Transcript Carnegie Mellon

Discrete and Continuous Optimization Models
for the Design and Operation of
Sustainable and Robust Process Systems
Ignacio E. Grossmann
Center of Advanced Process Decision-making
Carnegie Mellon University
Pittsburgh, PA 15217
U.S.A.
BUAP, Puebla
Enero 13, 2012
1
Carnegie Mellon
Motivation
1. Increasing interest in energy systems and supply chains
2. Need to address design of sustainable chemical processes
- Minimize energy use
- Minimize water consumption
3. Need to introduce robustness to account for uncertainties
Goal: Systematic Optimization Approaches for Sustainable and Robust
Optimization Process Design and PlanningOperations Problems
Challenges: Nonconvexities in MINLP/GDP models
Large-scale stochastic optimization problems
2
Carnegie Mellon
Water scarcity
Two-thirds of the world population will face water stress by year 2025
Carnegie Mellon
3
Conventional
System
Raw Water
Raw Water
Treatment
Water-using
unit 1
Freshwater
Storm
Water
Wastewater
Water-using
unit 2
Water-using
unit 3
Boiler Feedwater
treatment
Condensate
Steam Losses
System
Steam
Boiler
Boiler Blowdown
Wastewater
Water Loss by Evaporation
Cooling
Tower
Cooling Tower Blowdown
Other Uses
(Housekeeping)
Wastewater
Wastewater
Treatment
4
Carnegie Mellon
Discharge
Synthesis of Integrated Process Water Networks
• Given is:
– a set of single/multiple water sources with/without contaminants,
– a set of water-using, water pre-treatment, and wastewater
treatment operations, sinks and sources of water
• Synthesize an integrated process water network
–
–
–
interconnection of process and treatment units (reuse, recycle)
the flow rates and contaminants concentration of each stream
minimum total annual cost of water network
•
Synthesis Integrated Process Water Networks
Pinch analysis and mathematical programming models
•
Reviews in Bagajewicz (2000), Ježowski (2008), Bagajewicz and Faria (2009), and Foo (2009).
Approach: Global NLP or MINLP superstructure optimization model
5
Carnegie Mellon
Superstructure for water networks for water reuse,
recycle, treatment, and with sinks/sources water
Ahmetovic, Grossmann (2010)
Freshwater
Process Unit
Sink
Main features:
- Multiple feeds
- Source/Sink units
- Local recycles
- All possible
interconnections
-Fixed and variable flows
through process units
Source
6
Carnegie Mellon
Treatment Unit
Optimization Model
Nonconvex NLP or MINLP
Objective function:
min Cost
Subject to:
Splitter mass balances
Mixer mass balances (bilinear)
Process units mass balances
Treatment units mass balances
Design constraints
0-1 variables for piping sections
Model can be solved to global optimality
Carnegie Mellon
7
Splitter
FWs  FIFs 
linear
 FIP

s, p
pPU
 FID
s ,d
d DU

 FIT
FIFsL  yFIFs  FIFs  FIFsU  yFIFs
s  SW
FIPsL, p  yFIPs , p  FIPs , p  FIPsU, p  yFIPs , p
0-1
optional
s  SW , p  PU
FIDsL,d  yFIDs ,d  FIDs ,d  FIDsU,d  yFIDs ,d
FITsL,t  yFITs ,t  FITs ,t  FITsU,t  yFITs ,t
s  SW
s ,t
tTU
s  SW , d  DU
s  SW , t  TU
Mixer
FPU inp 
 FSP
rSU
r, p
FPU inp  xPU inp , j 
bilinear

