Real-Time Optimization

Transcription

1 Real-Time Optimization w Plant APC RTO c(x, u, p) = 0 On line optimization Steady state model for states (x) Supply setpoints (u) to APC (control system) Model mismatch, measured and unmeasured disturbances (w) u Min u F(x, u, w) s.t. c(x, u, p, w) = 0 x X, u U p y DR-PE c(x, u, p) = 0 Data Reconciliation & Parameter Identification Estimation problem formulations Steady state model Maximum likelihood objective functions considered to get parameters (p) Min p Φ(x, y, p, w) s.t. c(x, u, p, w) = 0 x X, p P 9 RTO Characteristics w APC Plant u RTO c(x, u, p) = 0 p y DR-PE c(x, u, p) = 0 Data reconciliation identify gross errors and consistency in data Periodic update of process model identification Usually requires APC loops (MPC, DMC, etc.) RTO/APC interactions: Assume decomposition of time scales APC to handle disturbances and fast dynamics RTO to handle static operations Typical cycle: 1-2 hours, closed loop 10 1

2 How simple a model is simple? Plant and RTO model must be feasible for measurements (y), parameters (p) and setpoints (u) Plant and RTO model must recognize (close to) same optimum (u*) => satisfy same KKT conditions Can RTO model be tuned parametrically to do this? RTO Consistency (Marlin and coworkers) 11 RTO Stability (Marlin and coworkers) Stability of APC loop is different from RTO loop Is the RTO loop stable to disturbances and input changes? How do DR-PE and RTO interact? Can they cycle? Interactions with APC and plant? Stability theory based on small gain in loop < 1. Can always be guaranteed by updating process sufficiently slowly. APC u w RTO c(x, u, p) = 0 p Plant y DR-PE c(x, u, p) =

3 RTO Robustness (Marlin and coworkers) What is sensitivity of the optimum to disturbances and model mismatch? => NLP sensitivity Are we optimizing on the noise? Has the process really changed? Statistical test on objective function => change is within a confidence region satisfying a χ 2 distribution Implement new RTO solution only when the change is significant Eliminate ping-ponging 13 Real-time Optimization with rsqp Sunoco Hydrocracker Fractionation Plant (Bailey et al, 1993) Existing process, optimization on-line at regular intervals: 17 hydrocarbon components, 8 heat exchangers, absorber/stripper (30 trays), debutanizer (20 trays), C3/C4 splitter (20 trays) and deisobutanizer (33 trays). REACT OR E FFL UENT FROM LOW PRESSURE SEPARATOR FUE L GAS ABSORBER/ STRIPPER C3 PREFLASH LIGHT NAPHT HA MIXED LPG C3/C4 SPLITTER ic4 MAIN FRAC. REFORM ER NAPHT HA DEBUTANIZER DIB REC YC LE OIL n C4 square parameter case to fit the model to operating data. optimization to determine best operating conditions 6 3

4 Optimization Case Study Characteristics Model consists of 2836 equality constraints and only ten independent variables. It is also reasonably sparse and contains nonzero Jacobian elements. G E P P = " z i C i + " z i C i + " z i C m i # U i!g i!e m=1 NP Cases Considered: 1. Normal Base Case Operation 2. Simulate fouling by reducing the heat exchange coefficients for the debutanizer 3. Simulate fouling by reducing the heat exchange coefficients for splitter feed/bottoms exchangers 4. Increase price for propane 5. Increase base price for gasoline together with an increase in the octane credit 7 8 4

5 Real-time Optimization: Components w Plant APC u y RTO c(x, u, p) = 0 p DR-PE c(x, u, p) = 0 Data reconciliation identify gross errors and inconsistency in data Periodic update of process model identification Usually requires APC loops (MPC, DMC, etc.) RTO/APC interactions: Assume decomposition of time scales APC to handle disturbances and fast dynamics RTO to handle static operations Typical cycle: 1-2 hours, closed loop What if steady state and dynamic models are inconsistent? Dynamic On-line Optimization? d PC m Plant u y D-RTO c(x, x, u, p) = 0 p DR-PE c(x, x, u, p) = 0 Goal: Integrate On-line Optimization with Advanced Process Control +Consistent, first-principle dynamic models +Prediction and extrapolation not fitting and interpolation +Links to Off-line Decision-making - Leads to increase in computational complexity - Requires time-critical calculations Essential for: Inherently Dynamic processes Nonstandard operations Optimal disturbance rejection 10 5

