May 10, 2017 - Processes using PMP and Parsimonious Parameterization. 1 Max Planck Institute for Dynamics of Complex Technical Systems. 2 Laboratoire ...
Dynamic Optimization of Semi-batch Processes using PMP v
and Parsimonious Parameterization E. Aydin1,4 , D. Bonvin2 , K. Sundmacher1,3 1 Max
Planck Institute for Dynamics of Complex Technical Systems
2 Laboratoire 3 Chair 4
d’Automatique, École Polytechnique Fédérale de Lausanne
for Process Systems Engineering, Otto-von-Guericke University
International Max Planck Research School (IMPRS) for Advanced Methods in Process and Systems Engineering ESCAPE, Barcelona 05.10.2017
Outline •
Dynamic optimization & available methods
•
Proposed indirect–based parsimonious algorithm
•
Case Studies:
I.
Binary batch distillation column
II. Semi-batch hydroformylation reactor E. Aydin – Dynamic optimization using indirect methods
Escape 2017
2
Outline •
Dynamic optimization & available methods
•
Proposed indirect–based parsimonious algorithm
•
Case Studies:
I.
Binary batch distillation column
II. Semi-batch hydroformylation reactor E. Aydin – Dynamic optimization using indirect methods
Escape 2017
3
Dynamic Optimization
min 𝐽 = 𝜙(𝑥(𝑡𝑓 ))
: cost function
𝑡𝑓 ,𝑢(𝑡)
s.t. 𝑥 = 𝐹 𝑥, 𝑢 ,
𝑆 𝑥(𝑡) ≤ 0, path
𝑥 0 = 𝑥0
: system equations
𝑇(𝑥(𝑡𝑓 )) ≤ 0
: constraints
terminal
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
4
Dynamic Optimization
Solution Methods
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
5
Numerical Solution Methods Direct Methods
E. Aydin – Dynamic optimization using indirect methods
Indirect Methods (PMP)
Escape 2017
6
Numerical Solution Methods Direct Methods Sequential
Simultaneous
- states integrated
- states discretized
- expensive for path constraints
- trade-off between approx. & optim.
Indirect Methods (PMP)
- efficient for large-scale problems
Srinivasan, B., Palanki, S., & Bonvin, D. (2003). Dynamic optimization of batch processes: I. Characterization of the nominal solution. Computers & Chemical Engineering, 27, 1-26. E. Aydin – Dynamic optimization using indirect methods
Escape 2017
7
Numerical Solution Methods Direct Methods
Indirect Methods (PMP)
Sequential
Simultaneous
Shooting
- states integrated
- states discretized
- requires good initial guesses
- expensive for path constraints
- trade-off between approx. & optim.
- no fast convergent - efficient for method large-scale problems available
Gradient-based - convergence problems for path constraints - no fast convergent method available
Srinivasan, B., Palanki, S., & Bonvin, D. (2003). Dynamic optimization of batch processes: I. Characterization of the nominal solution. Computers & Chemical Engineering, 27, 1-26. E. Aydin – Dynamic optimization using indirect methods
Escape 2017
8
Numerical Solution Methods Direct Methods
Indirect Methods (PMP)
Sequential
Simultaneous
Shooting
- states integrated
- states discretized
- requires good initial guesses
- expensive for path constraints
- trade-off between approx. & optim.
- no fast convergent - efficient for method large-scale problems available
Gradient-based - convergence problems for path constraints - no fast convergent method available
Srinivasan, B., Palanki, S., & Bonvin, D. (2003). Dynamic optimization of batch processes: I. Characterization of the nominal solution. Computers & Chemical Engineering, 27, 1-26. E. Aydin – Dynamic optimization using indirect methods
Escape 2017
9
Outline •
Dynamic optimization & available methods
•
Proposed indirect–based parsimonious algorithm
•
Case Studies:
I.
Binary batch distillation column
II. Semi-batch hydroformylation reactor E. Aydin – Dynamic optimization using indirect methods
Escape 2017
10
Pontryagin’s Minimum Principle
Lev Pontryagin (1908-1988) E. Aydin – Dynamic optimization using indirect methods
Escape 2017
11
Pontryagin’s Minimum Principle min 𝐻 𝑡 = 𝜆𝑇 𝐹 𝑥, 𝑢 + 𝜇𝑇 𝑆(𝑥)
Hamiltonian
𝑡𝑓 ,𝑢(𝑡) co-states Lagrange multipliers for path constraints
s.t. 𝜆𝑇
=
𝑥 = 𝐹 𝑥, 𝑢 ; 𝑥 0 = 𝑥0 ; 𝜕𝐻 − , 𝜕𝑥
𝜆𝑇
𝑡𝑓 =
𝜕𝜙 𝜕𝑥 𝑡𝑓
+
system equations
𝜕𝑇 𝑇 𝜐 ; 𝜕𝑥 𝑡𝑓
co-states
Bryson, A. E. (1975). Applied Optimal Control: Optimization, Estimation and Control: CRC Press.
