1 Dynamic Real-Time Optimization: Concepts in Modeling, Algorithms and Properties L. T. Biegler Chemical Engineering Department Carnegie Mellon University Pittsburgh, PA November 28, 2007 2 I Introduction Typical Applications Problem Statement II Dynamic Optimization Sequential Methods Multiple Shooting Simultaneous Methods III Off-line Case Studies Unstable Grade Transitions Simulated Moving Beds Parameter Estimation – Reactor Models IV On-line Optimization NMPC Case Study Advanced Step NMPC Moving Horizon Estimation V Conclusions Summary References Dynamic Optimization Outline
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
1
Dynamic Real-Time Optimization: Concepts in Modeling, Algorithms and
Properties
L. T. BieglerChemical Engineering Department
Carnegie Mellon UniversityPittsburgh, PA
November 28, 2007
2
I IntroductionTypical ApplicationsProblem Statement
II Dynamic OptimizationSequential MethodsMultiple ShootingSimultaneous Methods
III Off-line Case StudiesUnstable Grade TransitionsSimulated Moving BedsParameter Estimation – Reactor Models
IV On-line OptimizationNMPC Case StudyAdvanced Step NMPCMoving Horizon Estimation
Algebraic EquationsConstitutive Equations, Equilibrium (physical properties, hydraulics, rate laws)Semi-explicit formAssume to be index one (i.e., algebraic variables can be solved uniquely by algebraic equations)If not, DAE can be reformulated to index one (see Ascher and Petzold)
CharacteristicsLarge-scale models – not easily scaledSparse but no regular structureDirect linear solvers widely usedCoarse-grained decomposition of linear algebra
Results:Piecewise Linear Approximation with Variable Time ElementsOptimum B/A: 0.5726Equivalent # of ODE solutions: 32
Batch Reactor Optimal Temperature Program Piecewise Linear
13
25
Op
tim
al P
rofile
, u
(t)
0. 0.2 0.4 0.6 0.8 1.0
2
4
6
Time, h
Results:Control Vector Iteration with Conjugate GradientsOptimum (B/A): 0.5732Equivalent # of ODE solutions: 58
Batch Reactor Optimal Temperature Program
Indirect Approach
26
Results of Optimal Temperature Program Batch Reactor (Revisited)
Results- NLP with Orthogonal CollocationOptimum B/A - 0.5728# of ODE Solutions - 0.7(Equivalent)
14
27
Dynamic Optimization Engines
Evolution of NLP Solvers:
Î for dynamic optimization, control and estimation
E.g., NPSOL and Sequential Dynamic Optimization - over 100 variables and constraints
SQP
28
Dynamic Optimization Engines
Evolution of NLP Solvers:
Î for dynamic optimization, control and estimation
E.g, SNOPT and Multiple Shooting - over 100 d.f.s but over 105 variables and constraints
SQP rSQP
15
29
Dynamic Optimization Engines
Evolution of NLP Solvers:
Î for dynamic optimization, control and estimation
E.g., NPSOL and Sequential Dynamic Optimization - over 100 variables and constraints E.g, SNOPT and Multiple Shooting - over 100 d.f.s but over 105 variables and constraintsE.g., IPOPT - Simultaneous dynamic optimizationover 1 000 000 variables and constraints
SQP rSQP Full-spaceBarrier
Object Oriented Codes tailored to structure, sparse linearalgebra and computer architecture (e.g., IPOPT 3.3)
30
Barrier Methods for Large-Scale Nonlinear Programming
0
0)(s.t
)(min
≥=
ℜ∈
x
xc
xfnx
Original Formulation
0)(s.t
ln)()( min1
=
−= ∑=ℜ∈
xc
xxfxn
ii
x nµϕµBarrier Approach
Can generalize for
bxa ≤≤
⇒ As µ Î 0, x*(µ) Î x* Fiacco and McCormick (1968)
- Filter method (adapted and extended from Fletcher and Leyffer)
Hessian Calculation
- BFGS (full/LM and reduced space)