6 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 complexity Time-critical calculations Applications Batch processes Grade transitions Cyclic reactors (coking, regeneration ) Cyclic processes (PSA, SMB ) Continuous processes are never in steady state: Feed changes Nonstandard operations Optimal disturbance rejections Simulation environments (e.g., ACM, gproms) and first principle dynamic models are widely used for off-line studies 8 11 Some DRTO Case Studies Integrated grade transitions MINLP of scheduling with dynamics (Flores & Grossmann, 2006, Prata et al., 2007) Significant reduction in transition times Dynamic Predictive Scheduling Processes and supply chains need to optimally respond to disturbances through dynamic models Reduction in energy cost by factor of two Cyclic Process Optimization Decoking scheduling SMB optimization PSA optimization Productivity increases by factor of

7 On-line Issues: Model Predictive Control (NMPC) NMPC Estimation and Control d : disturbances u : manipulated variables Process y sp : set points z : differential states y : algebraic states NMPC Controller z = F( z, y, u, p, d ) 0 = G( z, y, u, p, d ) Model Updater z = F( z, y, u, p, d ) 0 = G( z, y, u, p, d ) NMPC Subproblem N # min J(x(k)) = "(z l,u l )+F(z N ) u s.t. l= 0 z l +1 = f (z l,u l ) z 0 = x(k) Bounds 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) 13 MPC - Background Motivate: embed dynamic model in moving horizon framework to drive process to desired state Generic MIMO controller Direct handling of input and output constraints Slow time-scales in chemical processes consistent with dynamic operating policies Different types Linear Models: Step Response (DMC) and State-space Empirical Models: Neural Nets, Volterra Series Hybrid Models: linear with binary variables, multi-models First Principle Models direct link to off-line planning NMPC Pros and Cons + Operate process over wide range (e.g., startup and shutdown) + Vehicle for Dynamic Real-time Optimization - Need Fast NLP Solver for Time-critical, on-line optimization - Computational Delay from On-line Optimization degrades performance 14 7

8 6/24/11 Tennessee Eastman Process Unstable Reactor 11 Controls; Product, Purge streams Model extended with energy balances 15 Tennessee Eastman Challenge Process DAE Model Number of differential equations Number of algebraic variables NLP Optimization problem Number of algebraic equations 141 Difference (control variables) 11 Number of variables of which are fixed Number of constraints Number of lower bounds Number of upper bounds Number of nonzeros in Jacobian Number of nonzeros in Hessian Method of Full Discretization of State and Control Variables Large-scale Sparse block-diagonal NLP 16 8

9 Setpoint change studies Process variable Type Magnitude Production rate change Reactor operating pressure change Purge gas composition of component B change Step Step Step -15% Make a step change to the variable(s) used to set the process production rate so that the product flow leaving the stripper column base changes from 14,228 to 12,094 kg h kpa Make a step change so that the reactor operating pressure changes from 2805 to 2745 kpa +2% Make a step change so that the composition of component B in the gas purge changes from to 15.82% Setpoint changes for the base case [Downs & Vogel] 17 Case Study: Change Reactor pressure by 60 kpa Control profiles All profiles return to their base case values Same production rate Same product quality Same control profile Lower pressure leads to larger gas phase (reactor) volume Less compressor load 18 9

10 TE Case Study Results I Shift in TE process Same production rate More volume for reaction Same reactor temperature Initially less cooling water flow (more evaporation) 19 Case Study- Results II Shift in TE process Shift in reactor effluent to more condensables Increase cooling water flow Increase stripper steam to ensure same purity Less compressor work 20 10

