discrete and continuous optimization models for the design

1

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.

OSE, Abo Akademi, Turku December 8, 2011

2

Carnegie Mellon

Motivation

2. Need to address design of sustainable chemical processes - Minimize energy use - Minimize water consumption

3. Need to introduce robustness to account for uncertainties

1. Increasing interest in energy systems and supply chains

Challenges: Nonconvexities in MINLP/GDP models Large-scale stochastic optimization problems

Goal: Systematic Optimization Approaches for Sustainable and Robust Optimization Process Design and PlanningOperations Problems

3

Carnegie Mellon

Water scarcity

Two-thirds of the world population will face water stress by year 2025

4

Carnegie Mellon

Water-using unit 1

Water-using unit 2

Water-using unit 3

Raw Water Raw Water Treatment

Boiler Feedwater treatment

Steam System

Wastewater Boiler Blowdown

Steam Condensate Losses

Boiler

Freshwater Wastewater

Cooling Tower

Cooling Tower Blowdown

Water Loss by Evaporation

Discharge

Wastewater Treatment

Storm Water

Other Uses (Housekeeping)

Wastewater

Conventional System

5

Carnegie Mellon

• 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

Approach: Global NLP or MINLP superstructure optimization model

Synthesis of Integrated Process Water Networks

Synthesis Integrated Process Water Networks • Pinch analysis and mathematical programming models • Reviews in Bagajewicz (2000), Ježowski (2008), Bagajewicz and Faria (2009), and Foo (2009).

6

Carnegie Mellon

Superstructure for water networks for water reuse, recycle, treatment, and with sinks/sources water

Freshwater

Process Unit

Treatment Unit

Sink

Source

Ahmetovic, Grossmann (2010)

Main features: - Multiple feeds - Source/Sink units - Local recycles - All possible interconnections -Fixed and variable flows through process units

7

Carnegie Mellon

Objective function: min Cost

Subject to: Splitter mass balances Mixer mass balances (bilinear) Process units mass balances Treatment units mass balances Design constraints

Nonconvex NLP or MINLP

Optimization Model

0-1 variables for piping sections

Model can be solved to global optimality

8

Carnegie Mellon

SWsFITFIDFIPFIFFWTUt

ts

DUd

ds

PUp

psss

,,,

Splitter

linear

SWsyFIFFIFyFIFss FIF

U

ssFIF

L

s

PUpSWsyFIPFIPyFIPpsps FIP

U

pspsFIP

L

ps ,,, ,,,

DUdSWsyFIDFIDyFIDdsds FID

U

dsdsFID

L

ds ,,, ,,,

TUtSWsyFITFITyFITtsts FIT

U

tstsFIT

L

ts ,,, ,,,

0-1 optional

bilinear if the flow treated as cont. variable

PUpFPFPFIPFTPFSPFPU

pp RPUp

pp

RppPUp

pp

SWs

ps

TUt

pt

SUr

pr

in

p

,1

','

0,''

,',,,

jPUpxSPUFPxSPUFP

xWFIPxSTUFTPxSUFSPxPUFPU

out

jp

RPUp

pp

out

jp

RppPUp

pp

in

js

SWsTUt

out

jtpt

out

jr

SUr

pr

in

jp

in

p

pp

,,,'

1'

,','

0,''

,'

,,,,,,

Mixer

bilinear

PUpFPUFPU out

p

in

p

jPUpxPUFPULPUxPUFPU out

jp

out

pjp

in

jp

in

p ,10 ,3

,,linear if flowrate is fixed

Process unit

9

Carnegie Mellon

TUt

out

tt

TUt

out

tts

SWs

s FTUOCHFTUICARCFWFWHZmin

Cost function linear in feedwater, concave in treatment unit, linear in operating cost, pipe section fixed charge (0-1)

10

Carnegie Mellon

Convexification of Non-convex functions

iL

j

iUiiL

j

i

j

iUi

j

iU

j

iLiiU

j

i

j

iLi

j

iU

j

iUiiU

j

i

j

iUi

j

iL

j

iLiiL

j

i

j

iLi

j

CFFCCFf

CFFCCFf

CFFCCFf

CFFCCFf

McCormick (1976 )

Under- and over-estimators ( Linear Inequalities )