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
12
Pontryagin’s Minimum Principle min 𝐻 𝑡 = 𝜆𝑇 𝐹 𝑥, 𝑢 + 𝜇𝑇 𝑆(𝑥)
Hamiltonian
𝑡𝑓 ,𝑢(𝑡) co-states Lagrange multipliers for path constraints
s.t. 𝜆𝑇
=
𝑥 = 𝐹 𝑥, 𝑢 ; 𝑥 0 = 𝑥0 ; 𝜕𝐻 − , 𝜕𝑥
𝜆𝑇
𝑡𝑓 =
𝜇𝑇 𝑆 = 0;
𝜕𝜙 𝜕𝑥 𝑡𝑓
+
system equations
𝜕𝑇 𝑇 𝜐 ; 𝜕𝑥 𝑡𝑓
𝜐𝑇 𝑇 = 0
co-states slackness conditions
Lagrange multipliers
𝜕𝐻(𝑡) 𝜕𝐹 𝜕𝑆 𝑇 𝑇 =𝜆 +𝜇 =0 𝜕𝑢 𝜕𝑢 𝜕𝑢
stationarity conditions
Bryson, A. E. (1975). Applied Optimal Control: Optimization, Estimation and Control: CRC Press.
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
13
Pontryagin’s Minimum Principle min 𝐻 𝑡 = 𝜆𝑇 𝐹 𝑥, 𝑢 + 𝜇𝑇 𝑆(𝑥)
𝑡𝑓 ,𝑢(𝑡)
Lagrange multipliers for path constraints
s.t. 𝜆𝑇
=
𝑥 = 𝐹 𝑥, 𝑢 ; 𝑥 0 = 𝑥0 ; 𝜕𝐻 − , 𝜕𝑥
𝜆𝑇
𝑡𝑓 =
𝜇𝑇 𝑆 = 0;
𝜕𝜙 𝜕𝑥 𝑡𝑓
+
update using
𝜕𝑇 𝑇 𝜐 ; 𝜕𝑥 𝑡𝑓
𝜐𝑇 𝑇 = 0
shooting & optimization: COSTLY!
Lagrange multipliers
𝜕𝐻(𝑡) 𝜕𝐹 𝜕𝑆 𝑇 𝑇 =𝜆 +𝜇 =0 𝜕𝑢 𝜕𝑢 𝜕𝑢
may not even converge!
Bryson, A. E. (1975). Applied Optimal Control: Optimization, Estimation and Control: CRC Press. Chachuat, B. (2007). Nonlinear and Dynamic Optimization: From Theory to Practice. Lecture Notes.
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
14
Proposed Method To deal with path constraints: min 𝐻 𝑡 = 𝜆𝑇 𝐹 𝑥, 𝑢 + 𝜇𝑇 𝑆(𝑥)
𝑡𝑓 ,𝑢(𝑡)
Aydin, E., Bonvin, D., & Sundmacher, K. (2017). Dynamic optimization of constrained semi-batch processes using Pontryagin’s minimum principle—An effective quasi-Newton approach. Computers & Chemical Engineering, 99, 135-144.
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
15
Proposed Method To deal with path constraints: min 𝐻 𝑡 = 𝜆𝑇 𝐹 𝑥, 𝑢 + 𝜇𝑇 𝑆(𝑥)
𝑡𝑓 ,𝑢(𝑡)
1) 𝜇 as penalty term that ensures feasibility! (instead of parametrizing & shooting for 𝜇)
Aydin, E., Bonvin, D., & Sundmacher, K. (2017). Dynamic optimization of constrained semi-batch processes using Pontryagin’s minimum principle—An effective quasi-Newton approach. Computers & Chemical Engineering, 99, 135-144.
