Constructive Nonlinear Dynamics in Process Systems Engineering … · 2019. 8. 15. · Constructive...

Constructive Nonlinear Dynamicsin

Process Systems Engineering

Wolfgang Marquardt and Martin Mönnigmann

Process Systems EngineeringRWTH Aachen University

ESCAPE-14, Lisbon, May 16-19, 2004

1Constructive Nonlinear Dynamics in Process Systems Engineering

Nonlinear Analysis of Chemical Process Systems

• Chemical reactorsBilous & Amundsen (1955), van Heerden (1958),Aris & Amundsen (1958), Razon & Schmitz (1987), ...Altimari et al. (2004)

• Distillation columnsPetlyuk & Avet'yan (1971), Michelsen & Villadsen (1979), Kienle & Marquardt (1991), Jacobsen & Skogestad (1991), Bekiaris et al. (1993), Kienle et al. (1994), ..., Li et al. (2004)

• Reactive distillation columnsPisarenko et al. (1987), Jacobs & Krishna (1993),Nijhuis et al. (1993), Ciric & Miao (1994), ...

• Reactor-Separator processes with recycleBildea & Dimian (1998), Kiss & Bildea (2002, 2003),Zeyer et al. (2003), Balasubramanian et al. (2003), ... , Bildea et al. (2004), Schmidt & Jacobsen (2004)


Example: Ammonia Reactor

Zeit [min]

Tem

pera

tur [

C]

1916:first industrial process(Bosch, BASF)

1997:global production>100 mio t/year

Nonlinear modelwith temperature coupling

Simulatedtemperature waves

Understandingnonlinear dynamics

>80 years later

Influence of preheateron root locus

Morud, Skogestad(1998)

Large scaleindustrial process

Temperature recordings ofindustrial ammonia reactor


Example: Decanting Reactor

D

Phase IPhase II C

A

DB E

EF

B

E

++

+

+

C

A, B : Educts

D : Main productC : Catalyst

E, F : By-product

Liquid/LiquidReactor

ComplexNonlinear Dynamics

H: HopfHV: HysteresisDL: Double LimitBL: Boundary LimitDH: Degenerate HopfHH: Hopf-HopfDZ: Double ZeroBH: Boundary Hopf HL: Hopf-Limit

Coolant average temperature Θm[-]

Coo

ling

capa

city

∆ [−

]

Phase I : A, B, C Phase II : B, D, E

B, C, D, E, FLC

Phase II

TC

(b) periodicoscillations

(as one example)

Luss et al.(1998)

(a) multiple steady states

Rea

ctor

tem

pera

ture

Θ [−

]

Coolant average temperature Θm[-]

„simple" process


Example: Heteroazeotropic Rectification

0 40 800.8

0.85

0.9

0.95

D [ml/h]120

xB,E ExperimentSimulation

A

B

Column with decanterethanole / water / cyclohexane

DF

xB,E

0

333

338

343

348

Tem

pera

tur [

K]

20 40 60 80 Zeit [h]

T8

T4

T1

A B A

353

• SimulationMagnussen et al. (1979)

• TheoryPetlyuk and Avet'yan (1971)Bekiaris et al. (1996)

• Experimental verificationMüller and Marquardt (1997)


Example: Reactor-Separator Recycle ProcessKiss and Bildea (2002, 2003)

• unique steady-state for isothermalstand-alone reactor

• feasibility constraint, Da > Dacr,for isothermal reactor-separatorrecycle process

stand alone CSTR

CSTR-seperator-recylce

Rea

ctio

nco

nver

sion

Damkoehler number

Dacr

Rea

ctio

nco

nver

sion

Damkoehler number

Monomerfeed

Recycle

TC

Sepa

ratio

n

CC

CC

Product

Coolant

CC

Initiatorfeed

PFR

Multiplicity for non-isothermal polymerization process wit PFR-separator recycle


Bifurcation Analysis

continuation

stability analysis

(steady-state) simulation

continuation and local stability analysis

singularity analysis and unfolding

||x||

pi

pj

• large-scale DAE systems• two-parameter continuation• stability analysis via test functions• ...but ...• only few parameters• not part of process modelling software• not constructive

analytical and numerical techniques ... dynamicsimulation


From Analysis to Synthesis

Ramirez & Gani (2004)

Process synthesis

Find (process structure,) designparameters and operating point such that

- profit is maximized

- quality and safety constraintsare fulfilled

How to considerNonlinear Dynamics?

