Optimization of Differential- Algebraic Equation...

Optimization of Differential-Algebraic Equation Systems

L. T. BieglerChemical Engineering Department

Carnegie Mellon UniversityPittsburgh, PA

2

I IntroductionProcess Examples

II ParametricOptimization- Gradient Methods

• Perturbation• Direct - Sensitivity Equations• Adjoint Equations

III Optimal ControlProblems- Optimality Conditions- Model Algorithms

• Sequential Methods• Multiple Shooting• Indirect Methods

IV SimultaneousSolutionStrategies- Formulation and Properties- Process Case Studies- Software Demonstration

DAE Optimization Outline

3

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.

4

DAE Models in Process Engineering

Differential Equations•Conservation Laws (Mass, Energy, Momentum)

Algebraic Equations•Constitutive Equations, Equilibrium (physical properties, hydraulics, rate laws)•Semi-explicit form•Assume 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)

Characteristics•Large-scale models – not easily scaled•Sparse but no regular structure•Direct linear solvers widely used•Coarse-grained decomposition of linear algebra

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

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"

$ %

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

sConstraintOther

sConstraint Bound

)()),(),(),((0

)),(),(),(()(..

||))||||)(||min

init

1sp

ztztttytzG

tttytzFtzts

yty uy Q

kk

Q

==

=′

−+−∑ ∑ −

u

u

u(tu(tu

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

,,,,

==′

8

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 1.25Tim e (h r.)

Dyn a mic

C o nsta nt

Dyn a mic

C o nsta nt

optimal reactor temperature policy optimal column reflux ratio

Batch Process Optimization

9

FexitC H2 4

Texit ≤ 1180K

C2H CH

6 32→ • CH CH CH CH

3 2 6 4 2 5•+ → + •

C2H CH H

5 2 4•→ + •H CH H CH•+ → + •

2 6 2 2 52C

2H CH

5 4 10•→C

2H CH CH CH

5 2 4 3 6 3•+ → + •

H CH CH•+ → •2 4 2 5

0

1

2

3

4

5

6

0 2 4 6 8 10

Length m

Flo

w r

ate

mol

/s

0

500

1000

1500

2000

2500

Hea

t flu

x kJ

/m2s

C2H4 C2H6 log(H2)+12 q

Reactor Design ExamplePlug Flow Reactor Optimization

The cracking furnace is an important example in the olefin production industry, where various hydrocarbon feedstocks react. Consider a simplified model for ethane cracking (Chen et al., 1996). The objective is to find an optimal profile for the heat flux along the reactor in order to maximize the production of ethylene.

Max s.t. DAE

The reaction system includes six molecules, three free radicals, and seven reactions. The model also includes the heat balance and the pressure drop equation. This gives a total of eleven differential equations.

Concentration and Heat Addition Profile

10

Dynamic Optimization Approaches

DAE Optimization Problem

Sequential Approach

Vassiliadis(1994)Discretize controls

Variational Approach

Pontryagin(1962)

Inefficient for constrained problems

Apply a NLP solver

Efficient for constrained problems

11

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)

12

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)

13

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

14

Example: Sensitivity Equations

0)0(,0

22

1)0(,0

22

2,1,/)()(,/)()(

)0(,5

2,1,

1,1,22,112,

2,21,11,

2,1,

1,1,22,12,

2,21,11,

,,

21

1212

22

211

==

+++=′

+=′

==

++=′

+=′

=∂∂=∂∂===

+=′+=′

bb

bbbbb

bbb

aa

baaaa

aaa

bjjbajja

a

b

ss

psszszzs

szszs

ss

psszszs

szszs

jptztsptzts

pzz

pzzzz

zzz

15

Adjoint Sensitivity

Adjoint or Dual approach to sensitivity

Adjoin model to objective function orconstraint

(ψ = Φ,g or h)

(λ(t)) serve as multipliers on ODE’s)

Now, integrate by parts

and find dψ/dp subject to feasibility of ODE’s

Now, set all terms notin dp to zero.

∫ −′−=ft

Tf dttpzfzt

0

)),,(()( λψψ

∫ +′+−+=ft

TTf

Tf

Tf dttpzFztztpzt

0

0 )),,()()()()0()( λλλλψψ

00

∫

∂∂+

∂∂+′+

∂

∂+

−

∂∂

=ft TTT

fff

f dtdpp

ftz

z

fdp

p

pztzt

tz

tzd

0

0 )()0()(

)()()(

))((λδλλλδλ

ψψ

16

Adjoint System

Integrate model equations forward

Integrate adjoint equations backward and evaluate integral and sensitivities.