11 Case Study: Change Reactor Pressure by 60 kpa Optimization with IPOPT 1000 Optimization Cycles 5-7 CPU seconds Iterations Optimization with SNOPT Often failed due to poor conditioning Could not be solved within sampling times > 100 Iterations 21 What about Fast NMPC? Fast NMPC is not just NMPC with a fast solver Computational delay between receipt of process measurement and injection of control, determined by cost of dynamic optimization Leads to loss of performance and stability (see Findeisen and Allgöwer, 2004; Santos et al., 2001) As larger NLPs are considered for NMPC, can computational delay be overcome? 22 11

12 NMPC Can we avoid on-line optimization? Divide Dynamic Optimization Problem: preparation, feedback response and transition stages solve complete NLP in background ( between sampling times) as part of preparation and transition stages solve perturbed problem on-line > two orders of magnitude reduction in on-line computation Based on NLP sensitivity of z 0 for dynamic systems Extended to Collocation approach Zavala et al. (2008, 2009) Similar approach for MH State and Parameter Estimation Zavala et al. (2008) Stability Properties (Zavala et al., 2009) Nominal stability no disturbances nor model mismatch Lyapunov-based analysis for NMPC Robust stability some degree of mismatch Input to State Stability (ISS) from Magni et al. (2005) Extension to economic objective functions 23 NLP Sensitivity Parametric Programming Solution Triplet Optimality Conditions NLP Sensitivity Rely upon Existence and Differentiability of Path Main Idea: Obtain and find by Taylor Series Expansion 24 12

13 NLP Sensitivity Obtaining Optimality Conditions of Apply Implicit Function Theorem to around KKT Matrix IPOPT Already Factored at Solution Sensitivity Calculation from Single Backsolve Approximate Solution Retains Active Set 25 Advanced Step Nonlinear MPC (Zavala, B., 2008) Solve NLP in background (between steps, not on line) Update using sensihvity on line x(k) x k+1 k u(k) t k t k+1 t k+2 k +N#1 min J(x(k), u(k)) = F(x k +N k ) + "(x l k,v l k ) l= k +1 s.t. x k +1 k = f (x(k),u(k)) x l +1 k = f (x l k,v l k ), l = k +1,...k + N -1 x l k % X, v l k % U, x k +N k % X f Solve NLP(k) in background (between t k and t k+1 ) $ t k+n 26 13

14 Advanced Step Nonlinear MPC (Zavala, B., 2008) Solve NLP in background (between steps, not on line) Update using sensihvity on line x k+1 k x(k) x(k+1) u(k+1) u(k) t k t k+1 t k+2 t k+n # W k A k "I& % T ( % A k 0 0 ( $ % Z k 0 X k '( #)x& % ( % )* ( = $ %)z'( # 0 & % ( %! ( % x k +1 k " x(k +1) ( % 0 ( $ % ' ( Solve NLP(k) in background (between t k and t k+1 ) SensiHvity to update problem on line to get (u(k+1)) 27 Advanced Step Nonlinear MPC (Zavala, B., 2008) Solve NLP in background (between steps, not on line) Update using sensihvity on line x k+2 k+1 x(k) x(k+1) u(k+1) u(k) t k t k+1 t k+2 t k +N k+n min J(x(k +1), u(k +1)) = F(x k +N +1 k +1 ) + "(x l k +1,v l k +1 ) s.t. x k +2 k +1 = f (x(k +1),u(k +1)) x l +1 k +1 = f (x l k,v l k ), l = k +2,...k + N x l k +1 $ X, v l k +1 $ U, x k +N +1 k +1 $ X f # l= k +2 Solve NLP(k) in background (between t k and t k+1 ) SensiHvity to update problem on line to get (u(k+1)) Solve NLP(k+1) in background (between t k+1 and t k+2 ) 28 14