),(0),(0..

),(max,

αα

αφα

xgxfts

xx

≤=

Nonlinear programming problem

Synthesis

Analysis


Conceptual Problem Formulation (1)

• model of open-loop or closed loop process system with given structure

- large-scale system of (index one) DAEs- time-varying inputs u(t), references r(t), or disturbances d(t)- process, equipment, model ... parameters, subject to uncertainty

• simplifying assumptions

- only differential equations (for this presentation)- u(t), r(t), d(t) vary much slower than plant dynamics

0)0(,),( xxxfx == η&

... a parametric dynamic process model


Conceptual Problem Formulation (2)

Steady-state process design by optimization

• parameter space with different types of regions• regions separated by critical manifolds• a design is a point in parameter space

• formulate and solve optimizationproblem with cost function, processmodel and inequality constraints(feasibility, stability etc.)

• optimal solution: η = η∗, x=x*

),(0..

),(min,

η

ηφη

xfts

xx

=

unstable

stable

η1

η2

feasible

infeasibleη1

η2 η1

η2P

(i) stability boundary

(ii) feasibility boundary

(i) + (ii)


Optimization Problem Formulations

Optimization with respect to cost function φ, model and ...

without stability or feasibility constraints

),(0..

),(min,

η

ηφη

xfts

xx

=

η1

η2

optimum

feasible but not stable

with stability or feasibility constraints

Pxfts

xx

∈=

ηη

ηφη

),(0..

),(min,

η1

η2P

optimum

stable and feasible, but not robust to parametric

uncertainty

PRxfts

xx

⊆= ),(0..

),(min,

η

ηφη

with robust stability and feasibility constraints

stable, feasible and robust to parametric

uncertainty

η1 = α2

η2 = α2

P

R optimum

profit loss


Leveraging the Profit Loss

• Quantification of loss

– to specific parametric uncertainty,

– to specific stability or feasibility constraint

• Reduction of the loss by structural modifications

– implementation or modification of feedback control system

– modification of the process structure

• Reduction of the loss by reduction of parametric uncertainty


Parameterization of Uncertainty ... a deterministic rather than a stochastic setting

Approximated robustness box • Ellipsoid overestimates

parametric uncertainty• Kreisselmeier-Steiner function

underestimatesparametric uncertainty

• Biegler, Rooney (2001)

∆α2

∆α1

α1

r

α2specialization and approximation

Robustness manifold• arbitrary connected smooth manifold• r normal to both, the critical and the

robustness manifold• high computational effort

rα2

α1


Parametric Distance to a Critical Manifold

x

α1 α2α1

α2

α2(0)

α1(0)

∆α1

∆α2

• Normal vector to nearest saddle-node and Hopf bifurcation points (Dobson, 1993)

• Normal vector to general critical manifolds, simplificationof defining equations (Mönnigmann & Marquardt, 2002)


Normal Vector Equation SystemsSaddle-node bifurcation

augmented system

vfrvvvf

f

T

T

x

α−=−=

==

010

00

Hopf bifurcation

augmented system

)2()2()1()1(

)2()2()1()1(

)1()2()2()1(

)2()2()1()1(

)1(2

)2(1

)1()2(

)2(2

)1(1

)2()1(

)2()1(

)1()2(

)2()1(

000

100000000

wfvwfvufrwfvwfvuf

wvwvwvwv

wwvvfwwvvf

wwww

wwfwwf

f

xT

xTT

xxT

xxTT

x

TT

TT

Tx

Tx

T

T

x

x

ααα

γγωγγω

ωω

++−=++=

−=−+=

+++=−+−=

==

−=+=

=

2n +2m+ 2n + 1Feasiblityconstraint

2n + m + 12n + 2Isola

4n + m + 43n + 3Cusp

6n + m + 43n + 2Hopf

2n + m + 12n + 1Saddle node

normal vectorsystem

augmentedsystem


Optimization under Uncertainty – the General Case

number and types of critical manifolds:• feasibility constraints (e.g. safety, quality, equipment ...)• stability boundaries (Hopf and saddle-node bifurcations)• performance constraints (eigenvalue sectors)• higher codimension bifurcations

(cusp, isola, non-transversal Hopf, ... bifurcations)

