H2 ... with alternative order of placement. C1 Streams may be split as well. Split C2 Mix Result is highly combinatorial. Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 5 / 30
- 7. Heat exchanger networks. Combinatorial aspects H1 Many possible alternative matches ... H2 ... with alternative order of placement. C1 Streams may be split as well. Split C2 Mix Result is highly combinatorial. Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 5 / 30
- 8. An evolutionary strategy. Outline Heat exchanger networks 1 An evolutionary strategy 2 Implementation 3 Results 4 Conclusions 5 Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 6 / 30
- 9. An evolutionary strategy. Evolving a HEN structure Biological analogue† : Genotype: the plan Phenotype: the instance For HEN, the aim is to use a genotype to represent overall structure: possible matches and stream splits. The phenotype is an instantiation of the genotype with speciﬁc matches. Question becomes one of representation for the genotype and how this can be manipulated in an evolutionary manner. † De Jong (2006). Proc. ACDM 2006, I C Parmee (ed.), 23-25. Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 7 / 30
- 10. An evolutionary strategy. Lindenmayer Systems Prusinkiewicz & Lindenmayer† presented a Lindenmayer system, often known as an L-system: The central concept of L-systems is that of rewriting. In general, rewriting is a technique for deﬁning complex objects by successively replacing parts of a simple initial object using a set of rewriting rules or Image courtesy Solkoll @ wikipedia. productions. We wish to apply this concept of rewriting to streams in heat exchanger network synthesis problems to create potential integration structures. † Prusinkiewicz & Lindenmayer (1990). “The algorithmic beauty of plants,” Springer-Verlag. Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 8 / 30
- 11. An evolutionary strategy. L-system deﬁnition An L-system is deﬁned by a tuple, G = V , ω, P where V the alphabet or set of symbols which can be replaced in a string by speciﬁc combinations symbols from the same set. ω the initial conﬁguration (set of strings). P (⊂ V × V ∗ ) is the set of replacement rules. For the heat exchanger network synthesis problem, we deﬁne an L-system, GHEN . Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 9 / 30
- 12. An evolutionary strategy. GHEN – 1. Symbols V The alphabet includes: +, - denote the heating and cooling requirements of each stream; S, E the start and end of each stream; x indication of exchange; s, m split and mix; and, [, ] start and end of split stream segments. The full alphabet, therefore, is V ≡ {−, +, S, E, s, m, [, ]} Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 10 / 30
- 13. An evolutionary strategy. GHEN – 2. Starting representation, ω The starting set of symbols is a set of strings, one for each stream in the network problem deﬁnition. Each cold stream is represented initially by the string S-E. Each hot stream by E+S. The hot and cold streams are written in opposite order to indicate the use of counter-current heat exchangers. Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 11 / 30
- 14. An evolutionary strategy. GHEN – 2. Starting representation, ω The starting set of symbols is a set of strings, one for each stream in the network problem deﬁnition. Each cold stream is represented initially by the string S-E. Each hot stream by E+S. The hot and cold streams are written in opposite order to indicate the use of counter-current heat exchangers. H1:E+S, H2:E+S, For our example, ω = . C1:S-E, C2:S-E; Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 11 / 30
- 15. An evolutionary strategy. GHEN – 3. Rule set, P Target → Replacement Rule Description - → x- R1 Add an exchanger to a cold stream - → s[x-][x-]m- R2 Split a cold stream + → x+ R3 Add an exchanger to a hot stream + → m]x+[]x+[s+ R4 Split a hot stream S→S R5 A do-nothing rule Note: the rule for splitting a hot stream creates a structure that is the reverse of that created by a cold stream split rule. Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 12 / 30
