Constructive Nonlinear Dynamics in Process ... - Semantic Scholar

Constructive Nonlinear Dynamics in Process Systems Engineering Wolfgang Marquardt and Martin Mönnigmann Process Systems Engineering RWTH Aachen University

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

Nonlinear Analysis of Chemical Process Systems • Chemical reactors Bilous & Amundsen (1955), van Heerden (1958), Aris & Amundsen (1958), Razon & Schmitz (1987), ... Altimari et al. (2004) • Distillation columns Petlyuk & 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 columns Pisarenko et al. (1987), Jacobs & Krishna (1993), Nijhuis et al. (1993), Ciric & Miao (1994), ... • Reactor-Separator processes with recycle Bildea & Dimian (1998), Kiss & Bildea (2002, 2003), Zeyer et al. (2003), Balasubramanian et al. (2003), ... , Bildea et al. (2004), Schmidt & Jacobsen (2004) Constructive Nonlinear Dynamics in Process Systems Engineering

1

Example: Ammonia Reactor Large scale industrial process

>80 years later

Understanding nonlinear dynamics

Nonlinear model with temperature coupling

1916: first industrial process (Bosch, BASF)

Simulated temperature waves

1997: global production >100 mio t/year Influence of preheater on root locus

Temperatur [C]

Temperature recordings of industrial ammonia reactor

Morud, Skogestad (1998) Zeit [min]

Constructive Nonlinear Dynamics in Process Systems Engineering

2

Example: Decanting Reactor Liquid/Liquid Reactor

„simple" process

Phase II : B, D, E

TC

Coolant average temperature Θm[-]

Phase II

E

B

+

Phase I

D D + E

B + A

A, B : Educts C : Catalyst D : Main product E, F : By-product

Reactor temperature Θ [−]

F

C

C

E

(a) multiple steady states

B, C, D, E, F

LC

+

H: Hopf HV: Hysteresis DL: Double Limit BL: Boundary Limit DH: Degenerate Hopf HH: Hopf-Hopf DZ: Double Zero BH: Boundary Hopf HL: Hopf-Limit

Cooling capacity ∆ [−]

Phase I : A, B, C

Phase II

Complex Nonlinear Dynamics

(b) periodic oscillations (as one example)

Coolant average temperature Θm[-]


Luss et al. (1998) 3

Example: Heteroazeotropic Rectification Column with decanter ethanole / water / cyclohexane

xB,E

Experiment Simulation

0.95

A

0.9 B 0.85

D F

0.8

xB,E

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

40

80

120

348 343

T8

338

T4 T1

333

• Experimental verification Müller and Marquardt (1997)

D [ml/h]

353

Temperatur [K]

• Simulation Magnussen et al. (1979)

0

A 0

B 20


40

A 60

80 Zeit [h]

4

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

steady-state for isothermal stand-alone reactor

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

Reaction conversion

• unique

Damkoehler number

CC

PFR

Reaction conversion

TC

CC

Separation

Monomer feed

CSTR-seperator-recylce

Dacr

Recycle Initiator feed CC

stand alone CSTR

Product Damkoehler number

Coolant

Multiplicity for non-isothermal polymerization process wit PFR-separator recycle Constructive Nonlinear Dynamics in Process Systems Engineering

5

Bifurcation Analysis analytical and numerical techniques ...

dynamic simulation

(steady-state) simulation

||x||

continuation and local stability analysis singularity analysis and unfolding pi

pj

stability analysis

continuation

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


6

From Analysis to Synthesis Process synthesis Find (process structure,) design parameters and operating point such that - profit is maximized

How to consider Nonlinear Dynamics?

- quality and safety constraints are fulfilled Nonlinear programming problem

max φ ( x,α ) x ,α

s.t. 0 = f ( x,α ) 0 ≤ g ( x, α ) Constructive Nonlinear Dynamics in Process Systems Engineering

Synthesis Ramirez & Gani (2004) Analysis 7

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

x& = f ( x,η ) , x(0) = x0 ... a parametric dynamic process model Constructive Nonlinear Dynamics in Process Systems Engineering

8

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 optimization problem with cost function, process model and inequality constraints (feasibility, stability etc.)

