dynamic real-time optimization: linking off-line planning...
TRANSCRIPT
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April 12, 2007
Dynamic Real-Time Optimization: Linking Off-line Planning with
On-line Optimization
L. T. Biegler and V. ZavalaChemical Engineering Department
Carnegie Mellon University Pittsburgh, PA 15213
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OverviewIntroduction• Pyramid of operations• Off-line vs On-line Tasks• Steady state vs Dynamics• Enablers and open questions
On-line Issues• Model predictive control• Treatment of nonlinear model and uncertainty• Benefits and successes
Advanced Step Controller• Basic concepts and properties• Performance for examples
Future Work
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Goal: Bridge between planning, logistics (linear, discrete problems) and detailed process models (nonlinear, spatial, dynamic)
Planning and Scheduling• Many Discrete Decisions• Few Nonlinearities
Planning
Scheduling
Site-wide Optimization
Real-time Optimization
Model Predictive Control
Regulatory Control
Feas
ibili
ty
Per
form
ance
Decision Pyramid for Process Operations
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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
Scheduling
Site-wide Optimization
Real-time Optimization
Model Predictive Control
Regulatory Control
Feas
ibili
ty
Per
form
ance
Decision Pyramid for Process Operations
APCMPC ⊂
Off-line (open loop)
On-line (closed loop)
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Steady State RTO
Data Reconciliation & Parameter Identification
•Estimation problem formulations•Steady state model•Maximum likelihood objective functions considered to get parameters (p)
Minp Φ(x, y, p, w)s.t. c(x, u, p, w) = 0
x ∈ X, p ∈ P
Plant
DR-PEc(x, u, p) = 0
RTOc(x, u, p) = 0
APC
y
p
u
w
On line optimization•Nonlinear steady state model for states (x)•Supply setpoints (u) to APC (control system) withLinear Dynamic Models•Model mismatch, measured and unmeasured disturbances (w)
Minu F(x, u, w)s.t. c(x, u, p, w) = 0x ∈ X, u ∈ U 5
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Steady-state On-line Optimization: Components
Plant
DR-PEc(x, u, p) = 0
RTOc(x, u, p) = 0
APC
y
p
u
w
•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? 6
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Steady-state On-line Optimization: Properties(Marlin and coworkers)
• Consistency of models and Stability of steady-state RTO
• Sensitivity of the optimum to disturbances and model mismatch? => NLP sensitivity
• Steady state test? Has the process changed?
– Statistical test on objective function => change is within a confidence regionsatisfying a χ2 distribution
– Implement new RTO solution only when the change is significant
– Assumes accurate model
• Are we optimizing on the noise? Model mismatch? Dynamics?
• Can lead to ping-ponging
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Why Dynamic RTO?
• 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
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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 2-3
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tf, final timeu, control variablesp, time independent parameters
t, timez, differential variablesy, algebraic variables
Dynamic Optimization Problem
( )ftp,u(t),y(t),z(t), ψmin
( )pttutytzFdt
tdz ,),(),(),()(=
( ) 0,),(),(),( =pttutytzG
ul
ul
ul
ul
o
ppputuuytyy
ztzzzz
≤≤
≤≤
≤≤
≤≤
=
)()(
)()0(
s.t.
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Dynamic Optimization Approaches
DAE Optimization Problem
Multiple Shooting
+Embeds DAE Solvers/Sensitivity- Dense Sensitivity Blocks
+Handles instabilities
Single Shooting
Sullivan (1977)
+Small NLP - No instabilities
Discretize controls
Collocation
Large/Sparse NLP
Direct NLP solutionEfficient for constrained problems
Simultaneous Approach
-Larger NLPs
Discretize states and controls
Indirect/Variational
Pontryagin(1962)
- Inefficient for large, constrained problems
Bock, Plitt (1984)
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Dynamic Optimization Engines
Evolution of NLP Solvers:
for dynamic optimization, control and estimation
E.g., NPSOL and Sequential Dynamic Optimization - over 100 variables and constraints E.g, SNOPT and Multiple Shooting - over 100 d.f.s but over 105 variables and constraintsE.g., IPOPT - Simultaneous dynamic optimizationover 1 000 000 variables and constraints
SQP rSQP Full-spaceBarrier
Object Oriented Codes tailored to structure, sparse linearalgebra and computer architecture (e.g., IPOPT 3.2)
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Hierarchy of Nonlinear Programming for Dynamic Optimization Formulations
Variables/Constraints102 104 106
Black Box
Direct SensitivitiesSingle Shooting
Multiple ShootingAdjoint Sensitivity
Simultaneous Full Space Formulation
100
SQPSQP
rSQPrSQP
Interior PointInterior Point
DFODFO
Com
putational Efficiency
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Comparison of Computational Complexity(α ∈ [2, 3], β ∈ [1, 2], nw, nu - assume Nm = O(N))
((nu + nw)N)------Backsolve
((nu + nw)N)β(nu N)α(nu N)αStep Determination
---nw3 N---NLP Decomposition
N (nu + nw)(nw N) (nu + nw)2(nw N) (nu N)2Exact Hessian
N (nu + nw)(nw N) (nu + nw)(nw N) (nu N)Sensitivity
---nwβ Nnw
β NDAE Integration
SimultaneousMultiple Shooting
Single Shooting
O((nuN)α + N2nwnu
+ N3nwnu2)
O((nuN)α + N nw3
+ N nw (nw +nu)2)O((nu + nw)N)β
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On-line Issues: Model Predictive Control (NMPC)
Process
NMPC Controller
d : disturbancesz : differential statesy : algebraic states
u : manipulatedvariables
ysp : set points
( )( )dpuyzG
dpuyzFz,,,,0,,,,
==′
NMPC Estimation and Control
Boundskxz
uzfzts
zEuzkxJ
lll
N
lNll
)()),(
..