KkIi

l

mrl

rl

rxF

xf

x

i

ii

iieqi

iiiik

eqeq

eqeq

lx ieqeq

∈=≥

≥

+=

=

=

)(

)(

,,10

),,(0

),(0

,max

)(

)()(

)()()(

)()()()(

)()(

)()(

,, )()()(

K

ααα

α

αφα

cost function

normal vector r(i) to critical manifold i

normal distance between designand critical manifold i

steady state (x(eq), α(eq) )

minimal normal back-off


Complicated Real Situations

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test function may fail to detect crossing of critical manifold

critical manifolds have to be detected as the optimization proceeds

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different critical manifolds may result in more than one normal vector constraint








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(i) (ii)


Overview on Algorithm

(i) initialization

(ii) update set of locally closest points

initialization with steady state which is feasible and has desired nonlinear dynamic properties

(a) find locally closest points and corresponding normal vector of known critical manifolds(b) remove normal vector constraints for which distance exceeds specified value

run optimization with active normal vector constraintssearch for new critical points along the linear connection between starting and end point of optimization

return to step (ii) if new critical point is found

check for critical points with interval arithmetics in the robustness region

optimal and steady state is found – parametrically robust with respect to feasiblity and nonlinear dynamics constraints

add new critical point

to set of critical points

(iii) optimization

(iv) analysis

(v) rigorous search

robust optimum found

new critical point?yes

yes

no

new critical point?

no


Software Implementation – the Status

Augmented process model involves higher order derivatives of process model equations (normal vector constraints)

• process model is coded in MAPLE (Monagan et al. 2000)

• normal vector constraints are calculated by symbolic differentiation with MAPLE (Monagon et al. 2000) to augment process model

• first order derivates for numerical solution of NLP calculated by automatic differentiation with ADIFOR (Bischof et al. 1998)

• optimization by standard NLP solverso should use feasible path solver (e.g. FSQP, Lawrence &

Tits (2001)) to properly apply test functionso using NPSOL (Gill et al., 1986) and apply test functions along

the linear connection between starting and end point

• rigorous search with interval mathematics (Belitz et al., 2004) limited to small models, alternatively carefully selected test points


Applications: Three Problem Classes

• design under uncertaintyContinuous fermenter

(Mönnigmann & Marquardt, 2002)Continuous vinylacetate polymerization

(Mönnigmann & Marquardt, 2003)

• controller tuning and robustness analysisCSTR with unmodelled dynamics

(Hahn et al. 2003, Gerhard et al., 2004)

• integration of design and controlMSMPR crystallizer

(Grosch et al., 2003)Reaction section of HDA plant

(Mönnigmann & Marquardt, 2004)

Halemane & Grossmann (1982)Swaney & Grossmann (1986) Kokossis & Floudas (1994)Bahri et al. (1996) ...

Ackermann (1980)Cibrario & Levine (1991)Giona & Paladino (1994) ...

Brengel &Seider (1992)Lewin & Bogle (1996)Mohideen et al. (1996, 1997)Bahri et al. (1997) ...



design under uncertainty- open-loop process system- operating point & equipment parameters- stability & feasibility- process & model uncertainty

controller tuning and robustness analysis- closed-loop process system- control system design parameters - stability (and performance) in a large

region of operating conditions- process & model uncertainty

integration of design and control- closed-loop process system - operating point, equipment parameters

& control system parameters- stability & feasibility- process & model uncertainty


Design Under Uncertainty: Fermenter (1)

φ = cost of the substrate− profit from produced cells

uncertain parameters• Damköhler Number Da = µ(SF) V/F • substrate feed concentration SF

degrees of freedom• SF, Da

robustness w.r.t. stability boundaries

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0.3

0.4

0.5

0.6

0.7

2 3 4 5 6 7

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Hopf saddle–nodeSF [kmol m–3]

F [kg s–1]

0.8

two stability boundaries:Hopf and saddle-node bifurcations

Fermenter(Agrawal et al., 1982)

state variablesS, substrate concentrationX, biomass concentration

F, SF

S, X

V


Design under Uncertainty: Fermenter (2)

0

0.05

0.1

0.15

0.2

0.25

0.5 0.6 0.7 0.8 0.9 1

stableunstable

Hopfsaddle-node

Da

x1

0.2

0.25

0.3

0.35

0.4

0.45

0.5 0.6 0.7

saddle-nodeHopf

Da

SF

0

0.2

0.4

0.6

0.8

1

0.5 0.6 0.7 0.9 10.8 Da

φ

Optimization• without normal

constraints• with normal

vector constraints for robust stability

0.2

0.25

0.3

0.35

0.4

0.45

0.5 0.6 0.7 0.8Da

SF


Design Under Uncertainty: VA Polymerization (1)

