A Brief Perspective on Process Optimization

Transcription

1 A Brief Perspective on Process Optimization L. T. Biegler Carnegie Mellon University November, 2007 Annual AIChE Meeting Salt Lake City, UT

2 Overview Introduction Main Themes Modeling Levels Closed Models PDEs and ROMs NLPs and Model Integration LDPE Eample Fully Open Models Challenge Problem: NMPC Conclusions 2

3 Optimization in Design, Logistics and Control MILP MINLP Global LP,QP NLP HENS MENS Separations Reactors Equipment Design Process Flowsheeting Planning Scheduling Real-time optimization Linear MPC Nonlinear MPC Hybrid 3

4 Hierarchy for Optimization Models 1. Fully Open Completely declarative smooth models First, second derivatives cheap to calculate Solved with Newton-type type methods 2. Semi-closed procedural models solved efficiently (e.g., DAE integrators) can be differentiated efficiently 3. Closed Procedural models, comple, special purpose solvers Smooth Difficult to differentiate 4

5 Hierarchy of Nonlinear Programming Formulations and Model Intrusion Full Space Formulation Open Interior Point Adjoint Sens & SAND Adjoint Compute Efficiency Semi-Closed Direct Sensitivities Finite Differences SQP rsqp Black Bo DFO Closed Variables/Constraints 5

6 Closed Optimization Models Only function information available. Gradient and Hessian information cannot be obtained eactly Nested iterative calculations in model (round-off error due to convergence tolerences) How can these models be used for optimization? Derivative Free Optimization NOMAD, DFO, UOBYQA Strong convergence theory, successful on difficult problems Reduced order models convert to approimate models in analytic form (manageable size) Closed Detailed Models Î Fully Open Simple Models allows integration with models from other levels 6

7 Challenge for Process Optimization FutureGen Power Cycle (DOE/NETL) PDE/CFD: Gas Turbine, Gasifier DAE: H2/CO2 Separation: PSA Algebraic: Rest of Flowsheet 7

8 Advanced Process Engineering Cosimulation (Zitney and Syamlal, 2002) Detailed CFD Models Multiphase, reactive flow Specialized solution procedures Model requires ~ CPU h Smooth responses epected but derivatives not available Few (<10) optimization variables Goal: ROM for integration with other model equations 8

9 Optimal Operation for Gas Separation (Jiang, Fo, B., 2004) Pressure Swing Adsorption in FutureGen Cycle Need for high purity H2 Possible technology to capture CO2 Respond quickly to changes in process demand Large, highly nonlinear dynamic separation Eamples Large-scale H2/CO Separation 5 beds, 11 steps, > 10 4 variables with dense Jacobians,, 10 4 DAEs Simulated Moving Bed Separation 6 columns, 4 zones, > 100 variables/ DAEs with dense Jacobians 9

10 Black Bo Optimization (Closed Models) + Easy to interface + Easily applied to legacy codes, closed models - Repeated model solutions - Failure prone, no constraint handling in model - Derivatives by finite difference (if at all) Optimization Min Φ() s.t. g() 0 p u(t) (PD)AE Model F(z, z, u, p, t) = 0 - (Fully Open) Reduced Order Models (ROMs) for optimization formulations Proper Orthogonal Decomposition (Karhunen, Loeve ) PCA-based ROMs (Willco( Willco, ) 10

11 PSA Optimization 4 step Methane/Hydrogen Separation PSA Model: 3200 DAEs (several CPU hrs to optimize) POD Model Reduction: reduced to just 200 DAEs DAEs in ROM were converted to algebraic equations by temporal discretization (simultaneous approach) The resultant non-linear program (NLP) was solved using IPOPT solver in AMPL: optimum obtained in 3 CPU min 11

12 PSA/ROM Optimization Results Profiles for gas-phase methane mole fraction Adsorption Depressurization Rigorous Model ROM Rigorous Model ROM Desorption Hydrogen recovery increased from 10.9% to 16.3% Performance criteria from rigorous model match ROM (high product purities) Pressurization Rigorous Model ROM Rigorous Model ROM 12

13 ROM Sufficiently Accurate for Optimization? Create ROM for CFD Model Solve ROM-based optimization s.t.. Trust Region Sufficient Decrease In CFD Model? Converged? Stop Tighten Trust Region Apply ROM-Based Trust Region Strategy (Fahl-Sachs, Kunisch-Volkwein Volkwein, Gratton- Sartenaer-Toint Toint ) TR model based on ROM optimization ROM matches functions/gradients 13

14 Fully Open Optimization Models Eact first (and higher!) derivatives available Single model solution no nested calculations Basis for etending NLP strategies to MINLP, global optimization NLP Min Φ() s.t. c() = 0 L U Which NLP algorithms should be used? Fast convergence properties Low compleity in dealing with etending to large-scale systems Eploit structures of model and optimization formulations Essential for many time-critical optimizations (RTO, NMPC, DRTO) 14