FiL ≤ Fi ≤ FiU CjiL ≤ Cj

i ≤ CjiU

C

CL

FL

CU

FU

F

Underestimators

Overestimators

Bilinear term

Convex Envelopes for Bilinear Terms F*C

Underestimation of Concave functions

F

(FL)

FL

(FU)

FU

F

Underestimator

Concave term

iLi

iLiU

iLiUiLi

FFFF

FFFF

FiL ≤ Fi ≤ FiU

( Secant line )

11

Carnegie Mellon

bilinear terms for the treatment units and final mixing points

jxDUFDUxF

xTUFTUxSUFSULPUxWFW

in

jd

DUd

in

d

out

j

out

in

jt

TUt

in

tjtTU

out

jr

SUr

out

r

PUp

jp

in

js

SWs

s

,

,,,3

,, )1(10

Cut is redundant for original problem Non-redundant for relaxation problem

•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

•Tight bounds on the variables are expressed as general equations obtained by physical inspection of the superstructure and using logic specifications

12

Carnegie Mellon

Superstructure of the integrated water network

MINLP: 72 0-1 vars, 233 cont var, 251 constr BARON optcr=0.01 197.5 CPUsec

1 feed, 5 process units, 3 treatment units, 3 contaminants

13

Carnegie Mellon

Optimal design of the simplified water network with 13 removable connections

Optimal Freshwater Consumption

40 t/h vs

300 t/h conventional

14

Carnegie Mellon

US Energy Sources

Biomass emerging as important renewable

15

Carnegie Mellon

Process Design Challenges in Bioethanol Energy consumption corn-based process level:

Water 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

Author (year) 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

16

Carnegie Mellon

Proposed Design Strategy for Energy and Water Optimization

Energy optimization Issue: fermentation reactions at modest temperatures

Multieffect distillation followed by heat integration process streams

=> No source of heat at high temperature as in petrochemicals

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

17

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

18

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

19

Carnegie Mellon

60 M gallon /yr plant

Ethanol losses : 0.5%

Equipment cost = M$ 20.7 Steam cost = M$ 7.1/yr (-66%) Prod. cost = 1.28 $/gal

Wash1 Grind1

Src2

Src1

Feedstock

Washing water

Premix1 Col1 Liq1 Sac1Mix2

Src4

Jet1

HX1 HX2

Superheated steam

Src6Src5

Enzyme Enzyme

Fer1

HX3

Mix3 Src7

Yeast, Urea, Water

Snk1

Vent gasCO2, O2

Str1

Str2

Spl2 Ads1 Spl5

Spl4

Mix7 Snk5

Snk4

Snk3

Ethanol

Adsorbant

Snk2

HX4

Spl1

BC1

WWT1

Water

Spl3

Mix4

Mix5

Mix6

MecP1

HX5Cond1

Spl6

Src9

Adsorbant: Corn grits

HX6

MS1 MS2 Src8

HX7

HX8

HX9

Flot1

Dry1

Snk6Snk7

Snk8

Water Biogas

Dry DDGS Solids

Proteins

Spl7Snk9

HX10

VOC removal

Rec1

To discharge/ re-use

Dry air

Storage tank

Storage tank

Humid air

HX11

Cond2

100% Ethanol

72% Ethanol

95% Ethanol

97.7% Ethanol

10.8% Ethanol

Heat Integration and Multieffect Columns

Reduction from $1.50/gal (base case) to $1.28/gal !

Energy Optimal Design

20

Carnegie Mellon

Energy Profiles in Multieffect Columns Beer Column

Rectification Column

Single column

Single column

Triple effect column

Double effect column

24,918 Btu/gal vs 38,323 Btu/gal

21

Carnegie Mellon

Remarks

21

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)

22

Carnegie Mellon

Challenge: Many alternative flowsheets

b) Hydrolysis Process (fermentation)

Wastewater Power-Heat

Biomass Pretreatment

Cellulosic Hydrolysis

Sugar Fermentation

Ethanol Recovery

Electricity

Feed Ethanol

a) Thermochemical Process (gasification)

Power-Heat Gasification Gas clean-up

Fermentation or Catalytic

Ethanol Recovery

Electricity

Feed Ethanol

Lignocellulosic Bioethanol

23

Carnegie Mellon

Superstructure Thermochemical Bioethanol

Process Design Alternatives: Gasification Indirect Low pressure Direct high Pressure Reforming. Steam reforming Partial oxidation 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