Notes:

nz (ng + nh + 1) adjoint equations must be solved backward (one for each objective and constraint function)

for implicit ODE solvers, profiles (and even matrices) can be stored and carried backward after solving forward for z as in DASPK/Adjoint (Li and Petzold)

more efficient on problems where: np > 1 + ng + nh

∫

∂∂+

∂∂=

∂∂

=∂∂−=′

ft

f

ff

dttp

f

p

pz

dp

d

tz

tztt

z

f

0

0 )()0()(

)(

))(()( ),(

λλψ

ψλλλ

17

Example: Adjoint Equations

dttztdp

td

dp

td

tz

ttzz

tz

ttpzz

dttp

f

p

pz

dp

d

tz

tztt

z

f

pzzzzztpzf

pzz

pzzzz

zzz

f

f

t

b

f

a

f

f

ff

f

ffb

t

f

ff

bT

a

b

)()()(

)0()(

)(

)()( ,2

)(

)()( ),(2

:becomesthen

)()0()(

)(

))(()( ),(

)()(),,( Form

)0(,5

1

0

2

1

2212212

1122111

0

0

121222

211

21

1211

22

211

∫

∫

=

=

∂∂

=−−=′

∂∂

=+−−=′

∂∂+

∂∂=

∂∂

=∂∂−=′

+++=

==

+=′+=′

λψ

λψ

ψλλλλ

ψλλλλ

λλψ

ψλλλ

λλλ

18

A + 3B --> C + 3D

L

T s

T R

T P

3:1 B/A

383 K

TP = specified product temperatureTR = reactor inlet, reference temperatureL = reactor lengthTs = steam sink temperatureq(t) = reactor conversion profileT(t) = normalized reactor temperature profile

Cases considered:• Hot Spot - no state variable constraints• Hot Spot with T(t) � 1.45

Example: Hot Spot Reactor

Roo

P

Pproducto

Rfeed

RS

L

RSTLTT

C/T C, T(L) T

, T(L)) (THC) -,(T+

Tdt

dqTTtT

dt

dT

qtTtqdt

dqts

dtTTtTLMinSRP

101120

0110

1)0( ,3/2)/)((5.1

0)0( )],(/2020exp[))(1(3.0 ..

)/)(( 0

,,,

+==

=∆

=+−−=

=−−=

−−=Φ ∫

19

1.51.00.50.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Normalized Length

Con

vers

ion,

q

1.51.00.50.0

1.0

1.1

1.2

1.3

1.4

1.5

1.6

Normalized Length

Nor

mal

ized

Tem

pera

ture

Method: SQP (perturbation derivatives)

L(norm) TR(K) TS(K) TP(K)Initial: 1.0 462.23 425.26 250Optimal: 1.25 500 470.1 188.413 SQP iterations / 2.67 CPU min. (µVax II)

Constrained Temperature Case: could not be solved with sequential method

Hot Spot Reactor: Unconstrained Case

20

Variable Final Time(Miele, 1980)

Define t = pn+1 τ, 0 ≤ τ ≤ 1, pn+1 = tf

Let dz/dt = (1/ pn+1) dz/dτ = f(z, p) ⇒ dz/dτ = (pn+1) f(z, p)

Converting Path Constraints to Final Time

Define measure of infeasibility as a new variable, znz+1(t) (Sargent & Sullivan, 1977):

Tricks to generalize classes of problems

)degenerate is constraint (however, )( Enforce

0)0( , ))(),((,0max()(

))(),((,0max()(

1

12

1

0

21

ε≤

==

=

+

++

+

∑

∑∫

fnz

nzj

jnz

j

t

jfnz

tz

ztutzgtzor

dttutzgtzf

�

21

Profile Optimization - (Optimal Control)

Optimal Feed Strategy (Schedule) in Batch Reactor

Optimal Startup and Shutdown Policy

Optimal Control of Transients and Upsets

Sequential Approach: Approximate control profile as through parameters (piecewise constant, linear, polynomial, etc.)

Apply NLP to discretization as with parametric optimization

Obtain gradients through adjoints (Hasdorff; Sargent and Sullivan; Goh and Teo) or sensitivity equations (Vassiliadis, Pantelides and Sargent; Gill, Petzold et al.)

Variational (Indirect) approach: Apply optimality conditions and solve as boundary value problem

22

Optimality Conditions(Bound constraints on u(t))

Min φ(z(tf))s.t. dz/dt = f(z, u), z (0) = z0

g (z(tf)) � 0h (z(tf)) = 0a � u(t) � b

Form Lagrange function - adjoin objective function and constraints:

Derivation of Variational Conditions Indirect Approach

dtbtutuauzfz

tztzvtzhtzgt

dtbtutuauzfz

vtzhtzgt

Tb

t Ta

TT

ffTTT

fT

ff

Tb

t Ta

T

Tf

Tff

f

f

))(())((),(

)()()0()0())(())(()(

:partsby Integrate

))(())(()),((

))(())(()(

0

0

−+−+++

−+++=

−+−+−−

++=

∫

∫

ααλλ

λλµφφ

ααλ

µφφ

�

�

23

λ ft( )=∂φ∂z

+∂g

∂zµ +

∂h

∂zγ

ft=t

∂f

∂uλ =

∂H

∂u= 0

∂ H

∂u= α a − α b

α aT (a − u(t))

α bT (u(t) − b)

ua ≤ u(t) ≤ ub

α a ≥ 0,α b ≥ 0

∂H

∂u= −α b ≤ 0

∂H

∂u= α a ≥ 0

At optimum, δφ ≥0. Since u is the control variable, let all other terms vanish.⇒ δ z(tf):

δz(0): λ(0) = 0 (if z(0) is not specified)δz(t):

Define Hamiltonian, H = λTf(z,u)For u notat bound:

For u atbounds:

Upper bound, u(t) = b, Lower bound, u(t) = a,

Derivation of Variational Conditions

λλz

f

z

H

∂∂−=

∂∂−=�

0 )(),(

)(),(

)0()0()(

0≥

−+

∂∂+

∂∂++

+

−

∂∂+

∂∂+

∂∂=

∫ dttuu

uzftz

z

uzf

ztzvz

h

z

g

z

ftT

ab

T

Tf

T

δααλδλλ

δλδλµφδφ

�

24

Car ProblemTravel a fixed distance (rest-to-rest) in minimum time.