15 Stability Properties of asnmpc x(k+1) = f(x(k), u(k)) plant and model identical Nominal Stability Theorem (Zavala, B., 2008) Assume that the NLP can be solved within one sampling time, nominal stability assumptions hold for ideal NMPC (Mayne, 2000), and the nonlinear model is perfect without measurement noise. Then ideal NMPC controller performance and asnmpc controller performance are identical. Ideal NMPC stability asnmpc stability. x(k+1) = f(x(k), u(k)) + g(x(k), u(k), w(k)) plant and model not identical Robust Stability Theorem (Zavala, B., 2008) Assume that the NLP can be solved within one sampling time, and that robust stability assumptions hold for ideal NMPC (Magni, Scattolini, 2007). Then there exist bounds on the noise, w, and model mismatch, g, for which the cost function J N+1 (x), obtained from the asnmpc strategy, is an input-to-state (ISS)- Lyapunov function and the resulting closed-loop system is ISS stable. 29 NMPC for High Purity Distillation Air Separation Unit in IGCC-based Power Plants Need for high purity O2 Respond quickly to changes in process demand Large, highly nonlinear dynamic separation (MESH) models Methanol distillation (Diehl, Bock et al., 2005) 40 trays, 210 DAEs, discretized equations Argon Recovery Column 50 trays, 260 DAEs, discretized equations Double Column ASU Case Study 80 trays, 1520 DAEs, 116,900 discretized equations 30 15

16 Nonlinear Model Predictive Control: Air Sep n Unit (Huang, B., 2008) Objective: force the production rates to follow the set-points, while main their purities. 4 manipulated variables. 4 output variables. Horizon: 100 minutes in 20 finite elements. Sampling time: 5 minutes. Air Feed C rude Gas Nitrogen U4 (GN) Heat exchanger & compressor U1 (EA ) Pure oxygen Y 1 (POX ) Pure nitrogen Y 2 (PNI) Temperature at the 15 th tray Y 4 (Th15) Gas Nitrogen L P C olumn Reboiler C ondenser Temperature at the 30 th tray Y 3 (Tl30) Liquid Nitrogen U3 (L N) U2 (MA ) 40 HP Column C rude oxygen 31 Case Study: Basic Air Separation Unit Mesh Equations for Distillation Column Assumption: Vapor holdups are negligible. Index 2 system. Li-1 Vi Ideal vapor phases. Well mixed entering streams. Fi Mi Constant pressure drop. Equilibrium stage model. Vi+1 Li Mass balance: Component balance: Energy balance: Phase equilibrium: Summation: Reformulated index 1 system contains 320 ODEs, 1200 AEs. Hydrodynamics : 32 16

17 ASU Nonlinear MPC - Case 1 t = min, product rates are ramped down by 30%. t = min, they are ramped back. AS-NMPC is compared to MPC with linear input-output empirical model. Output Variables Manipulated Variables The green dot-dashed lines are the set-points, the blue dashed lines are the linear controller profiles and red solid lines are AS-NMPC profile. All the tuning parameters are favored to the linear controller. 33 NMPC of Air Separation Unit Case 2 (Huang, B., 2009) At t = min, product rates are ramped down by 40%. At t = min, they are ramped back. 5% disturbance is added to M i. N = 20, K = ODEs, 1200 AEs. Variables: 117,140 Constraints: 116, NLPs solved Background: 200 CPUs, 6 iters. Online: 1 CPUs Computational Feedback Delay Reduced from second. Blue dashed lines are ideal NMPC profile Red lines are AS-NMPC profile. In contrast, linearized controller is unstable 34 17

18 Parameter/State Estimation d : disturbances NMPC Estimation and Control u : manipulated variables y sp : set points Process z : differential states y : algebraic states NMPC Controller z = F( z, y, u, p, d ) 0 = G( z, y, u, p, d ) Model Updater z = F( z, y, u, p, d ) 0 = G( z, y, u, p, d ) Moving Horizon Estimation? Estimate a finite number of states and model parameters (unmeasured disturbances, rate constants, transport parameters) Compensate for process drifts and slowly changing conditions Allow better controller performance Parameter Estimation Subproblem 35 Moving Horizon Estimation (MHE) Linear Systems, No inequalities Kalman Filter for State Estimation Nonlinear Systems: Extended Kalman Filters in Practice Moving Horizon Estimation: + directly captures nonlinear dynamics, statistical behavior - need to solve NLP on-line Can MHE be improved? Large State Dimensionality Degrees of Freedom Highly Nonlinear, Ill-Conditioned? Solution Time Order of Minutes 36 18

