dynamic real-time optimization: concepts in modeling, algorithms … · dynamic real-time...

1

Dynamic Real-Time Optimization: Concepts in Modeling, Algorithms and

Properties

L. T. BieglerChemical Engineering Department

Carnegie Mellon UniversityPittsburgh, PA

November 28, 2007

2

I IntroductionTypical ApplicationsProblem Statement

II Dynamic OptimizationSequential MethodsMultiple ShootingSimultaneous Methods

III Off-line Case StudiesUnstable Grade TransitionsSimulated Moving BedsParameter Estimation – Reactor Models

IV On-line OptimizationNMPC Case StudyAdvanced Step NMPCMoving Horizon Estimation

V ConclusionsSummary References

Dynamic Optimization Outline

2

3

DAE Models in Process Engineering

Differential EquationsConservation Laws (Mass, Energy, Momentum)

Algebraic EquationsConstitutive Equations, Equilibrium (physical properties, hydraulics, rate laws)Semi-explicit formAssume to be index one (i.e., algebraic variables can be solved uniquely by algebraic equations)If not, DAE can be reformulated to index one (see Ascher and Petzold)

CharacteristicsLarge-scale models – not easily scaledSparse but no regular structureDirect linear solvers widely usedCoarse-grained decomposition of linear algebra

4

Batch Distillation Multi-product Operating Policies

5XQ�EHWZHHQ�GLVWLOODWLRQ�EDWFKHV7UHDW�DV�ERXQGDU\�YDOXH�RSWLPL]DWLRQ�SUREOHP

:KHQ�WR�VZLWFK�IURP�$�WR�RIIFXW WR�%"+RZ�PXFK�RIIFXW WR�UHF\FOH"5HIOX["%RLOXS 5DWH"2SHUDWLQJ�7LPH"

$ %

3

5

Catalytic Cracking of Gasoil (Tjoa, 1991)

number of states and ODEs: 2number of parameters:3no control profilesconstraints: pL � p � pU

Objective Function: Ordinary Least Squares

(p1, p2, p3)0 = (6, 4, 1)(p1, p2, p3)* = (11.95, 7.99, 2.02)(p1, p2, p3)true = (12, 8, 2)

1.00.80.60.40.20.0

0.0

0.2

0.4

0.6

0.8

1.0

YA_data

YQ_data

YA_estimate

YQ_estimate

t

Yi

Parameter Estimation

0)0( ,1)0(

)(

, ,

22

1

231

321

==−−=

+−=

→→→

qa

qpapq

appa

SASQQA ppp

�

�

6

Optimization of dynamic batch process operation resulting from reactor and distillation column

DAE models:z’ = f(z, y, u, p)g(z, y, u, p) = 0

number of states and DAEs: nz + nyparameters for equipment design (reactor, column)nu control profiles for optimal operation

Constraints: uL � u(t) � uU zL � z(t) � zU

yL � y(t) � yU pL � p � pU

Objective Function: amortized economic function at end of cycle time tf

zi,I0 zi,II

0zi,III0 zi,IV

0

zi,IVf

zi,If zi,II

f zi,IIIf

Bi

A+B→C

C+B→P+E

P+C→G

5

10

15

20

25

580

590

600

610

620

630

640

0 0.5 1 1.5 2 2.50 0.25 0.5 0.75 1 1.25Time (h r.)

Dyn a m ic

C o nsta nt

Dyn a mic

C o nsta nt

optimal reactor temperature policy optimal column reflux ratio

Batch Process Optimization

4

7

Nonlinear 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

NMPC Subproblem

Why NMPC?

� Track a profile

� Severe nonlinear dynamics (e.g, sign changes in gains)

� Operate process over wide range (e.g., startup and shutdown)

Model Updater( )( )dpuyzG

dpuyzFz

,,,,0

,,,,

==′

sConstraintOther

sConstraint Bound

0init

212sp

z)t(z)t),t(),t(y),t(z(G

)t),t(),t(y),t(z(F)t(z.t.s

||))||||y)t(y||minuy Q

kkQ

==

=′

−+−∑ ∑ −

u

u

u(tu(tu

8

tf, final timeu, control variablesp, time independent parameters

t, timez, differential variablesy, algebraic variables

Dynamic Optimization Dynamic Optimization ProblemProblem

( )ftp,u(t),y(t),z(t), min Φ

( )pttutytzfdt

tdz,),(),(),(

)( =

( ) 0,),(),(),( =pttutytzg

ul

ul

ul

ul

o

ppp

utuu

ytyy

ztzz

zz

dd

dd

dd

dd

)(

)(

)(

)0(

s.t.

5

9

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 solution

Efficient for constrained problems

Simultaneous Approach

-Larger NLPs

Discretize states and controls

Indirect/Variational

Pontryagin(1962)

- Inefficient for large, constrained problems

Bock, Plitt (1984)

10

Sequential Approaches - Parameter Optimization

Consider a simpler problem without control profiles:

e.g., equipment design with DAE models - reactors, absorbers, heat exchangers

Min Φ (z(tf))

z’ = f(z, p), z (0) = z0

g(z(tf)) � 0, h(z(tf)) = 0

By treating the ODE model as a "black-box" a sequential algorithm can be constructed that can be treated as a nonlinear program.