)(),())((min
0
1
0
==
+=
+
=∑ϕ
u
NMPC Subproblem
Why NMPC?ν Track a profile – evolve from
linear dynamic models (MPC)ν Severe nonlinear dynamics (e.g,
sign changes in gains)ν Operate process over wide range
(e.g., startup and shutdown)
Model Updater( )( )dpuyzG
dpuyzFz,,,,0,,,,
==′
Planning
Scheduling
Site-wide Optimization
Real-time Optimization
Model Predictive ControlRegulatory Control
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MPC - BackgroundMotivate: 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
Stability properties• Nominal – remain bounded and eventually achieve desired state
without noise and with perfect model• Robust – remain stable with model mismatch and noise
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On-line Issues: Stability is Forever
Discrete Lyapunov stability proof
•Nominal case – no noise: perfect model•General formulation with local asymptotic controller for t ∞•Robust case – keep (Jk-1 – Jk) sufficiently positive in presence of noise/mismatch
{ }
0)(,0)(
))1(),1(()(
))1(),1((
)(),()())1(),1((
1110
|||1|1
→→⇒
−−=−≥
−−≥
−−+−−≥−
∑∑∞
=
∞
=−
−−
kukx
kukxJJJ
kukx
zEvzzEkukxJJ
kkkk
kNkNkNkNkk
ϕ
ϕ
ϕϕ
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Boundskxz
uzfzts
zEuzJ
lll
N
lNllk
)()),(
..
)(),(min
0
1
0
==
+=
+
=∑ϕ
u
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NMPC for Tennessee Eastman Process(Jockenhövel, Wächter, B., 2003)
Unstable Reactor
11 Controls; Product, Purge streams
171 DAEs: Model extended with energy balances
After discretization10920 variables660 degrees of freedom
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NMPC Results – Tennessee Eastman Problem
Optimization with IPOPT
Warm start with μ = 10-4
350 Optimizations
5-7 CPU seconds
11-14 Iterations
Optimization with SNOPT
Uses approximate (dense) reduced Hessian updates
Could not be solved within sampling times
> 100 Iterations
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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 As larger NLPsNLPs are considered for NMPC, can are considered for NMPC, can computational delay be overcome?computational delay be overcome?
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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 z0 for dynamic systems• Extended to Collocation approach – Zavala et al. (2006)• Three computational variants developed• Similar approach for MH State and Parameter Estimation – Zavala et al.
(2007)
Stability Properties for Advanced Step Controller• Nominal stability – can be shown from standard Lyapunov analysis• Robust stability – apply results for input to state stability (ISS) from Magni
et al. (2005)
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Nonlinear Model Predictive Control Nonlinear Model Predictive Control ––Parametric ProblemParametric Problem
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NLP SensitivityNLP SensitivityParametric Programming
NLP Sensitivity Rely upon Existence and Differentiability of Path
Main Idea: Obtain and find by Taylor Series Expansion
Optimality Conditions
Solution Triplet
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NLP SensitivityNLP Sensitivity
Optimality Conditions of
Obtaining
Already Factored at Solution
Sensitivity Calculation from Single Backsolve
Approximate Solution Retains Active Set
KKT Matrix IPOPT
Apply Implicit Function Theorem to around
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NMPC: Direct Sensitivity VariantNMPC: Direct Sensitivity Variant
Optimal Solution
Optimal Solution
Approximate by Perturbing
NLP Sensitivity Approximate Solution Retains Active Set of Nominal Problem
This is fast but active set is inconsistent because it does not shift
Assume:
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Nonlinear Model Predictive Control Nonlinear Model Predictive Control ––Equivalence to Previously Solved ProblemEquivalence to Previously Solved Problem
Solutions to both problems are equivalent in nominal case
(ideal models, no disturbances)
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Advanced Step NMPCAdvanced Step NMPCCombine delay concept with sensitivity to solve NLP in background
(between steps) – not on-line
Solve P(zℓ) in backgroundSensitivity to updated problem to get (z0, u0)Solve P(zℓ+1) in background with new (z0, u0)
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CSTR NMPC Example (Hicks and Ray)
Maintain unstable setpointClose to bound constraintFinal time constraint for stability (N is short)
Study effects of:Computational DelayAdvanced Step NMPCMeasurement NoiseModel Mismatch
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CSTR NMPC Example – Nominal Case
•NMPC applied with N = 10, τ = 0.5 sampling time•Unstable steady state•Steady state u close to upper bound•Computational delay = 0.5, leads to instabilities•Advanced Step Controller identical to ideal NMPC controller
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CSTR NMPC Example – Model Mismatch
Advanced Step NMPC not as robust as ideal - suboptimal selection of u(k)
Better than Direct Variant – due to better active set preservation