- 16. An evolutionary strategy. GHEN summary The L-system for HEN design is context free. It is non-deterministic, perfect for an evolutionary algorithm. The majority of strings (words generated from V ∗ starting with ω) represent a valid genotype for the HEN design problem. The genotype describes a conﬁguration with locations of splits and locations for integrated exchangers. Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 13 / 30
- 17. Implementation. Outline Heat exchanger networks 1 An evolutionary strategy 2 Implementation 3 Results 4 Conclusions 5 Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 14 / 30
- 18. Implementation. The evolutionary algorithm Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 15 / 30
- 19. Implementation. The evolutionary algorithm Given: population size, np ; number of generations, ng ; the L-system, GHEN = V , ω, P . Outputs: Best solution found. p ← ω {Initialise population} 1: return best solution in p 9: Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 15 / 30
- 20. Implementation. The evolutionary algorithm Given: population size, np ; number of generations, ng ; the L-system, GHEN = V , ω, P . Outputs: Best solution found. p ← ω {Initialise population} 1: for i = 1, . . . , ng do 2: return best solution in p 9: Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 15 / 30
- 21. Implementation. The evolutionary algorithm Given: population size, np ; number of generations, ng ; the L-system, GHEN = V , ω, P . Outputs: Best solution found. p ← ω {Initialise population} 1: for i = 1, . . . , ng do 2: Select g from population p {Random or ﬁtness based} 3: Choose rule, r ∈ P {Random} 4: return best solution in p 9: Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 15 / 30
- 22. Implementation. The evolutionary algorithm Given: population size, np ; number of generations, ng ; the L-system, GHEN = V , ω, P . Outputs: Best solution found. p ← ω {Initialise population} 1: for i = 1, . . . , ng do 2: Select g from population p {Random or ﬁtness based} 3: Choose rule, r ∈ P {Random} 4: r g → g {Apply rule} 5: return best solution in p 9: Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 15 / 30
- 23. Implementation. The evolutionary algorithm Given: population size, np ; number of generations, ng ; the L-system, GHEN = V , ω, P . Outputs: Best solution found. p ← ω {Initialise population} 1: for i = 1, . . . , ng do 2: Select g from population p {Random or ﬁtness based} 3: Choose rule, r ∈ P {Random} 4: r g → g {Apply rule} 5: Evaluate g {Instantiate phenotype from g } 6: return best solution in p 9: Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 15 / 30
- 24. Implementation. The evolutionary algorithm Given: population size, np ; number of generations, ng ; the L-system, GHEN = V , ω, P . Outputs: Best solution found. p ← ω {Initialise population} 1: for i = 1, . . . , ng do 2: Select g from population p {Random or ﬁtness based} 3: Choose rule, r ∈ P {Random} 4: r g → g {Apply rule} 5: Evaluate g {Instantiate phenotype from g } 6: Insert g into population subject to diversity constraint 7: Shrink p if necessary {so that |p| ≤ np } 8: 9: return best solution in p Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 15 / 30
- 25. Implementation. Phenotype instantiation A genotype describes the overall structure of a HEN. To instantiate a phenotype from a genotype: link the integrated exchangers and 1 deﬁne the appropriate optimisation problem to size exchangers 2 and determine split factors. Linking exchangers is non-deterministic so a single genotype may lead to diﬀerent phenotypes. The do-nothing rule, R5, allows for multiple instances of the same genotype in a population. Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 16 / 30
- 26. Implementation. Exchanger matching for phenotype instantiation Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 17 / 30
- 27. Implementation. Exchanger matching for phenotype instantiation Given: g, the genotype to instantiate. Outputs: ϕ, a phenotype instantiation of the genotype Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 17 / 30
- 28. Implementation. Exchanger matching for phenotype instantiation Given: g, the genotype to instantiate. Outputs: ϕ, a phenotype instantiation of the genotype 1: ϕ ← g {Phenotype is initially the genotype.} return ϕ 10: Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 17 / 30