Task: How are gradients calculated for optimizer?

NLPSolver

ODEModel

GradientCalculation

P

φ,g,h

z (t)

6

11

Gradient Calculation

Perturbation

Sensitivity Equations

Adjoint Equations

Perturbation

Calculate approximate gradient by solving ODE model (np + 1) times

Let ψ = Φ, g and h (at t = tf)

dψ/dpi = {ψ (pi + ̈ pi) - ψ (pi)}/ ¨pi

Very simple to set up

Leads to poor performance of optimizer and poor detection of optimum unless roundoff error (O(1/¨pi) and truncation error (O(̈pi)) are small.

Work is proportional to np (expensive)

12

Direct Sensitivity

From ODE model:

(nz x np sensitivity equations)

• z and si , i = 1,…np, an be integrated forward simultaneously.

• for implicit ODE solvers, si(t) can be carried forward in time after converging on z

• linear sensitivity equations exploited in ODESSA, DASSAC, DASPK, DSL48s and a number of other DAE solvers

Sensitivity equations are efficient for problems with many more constraints than parameters (1 + ng + nh > np)

{ }

iii

T

iii

ii

p

zss

z

f

p

fs

dt

ds

ip

tzts

pzztpzfzp

∂∂=

∂∂+

∂∂==′

=∂

∂=

==′∂∂

)0()0( ,)(

...np 1, )(

)( define

)()0(),,,( 0

7

13

Multiple Shooting for Dynamic Optimization

Divide time domain into separate regions

Integrate DAEs state equations over each region

Evaluate sensitivities in each region as in sequential approach wrt uij, p and zj

Impose matching constraints in NLP for state variables over each region

Variables in NLP are due to control profiles as well as initial conditions in each region

14

Multiple ShootingNonlinear Programming Problem

uL

x

xxx

xc

xfn

≤≤

=

ℜ∈

0)(s.t

)(min

( ))(),( min,,

ffpu

tytzji

ψ

( ) z)z(tpuyzfdt

dzjjji ==

,,,, ,

( ) 0,ji, =pz,y,ug

ul

uiji

li

ukkjij

lk

ukkjij

lk

jjjij

ppp

uuu

ytpuzyy

ztpuzzz

ztpuzz

≤≤

≤≤

≤≤

≤≤

=− ++

,

,

,

11,

),,,(

),,,(

0),,,(s.t.

(0)0 zzo = Solved Implicitly

8

15

Dynamic Optimization – Multiple Shooting Strategies

Larger NLP problem O(np+nu+NE nz) • Use SNOPT, MINOS, etc.• Second derivatives difficult to get

Repeated solution of DAE model and sensitivity/adjoint equations, scales with nz and np

• Dominant computational cost• May fail at intermediate points

Multiple shooting can deal with unstable systems with sufficient time elements.

Discretize control profiles to parameters (at what level?)

Path constraints are difficult to handle exactly for NLP approach

Block elements for each element are dense!

Extensive developments and applications by Bock and coworkers using MUSCOD code

16

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 solution

Efficient for constrained problems

Simultaneous Approach

-Larger NLPs

Discretize states and controls

Indirect/Variational

Pontryagin(1962)

- Inefficient for large, constrained problems

Bock, Plitt (1984)

9

17

Nonlinear DynamicOptimization Problem

Collocation onfinite Elements

Continuous variables

Nonlinear ProgrammingProblem (NLP)

Discretized variables

Nonlinear Programming Formulation

18

Discretization of Differential Equations Orthogonal Collocation

Given:dz/dt = f(z, u, p), z(0)=given

Approximate z and u by Lagrange interpolation polynomials (order K+1 and K, respectively) with interpolation points, tk

kkNjk

jK

kjj

k

K

kkkK

kkNjk

jK

kjj

k

K

kkkK

ututt

ttttutu

ztztt

ttttztz

===>−−

∏==

===>−−

∏==

≠==

+

≠==

+

∑

∑

)()(

)()(,)()(

)()(

)()(,)()(

11

100

1

""

""

Substitute zN+1 and uN into ODE and apply equations at tk.Substitute zN+1 and uN into ODE and apply equations at tk.

Kkuzftztr kk

K

jkjjk ,...1 ,0),()()(

0

==−= ∑=

"�

10

19

z(t)

z N+1(t)

Sta

te P

rofile

t ft 1 t 2 t 3

r(t)

t 1 t 2 t 3

Min φ(z(tf))s.t. z’ = f(z, u, p), z(0)=z0

g(z(t), u(t), p) � 0h(z(t), u(t), p) = 0

to Nonlinear Program

How accurate is approximation

Converted Optimal Control Problem

Using Collocation

0)1(

,...1

0

0

z(0) ,0),()(

0

00

=−

=

=≤

==−

∑

∑

=

=

f

K

jjj

kk

kk

kk

K

jkjj

f

zz

Kk

),uh(z

),ug(z

zuzftz

)(z Min

"

"�

φ

20

to tf

u u u u

Collocation points

• ••• •

•• •

•••

•

Polynomials

u uu u

•

Finite element, i

ti

Mesh points

hi

u u u u

∑=

=K

qiqq(t) zz(t)