15 Decision Pyramid for Process Operations Real-time Optimization and Advanced Process Control Fewer discrete decisions Many nonlinearities Frequent, on-line time-critical solutions Higher level decisions must be feasible Performance communicated for higher level decisions Planning Feasibility Scheduling Site-wide Optimization Real-time Optimization Model Predictive Control Regulatory Control Performance MPC APC Off-line (open loop) On-line (closed loop) 15

16 Dynamic Real-time Optimization Integrate On-line Optimization/Control with Off-line Planning Consistent, first-principle models Consistent, long-range, multi-stage planning Increase in computational compleity Time-critical calculations Applications Batch processes Grade transitions Cyclic reactors (coking, regeneration ) Cyclic processes (PSA, SMB ) Feasibility Planning Scheduling Site-wide Optimization Performance Continuous processes are never in steady state: Real-time Optimization Feed changes Model Predictive Control Nonstandard operations Regulatory Control Optimal disturbance rejections Simulation environments and first principle dynamic models are widely used for off-line studies Can these results be implemented directly on-line for large-scale systems? 8 16

17 Dynamic Optimization Engines Evolution of NLP Solvers: Î for dynamic optimization, control and estimation SQP rsqp Full-space Barrier E.g., E.g, SNOPT IPOPT NPSOL- and Simultaneous Multiple Sequential Shooting dynamic Dynamic - over optimization 100 over Optimization d.f.s 1 but 000 over over variables variables and and constraints and constraints Object Oriented Codes tailored to structure, sparse linear algebra and computer architecture (e.g., IPOPT 3.3) 17

18 LD P E Plant- Multistage Dynamic Optimization E t h y l e n e 2 7 b a r s / 2 5 C 1. 5 b a r s / 4 0 C T o r c h 6 b a r s / 4 0 C P u r g e B u t a n e 6 b a r s / 4 0 C P R I M A R Y C O M P R E S S O R b a r s / 4 0 C 5 6 b a r s / 4 0 C H Y P E R - C O M P R E S S O R 2 4 b a r s / 4 0 C L O W P R E S S U R E R E C Y C L E b a r s 4 0 C H I G H P R E S S U R E R E C Y C L E b a r s 4 0 C b a r s 8 5 C T U B U L A R R E A C T O R P r e h e a t i n g Z o n e R e a c t i o n Z o n e b a r s C L D V a l v e C o o l i n g Z o n e C Planning Scheduling Site-wide Optimization Real-time Optimization Model Predictive Control Regulatory Control 1. 5 b a r s 4 0 C b a r s 4 0 C W a e s O i l s 2. 5 b a r s C L D P H i g h P r e s s u r e S e p a r a t o r L o w P r e s s u r e S e p a r a t o r High pressure reaction (>2000 atm) through mile-long reactor coil Highly nonlinear model with 1000 s of chemical species (moment models) 18

19 Large-Scale Parameter Estimation Initiator(s) Monomer Comonomer NZ z z z z Material & Energy Physical Properties Zone Transitions 500 ODEs 1000 AEs 8 Stiffness + Highly Nonlinear + Parametric Sensitivity + Algebraic Coupling 19

20 Large-Scale Parameter Estimation Comple Kinetic Mechanisms ~ 35 Elementary Reactions ~100 Kinetic Parameters 20

21 Large-Scale Parameter Estimation Parameter Estimation for Industrial Applications Use Rigorous Model to Match Plant Data Directly Start with Standard Least-Squares Formulation Least-Squares Rigorous Reactor Model Special Case of Multi-Stage Dynamic Optimization Problem Solve using Simultaneous Collocation-Based Approach 1 data set 6 data sets 500 ODEs ODEs 1000 AEs 6000 AEs21

22 Multi-zone Reactor Parameter Estimation Data Sets: Operating Conditions and Properties for Different Grades Match: Temperature Profiles and Product Properties On-line Adjusting Parameters Æ Track Evolution of Disturbances Kinetic Parameters Æ Apply to Rigorous Models Add Data for Process Inputs (EVM) Æ remove additional uncertainties Single Data Set (On-line Adjusting Parameters) Multiple Data Sets (On-line Adjusting Parameters + Kinetics) EVM (On-line Adjusting Parameters + Kinetics + Process Inputs) 22

23 Large-Scale Parameter Estimation Improved Match of Reactor Temperatures Profile 23

24 Industrial Case Study Results - Reactor Overall Monomer Conversion ( up to 20 Different Grades ) Plant Conversion (%) Predicted Conversion (%) Avg. Conversion Deviation Base Model 12.1 % New Model 2.5 % EVM Results % 24