___ Ideal____ Adv. Step NMPC- - - - Direct Variant
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CSTR Example: Mismatch + Noise
___ Ideal____ Adv. Step NMPC- - - - Direct Variant
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Industrial Case Study – Grade Transition Control
Simultaneous Collocation-BasedApproach
27,135 constraints, 9630 LB & UB
Off-line Solution with IPOPT
Feedback Every 6 min
Process Model: 289 ODEs, 100 AEs
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Nonlinear Model Predictive ControlNonlinear Model Predictive ControlOptimal Feedback Policy (On-line Computation 351 CPU s)
Ideal NMPC controller - computational delay not consideredTime delays as disturbances in NMPC
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Nonlinear Model Predictive ControlNonlinear Model Predictive ControlOptimal Policy vs. NLP Sensitivity -Shifted (On-line Computation 1.04 CPU s)
Very Fast Close-to-Optimal FeedbackLarge-Scale Rigorous Models
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Future Work: MultiFuture Work: Multi--Stage Dynamic Stage Dynamic OptimizationOptimization
DAE Model
Stage Transitions
Bounds
Path ConstraintsBoundary Conditions
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Multi-stage OptimizationDetermine Off-line policy• Tracking problem for NMPC• Nominal policy or incorporate uncertainty in conservative
way (no recourse due to policy tracking)
Determine On-line policy• Solve consistent economic objective through on-line
DRTO (NMPC that incorporates multi-stage with long horizons)
• Exploit uncertainty through recourse on trajectory optimization
– Retain stability and robustness– Incorporate feedforward measurements
• Requires state and parameter estimation from plant– Use nonlinear model– Adopt Moving Horizon Estimation Formulation
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Moving Horizon EstimationMoving Horizon Estimation
Process
NMPC Controller
d : disturbancesz : differential statesy : algebraic states
u : manipulatedvariables
ysp : set points
( )( )dpuyzG
dpuyzFz,,,,0,,,,
==′
Model Updater( )( )dpuyzG
dpuyzFz,,,,0,,,,
==′
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RealReal--Time Moving Horizon EstimationTime Moving Horizon Estimation
On-Line Calculation
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Industrial Case StudyIndustrial Case Study
Single Measurement + Gaussian NoiseComposition of Recycle Gas
27,121 Constraints, 9330 Bounds294 Degrees of Freedom
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RealReal--Time Moving Horizon EstimationTime Moving Horizon Estimation
Correct Inertia at Solution – System Observable
On-Line Calculation
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Summary• RTO and MPC widely used for refineries, ethylene and,
more recently, chemical plants – Inconsistency in models– Can lead to operating problems
• Off-line dynamic optimization is widely used – Polymer processes (especially grade transitions)– Batch processes– Periodic processes
• NMPC provides link for off-line and on-line optimization– Stability and robustness properties– Advanced step controller leads to very fast calculations
• Analogous stability and robustness properties• On-line cost is negligible
• Multi-stage planning and on-line switches– Leads to exploitation of uncertainty (on-line recourse) – Avoids conservative performance– Update model with MHE– Evolve from regulatory NMPC to Large-scale DRTO
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ReferencesF. 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, to appear
Biegler Homepage: http://dynopt.cheme.cmu.edu/papers.htm
Forbes, J. F. and Marlin, T. E.. Model Accuracy for Economic Optimizing Controllers: The Bias Update Case. Ind.Eng.Chem.Res. 33, 1919-1929. 1994
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], 717-734. 1996
Grossmann Homepage: http://egon.cheme.cmu.edu/papers.html
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], 1345-1362. 2002.
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References – Some DRTO Case Studies
Busch, J.; Oldenburg, J.; Santos, M.; Cruse, A.; Marquardt, W. Dynamicpredictive scheduling of operational strategies for continuous processes using mixed-logic dynamic optimization, Comput. Chem. Eng., 2007, 31, 574-587.
Flores-Tlacuahuac, A.; Grossmann, I.E. Simultaneous cyclic scheduling andcontrol of a multiproduct CSTR, Ind. Eng. Chem. Res., 2006, 27, 6698-6712.
Kadam, J.; Srinivasan, B., Bonvin, D., Marquardt, W. Optimal grade transition in industrial polymerization processes via NCO tracking. AIChE J., 2007, 53, 3, 627-639.
Oldenburg, J.; Marquardt, W.; Heinz D.; Leineweber, D. B., Mixed-logic dynamic optimization applied to batch distillation process design, AIChE J. 2003, 48(11), 2900- 2917.
E Perea, B E Ydstie and I E Grossmann, A model predictive control strategy for supply chain optimization, Comput. Chem. Eng., 2003, 27, 1201-1218.
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)
See also Biegler homepage
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