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Flexibility index analysis using geometric characteristics and Genetic Algorithm

LPPD Group seminar. Flexibility index analysis using geometric characteristics and Genetic Algorithm. J. Moon, L. Zhang And A.A. Linninger April 5 th , 2007 Laboratory for Product and Process Design, Department of Chemical Engineering, University of Illinois, Chicago, IL 60607, U.S.A.

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Flexibility index analysis using geometric characteristics and Genetic Algorithm

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  1. LPPD Group seminar Flexibility index analysis using geometric characteristics and Genetic Algorithm J. Moon, L. Zhang And A.A. Linninger April 5th , 2007 Laboratory for Product and Process Design, Department of Chemical Engineering, University of Illinois, Chicago, IL 60607, U.S.A.

  2. Contents • Introduction • Flexibility analysis • Genetic algorithm • Part I – FI without control variables • Cost function definition • Repairing procedure • Case studies • Part II – FI with control variables • Characteristic of geometry with control variables • Framework for flexibility index problem with control variable • Repair procedure • Case study: • Heat Exchanger network[5][6] • Pump and pipe[3][5][6] • Reactor-cooler system • conclusion • Future work

  3. Introduction • Flexibility index problem (analysis) • determine the maximum parameter (q) range that a design can tolerate for feasible operation • Applying traditional method for solving this problem is not efficient! • Gradient based method can find solution only with good initial guess-local solution • Maybe stochastic method (Genetic algorithm ,etc) is efficient

  4. Introduction • Genetic algorithm • Stochastic optimization method which mimics natural selection and principles of genetics . • It is globally convergent, so good initial guess is not necessary! • One of most important aspects of GA is “to define exact cost function” which represents each chromosome’s value. • This presentation shows how Cost Function of flexibility problem is defined! START Define cost function & parameters Generate initial population Find cost for each chromosome Yes Is convergent? No Natural Selection Mating Mutation END General procedure of genetic algorithm

  5. General concept of Repairing procedure • The ways to treat individuals not in a search space • Rejection (death Penalty): fitness =0 • Penalty : fitness- penalty: • Repairing: move illegal individuals to feasible search space • The procedure of repairing • (Local search for closet solution)-Baldwin effect • is a problem dependent (weakness) • Example • Salesman problem, knapsack problem, set covering problem • GENOCOP (GEnetic algorithm for Numerical Optimization for COnstrained Problems) Search space Repairing of infeasible individual

  6. Part I – without Control variables

  7. Nominal point Critical point Cost Function-definition • The Objective : we want to find out ‘critical point’ which : • lies on the constraints.(1st term) • Search space of GA is the boundaries of constraints • Individuals which are not on the constraints must • be killed ,get the penalty ,or be moved on one of constraints • has minimum flexibility index. (2nd term) • Cost ≈ & Penalty corresponding to distance from constraint Flexibility Index Search space + = 1st Term 2nd Term

  8. 1st Term: Repairing Procedure • 1st Term is reduced by Repairing procedure • Replacement of individuals using distance from constraint • Calculate all distances and closest points from every constraint. • Then select minimum one. • Move individual to the closest point.

  9. 1st Term: Repairing Procedure • 1st Term is reduced by Repairing process • Select dominant parameter and direction.(+-) • Move chromosome until it meets any constraint. • Dominant parameter is selected by • Direction is selected by After Repairing procedure, 1st term is diminished!

  10. Case Study 1: non-convex two dimensional 4y+9x-198≤0 20y-(x-4)(x-8)(x-12)-240 ≤0 -y ≤0 4y+9x-198-4sin(2x)≤0 20y-(x-4)(x-8)(x-12)-240 ≤0 -y ≤0 4y+9x-198-4sin(2x)≤0 20y-(x-4)(x-8)(x-12)-240 ≤0 -y ≤0 Pop size :400,Iteration :300,Selection ratio :0.5,Mutation ratio :0.1

  11. Case Study 2 • Convex 3D PN=(0.5,0.5,0.5) PN=(0.4,0.7,0.4)

  12. Case Study 3 • Non-convex 3D PN=(0.8,0.5,0.8) PN=(0.8,0.5,0.8) PN=(0.5,0.5,0.5) PN=(0.5,0.5,0.5)

  13. Part II – With Control variable

  14. (-1,0) (1.5,0) (2.1,0) Actual feasible region Characteristic of geometry with control variables • Movable constraints & actual feasible region θ2 The constraints move! Z=0.0 Z=0.5 Z=1.0 θ1 z θ2 θ1 If we know the explicit expression of actual feasible region, critical point can be obtained easily! (same with problem which does not have control variables )