0

"

u uu

u

element i

q = 1q = 1

q = 2 q = 2

uuuuContinuous Differential variables

Discontinuous Algebraic and

Control variables

u

u

u u

Collocation on Finite ElementsCollocation on Finite Elements

∑=

=K

qiqq(t) yy(t)

1

" ∑=

=K

qiqq(t) uu(t)

1

"

τd

dz

hdt

dz

i

1=

),( uzfhd

dzi=

τ

NE 1,.. i 1,..K,k ,0),,())(()(0

===−= ∑=

K

jikikikjijik puzfhztr τ"�

]1,0[,1

1’’ ∈+= ∑

−

=

ττ ji

i

iiij hht

11

21

Nonlinear Programming ProblemNonlinear Programming Problem

uL

x

xxx

xc

xfn

≤≤

=

ℜ∈

0)(s.t

)(min( )fzψ min

( ) 0,, ,,, =p,uyzg kikiki

ul

ujiji

lji

uji

lji

ul

ppp

uuu

yyy

zzz

≤≤

≤≤

≤≤

≤≤

,,,

,ji,,

ji, ji,ji,

s.t. ∑=

=−K

jikikikjij puzfhz

0

0),,())(( τ"�

)0( ,0))1((

,..2 ,0))1((

100

,

00,1

zzzz

NEizz

K

jfjjNE

K

jijji

==−

==−

∑

∑

=

=−

"

"

Finite elements,hi, can also be variable to determine break points for u(t).

Add hu � hi � 0, Σ hi=tf

Can add constraints g(h, z, u) � ε for approximation error

22

Theoretical Properties of Simultaneous Method

A. Stability and Accuracy of Orthogonal Collocation

• Equivalent to performing a fully implicit Runge-Kuttaintegration of DAE models at Gaussian (Radau) points

• 2K order (2K-1) method which uses K collocation points• Algebraically stable (i.e., possesses A, B, AN and BN stability)

B. Analysis of the Optimality Conditions (Kameswaran, B., 2007)

• An equivalence has been established between the KKT conditions of NLP and the variational necessary conditions

• Rates of convergence have been established for the NLP method

12

23

A B

C

u

u /22

u(T(t))

Example: Batch reactor - temperature profile

Maximize yield of B after one hour’s operation by manipulating a transformed temperature, u(t).

⇒ Minimize -zB(1.0)s.t.

z’A = -(u+u2/2) zA

z’B = u zA

zA(0) = 1zB(0) = 00 � u(t) � 5

Optimality conditions:H = -λA(u+u2/2) zA + λB u zA

∂H/∂u = λA (1+u) zA + λB zAλ’A = λA(u+u2/2) - λB u, λA(1.0) = 0λ’B = 0, λB(1.0) = -1

24

Op

tim

al P

rofile

, u

(t)

0. 0.2 0.4 0.6 0.8 1.0

2

4

6

Time, h

Results:Piecewise Linear Approximation with Variable Time ElementsOptimum B/A: 0.5726Equivalent # of ODE solutions: 32

Batch Reactor Optimal Temperature Program Piecewise Linear

13

25

Op

tim

al P

rofile

, u

(t)

0. 0.2 0.4 0.6 0.8 1.0

2

4

6

Time, h

Results:Control Vector Iteration with Conjugate GradientsOptimum (B/A): 0.5732Equivalent # of ODE solutions: 58

Batch Reactor Optimal Temperature Program

Indirect Approach

26

Results of Optimal Temperature Program Batch Reactor (Revisited)

Results- NLP with Orthogonal CollocationOptimum B/A - 0.5728# of ODE Solutions - 0.7(Equivalent)

14

27

Dynamic Optimization Engines

Evolution of NLP Solvers:

Î for dynamic optimization, control and estimation

E.g., NPSOL and Sequential Dynamic Optimization - over 100 variables and constraints

SQP

28




E.g, SNOPT and Multiple Shooting - over 100 d.f.s but over 105 variables and constraints

SQP rSQP

15

29




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 3.3)

30

Barrier Methods for Large-Scale Nonlinear Programming

0

0)(s.t

)(min

≥=

ℜ∈

x

xc

xfnx

Original Formulation

0)(s.t

ln)()( min1

=

−= ∑=ℜ∈

xc

xxfxn

ii

x nµϕµBarrier Approach

Can generalize for

bxa ≤≤

⇒ As µ Î 0, x*(µ) Î x* Fiacco and McCormick (1968)

16

31

Solution of the Barrier Problem

⇒ Newton Directions (KKT System)

0 )(

0

0 )()(

==−=−+∇

xc

eXv

vxAxf

µλ

⇒ Solve

−

−+∇−=

−

eXv

c

vAf

d

d

d

XV

A

IAW x

T

0

00

µ

λ

ν

λ

IPOPT Code IPOPT Code –– www.coinwww.coin --or.orgor.org

)(

...]1 ,1 ,1[

xdiagX

eT

==

32

Solution of the Barrier Problem

⇒ Newton Directions (KKT System)