- SR1 (full/LM and reduced space)
- Exact full Hessian (direct)
- Exact reduced Hessian (direct)
- Preconditioned CG
Algorithmic PropertiesGlobally, superlinearlyconvergent (Wächter and B., 2005)
Easily tailored to different problem structures
Freely AvailableCPL License and COIN-OR distribution: http://www.coin-or.org
IPOPT 3.x recently rewritten in C++
Solved on thousands of test problems and applications
34
Comparison of NLP Solvers: Data Reconciliation(Poku, Kelly, B. (2004))
0.01
0.1
1
10
100
0 200 400 600
Degrees of Freedom
CP
U T
ime
(s,
norm
.) LANCELOT
MINOS
SNOPT
KNITRO
LOQO
IPOPT
0
200
400
600
800
1000
0 200 400 600Degrees of Freedom
Itera
tions
LANCELOT
MINOS
SNOPT
KNITRO
LOQO
IPOPT
18
35
Comparison of Computational Complexity(α ∈ [2, 3], β ∈ [1, 2], nw, nu - assume Nm = O(N))
((nu + nw)N)------Backsolve
((nu + nw)N)β(nu N)α(nu N)αStep Determination
---nw3 N---NLP Decomposition
N (nu + nw)(nw N) (nu + nw)2(nw N) (nu N)2Exact Hessian
N (nu + nw)(nw N) (nu + nw)(nw N) (nu N)Sensitivity
---nwβ Nnw
β NDAE Integration
SimultaneousMultiple Shooting
Single Shooting
O((nuN)α + N2nwnu
+ N3nwnu2)
O((nuN)α + N nw3
+ N nw (nw +nu)2)
O((nu + nw)N)β
36
Case Studies• Reactor - Based Flowsheets• Fed-Batch Penicillin Fermenter• Temperature Profiles for Batch Reactors• Parameter Estimation of Batch Data• Synthesis of Reactor Networks• Batch Crystallization Temperature Profiles• Ramping for Continuous Columns• Reflux Profiles for Batch Distillation and Column Design• Air Traffic Conflict Resolution• Satellite Trajectories in Astronautics• Batch Process Integration• Source Detection for Municipal Water Networks• Optimization of Simulated Moving Beds• Grade Transition of Polymerization Processes• Parameter Estimation of Tubular Reactors• Nonlinear MPC
Simultaneous DAE Optimization
19
37
Production of High Impact Polystyrene (HIPS)Startup and Transition Policies (Flores et al., 2005a)
Catalyst
Monomer, Transfer/Term. agents
Coolant
Polymer
38
Upper Steady−State
Bifurcation Parameter
System State
Lower Steady−State
Medium Steady−State
Phase Diagram of Steady States
Transitions considered among all steady state pairs
20
39
Upper Steady−State
Bifurcation Parameter
System State
Lower Steady−State
Medium Steady−State
Phase Diagram of Steady States
Transitions considered among all steady state pairs
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
300
350
400
450
500
550
600
N1
N2
N3
A1
A4
A2
A3
A5
Cooling water flowrate (L/s)
Te
mp
era
ture
(K
)
1: Qi = 0.00152: Qi = 0.00253: Qi = 0.0040
3
2
1
0 0.02 0.04 0.06 0.08 0.1 0.12
300
350
400
450
500
550
600
650
Initiator flowrate (L/s)
Te
mp
era
ture
(K
)
1: Qcw = 102: Qcw = 1.03: Qcw = 0.1
32 1
3
2
1
N2
B1 B2 B3
C1C2
40
0 0.5 1 1.5 20
0.5
1x 10
−3
Time [h]
Initi
ator
Con
c. [m
ol/l]
0 0.5 1 1.5 24
6
8
10
Time [h]
Mon
omer
Con
c. [m
ol/l]
0 0.5 1 1.5 2300
350
400
Time [h]
Rea
ctor
Tem
p. [
K]
0 0.5 1 1.5 2290
300
310
320
Time [h]
Jack
et T
emp.
[K
]
0 20 40 60 80 1000
0.5
1
1.5x 10
−3
Time [h]
Initi
ator
Flo
w. [
l/sec
]
0 0.5 1 1.5 20
0.5
1
Time [h]Coo
ling
wat
er F
low
. [l/s
ec]
0 0.5 1 1.5 20
1
2
3
Time [h]
Fee
drat
e F
low
. [l/s
ec]
• 926 variables• 476 constraints• 36 iters. / 0.95 CPU s (P4)
Startup to Unstable Steady State
21
41
HIPS Process Plant (Flores et al., 2005b)
•Many grade transitions considered with stable/unstable pairs
•1-6 CPU min (P4) with IPOPT
•Study shows benefit for sequence of grade changes to achieve wide range of grade transitions.