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
16
Proposed Method To deal with path constraints: min 𝐻 𝑡 = 𝜆𝑇 𝐹 𝑥, 𝑢 + 𝜇𝑇 𝑆(𝑥)
𝑡𝑓 ,𝑢(𝑡)
1) 𝜇 as penalty term that ensures feasibility! (instead of parametrizing & shooting for 𝜇) If feasible iteration => set 𝜇 = 0 ; 𝜐 = 0 else set 𝜇 = K ; 𝜐 = K
(penalty term)
Aydin, E., Bonvin, D., & Sundmacher, K. (2017). Dynamic optimization of constrained semi-batch processes using Pontryagin’s minimum principle—An effective quasi-Newton approach. Computers & Chemical Engineering, 99, 135-144.
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
17
Proposed Method To deal with path constraints: min 𝐻 𝑡 = 𝜆𝑇 𝐹 𝑥, 𝑢 + 𝜇𝑇 𝑆 𝑥 𝑡𝑓 ,𝑢(𝑡)
1) 𝜇 as penalty term that ensures feasibility!
𝜇𝑇 𝑆 = 0
2) indirect adjoining! 𝑆 𝑥 : = 𝑆 (𝑛) 𝑥, 𝑢
time derivation until explicit in u
Aydin, E., Bonvin, D., & Sundmacher, K. (2017). Dynamic optimization of constrained semi-batch processes using Pontryagin’s minimum principle—An effective quasi-Newton approach. Computers & Chemical Engineering, 99, 135-144.
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
18
Proposed Method To deal with path constraints: min 𝐻′ 𝑡 = 𝜆𝑇 𝐹 𝑥, 𝑢 + 𝜇𝑇 𝑆 (𝑛) 𝑥, 𝑢
𝑡𝑓 ,𝑢(𝑡)
1) 𝜇 as penalty term that ensures feasibility!
𝜇𝑇 𝑆 = 0
2) indirect adjoining! 𝑆 𝑥 : = 𝑆 (𝑛) 𝑥, 𝑢
Aydin, E., Bonvin, D., & Sundmacher, K. (2017). Dynamic optimization of constrained semi-batch processes using Pontryagin’s minimum principle—An effective quasi-Newton approach. Computers & Chemical Engineering, 99, 135-144.
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
19
Proposed Method To deal with path constraints: min 𝐻′ 𝑡 = 𝜆𝑇 𝐹 𝑥, 𝑢 + 𝜇𝑇 𝑆 (𝑛) 𝑥, 𝑢
𝑡𝑓 ,𝑢(𝑡)
1) 𝜇 as penalty term that ensures feasibility!
𝜇𝑇 𝑆 = 0
2) indirect adjoining! 3) if infeasible iteration in terms of 𝑺 𝒙 => compute 𝒖 that makes 𝑆 (𝑛) 𝑥, 𝑢 =0 (enforce active path constraints)
Aydin, E., Bonvin, D., & Sundmacher, K. (2017). Dynamic optimization of constrained semi-batch processes using Pontryagin’s minimum principle—An effective quasi-Newton approach. Computers & Chemical Engineering, 99, 135-144.
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
20
Proposed Method To illustrate: min 𝐻 𝑡 = 𝜆𝑇 𝐹 𝑥, 𝑢 + 𝜇𝑇 𝑆 𝑥
𝑡𝑓 ,𝑢(𝑡)
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
21
Proposed Method To illustrate: min 𝐻′ 𝑡 = 𝜆𝑇 𝐹 𝑥, 𝑢 + 𝜇𝑇 𝑆 (𝑛) 𝑥, 𝑢
𝑡𝑓 ,𝑢(𝑡)
indirect adjoining
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
22
Proposed Method To illustrate: min 𝐻′ 𝑡 = 𝜆𝑇 𝐹 𝑥, 𝑢 + 𝜇𝑇 𝑆 (𝑛) 𝑥, 𝑢
𝑡𝑓 ,𝑢(𝑡)
th
n iteration (infeasible)
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
23
Proposed Method To illustrate: min 𝐻′ 𝑡 = 𝜆𝑇 𝐹 𝑥, 𝑢 + 𝜇𝑇 𝑆 (𝑛) 𝑥, 𝑢
𝑡𝑓 ,𝑢(𝑡)
th
n iteration (infeasible)
activate path constraint {𝑆 (𝑛) 𝑥, 𝑢 = 0}
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