0

0.02

0.04

0.06

0.08

0.1

0 50 100 150 200residence time [min]

20406080

100120140160180

temperature

0 50 100 150 200residence time[min]

stableinstableoptimal

saddle-node

Hopf

500

conc

entra

tion

of in

itiat

or[m

ol/l]

VA Polymerisation(Teymour & Ray, 1992)

state variablesM, monomer conc.I, initiator conc.P, polymer conc.T, reactor temperature

F, IF, MF

M, I, P, T

V

• small model• experimentally validated• critical (stability) manifoldsare known


Design Under Uncertainty: VA Polymerization (1)

VA Polymerisation(Teymour & Ray, 1992)

state variablesM, monomer conc.I, initiator conc.P, polymer conc.T, reactor temperature

F, IF, MF

M, I, P, T

V

• small model• experimentally validated• critical (stability) manifoldsare known

φ = profit from polymer− cost of initiator− cost of monomer− cost of solvent

uncertain parameters• residence time θ = V/F • initiator feed concentration IF

degrees of freedom• F, IF, MF, θ

robustness w.r.t• stability boundaries (Hopf and

saddle-node)• feasibility constraints

(avoid boiling, T≤100°C)


Robust Design: Stability and Feasibility (2)

residence time [min]

0

0.02

0.04

0.06

0.08

0.1

0 50 100 150 200

20406080

100120140160180

temperature

0 50 100 150 200residence time [min]


saddlenode

Hopf

conc

. of i

nitia

tor [

mol

/l]

saddle-node bifurcation only

20406080

100120140160180

temperature

0 50 100 150 200residence time[min]


saddlenode

Hopf

temp.constr.

residence time [min]0 50 100 150 200

0

0.02

0.04

0.06

0.08

0.1

conc

.on

of In

itiat

or [m

ol/l]

saddle-node and Hopf bifurcationsand temperature bound


Tuning and Robustness Analysis: CSTR (1)

CSTR, exothermic 1st order reaction A -> B, with unmodeledcooling jacket dynamics(Hahn et al. 2003)

• linearizing feedback (and PID) temperature control, parameter ε

• unmodeled dynamics in inner cascade control loop (TC2) parameterized by 2nd order dynamics

TC1

Coolant

TC2

Tc

q, CAF, Tf

CA, T

state variablesCA, conc. of AT, reactor temp.

max φ = yield of product B

find control parameters to guarantee stability for all set-points Tsp

non-transversal Hopf (NTH) manifoldsplits parameter space into • region with stable behavior for all

values of Tsp• region with unstable behavior for

some values of Tsp

uncertain parameters• feed rate q• time constant εv of unmodeled

dynamics

degrees of freedom: Tsp, q, ε


Tuning and Robustness Analysis: CSTR (2)

unstable for some values of Tspwithout normal vector constraints

0 510

15

q /∆

q

εv/ ∆εv 300 350 4000

0.5

C A

Tsp

stableunstableHopf

setp

oint

tem

pera

ture

Tsp

Control parameter ε

stable forall Tsp

unstable

nontransversalHopf

stable for all values of Tspwith normal vector constraints

0 510

15

q /∆

q

εv/ ∆εv 300 350 4000

0.5

CA

Tsp


Integrated Design & Control: MSMPR Crystallizer (1)

F=τ • V

c0 ε = suspension density

ideal controlT, V

Kp, Ti

F=τ • V


ideal controlT, V

F=τ • V


ideal controlT, V

Kp, Ti

discretized population balance, solute balance, controller

PI control of suspension density εvia feed concentration c0

state variables mi,momentsc, concentrationεi, suspension

density 990 992 994 996 998 10000

0.5

1

1.5

2

2.5

3x 10-3

c0 [kg/m3]

m3 stableunstable

τ=1h τ=2h

τ=3h

990 992 994 996 998 10000

1

2

3

4

c0 [kg/m3]

τ [h]

σ = 0 1/h

σ = -0.1 1/h

σ = -0.5 1/h

unstable

stable

stability boundaryperformance constr.

open-loop behaviorMSMPR crystallizer(Jerauld, Doherty, 1982)



max φ = mass production rate

find design and controller tuning that guarantee dynamic performance

uncertain parameters• residence time τ• feed concentration c0 (open loop)

degrees of freedom• τ, Kp, Ti

use normal vector constraints on eigenvalue bounds to enforce performance

MSMPR crystallizer(Jerauld, Doherty, 1982)