 FTP
t, p
tTU
 FSP
rSU

r, p
 FIP
p ', p
s, p
sSW
 xSUrout
,j 
 FP
p 'PU
p  p ', R p 0

 xSPU
 FP

p 'PU
p  p ', R p 0
 FTP
tTU
out
p ', j

t, p

 FP
p ', p
p ', p
p 'PU
R p 1
 xSTUtout
,j 
 FP
p 'PU
R p 1
p ', p
 xSPU
,
p  PU
 FIP  xW
sSW
out
p ', j
,
in
s, j
p  PU , j
Process unit
FPU inp  FPU pout
p  PU
linear if FPU inp  xPUinp, j  LPU p, j 103  FPU pout  xPU pout, j
flowrate is fixed
p  PU , j
bilinear if the flow treated as cont. variable
8
Carnegie Mellon
Cost function linear in feedwater, concave in treatment unit,
linear in operating cost, pipe section fixed charge (0-1)
min Z  H 

out
FW

CFW

AR

IC

FTU
 s
 t
s
t
sSW
tTU


 H   OCt  FTU tout
tTU
9
Carnegie Mellon
Convexification of Non-convex functions
Convex Envelopes for Bilinear Terms F*C
Bilinear term
f ji  F iLC ij  C iLj F i  F iLC iLj
CU
Overestimators
McCormick (1976 )
i
iU iU
f ji  F iU C ij  C iU
j F  F Cj
i
iL iU
f ji  F iLC ij  C iU
j F  F Cj
C
f  F C C F F C
i
j
iU
i
j
iL
j
i
iU
Under- and over-estimators
( Linear Inequalities )
iL
j
CL
FL
FiL ≤ Fi ≤ FiU
FU
CjiL ≤ Cji ≤ CjiU
F
Underestimators
Underestimation of Concave functions
Concave term

Fi  
 
 
(FU)
F

 

F    F F   FF    F
iU
iL
iL

iU
iL

i
 F iL

( Secant line )


Underestimator
(FL)
FL
F
FiL ≤ Fi ≤ FiU
FU
10
Carnegie Mellon
•The cut proposed by Karuppiah and Grossmann (2006) is incorporated to
significantly improve the strength of the lower bound for the global optimum:
contaminant flow balances for the overall water network system
 FW  xW
sSW
s
 F out  x out
j 
in
s, j
 LPU

