Advanced Optimization Strategies for Integrated Dynamic Process Operations

L. T. BieglerCarnegie Mellon University

withY.Nie, C. M. Villa, J. M. Wassick

T-Y. Jung, H. Jang, J. H. Lee

Overview • Introduction to Dynamic Optimization

– Roadmap of Methods

• Off-line Reactor Optimization– Optimal design of runaway reactors– Polymerization Recipe Optimization

• Integrated Dynamic Optimization– Embedded recipe optimization – Combined optimization and scheduling

• Optimization Under Uncertainty– Back-off vs. Multi-stage stochastic programming– Polymer Case Study

• Conclusions

zi,I0 zi,II

0 zi,III0 zi,IV

0

zi,IVf

zi,If zi,II

f zi,IIIf

Bi

A + B → CC + B → P + EP + C → G

Impact and Challenges of Dynamic Optimization in Chemical Engineering?

• Design – Reactors– Separations– Batch process recipes

• Control– Optimal reference trajectories– Real-time optimization

• Operations– Transitions– Upsets– Integration with logistics

3

tf, final timeu(t), control variablesp, time independent parameters

t, timez(t), differential variablesy(t), algebraic variables

Dynamic Optimization Problem

s.t.

Dynamic Optimization Approaches

DAE Optimization Problem

Multiple Shooting

Embeds DAE Solvers/SensitivityHandles instabilities

Single Shooting

Hasdorff (1977), Sullivan (1977), Vassiliadis (1994)…Discretize controls

Simultaneous Collocation(Direct Transcription)

Large/Sparse NLP - Betts; B…

Apply a NLP solver

Efficient for constrained problems

Simultaneous Approach

Larger NLP

Discretize state, control variables

Variational Approach

Pontryagin et al.(1956)

Bock and coworkers

Take Full Advantage of Open Structure• Many Degrees of Freedom• Periodic Boundary Conditions• Multi-stage Formulations…

Nonlinear DynamicOptimization Problem

Collocation onfinite Elements

(Piecewise)Continuous profiles

Nonlinear ProgrammingProblem (NLP)

Discretized variables

Nonlinear Programming Formulation

• Fully Equation-based formulations• Exact gradients and Hessians • BVP (not IVP) stability properties • Sparse, large-scale structures fully exploited

7

Design Optimization of Runaway Reactors(Hagan et al., 1988; Koning, et al., 2006; Nie et al., 2013)

8

Design Optimization of Runaway Reactors

9

Dynamic Optimization Solution Strategy

10

Catalyst Distribution Profiles

Zones Max. Product1 79.333 105.5110 106.63∞ 106.64

11

Recipe OptimizationSemi-Batch Polymer Process (Nie et al., 2013)

12

Semi-batch polyether polyol process (Nie et al., 2014)

• Comprehensive population balance models for MWD properties• Moment models implemented and compared• Operating strategies validated in plant

13

Polyol Dynamic Process Validation

14

Process Recipe Optimization

15

Recipe Optimization Results

16

Optimal Constraint Profiles

17

Satisfaction of Product Specifications

18

Goal: Bridge between planning, logistics (linear, discrete problems) and detailed process models (nonlinear, spatial, dynamic)

Planning

Scheduling

Site-wide Optimization

Real-time Optimization

Model Predictive Control

Regulatory Control

Feas

ibili

ty

Per

form

ance

Corporate Decision Pyramid for Process Operations

19

• Coordinate batch and continuous unit operations• Minimize wait times• NLP models for dynamic optimization• MILP models for process/product scheduling• Manage shared resources and inventories

Planning

Scheduling

Site-wide Optimization

Real-time Optimization

Model Predictive Control

Regulatory Control

Feas

ibili

ty

Per

form

ance

Synchronizing Dynamic Optimization and Scheduling

Integration with Process Scheduling

Integration with Process Scheduling(State-Space RTN Formulation)

22

Integrated RTN/Optimization Framework

Reactor Opt.