0)(’,0)0(’

)(,0)0(

)(

" ..

==

==≤≤

=

f

f

f

txx

Ltxx

btua

uxts

tMin

0)(,0)0(

)(,0)0(

)(

1’

’

’ ..

22

11

3

2

21

3

==

==≤≤

===

f

f

f

txx

Ltxx

btua

x

ux

xxts

)(tMin x

s

f

f

f

f

f

tt

auctt

buctctttcc

u

H

tt

ttcct

ct

uxH

==

=>==>+=

−+==∂∂

====>=

−+===>−=

===>=

++=

at occurs )0(Crossover

,0,

,0,0)(

1)( ,1)(0

)()(

)(0 :Adjoints

:n Hamiltonia

2

2

21122

333

12212

111

3221

λ

λ

λλλ

λλλλλ

λλλ

�

�

�

25

t f

u(t)

b

a

t s

1 / 2 bt2,t < ts

1 / 2 bts2 - a ts - tf( )2( ), t ≥ ts

bt, t < ts

bts + a t - ts( ), t ≥ ts

2L

b 1- b / a( )

1/2

(1− b / a)2L

b 1 - b / a( )

1/2

Optimal Profile

From state equations:

x1(t) =

x2 (t) =

Apply boundary conditions at t = tf:x1(tf) = 1/2 (b ts

2 - a (ts - tf)2) = L

x2(tf) = bts + a (tf - ts) = 0⇒ ts =

tf =

•Problem is linear in u(t). Frequently these problems have "bang-bang" character.•For nonlinear and larger problems, the variational conditions can be solved numerically as boundary value problems.

Car ProblemAnalytic Variational Solution

26

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

Adjoint Equations: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

Cases Considered1. NLP Approach- piecewise constant and linear profiles.2. Control Vector Iteration

27

Batch Reactor Optimal Temperature Program Piecewise Constant

Op

tim

al P

rofile

, u

(t)

0. 0.2 0.4 0.6 0.8 1.0

2

4

6

Time, h

ResultsPiecewise Constant Approximation with Variable Time ElementsOptimum B/A: 0.57105

28

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

29

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

30

Dynamic Optimization - Sequential Strategies

Small NLP problem, O(np+nu) (large-scale NLP solver not required) • Use NPSOL, NLPQL, 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

Sequential optimization is not recommended for unstable systems. State variables blow up at intermediate iterations for control variables and parameters.

Discretize control profiles to parameters (at what level?)

Path constraints are difficult to handle exactly for NLP approach

31

Instabilities in DAE ModelsThis example cannot be solved with sequential methods (Bock, 1983):

dy1/dt = y2

dy2/dt = τ2 y1 + (π2 − τ2) sin (π t)

The characteristic solution to these equations is given by:

y1(t) = sin (π t) + c1 exp(-τ t) + c2 exp(τ t)

y2 (t) = πcos (π t) - c1 τ exp(-τ t) + c2 τ exp(τ t)

Both c1 and c2 can be set to zero by either of the following equivalent conditions:

IVP y1(0) = 0, y2 (0) = π

BVP y1(0) = 0, y1(1) = 0

32

IVP SolutionIf we now add roundoff errors e1 and e2 to the IVP and BVP conditions, we see significant differences in the sensitivities of the solutions.

For the IVP case, the sensitivity to the analytic solution profile is seen by large changes in the profiles y1(t) and y2(t) given by:

y1(t) = sin (π t) + (e1 - e2/τ) exp(-τ t)/2

+(e1 + e2/τ) exp(τ t)/2

y2 (t) = πcos (π t) - (τ e1 - e2) exp(-τ t)/2

+ (τ e1 + e2) exp(τ t)/2

Therefore, even if e1 and e2 are at the level of machine precision (< 10-13), a large value of τ and t will lead to unbounded solution profiles.

33

BVP Solution

On the other hand, for the boundary value problem, the errors affect the analytic solution profiles in the following way:

y1(t) = sin (π t) + [e1 exp(τ)- e2] exp(-τ t)/[exp(τ) - exp(-τ)]

+ [e1 exp(-τ) - e2] exp(τ t)/[exp(τ) - exp(-τ)]

y2(t) = πcos (π t) – τ [e1 exp(τ)- e2] exp(-τ t)/[exp(τ) - exp(-τ)]

+ τ [e1 exp(-τ) - e2] exp(τ t)/[exp(τ) - exp(-τ)]

Errors in these profiles never exceed t (e1 + e2), and as a result a solution to the BVP is readily obtained.