Ethanol via gasification

-Martin, M. Grossmann, I. E (2010) Aiche J. Submitted

Gasification Reforming Clean up CO/H2 Adj. Sour gases removal

Fermentation

Catalysis

Martin, Grossmann (2010)

24

Carnegie Mellon

Superstructure

Gasifier 1 Gasifier 2

Partialoxidation

Steam reforming

Partialoxidation

Steam reforming

Catalysis Fermentation Catalysis Fermentation Catalysis Fermentation Catalysis Fermentation

Problem

Subproblem

(A) (B) (C) (D) (E) (F) (G) (H)

Solution Strategy Energy Optimization

Decomposition of GDP in 8 subproblems Decision levels: Gasifier Removal HCs Reaction of Syn Gas

Heat integration and economic evaluation

25

Carnegie Mellon

Ethanol: $0.81 /gal (no H2 credits) $ 0.42/gal (H2 credits)

Optimal Design of Lignocellulosic Ethanol Plant

Low cost is due to H2 production Each NLP subproblem: 7000 eqs., 8000 var ~25 min to solve

$67.5 Million/yr

1,996 Btu/gal (< 1/10th of corn!)

26

Carnegie Mellon

Freshwater

Dist. Colums.

Fermentor

Washing

Discharge

Solids removal

Organics removal

Optimal Water Network: Corn Ethanol

TDS removal

-Ahmetović , E., Martin, M. Grossmann, (2009) I&ECR. 2010, 49, 7972–7982

Gal. Water/Gal. Ethanol = 1.5

1.5 vs 3.4

27

Carnegie Mellon -Martin, M. , Ahmetović, E., Grossmann, I. E (2010) I&ECR ASAP

Cellulosic Bioethanol via Gasification

Freshwater

Wastewater

Gasifier

Washing Solids removal

Organics removal

Optimal Water Network: Lignocellulosic Ethanol

Gal. Water/Gal. Ethanol = 4.2

28

Carnegie Mellon

Table Summary of results [6-10]

A C B D F E

[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

(*) kg instead of gal

29

Carnegie Mellon

Design and Planning under Uncertainty

Design and Planning of Offshore Oilfields Uncertain fields size, deliverability, water

Maximize expected flexibility/Minimize Cost Multi-stage programming MINLP

Goal: robustness in decisions

Design of Responsive Supply Chains Uncertain demands

Maximize NPV/Minimize responsiveness Chance constrained MINLP

Optimal Design of Responsive Process Supply Chains

Background

Fengqi You

Objective: design supply chains under responsive and economic criteria with consideration of inventory management and demand uncertainty

Problem Statement

Production Network Costs and prices Production and transportation time Demand information

Suppliers Plants DCs Customers

Safety Stock

Target DemandMax: Net present value

Max: Responsiveness

Network Structure

Operational Plan

Production Schedule

Where? What? When?

Background

II

III

SPS - 3

EPS - 1

SPS - 2

SPS - 1

EPS - 2

I

Ethylene

Benzene

Styrene

Production Network of Polystyrene Resins

Source: Data Courtesy Nova Chemical Inc. http://www.novachem.com/

Three types of plants:

Basic Production Network

Single Product

Multi Product

Multi Product

Plant I: Ethylene + Benzene Styrene (1 products)

Plant II: Styrene Solid Polystyrene (SPS) (3 products)

Plant III: Styrene Expandable Polystyrene (EPS) (2 products)

Example

Possible Plant Site Supplier Location

Distribution Center Customer Location

Location Map

Example

IL

TXI

II

III

III

II

I

CAEthylene

Ethylene

Benzene

Benzene

Styrene

Styrene Styrene

SPS

SPS

EPS

EPS

AZ

OK

Plant Site MI

Plant Site TX Plant Site CA

NV

IEthylene

Benzene

Styrene

Plant Site LA

TX

GA

PA

NC

FL

OH

MA

MN

WA

IA

TX

MS

LA

AL

IIIEPS

Suppliers Plant Sites Distribution Centers Customers

Potential Network Superstructure

Example

Responsiveness - Lead Time

Lead Time: The time of a supply chain network to respond to customer demands and preferences in the worst case

Lead Time is a measure of responsiveness in SCs

Model & Algorithm

Responsiveness

Lead Time

• A supply chain network = ∑Linear supply chains Assume information transfer instantaneously