19 Advanced Step Moving Horizon Estimation (as-mhe) Given y l, z l, set fake measurement z l +1 = f (z l,w l ), y l +1 " #(z l +1 ) Solve M N in background (between t l and t l+1 ) Obtain measurement at t l+1, use KKT sensihvity on line to get z l+1 Solve P(l+1) in background (between t l+1 and t l+2 ) 37 Combining MHE & NMPC (Huang, Patwardhan, B., 2010) Offset-free Formulation Apply MHE results as state and output corrections for NMPC problem Modify with an advanced step approach as-mhe 38 19

20 Combined MHE and NMPC for ASU (Huang, B., 2010) Change in model mismatch (hydraulic parameter) Setpoint MHE/NMPC with correction NMPC without correction AMPL/IPOPT (DuoCore 2.4 GHz) NMPC background (6 iter/200 CPUs) MHE background (15 iter/90 CPUs) N=5, K=3 29,285 constraints 30,885 variables Online NMPC+MHE ~ 2 CPUs 39 D-RTO with Economic Objectives Beyond NMPC Tracking APC w Plant Benefits of combining RTO with NMPC? Direct, dynamic production maximization Remove artificial tracking objective Remove artificial steady state problem u y RTO c(x, u, p) = 0 p DR-PE c(x, u, p) = 0 PC m Plant u y D-RTO c(x, x, u, p) = 0 p DR-PE c(x, x, u, p) =

21 Challenges with D-RTO Replace regulation objective with economic objective in NMPC? Active ongoing activity: Bartusiak (2007) Chachuat et al. (2008) Dadhe and Engell (2008), Engell (2007, 2009) Diehl and Rawlings (2009), Rawlings and Amrit (2009) Kadam et al. (2008) Zavala, B. (2009) Swartz et al. (2010) "(z,u) must be a K function Sufficient Condition: # Min { w l "(z l,u l )} + F(z N ) Satisfy weak controllability condition Regularize economic objective so that NLP satisfies SSOC (and LICQ, SC) Many Open Questions Still Remain l - need to assume an optimal steady state profit? - how to consider periodic problems? 41 Low-Density Polyethylene Process (Zavala, B., 2009) Centralized Model-Based Control Initiators Initiators Initiators Initiators Ethylene Cold-Shots Flowrate Inlet Temperatures Low-Pressure Recycle Hyper-Pressure Recycle Ethylene Recycle System and Flash Separation Chain-Transfer Agent LDPE Centralized Control Objectives: + Capture Complex Multivariable Interactions Between Zones Control of Product Quality, Fouling Management + Perform Fast Grade Transitions (Highly Frequent) 42 21

22 Complex Kinetic Mechanisms LDPE Kinetic Model ~ 35 Elementary Reactions ~100 Kinetic Parameters 43 Fouling Case Study: Low-Density Polyethylene Process Cannot Remove Heat of Reaction - Drop Production to Avoid Runaway Defouling Heat Transfer Coefficient Time (Days) Persistent Dynamic Disturbances Strong Effect on Profitability Potential Economic Benefits + 1% Production 0.01 x (300,000 Ton/yr) x (1,500 $/Ton) = +4,500,000 $/yr 44 22

23 LDPE Process Model Initiator(s) Cw Cw Initiator(s) Cw Ethylene Reaction Cooling Reaction 1 2 NZ LDPE Cw Cw Cw States Inputs Parameters Measured Outputs Fouling must be estimated (using MHE) Reference tracking inadequate! Conservation Mass, Energy, Momentum Thermodynamic & Transport Properties Output Mapping Initial Conditions Boundary Conditions 45 NMPC-MHE Scenario LDPE NMPC Computational Performance - Full-Discretization + IPOPT (MA57) - NLP ~ 50,000 Constraints, 300 Degrees of Freedom (DOF) Sampling Time = 2 min Reordered Original - Scale-Up With Prediction Horizon and Effect of KKT Matrix Reordering Nested Dissection Minimum Degree NLP with 350,000 Constraints and 1,000 DOF Solved in ~ 2-3 Minutes 46 23