24
Proposed Method
• Original problem converted to an unconstrained optimization problem! • Solve using a Quasi-Newton algorithm
E. Aydin – Dynamic optimization using indirect methods
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25
Parsimonious Solution Model A solution arc can be: - either on a lower or upper bound (𝒖𝒎𝒊𝒏 or 𝒖𝒎𝒂𝒙 )
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
26
Parsimonious Solution Model A solution arc can be: - either on a lower or upper bound (𝒖𝒎𝒊𝒏 or 𝒖𝒎𝒂𝒙 ) - on a path constraint (𝒖𝒑𝒂𝒕𝒉 ) , can be computed via system equations
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
27
Parsimonious Solution Model A solution arc can be: - either on a lower or upper bound (𝒖𝒎𝒊𝒏 or 𝒖𝒎𝒂𝒙 ) - on a path constraint (𝒖𝒑𝒂𝒕𝒉 ) , can be computed via system equations - or inside the feasible region as a sensitivity-seeking arc (𝒖𝒔𝒆𝒏𝒔 )
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
28
Parsimonious Solution Model A solution arc can be: - either on a lower or upper bound (𝒖𝒎𝒊𝒏 or 𝒖𝒎𝒂𝒙 ) - on a path constraint (𝒖𝒑𝒂𝒕𝒉 ) - or inside the feasible region as a sensitivity-seeking arc (𝒖𝒔𝒆𝒏𝒔 )
Fine shape of 𝒖𝒔𝒆𝒏𝒔 :
- hard to compute accurately
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
29
Parsimonious Solution Model A solution arc can be: - either on a lower or upper bound (𝒖𝒎𝒊𝒏 or 𝒖𝒎𝒂𝒙 ) - on a path constraint (𝒖𝒑𝒂𝒕𝒉 ) - or inside the feasible region as a sensitivity-seeking arc (𝒖𝒔𝒆𝒏𝒔 )
Fine shape of 𝒖𝒔𝒆𝒏𝒔 :
- hard to compute accurately - often negligible effect on optimal cost! E. Aydin – Dynamic optimization using indirect methods
Escape 2017
30
Parsimonious Solution Model A solution arc can be: - either on a lower or upper bound (𝒖𝒎𝒊𝒏 or 𝒖𝒎𝒂𝒙 ) - on a path constraint (𝒖𝒑𝒂𝒕𝒉 ) - or inside the feasible region as a sensitivity-seeking arc (𝒖𝒔𝒆𝒏𝒔 )
Parsimonious parameterization
Fine shape of 𝒖𝒔𝒆𝒏𝒔 :
- hard to compute accurately
using prior solution - often negligible effect on optimal cost! E. Aydin – Dynamic optimization using indirect methods
Escape 2017
31
Parsimonious Solution Model Parsimonious parameterization of 3-arc solution : 𝑢𝑚𝑎𝑥
- 𝑢𝑠𝑒𝑛𝑠 - 𝑢𝑚𝑖𝑛
PMP : Standard parameterization min 𝐻 𝑡 = 𝜆𝑇 𝐹 𝑥, 𝑢 + 𝜇𝑇 𝑆(𝑥, 𝑢) 𝑢(𝑡)
s.t. 𝑥 = 𝐹 𝑥, 𝑢 ; 𝑥 0 = 𝑥0 ; 𝜕𝐻 𝜕𝜙 𝜕𝑇 𝜆𝑇 = − , 𝜆𝑇 𝑡𝑓 = + 𝜐𝑇 𝜕𝑥
𝜕𝑥 𝑡𝑓
𝜕𝑥 𝑡𝑓
;
𝜇𝑇 𝑆(𝑥, 𝑢) = 0; 𝜐 𝑇 𝑇(𝑥(𝑡𝑓 )) = 0 ; 𝜕𝐻(𝑡) 𝜕𝐹 𝑥, 𝑢 𝜕𝑆 𝑥, 𝑢 = 𝜆𝑇 + 𝜇𝑇 =0 𝜕𝑢 𝜕𝑢 𝜕𝑢
E. Aydin – Dynamic optimization using indirect methods
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32
Parsimonious Solution Model Parsimonious parameterization of 3-arc solution : 𝑢𝑚𝑎𝑥 PMP : Standard parameterization
PMP : Pars. parameterization 𝑚𝑖𝑛 𝐻 𝑡 = 𝜆𝑇 𝐹 𝑥, 𝜋 + 𝜇𝑇 𝑆(𝑥, 𝜋)
min 𝐻 𝑡 = 𝜆𝑇 𝐹 𝑥, 𝑢 + 𝜇𝑇 𝑆(𝑥, 𝑢)
𝝅
𝒖(𝑡)
s.t. 𝑥 = 𝐹 𝑥, 𝑢 ; 𝑥 0 = 𝑥0 ; 𝜕𝐻 𝜕𝜙 𝜕𝑇 𝜆𝑇 = − , 𝜆𝑇 𝑡𝑓 = + 𝜐𝑇 𝜕𝑥
𝜕𝑥 𝑡𝑓
- 𝑢𝑠𝑒𝑛𝑠 - 𝑢𝑚𝑖𝑛
s.t.
