High Performance Processes Design under Uncertainty

1. High Performance Processes Design under Uncertainty Andrés Malcolm Advisor : Andreas A. Linninger University of Illinois at Chicago, Chicago, IL, September 20, 2006 Laboratory for Product and Process Design, Department of Chemical Engineering, University of Illinois, Chicago, IL 60607, U.S.A.

2. Detailed Chemical Process Design under Uncertainty Empirical Overdesign Design for High Performance Guarantee safe operation under all conditions

3. Arbitrary overdesign Flexible process? Model Optimize Control Integrated Design & Control Simultaneous Design and control optimization Model Exact Metrics of Flexibility Classical Design Approach • MODEL: • Make Steady State Model of the Process • DESIGN OPTIMIZATION: • Nominal operating conditions and specifications • Optimization and validation • DESIGN FLEXIBILITY • Empirical overdesign to accommodate uncertainty • DYNAMICS and OPERABILITY • Controller design to accommodate disturbances and uncertainty Arbitrary overdesign Flexible process? Model Optimize Control

4. Design constraint Steady State Analysis State Variable Dynamic Analysis   time Feasible Design Infeasible Design Dynamic Uncertainty Analysis

5. Feedback Existing Models Input Output Unit 1 Unit 2 Unit 3 Unit 4 Uncertainty propagation without feedback Real Practice Input Output Unit 1 Unit 2 Unit 3 Unit 4 • Uncertainty propagation with feedback • Quality Control • Chemical processes ARE feedback controlled. • Design needs to consider control influence.

6. Trade-off between design and control Total Cost (Capital + Operating) Design Decisions Optimal integrated design and control Control Complexity Optimal control for a design

7. Exploring the Control Dimension • Trade-off between design and control decisions • MIMO systems with high value product complex control schemes identified to be optimal • SISO systems with low value product: simpler control schemes identified to be optimal

8. High-Performance Chemical Process Design Under Uncertainty • Dynamic uncertainty analysis • To ensure flexibility we NEED a dynamic analysis • Consider Feedback • Dynamic analysis is INSEPARABLE from control Challenges • Design and control are traditionally separated • Integration renders non-polynomial (NP) hard non-convex MINLPs • No mathematical programming methodology can address complex integrated problems

9. θNEW, ξNEW(t) Not Flexible Update Critical Scenarios Min Cost Flexible Controlled Design Flexible Optimal d Problem Decomposition Initial d Sample Uncertain Space θ, ξ(t) Min Expected Cost Optimal Design and Control d Min Cost Controlled Design Rigorous Dynamic Flexibility Test

10. THOTIN Process Constraints 0 Control Design sp MCOLD≤ MMAX IN TCOLDIN TC Uncertainty Sources TT MHOTIN Design Variable = Area THOT≤ TMAX OUT End-Point Constraints Heat Transfer, U V-L Data T COLD out Identification of Structural Decisions • Mass, heat and momentum balances • DAE systems • Explore different equipment configurations find process constraints • Integer decisions • Identify controllable and manipulated variables • Integer decisions

11. Dynamic uncertainty ξ(t) time P() Time-Invariant uncertainty θ q2 time q1 Uncertainty Modeling Scenario sampling: by LHS techniques Simple method to compute expected performance

12. Simultaneous Design and Control • Stochastic dynamic optimization • Defined over the finite sample set • Optimizes design and control decisions for minimum expected cost Minimize Total Expected Cost Conservational Laws Control Algorithm Process and Product Constraints Process Design, Control Tuning and Structure in Objective Dynamic Constraints

13. Is Simultaneous Design and Control Solvable? Simultaneous d,c Optimization ... d2 d1 dm-1 dm c1 c1 c1 c1 c2 c2 c2 c2 ... ... ... ... cn-1 cn-1 cn-1 cn-1 cn cn cn cn Combinatorial Explosion of Design and Control Decisions NP-problem, dynamic non convex constraints Existing mathematic programming solutions do not work Not Solvable for industrial applications

14. Solution Approach : Problem Reformulation • Problem Reformulation: Embedded Control Optimization • Massive reduction of optimization problem search space • Find a control formulation that is implicitly related to the design • Avoid Combinatorial Explosion • Implicit relation should give optimal control action • Problem solvable for industrial applications Master Design Optimization (d) d1 d2 dm-1 dm ... c=f(d1) c=f(d2) c=f(dm-1) c=f(dm) Tractable Design Search Space

15. Embedded Control Optimization Embedded Control • Dynamic Linearization : Adapts to complex system dynamics. • Identify the system : Sequential identification. Fast convergence. • Estimate the full state : Accurate state prediction. Simple to solve. • Optimal control action: Optimal control action with algebraic solution. Avoids pairing problem.

16. 1. Dynamic Model Identification Adaptive system identification: Project existing design dynamics onto a set of linear ODEs. y u System (d) Identification Using the Sequential Least Squares (sLS) identification technique we can systematically obtain the i andi parameters

17. 1. Sequential Least Squares

18. 1. Sequential Least Squares (cont) Matrix Inversion Lemma No numerical Approximation!