34

BVP and IVP Profiles

e1, e2 = 10-9

Linear BVP solves easily

IVP blows up before midpoint

35

Dynamic Optimization Dynamic Optimization ApproachesApproaches

DAE Optimization Problem

Multiple Shooting

Sequential Approach

Vassiliadis(1994)

Can not handle instabilities properlySmall NLP

Handles instabilities Larger NLP

Discretize some state variables

Discretize controls

Variational Approach

Pontryagin(1962)

Inefficient for constrained problems

Apply a NLP solver

Efficient for constrained problems

36

Multiple Shooting for Dynamic OptimizationDivide 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

37

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

38

BVP Problem Decomposition

Consider: Jacobian of Constraint Matrix for NLP

• bound unstable modes with boundary conditions (dichotomy)

• can be done implicitly by determining stable pivot sequences in multiple shooting constraints approach

• well-conditioned problem implies dichotomy in BVP problem (deHoog and Mattheij)

Bock Problem (with t = 50)

• Sequential approach blows up (starting within 10-9 of optimum)

• Multiple Shooting optimization requires 4 SQP iterations

B1 A1

A2

A3

A4

AN

B2

B3

B4

BN

IC

FC

39

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

40

Dynamic Optimization Dynamic Optimization ApproachesApproaches

DAE Optimization Problem

Simultaneous Approach

Sequential Approach

Vassiliadis(1994)

Can not handle instabilities properlySmall NLP

Handles instabilities Large NLP

Discretize all state variables

Discretize controls

Variational Approach

Pontryagin(1962)

Inefficient for constrained problems

Apply a NLP solver

Efficient for constrained problems

41

Nonlinear DynamicOptimization Problem

Collocation onfinite Elements

Continuous variablesContinuous variables

Nonlinear ProgrammingProblem (NLP)

Discretized variablesDiscretized variables

Nonlinear Programming Formulation

42

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.

Kkuzftztr kk

K

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

0

==−= ∑=

"�

43

Collocation Example

kkNjk

jK

kjj

k

K

kkkK ztz

tt

ttttztz ===>

−−

∏== +

≠==

+ ∑ )()(

)()(,)()( 1

001 ""

2

210

22

221

22

2222211200

12

121

12

1122111100

0

2

22

11

02

0

210

76303053371

706073840319029100

2334786

23

2346412098572

23

0

0023

4641203924 ,464102196252

391644836 ,4483621958

613 ,61

7886802113200

t. t - . z(t)

). (. ), z. (. , z z

) z - z z z.(-

z - z) (t z) (t z) (tz

) z - z z. z.(

z - z) (t z) (t z) (tz

z

) , z( z - zSolve z’

.t - . (t) t. - t. (t)

t. - . (t) t. t. -(t)

/ - t/(t) )/ - t (t(t)

. , t. , t t

=

===+=+

+=++

+=+

+=++

===>=+=

==

=+=

=+=

===

"�"�"�

"�"�"�

"�"

"�"

"�"

44

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

45

Results of Optimal Temperature Program Batch Reactor (Revisited)

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

46

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

uelement 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

47

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

48

A + 3B --> C + 3D

L

T s

T R

T P

3:1 B/A

383 K

TP = specified product temperatureTR = reactor inlet, reference temperatureL = reactor lengthTs = steam sink temperatureq(t) = reactor conversion profileT(t) = normalized reactor temperature profile

Cases considered:• Hot Spot - no state variable constraints• Hot Spot with T(t) � 1.45

Hot Spot Reactor Revisited

Roo

P

Pproducto

Rfeed

RS

L

RSTLTT

C/T C, T(L) T

, T(L)) (THC) -,(T+

Tdt

dqTTtT

dt

dT

qtTtqdt

dqts

dtTTtTLMinSRP

101120

0110

1)0( ,3/2)/)((5.1

0)0( )],(/2020exp[))(1(3.0 ..

)/)(( 0

,,,

+==

=∆

=+−−=

=−−=

−−=Φ ∫

49

1.21.00.80.60.40.20.00

1

2

integrated profile

collocation

Normalized Length

Con

vers

ion

1.21.00.80.60.40.20.01.0

1.2

1.4

1.6

1.8

integrated profile

collocation

Normalized Length

Tem

pera

ture

Base Case SimulationMethod: OCFE at initial point with 6 equally spaced elements

L(norm) TR(K) TS(K) TP(K)Base Case: 1.0 462.23 425.26 250

50

1.51.00.50.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Normalized Length

Con

vers

ion,

q

1.51.00.50.01.0

1.1

1.2

1.3

1.4

1.5

1.6

Normalized Length

Nor

mal

ized

Tem

pera

ture

Unconstrained CaseMethod: OCFE combined formulation with rSQP

identical to integrated profiles at optimum L(norm) TR(K) TS(K) TP(K)

Initial: 1.0 462.23 425.26 250Optimal: 1.25 500 470.1 188.4

123 CPU s. (µVax II)φ∗ = -171.5

51

1.51.00.50.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Normalized Length

Con

vers

ion

1.51.00.50.01.0

1.1

1.2

1.3

1.4

1.5

Normalized Length

Tem

pera

ture

Temperature Constrained CaseT(t) � 1.45

Method: OCFE combined formulation with rSQP, identical to integrated profiles at optimum

