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AIChE 200 7 Annual Meeting Sa lt Lake , UT November 04 – 09 , 200 7

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### Solving Flexibility Index Problem Using Combined Stochastic Method and Reduced Space Search Algorithms

AIChE 2007 Annual Meeting

Salt Lake, UT

November 04 – 09, 2007

Session #288: Design, Analysis and Operations Under Uncertainty (10A03)

Paper 288e

Jeonghwa Moon, Kedar Kulkarini, Libin Zhang and Andreas A. Linninger

11/06/2007

Laboratory for Product and Process Design,

Department of Chemical Engineering, University of Illinois,

Chicago, IL 60607, U.S.A.

Introduction

- Flexibility index problem (Swaney and Grossman, 1985)
- determine the maximum parameter (q) range that a design can tolerate for feasible operation
- Previous work
- Exhaustive enumeration (Swaney and Grossmann, 1985a)
- Implicit vertex enumeration and heuristic vertex search (Swaney and Grossmann, 1985b)
- Active constraint formulation (Grossmann and Floudas, 1987)
- Convex Hull methods (Ierapetritou et al, 2001)
- Deterministic global optimization methods – a-BB algorithm (Floudas et al, 2001)

FI : defined as the largest scaled deviation of any of the expected deviations

We suggest a

New hybrid algorithm (genetic algorithms+ line search for handling constraints)

Characteristics of FI problem- Why not deterministic way?

(a)

(b)

Non-differentiable and discontinuous

Global optimization problem

Search space

Search space

Search space

Rejection

Penalization

Repairing

Genetic Algorithm for constraint problems- Features of genetic algorithm
- Stochastic optimization method which mimics natural selection and principles of genetics .
- It is global optimizer
- No sensitivity information is required
- The ways to treat individuals not in a search space

Framework

START

Cost function, Search space

Initial population

Repair individuals

Cost evaluation

Natural Selection

No

Mating

Mutation

Is convergent?

Yes

END

Framework

START

Cost function, Search space

Initial population

Repair procedure

Cost evaluation

Natural Selection

No

Mating

Special algorithm is needed

Mutation

Is convergent?

Yes

END

Framework

START

Cost function, Search space

Initial population

Repair procedure

Cost evaluation

Natural Selection

No

Mating

Some of superior individuals are selected

Some of inferior individuals are removed

Mutation

Is convergent?

Yes

END

Framework

START

Cost function, Search space

Initial population

Repair procedure

Cost evaluation

Natural Selection

No

Mating

Mutation

Is convergent?

Yes

END

Framework

START

Cost function, Search space

Initial population

Repair procedure

Cost evaluation

Natural Selection

No

Mating

Mutation

Some of individuals have new values randomly

Is convergent?

Yes

END

Framework

START

Cost function, Search space

Initial population

Repair procedure

Cost evaluation

Natural Selection

No

Mating

Mutation

Is convergent?

Yes

END

Framework

START

Critical point will be found in several iterations

Cost function, Search space

Initial population

Repair procedure

Cost evaluation

Natural Selection

No

Mating

Mutation

This loop is repeated

Is convergent?

Yes

END

i=1

(+)

i=2

Choose direction (+/-)

Θ is feasible

i=2

(+)

(-)

i=1

Θ is infeasible

(+)

i=2

Move individual until it meets boundary

No

Select parameter to move

Yes

End

Repair procedure: How to move ?The line search- (without control variable)

Step size

Feasibility function when z exists

- Movable constraints & actual feasible region

θ2

The constraints vary!

Z=0.0

Z=0.5

Z=1.0

θ1

z

θ2

θ1

Select default z value

z

Zigzag movement

θi= θi+ α ∆ θi

Repaired individual

1.0

Select zk

0.5

Decide b(+/-) and ∆z

Repeat until it reaches boundary

zk=zk+bdz

0.0

θ1

:extended

Extended line search- How to get new replacement of individuals when control variables exist ?
- Extended line search:
- Changes z values to have θ go as far as possible
- Zigzag movement in z and θ space

z1

θ

θ

0.0

0.75

1

z1

0.4

0.0

0.0

z2

0.5

0.5

0.0

Multiple control variables- When many control variables exist
- Select one z and do extended line search
- Then swap z and do again
- Repeat this until every z is involved in search

z1=0

z2

z2=0.5

z1

θ

θ

Select z2 and do line search

Select z1 and do line search

Case studies

- Several studies are done

Reactor-cooler (Floudas2001)

Heat Exchanger network-1 (Swaney, 1985)

Pump and pipe( Grossmann 1987, Floudas 2001)

Polymerization Reactor

- Average of molecular weight should be kept between 52,569g/mol and 64,361/g/mol
- Temperature of reactor must be less than 423K

Control variables : QI,Qc

Conclusion

- It’s new approach for flexibility index problem.
- Uses stochastic method with line search algorithms.
- Uses geometric characters of uncertain parameters and control variables spaces.
- Works regardless of convexity of feasible region.
- Easy to formulate.
- Proper parameter values of GA are needed. (population size , maximum iteration, selection ratio, mutation ratio)
- Our research provides another option for flexibility index problem.
- Future work
- Genetic algorithm is the best choice for this problem? (PSO, SA)
- Another novel repair procedure?
- Mating method, termination condition should be studied

Reference

- Perkins, J.D. & S.P.K. Walsh, 1994, Optimization as a Tool for Design/Control Integration. Proceedings of the IFAC workshop on Integration of Process Design and Control CE. 7_afiriou Ed.), 1.
- I. E Grossmann and M. Morari, Operability, resiliency and flexibility : process design objectives for a changing world proc 2nd Int. Conf Foundations Computer Aided Process Design (Weterberg and Chien Eds). CACHE 937 (1984)
- Halemane K, Grossmann IE. Optimal Process Design under Uncertainty. AIChE J. 1983; 29(3):425-432.
- Swaney R, Grossmann IE. An Index for Flexibility in Chemical Process Design Part I. AIChE J. 1985; 31(4): 621-630.
- Grossmann IE, Floudas CA. Active Constraint Strategy for Flexible Analysis in Chemical processes. Comp. Chem. Eng. 1986;11(6):675-693.
- Floudas CA, Gumus ZH, Ierapetritou MG. Global Optimization in Design under Uncertainty: Feasibility Test and Flexibility Index Problems. Ind. Eng Chem Res. 2001; 40: 4267-4282.
- Ierapetritou MG. New Approach for Quantifying Process Feasibility: Convex and 1-D Quasi-Convex Regions
- Holland JH. Adaptation in Natural and Artificial Systems. Ann Arbor: University of Michigan Press, 1975.
- Goldberg DE. Genetic Algorithms in Search, Optimization, and Machine Learning. Reading ,MA: Addison-Wesley, 1989.
- Michalewicz Z. Genetic Algorithms + Data Structures = Evolution Programs, New York, NY: Springer-Verlag, Second Edition, 1994.
- Radcliff, N.J. 1991. Forma analysis and random respectful recombination. In proc 4th Int. Conf. on genetic algorithms, San Mateo, CA: Morgan Kauffman.
- Maner BR, Doyle FJ, Ogunnaike BA, Pearson RK. Nonlinear model predictive control of a simulated multivariable polymerization reactor using second order Volterra models. Automatica, 32(9):1285-1301, 1996.

Four infeasible areas are found simultaneously

Future work-feasibility test- Feasibility test problem

- Practical and robust
- less expensive than flexibility index problem
- Multiple infeasible areas can be found!
- Using niche technic

Finding multiple solutions in feasibility test problem using fitness sharing

θ1

Z=0.0->0.5

θ2

θ2

θ2

z

Zigzag movement

1.0

θ1

θ1

θ1

0.5

Z=0.5

Z=0.5->1.0

Z=1.0

0.0

θ1

Example of extended line searchθ2

θ1

Z=0.0

θ1

Z=0.0->0.5

θ2

θ2

θ2

z

Zigzag movement

1.0

θ1

θ1

θ1

0.5

Z=0.5

Z=0.5->1.0

Z=1.0

0.0

θ1

Example of extended line searchθ2

θ1

Z=0.0

z1

θ

z2=0.5

z1

θ

z1=0

z2

θ

Multiple control variablesm=2

k=1

Line search with (θi,zk)

Local termination?

Yes

k=1

k=1?

No

No

Yes

Any movement of θi is done?

k=2

Yes

No

k=1

k=k+1

k=1

k>m

Yes (global termination)

k=2

End

k=3 : globally terminated

m : no. of control variables

Reference

- Grossmann,I.;Halemane,K.;Swaney, R., “Optimization Strategies For Flexible Chemical Processes” ,Computers and Chemical Eng.,Vol.7(4),439-462,1983.
- Halemane,K.; Grossmann,I., “Optimal Process Design under Uncertainty” , AIChE Journal,Vol.29(3),425-432,1983.
- Grossmann,I.; Swaney, R., “An Index for Flexibility in Chemical Process Design Part I” ,AIChE Journal,Vol.31(4),621-630,1985.
- Grossmann,I.; Swaney, R., “An Index for Flexibility in Chemical Process Design Part II” ,AIChE Journal,Vol.31(4),631-641,1985.
- Grossmann,I.; Floudas,C., “Active Constraint Strategy for Flexible Analysis in Chemical Processes” , Computers and Chemical Eng.,Vol.11(6),675-693,1986.

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

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

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

?

?

Move θ

θ1

?

Move z

?

Move z

Move θ

Move θ

Individual movement using repairing algorithm

Repair procedure- How to get new replacement of individuals ?
- Extended line search:
- Changes z values to have θ go as far as it can
- Zigzag movement in z and θ space
- Local optimization method

Select θi and α

Select default z value

θi= θi+ α h

Select zk

Decide b(+/-) and ∆z

Repeat until it reaches boundary

zk=zk+bdz

:extended

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