RTNSchedule

23

Integration of Dynamic Optimization and RTN Scheduling

23

12% higher profit

24

Presented at Annual AIChE Meeting, Salt Lake City, Nov. 2015

25

Dynamic Real-Time Optimization of a Mixed Mode Process (Nie, Lin, Hayot, Wassick et al., 2015)

Reactor Model extensively validated against internal models and operating data

26

Integrated NMPC and On-Line Optimization: An Initial Study

D-RTO through NMPC• Retain dynamic model • Shrinking batch horizon• Variable batch time• Economic NMPC (tf)• Key Issues

– Uncertainty-tolerant optimization formulations

– Time critical optimization(efficient large-scale NLP)

– Accelerate with advanced step (sensitivity-based) strategies

Incorporation of Uncertainty• From states, parameters,

measurement noise • State estimation with expanding

batch horizon, …• Back-off constraints for noise, …• Stochastic, Multi-stage Programs

within MPC for model errors and disturbances with delayed realization (Lucia, Finkler, & Engell, 2013)

T.-Y. Jung, Y. Nie, J. H. Lee, L. T. B., IFAC-PapersOnLine, 48 (8), 164-169 (2015)H. Jang, J. H. Lee, L. T. B., pp. 37-42, IFAC Papers Online, (2016)

• Minimize operating time of polymerization under uncertainty• Consider the operating time as a manipulated variable• Shrinking Horizon NMPC combined with Expanding Horizon LSE

• Based on the rigorous DAE model used in recipe optimization• Over operation window → Minimization of approximated error

Recipe Optimization

Estimator (LSE)

Controller (NMPC)

Plant/Simulator

Framework for On-line Dynamic Optimization

• Minimize operating time of polymerization under uncertainty• Consider the operating time as a manipulated variable• Shrinking Horizon NMPC combined with Expanding Horizon LSE

• Based on the rigorous DAE model used in recipe optimization• Over operation window → Minimization of approximated error

Recipe Optimization

Estimator (LSE)

Controller (NMPC)

Plant/Simulator

Framework for On-line Dynamic Optimization

• Minimize operating time of polymerization under uncertainty• Consider the operating time as a manipulated variable• Shrinking Horizon NMPC combined with Expanding Horizon LSE

• Based on the rigorous DAE model used in recipe optimization• Over operation window → Minimization of approximated error

Recipe Optimization

Estimator (LSE)

Controller (NMPC)

Plant/Simulator

Framework for On-line Dynamic Optimization

Batch LSE formulation(Analogous to MHE)

min𝑥𝑥1,𝑤𝑤𝑘𝑘𝑘𝑘=1,…,𝐾𝐾−1

�𝑥𝑥1𝑒𝑒 𝑇𝑇 �𝑃𝑃1−1 �𝑥𝑥1𝑒𝑒 + ∑𝑘𝑘=1𝐾𝐾 𝑣𝑣𝑘𝑘𝑇𝑇𝑅𝑅−1𝑣𝑣𝑘𝑘 + ∑𝑘𝑘=1𝐾𝐾−1𝑤𝑤𝑘𝑘𝑇𝑇𝑄𝑄−1𝑤𝑤𝑘𝑘

�𝑥𝑥𝑘𝑘 = 𝑥𝑥𝑘𝑘𝑇𝑇 𝑑𝑑𝑘𝑘𝑇𝑇 𝑇𝑇

𝑥𝑥1𝑒𝑒 = �𝑥𝑥1 − �̅�𝑥1𝑣𝑣𝑘𝑘 = 𝑦𝑦𝑘𝑘 − ℎ 𝑥𝑥𝑘𝑘 ,𝑢𝑢𝑘𝑘

𝑤𝑤𝑘𝑘 = 𝑥𝑥𝑘𝑘+1 − 𝑓𝑓 𝑥𝑥𝑘𝑘 ,𝑢𝑢𝑘𝑘 , 𝑑𝑑𝑘𝑘𝑑𝑑𝑘𝑘+1 − 𝑑𝑑𝑘𝑘

�𝑥𝑥𝑘𝑘 ∈ 𝕏𝕏𝕏, 𝑢𝑢𝑘𝑘 ∈ 𝕌𝕌, 𝑦𝑦𝑘𝑘 ∈ 𝕐𝕐

Augmented state vector

Measurement error vector

Augmented state error vector

Feasible domain

Initial error

: the time index

• Using full state information contained in measurements

Adding back-off constraints(see Srinivasan, Bonvin et al., 2003)