0 )(

0

0 )()(

==−=−+∇

xc

eXv

vxAxf

µλ

⇒ Solve

−

−+∇−=

−

eXv

c

vAf

d

d

d

XV

A

IAW x

T

0

00

µ

λ

ν

λ

⇒ Reducing the Systemxv VdXveXd 11 −− −−=µ

∇−=

Σ++ c

d

A

AW x

T

µϕλ

0 VX 1−=Σ

IPOPT Code IPOPT Code –– www.coinwww.coin --or.orgor.org

)(

...]1 ,1 ,1[

xdiagX

eT

==

17

33

IPOPT Algorithm – Features

Line Search Strategies for Globalization

- l2 exact penalty merit function

- augmented Lagrangian merit function

- Filter method (adapted and extended from Fletcher and Leyffer)

Hessian Calculation

- BFGS (full/LM and reduced space)

- SR1 (full/LM and reduced space)

- Exact full Hessian (direct)

- Exact reduced Hessian (direct)

- Preconditioned CG

Algorithmic PropertiesGlobally, superlinearlyconvergent (Wächter and B., 2005)

Easily tailored to different problem structures

Freely AvailableCPL License and COIN-OR distribution: http://www.coin-or.org

IPOPT 3.x recently rewritten in C++

Solved on thousands of test problems and applications

34

Comparison of NLP Solvers: Data Reconciliation(Poku, Kelly, B. (2004))

0.01

0.1

1

10

100

0 200 400 600

Degrees of Freedom

CP

U T

ime

(s,

norm

.) LANCELOT

MINOS

SNOPT

KNITRO

LOQO

IPOPT

0

200

400

600

800

1000

0 200 400 600Degrees of Freedom

Itera

tions

LANCELOT

MINOS

SNOPT

KNITRO

LOQO

IPOPT

18

35

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

36

Case Studies• Reactor - Based Flowsheets• Fed-Batch Penicillin Fermenter• Temperature Profiles for Batch Reactors• Parameter Estimation of Batch Data• Synthesis of Reactor Networks• Batch Crystallization Temperature Profiles• Ramping for Continuous Columns• Reflux Profiles for Batch Distillation and Column Design• Air Traffic Conflict Resolution• Satellite Trajectories in Astronautics• Batch Process Integration• Source Detection for Municipal Water Networks• Optimization of Simulated Moving Beds• Grade Transition of Polymerization Processes• Parameter Estimation of Tubular Reactors• Nonlinear MPC

Simultaneous DAE Optimization

19

37

Production of High Impact Polystyrene (HIPS)Startup and Transition Policies (Flores et al., 2005a)

Catalyst

Monomer, Transfer/Term. agents

Coolant

Polymer

38

Upper Steady−State

Bifurcation Parameter

System State

Lower Steady−State

Medium Steady−State

Phase Diagram of Steady States

Transitions considered among all steady state pairs

20

39

Upper Steady−State

Bifurcation Parameter

System State

Lower Steady−State

Medium Steady−State

Phase Diagram of Steady States

Transitions considered among all steady state pairs

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

300

350

400

450

500

550

600

N1

N2

N3

A1

A4

A2

A3

A5

Cooling water flowrate (L/s)

Te

mp

era

ture

(K

)

1: Qi = 0.00152: Qi = 0.00253: Qi = 0.0040

3

2

1

0 0.02 0.04 0.06 0.08 0.1 0.12

300

350

400

450

500

550

600

650

Initiator flowrate (L/s)

Te

mp

era

ture

(K

)

1: Qcw = 102: Qcw = 1.03: Qcw = 0.1

32 1

3

2

1

N2

B1 B2 B3

C1C2

40

0 0.5 1 1.5 20

0.5

1x 10

−3

Time [h]

Initi

ator

Con

c. [m

ol/l]

0 0.5 1 1.5 24

6

8

10

Time [h]

Mon

omer

Con

c. [m

ol/l]

0 0.5 1 1.5 2300

350

400

Time [h]

Rea

ctor

Tem

p. [

K]

0 0.5 1 1.5 2290

300

310

320

Time [h]

Jack

et T

emp.

[K

]

0 20 40 60 80 1000

0.5

1

1.5x 10

−3

Time [h]

Initi

ator

Flo

w. [

l/sec

]

0 0.5 1 1.5 20

0.5

1

Time [h]Coo

ling

wat

er F

low

. [l/s

ec]

0 0.5 1 1.5 20

1

2

3

Time [h]

Fee

drat

e F

low

. [l/s

ec]

• 926 variables• 476 constraints• 36 iters. / 0.95 CPU s (P4)

Startup to Unstable Steady State

21

41

HIPS Process Plant (Flores et al., 2005b)

•Many grade transitions considered with stable/unstable pairs

•1-6 CPU min (P4) with IPOPT

•Study shows benefit for sequence of grade changes to achieve wide range of grade transitions.