  15. Characteristic of geometry with control variables • Fig 1, Flexibility index region is represented as hyper cube • Fig1, Nominal condition is represented as point • Fig 2, Flexibility index region is not represented as formal shapes • Fig 2, Nominal condition is represented as hyper cube. (not point) θ2 z F F ∆θ-=0 ∆θ-=0 θ1 θ1 Fig 1.Domain space without control variable Fig2. Domain space with control variable

  16. θ z Θ’ projection θ1 Framework for flexibility index problem with control variable • Main optimization problem is solved by Genetic algorithm • Every individual is moved to new place using repairing procedure START Define cost function & parameters Generate initial population Θ’ Repair chromosomes z3 z2 Find cost for each chromosome z1 θ2 Yes Is convergent? θ No θN Natural Selection Mating Mutation θ1 END

  17. Repair procedure • How to get new replacement of individuals ? • Other stochastic method (GA) • rSQP • … • Suggest new algorithm which is: • Zigzag movement • Does not need sensitivity information • Local optimization method • Multidimensional z, Unidimensional θ Move θ Select θ and direction Move z Move z Move θ (Initial step) Move θ Move θ Check moving direction of z Repeat until it is terminated Move z Individual movement using repairing algorithm Move θ

  18. g2 g1 Zi θ g2 Zi g1 θ g2 Zi g1 θ Checking moving direction of Z • Procedure • G+=Max(gi(θ, zi+h)) and G-=Max(gi(θ, zi-h)) • When h <0 • Terminate it. G+<0 G- >0 G+<0, G- <0 Zi g1 θ G+>0, G- >0 h gets less than 0

  19. z2 g2 znew Zi z1 g1 Moving direction :+ θ Movement of z & θ • Move control variable (z) • Find z2 moving z from z1 with direction until it meets constraint. • Change z value • znew=(z1+z2)/2 • Move uncertain parameter • Go until it meets constraint. • θ(i+1)= θ(i)+∆h • Two types of moving • From feasible region • From infeasible region g2 θ θnew θnew θ Zi g1 Moving direction :+ θ

  20. Multiple starting points • This repairing algorithm is a local optimizer, so multiple starting points are needed! It gets a local solution

  21. Termination condition • Local termination • No more movement of theta with changing one z • In case of h<0 (checking moving direction step) • Go to next z or the first z • Global termination • No more movement of θ with changing any z • θ meets boundary constraint

  22. Multiple control variables (z) • When moving point meets local termination condition with zi • When i is 1 (first) , then i=i+1 • When i is not 1 • When any movement of θ is done with zi :i=1; • When not , i=i+1 • So movement is done with most sensitive z at first • order of z is important • More case studies about Multiple z must be done!

  23. Flowchart of whole procedure of flexibility problem with control variables First step Move Define cost function & parameters Select zi, i=0 Generate initial population Check moving direction zi+ or zi- Repairing Y Y Is terminated? i=0 Find cost for each chromosome N N Is moved? Move z Natural Selection Y N Move θ Mating i=i+1 i=0 Is moved? Mutation No i>n Y: bmove=TRUE N:bmove=FALSE Is convergent? N Y Yes End End Repairing procedure

  24. H2,2kW/K H1,FH1 583K 723K 1 2 T1 C2 388K 563K 2kW/K T2 Qc 3 393K C1 313K 3kW/K T3≤323K Case study1-Heat Exchanger network[5][6] • Uncertain parameter • Heat flow rate of stream, H1 :FH1 (kW/K) • Control variable • Cooling load :Qc (K) • Nominal Value: • FNH1=1kW/K • Expected deviation • ∆F+H1=0.8kW/K, ∆ F-H1=0kW/K • Constraints after eliminating the state variables Network of case study with uncertain flowrate FH1 [5]

  25. Case study1-Heat Exchanger network[5][6] • One uncertain parameter FH1, and one control variable Qc • Starting point (1,15) Results of case study Feasible regions Repairing procedure for finding the critical point Feasible region of case study 1

