Constructive Nonlinear Dynamics in Process Systems Engineering. Nonlinear Analysis of Chemical Process Systems. ⢠Chemical reactors. Bilous & Amundsen ...
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
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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[-]
Constructive Nonlinear Dynamics in Process Systems Engineering
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
Constructive Nonlinear Dynamics in Process Systems Engineering
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
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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
Constructive Nonlinear Dynamics in Process Systems Engineering
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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
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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*
Constructive Nonlinear Dynamics in Process Systems Engineering
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
Constructive Nonlinear Dynamics in Process Systems Engineering
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
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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)
Constructive Nonlinear Dynamics in Process Systems Engineering
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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 )
Constructive Nonlinear Dynamics in Process Systems Engineering
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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)
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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
Constructive Nonlinear Dynamics in Process Systems Engineering
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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) ...
Constructive Nonlinear Dynamics in Process Systems Engineering
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
Constructive Nonlinear Dynamics in Process Systems Engineering
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
Constructive Nonlinear Dynamics in Process Systems Engineering
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
Constructive Nonlinear Dynamics in Process Systems Engineering
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)
Constructive Nonlinear Dynamics in Process Systems Engineering
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
Constructive Nonlinear Dynamics in Process Systems Engineering
0 100 150 200 residence time [min]
25
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
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,
Constructive Nonlinear Dynamics in Process Systems Engineering
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
Constructive Nonlinear Dynamics in Process Systems Engineering
5
εv / ∆εv
0 300 350 400 Tsp
stable for all values of Tsp with normal vector constraints 28
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
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
Constructive Nonlinear Dynamics in Process Systems Engineering
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
Constructive Nonlinear Dynamics in Process Systems Engineering
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
Constructive Nonlinear Dynamics in Process Systems Engineering
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
Constructive Nonlinear Dynamics in Process Systems Engineering
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]
Constructive Nonlinear Dynamics in Process Systems Engineering
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, ...
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