42

Simulated Moving Bed Optimization(Kawajiri, B., 2005-2007)

Direction of liquid flowand valve switching

Feed Raffinate

DesorbentExtract

22

43



Feed

Raffinate

Desorbent

ExtractRepeats exactly

the same operation

(Symmetric)

44



Repeats exactly

the same operation

(Symmetric)

Operating parameters:

4 Zone velocities

+

Step time

Zone 4 Zone 2

Zone 3

Zone 1

23

45

Formulation of Optimization Problem

Zone velocities Step time

(Maximize average feed velocity)

Bounds on liquid velocities

Product requirements

CSS constraintSMB model

46

Treatment of PDEs: Simultaneous Approach

t

x

(Orthogonal Collocation on Finite Elements)

k=1

k=2k=3

Algebraic equations PDE

Step size is determined a priori

Step size is determined a priori

tHuge number of variables (handled by optimizer)

C(xi,t)

24

47

Comparison of two approaches

CPU Time*

Sequential Approach 111.8 min

1.53 minSimultaneous Approach

# of iteration

49

47

0 1 2 3 4 5 6 7 8−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x [m]

Nor

mal

ized

Con

cenr

atio

n, C

i(x,t st

ep)/

CF

,i

Comp.1 Single discretizationComp.2 Single discretizationComp.1 Full discretizationComp.2 Full discretization Sequential and Simultaneous

methods find same optimal solution

# of variables

33999

644Implemented on gPROMS, solved using SRQPDImplemented on gPROMS, solved using SRQPD

Implemented on AMPL, solved using IPOPTImplemented on AMPL, solved using IPOPT

*On Pentium IV 2.8GHz

(89% spent by integrator)

(Linear isotherm, fructose/glucose separation)

Initial feed velocity: 0.01 m/h

Optimal feed velocity: 0.52 m/h

Optimization

48

z Standard SMB

Nonstandard SMB: Addressed by Extended Superstructure NLP

z Three Zone

(Circulation loop is cut open)

z VARICOL

(Asynchronous switching)

25

49

0 2 4 6 80

0.5

1

x [m]

Ci(x

,t)/C

F,i →

8.000 m/h

⇓u

D1 =8.000m/h

t/tstep

= 0.142

→8.000 m/h

t/tstep

= 0.142

→8.000 m/h

⇓ uR3 =8.000m/h

t/tstep

= 0.142

→8.000 m/h

⇓u

D4 =8.000m/h

⇓ uE4=8.000m/h

t/tstep

= 0.142

0 2 4 6 80

0.5

1

x [m]

Ci(x

,t)/C

F,i →

7.916 m/h

⇓u

D1 =7.916m/h

t/tstep

= 0.334

→7.916 m/h

t/tstep

= 0.334

→7.916 m/h

⇓ uR3 =7.916m/h

t/tstep

= 0.334

→8.000 m/h

⇓u

D4 =8.000m/h

⇓ uE4=8.000m/h

t/tstep

= 0.334

0 2 4 6 80

0.5

1

x [m]

Ci(x

,t)/C

F,i →

8.000 m/h

⇓u

D1 =8.000m/h

t/tstep

= 0.434

→8.000 m/h

t/tstep

= 0.434

→8.000 m/h

t/tstep

= 0.434

→8.000 m/h

⇓ uE4=8.000m/h

t/tstep

= 0.434

0 2 4 6 80

0.5

1

x [m]

Ci(x

,t)/C

F,i →

6.332 m/h

⇓u

D1 =6.332m/h

t/tstep

= 0.800

→6.332 m/h

t/tstep

= 0.800

→8.000 m/h

⇓u

F3=1.668m/h

t/tstep

= 0.800

→8.000 m/h

⇓ uE4=8.000m/h

t/tstep

= 0.800

0 2 4 6 80

0.5

1

x [m]

Ci(x

,t)/C

F,i →

8.000 m/h

⇓u

D1 =8.000m/h

⇓ uE1=3.757m/h

t/tstep

= 1.000

→4.243 m/h

t/tstep

= 1.000

→8.000 m/h

⇓u

F3=3.757m/h

t/tstep

= 1.000

→8.000 m/h

⇓ uR4 =8.000m/h

t/tstep

= 1.000

Optimal Operating Scheme:Result of Superstructure Optimization

Standard SMB

PowerFeed Super -Structure

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Op

tim

al

Th

rou

gh

pu

t [m

/h]

CPU Time for optimization: 9.03 min*

34098 variables, 34013 equations *on Xeon 3.2 GHz

50

Parameter Estimation

Model Development - Kinetics

Product Properties Correlations

On-line Implementation - DisturbancesHeat-transfer, Impurities

},...,,,,,,,{ dcpsmi ttfffpIik ∈

LargeLarge --Scale Rigorous Reactor ModelScale Rigorous Reactor Model

Reactor Operating ConditionsReactor Operating Conditions

Polymer Properties (Melt Index)Polymer Properties (Melt Index)

Kinetic Mechanism (Free-Radical)

Reaction Rates

Method of Moments Monomer(s)

Initiator(s)

Solvent(s), CTA(s)

Moments NCLD

Material & Energy ODEsPDEs

Type & ConfigurationSteady-State vs. Dynamic

RigorousPTT Properties

ViscosityHeat Capacity

DensityTransport

MonomerComonomer

Initiator(s)

1 2 3 NZ

LargeLarge --Scale Parameter EstimationScale Parameter EstimationPolymerization Reactor Polymerization Reactor (Zavala, B., 2006)(Zavala, B., 2006)