(i) stability boundary unstable

η2

(i) + (ii) stable

η2 η1

min φ ( x,η ) x ,η

s.t. 0 = f ( x,η ) • optimal solution: η

P

(ii) feasibility boundary η2

η1 feasible

= η∗, x=x*


infeasible

η1

9

Optimization Problem Formulations Optimization with respect to cost function φ, model and ... without stability or feasibility constraints

with stability or feasibility constraints

with robust stability and feasibility constraints

min φ ( x,η )

min φ ( x,η )

min φ ( x ,η )

s.t. 0 = f ( x,η )

s.t. 0 = f ( x,η )

s.t . 0 = f ( x ,η )

x ,η

x ,η

x ,η

R⊆P

profit ηloss ∈P

feasible but not stable

η2

stable and feasible, but not robust to parametric uncertainty η2

optimum

η 2 = α2 P optimum

η1

stable, feasible and robust to parametric uncertainty

η1


R

optimum P

η 1 = α2 10

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


11

Parameterization of Uncertainty ... a deterministic rather than a stochastic setting α2

r

Robustness manifold • arbitrary connected smooth manifold • r normal to both, the critical and the robustness manifold • high computational effort α1

α2

∆α2

specialization and approximation r

∆α1

α1

Approximated robustness box • Ellipsoid overestimates parametric uncertainty • Kreisselmeier-Steiner function underestimates parametric uncertainty • Biegler, Rooney (2001)


12

Parametric Distance to a Critical Manifold

x

α2

∆α2 ∆α1

α2(0)

α1

α2

α1(0)

α1

• Normal vector to nearest saddle-node and Hopf bifurcation points (Dobson, 1993) • Normal vector to general critical manifolds, simplification of defining equations (Mönnigmann & Marquardt, 2002) Constructive Nonlinear Dynamics in Process Systems Engineering

13

Normal Vector Equation Systems Saddle-node bifurcation

Hopf bifurcation

augmented system

augmented system

0= f

0= f

0 = f xv

0 = f x w(1) + ωw( 2 )

0 = vT v − 1

0 = f x w( 2 ) − ωw(1)

0 = r − fαT v

0 = wT w 0 = w(1)T w( 2 )

augmented system

normal vector system

Saddle node

2n + 1

2n + m + 1

0 = f xT v ( 2 ) + ωv (1) + γ 1 w( 2 ) + γ 2 w(1)

Hopf

3n + 2

6n + m + 4

Cusp

3n + 3

4n + m + 4

0 = v (1)T w(1) + v ( 2 )T w( 2 ) − 1

Isola

2n + 2

2n + m + 1

n+1

2n +2m+ 2

Feasiblity constraint

0 = f xT v (1) − ωv ( 2 ) + γ 1 w(1) − γ 2 w( 2 )

0 = v (1)T w( 2 ) − v ( 2 )T w(1) 0 = f xT u + v (1)T f xx w(1) + v ( 2 )T f xx w( 2 ) 0 = r − fαT u + v (1)T f xα w(1) + v ( 2 )T f xα w( 2 )


14

Optimization under Uncertainty – the General Case max

x ( eq ) ,α ( eq ) , l ( i )

φ ( x ( eq ) ,α ( eq ) )

0 = f ( x ( eq ) , α ( eq ) )

steady state (x(eq), α(eq) )

0 = Fk( i ) ( x (i ) , α (i ) , r (i ) )

normal vector r(i) to critical manifold i

α ( i ) = α ( eq ) + l (i ) r (i ) l (i ) r (i ) ≥ m l

cost function

(i )

normal distance between design and critical manifold i minimal normal back-off

≥0

i = 1, K , I k∈K 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)


15

Complicated Real Situations location of critical manifolds usually unknown a priori

ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ (i) (i) a ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ r a (0) ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ a1 ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ a2

?

?

?

test function may fail to detect crossing of critical manifold a2

ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ a ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ1

critical manifolds have to be detected as the optimization proceeds (i)

(ii)

ÇÇÇÇ a 2ÇÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇ M (c,1) ÇÇÇÇ ÇÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇ 11 ÇÇÇÇ ÇÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇ 1 a (0) ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇ 2 ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇ ÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇ ÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇ ÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇ ÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇ ÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇ 2 ÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇ ÇÇÇÇ a1 a 1 ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇ

a2

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

(ii)

ÇÇÇÇ ÇÇÇÇ a2 ÇÇÇÇ ÇÇÇÇ ÇÇÇÇ ÇÇÇÇ ÇÇÇÇ ÇÇÇÇ ÇÇÇÇ ÇÇÇÇ ÇÇÇÇ ÇÇÇÇ 1 1 ÇÇÇÇ ÇÇÇÇ ÇÇÇÇ ÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ 2 2 ÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇ ÇÇÇÇ ÇÇÇÇ ÇÇÇÇ ÇÇÇÇ ÇÇÇÇ ÇÇÇÇ ÇÇÇÇ ÇÇÇÇ ÇÇÇÇ ÇÇÇÇ ÇÇÇÇ ÇÇÇÇ ÇÇÇÇ ÇÇÇÇ ÇÇÇÇ ÇÇÇÇ ÇÇÇÇ a1 a1

a2


16

Overview on Algorithm (i) initialization (ii) update set of locally closest points add new critical point to set of critical points

(iii) optimization (iv) analysis new critical point?