42
Simulated Moving Bed Optimization(Kawajiri, B., 2005-2007)
Direction of liquid flowand valve switching
Feed Raffinate
DesorbentExtract
22
43
Simulated Moving Bed Optimization(Kawajiri, B., 2005-2007)
Direction of liquid flowand valve switching
Feed
Raffinate
Desorbent
ExtractRepeats exactly
the same operation
(Symmetric)
44
Simulated Moving Bed Optimization(Kawajiri, B., 2005-2007)
Direction of liquid flowand valve switching
Repeats exactly
the same operation
(Symmetric)
Operating parameters:
4 Zone velocities
+
Step time
Zone 4 Zone 2
Zone 3
Zone 1
23
45
Formulation of Optimization Problem
Zone velocities Step time
(Maximize average feed velocity)
Bounds on liquid velocities
Product requirements
CSS constraintSMB model
46
Treatment of PDEs: Simultaneous Approach
t
x
(Orthogonal Collocation on Finite Elements)
k=1
k=2k=3
Algebraic equations PDE
Step size is determined a priori
Step size is determined a priori
tHuge number of variables (handled by optimizer)
C(xi,t)
24
47
Comparison of two approaches
CPU Time*
Sequential Approach 111.8 min
1.53 minSimultaneous Approach
# of iteration
49
47
0 1 2 3 4 5 6 7 8−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
x [m]
Nor
mal
ized
Con
cenr
atio
n, C
i(x,t st
ep)/
CF
,i
Comp.1 Single discretizationComp.2 Single discretizationComp.1 Full discretizationComp.2 Full discretization Sequential and Simultaneous
methods find same optimal solution
# of variables
33999
644Implemented on gPROMS, solved using SRQPDImplemented on gPROMS, solved using SRQPD
Implemented on AMPL, solved using IPOPTImplemented on AMPL, solved using IPOPT
*On Pentium IV 2.8GHz
(89% spent by integrator)
(Linear isotherm, fructose/glucose separation)
Initial feed velocity: 0.01 m/h
Optimal feed velocity: 0.52 m/h
Optimization
48
z Standard SMB
Nonstandard SMB: Addressed by Extended Superstructure NLP
z Three Zone
(Circulation loop is cut open)
z VARICOL
(Asynchronous switching)
25
49
0 2 4 6 80
0.5
1
x [m]
Ci(x
,t)/C
F,i →
8.000 m/h
⇓u
D1 =8.000m/h
t/tstep
= 0.142
→8.000 m/h
t/tstep
= 0.142
→8.000 m/h
⇓ uR3 =8.000m/h
t/tstep
= 0.142
→8.000 m/h
⇓u
D4 =8.000m/h
⇓ uE4=8.000m/h
t/tstep
= 0.142
0 2 4 6 80
0.5
1
x [m]
Ci(x
,t)/C
F,i →
7.916 m/h
⇓u
D1 =7.916m/h
t/tstep
= 0.334
→7.916 m/h
t/tstep
= 0.334
→7.916 m/h
⇓ uR3 =7.916m/h
t/tstep
= 0.334
→8.000 m/h
⇓u
D4 =8.000m/h
⇓ uE4=8.000m/h
t/tstep
= 0.334
0 2 4 6 80
0.5
1
x [m]
Ci(x
,t)/C
F,i →
8.000 m/h
⇓u
D1 =8.000m/h
t/tstep
= 0.434
→8.000 m/h
t/tstep
= 0.434
→8.000 m/h
t/tstep
= 0.434
→8.000 m/h
⇓ uE4=8.000m/h
t/tstep
= 0.434
0 2 4 6 80
0.5
1
x [m]
Ci(x
,t)/C
F,i →
6.332 m/h
⇓u
D1 =6.332m/h
t/tstep
= 0.800
→6.332 m/h
t/tstep
= 0.800
→8.000 m/h
⇓u
F3=1.668m/h
t/tstep
= 0.800
→8.000 m/h
⇓ uE4=8.000m/h
t/tstep
= 0.800
0 2 4 6 80
0.5
1
x [m]
Ci(x
,t)/C
F,i →
8.000 m/h
⇓u
D1 =8.000m/h
⇓ uE1=3.757m/h
t/tstep
= 1.000
→4.243 m/h
t/tstep
= 1.000
→8.000 m/h
⇓u
F3=3.757m/h
t/tstep
= 1.000
→8.000 m/h
⇓ uR4 =8.000m/h
t/tstep
= 1.000
Optimal Operating Scheme:Result of Superstructure Optimization
58Computational Results – LDPE Reactor EVM Problem