24 NMPC Case Study Economic Objective Minimize Transition Time Maximize Production Reference Profile Core Temperature Economics-Oriented NMPC Moves Away from Suboptimal Reference Profile Overall Production 3% More Production Economics-Oriented Tracking Distributes Production Efficiently - More Production in Less Fouled Zones Purely Economic Objective Leads to Ill-Posed Problems Need Regularization Term Due to Guarantee Stability 47 Conclusions Dynamic optimization essential for many processes Batch processes Polymer processes (especially grade transitions) Periodic adsorption processes Chemical Process Operations: RTO D-RTO Need for First-Principles Dynamic Models Extension to On-Line Economic Decision-Making NMPC and MHE Computational Strategies Full-Discretization + Fast Sensitivity Calculations Large Scale Models Basic ASU process with DAE model LDPE Process with NMPC and RTO Advantages over linear MPC Extension to Uncertainties NMPC + MHE Formulations 48 24

25 References F. Allgöwer and A. Zheng (eds.), Nonlinear Model Predictive Control, Birkhaeuser, Basel (2000) R. D. Bartusiak, NLMPC: A platform for optimal control of feed- or product-flexible manufacturing, in Nonlinear Model Predictive Control 05, Allgower, Findeisen, Biegler (eds.), Springer (2007) Biegler Homepage: Biegler, L. T., Nonlinear Programming: Concepts, Algorithms and Applications to Chemical Engineering, SIAM (2010) Forbes, J. F. and Marlin, T. E.. Model Accuracy for Economic Optimizing Controllers: The Bias Update Case. Ind.Eng.Chem.Res. 33, Forbes, J. F. and Marlin, T. E.. Design Cost: A Systematic Approach to Technology Selection for Model-Based Real-Time Optimization Systems, Computers Chem.Engng. 20[6/7], M. Grötschel, S. Krumke, J. Rambau (eds.), Online Optimization of Large Systems, Springer, Berlin (2001) K. Naidoo, J. Guiver, P. Turner, M. Keenan, M. Harmse Experiences with Nonlinear MPC in Polymer Manufacturing, in Nonlinear Model Predictive Control 05, Allgower, Findeisen, Biegler (eds.), Springer, to appear Yip, W. S. and Marlin, T. E. Multiple Data Sets for Model Updating in Real-Time Operations Optimization, Computers Chem.Engng. 26[10], References Some DRTO Case Studies Busch, J.; Oldenburg, J.; Santos, M.; Cruse, A.; Marquardt, W. Dynamic predictive scheduling of operational strategies for continuous processes using mixed-logic dynamic optimization, Comput. Chem. Eng., 2007, 31, Flores-Tlacuahuac, A.; Grossmann, I.E. Simultaneous cyclic scheduling and control of a multiproduct CSTR, Ind. Eng. Chem. Res., 2006, 27, Kadam, J.; Srinivasan, B., Bonvin, D., Marquardt, W. Optimal grade transition in industrial polymerization processes via NCO tracking. AIChE J., 2007, 53, 3, Oldenburg, J.; Marquardt, W.; Heinz D.; Leineweber, D. B., Mixed-logic dynamic optimization applied to batch distillation process design, AIChE J. 2003, 48(11), E Perea, B E Ydstie and I E Grossmann, A model predictive control strategy for supply chain optimization, Comput. Chem. Eng., 2003, 27, M. Liepelt, K Schittkowski, Optimal control of distributed systems with breakpoints, p. 271 in M. Grötschel, S. Krumke, J. Rambau (eds.), Online Optimization of Large Systems, Springer, Berlin (2001) V. M. Zavala, and L.T.Biegler, Optimization-Based Strategies for the Operation of Low- Density Polyethylene Tubular Reactors: Nonlinear Model Predictive Control," Computers and Chemical Engineering, 33, pp (2009) See also Biegler homepage 43 25

26 For more details MOS-SIAM Series on Optimization NLP Theory and Algorithms Steady State Process Optimization Dynamic Process Optimization Optimal Control Sequential Approaches Simultaneous Approaches Mathematical Programs with Complementarity Constraints For more information: