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An overview of the active constraint methodology for flexibility analysis

An overview of the active constraint methodology for flexibility analysis. Kedar Kulkarni Advisor: Prof. Andreas A. Linninger Laboratory for Product and Process Design, Department of Bioengineering, University of Illinois, Chicago, IL 60607, U.S.A. Motivation.

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An overview of the active constraint methodology for flexibility analysis

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  1. An overview of the active constraint methodology for flexibility analysis Kedar Kulkarni Advisor: Prof. Andreas A. Linninger Laboratory for Product and Process Design, Department of Bioengineering, University of Illinois, Chicago, IL 60607, U.S.A.

  2. Motivation • To really understand various approaches of uncertainty analysis as employed by Grossmann et al. (1980-87) and apply them to design of distributed systems under uncertainty Outline of the talk • The active constraint methodology for flexibility analysis • A note on Lagrangian Multipliers

  3. Process adjustments Control variables, z Specifications Changing constraints Fixed conditions Design £ Uncertain f (d,z,q) 0 d q parameters, FEASIBLE OPERATION??? Flexibility Analysis

  4. Mathematical background Chemical process: System equations:h(d,z,x,q) = 0 (Mass or energy balances) Constraints:g(d,z,x,q)  0 (For feasible operation) Feasibility condition (Semantic): { q  T { z {s.t.  j  J, hj(d,z,x,q)  0}}} Design ‘just’ feasible

  5. Feasibility test: the formulation Feasibility condition (Mathematical): Calculate • x(d) could be regarded as a feasibility measure for a given • design d • * x(d)  0 => feasibility ensured q  T • * x(d) > 0 => design infeasible for at least one value of q Rewrite:

  6. The formulation

  7. The formulation KKT: Single level optimization problem solvable by available methods

  8. Flexibility index: the formulation Observing that to calculate the flexibility index we need to enforce that the feasibility test condition holds as an equality Thus

  9. Flexibility index: the formulation Thus

  10. Flexibility index: the formulation Thus

  11. Case study - I

  12. MILP formulation: Solution

  13. Case study – II (nz = 1)

  14. MILP formulation:

  15. A note on Lagrangian multipliers:

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