Parameter Estimation in Parallel ArchitecturesParameter Estimation in Parallel Architectures(Zavala, Laird, B., 2007)
30
59
Supply Chain, Planning and Scheduling• Large LP and MILP models• Many Discrete Decisions• Few Nonlinearities• Essential link needed to process models• Decisions need to be feasible at lower levels
Planning
Scheduling
Site-wide Optimization
Real-time Optimization
Model Predictive Control
Regulatory Control
Fe
asi
bili
ty
Pe
rfo
rma
nc
e
Decision-making in Chemical Industries
60
Real-time Optimization and Advanced Process Control• Fewer discrete decisions• Many nonlinearities• Frequent, “on-line” time-critical solutions• Higher level decisions must be feasible• Performance communicated for higher level decisions
Method of Full Discretization of State and Control Variables
Large-scale Sparse block-diagonal NLP
11Difference (control variables)
141Number of algebraic equations
152Number of algebraic variables
30Number of differential equations
DAE Model
14700Number of nonzeros in Hessian
49230Number of nonzeros in Jacobian
540Number of upper bounds
780Number of lower bounds
10260Number of constraints
109200
Number of variablesof which are fixed
NLP Optimization problem
66
Case Study:Change Reactor pressure by 60 kPa
Control profiles
All profiles return to their base case values
Same production rate
Same product quality
Same control profile
Lower pressure – leads to larger gas phase (reactor) volume
Less compressor load
34
67
Case Study: Change Reactor Pressure by 60 kPa
Optimization with IPOPT
1000 Optimization Cycles
5-7 CPU seconds
11-14 Iterations
Optimization with SNOPT
Often failed due to poor conditioning
Could not be solved within sampling times
> 100 Iterations
68
Limitations to NMPC Implementation
Issues: time-critical, more complex models, fast NLP solvers.
Computational delay – between receipt of process measurement and injection of control, determined by cost of dynamic optimization
Leads to loss of performance and stability (see Findeisen and Allgöwer, 2004; Santos et al., 2001)
As larger As larger NLPsNLPs are considered for NMPC, can are considered for NMPC, can computational delay be overcome?computational delay be overcome?
35
69
Avoid computational delay due to on-line optimization?
Real-time Iteration• preparation, feedback response and transition stages
• solve perturbed (linearized) problem on-line
– Li, de Oliveira, Santos, B. (1990+) – Diehl, Findeisen, Bock, Allgöwer et al. (2000+)– > two orders of magnitude reduction in on-line computation
• solve complete NLP in background (‘between’ sampling times as part of preparation and transition stages
Based on NLP sensitivity for dynamic systems• Extended to Simultaneous Collocation approach – Zavala et al.
(2007)
• Develop Advanced Step NMPC
• Related to MPC with linearization constantly updated one step behind
70
Nonlinear Model Predictive Control Nonlinear Model Predictive Control ––Parametric Problem (Zavala, Laird, B.)Parametric Problem (Zavala, Laird, B.)
36
71
Nonlinear Model Predictive Control Nonlinear Model Predictive Control ––Parametric Problem (Zavala, Laird, B.)Parametric Problem (Zavala, Laird, B.)