,

Uncertainties Original constraint

Back-off constraint

Optimization w/o Uncertainty

Monte Carlo Simulation

Data ProcessingApproximate backoff 𝛽𝛽0

Optimization w/ Backoff Constraints

Monte Carlo SimulationApproximate backoff 𝛽𝛽𝑘𝑘

Optimal control profiles 𝑢𝑢∗

State and output profiles

Backoff𝛽𝛽0

Optimal control profiles 𝑢𝑢𝑘𝑘∗

No

Converged

Tighten path constraints and product specifications with back-off terms, β

Multi-stage MPC – Stochastic Programming Formulation

Lucia, Engell et al., J. Process Control, 2013

Scenario branching: effect of uncertainty while optimizing control input

Non-anticipativity: control inputs from same node set equal until uncertainty is realized

xk

xk+1

xk+2

xk+3

(dk, uk)

(dk+1, uk+1)(dk+2, uk+2)

Simplified Multi-stage MPC

Robust horizon = 1

Lucia, Engell et al., J. Process Control, 2013

Non-anticipativity: control inputs from same node set equal until uncertainty is realized

Multi-stage, economic MPC formulation

min𝑥𝑥𝑘𝑘+1𝑗𝑗 ,𝑢𝑢𝑘𝑘

𝑗𝑗 ,𝑘𝑘=𝐾𝐾,…,𝑁𝑁𝑇𝑇−1,𝑗𝑗=1,…,𝑁𝑁𝑆𝑆 𝑘𝑘∑𝑘𝑘=𝐾𝐾𝑁𝑁𝑇𝑇−1 ∑𝑗𝑗=1

𝑁𝑁𝑆𝑆(𝑘𝑘) ℒ 𝑥𝑥𝑘𝑘+1𝑗𝑗 ,𝑢𝑢𝑘𝑘

𝑗𝑗

𝑁𝑁𝑆𝑆(𝑘𝑘)

ℒ 𝑥𝑥𝑘𝑘+1𝑗𝑗 ,𝑢𝑢𝑘𝑘

𝑗𝑗 = Δ𝑡𝑡𝑏𝑏 (+𝜇𝜇𝑃𝑃)

𝑥𝑥𝑘𝑘+1𝑗𝑗 = 𝑓𝑓 𝑥𝑥𝑘𝑘

𝑝𝑝 𝑗𝑗 ,𝑢𝑢𝑘𝑘𝑗𝑗 ,𝑑𝑑𝑘𝑘

𝑗𝑗

𝑔𝑔 𝑥𝑥𝑘𝑘𝑗𝑗 ,𝑢𝑢𝑘𝑘

𝑗𝑗 = 0

𝑐𝑐𝑝𝑝 𝑥𝑥𝑘𝑘𝑗𝑗 ,𝑢𝑢𝑘𝑘

𝑗𝑗 + 𝜆𝜆 β ≤ 0

𝑐𝑐𝑒𝑒 𝑥𝑥𝑁𝑁𝑇𝑇𝑗𝑗 ,𝑢𝑢𝑁𝑁𝑇𝑇−1

𝑗𝑗 + 𝜆𝜆 β ≤ 𝑃𝑃,𝑃𝑃 ≥ 0

𝑥𝑥𝑘𝑘𝑗𝑗 ∈ 𝕏𝕏, 𝑢𝑢𝑘𝑘

𝑗𝑗 ∈ 𝕌𝕌

𝑢𝑢𝑘𝑘𝑗𝑗 = 𝑢𝑢𝑘𝑘𝑙𝑙 if 𝑥𝑥𝑘𝑘

𝑝𝑝 𝑗𝑗 = 𝑥𝑥𝑘𝑘𝑝𝑝 𝑙𝑙

Economic cost function (processing time)