pPU
 FDU
d DU
in
d
p, j
103 
 xDU din, j
 FSU
rSU
out
r
 xSUrout
,j 
 (1  
tTU
TU t , j
)  FTU tin  xTUtin, j
j
bilinear terms for the treatment units
and final mixing points
Cut is redundant for original problem
Non-redundant for relaxation problem
•Tight bounds on the variables are expressed as general equations
obtained by physical inspection of the superstructure and using logic specifications
11
Carnegie Mellon
Superstructure of the integrated water network
1 feed, 5 process units, 3 treatment units, 3 contaminants
Carnegie Mellon
MINLP: 72 0-1 vars, 233 cont var, 251 constr
BARON
optcr=0.01
197.5 CPUsec
12
Optimal design of the simplified water network
with 13 removable connections
Optimal Freshwater
Consumption
40 t/h
vs
300 t/h
conventional
13
Carnegie Mellon
Biomass emerging as important renewable
US Energy Sources
14
Carnegie Mellon
Process Design Challenges in Bioethanol
Energy consumption corn-based process level:
Author (year)
Energy consumption
(Btu/gal)
Pimentel (2001)
75,118
Keeney and DeLuca
(1992)
48,470
Wang et al. (1999)
40,850
Shapouri et al. (2002)
51,779
Wang et al (2007)
38,323
Water consumption corn based - process level:
Author (year)
Carnegie Mellon
Water consumption
( gal/gal ethanol)
Gallager (2005) First
plants
11
Philips (1998)
5.8
MATP (2008)
Old plants in 2006
4.6
MATP (2008)
New plants
3.4
15
Proposed Design Strategy
for Energy and Water Optimization
Energy optimization
Issue: fermentation reactions at modest temperatures
=> No source of heat at high temperature as in petrochemicals
Multieffect distillation followed by heat integration process streams
Water optimization
Issue: cost contribution is currently still very small
(freshwater contribution < 0. 1%)
=> Total cost optimization is unlikely to promote water conservation
Optimal process water networks for minimum energy consumption
16
Carnegie Mellon
Energy Optimization of Corn-based Bioethanol
Peschel, Martin, Karuppiah, Grossmann, Zullo, Martinson (2007)
60 M gallon /yr plant
Equipment cost = M$ 18.4
Steam cost = M$ 21/yr
Prod. cost = 1.50 $/gal
17
Carnegie Mellon
Alternatives for Energy Reduction
Heat Integration process streams:
Multieffect columns:
Low Pressure
column
High Pressure
column
GDP model comprises mass, energy balances, design equations (short cut)
2,922 variables (2 Boolean) 2,231 constraints
18
Carnegie Mellon
Energy Optimal Design
60 M gallon /yr plant
Superheated steam
Washing water
Src2
Src4
Enzyme
Src5
Src6
HX1
Feedstock
Src1
Enzyme
Wash1
Grind1
HX2
Premix1
Mix2
Col1
Jet1
Liq1
Sac1
HX3
To discharge/ re-use
Vent gas
CO2, O2
Fer1
Snk1
VOC removal
Snk9
Storage tank
Yeast, Urea, Water
Str1
Mix3
Spl7
Src7
95%
Ethanol
Cond2
Str2
Dry DDGS
Snk8
HX10
Dry1
MecP1
Proteins
Storage tank
Rec1
Mix4
Spl4
10.8%
Ethanol
Solids
Spl1
72%
Ethanol
Snk2
Adsorbant:
Corn grits
Flot
1
BC1
HX6
100%
Ethanol
Src9
97.7%
Ethanol
Ethanol
HX4
Spl2
Spl3
Mix5
Ads1
Spl5
Mix7
Snk5
Cond1
Water
Spl6
Heat Integration and
Multieffect Columns
HX11
Snk3
Adsorbant
HX8
MS1
Mix6
Snk7
MS2
Src8
Humid air
Snk6
HX9
Water
Snk4
Biogas
Equipment cost = M$ 20.7 Steam cost = M$ 7.1/yr (-66%)
Prod. cost = 1.28 $/gal
Reduction from $1.50/gal (base case) to $1.28/gal !
Carnegie Mellon
Dry air
WWT1
HX7
Ethanol losses : 0.5%
HX5
19
Energy Profiles in Multieffect Columns
Single column
Beer Column
Triple effect column
Rectification Column
Single column
24,918 Btu/gal vs 38,323 Btu/gal