H I G HP R E S S U R ER E C Y C L E

L O WP R E S S U R ER E C Y C L E

E t h y l e n e

B u t a n eP u r g e

T o r c h

P R I M A R YC O M P R E S S O R

H Y P E R - C O M P R E S S O R

T U B U L A RR E A C T O R

L D P

W a x e s

2 7 b a r s / 2 5 C

6 b a r s / 4 0 C1 . 5 b a r s / 4 0 C

6 b a r s / 4 0 C

2 4 b a r s / 4 0 C5 6 b a r s / 4 0 C1 0 3 b a r s / 4 0 C

2 5 5 b a r s4 0 C

1 0 1 0 b a r s4 0 C

2 2 0 0 b a r s8 5 C

P r e h e a t i n gZ o n e

R e a c t i o nZ o n e

C o o l i n gZ o n e

2 . 5 b a r s2 2 0 C

1 . 5 b a r s4 0 C

2 8 5 b a r s4 0 C

L D V a l v e

2 7 4 C

3 5 0 b a r s2 7 0 C

O i l s

H i g h P r e s s u r eS e p a r a t o r

L o w P r e s s u r eS e p a r a t o r

26

51

MonomerComonomer

Initiator(s)

1 2 3 NZ

z z z z

Material & Energy

Physical Properties

Zone Transitions

8 Stiffness + Highly Nonlinear + Parametric Sensit ivity + Algebraic Coupling

500 ODEs1000 AEs

LargeLarge --Scale Parameter EstimationScale Parameter Estimation

52


~ 35 Elementary Reactions~100 Kinetic Parameters

� Complex Kinetic Mechanisms

27

53


� Parameter Estimation for Industrial Applications

� Use Rigorous Model to Match Plant Data Directly

� Start with Standard Least-Squares Formulation

Rigorous Reactor Model

� Special Case of Multi-Stage Dynamic Optimization Pr oblem

� Solve using Simultaneous Collocation-Based Approach

Least-Squares

1 data set 6 data setsx 6500 ODEs

1000 AEs3000 ODEs6000 AEs

54

� Multi-Zone Tubular Reactor – Quasi Steady-State

� 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 Development and Discrimination among Rigorous Models

� Results

� Single Data Set (On-line Adjusting Parameters)

� Multiple Data Sets (On-line Adjusting Parameters + Kinetics)

Bottleneck (Memory Requirements) Factorization Step


28

55

Improved Match of Reactor Temperatures Profile


56

Industrial Case StudyIndustrial Case Study

� Results - Reactor Overall Monomer Conversion ( up to 20 Different Grades )( up to 20 Different Grades )

Avg. Conversion DeviationBase Model – 12.1 %New Model – 2.5 %

EVM Results - 0.12 %

Predicted Conversion (%)

Pla

nt C

onve

rsio

n (%

)

29

57

Exploit Structure of KKT Matrix – Laird, B. 2006

Parameter Estimation in Parallel Architectures Parameter Estimation in Parallel Architectures

Direct Factorization MA27Memory Bottlenecks

Factorization Time Scales Superlinearly with Data sets

Block-bordered Diagonal Structure

Coarse-Grained Parallelization using Schur Complement Decomposition

IPOPT 3.x architecture supports tailored structured decompositions

58Computational Results – LDPE Reactor EVM Problem

Parameter Estimation in Parallel ArchitecturesParameter Estimation in Parallel Architectures(Zavala, Laird, B., 2007)

30

59

Supply Chain, Planning and Scheduling• Large LP and MILP models• Many Discrete Decisions• Few Nonlinearities• Essential link needed to process models• Decisions need to be feasible at lower levels

Planning

Scheduling

Site-wide Optimization

Real-time Optimization

Model Predictive Control

Regulatory Control

Fe

asi

bili

ty

Pe

rfo

rma

nc

e

Decision-making in Chemical Industries

60

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




Regulatory Control

Fe

asi

bili

ty

Pe

rfo

rma

nc

e

Decision Pyramid for Process Operations

APCMPC ⊂

Off-line (open loop)

On-line (closed loop)

31

61

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 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

Planning

Scheduling




Regulatory Control

Fe

asi

bili

ty

Pe

rfo

rma

nc

e

62


Process

NMPC Controller



ysp : set points

( )( )dpuyzG

dpuyzFz

,,,,0

,,,,

==′


sConstraintOther

sConstraint Bound

0init

212sp




kkQ

==

=′

−+−∑ ∑ −

u

u

u(tu(tu

NMPC Subproblem

Why NMPC?

� Track a profile




dpuyzFz

,,,,0

,,,,

==′

32

63


Process

NMPC Controller



ysp : set points

( )( )dpuyzG

dpuyzFz

,,,,0

,,,,

==′


sConstraintOther

sConstraint Bound

0init

212sp




kkQ

==

=′

−+−∑ ∑ −

u

u

u(tu(tu

NMPC Subproblem

Why NMPC?