L(norm) TR(K) TS(K) TP(K)Initial: 1.0 462.23 425.26 250Optimal: 1.25 500 450.5 232.1

57 CPU s. (µVax II), φ∗ = -148.5

52

Theoretical Properties of Simultaneous Method

A. Stability and Accuracy of Orthogonal Collocation

• Equivalent to performing a fully implicit Runge-Kutta integration of the 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

• An equivalence has been established between the Kuhn-Tucker conditions of NLP and the variational necessary conditions

• Rates of convergence have been established for the NLP method

53

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• Grade Transition for LDPE Process• Ramping for Continuous Columns• Reflux Profiles for Batch Distillation and Column Design• Source Detection for Municipal Water Networks• Air Traffic Conflict Resolution• Satellite Trajectories in Astronautics• Batch Process Integration• Optimization of Simulated Moving Beds

Simultaneous DAE Optimization

54

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

Catalyst

Monomer, Transfer/Term. agents

Coolant

Polymer

55

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

56

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

57

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.

58

Batch Distillation – Optimization Case Study - 1

D(t), x d

V(t)

R(t)

xb

[ ]

[ ]

1+R

V-=

dt

dS

x-xS

V=

dt

dx

x-yH

V=

dt

dx

i,0i,i,0

i,i,cond

di,

d

dN

*DXJH�HIIHFW�RI�FROXPQ�KROGXSV�2YHUDOO�SURILW�PD[LPL]DWLRQ0DNH�7UD\�&RXQW�&RQWLQXRXV

59

2SWLPL]DWLRQ�&DVH�6WXG\�� �0RGHOLQJ�$VVXPSWLRQV,GHDO�7KHUPRG\QDPLFV1R�/LTXLG�7UD\�+ROGXS1R�9DSRU�+ROGXS)RXU�FRPSRQHQW�PL[WXUH��α �����������������6KRUWFXW�VWHDG\�VWDWH�WUD\�PRGHO��)HQVNH�8QGHUZRRG�*LOOLODQG�

&DQ�EH�VXEVWLWXWHG�E\�PRUH�GHWDLOHG�VWHDG\�VWDWH�PRGHOV�)UHGHQVOXQG DQG�*DOLQGH]���������$OND\D�������

2SWLPL]DWLRQ�3UREOHPV�&RQVLGHUHG(IIHFW�RI�&ROXPQ�+ROGXS��+FRQG�7RWDO�3URILW�0D[LPL]DWLRQ

60

0.0 0.2 0.4 0.6 0.8 1.0

0

10

20

30

With Holdup

No Holdup

Time ( hours )

Re

flu

x R

atio

0.0 0.2 0.4 0.6 0.8 1.0

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

With Holdup

No Holdup

Time ( hours )

Dis

tilla

te C

om

po

sitio

n

0D[LPXP�'LVWLOODWH�3UREOHP�

&RPSDULVRQ�RI�RSWLPDO�UHIOX[�SURILOHV�ZLWK�DQG�ZLWKRXW�KROGXS��+FRQG��

&RPSDULVRQ�RI�GLVWLOODWH�SURILOHV�ZLWK�DQG�ZLWKRXW�KROGXS��+FRQG��DW�������RYHUDOO�SXULW\�

61

0 1 2 3 4

0

10

20

30

N = 20

N = 33.7

Time ( hours )

Re

flu

x R

atio

%DWFK�'LVWLOODWLRQ�3URILW�0D[LPL]DWLRQMax {Net Sales(D, S0)/(tf +Tset) – TAC(N, V)}

N = 20 trays, Tsetup= 1 hourxd = 0.98, xfeed= 0.50, α = 2Cprod/Cfeed= 4.1V = 120 moles/hr, S0 = 100 moles.

62

%DWFK�'LVWLOODWLRQ� 2SWLPL]DWLRQ�&DVH�6WXG\�� �

D(t), x d

V(t)

R(t)

xb

Ideal Equilibrium Equationsyi,p = Ki,p xi,p

Binary Column (55/45, Cyclohexane, Toluene)S0 = 200, V = 120, Hp = 1, N = 10, ~8000 variables, < 2 CPU hrs. (Vaxstation 3200)

dx1,N+1

dt=

VH N+1

y1,N - x1,N+1[ ]dx1,p

dt=

V

Hp

y1, p-1- y1,p +R

R+1x1,p+1- x1,p

[ ], p = 1,...,N

dx1,0

dt=

V

Sx1,0 - y1,0 + R

R+1x1,1 - x1,0

[ ]

dDdt

=V

R +1

xi,p1

C∑ = 1.0 yi,p

1

C∑ =1.0

S0x i,00 = S0 - Hp

p=1

N+1∑

x i,0 + Hp

p=1

N+1∑ x i,p

63

2SWLPL]DWLRQ�&DVH�6WXG\�� �

&DVHV�&RQVLGHUHG&RQVWDQW�&RPSRVLWLRQ�RYHU�7LPH6SHFLILHG�&RPSRVLWLRQ�DW�)LQDO�7LPH%HVW�&RQVWDQW�5HIOX[�3ROLF\3LHFHZLVH�&RQVWDQW�5HIOX[�3ROLF\

0RGHOLQJ�$VVXPSWLRQV,GHDO�7KHUPRG\QDPLFV&RQVWDQW�7UD\�+ROGXS1R�9DSRU�+ROGXS%LQDU\�0L[WXUH�����WROXHQH����F\FORKH[DQH���KRXU�RSHUDWLRQ7RWDO�5HIOX[�,QLWLDO�&RQGLWLRQ

64

1.00.80.60.40.20.00

1

2

3

4

5

6

0.000

0.001

0.002

0.003

0.004

0.005

0.006

Reflux

Purity

Time (hrs.)