𝜕𝑥 𝑡𝑓
;
𝜇𝑇 𝑆(𝑥, 𝑢) = 0; 𝜐 𝑇 𝑇(𝑥(𝑡𝑓 )) = 0 ; 𝜕𝐻(𝑡) =0 𝜕𝑢
𝜆𝑇 = −
𝒙 = 𝑭 𝒙, 𝝅 ; 𝑥 0 = 𝑥0 ;
𝜕𝐻 , 𝜕𝑥
𝜕𝜙 𝜕𝑥 𝑡𝑓 𝑇
𝜆𝑇 𝑡𝑓 =
+ 𝜐𝑇
𝜕𝑇 ; 𝜕𝑥 𝑡𝑓
𝝁𝑻 𝑺(𝒙, 𝝅) = 𝟎; 𝜐 𝑇(𝑥(𝑡𝑓 )) = 0; 𝝏𝑯(𝒕) =𝟎 𝝏𝝅
𝑢(𝑡) ≈ 𝑈(𝜋)
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
33
Parsimonious Solution Model Parsimonious parameterization of 3-arc solution : 𝑢𝑚𝑎𝑥
- 𝑢𝑠𝑒𝑛𝑠 - 𝑢𝑚𝑖𝑛
𝑢(𝑡) ≈ 𝑈(𝜋) 𝑢max 𝑈 𝜋 =
𝒖𝒔𝒆𝒏𝒔 𝒕 = 𝑢𝑚𝑎𝑥 + 𝑢min
𝑢𝑚𝑖𝑛 − 𝑢𝑚𝑎𝑥 (𝑡 − 𝑡1 ) 𝑡2 − 𝑡1
𝑖𝑓 0 ≤ 𝑡 < 𝑡1 ; 𝑖𝑓 𝑡1 ≤ 𝑡 < 𝑡2 ; 𝑖𝑓 𝑡2 ≤ 𝑡 < 𝑡𝑓
𝝅 = (𝒕𝟏 , 𝒕𝟐 )
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
34
Parsimonious Solution Model Parsimonious parameterization of 3-arc solution : 𝑢𝑚𝑎𝑥
- 𝑢𝑠𝑒𝑛𝑠 - 𝑢𝑚𝑖𝑛
𝑢(𝑡) ≈ 𝑈(𝜋) 𝑢max 𝑈 𝜋 =
𝒖𝒔𝒆𝒏𝒔 𝒕 = 𝑢𝑚𝑎𝑥 + 𝑢min
𝑢𝑚𝑖𝑛 − 𝑢𝑚𝑎𝑥 (𝑡 − 𝑡1 ) 𝑡2 − 𝑡1
𝑖𝑓 0 ≤ 𝑡 < 𝑡1 ; 𝑖𝑓 𝑡1 ≤ 𝑡 < 𝑡2 ; 𝑖𝑓 𝑡2 ≤ 𝑡 < 𝑡𝑓
𝝅 = (𝒕𝟏 , 𝒕𝟐 )
Apply the same solution strategy!
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
35
Outline •
Dynamic optimization & available methods
•
Proposed indirect–based parsimonious algorithm
•
Case Studies:
I.
Binary batch distillation column
II. Semi-batch hydroformylation reactor E. Aydin – Dynamic optimization using indirect methods
Escape 2017
36
Case Study : Binary Batch Distillation input: 𝒓 = L/V V, y3
V-L, y3
𝒎𝒂𝒙 𝑱 = 𝑫(𝒕𝒇 ) 𝒓(𝒕)
s.t.
L, y3
3
D, xD
• dynamic system equations, initial conditions
2
• physical constraints; 𝑡𝑓 = 3 [ℎ]
1
• input constraints:
0≤𝒓 𝒕 ≤1
• terminal purity constraints: 𝑥𝐷 𝑡𝑓 ≥ 0.8 𝑥B 𝑡𝑓 ≤ 0.2
B, xB
Aydin, E., Bonvin, D., & Sundmacher, K. (2017). Dynamic optimization of constrained semi-batch processes using Pontryagin’s minimum principle—An effective quasi-Newton approach. Computers & Chemical Engineering, 99, 135-144.