NLP Sensitivity Æ Rely upon Existence and Differentiability of Path
Æ Main Idea: Obtain and find b y Taylor Series Expansion
Optimality Conditions
Solution Triplet
37
73
NLP SensitivityNLP Sensitivity
Optimality Conditions of
Obtaining
Æ Already Factored at Solution
Æ Sensitivity Calculation from Single Backsolve
Æ Approximate Solution Retains Active Set
KKT Matrix IPOPT
Apply Implicit Function Theorem to around
74
Key Concept Key Concept –– Relate to Previous HorizonRelate to Previous Horizon
Solutions to both problems are equivalent in nominal case
(ideal plant model, no disturbances)
38
75
Advanced Step NMPCAdvanced Step NMPCCombine advanced step with sensitivity to solve NLP in background
(between steps) – not on-line
Solve P(z ) in background (between t0 and t1)
υνλ ∆=
∆∆∆
−K
XV
A
IAW
kk
Tk
kk [
0
00
76
Advanced Step NMPCAdvanced Step NMPCCombine advanced step with sensitivity to solve NLP in background
(between steps) – not on-line
Solve P(z ) in background (between t0 and t1)
Sensitivity to updated problem to get (z0, u0)
υνλ ∆=
∆∆∆
−K
XV
A
IAW
kk
Tk
kk [
0
00
39
77
Advanced Step NMPCAdvanced Step NMPCCombine advanced step with sensitivity to solve NLP in background
(between steps) – not on-line
Solve P(z ) in background (between t0 and t1)
Sensitivity to updated problem to get (z0, u0)Solve P(z +1) in background with new (z0, u0)
υνλ ∆=
∆∆∆
−K
XV
A
IAW
kk
Tk
kk [
0
00
78
AS-NMPC Stability Analysis
Nominal NMPC stability proof
•Nominal case – no noise: perfect model•General formulation with local asymptotic controller for t Æ �•Advanced step controller satisfies same relations, has same input sequence
Æ shares identical stability property
klll
kkJkkk
kkkkkllk
xz)),u,z(fz
||)w(|| ||x||L ||))w,u,x(g||
))w,u,x(g))u,x(f))u,x(fx
==
+≤+==
+
+
01
1
σ
Robust Stability Margins
• Analysis similar to Limon, Alamo, Camacho (2004), Magni and Scattolini (2005)• Advanced step NMPC is ISS and tolerates some model mismatch• ISS property (Jiang and Wang, 2001; Magni and Scattolini, 2005) • Advanced step NMPC has smaller margin than Ideal NMPC,
Æ but can be implemented without computational delay
Plant
Model
40
79
CSTR NMPC Example (Hicks and Ray)
• Maintain unstable setpoint• Close to bound constraint• Final time constraint for stability
• NMPC applied with N = 10, τ = 0.5 sampling time• Stable (z = 0) and unstable (z = 0.1) steady states• u2* close to upper bound• Computational delay = 0.5, leads to instabilities
41
81
CSTR NMPC Example – Model Mismatch
Advanced Step NMPC not as robust as ideal - suboptimal selection of u(k)
Better than Direct Variant – due to better active set preservation
0 5 10 15 20 25 30 35 40 45 50
0
0.05
0.1 θ = θnom
− 55%
z c [−]
Time [s]
0 5 10 15 20 25 30 35 40 45 50
0
0.05
0.1 θ = θnom
− 50%
z c [−]
0 5 10 15 20 25 30 35 40 45 50
0
0.05
0.1 θ = θnom
− 40%z c [−
]
0 5 10 15 20 25 30 35 40 45 50
0
0.05
0.1 θ = θnom
− 25%
z c [−]
IdealAdvanced StepDirect
82
CSTR Example: Mismatch + Noise
0 10 20 30 40 50 60 70
0
0.05
0.1σ = 2.5%
θ = θnom
− 50%
z c [−]
IdealAdvanced StepDirect
0 10 20 30 40 50 60 70
0
0.05
0.1σ = 5.0%
z c [−]
0 10 20 30 40 50 60 70
0
0.05
0.1
0.15
σ = 7.5%
z c [−]
Time [s]
42
83
Industrial Case Study – Grade Transition Control
Simultaneous Collocation-BasedApproach
27,135 constraints, 9630 LB & UB
Off-line Solution with IPOPT
Feedback Every 6 min
Process Model: 289 ODEs, 100 AEs
84
NMPC Case Study� Optimal Feedback Policy Æ (On-line Computation 351 CPU s)
Ideal NMPC controller - computational delay not considered
Time delays as disturbances in NMPC
43
85
NMPC Case Study� Optimal Policy vs. NLP Sensitivity -Shifted Æ (On-line Computation 1.04 CPU s)
Very Fast Close-to-Optimal FeedbackLarge-Scale Rigorous Models
86
Moving Horizon Estimation
Large State Dimensionality
Degrees of Freedom
Linear Systems, No inequalities Æ Kalman Filter for State Estimation