- 29. Implementation. Exchanger matching for phenotype instantiation Given: g, the genotype to instantiate. Outputs: ϕ, a phenotype instantiation of the genotype 1: ϕ ← g {Phenotype is initially the genotype.} 2: while ∃ unassigned exchange term in ϕ do return ϕ 10: Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 17 / 30
- 30. Implementation. Exchanger matching for phenotype instantiation Given: g, the genotype to instantiate. Outputs: ϕ, a phenotype instantiation of the genotype 1: ϕ ← g {Phenotype is initially the genotype.} 2: while ∃ unassigned exchange term in ϕ do x1 ← random unassigned exchange term in ϕ 3: x2 ← complementary random unassigned exchange in ϕ 4: return ϕ 10: Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 17 / 30
- 31. Implementation. Exchanger matching for phenotype instantiation Given: g, the genotype to instantiate. Outputs: ϕ, a phenotype instantiation of the genotype 1: ϕ ← g {Phenotype is initially the genotype.} 2: while ∃ unassigned exchange term in ϕ do x1 ← random unassigned exchange term in ϕ 3: x2 ← complementary random unassigned exchange in ϕ 4: if x2 then 5: return ϕ 10: Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 17 / 30
- 32. Implementation. Exchanger matching for phenotype instantiation Given: g, the genotype to instantiate. Outputs: ϕ, a phenotype instantiation of the genotype 1: ϕ ← g {Phenotype is initially the genotype.} 2: while ∃ unassigned exchange term in ϕ do x1 ← random unassigned exchange term in ϕ 3: x2 ← complementary random unassigned exchange in ϕ 4: if x2 then 5: x2 ← new random complementary exchange term 6: return ϕ 10: Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 17 / 30
- 33. Implementation. Exchanger matching for phenotype instantiation Given: g, the genotype to instantiate. Outputs: ϕ, a phenotype instantiation of the genotype 1: ϕ ← g {Phenotype is initially the genotype.} 2: while ∃ unassigned exchange term in ϕ do x1 ← random unassigned exchange term in ϕ 3: x2 ← complementary random unassigned exchange in ϕ 4: if x2 then 5: x2 ← new random complementary exchange term 6: if x2 then {Match might not be possible} 7: return φ {Infeasible instantiation} 8: return ϕ 10: Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 17 / 30
- 34. Implementation. Exchanger matching for phenotype instantiation Given: g, the genotype to instantiate. Outputs: ϕ, a phenotype instantiation of the genotype 1: ϕ ← g {Phenotype is initially the genotype.} 2: while ∃ unassigned exchange term in ϕ do x1 ← random unassigned exchange term in ϕ 3: x2 ← complementary random unassigned exchange in ϕ 4: if x2 then 5: x2 ← new random complementary exchange term 6: if x2 then {Match might not be possible} 7: return φ {Infeasible instantiation} 8: create heat exchange link between x1 and x2 9: 10: return ϕ Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 17 / 30
- 35. Implementation. Example evolution with phenotype instantiation Initial: { H1:E+S, H2:E+S, C1:S-E, C2:S-E; } Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 18 / 30
- 36. Implementation. Example evolution with phenotype instantiation Initial: { H1:E+S, H2:E+S, C1:S-E, C2:S-E; } H1 H2 C1 C2 Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 18 / 30
- 37. Implementation. Example evolution with phenotype instantiation Initial: { H1:E+S, H2:E+S, C1:S-E, C2:S-E; } +R1: { H1:E+S, H2:E+S, C1:Sx-E, C2:S-E; } H1 H2 C1 C2 Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 18 / 30
- 38. Implementation. Example evolution with phenotype instantiation Initial: { H1:E+S, H2:E+S, C1:S-E, C2:S-E; } +R1: { H1:E+S, H2:E+S, C1:Sx-E, C2:S-E; } H1 H2 C1 C2 H1:E+S, H2:Ex(C1)+S, C1:Sx(H2)-E, C2:S-E; Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 18 / 30
- 39. Implementation. Example evolution with phenotype instantiation Initial: { H1:E+S, H2:E+S, C1:S-E, C2:S-E; } +R1: { H1:E+S, H2:E+S, C1:Sx-E, C2:S-E; } H1 H2 C1 C2 H1:Ex(C1)+S, H2:E+S, C1:Sx(H1)-E, C2:S-E; Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 18 / 30
- 40. Implementation. Example evolution with phenotype instantiation Initial: { H1:E+S, H2:E+S, C1:S-E, C2:S-E; } +R1: { H1:E+S, H2:E+S, C1:Sx-E, C2:S-E; } +R2: { H1:E+S, H2:E+S, C1:Sx-E, C2:Ss[x-][x-]m-E; } H1 H2 C1 Split C2 Mix Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 18 / 30