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
37
Case Study : Binary Batch Distillation Distillate, D (kmol)
Fully-parameterized solution: N=500, direct simultaneous Reflux ratio, r
1 0.8 0.6 0.4 0.2 Full par. DS 0
0
1
2
50
30 20 10 0
3
J=40.31
40
0
1
0.88 0.86 0.84 0.82 0.8
0
1
3
2
3
Time (h)
2
3
Bottoms composition, xB
Distillate composition, xD
Time (h)
2
0.4 0.35 0.3 0.25 0.2
0
Time (h)
1
Time (h)
Aydin, E., Bonvin, D., & Sundmacher, K. (2017). Dynamic optimization of constrained semi-batch processes using Pontryagin’s minimum principle—An effective quasi-Newton approach. Computers & Chemical Engineering, 99, 135-144.
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
38
Case Study : Binary Batch Distillation Distillate, D (kmol)
Fully-parameterized solution: N=500, direct simultaneous Reflux ratio, r
1 0.8 0.6
𝒓𝒎𝒂𝒙
0.4
𝒓𝒔𝒆𝒏𝒔
𝒓𝒎𝒊𝒏
0.2 Full par. DS 0
0
1
2
50
30 20 10 0
3
J=40.31
40
0
1
0.88 0.86 0.84 0.82 0.8
0
1
3
2
3
Time (h)
2
3
Bottoms composition, xB
Distillate composition, xD
Time (h)
2
0.4 0.35 0.3 0.25 0.2
0
Time (h)
1
Time (h)
Aydin, E., Bonvin, D., & Sundmacher, K. (2017). Dynamic optimization of constrained semi-batch processes using Pontryagin’s minimum principle—An effective quasi-Newton approach. Computers & Chemical Engineering, 99, 135-144.
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
39
Case Study : Binary Batch Distillation 𝜋 = (𝑡1 , 𝑡2 , 𝑟𝑏 )𝑇
Distillate, D (kmol)
Parsimonious solution: constant 𝑟𝑠𝑒𝑛𝑠 (t) = rb Reflux ratio, r
1 0.8 0.6 0.4 Full par. DS Pars. PMP constant rsens
0.2 0
0
1
2
50
30 20 10 0
3
J=40.31 J=40.18
40
0
1
0.88 0.86 0.84 0.82 0.8
0
1
3
2
3
Time (h)
2
3
Bottoms composition, xB
Distillate composition, xD
Time (h)
2
0.4 0.35 0.3 0.25 0.2 0
Time (h)
E. Aydin – Dynamic optimization using indirect methods
1
Time (h)
Escape 2017
40
Case Study : Binary Batch Distillation 𝜋 = (𝑡1 , 𝑡2 , 𝑟𝑏 )𝑇
Reflux ratio, r
1
Distillate, D (kmol)
Parsimonious solution: constant 𝑟𝑠𝑒𝑛𝑠 (t) = rb 𝑟𝑏
0.8
𝑡2
𝑡1
0.6 0.4
Full par. DS Pars. PMP constant rsens
0.2 0
0
1
2
50
30 20 10 0
3
J=40.31 J=40.18
40
0
1
0.88 0.86 0.84 0.82 0.8
0
1
3
2
3
Time (h)
2
3
Bottoms composition, xB
Distillate composition, xD
Time (h)
2
0.4 0.35 0.3 0.25 0.2 0
Time (h)
E. Aydin – Dynamic optimization using indirect methods
1
Time (h)
Escape 2017
41
Case Study : Binary Batch Distillation Parsimonious solution: linear 𝑟𝑠𝑒𝑛𝑠 𝑡 = 𝑟𝑏1 +
E. Aydin – Dynamic optimization using indirect methods
𝑟𝑏2−𝑟𝑏1 (𝑡 𝑡2 −𝑡1
Escape 2017
− 𝑡1 )
𝜋 = (𝑡1 , 𝑡2 , 𝑟𝑏1 , 𝑟𝑏2 )𝑇
42