� Track a profile




dpuyzFz

,,,,0

,,,,

==′

64

Tennessee Eastman Process(Downs and Vogel, 1993)

Unstable Reactor

11 Controls; Product, Purge streams

Model extended with energy balances

33

65

Tennessee Eastman NMPC Model(Jockenhövel, Wächter, B., 2003)

Method of Full Discretization of State and Control Variables

Large-scale Sparse block-diagonal NLP

11Difference (control variables)

141Number of algebraic equations

152Number of algebraic variables

30Number of differential equations

DAE Model

14700Number of nonzeros in Hessian

49230Number of nonzeros in Jacobian

540Number of upper bounds

780Number of lower bounds

10260Number of constraints

109200

Number of variablesof which are fixed

NLP Optimization problem

66

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

34

67

Case Study: Change Reactor Pressure by 60 kPa

Optimization with IPOPT

1000 Optimization Cycles

5-7 CPU seconds

11-14 Iterations

Optimization with SNOPT

Often failed due to poor conditioning

Could not be solved within sampling times

> 100 Iterations

68

Limitations to NMPC Implementation

Issues: time-critical, more complex models, fast NLP solvers.

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?

35

69

Avoid computational delay due to on-line optimization?

Real-time Iteration• preparation, feedback response and transition stages

• solve perturbed (linearized) problem on-line

– Li, de Oliveira, Santos, B. (1990+) – Diehl, Findeisen, Bock, Allgöwer et al. (2000+)– > two orders of magnitude reduction in on-line computation

• solve complete NLP in background (‘between’ sampling times as part of preparation and transition stages

Based on NLP sensitivity for dynamic systems• Extended to Simultaneous Collocation approach – Zavala et al.

(2007)

• Develop Advanced Step NMPC

• Related to MPC with linearization constantly updated one step behind

70

Nonlinear Model Predictive Control Nonlinear Model Predictive Control ––Parametric Problem (Zavala, Laird, B.)Parametric Problem (Zavala, Laird, B.)

36

71

Nonlinear Model Predictive Control Nonlinear Model Predictive Control ––Parametric Problem (Zavala, Laird, B.)Parametric Problem (Zavala, Laird, B.)

72

NLP SensitivityNLP SensitivityParametric Programming

NLP Sensitivity Æ Rely upon Existence and Differentiability of Path

Æ Main Idea: Obtain and find b y Taylor Series Expansion

Optimality Conditions

Solution Triplet

37

73

NLP SensitivityNLP Sensitivity

Optimality Conditions of

Obtaining

Æ Already Factored at Solution

Æ Sensitivity Calculation from Single Backsolve

Æ Approximate Solution Retains Active Set

KKT Matrix IPOPT

Apply Implicit Function Theorem to around

74

Key Concept Key Concept –– Relate to Previous HorizonRelate to Previous Horizon

Solutions to both problems are equivalent in nominal case

(ideal plant model, no disturbances)

38

75

Advanced Step NMPCAdvanced Step NMPCCombine advanced step with sensitivity to solve NLP in background

(between steps) – not on-line

Solve P(z ) in background (between t0 and t1)

υνλ ∆=

∆∆∆

−K

XV

A

IAW

kk

Tk

kk [

0

00

76




Sensitivity to updated problem to get (z0, u0)

υνλ ∆=

∆∆∆

−K

XV

A

IAW

kk

Tk

kk [

0

00

39

77




Sensitivity to updated problem to get (z0, u0)Solve P(z +1) in background with new (z0, u0)

υνλ ∆=

∆∆∆

−K

XV

A

IAW

kk

Tk

kk [

0

00

78

AS-NMPC Stability Analysis

Nominal NMPC stability proof

•Nominal case – no noise: perfect model•General formulation with local asymptotic controller for t Æ �•Advanced step controller satisfies same relations, has same input sequence

Æ shares identical stability property

klll

kkJkkk

kkkkkllk

xz)),u,z(fz

||)w(|| ||x||L ||))w,u,x(g||

))w,u,x(g))u,x(f))u,x(fx

==

+≤+==

+

+

01

1

σ

Robust Stability Margins

• Analysis similar to Limon, Alamo, Camacho (2004), Magni and Scattolini (2005)• Advanced step NMPC is ISS and tolerates some model mismatch• ISS property (Jiang and Wang, 2001; Magni and Scattolini, 2005) • Advanced step NMPC has smaller margin than Ideal NMPC,

Æ but can be implemented without computational delay

Plant

Model

40

79

CSTR NMPC Example (Hicks and Ray)

• Maintain unstable setpoint• Close to bound constraint• Final time constraint for stability

Effects of:• Computational Delay• Measurement Noise• Model Mismatch• Advanced Step NMPC

80

CSTR NMPC Example – Nominal Case

• NMPC applied with N = 10, τ = 0.5 sampling time• Stable (z = 0) and unstable (z = 0.1) steady states• u2* close to upper bound• Computational delay = 0.5, leads to instabilities

41

81

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

0 5 10 15 20 25 30 35 40 45 50

0

0.05

0.1 θ = θnom

− 55%

z c [−]

Time [s]

0 5 10 15 20 25 30 35 40 45 50

0

0.05

0.1 θ = θnom

− 50%

z c [−]

0 5 10 15 20 25 30 35 40 45 50

0

0.05

0.1 θ = θnom

− 40%z c [−

]