Nonlinear dynamic model

Algebraic constraints

Back-off path constraint

Back-off end-point constraint (relaxed)

Feasible domain

Non-anticipativity constraints

k : time indexj : scenario index

• Tools: GAMS implementation using IPOPT• Numerical strategy: Simultaneous collocation approach

• 24 finite elements (sampling time > 10 min) and three Gauss-Radaucollocation points reactor model instance: 3000 variables, 150 dofs

• Hypothetical Uncertainties for Demonstration• Three branches for parameter error (Kp): min (-10%), nominal, max

(+10%) • ±2% errors in measurements and initial states

eNMPC: ± 2% initial state error and measurement noise

eNMPC+ batch LSE Performance (410 min)

Inpu

tsPa

th co

nstr

aint

s

• Stabilized control inputs• Adiabatic temperature bounded

• Oscillatory control inputs• Adiabatic temperature violated

eNMPC Performance(nominal time = 380 min)

Multi-stage eNMPC PerformanceModel Parameter Error (10%) (530 min)

• Stabilized control inputs• Adiabatic temperature bounded

Inpu

tsPa

th co

nstr

aint

sMulti-Stage eNMPC

Performance ComparisonMulti-stage eNMPC + LSE

Model , State, Measurement Errors (535 min)

• Violation of adiabatic end temperature constraint

Multi-stage MPC and Back-off eNMPC + batch LSE + back-off (all errors)

(589 min)

• Avg. computation time= 8.77 CPU seconds (max. 33.48 CPU sec)

• > 10*(nominal and back-off cases, ~0.7 CPUs) • << sampling time (10 ~ 20 min)

Inpu

tsPa

th co

nstr

aint

s

Multi-stage eNMPC + batch LSE + back-off (539 min)

Dynamic Optimization widely applicable in: • Process design• Process control• Integrated Process Operations

Simultaneous Collocation Approach• Efficient, accurate and robust optimization approach• Deals with unstable processes, state constraints

Significant Impact Observed with Simultaneous Optimization• Optimization of Runaway Reactors• Recipe Optimization• Integrated optimization for process operation and scheduling• Optimization under Uncertainty

Broad Scope to Handle On-line Optimization• Improved Formulations to Handle Uncertainty• Parallel Decomposition for Dynamic Optimization• Fast NMPC Formulations for Time Critical Solutions

Conclusions

to tf

× × × ×

Collocation points

• ••• •

••

••

••

•

Polynomials

× ×× ×

•

Finite element, i

ti

Mesh pointshi

× × × ×

∑=

=K

qiqq(t) zz(t)

0

× ××

×element i

q = 1q = 2

××××Continuous Differential variables

Discontinuous Algebraic and Control variables

×

×

× ×

Collocation on Finite Elements

∑=

=K

qiqq(t) yy(t)

1 ∑

=

=K

qiqq(t) uu(t)

1

τddz

hdtdz

i

1=

),( uzfhddz

i=τ

]1,0[,1

1'' ∈+= ∑

−

=

ττ ji

i

iiij hht

41

Integrated Approach (Dynamics/Scheduling Optimized Simultaneously)

• MINLP Solution via GBD• 3 major iterations• 393 CPUs (MILP: 20 CPUs, NLP: 373 CPUs)• 12% higher profit

Integrated Optimization: Scheduling and Dynamics

• Both approaches have optimized reactor recipes• Integrated recipes are re-optimized over time• Re-optimized recipe squeezes an extra batch with lower utility• 11.7% higher profit in integrated case (validated in industry)

End-point constraints

Method NAMW [g/mol] Unsat. value [mmol/g polyol]

Unreacted PO [ppm]

Limit 3027.74 (lower) 0.02 (upper) 2000 (upper)Standard 3017.22 0.0202 2508.40

Back-off only 3051.09 0.0198 1493.78Multi-stage +

Back-off 3029.66 0.0199 1893.36