Case Study : Binary Batch Distillation Distillate, D (kmol)
Parsimonious solution: linear 𝑟𝑠𝑒𝑛𝑠 𝑡 = 𝑟𝑏1 + Reflux ratio, r
1 0.8 0.6 0.4
Full par. DS Pars. PMP constant r sens
0.2 0
Pars. PMP linear r sens
0
1
2
𝑟𝑏2−𝑟𝑏1 (𝑡 𝑡2 −𝑡1
J=40.31
40
J=40.18 J=40.25
30 20 10 0
1
0.88 0.86 0.84 0.82
0
1
2
3
2
3
Time (h)
2
3
Bottoms composition, xB
Distillate composition, xD
Time (h)
0.8
𝜋 = (𝑡1 , 𝑡2 , 𝑟𝑏1 , 𝑟𝑏2 )𝑇
50
0
3
− 𝑡1 )
0.4 0.35 0.3 0.25 0.2 0
Time (h)
E. Aydin – Dynamic optimization using indirect methods
1
Time (h)
Escape 2017
43
Case Study : Binary Batch Distillation
Reflux ratio, r
1
𝑟𝑏2
0.8
𝑡1
0.6 0.4
Full par. DS Pars. PMP constant r sens
0.2 0
𝑡2
𝑟𝑏1 Pars. PMP linear r sens
0
1
2
Distillate, D (kmol)
Parsimonious solution: linear 𝑟𝑠𝑒𝑛𝑠 𝑡 = 𝑟𝑏1 +
𝑟𝑏2−𝑟𝑏1 (𝑡 𝑡2 −𝑡1
J=40.31
40
J=40.18 J=40.25
30 20 10 0
1
0.88 0.86 0.84 0.82
0
1
2
3
2
3
Time (h)
2
3
Bottoms composition, xB
Distillate composition, xD
Time (h)
0.8
𝜋 = (𝑡1 , 𝑡2 , 𝑟𝑏1 , 𝑟𝑏2 )𝑇
50
0
3
− 𝑡1 )
0.4 0.35 0.3 0.25 0.2 0
Time (h)
1
Time (h)
0.15 % gap E. Aydin – Dynamic optimization using indirect methods
Escape 2017
44
Case Study : Binary Batch Distillation Full par. DS Pars. PMP constant rsens Pars. PMP linear rsens
CPU Time [s]
1
10
0
10
0
50
100
150
200
250
300
350
400
450
500
Discretization Level
Full par. DS:
40.31 kmol
Pars. PMP : constant 𝑟𝑠𝑒𝑛𝑠
40.18 kmol
Pars. PMP : linear 𝑟𝑠𝑒𝑛𝑠
40.25 kmol
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
45
Outline •
Dynamic optimization & available methods
•
Proposed indirect–based parsimonious algorithm
•
Case Studies:
I.
Binary batch distillation column
II. Semi-batch hydroformylation reactor E. Aydin – Dynamic optimization using indirect methods
Escape 2017
46
Case Study : Hydroformylation 𝒖 𝒕 (𝑠𝑦𝑛𝑔𝑎𝑠)
𝑻(𝒕) 𝒎𝒂𝒙 𝑱 = 𝒄𝒏𝒄𝟏𝟑𝒂𝒍 (𝒕𝒇 )
𝒖 𝒕 ,𝑻(𝒕)
s.t. • dynamic system equations, balances, physical constraints, 𝑡𝑓 = 80 [𝑚𝑖𝑛] • gas-liquid mass-transfer equations • input constraints:
0≤𝒖 𝒕 ;
368.15 K ≤ 𝑻 𝒕 ≤ 388.15 K
• total partial pressure constraint: 1 𝑏𝑎𝑟 ≤ 𝒑𝒕𝒐𝒕𝒂𝒍 (𝒕) ≤ 20 𝑏𝑎𝑟 - Hentschel, B., Kiedorf, G., Gerlach, M., Hamel, C., Seidel-Morgenstern, A., Freund, H., & K. Sundmacher. (2015). Model-based identification and experimental validation of the optimal reaction route for the Hydroformylation of 1-dodecene. Industrial & Engineering Chemistry Research, 54, 1755-1765. - Aydin, E., Bonvin, D., & Sundmacher, K. (2017). Dynamic optimization of constrained semi-batch processes using Pontryagin’s minimum principle—An effective quasiNewton approach. Computers & Chemical Engineering, 99, 135-144.
