A Two-Stage Algorithm for Multi-Scenario Dynamic Optimization Problem

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1 A Two-Stage Algorithm for Multi-Scenario Dynamic Optimization Problem Weijie Lin, Lorenz T Biegler, Annette M. Jacobson March 8, 2011 EWO Annual Meeting

2 Outline Project review and problem introduction A Two-stage Algorithm Parameter estimation from multiple data sets Optimization under uncertainty with multi-scenario formulation An Illustrative example Current direction Summary 2

3 Project Review (1) Strong & flexible Low productivity & difficult to control Monomer / Initiator Suspension reactor SIPN (Semi-Interpenetrating Polymer Network) Aqueous media Seed particle Monomer droplet 3

Project Review (2) Process Stages Features Modeling Control variables Optimization Approach Swelling Polymerization Complex diffusion; single component reaction Particle Growth model Monomer feeding

4 Project Review (2) Process Stages Features Modeling Control variables Optimization Approach Swelling Polymerization Complex diffusion; single component reaction Particle Growth model Monomer feeding rate Initiator feeding rate Dynamic Optimization Crosslinking Complex composite networking reaction Semi-IPN kinetic model Initial polymer Monomer concentration Initiator concentration Holding temperature Holding duration Surrogate Model 4

5 Project Review (2) Process Stages Features Modeling Control variables Optimization Approach Swelling Polymerization Complex diffusion; single component reaction Particle Growth model Stage I Monomer feeding rate Initiator feeding rate Dynamic Optimization Crosslinking Complex composite networking reaction Semi-IPN kinetic model Initial polymer Monomer concentration Initiator concentration Holding temperature Holding duration Surrogate Model 5

6 Project Review (2) Process Stages Features Modeling Control variables Optimization Approach Swelling Polymerization Complex diffusion; single component reaction Particle Growth model Stage I Monomer feeding rate Initiator feeding rate Dynamic Optimization Crosslinking Complex composite networking reaction Semi-IPN kinetic model Initial polymer Monomer concentration Initiator concentration Holding temperature Holding duration Surrogate Model 6 Stage II

7 Project Review (2) Process Stages Features Modeling Control variables Optimization Approach Swelling Polymerization Complex diffusion; single component reaction Particle Growth model Stage I Monomer feeding rate Initiator feeding rate Dynamic Optimization Crosslinking Complex composite networking reaction Semi-IPN kinetic model Initial polymer Monomer concentration Initiator concentration Holding temperature Holding duration Surrogate Model 7 Stage II

8 New Challenges Continuous effect for process improvement Improve model reliability Additional information acquisition Update model / parameters Improve solution robustness Uncertainty consideration Optimization under uncertainty 8

9 Multi-scenario Dynamic Optimization Parameter estimation from multiple data sets min μ XNS i=1 s:t: y i = f i (x i ; μ) h i (x i ; μ) =0 (y i y m i ) T i (y i y m i ) Dynamic optimization under uncertainty Z max E μf ( _x; x; y; u; º; ; μ)g =max ª(μ) ( _x; x; y; u; º; ; μ)dμ u;v; u;v; μ2 S.t. J 0 (_x(0);x(0);y;u(0);º; ; μ) =0 h( _ (x);x;y;u;v;t; μ) =0; g( _ (x);x;y;u;v;t; μ) 0; 9

10 Current Researches (1) Sequential approaches ( Faver 2003 ): Computationally expensive for derivative evaluation Upper Stage NLP Middle Stage Sub-NLP 1 Sub-NLP n Lower Stage Simulation 1 Simulation n [Anderson 1978], [Rod 1980], [Reilly 1981], [Dovi 1989], [Kim 1990], 10

11 Current Researches (2) Simultaneous approach (Zavala and Biegler 2007) W 1 A 1 W 2 A 2 W 3 A W NS A NS 5 4 A T 1 A T 2 A T 3 A T NS ± 1 I v 1 v 2 v 3. v NS d 3 2 = r 1 r 2 r 3. r NS r d Difficult for highly nonlinear, ill-condition problem Wk = 2 4 Hl k + ± 1I r xk c l k D T k (r xk c l k )T ± 2 I 0 D k 0 ± 2 I where r T k = [(r x k L l k )T ; (c l k )T ; (D k x l k ¹D k d l ) T ], v T k =[ xt k Tk ¾T k ], AT k =[00 ¹D T k ], 3 5 Schur complement [± 1 I XNS k=1 A T k (W k) 1 A k ] d = r d XNS k=1 A T k (W k) 1 r k [Tjoa and Biegler1991,1992] [Gondzio and Gothrey 2005, Gondzio and Sarkissian 2003] 11

12 A Two-Stage Algorithm μ L (k) df j dμ L (k) ; d 2 f j d 2 μ L (k) min μ L F (μ L )= XNS j=1 f j (μ L ) min f 1 (μ S 1 ; μ(k)) L min f 2 (μ S 2 ; μ(k)) L min μ S 1 μ S 2 μ S n s:t: M 1 s:t: M 2 s:t: f n (μ S n ; μ L (k)) M n Efficient algorithm for better behaved large inner problem Robust solver for well-conditioned small outer problem 12

13 Sensitivity from Inner Optimization Problem As-NMPC Features: NLP sensitivity evaluation At the solution point, the primal-dual system satisfies Á(s ( ); ) =0 Applying the implicit function theorem ¹K ( 0); Substitute the right hand size with I at the desired parameter constraints Exact gradient information is conveniently available at the optimal point 13

14 Sensitivity from Inner Optimization Problem Hessian evaluation When Hessian information is required, Hessianvector product is computed Forward difference Central difference Exact Hessian-vector product (Pearlmutter, 1994) Operator Apply R{} to Gradient equation 14

15 Solvers: Outer Optimization Problem Bound constrained optimization algorithms L-BFGS-B A limited-memory quasi-newton code for boundconstrained optimization TRON Trust region Newton method for the solution of boundconstrained optimization problems. ACO Adaptive cubic overestimation 15

16 An Illustrative Example Parameter Estimation from Multiple data sets First-order Irreversible Chain reaction A k 1! B k 2! C Assume k2 is a Linking parameter, k1 is a separate parameter. 20 data sets were generated from model simulation. Outer problem solved in TRON, converged in 3 iterations. Inner problem solved in As-NMPC converged in 6 iterations in average. The same optimal solution is found at the optimal. 16

17 Current Direction Reduce kinetic parameter uncertainty by multiscenario parameter estimation Optimization of operation condition under uncertainty Investigation of efficient algorithm for outer optimization problem Pilot plant study for optimal solution Extension of model application for broader products 17

18 Summary Multi-scenario optimization for dynamic system is often desired but challenging. Current sequential and simultaneous algorithms have limitations in terms of efficiency and robustness. A two-stage algorithm is proposed which takes advantage of efficient interior-point method and robust bound constraint algorithm. Small test problems are studied. Application to the process model is planned. 18

19 Thank You! 19

Kinetic Modeling for a Semi-Interpenetrating Polymer Network (SIPN) Process

Kinetic Modeling for a Semi-Interpenetrating Polymer Network (SIPN) Process Weijie Lin Advisor: Lorenz T. Biegler, Annette Jacobson Chemical Engineering Department Carnegie Mellon University, Pittsburgh,