  26. ∆P,k P2 P1 m D Cv ρ H,W η Case study2 –Pump and pipe[3][5][6] Design Variables Uncertain parameter

  27. Case study2 –Pump and pipe[3][5][6] • Modifications of constraints • Constraint g3 is max range of m -> Remove • Modify control variable • Constraint g4,g5 is min and max range of control variable • Finally,

  28. Case Study2 –Pump and pipe[3][6] Expected critical point m Nominal Point Cv’ P2 P2 m

  29. Case Study2 –Pump and pipe[3][6] • GA parameters • pop size=800,iteration =100 • Mating method=simplex mating • h=range/100,dh=h/100 • Critical point found • (P2,m,C’v)=915.4706,11.2749,0.3036 • FI=0.6375 -similar to Flodaus[6] (0.618) C’v=0.3 C’v=120

  30. Case Study2 –Pump and pipe[3][6] • Snapshots of repairing procedure Starting point boundary Starting point boundary

  31. Case Study3 –Pump and pipe[5][6] ∆P,k P2 P1 m D Cv ρ H,W η Design Variables Uncertain parameter

  32. Case Study3 –Pump and pipe[3][5][6] • Modification of constraint • Modify control variable • Constraint g4,g5 is min and max range of control variable • Finally, iteration :50 pop size=800 mating =simplex sub pop=4 h=100 dh=100

  33. Case study4-Heat Exchanger network[5] • Energy Balance • Inequality constraints • By eliminating equality constraints • Deviation :+-10 H1,1.5kW/K H2,1kW/K T1(TN1=620K) T5(TN5=583K) 1 2 563K C1 388K 2kW/K T4 T3(TN3=388K) T2 T6 Qc 3 C2 T8(TN8=313K) 393K 3kW/K T7≤323K Network of case study with uncertain flowrate FH1 [5]

  34. Case study4-Heat Exchanger network[5] • GA parameters • pop size=800 • iteration 50 • mating =simplex • h=ranges/50 dh=h/50

  35. Case study5: Reactor-cooler system • Mass & Energy Balance • Inequality constraints A->B T1 F0 CA V T1 CA0 T0 T4 T2 F1 T1 A T1 Fw Tw1 Tw2 Reactor-cooler system[6] Control variable: T1,T2,Tw2

  36. Case study5: Reactor-cooler system • Eliminating state variables Feasible region g1 FI=1 g2 g1 Feasible region when T1=389K,T2=350K ,Tw2=311.1K

  37. Conclusion • It’s new approach for flexibility index problem. • Uses geometric characters of uncertain parameters and control variables spaces. • Uses stochastic algorithm with intelligent repairing algorithm. • Works regardless of convexity of feasible region. • Proper parameter values of GA are needed. (population size , maximum iteration, selection ratio, mutation ratio and step size for RP) • It is not superior than deterministic method, (and vise versa) but our research provides another option for flexibility index problem. • More research is needed. • Genetic algorithm is the best choice for this problem? (PSO, AH) • Another novel repairing method? • What is best mating method for this problem?

  38. Future work • More case studies with multiple control variables • More research of movement from infeasible region (RP) • Another method, instead of current repairing procedure

  39. Reference • Grossmann,I.; Floudas,C., “Active Constraint Strategy for Flexible Analysis in Chemical Processes” , Computers and Chemical Eng.,Vol.11(6),675-693,1986. • Floudas CA,Gumus ZH, Ierapetritou,M G “Global Optimization in Design under Uncertainty: Feasibility Test and Flexibility Index Problems” Ind. Eng Chem Res 40, 4267-4282,2001

  40. θ z projection θ1 Framework for flexibility index problem with control variable • Main optimization problem is solved by Genetic algorithm • Every individual is moved to new place using repairing procedure START Define cost function & parameters Generate initial population Θ’ Θ’ Repair chromosomes z3 z2 Find cost for each chromosome z1 θ2 Yes Is convergent? θ No θN Natural Selection Mating Mutation θ1 END

  41. Genetic algorithm with repairing procedure START Define cost function & parameters Generate initial population Repair chromosomes Find cost for each chromosome Is converged? Yes No Natural Selection Mating Mutation END

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