Ref

lux

Pol

icy

Top

Pla

te P

urity

, (1-

x)

1.00.80.60.40.20.00

2

4

6

8

0.000

0.002

0.004

0.006

0.008

0.010

Reflux

Purity

Time (hrs.)

Ref

lux

Pol

icy

Dis

tilla

te P

urit

y, (

1-x)

5HIOX[�2SWLPL]DWLRQ�&DVHV

2YHUDOO�'LVWLOODWH�3XULW\∫[G�W��9��5����GW���'�WI���� �����

' �WI�� ������6KRUWFXW�&RPSDULVRQ

' �WI�� ������

&RQVWDQW�3XULW\�RYHU�7LPH[��W��� �����' �WI�� ������

65

5HIOX[�2SWLPL]DWLRQ�&DVHV

3LHFHZLVH�&RQVWDQW�5HIOX[�RYHU�7LPH∫[G�W��9��5����GW���'�WI���� �����

' �WI�� ������

&RQVWDQW�5HIOX[�RYHU�7LPH[G�W��9��5����GW���'�WI���� �����' �WI�� �����

1.00.80.60.40.20.00.20

0.22

0.24

0.26

0.28

0.30

0.000

0.001

0.002

0.003

0.004

Reflux

Purity

Time (hrs.)

Ref

lux

Pol

icy

Dis

tilla

te P

urit

y, (

1-x)

1.00.80.60.40.20.00.000

0.002

0.004

0.006

0.008

0.010

0

1

2

3

4

5

6

Purity

Reflux

Time (hrs.)

Ref

lux

Pol

icy

Dis

tilla

te P

urit

y, (

1-x)

66

Batch Reactive Distillation – Case Study 3

max Ddtt f

0

1=∫max Ddtt f

0

1=∫

s.t. DAE

xDEster ≥ 0 4600.xD

Ester ≥ 0 4600.

Reversible reaction between acetic acid and ethanolReversible reaction between acetic acid and ethanol

CH3COOH + CH3CH2OH l CH3COOCH2CH3 + H2O

t = 0, x = 0.25for all components

Wajde & Reklaitis (1995)

D(t), x d

V(t)

R(t)

xb

67

2SWLPL]DWLRQ�&DVH�6WXG\�� �

&DVHV�&RQVLGHUHG6SHFLILHG�&RPSRVLWLRQ�DW�)LQDO�7LPH2SWLPXP�5HIOX[�3ROLF\9DULRXV�7UD\V�&RQVLGHUHG��������������KRXU�RSHUDWLRQ

0RGHOLQJ�$VVXPSWLRQV,GHDO�7KHUPRG\QDPLFV&RQVWDQW�7UD\�+ROGXS1R�9DSRU�+ROGXS7HUWLDU\�0L[WXUH��(W2+��+2$F��(7$F��+�2�&ROG�6WDUW�,QLWLDO�&RQGLWLRQ

68

Batch Reactive Distillation

Distillate Composition

0.25

0.35

0.45

0.0 0.2 0.4 0.6 0.8 1.0time (hr)

Eth

yl A

ceta

te

8 trays 15 trays 25 trays

Optimal Reflux Profiles

0

20

40

60

80

100

0.0 0.2 0.4 0.6 0.8 1.0

time (hr)

Ref

lux

Rat

io

8 trays 15 trays 25 trays

0.0 0.2 0.4 0.6 0.8 1.0

0.00

0.10

0.20

0.30

0.40

0.50

M. f

ract

ion

time(hr)

Condenser Composition (8 trays)

Ester Eth. Wat. A. A.

�������YDULDEOHV������'$(V���GHJUHHV�RI�IUHHGRP���ILQLWH�HOHPHQWV�����,3237�LWHUDWLRQV�����&38�PLQXWHV

69

CPU Decomposition Time

1

6

11

16

21

90 130 170 210 250

DAEs

CP

U(s

)/ite

r

DiscretizedVariables

Iterations CPU (s)

3048

1788

245.732

DAEsTrays

168

8 98

15

56.414

4678 1083.24525825

207.5

37.2

659.3

Global Elemental

Batch Reactive Distillation

70

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

sConstraintOther

sConstraint Bound

)()),(),(),((0

)),(),(),(()(..