- 41. Implementation. Example evolution with phenotype instantiation Initial: { H1:E+S, H2:E+S, C1:S-E, C2:S-E; } +R1: { H1:E+S, H2:E+S, C1:Sx-E, C2:S-E; } +R2: { H1:E+S, H2:E+S, C1:Sx-E, C2:Ss[x-][x-]m-E; } H1 H2 C1 Split C2 Mix H1:Ex(C2)+S, H2:Ex(C1)x(C2)+S, C1:Sx(H2)-E, C2:Ss[x(H2)-][x(H1)-]m-E; Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 18 / 30
- 42. Implementation. Example evolution with phenotype instantiation Initial: { H1:E+S, H2:E+S, C1:S-E, C2:S-E; } +R1: { H1:E+S, H2:E+S, C1:Sx-E, C2:S-E; } +R2: { H1:E+S, H2:E+S, C1:Sx-E, C2:Ss[x-][x-]m-E; } H1 H2 C1 Split C2 Mix H1:Ex(C2)+S, H2:Ex(C2)x(C1)+S, C1:Sx(H2)-E, C2:Ss[x(H2)-][x(H1)-]m-E; Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 18 / 30
- 43. Implementation. Embedded optimisation problem A network structure is created from the phenotype. The structure deﬁnes a nonlinear programme (NLP). The decision variables are xi ∈ [0, 1], the fraction to exchange, so that Qi = xi Qi,max , and xj ∈ [0, 1], the split fraction. Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 19 / 30
- 44. Implementation. Embedded optimisation problem A network structure is H1 created from the phenotype. H2 The structure deﬁnes a nonlinear programme x1 (NLP). x2 x3 The decision variables are C1 xi ∈ [0, 1], the fraction to exchange, so that x4 C2 Mix Qi = xi Qi,max , and xj ∈ [0, 1], the split fraction. Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 19 / 30
- 45. Implementation. Embedded optimisation problem A network structure is H1 created from the phenotype. H2 The structure deﬁnes a nonlinear programme x1 (NLP). x2 x3 The decision variables are C1 xi ∈ [0, 1], the fraction to exchange, so that x4 C2 Mix Qi = xi Qi,max , and xj ∈ [0, 1], the split fraction. The NLP represents a superstructure and is solved using a hybrid stochastic & direct search procedure† . † ˘ ESF (2006). In “Computer Aided Methods for Optimal Design and Operations”, Zilinskas & Bogle (ed.), 1-14. Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 19 / 30
- 46. Results. Outline Heat exchanger networks 1 An evolutionary strategy 2 Implementation 3 Results 4 Conclusions 5 Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 20 / 30
- 47. Results. Summary of performance A wide range of case studies has been considered: Objective function value (×103 $ y −1 ) Case study Best Mean Worst σ 1. 4SP 83.5 84.5 89.1 1.07 2. 10SP1 44.9 45.1 45.4 0.171 3. Morton 1620. 1680. 1720. 35. 4. Lewin A 573. 575. 594. 6.61 5. Aromatics 2940. 2960. 2980. 13. Best known solution obtained or bettered in all cases. Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 21 / 30
- 48. Results. 4SP Network viewer genotype:[R3(1), R4(1), R3(4), R2(1), R1(1), R1(4)] H1 H2 C1 C2 8.35E4 Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 22 / 30
- 49. Results. 4SP. Evolution of objective function Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 23 / 30
- 50. Results. 4SP. Variation in ﬁnal solution obtained Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 24 / 30
- 51. Results. 10SP1 Network viewer genotype:[R3(1), R3(4), R3(2), R3(5), R3(3), R4(3)] H2 H3 H5 H1 H4 C4 C3 C5 C1 C2 4.49E4 Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 25 / 30
- 52. Results. Morton Network viewer genotype:[R3(2), R3(3), R3(3), R4(1), R3(4), R1(2), R3(1)] H2 H3 H1 C3 C1 C2 1.62E6 6.19E5 Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 26 / 30
- 53. Results. Lewin A Network viewer genotype:[R2(1), R2(1), R1(4)] H3 H2 H5 H1 H4 C1 5.73E5 Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 27 / 30
- 54. Results. Aromatics Network viewer genotype:[R4(4), R3(5), R3(3), R3(2), R3(1), R3(1), R3(1), R3(1), R2(3)] H1 H2 H3 H4 C4 C3 C5 C1 C2 2.94E6 Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 28 / 30
- 55. Conclusions. Outline Heat exchanger networks 1 An evolutionary strategy 2 Implementation 3 Results 4 Conclusions 5 Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 29 / 30
- 56. Conclusions. Summary The challenging optimisation problem of HEN design with Network viewer genotype:[R3(1), R4(1), R2(1), R3(4), R1(4), R1(1), R3(3)] stream splitting has been H2 addressed through a simple H1 two-level algorithm with a Lindenmayer system for C1 evolving designs. C2 The result is a robust and 8.35E4 eﬀective tool for heat exchanger network design. http://www.homepages.ucl.ac.uk/~ucecesf/research/ Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 30 / 30

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