Carnegie Mellon
Double effect column
20
Remarks
Current ethanol from corn and sugar cane and biodiesel from vegetable oils
compete with the food chain.
U.S. Government policies support the production of lignocellulosic based
biofuels and the reuse of wastes and new sources (algae)
Carnegie Mellon
21
21
Lignocellulosic Bioethanol
a) Thermochemical Process (gasification)
Gasification
Gas
clean-up
Fermentation
or Catalytic
Feed
Ethanol
Recovery
Power-Heat
Electricity
Wastewater
Power-Heat
Electricity
Ethanol
b) Hydrolysis Process (fermentation)
Biomass
Pretreatment
Cellulosic
Hydrolysis
Sugar
Fermentation
Feed
Ethanol
Recovery
Ethanol
Challenge:
Many alternative flowsheets
22
Carnegie Mellon
Superstructure Thermochemical Bioethanol
Ethanol via gasification
Martin, Grossmann (2010)
CO/H2 Adj.
Gasification
Reforming
Sour gases removal
Clean up
Process Design Alternatives:
Gasification
Indirect Low pressure
Direct high Pressure
Reforming.
Steam reforming
Partial oxidation
Fermentation
CO/H2 adjustment
WGSR
Bypass
Membrane/PSA
Sour gases removal:
MEA
PSA
Membrane
Synthesis
Fermentation
Rectification
Adsorption Corn grits
Molecular sieves
Pervaporation
Catalytic
Direct Sequence
Indirect sequence
Catalysis
23
Carnegie Mellon
-Martin, M. Grossmann, I. E
(2010) Aiche J. Submitted
Solution Strategy Energy Optimization
Superstructure
Problem
Gasifier 2
Gasifier 1
Partial
oxidation
Catalysis
(A)
Partial
oxidation
Steam
reforming
Fermentation
Catalysis
Fermentation
(B)
(C)
(D)
Catalysis Fermentation
(E)
(F)
Steam
reforming
Catalysis
Fermentation
(G)
(H)
Subproblem
Decomposition of GDP in 8 subproblems
Decision levels: Gasifier
Removal HCs
Reaction of Syn Gas
Heat integration and economic evaluation
Carnegie Mellon
24
Optimal Design of Lignocellulosic Ethanol Plant
$67.5 Million/yr
1,996 Btu/gal (< 1/10th of corn!)
Ethanol: $0.81 /gal (no H2 credits)
$ 0.42/gal (H2 credits)
Each NLP subproblem: 7000 eqs., 8000 var
~25 min to solve
Low cost is due to H2 production
25
Carnegie Mellon
Optimal Water Network: Corn Ethanol
Gal. Water/Gal. Ethanol = 1.5
1.5 vs 3.4
Washing
Solids removal
Fermentor
Freshwater
Organics removal
Discharge
Dist. Colums.
TDS removal
26
Carnegie Mellon
-Ahmetović , E., Martin, M. Grossmann, (2009) I&ECR. 2010, 49, 7972–7982
Optimal Water Network: Lignocellulosic Ethanol
Cellulosic Bioethanol via Gasification
Gal. Water/Gal. Ethanol = 4.2
Gasifier
Washing
Solids removal
Freshwater
Wastewater
Organics removal
27
Carnegie Mellon
-Martin, M. , Ahmetović, E., Grossmann, I. E (2010) I&ECR ASAP
Table Summary of results [6-10]
A
C
D
B
F
E
(*) kg instead of gal
[6] Martín, M., Grossmann, I.E. (2011) AIChE J. DOI: 10.1002/aic.12544
[7] Martín, M., Grossmann, I.E. Energy optimization of Hydrogen production from biomass. Rev. Submited to Comp. Chem. Eng.
[8] Martín, M., Grossmann, I.E. Energy optimization of lignocellulosic bioethanol production via Hydrolysis to be submitted AIChE J.
[9] Martín, M., Grossmann, I.E. Process optimization of FT- Diesel production from biomass. To be submitted
[10] Martín, M., Grossmann, I.E. Process optimization bioDiesel production from cooking oil and Algae. To be submitted
28
Carnegie Mellon
Design and Planning under Uncertainty
Goal: robustness in decisions
Design of Responsive Supply Chains
Uncertain demands
Maximize NPV/Minimize responsiveness
Chance constrained MINLP
Design and Planning of Offshore Oilfields