yes

no

(a) find locally closest points initialization with steady state optimal and steady state is check for critical points with run search optimization for new critical with active points and corresponding normal which isvector feasible and found – parametrically robust interval arithmetics in has the normal along the linearconstraints connection vector ofnonlinear known critical desired dynamic with respect to feasiblity robustness region between starting and endand manifolds properties nonlinear dynamics point of optimization (b) remove normal vector constraints constraints for which distance exceeds specified return to step (ii) if value new critical point is found

(v) rigorous search new critical point?

yes

no robust optimum found Constructive Nonlinear Dynamics in Process Systems Engineering

17

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 solvers o should use feasible path solver (e.g. FSQP, Lawrence & Tits (2001)) to properly apply test functions o 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 Constructive Nonlinear Dynamics in Process Systems Engineering

18

Applications: Three Problem Classes

• design under uncertainty Continuous fermenter (Mönnigmann & Marquardt, 2002) Continuous vinylacetate polymerization (Mönnigmann & Marquardt, 2003)

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

• controller tuning and robustness analysis CSTR with unmodelled dynamics (Hahn et al. 2003, Gerhard et al., 2004)

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

• integration of design and control MSMPR crystallizer (Grosch et al., 2003) Reaction section of HDA plant (Mönnigmann & Marquardt, 2004)

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


19

Applications: Three Problem Classes 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 Constructive Nonlinear Dynamics in Process Systems Engineering

20

Design Under Uncertainty: Fermenter (1) Fermenter F, SF (Agrawal et al., 1982)

two stability boundaries: Hopf and saddle-node bifurcations

state variables S, substrate concentration X, biomass concentration

V S, X

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

SF [kmol m–3] 0.8

0.7 0.6 0.5 0.4 0.3 0.2

robustness w.r.t. stability boundaries


2

3

Hopf saddle–node

ÉÉÉ ÉÉÉÉÉÉÉÉÉ ÉÉÉ ÉÉÉÉÉÉÉÉÉ ÉÉÉ ÉÉÉÉÉÉÉÉÉ ÉÉÉ ÉÉÉÉÉÉÉÉÉ ÉÉÉ ÉÉÉÉÉÉÉÉÉ ÉÉÉ ÉÉÉÉÉÉÉÉÉ ÉÉÉ ÉÉÉÉÉÉÉÉÉ ÉÉÉ ÉÉÉÉÉÉÉÉÉ ÉÉÉ ÉÉÉÉÉÉÉÉÉ ÉÉÉ ÉÉÉÉÉÉÉÉÉ ÉÉÉ ÉÉÉÉÉÉÉÉÉ ÉÉÉ ÉÉÉÉÉÉÉÉÉ ÉÉÉ ÉÉÉÉÉÉÉÉÉ ÉÉÉ ÉÉÉÉÉÉÉÉÉ ÉÉÉ ÉÉÉÉÉÉÉÉÉ ÉÉÉ ÉÉÉÉÉÉÉÉÉ ÉÉÉ ÉÉÉÉÉÉÉÉÉ ÉÉÉ ÉÉÉÉÉÉÉÉÉ ÉÉÉ ÉÉÉ ÉÉÉ ÉÉÉ

4 5 6 F [kg s–1]

7

21

Design under Uncertainty: Fermenter (2) x1

stable unstable

0.25

Optimization • without normal constraints • with normal vector constraints for robust stability

0.2 0.15 0.1 0.05 0

0.5

saddle-node Hopf 0.6

0.7

0.8

0.9 Da 1

φ

SSFF

1

0.8

0.4

0.6

0.35

0.4 0.2

0.3

0

0.25 0.5

saddle-node Hopf

0.45

0.6

0.7

0.8

0.9 Da 1

0.2

0.5


0.5

0.6

0.6

0.7 Da

0.7 Da

0.8 22

Design Under Uncertainty: VA Polymerization (1) VA Polymerisation (Teymour & Ray, 1992)

temperature

160

stable

140

instable

120

optimal

100

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

80

Hopf

60

saddlenode

40 20

00

50

100

200

150

residence time[min] [m ol/ l]

F, IF, MF

V

180

0.1

M, I, P, T

co nc en tra tio no f in itia tor

0.08

0.06

• small model

• experimentally validated • critical (stability) manifolds are known

0.04

0.02

0

50


100

150

200

residence time [min]

0

23

Design Under Uncertainty: VA Polymerization (1) VA Polymerisation (Teymour & Ray, 1992) F, IF, MF

V

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

• small model

• experimentally validated • critical (stability) manifolds are 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)


24

Robust Design: Stability and Feasibility (2) saddle-node and Hopf bifurcations and temperature bound

180 160 140 120 100 80 60 40 20

temperature stable instable optimal Hopf saddle node 0

50

200 150 100 residence time [min]

180 160 140 120 100 80 60 40 20

temperature stable instable optimal Hopf saddle node temp. constr. 0

50

0.1

0.1

0.08

[m ol/ l]

0.08

0.06

co nc .o f in itia tor

0.06

0.04

0.04

0.02

0.02

0

50

200 100 150 residence time [min]

0

100 150 200 residence time[min]

co nc .on of Ini tia tor [m ol/ l]

saddle-node bifurcation only

0

50


0 100 150 200 residence time [min]

25


26

Tuning and Robustness Analysis: CSTR (1) CSTR, exothermic 1st order reaction A -> B, with unmodeled cooling jacket dynamics (Hahn et al. 2003) q, CAF, Tf

state variables CA, conc. of A T, reactor temp.