25 IPOPT Factorization Byproduct W k + Σ A k T k + δ I 1 A k δ I 2 KKT matri factored by indefinite symmetric factorization Without regularization at solution Î sufficient second order conditions and uniquely estimated parameters Reduced Hessian cheaply available to calculate confidence regions 25

26 Parameter Estimation in Parallel Architectures Eploit Structure of KKT Matri Laird, B Direct Factorization MA27 Memory Bottlenecks Factorization Time Scales Superlinearly with Data sets Block-bordered Diagonal Structure Coarse-Grained Parallelization using Schur Complement Decomposition IPOPT 3. architecture supports tailored structured decompositions 26

27 Parameter Estimation in Parallel Architectures Computational Results LDPE Reactor EVM Problem 27

28 On-line Issues: Model Predictive Control (NMPC) Zavala, Laird, B. (2006, 2007) NMPC Estimation and Control d : disturbances u : manipulated variables y sp : set points Process NMPC Controller z = F( z, y, u, p, d) 0 = G( z, y, u, p, d) z : differential states y : algebraic states Model Updater z = F( z, y, u, p, d) 0 = G( z, y, u, p, d) Why NMPC? Track a profile evolve from linear dynamic models (MPC) Severe nonlinear dynamics (e.g, sign changes in gains) Operate process over wide range (e.g., startup and shutdown) NMPC Subproblem min u s.t. J z ( l + 1 ( = k z 0 = ( Bounds f )) ( k = z ) l N l = 0,u l ϕ ( )) z l,u l ) + E ( Challenge: Computational Delay Æ Performance and Stability? z N ) 28

29 Nonlinear Model Predictive Control Parametric Problem 29

30 30 Advanced Step NMPC Advanced Step NMPC Combine advanced step with sensitivity to solve NLP in background (between steps) not on-line Solve P(z ) in background (between t 0 and t 1 ) Sensitivity to updated problem to get (z 0, u 0 ) Solve P(z +1 ) in background with new (z 0, u 0 ) υ ν λ = K X V A I A W k k T k k k [ 0 0 0

31 Industrial Case Study Grade Transition Control Process Model: 289 ODEs, 100 AEs Simultaneous Collocation-Based Approach 27,135 constraints, 9630 LB & UB Off-line Solution with IPOPT Feedback Every 6 min 31

32 Nonlinear Model Predictive Control Grade Transition to Reduce MW of LDPE Ideal NMPC controller - computational delay not considered Time delays as disturbances in NMPC Optimal Feedback Policy Æ (On-line Computation 351 CPU s) 32

33 Nonlinear Model Predictive Control Optimal Policy vs. NLP Sensitivity -Shifted Æ (On-line Computation 1.04 CPU s) Very Fast Close-to-Optimal Feedback Large-Scale Rigorous Models Analogous Strategy Developed for Moving Horizon Estimation 33

34 Summary Optimization plays a central role in all aspects of chemical process engineering Challenging Real-World Applications Modeling-design-control-operations Conflict between detailed models (off-line analysis) and time critical computation (on-line optimization) Challenge to develop efficient optimization strategies with multiple model levels Closed -- Semi-closed Fully Open Advanced algorithms needed to integrate optimization models Across process systems Across operating time scales Across functionalities 34

35 Related Papers at this Conference (and more details) Reduced Order Optimization Models 429a Reduced-Order Model for Dynamic Optimization of Pressure Swing Adsorption A. Agarwal, L. T. Biegler, S. E. Zitney 398c Advanced Process Engineering Co-Simulation Using CFD-Based Reduced Order Models, Y-D Lang, L. T. Biegler, S. Munteanu, J. Madsen, S. E. Zitney Open Optimization Models and Parallel Decomposition 149d Efficient Parallel Solution of Dae Constrained Optimization Problems with Loosely Coupled Algebraic Components, C. D. Laird, V. M. Zavala, L. T. Biegler 531a Modeling and Optimization of Polymer Electrolyte Membrane Fuel Cells, P. Jain, L. T. Biegler, M. S. Jhon Fast NMPC and MHE 149b A Moving Horizon Estimation Algorithm Based on NLP Sensitivity, V. M. Zavala, L. T. Biegler 256a Stability and Performance Analysis of Nlp Sensitivity-Based NMPC Controllers, V. M. Zavala, L. T. Biegler 35

36 Acknowledgements Funding US Department of Energy EonMobil Upstream Research Corporation US National Science Foundation Center for Advanced Process Decision-making (CMU) Research Colleagues Anshul Agarwal Parag Jain Carl Laird Yi-dong Lang Andreas Wächter Victor Zavala Steve Zitney 36