19. 1. Model Identification: Sequential LS Algorithm Initialize the states y(-1)= y(0)=0, u(-1)= u(0)=0 Initialize the parameters r=0 r=r+1 Estimate the parameters r=rmax Estimated parameters

20. 1. Model Identification: Results Two states x two controls system identification 0.8 seconds to identify 3000 times steps

21. 2. State Estimation Having a model of the system and a set of measurements: Can I predict all system states minimizing the prediction error? Estimation Measured variable Current measurement Past “identified” data time Solutions: Direct extrapolation form the linearized model Luenberger full state observer (Ensures stable prediction) Need to solve eigenvalue problem Kalman Filter (Predictor – Corrector Recursive Estimator) Need to know an estimate of the noise covariance

22. 2. Kalman State Estimation Kalman Filter (Predictor – Corrector Recursive Estimator) d Having the system model: and the measurement y: The random variables process wk and measurement noise vk are assumed to be independent with normal probability distributions with covariance Q and R respectively.

23. 2. Kalman Filter State Estimation a priori prediction error prediction error a priori error covariance error covariance a priori estimation a posteriori estimation Find Kkthat minimizes Pk Completely trust the measurement Completely trust the prediction

24. Initial estimates for and 2. State Estimation recap Measurement Update (Correct) Time Update (Predict)

25. 3. Optimal Control Action Linear Quadratic Regulator (LQR) Analytical solution to optimal control move With R > 0, and Q = CTC, where (A;C) observable and (A;B) is controllable, a solution to the steady-state LQR exists Solution : Riccati Equation Vector of optimal control Gains solution of an algebraic problem

26. 3. Optimal Control Action

27. 3. Summary: Embedded Control Optimization Master Control Optimization d1 d2 dm-1 dm ... c=f(d1) c=f(d2) c=f(dm-1) c=f(dm) Tractable Design Search Space Minimize Total Expected Cost • Avoids Combinatorial Explosion • No Matching Problem • No controller tuning • Smooth control action • Fast optimal control actions elaboration Conservational Laws Process and Product Constraints Embedded Control Algorithm Simple and fast to solve

28. Distance to closest constraint Flexibility test Active Constraint Solution s.t. Rigorous Flexibility Test • Ensure constraints satisfaction for ALL uncertain realizations • Find critical scenarios • Flexibility test

29. 1)Initiation 2)Propagation 3)Termination Integrated Design and Control of a Polymerization Reactor Polymerization Reactor (Ogunnaike, AIChE J. 1999) Non-Linear Stiff ODE system with multiple steady states

30. Identification of Structural Decisions • Design variables, controls, and uncertainty sources • Design: Reactor Volume, Monomer Concentration • Control: Coolant Flow and Initiator Flow • Uncertainty Sources: Coolant Inlet Temperature, Heat Transfer Coef. (U) • Kinetic Parameters (k, E)

31. dA dB dC Cost7 Cost6 Cost1 Cost2 Cost3 Cost4 Cost5 ω7 ω6 ω1 ω4 ω5 ω2 ω3 + Exp Cost A Optimal Design with Embedded Control Master Optimization Embedded Control Optimization θ1 Embedded Control Optimization Embedded Control Optimization θ2 Embedded Control Optimization θ3 Embedded Control Optimization θ4 Embedded Control Optimization θ5 Embedded Control Optimization θ6 θ7

32. Optimal Reactor Design with Embedded Control: Results Minimize Total Expected Cost Conservational Laws Process and Product Constraints Embedded Control Algorithm Minimum Expected Cost Design V=0.83 m3 Cmi=4.52 mol/l Tsp=295.2 K Psp=24,950

33. Rigorous Flexibility Test • Ensure constraints satisfaction for ALL uncertain realizations • Find critical scenarios • Flexibility testActive Constraint Solution s.t. Critical scenario detected δ=0.89 Tcw= 283 K U=712 kJ·/(h·K·m2)

34. Simulation of the Reactor under Worst Combination of Uncertain Variables Constraint Violation AMW > 25,500 Identification Identification + Control

35. UUCL quality UCL SP LCL LLCL Time Integrated Polymerization Reactor Design and Control Results Optimal Design V=0.88 m3 Cmi=5.02 mol/l Tsp=293 K Psp=24,750 • The optimal design proposes a small reactor • Smaller reactors are more sensitive to input uncertainty • Smaller resident time leads to a faster control • Bigger overshoots but for shorter times

36. Simulation of the Design under Worst Combination of Uncertain Variables No violations!! Identification Identification + Control

37. Integrated Distillation Column Design and Control • Distillation Column Dynamic Model • Accounts for hold-ups • Multi-Input Multi-Output Control • DAE non-linear system Distillate Spec. xD≥0.995 F Uncertain Thermo Parameters zF Time Varying Feed Composition Bottom Spec. xB ≤0.02

38. Conclusions • A dynamic approach needed to guarantee flexibility. • Feedback cannot be ignored in design under uncertainty. • Embedded control: programming solution approach to design under uncertainty considering control. • Problem size reduction and convexity improved. • Solvable with current MINLPs solvers. • Systematic decision hierarchy for integrated D&C was successfully applied to polymerization process.

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