||))||||)(||min

init

1sp

ztztttytzG

tttytzFtzts

yty uy Q

kk

Q

==

=′

−+−∑ ∑ −

u

u

u(tu(tu

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

,,,,

==′

71

Dynamic optimization in a MATLAB Framework

Dynamic Optimization Problem

Process Model

Inequality Constraints

Initial Conditions

Constraints at Final Time

Objective Function

( ) 0puyxxf =′ t,,,,,

( ) 0=t,,,, puyxg

0),,,,,( ≤′ tpuyxxh

0xx =)( 0t

( ) 0,,,)(,)(,)(,)( 0 =′ fffff ttttt pxuyxxϕ

( )ffft

tttf

,,),(t,)(,)( P 0f,,(t),

min0

xpuyxxpu

NLP Optimization Problem

Process Model

Inequality Constraints

Constraints at Final Time

Objective Function

( ) 0xpuyxf 0 =,,,ˆ,ˆ,ˆˆft

( ) 0puyxg =ft,,ˆ,ˆ,ˆˆ

0),,ˆ,ˆ,ˆ(ˆ ≤ftpuyxh

( ) 0,,,, =fNNN tttt

puyxϕ

( )fNNN tttt

,,,, Pmin puyx

Full Discretization of State and

Control Variables

Discretization Method

No. of Time Elements

Collocation Order

Saturator-System

copy ofDesign

WärmeschaltplanNr. -F Ref T - Erlangen, 13.Oct.1999

SIEMENS AGF Ref T In Bearbeitung

P..Druck..bar

M..Massenstrom..kg/s

PHI..Luft-Feuchte..%

H..Enthalpie..kJ/kg

T..Temperatur..°C

bar kJ/kgkg/s °C (X)

JOBKENNUNG : C:\Krawal-modular\IGCC\IGCC_Puertol lano_komplett.gek

Alle Drücke sind absolut

Dynamic Process

72

Tennessee Eastman Process

Unstable Reactor

11 Controls; Product, Purge streams

Model extended with energy balances

73

Tennessee Eastman Challenge Process

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

74

Setpoint change studies

Setpoint changes for the base case [Downs & Vogel]

+2%Make a step change so that the composition of component B in the gas purge changes from 13.82 to 15.82%

StepPurge gas composition of component B change

-60 kPaMake a step change so that the reactor operating pressure changes from 2805 to 2745 kPa

StepReactor operating pressure change

-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-1

StepProduction rate change

MagnitudeTypeProcess variable

75

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

76

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)

77

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

78

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

79

+ Directly handles interactions, multiple conditions+ Trade-offs unambiguous, quantitative

- Larger problems to solve - Need to consider a diverse process models

Research Questions

How should diverse models be integrated?Is further algorithmic development needed?

Optimization as a Framework for IntegrationOptimization as a Framework for Integration

80

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Batch Integration Case Study

Production PlanningStage 1

Stage 2

A

A B

B C

C

Plant Design

TR

Time Time

Dynamic Processing

81

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Best transient Best constant

Higher conversion in same time

T

Time

Con

v.

Time

Fewer product batches

TSame conversion in reduced time

Time

Con

v.

Time

Shorter processing times

Dynamic Processing

Production PlanningStage 1

Stage 2

A

A B

B C

C

Shorter Planning Horizon

82

Scheduling Formulation

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stage 2

stage I

stage 2

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Case Study ExampleCase Study Example

z i ,I0 zi ,II

0z i ,III

0 zi ,IV0

z i ,IVf

zi ,If z i ,II

f z i ,IIIf

Bi

A + B → C

C + B → P + E

P + C → G

4 stages, 3 products of different purity Dynamic reactor - temperature profileDynamic column - reflux profile

Process Optimization Cases

SQ - Sequential Design - Scheduling - DynamicsSM - Simultaneous Design and Scheduling

Dynamics with endpoints fixed . SM* - Simultaneous Design, Scheduling and Dynamics

84

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 1.25Tim e (h r.)

Dyn a mic

C o nsta nt

Dyn a mic

C o nsta nt

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Results for Simultaneous Cases

R0/R1 best constant / optimal temperature

C0/C1best constant / optimal reflux ratio

0

5 0 0 0

1 0 0 0 0

1 5 0 0 0

2 0 0 0 0

Pro

fit (

$)

R 0 C 0 R 1 C 0 R 0 C 1 R 1 C 1

C a s e s

S Im u lt a n e o u s w it h F R E E s t a t e s

S im u lt a n e o u s w it h F I X E D s t a t e sS e q u e n t ia l

I

I I

I I I

I V

I

I I

I I I

I V

I

I I

I I I

I V

- ZW schedule becomes tighter

- less dependent on product sequences

64

60

60

86

SummarySequential Approaches- Parameter Optimization

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

• Variational Methods• NLP-Based Methods

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

Simultaneous Approach- Discretize ODE's using orthogonal collocation on finite elements (solve larger optimization problem)- Accurate approximation of states, location of control discontinuities through element placement.- Straightforward addition of state constraints.- Deals with unstable systems

Simultaneous Strategies are Effective- Directly enforce constraints- 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

87

ReferencesBryson, A.E. and Y.C. Ho, Applied Optimal Control, Ginn/Blaisdell, (1968).Himmelblau, D.M., T.F. Edgar and L. Lasdon, Optimization of Chemical Processes, McGraw-Hill, (2001).Ray. W.H., Advanced Process Control, McGraw-Hill, (1981).