TC1 Tc TC2 Coolant

CA, T

• linearizing feedback (and PID) temperature control, parameter ε • unmodeled dynamics in inner cascade control loop (TC2) parameterized by 2nd order dynamics

max φ = yield of product B find control parameters to guarantee stability for all set-points Tsp non-transversal Hopf (NTH) manifold splits 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 27

CA

stable unstable Hopf

q /∆ q

Tuning and Robustness Analysis: CSTR (2)

15

0.5

10

0

5

0 300 350 400Tsp

εv / ∆εv

unstable

A

C

nontransversal Hopf

q /∆q

setpoint temperature Tsp

unstable for some values of Tsp without normal vector constraints

15

0.5

stable for all Tsp 10

0

Control parameter ε


5

εv / ∆εv

0 300 350 400 Tsp

stable for all values of Tsp with normal vector constraints 28


29

Integrated Design & Control: MSMPR Crystallizer (1) MSMPR crystallizer

open-loop behavior

(Jerauld, Doherty, 1982) F=τ • V

x 10-3

3 m3 2.5

ideal control T, V

2

state variables mi,moments c, concentration εi, suspension density c0

stable unstable

ε = suspension density Kp, Ti

τ=3h τ=2h τ=1h

1.5 1 0.5 0 990 4 τ [h]

992

3

994 996 998 3 c0 [kg/m ] stability boundary performance constr. σ = 0 1/h

discretized population balance, solute balance, controller

2

PI control of suspension density ε via feed concentration c0

stable σ = -0.5 1/h 0 990 992 994 996 c0 [kg/m3]

1000

unstable

σ = -0.1 1/h 1


998

1000 30

Integrated Design & Control: MSMPR Crystallizer (1) MSMPR crystallizer (Jerauld, Doherty, 1982) F=τ • V

max

find design and controller tuning that guarantee dynamic performance

ideal control T, V

state variables mi,moments c, concentration εi, suspension density c0

φ = mass production rate

ε = suspension density Kp, Ti

discretized population balance, solute balance, controller PI control of suspension density ε via feed concentration c0

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


31

Integrated Design & Control: MSMPR Crystallizer (2) Optimization of the closed-loop process with guaranteed performance 0.0372

2.5 performance constraint constraints robustness ellipse

1.5 1

0.037 m3 [-]

τ [h]

2

σο = 0 h

σ = -0.1 1/h

0.0368

non-performant

0.0366

performant

σο = −0.2 h

0.5 0.2 990

995

1000 1005 3 c0 [kg/m ]

1010

1015

productivity – open-loop stable − closed-loop

0.0364 0

t [h]

30

φ = 12.0 kg/m3/h φ = 46.0 kg/m3/h


32

Integrated Design & Control – HDA Process (1) HDA process (Douglas, 1988)

purge compressor mixer

Reaction section & simplified separation section

TC

purge furnace

reactor

heat exchanger PC

• • •

TC

fuel H2 tolouene

flash

splitter

methane benzene

TC

• •

LC

tolouene

8 units 5 PI controllers large-scale model - 100 differential eqs. - 370 algebraic eqs. 12 uncertain parameters no knowledge on nonlinear dynamics

dyphenyl


33

Integrated Design & Control – HDA Process (2) 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

10 5ĂĂĂQ Furnace [kJ/min]

benzene prod. rate [kmol/min] 2.3

3.9

2.2

3.8

2.1

3.7

2.0

3.6 0

30

60

90

0

120 150 180

10 5ĂĂĂQ Cooler [kJ/min]

30

60

90

120 150 180

10 5ĂĂĂQ Flash [kJ/min] –3.70

–1.05 –3.80

–1.10 –1.15

–3.90

–1.20

–4.00 0

4.2

30

60

90

0

120 150 180

30

60

90

120 150 180

F PC,ĂĂMixer [kJ/min]

L LC,ĂĂFlash [kJ/min] 3.7

4.1 4.0

3.6

3.9

3.5

3.8

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

3.4

3.7

3.3

3.6 0

30

60

90 120 150 180 time [min]


0

30

60

90 120 150 180 time [min]

34

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 Constructive Nonlinear Dynamics in Process Systems Engineering

applications in design & control of process systems, vehicles, ...

35