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
47
Case Study : Hydroformylation 390
T (K)
385 380 375 370 365 0
10
20
30
40
50
60
70
80
u (mol/min)
4 Full par. PMP, J=0.595 mol/L Full par. DS , J=0.596 mol/L
3 2 1 0 0
10
20
30
E. Aydin – Dynamic optimization using indirect methods
40 Time (h)
50
Escape 2017
60
70
80
48
Case Study : Hydroformylation 390
T (K)
385 380 375
𝑻𝒎𝒊𝒏 - 𝑻𝒔𝒆𝒏𝒔 - 𝑻𝒎𝒂𝒙
370 365 0
10
20
30
40
50
60
70
80
u (mol/min)
4 Full par. PMP, J=0.595 mol/L Full par. DS , J=0.596 mol/L
3 2
𝒖𝒑𝒂𝒕𝒉
1 0 0
10
20
30
E. Aydin – Dynamic optimization using indirect methods
40 Time (h)
50
Escape 2017
60
70
80
49
Case Study : Hydroformylation 390
T (K)
385 380 375
𝑻𝒎𝒊𝒏 - 𝑻𝒔𝒆𝒏𝒔 - 𝑻𝒎𝒂𝒙
370 365 0
10
20
30
40
50
60
70
80
u (mol/min)
4 Full par. PMP, J=0.595 mol/L Full par. DS , J=0.596 mol/L
3 2
𝒖𝒑𝒂𝒕𝒉
1 0 0
10
20
30
𝒖 = 𝒖 𝒑𝒂𝒕𝒉
40 Time (h)
50
60
70
80
𝝅 = (𝒕𝟏 , 𝒕𝟐 )𝑻 (via system equations)
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
50
Case Study : Hydroformylation 390
T (K)
385
𝑡2
380 375
𝑡1
370 365 0
10
20
30
40
50
60
70
80
u (mol/min)
4 3
Pars. par. PMP, J=0.594 mol/L Full par. PMP, J=0.595 mol/L Full par. DS , J=0.596 mol/L
2 1 0 0
10
20
30
𝒖 = 𝒖 𝒑𝒂𝒕𝒉
40 Time (h)
50
𝝅 = (𝒕𝟏 , 𝒕𝟐 )𝑻 (via system equations)
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
60
70
80
0.33 % gap 51
Case Study : Hydroformylation 70 Full par. DS Full par. PMP 60
CPU Time [s]
50
40
30
20
10
0
100
150
200
250 300 Discretization Level
E. Aydin – Dynamic optimization using indirect methods
350
Escape 2017
400
450
500
52
Case Study : Hydroformylation 70
60
Full par. DS Full par. PMP Pars. par. PMP
CPU Time [s]
50
40
30
20
10
0
100
150
200
250 300 Discretization Level
E. Aydin – Dynamic optimization using indirect methods
350
Escape 2017
400
450
500
53
Conclusions and Outlook • An
alternative
PMP-based
parsimonious
algorithm
is
proposed for solving constrained dynamic optimization of semi-batch processes.
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
54
Conclusions and Outlook • An
alternative
PMP-based
parsimonious
algorithm
is
proposed for solving constrained dynamic optimization of semi-batch processes. • The method requires a prior solution.
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
55
Conclusions and Outlook • An
alternative
PMP-based
parsimonious
algorithm
is
proposed for solving constrained dynamic optimization of semi-batch processes. • The method requires a prior solution. • Drastic computational reduction is observed through the use of parsimonious models.
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
56
Conclusions and Outlook • An
alternative
PMP-based
parsimonious
algorithm
is
proposed for solving constrained dynamic optimization of semi-batch processes. • The method requires a prior solution. • Drastic computational reduction is observed through the use of parsimonious models. => can be extended to real-time model-based control.
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
57
Conclusions and Outlook => can be extended to real-time model-based control 4
Standard sh-NMPC Parsimonious sh-NMPC
3.5
CPU time [s]
3
2.5
2
1.5
1
0.5
0
0
10
20
30
40
50
60
70
Time (min) Aydin, E., Bonvin, D., & Sundmacher, K. Fast NMPC of semi-batch processes via simplified solution modelsThe parsimonious shrinking-horizon NMPC, Journal of Process Control. Under review. E. Aydin – Dynamic optimization using indirect methods
Escape 2017
58
Conclusions and Outlook • An
alternative
PMP-based
parsimonious
algorithm
is
proposed for solving constrained dynamic optimization of semi-batch processes. • The method requires a prior solution. • Drastic computational reduction is observed through the use of parsimonious models. => can be extended to real-time model-based control Aydin, E., Bonvin, D., & Sundmacher, K. Fast NMPC of semi-batch processes via simplified solution modelsThe parsimonious shrinking-horizon NMPC, Journal of Process Control. Under review.
E. Aydin – Dynamic optimization using indirect methods
Escape 2017
59