Nonlinear Systems: Æ Extended Kalman Filters in Practice
Sequential Approaches – Use DAE Integrators- Parameter Optimization
• Gradients by: Direct (and Adjoint) Sensitivity Equations- Optimal Control (Profile Optimization)
• Variational Methods• NLP-Based Methods - Single and Multiple Shooting
- Require Repeated Solution of Model- State Constraints are Difficult to Handle
Simultaneous Collocation Approach- Discretize ODE's using orthogonal collocation on finite elements - Straightforward addition of state constraints.- Deals with unstable systems- Solve model only once- Avoid difficulties at intermediate points
Large-Scale Extensions- Exploit structure of DAE discretization through decomposition- Large problems solved efficiently with IPOPT
48
95
Summary: On-line ExtensionsRTO and MPC widely used for refineries, ethylene and, more recently, chemical plants
NMPC provides link for off-line and on-line optimization• Stability and robustness properties• Advanced step controller leads to very fast calculations
– Analogous stability and robustness properties– On-line cost is negligible
Multi-stage planning and on-line switches• Avoids conservative performance• Update model with MHE• Evolve from regulatory NMPC to Large-scale DRTO
96
Acknowledgements
Funding• Department of Energy • National Science Foundation• Center for Advanced Process Decision-making (CMU)
Research Colleagues• Antonio Flores-Tlacuahuac• Tobias Jockenhövel• Shivakumar Kameswaran• Yoshiaki Kawajiri• Carl Laird• Yi-dong Lang• Andreas Wächter• Victor Zavala
http://dynopt.cheme.cmu.edu
49
97
References – Dynamic Optimization
F. Allgöwer and A. Zheng (eds.), Nonlinear Model Predictive Control, Birkhaeuser, Basel (2000)
R. D. Bartusiak, “NLMPC: A platform for optimal control of feed- or product-flexible manufacturing,” in Nonlinear Model Predictive Control 05, Allgower, Findeisen, Biegler (eds.), Springer, to appear
Forbes, J. F. and Marlin, T. E.. Model Accuracy for Economic Optimizing Controllers: The Bias Update Case. Ind.Eng.Chem.Res. 33, 1919-1929. 1994
Forbes, J. F. and Marlin, T. E.. “Design Cost: A Systematic Approach to Technology Selection for Model-Based Real-Time Optimization Systems,” Computers Chem.Engng. 20[6/7], 717-734. 1996
M. Grötschel, S. Krumke, J. Rambau (eds.), Online Optimization of Large Systems, Springer, Berlin (2001)
K. Naidoo, J. Guiver, P. Turner, M. Keenan, M. Harmse “Experiences with Nonlinear MPC in Polymer Manufacturing,” in Nonlinear Model Predictive Control 05, Allgower, Findeisen, Biegler (eds.), Springer, to appear
Yip, W. S. and Marlin, T. E. “Multiple Data Sets for Model Updating in Real-Time Operations Optimization,” Computers Chem.Engng. 26[10], 1345-1362. 2002.
98
References – Recent DRTO Case Studies
Busch, J.; Oldenburg, J.; Santos, M.; Cruse, A.; Marquardt, W. Dynamicpredictive scheduling of operational strategies for continuous processes using mixed-logic dynamic optimization, Comput. Chem. Eng., 2007, 31, 574-587.
Flores-Tlacuahuac, A.; Grossmann, I.E. Simultaneous cyclic scheduling andcontrol of a multiproduct CSTR, Ind. Eng. Chem. Res., 2006, 27, 6698-6712.
Kadam, J.; Srinivasan, B., Bonvin, D., Marquardt, W. Optimal grade transition in industrial polymerization processes via NCO tracking. AIChE J., 2007, 53, 3, 627-639.
Oldenburg, J.; Marquardt, W.; Heinz D.; Leineweber, D. B., Mixed-logic dynamic optimization applied to batch distillation process design, AIChE J. 2003, 48(11), 2900- 2917.
E Perea, B E Ydstie and I E Grossmann, A model predictive control strategy for supply chain optimization, Comput. Chem. Eng., 2003, 27, 1201-1218.
M. Liepelt, K Schittkowski, Optimal control of distributed systems with breakpoints, p. 271 in M. Grötschel, S. Krumke, J. Rambau (eds.), Online Optimization of Large Systems, Springer, Berlin (2001)