Software• Dynamic Optimization CodesACM – Aspen Custom ModelerDynoPC - simultaneous optimization code (CMU)COOPT - sequential optimization code (Petzold)gOPT - sequential code integrated into gProms (PSE)MUSCOD - multiple shooting optimization (Bock)NOVA - SQP and collocation code (DOT Products)• Sensitivity Codes for DAEsDASOLV - staggered direct method (PSE)DASPK 3.0 - various methods (Petzold)SDASAC - staggered direct method (sparse)DDASAC - staggered direct method (dense)

DAE Optimization Resources

88

Interface of DynoPC

Outputs/Graphic & TextsModel in FORTRAN

F2c/C++

ADOL_C

Preprocessor

DDASSL

COLDAE

Data/Config.

Simulator/Discretizer

F/DF

Starting Point Linearization

Optimal Solution

FILTER

Redued SQP

F/DFLine Search

Calc. of independent variable move

Interior PointFor QP

Decomposition

DynoPC – Windows Implementation

89

A B C1 2

da

atk

E

RTa= − − ⋅1

1exp( )

db

atk

E

RTa k

E

RTb= − ⋅ − − ⋅1

12

2exp( ) exp( )

a + b + c = 1

..

)(

ts

tbMax f

T

Example: Batch Reactor Temperature

90

Example: Car Problem

Min tfs.t. z1’ = z2

z2’ = uz2 � zmax

-2 � u � 1

subroutine model(nz,ny,nu,np,t,z,dmz,y,u,p,f)double precision t, z(nz),dmz(nz), y(ny),u(nu),p(np)

double precision f(nz+ny)

f(1) = p(1)*z(2) - dmz(1)f(2) = p(1)*u(1) - dmz(2)

returnend403020100

-3

-2

-1

0

1

2

0

2

4

6

8

10

12

Acceleration

Velocity

Time

Acc

eler

atio

n, u

(t)

Vel

ocit

y

91

Example: Crystallizer Temperature

Steam

Cooling water

TT TC

Control variable = Tjacket = f(t)?

Maximize crystal sizeat final time

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-03

3.5E-03

4.0E-03

4.5E-03

0 5 10 15 20

Time (hr)

Cry

stal

siz

e

0

10

20

30

40

50

60

T ja

cket

(C)

Crystal Size T jacket

SUBROUTINE model(nz,ny,nu,np,x,z,dmz,y,u,p,f)implicit double precision (a-h,o-z)double precision f(nz+ny),z(nz),dmz(nz),Y(ny),yp(4),u(1)double precision kgr, ln0, ls0, kc, ke, kex, lau, deltT, alphadimension a(0:3), b(0:3)data alpha/1.d-4/,a/-66.4309d0, 2.8604d0, -.022579d0, 6.7117d-5/,+ b/16.08852d0, -2.708263d0, .0670694d0, -3.5685d-4/, kgr/ 4.18d-3/,+ en / 1.1d0/, ln0/ 5.d-5/, Bn / 3.85d2/, em / 5.72/, ws0/ 2.d0/,+ Ls0/ 5.d-4 /, Kc / 35.d0 /, Kex/ 65.d0/, are/ 5.8d0 /,+ amt/ 60.d0 /, V0 / 1500.d0/, cw0/ 80.d0/,cw1/ 45.d0/,v1 /200.d0/,+ tm1/ 55.d0/,x6r/0.d0/, tem/ 0.15d0/,clau/ 1580.d0/,lau/1.35d0/,+ cp/ 0.4d0 /,cbata/ 1.2d0/, calfa/ .2d0 /, cwt/ 10.d0/

ke = kex*area x7i = cw0*lau/(100.d0-cw0)v = (1.d0 - cw0/100.d0)*v0 w = lau*v0 yp(1) = (deltT + dsqrt(deltT**2 + alpha**2))*0.5d0yp(2) = (a(0) + a(1)*yp(4) + a(2)*yp(4)**2 + a(3)*yp(4)**3)yp(3) = (b(0) + b(1)*yp(4) + b(2)*yp(4)**2 + b(3)*yp(4)**3)deltT = yp(2) - z(8) yp(4) = 100.d0*z(7)/(lau+z(7))

f(1) = Kgr*z(1)**0.5*yp(1)**en - dmz(1)f(2) = Bn*yp(1)**em*1.d-6 - dmz(2)f(3) = ((z(2)*dmz(1) + dmz(2) * Ln0)*1.d+6*1.d-4) -dmz(3)f(4) = (2.d0*cbata*z(3)*1.d+4*dmz(1)+dmz(2)*Ln0**2*1.d+6)-dmz(4)f(5) = (3.d0*calfa*z(4)*dmz(1)+dmz(2)*Ln0**3*1.d+6) - dmz(5)f(6) = (3.d0*Ws0/(Ls0**3)*z(1)**2*dmz(1)+clau*V*dmz(5))-dmz(6)f(7) = -dmz(6)/V - dmz(7)f(8) = (Kc*dmz(6) - Ke*(z(8) - u(1)))/(w*cp) - dmz(8)f(9) = y(1)+YP(3)- u(1) returnend