Uncertain fields size, deliverability, water
Maximize expected flexibility/Minimize Cost
Multi-stage programming MINLP
29
Carnegie Mellon
Background
Optimal Design of Responsive Process Supply Chains
Fengqi You
Objective: design supply chains under responsive and economic criteria
with consideration of inventory management and demand uncertainty
Background
Problem Statement
Where?
What?
When?
Suppliers
Plants
DCs
Production Network
Customers
Network Structure
Costs and prices
Operational Plan
Production and
transportation time
Target Demand
Demand information
Max: Net present value
Max: Responsiveness
Safety Stock
Production Schedule
Example
Production Network of Polystyrene Resins
Three types of plants:
Plant I: Ethylene + Benzene
Styrene (1 products)
Plant II: Styrene
Solid Polystyrene (SPS) (3 products)
Plant III: Styrene
Expandable Polystyrene (EPS) (2 products)
Basic Production Network
SPS - 1
II
Ethylene
Styrene
I
Benzene
Single Product
SPS - 2
Multi Product SPS - 3
III
EPS - 1
EPS - 2
Multi Product
Source: Data Courtesy Nova Chemical Inc. http://www.novachem.com/
Example
Location Map
Possible Plant Site
Supplier Location
Distribution Center
Customer Location
Example
Potential Network Superstructure
WA
IL
NV
Plant Site MI
Ethylene
I
TX
II
Styrene
III
Benzene
EPS
II
TX
CA
SPS
TX
SPS
AZ
OK
Ethylene
I
Benzene
MS
Styrene Styrene
Plant Site TX
GA
III
NC
EPS
Plant Site CA
FL
Ethylene
I
LA
PA
Styrene
Benzene
Plant Site LA
OH
III
EPS
IA
MA
AL
MN
Suppliers
Plant Sites
Distribution Centers
Customers
Model & Algorithm
Responsiveness - Lead Time
 Lead Time: The time of a supply chain network to respond to customer
demands and preferences in the worst case
Responsiveness
Lead Time
Lead Time is a measure of responsiveness in SCs
Model & Algorithm
Lead Time for A Linear Supply Chain
• A supply chain network = ∑Linear supply chains
 Assume information transfer instantaneously
Suppliers
Plants
Distribution Centers Customers
Information
Supplier ls
Plant i1 site k1
Plant i2 site k2
Plant i3 site k3
Distribution Center m
Customer ld
Model & Algorithm
Lead Time under Demand Uncertainty
Transporation
Transporation
Supplier ls
Plant i0 site k0
…
Transporation
Transporation
Plant in site kn
Production Lead Time (LP)
Distribution Center m
Customer ld
Delivery Lead Time (LD)
Inventory (Safety Stock)
Example
Expected Lead Time of SCN
•
Expected Lead time of a supply chain network (uncertain demand)
 The longest expected lead time for all the paths in the network (worst case)
 Example: A simple SC with all process are dedicated
P1=20%
1.2
1.8
1.5
Supplier
2
I
Plant Site 1
0.5
0.7
DC 1
II
Plant Site 2
2.1 days
Path 2
2.0 days
Customer 1
0.2
P2=20%
2.6
1.2
III
Path 1
0.5
DC 2
Plant Site 3
Customer 2
P
For Path 1: (2 + 1.5 + 0.5 + 1.2 + 1.8)×20% + 0.7 = 2.1 days
For Path 2: (2 + 1.5 + 0.2 + 2.6 + 1.2)×20% + 0.5 = 2.0 days
Expected Lead Time = max {2.1, 2.0} = 2.1 days
Safety Stock
Model & Algorithm
Bi-criterion Multiperiod MINLP Formulation
Choose Discrete (0-1), continuous variables
• Objective Function:
 Max: Net Present Value
 Min: Expected Lead time
Bi-criterion
• Constraints:
 Network structure constraints
Suppliers – plant sites Relationship
Plant sites – Distribution Center
 Cyclic scheduling constraints
Input and output relationship of a plant
Assignment constraint
Distribution Center – Customers
Sequence constraint
Cost constraint
