Dynamic Real-Time Optimization via Tracking of the Necessary Conditions of Optimality

Transcription

1 Dynamic Real-Time Optimization via Tracking of the Necessary Conditions of Optimality D. Bonvin and B. Srinivasan + Laboratoire d Automatique EPFL, Lausanne, Switzerland + Department of Chemical Engineering Ecole Polytechnique Montreal, Canada

2 D-RTO under Uncertainty Optimization under uncertainty Robust optimization Measurement-based optimization Explicit scheme (NMPC) Update a process model and repeat the optimization Implicit scheme () Use a solution model for updating the inputs directly 1

3 Outline Optimization and uncertainty Tracking the Necessary Conditions of Optimality Turn optimization problem into control problem Decentralized control scheme Simulation results Illustrative example Application projects Conclusions 2

4 Illustrative Example Semi-batch Chemical Reactor B k 1 A + B C 2 B k 2 D u(t) T A Exothermic reactions -- Isothermal T j Terminal objective: Maximize number of moles of C at t f by adjusting u(t) Safety path constraint: Heat removal limitation T j T j,min Selectivity terminal constraint: Number of moles of D at t f n Df n Df,max 3

5 Nominal Optimization Physical knowledge Modeling Process model Numerical Optimization Nominal solution 4

6 Nominal Optimal Solution

7 Implementation of the Nominal Solution Physical knowledge Modeling Process model Numerical Optimization Nominal solution Real Plant Uncertainty (disturbances, model mismatch) Feasibility? Optimality? 6

8 Measurement-based Optimization Physical knowledge Modeling Measurementbased Adjustments explicit implicit Numerical Optimization Process model Updated inputs Measurements Real Plant Uncertainty Feasibility OK Optimal performance OK 7

9 : Self-optimizing control structure Modeling Solution model Numerical Optimization Process model NCOtracking Controller Measurements Updated inputs Real Plant Uncertainty Feasibility OK Optimal performance OK Control problem 8

10 Solution Model Input representation capable of achieving (near) optimality Choice of decision variables Identify arcs and switching times that vary with uncertainty Introduce approximations and parameterization as needed Pairing for decentralized control Associate combinations of decision variables with active constraints Associate remaining decision variables with sensitivities 9

11 Input Dissection u u max 1 η 1 (t) 2 t 1 t 2 η 2 (t) 3 0 t Model Fixed part -- u max Variable parts -- t 1, t 2, η 1 (t), η 2 (t) Decision variables 10

12 NCO -- Solution Model A Decision variables: t 1, t 2, η 1 (t), η 2 (t) Pairing Constraints Sensitivities Path Objectives During the run T j = T j,min H/ η 2 = 0 H: Hamiltonian Terminal Objectives n D (t f ) = n Df,max End of the run - 11

13 Solution Model A Invariant part -- u max Path constraint -- T j = T j,min -- t 1, η 1 (t) Terminal constraint -- n D (t f ) = n Df,max -- t 2 Sensitivity -- H/ η 2 = 0 -- η 2 (t) u = % u ' max & K " (T j,min # T j ) ' ( G " ($H /$" 2 ) 0 ) t ) t 1 t 1 < t ) t 2 t 2 < t ) t f t 1 = t with T j (t) = T j,min and T j (t # ) > T j,min t 2 = R * (n Df,max # n D (t f )) 12

14 NCO-tracking Scheme for Solution Model A On-line T j,min n Df,max PI Controller I Controller PI Controller η 1 (t) t 2 η 2 (t) 2 3 u max 1 Input Generation u(t) Uncertainty System T j (t) x(t) n D (t f ) H η2 (t) Estimation of Sensitivity On-line Run-to-run 13

15 NCO -- Solution Model C Decision variables: t 1, t 2, η 1 (t), η 2 (t) Different pairing Constraints Sensitivities Path Objectives During the run T j = T j,min - Terminal Objectives End of the run n D (t f ) = n Df,max n C (t f )/ t 2 = 0 14

16 Solution Model C Invariant part -- u max Path constraints -- T j = T j,min -- t 1, η 1 (t) Terminal constraints -- n D (t f ) = n Df,max -- η 2 (t) Sensitivity -- n C (t f )/ t 2 = 0 -- t 2 u = $ u & max % K " (T j,min # T j ) & ' T " (n Df,max # n Df pred (t)) 0 ( t ( t 1 t 1 < t ( t 2 t 2 < t ( t f t 1 = t with T j (t) = T j,min and T j (t # ) > T j,min t 2 = G ) (*n C (t f ) /*t 2 ) 15

17 NCO-tracking Scheme for Solution Model C On-line T j,min 0 n Df,max - - PI η 1 (t) 2 Controller Interpolation I Controller n d D (t) - t 2 η 2 (t) 3 PI Controller u max 1 Input Generation Uncertainty u(t) System On-line T j (t) n D (t) n C (t f ) δn C (t f )/δt 2 Estimation of Sensitivity Run-to-run 16

18 NCO-tracking -- Input Profiles 17

19 NCO-tracking Results Model and run index Minimum jacket temperature 10 C Maximal final amount of D 100 mol Final amount of C Cost (mol) A A A C C C Ideal cost: Open-loop nominal cost:

20 in Practice Features of NCO tracking No need of a process model for implementation Need appropriate process measurements Approximations can (must) be introduced in solution model Practical observations Complexity depends on the number of inputs (not system order) For many terminal-time dynamic optimization problems, the solution is often determined by the constraints of the problem Practical extensions Measurement noise backoff Unknown active constraints superstructure 19

21 Projects Implementing On-line optimization Semi-batch reactor with safety constraint (Novartis Basel; Bayer-RWTH) Discontinuous wastewater treatment plant (Firmenich Geneva) Fed-batch fermenter (Bioengineering Lab EPFL) Batch distillation column (Engineering School Fribourg) Grade transitions in polymerization (Bayer-RWTH; CEPRI Thessaloniki) Run-to-run optimization Emulsion copolymerization reactor (Aqua+Tech Geneva) Electro-discharge machining (Charmilles Geneva) Batch reactive distillation column (INPT Toulouse) Fed-batch fermenter (Novozymes Denmark) Bold: experimental Italics: industrial 20

22 Conclusions Real-time optimization under uncertainty Turn optimization problem into control problem Considerable prior information goes into solution model How robust is this information wrt. uncertainty? solution model must remain valid over uncertainty Considerable potential for industrial applications Further work Rigorous mathematical framework Application to large-scale processes 21

23 F I N 22