â«On-line optimization. ⢠Semi-batch reactor with safety constraint (Novartis Basel; Bayer-RWTH). ⢠Discontinuous wastewater treatment plant (Firmenich Geneva).
Dynamic Real-Time Optimization via Tracking of the Necessary Conditions of Optimality D. Bonvin and B. Srinivasan+ Laboratoire d’Automatique EPFL, Lausanne, Switzerland + Department of Chemical Engineering Ecole Polytechnique Montreal, Canada
NCO Tracking
D-RTO under Uncertainty Optimization under uncertainty
Robust optimization
Measurement-based optimization
Explicit scheme (NMPC)
Implicit scheme (NCO Tracking)
Update a process model and repeat the optimization
Use a solution model for updating the inputs directly
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NCO Tracking
Illustrative example
Outline Optimization and uncertainty Tracking the Necessary Conditions of Optimality • Turn optimization problem into control problem • Decentralized control scheme • Simulation results • Application projects
Conclusions 2
NCO Tracking
Illustrative Example Semi-batch Chemical Reactor B k1
A+B → C k2 2B → D
u(t)
A T
Exothermic reactions -- Isothermal
Tj
Terminal objective: Maximize number of moles of C at tf by adjusting u(t) Safety path constraint: Heat removal limitation Tj ≥ Tj,min Selectivity terminal constraint: Number of moles of D at tf nDf ≤ nDf,max 3
NCO Tracking
Nominal Optimization Physical knowledge
Modeling Process model
Numerical Optimization Nominal solution
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NCO Tracking
Nominal Optimal Solution
1 2 3
5
NCO Tracking
Implementation of the Nominal Solution Physical knowledge
Modeling Process model
Numerical Optimization Nominal solution
Real Plant
Uncertainty (disturbances, model mismatch) Feasibility? Optimality? 6
NCO Tracking
Measurement-based Optimization Physical knowledge
Modeling explicit
Measurementbased Adjustments
Process model
Numerical Optimization implicit
Updated inputs
Real Plant Measurements
Uncertainty
Feasibility OK Optimal performance OK 7
NCO Tracking
NCO Tracking: Self-optimizing control structure
Modeling
NCOtracking Controller
Process model
Numerical Optimization Updated inputs
Real Plant
Control problem
Solution model
Uncertainty
Measurements Feasibility OK Optimal performance OK 8
NCO Tracking
Solution Model Input representation capable of achieving (near) optimality
Choice of decision variables • Identify arcs and switching times that vary with uncertainty • Introduce approximations and parameterization as needed
Pairing for decentralized control • Associate combinations of decision variables with active constraints • Associate remaining decision variables with sensitivities
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NCO Tracking
Input Dissection u
umax 1
η1(t) t1
2
t2
η2(t) 3
0
t Model
Fixed part -- umax Variable parts -- t1, t2, η1(t), η2(t)
Decision variables 10
NCO Tracking
NCO -- Solution Model A Decision variables: t1, t2, η1(t), η2(t) Pairing Constraints
Path Objectives During the run
Tj = Tj,min
Sensitivities
∂H/∂η2 = 0 H: Hamiltonian
Terminal Objectives End of the run
nD(tf) = nDf,max
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NCO Tracking
Solution Model A Invariant part -- umax Path constraint -- Tj = Tj,min -- t1, η1(t) Terminal constraint -- nD(tf) = nDf,max -- t2 Sensitivity -- ∂H/ ∂η2 = 0 -- η2(t)
% umax ' u = &K" (T j,min # T j ) ' G ($H /$" ) ( " 2 t1 = t
with
0 ) t ) t1 t1 < t ) t 2 t2 < t ) t f
T j (t) = T j,min
and
T j (t# ) > T j,min
t 2 = R* (n Df ,max # n D (t f )) 12
NCO Tracking
NCO-tracking Scheme for Solution Model A On-line
-
Tj,min nDf,max
PI Controller I Controller
0
-
PI Controller Hη2(t)
η1(t)
2
t2 η 2(t)
Uncertainty
umax 1 Input Generation
Tj(t)
u(t) System
x(t)
3
nD(tf)
Estimation of Sensitivity
On-line Run-to-run 13
NCO Tracking
NCO -- Solution Model C Decision variables: t1, t2, η1(t), η2(t) Different pairing Constraints
Path Objectives During the run
Terminal Objectives End of the run
Sensitivities
Tj = Tj,min
-
nD(tf) = nDf,max
∂nC(tf)/∂t2 = 0 14
NCO Tracking
Solution Model C Invariant part -- umax Path constraints -- Tj = Tj,min -- t1, η1(t) Terminal constraints -- nD(tf) = nDf,max -- η2(t) Sensitivity -- ∂nC(tf)/ ∂t2 = 0 -- t2
$ umax & u = % K" (T j,min # T j ) &T (n pred # n ' " Df ,max Df (t)) t1 = t
with
T j (t) = T j,min
0 ( t ( t1 t1 < t ( t 2 t2 < t ( t f and
T j (t# ) > T j,min
t 2 = G) (*nC (t f ) /*t 2 ) 15
NCO Tracking
NCO-tracking Scheme for Solution Model C On-line
-
Tj,min
PI Controller
0
nDf,max
Interpolation
I Controller nDd (t)
-
δnC(tf)/δt2
η1(t)
Uncertainty
umax 2
t2
1 Input Generation
Tj(t) u(t) System nD(t)
η2(t) 3 PI Controller
nC(tf)
On-line Estimation of Sensitivity
Run-to-run 16
NCO Tracking
NCO-tracking -- Input Profiles
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NCO Tracking
NCO-tracking Results Model and run index
Minimum jacket temperature 10 °C
Maximal final amount of D 100 mol
Final amount of C Cost (mol)
A1
10.0
79.6
376.4
A5
10.0
96.2
389.8
A 10
10.0
99.6
392.2
C1
10.0
100
391.8
C5 C 10
10.0 10.0
100 100
391.8 391.8
Ideal cost: 392.3
Open-loop nominal cost: 374.6 18
NCO Tracking
NCO Tracking in Practice Features of NCO tracking • No need of a process model for implementation • Need appropriate process measurements • Approximations can (must) be introduced in solution model
Practical observations • Complexity depends on the number of inputs (not system order) • For many terminal-time dynamic optimization problems, the solution is often determined by the constraints of the problem
Practical extensions • Measurement noise → backoff • Unknown active constraints → superstructure 19
NCO Tracking
Projects Implementing NCO Tracking On-line optimization • Semi-batch reactor with safety constraint (Novartis Basel; Bayer-RWTH) • Discontinuous wastewater treatment plant (Firmenich Geneva) • Fed-batch fermenter (Bioengineering Lab EPFL) • Batch distillation column (Engineering School Fribourg) • Grade transitions in polymerization (Bayer-RWTH; CEPRI Thessaloniki)
Run-to-run optimization • Emulsion copolymerization reactor (Aqua+Tech Geneva) • Electro-discharge machining (Charmilles Geneva) • Batch reactive distillation column (INPT Toulouse) • Fed-batch fermenter (Novozymes Denmark) Bold: experimental
Italics: industrial 20
NCO Tracking
Conclusions Real-time optimization under uncertainty • Turn optimization problem into control problem • Considerable prior information goes into solution model • How robust is this information wrt. uncertainty ? → solution model must remain valid over uncertainty • Considerable potential for industrial applications
Further work • Rigorous mathematical framework • Application to large-scale processes
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NCO Tracking
FIN
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