0 5 10 15 20 25 30 35 40 45 50

0

0.05

0.1 θ = θnom

− 25%

z c [−]

IdealAdvanced StepDirect

82

CSTR Example: Mismatch + Noise

0 10 20 30 40 50 60 70

0

0.05

0.1σ = 2.5%

θ = θnom

− 50%

z c [−]

IdealAdvanced StepDirect

0 10 20 30 40 50 60 70

0

0.05

0.1σ = 5.0%

z c [−]

0 10 20 30 40 50 60 70

0

0.05

0.1

0.15

σ = 7.5%

z c [−]

Time [s]

42

83

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

84

NMPC Case Study� Optimal Feedback Policy Æ (On-line Computation 351 CPU s)

Ideal NMPC controller - computational delay not considered

Time delays as disturbances in NMPC

43

85

NMPC Case Study� Optimal Policy vs. NLP Sensitivity -Shifted Æ (On-line Computation 1.04 CPU s)

Very Fast Close-to-Optimal FeedbackLarge-Scale Rigorous Models

86

Moving Horizon Estimation

Large State Dimensionality

Degrees of Freedom

Linear Systems, No inequalities Æ Kalman Filter for State Estimation

Nonlinear Systems: Æ Extended Kalman Filters in Practice

Moving Horizon Estimation:+ directly captures nonlinear dynamics, statistical behavior

- need to solve NLP on-line

Solution Time Æ Order of Minutes

Computational delay – between receipt of process measurement, estimation

and injection of new control

Leads to loss of performance and stability

Process

NMPC Controller



ysp : set points

( )( )dpuyzG

dpuyzFz

,,,,0

,,,,

==′


dpuyzFz

,,,,0

,,,,

==′

44

87


Large State Dimensionality

Degrees of Freedom

Linear Systems, No inequalities Æ Kalman Filter for State Estimation

Nonlinear Systems: Æ Extended Kalman Filters in Practice

Moving Horizon Estimation:+ directly captures nonlinear dynamics, statistical behavior

- need to solve NLP on-line

Solution Time Æ Order of Minutes

Computational delay – between receipt of process measurement, estimation

and injection of new control

Leads to loss of performance and stability

88


Computational Delay - MHE ImpracticalNLPs are Parametric

Solve in BackgroundFast Approximation to On-line

45

89

Fast Moving Horizon EstimationFast Moving Horizon Estimation

1) Solve Extended Problem Between and

Re-use KKT Matrix Available At Solution of Analyze Terms due to Extended Horizon

Dummy

Measurement

Dummy Measurement = Model Prediction at

90

Fast Moving Horizon EstimationFast Moving Horizon Estimation

2) At once we know

KKT System At Solution of

Augmented KKT System

Relax Multiplier

Force Dummy to Measurement

Plant-Model Mismatch

Find perturbation 'p that enforces

K is already factorizedSolve as Schur complement problem On-line Cost is a simple backsolve

46

91

MHE Case Study MHE Case Study (Zavala, Laird, B.)(Zavala, Laird, B.)

Single Measurement - Composition of Recycle Gas

Sampling Time ~ 6 minEstimation Horizon N = 15

27,121 Constraints, 9330 Bounds294 Degrees of Freedom

NLP Simultaneous Approach

Κ has correct Inertia at Solution – System Locally Observable

On-Line Calculation

Computational Delay

Controls

Meas

92

MHE Case StudyMHE Case StudySimulated Control Profiles

Measurement Profiles Gaussian Noise 5% SD

On-line NLP requires up to 4 CPU minutesLeads to Feedback Delay in Controller

47

93

MHE Case StudyMHE Case StudySimulated Control Profiles

Measurement Profiles Gaussian Noise 5% SD

On-line Update 1 SecondSensitivity Errors Negligible

94

Summary: Dynamic Optimization

Sequential Approaches – Use DAE Integrators- Parameter Optimization

• Gradients by: Direct (and Adjoint) Sensitivity Equations- Optimal Control (Profile Optimization)

• Variational Methods• NLP-Based Methods - Single and Multiple Shooting

- Require Repeated Solution of Model- State Constraints are Difficult to Handle

Simultaneous Collocation Approach- Discretize ODE's using orthogonal collocation on finite elements - Straightforward addition of state constraints.- Deals with unstable systems- Solve model only once- Avoid difficulties at intermediate points

Large-Scale Extensions- Exploit structure of DAE discretization through decomposition- Large problems solved efficiently with IPOPT

48

95

Summary: On-line ExtensionsRTO and MPC widely used for refineries, ethylene and, more recently, chemical plants

• Inconsistency in models Æ 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• Avoids conservative performance• Update model with MHE• Evolve from regulatory NMPC to Large-scale DRTO

96

Acknowledgements

Funding• Department of Energy • National Science Foundation• Center for Advanced Process Decision-making (CMU)

Research Colleagues• Antonio Flores-Tlacuahuac• Tobias Jockenhövel• Shivakumar Kameswaran• Yoshiaki Kawajiri• Carl Laird• Yi-dong Lang• Andreas Wächter• Victor Zavala

http://dynopt.cheme.cmu.edu

49

97

References – Dynamic Optimization

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, 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.

98

References – Recent 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)

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