Demand constraint
 Operation planning constraints
Production constraint
Capacity constraint
Production constraint
Cost constraint
 Probabilistic constraints
Target Demand
Mass balance constraint
Chance constraint for stock out
Demand constraint
(reformulations)
Upper bound constraint
dL
dM
Safety Stock
dU
Example
Case Study
WA
IL
NV
Plant Site MI
Ethylene
I
TX
II
Styrene
III
Benzene
EPS
II
TX
CA
SPS
TX
SPS
AZ
OK
Ethylene
I
Benzene
Styrene Styrene
Plant Site TX
MS
GA
III
NC
EPS
Plant Site CA
FL
Ethylene
I
LA
PA
Styrene
Benzene
Plant Site LA
OH
III
EPS
IA
MA
Distribution Centers
Customers
AL
Possible Plant Site
Supplier Location
•
Distribution Center
Customer Location
Problem Size:
 # of Discrete Variables: 215
 # of Continuous Variables: 8126
 # of Constraints: 14617
MN
Suppliers
•
Plant Sites
Solution Time:
 Solver: GAMS/BARON
 Direct Solution: > 2 weeks
 Proposed Algorithm: ~ 4 hours
Example
Pareto Curves – with and without safety stock
750
Best Choice
700
650
NPV (M$)
600
550
More Responsive
500
450
400
with safety stock
without safety stock
350
300
1.5
2
2.5
3
3.5
4
Expected Lead Time (day)
4.5
5
5.5
Example
Safety Stock Levels - Expected Lead Time
Safety Stock (10^4 T)
200
EPS in DC2
SPS in DC2
EPS in DC1
SPS in DC1
150
More inventory,
more responsive
100
50
0
Responsiveness
1.51
2.17
2.83
3.48
Expceted Lead Time (day)
4.14
4.8
Design of Responsive Chemical Supply Chains under Uncertainty
Network Structure at Location Map
43
Optimal Development Planning under Uncertainty
Offshore oilfield having several reservoirs under uncertainty
Tarhan, Grossmann (2009)
Maximize the expected net present value (ENPV) of the project
facilities
Decisions:
 Number and capacity of TLP/FPSO facilities
 Installation schedule for facilities
 Number of sub-sea/TLP wells to drill
 Oil production profile over time
TLP
FPSO
wells
Reservoirs
Uncertainty:
Initial productivity per well
Size
of reservoirs
Water breakthrough time
for reservoirs
44
Carnegie Mellon
Non-linear Reservoir Model
Single Well Oil and Water Rate (kbd)
Initial oil
production
Assumption: All wells in the same reservoir are identical.
Unconstrained
Maximum Oil Production
Water Rate
Tank Cumulative Oil (MBO)
Size of the reservoir
Uncertainty is represented by discrete distributions functions
45
Carnegie Mellon
Decision Dependent Scenario Trees
Assumption: Uncertainty in a field resolved as soon as WP installed at field
Invest in F in year 1
Invest in F in year 5
Invest in F
Size of F:
H
M
L
Invest in F
Size of F:
H
M
L
Scenario tree
Not unique: Depends on timing of investment at uncertain fields
Central to defining a Stochastic Programming Model
Carnegie Mellon
46
Stochastic Programming
ξ1=1
p=0.5
scenario
tree
ξ2=1
p=0.5
ξ2=2
p=0.5
ξ1=2
p=0.5
ξ2=1
p=0.5
t=1
ξ2=2
p=0.5
t=2
t=3
1
2
3
4
Alternative and equivalent scenario tree structure (Ruszczynski, 1997):
t=1
ξ1=1
p=0.25
ξ1=1
p=0.25
ξ1=2
p=0.25
ξ1=2
p=0.25
t=2
ξ2=1
p=1.00
ξ2=2
p=1.00
ξ2=1
p=1.00
ξ2=2
p=1.00
t=3
1
2
3
4
47
Carnegie Mellon
Stochastic Programming
ξ1=1
p=0.5
ξ2=1
p=0.5
ξ2=2
p=0.5
ξ1=2
p=0.5
ξ2=1
p=0.5
t=1
ξ2=2
p=0.5
t=2
t=3
1
2
3
4
Each scenario is represented by a set of unique nodes
t=1
ξ1=1
p=0.25
ξ1=1
p=0.25
ξ1=2
p=0.25
ξ1=2
p=0.25
t=2
ξ2=1
p=1.00
ξ2=2
p=1.00
ξ2=1
p=1.00
ξ2=2
p=1.00
t=3
1
2
3
4
48
Carnegie Mellon
Stochastic Programming
ξ1=1
p=0.5
ξ2=1
p=0.5
ξ2=2
p=0.5
ξ1=2
p=0.5
t=1
ξ2=1
p=0.5
ξ2=2
p=0.5
t=2
t=3
1
2
3
4