Model & Algorithm

Lead Time for A Linear Supply Chain

Information

Suppliers

Plants Distribution Centers Customers

Supplier ls Plant i1 site k1 Plant i3 site k3 Customer ldDistribution Center m

Plant i2 site k2

Lead Time under Demand Uncertainty

Model & Algorithm

Inventory (Safety Stock)

Production Lead Time (LP) Delivery Lead Time (LD)

Supplier ls Plant i0 site k0

…Plant in site kn Customer ldDistribution Center m

Transporation Transporation TransporationTransporation

Safety Stock

P

• 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

Expected Lead Time = max {2.1, 2.0} = 2.1 days

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 of SCN

P1=20%

Path 2 2.0 days

Path 1 2.1 days

P2=20%

Example

I

II

Supplier2

0.5

0.2

1.8

1.5

1.2

Plant Site 1

Plant Site 2 Customer 1

Customer 2

DC 1

III1.2

2.6

Plant Site 3

DC 2

0.7

0.5

• Objective Function: Max: Net Present Value

Min: Expected Lead time

• Constraints: Network structure constraints

Suppliers – plant sites Relationship Plant sites – Distribution Center Input and output relationship of a plant Distribution Center – Customers Cost constraint

Bi-criterion

Choose Discrete (0-1), continuous variables

Cyclic scheduling constraints Assignment constraint Sequence constraint Demand constraint Production constraint Cost constraint

Probabilistic constraints Chance constraint for stock out (reformulations)

Bi-criterion Multiperiod MINLP Formulation

d Md L d U

Safety Stock

Target Demand

Model & Algorithm

Operation planning constraints Production constraint Capacity constraint Mass balance constraint Demand constraint Upper bound constraint

Possible Plant SiteSupplier Location

Distribution CenterCustomer Location





IL

TXI

II

III

III

II

I

CAEthylene

Ethylene

Benzene

Benzene

Styrene

Styrene Styrene

SPS

SPS

EPS

EPS

AZ

OK

Plant Site MI

Plant Site TX Plant Site CA

NV

IEthylene

Benzene

Styrene

Plant Site LA

TX

GA

PA

NC

FL

OH

MA

MN

WA

IA

TX

MS

LA

AL

IIIEPS

Suppliers Plant Sites Distribution Centers Customers

Case Study

Example

• Problem Size: # of Discrete Variables: 215 # of Continuous Variables: 8126 # of Constraints: 14617

• Solution Time: Solver: GAMS/BARON Direct Solution: > 2 weeks Proposed Algorithm: ~ 4 hours

300

350

400

450

500

550

600

650

700

750

1.5 2 2.5 3 3.5 4 4.5 5 5.5Expected Lead Time (day)

NPV

(M$)

with safety stockwithout safety stock

Pareto Curves – with and without safety stock

Example

More Responsive

Best Choice

0

50

100

150

200

Safe

ty S

tock

(10^

4 T

)

1.51 2.17 2.83 3.48 4.14 4.8

Expceted Lead Time (day)

EPS in DC2SPS in DC2EPS in DC1SPS in DC1

Safety Stock Levels - Expected Lead Time

Example

More inventory, more responsive

Responsiveness

43

Network Structure at Location Map

Design of Responsive Chemical Supply Chains under Uncertainty

44

Carnegie Mellon

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

Reservoirs wells

facilities

Offshore oilfield having several reservoirs under uncertainty Maximize the expected net present value (ENPV) of the project

Tarhan, Grossmann (2009)

Optimal Development Planning under Uncertainty

Uncertainty: Initial productivity per well Size of reservoirs Water breakthrough time for reservoirs

TLP FPSO

45

Carnegie Mellon

Tank Cumulative Oil (MBO)

Sing

le W

ell O

il an

d W

ater

Rat

e (k

bd)

Unconstrained Maximum Oil Production

Water Rate

Initial oil production Assumption: All wells in the same reservoir are identical.