F=τ • V


ideal controlT, V

Kp, Ti

F=τ • V


ideal controlT, V

F=τ • V


ideal controlT, V

Kp, Ti

discretized population balance, solute balance, controller

PI control of suspension density εvia feed concentration c0

state variables mi,momentsc, concentrationεi, suspension

density



Optimization of the closed-loop process with guaranteed performance

990 995 1000 1005 1010 10150.20.5

1

1.5

2

2.5

c0 [kg/m3]

τ[h

]

0 300.0364

0.0366

0.0368

0.037

0.0372

t [h]

m3

[-]σ = -0.1 1/hnon-performant

performant

performance constraintconstraintsrobustness ellipse

σο = −0.2 h

σο = 0 h

productivity – open-loop stable φ = 12.0 kg/m3/h− closed-loop φ = 46.0 kg/m3/h


Integrated Design & Control – HDA Process (1)

HDA process (Douglas, 1988)

Reaction section & simplified separation section

• 8 units• 5 PI controllers• large-scale model

- 100 differential eqs.- 370 algebraic eqs.

• 12 uncertain parameters• no knowledge on nonlinear

dynamics

compressorpurge

purge

mixer

heat exchanger

furnace

toloueneH2

TC

TC

TC

LC

PC

fuel

flash splitter methane

benzene

dyphenyltolouene

reactor


Integrated Design & Control – HDA Process (2)

2.0

2.1

2.2

2.3

benzene prod. rate [kmol/min]

3.6

3.7

3.8

3.9

–1.20

–1.15

–1.10

–1.05

–4.00

–3.90

–3.80

–3.70

3.6

3.7

3.8

3.9

4.0

4.1

4.2

30 60 90 120 150 180

3.3

3.4

3.5

3.6

3.7

��

[kJ/min]

�� [kJ/min] ��

��

[kJ/min]

0

�� [kJ/min] �� [kJ/min]

30 60 90 120 150 1800time [min] time [min]

30 60 90 120 150 1800

30 60 90 120 150 1800

30 60 90 120 150 1800

30 60 90 120 150 1800

Optimization

min φ = Total annual costs =∑annual capital costs+ operating costs

+ costs of chemicals

uncertain process design & control parameters

parametric robustness w.r.t. performance, bounds on eigenvalues, σ0 ≤ 30 min

step response at optimal steady state, 10% increase of toluene feed rate


Summary and Future Perspectives

Constructive Nonlinear Dynamics: from science to engineering

• a unifying framework for the treatment of parametric uncertainty in process and control system design

• computationally feasible even for large-scale processes

• necessary extensions– time domain performance constraints– fast inputs– structural decisions and non-smooth models– processes with optimizing controllers– improvement of numerical methods

• software further development– large-scale problems– part of process modeling environments

applicationsin

design & controlof

process systems,vehicles, ...

Constructive Nonlinear Dynamicsin Process Systems EngineeringExample: Ammonia ReactorExample: Decanting ReactorExample: Heteroazeotropic RectificationExample: Reactor-Separator Recycle ProcessBifurcation AnalysisFrom Analysis to SynthesisConceptual Problem Formulation (1)Conceptual Problem Formulation (2)Optimization Problem FormulationsLeveraging the Profit LossParameterization of UncertaintyParametric Distance to a Critical ManifoldNormal Vector Equation SystemsOptimization under Uncertainty – the General CaseComplicated Real SituationsOverview on AlgorithmSoftware Implementation – the StatusApplications: Three Problem ClassesApplications: Three Problem ClassesDesign Under Uncertainty: Fermenter (1)Design under Uncertainty: Fermenter (2)Design Under Uncertainty: VA Polymerization (1)Design Under Uncertainty: VA Polymerization (1)Robust Design: Stability and Feasibility (2)Applications: Three Problem ClassesTuning and Robustness Analysis: CSTR (1)Tuning and Robustness Analysis: CSTR (2)Applications: Three Problem ClassesIntegrated Design & Control: MSMPR Crystallizer (1)Integrated Design & Control: MSMPR Crystallizer (1)Integrated Design & Control: MSMPR Crystallizer (2)Integrated Design & Control – HDA Process (1)Integrated Design & Control – HDA Process (2)Summary and Future Perspectives

Constructive Nonlinear Dynamics in Process Systems Engineering … · 2019. 8. 15. · Constructive...

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