Integrated Design and Control of Polymerization Reactor under Uncertainty

1. Integrated Design and Control of Polymerization Reactor under Uncertainty AIChE 2006 Annual Meeting San Francisco, California November 12 - November 17 Paper 662g Session 10A10 : Design, Analysis and Operations under Uncertainty Andrés Malcolm, Libin Zhang and Andreas A. Linninger 11/17/2006 Laboratory for Product and Process Design, Department of Chemical Engineering, University of Illinois, Chicago, IL 60607, U.S.A.

2. 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

3. 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.

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

5. 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

6. 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

7. 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

8. 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

9. 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 mathematical programming solutions do not work Not Solvable for industrial applications

10. 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.

11. 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

12. 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 from the linearized model A: Luenberger full state observer (Ensures stable prediction) Need to solve eigenvalue problem B: Kalman Filter (Predictor – Corrector Recursive Estimator) Need to know an estimate of the noise covariance

13. B. 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.

14. B. 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

15. B. Optimal Control Action Linear Quadratic Regulator (LQR) Analytical solution to optimal control move With T > 0, and S = 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

16. 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

17. 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

18. 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

19. 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)

20. 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

21. 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)

22. 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

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

24. 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.

25. Acknowledgments • Financial support from NSF Grant CBET-0626162. • Environmental Manufacturing Management (EvMM) fellowship. • Jamie Polan, REU student under the NSF Grant DMI-0328134