Size of the reservoir Uncertainty is represented by discrete distributions functions

Non-linear Reservoir Model

46

Carnegie Mellon

Decision Dependent Scenario Trees

Scenario tree Not unique: Depends on timing of investment at uncertain fields

Central to defining a Stochastic Programming Model

Invest in F in year 1

H

Invest in F

Size of F: M L

Assumption: Uncertainty in a field resolved as soon as WP installed at field

Invest in F in year 5

H Size of F: L

Invest in F

M

47

Carnegie Mellon

1 2 3 4

ξ2=1 p=0.5

ξ2=2 p=0.5

ξ2=1 p=0.5

ξ2=2 p=0.5

ξ1=1 p=0.5

ξ1=2 p=0.5

t=1

t=2

t=3

1 2 3 4

ξ1=1 p=0.25

ξ1=2 p=0.25

ξ1=2 p=0.25

ξ2=2 p=1.00

ξ2=1 p=1.00

ξ1=1 p=0.25

ξ2=2 p=1.00

ξ2=1 p=1.00

t=1

t=2

t=3

Alternative and equivalent scenario tree structure (Ruszczynski, 1997):

scenario tree

Stochastic Programming

48

Carnegie Mellon

1 2 3 4

ξ2=1 p=0.5

ξ2=2 p=0.5

ξ2=1 p=0.5

ξ2=2 p=0.5

ξ1=1 p=0.5

ξ1=2 p=0.5

t=1

t=2

t=3

1 2 3 4

ξ1=1 p=0.25

ξ1=2 p=0.25

ξ1=2 p=0.25

ξ2=2 p=1.00

ξ2=1 p=1.00

ξ1=1 p=0.25

ξ2=2 p=1.00

ξ2=1 p=1.00

t=1

t=2

t=3

Each scenario is represented by a set of unique nodes


49

Carnegie Mellon

1 2 3 4

ξ2=1 p=0.5

ξ2=2 p=0.5

ξ2=1 p=0.5

ξ2=2 p=0.5

ξ1=1 p=0.5

ξ1=2 p=0.5

t=1

t=2

t=3

ξ1=1 p=0.25

ξ1=2 p=0.25

ξ1=2 p=0.25

ξ2=2 p=1.00

ξ2=1 p=1.00

ξ1=1 p=0.25

ξ2=2 p=1.00

ξ2=1 p=1.00

1 2 3 4

t=1

t=2

t=3

Nodes have same amount of information Nodes are indistinguishable

Non-anticipativity constraints


50

Carnegie Mellon

t=1

t=2

t=3

1 2 3 4

t=4

t=1

t=2

t=3

1 2 3 4

t=4

1 2 3 4

1 2 3 4

Representation of Decision-Dependence Using Scenario Tree

51

Carnegie Mellon

Every scenario,

time period

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)

Every pair scenarios,

time period

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 Disjunctions (conditional constraints)

52

Carnegie Mellon

Formulation of Lagrangean dual

Relaxation • Relax disjunctions, logic

constraints • Penalty for equality

constraints

53

Carnegie Mellon

Uncertain Parameters (Discrete Values)

Scenarios

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

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

Construction Lead Time

(years)

Wells Facilities

TLP Sub-sea TLP Small FPSO Large FPSO

1 1 1 2 4

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.

One Reservoir Example

54

Carnegie Mellon

RS: Reservoir size IP: Initial Productivity BP: Breakthrough Parameter

E[NPV] = $4.92 x 109

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.

year 1

Build 2 small FPSO’s Drill 12 sub-sea wells

year 2 12 subsea wells

Low RS Low IP

High RS High IP

High RS Low IP

Low RS High IP

4 small FPSO’s, 5 TLP’s 12 subsea wells

5 small FPSO’s, 3 TLP’s


Mean RS Mean IP

Multistage Stochastic Programming Approach

55

Carnegie Mellon

RS: Reservoir size IP: Initial Productivity BP: Breakthrough Parameter

E[NPV] = $4.92 x 109

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.

year 1

year 2

Build 2 small FPSO’s Drill 12 sub-sea wells

12 subsea wells

Low RS Low IP

High RS High IP

High RS Low IP

Low RS High IP


5 small FPSO’s, 3 TLP’s


Mean RS Mean IP

8 9 1 2 6 7 3 4

year 3

year 4 High BP Low BP

6 subsea wells, 18 TLP wells

High BP Low BP High BP Low BP High BP Low BP

5

12 subsea wells, 30 TLP wells

12 subsea wells 6 subsea wells 6 subsea wells,

12 TLP wells

Mean BP

Multistage Stochastic Programming Approach

56

Carnegie Mellon

-2

0

2

4

6

8

10

1 2 3 4 5 6 7 8 9

Scenarios

Net P

rese

nt V

alue

($ x

109 )

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

Distribution of Net Present Value

57

Carnegie Mellon

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

Conclusions

discrete and continuous optimization models for the design

Documents