Nodes have same amount of information
Nodes are indistinguishable
t=1
ξ1=1
p=0.25
ξ1=1
p=0.25
ξ1=2
p=0.25
ξ1=2
p=0.25
t=2
ξ2=1
p=1.00
ξ2=2
p=1.00
ξ2=1
p=1.00
ξ2=2
p=1.00
t=3
1
2
3
Non-anticipativity constraints
Carnegie Mellon
4
49
Representation of Decision-Dependence Using Scenario Tree
t=1
t=2
t=3
t=4
1
2
3
4
1
2
3
4
1
2
3
4
t=1
t=2
t=3
t=4
1
2
3
4
50
Carnegie Mellon
Multi-stage Stochastic Nonconvex MINLP
Maximize.. Probability weighted average of NPV over uncertainty scenarios
subject to
 Equations about economics of the model
 Surface constraints
 Non-linear equations related to reservoir performance
 Logic constraints relating decisions
if there is a TLP available, a TLP well can be drilled
Non-anticipativity constraints
Non-anticipativity prevents a decision being taken now from
using information that will only become available in the future
Every
scenario,
time period
Every pair
scenarios,
time period
Disjunctions (conditional constraints)
Problem size MINLP increases
exponentially with number of time periods
and scenarios
Decomposition algorithm:
Lagrangean relaxation &
Branch and Bound
MILP Branch and cut: Colvin, Maravelias (2008)
Carnegie Mellon
51
Formulation of Lagrangean dual
Relaxation
• Relax disjunctions, logic
constraints
• Penalty for equality
constraints
52
Carnegie Mellon
One Reservoir Example
Optimize the planning decisions for an oilfield having single reservoir for 10 years.
Decisions:
Number, capacity and installation schedule of FPSO/TLP facilities
Number and drilling schedule of sub-sea/TLP wells
Oil production profile over time
Scenarios
Uncertain Parameters
(Discrete Values)
1
2
3
4
5
6
7
8
Initial Productivity per well (kbd)
10
10
20
20
10
10
20
20
Reservoir Size (Mbbl)
300
300
300
300
1500
1500
1500
1500
Water Breakthrough Time Parameter
5
2
5
2
5
2
5
2
Wells are drilled in groups of 3.
Maximum number of 12 sub-sea wells per year can be drilled.
Maximum of 6 TLP wells per year per TLP facility can be drilled.
Maximum of 30 TLP wells can be connected to a TLP facility.
Construction
Lead Time
(years)
Wells
Facilities
TLP
Sub-sea
TLP
Small FPSO
Large FPSO
1
1
1
2
4
53
Carnegie Mellon
Multistage Stochastic Programming Approach
RS: Reservoir size
IP: Initial Productivity
BP: Breakthrough Parameter
E[NPV] = $4.92 x 109
Build 2 small FPSO’s
Drill 12 sub-sea wells
12 subsea wells
Low RS
Low IP
Low RS
High IP
High RS
Mean RS
High RS
High IP
Mean IP
Low IP
2 small FPSO’s,
4 small FPSO’s,
2 TLP’s
5 TLP’s
3 subsea wells
12 subsea wells
year 1
5 small FPSO’s, year 2
3 TLP’s
Solution proposes building 2 small FPSO’s in the first year and then add
new facilities / drill wells (recourse action) depending on the positive or negative outcomes.
54
Carnegie Mellon
Multistage Stochastic Programming Approach
RS: Reservoir size
IP: Initial Productivity
BP: Breakthrough Parameter
E[NPV] = $4.92 x 109
Build 2 small FPSO’s
Drill 12 sub-sea wells
12 subsea wells
Low RS
Low IP
year 1
High RS
Mean RS
High RS
High IP
Mean IP
Low IP
2 small FPSO’s,
4 small FPSO’s,
2 TLP’s
5 TLP’s
3 subsea wells
12 subsea wells
Low RS
High IP
5 small FPSO’s, year 2
3 TLP’s
12 subsea wells
6 subsea wells,
12 TLP wells
6 subsea wells
High BP
Low BP
High BP Low BP
Mean BP
12 subsea wells,
30 TLP wells
High BP Low BP
6 subsea wells,
18 TLP wells
High BP
year 3
Low BP
year 4
1
2
3
4
5
6
7
8
9
Solution proposes building 2 small FPSO’s in the first year and then add
new facilities / drill wells (recourse action) depending on the positive or negative outcomes.
55
Carnegie Mellon
Distribution of Net Present Value
10
Net Present Value ($ x 10 9)
8
6
4
2
0
1
2
3
4
5
6
7
8
9
-2
Sce narios
Deterministic Mean Value = $4.38 x 109
Multistage Stoch Progr = $4.92 x 109 => 12% higher and more robust
Computation: Algorithm 1: 120 hrs; Algorithm 2: 5.2 hrs
Nonconvex MINLP: 1400 discrete vars, 970 cont vars, 8090 Constraints
56
Carnegie Mellon
Conclusions
1. Effective solution of nonconvex MINLP and GDP requires
tight lower bounds
Global optimization optimal water networks
2. Energy and water optimization yields sustainable designs of
biofuel plants
Optimization predicts lower energy and water targets
3. Robustness can be effectively introduced with stochastic
programming
Design of responsive supply chains, Multistage stochastic in oilfields
57
Carnegie Mellon