A Stochastic Programming Approach to Natural Gas Portfolio and Transpofrt Optimisation BSc

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A Stochastic Programming Approach to Natural Gas Portfolio and

Transport Optimization

Ketty Hua

A thesis submitted in partial fulfillment

of the requirements for the degree of

BACHELOR OF APPLIED SCIENCE

Supervisor: Roy H. Kwon

Department of Mechanical and Industrial Engineering

University of Toronto

March 2008

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Abstract

The purpose of this Thesis is to develop and compared linear programming models that will balance

and optimize a natural gas portfolio and transport problem. We have developed and compared

models using three approaches: deterministic, stochastic minimizing the expected cost, and

stochastic minimizing the Conditional Value-at-Risk. The Stochastic models implemented are based

on a two-stage stochastic model with integer recourse. Numerical examples using stylized data are

shown to illustrate the differences in the optimal decisions determined from the models.

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Acknowledgements

This thesis is supervised by Professor R.H. Kwon who provided me with guidance in research and

enlightenment through discussions.

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Table of Contents

Abstract ................................................................................................................................................................................ 2

Acknowledgements ............................................................................................................................................................. 3

List of Figures ...................................................................................................................................................................... 5

List of Tables ....................................................................................................................................................................... 6

CHAPTER 1 - INTRODUCTION ................................................................................................................................ 8

1.1 Natural Gas Market in North America ................................................................................................................ 8

1.2 Financial and Operations Risk Management ....................................................................................................... 9

1.3 Natural Gas Network and Portfolio ................................................................................................................... 11

1.4 Risk Measure Tools and Methodology ............................................................................................................... 14

1.4.1 Value-at-risk .................................................................................................................................................... 14

1.4.2 Conditional Value-at-risk (CVaR)................................................................................................................ 15

1.5 CVaR Minimization ............................................................................................................................................... 16

1.6 Stochastic Programming ....................................................................................................................................... 17

1.7 CVaR Minimization Approach in Stochastic Programming ........................................................................... 19

CHAPTER 2 PROBLEM FORMULATION ......................................................................................................... 20

2.1 Deterministic Approach ....................................................................................................................................... 20

2.2 Stochastic Approach Minimizing Expected Cost ......................................................................................... 28

2.3 Stochastic Approach Minimizing CVaR ......................................................................................................... 32

CHAPTER 3 COMPUTATIONAL RESULTS ..................................................................................................... 34

3.1 Deterministic Approach ....................................................................................................................................... 35

3.2 Expected Cost Approach ..................................................................................................................................... 43

3.2.1 Deterministic demand ................................................................................................................................... 43

3.2.2 Stochastic demand ........................................................................................................................................ 54

3.3 CVaR Approach ..................................................................................................................................................... 64

Conclusion ......................................................................................................................................................................... 76

APPENDIX : REFERENCES ...................................................................................................................................... 77

OPL format for Deterministic Model ........................................................................................................................... 78

OPL Format for Stochastic Model with minimizing Expected Cost ....................................................................... 80

OPL format for stochastic model with minimizing CVaR ........................................................................................ 82

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List of Figures

Figure 1 Source: NYMEX, Impact of events on Natural Gas Prices ........................................................................ 9

Figure 2 Source: Petroleum Encyclopedia, Oil and Gas Journal, Schematic Diagram of North American

Market Hubs ...................................................................................................................................................................... 11

Figure 3 Source: US Energy Policy, Schematic Diagram of North American Producing Basins ........................ 11

Figure 4 Source: CAPP, North American Transport Pipeline Network ................................................................. 12

Figure 5 Schematic diagram of the Flow of gas between locations within a gas network .................................... 20

Figure 6 Portfolio loss distribution of the Expected Cost Approach and CVaR Approach ................................ 65

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List of Tables

Table 1 Fixed and variable volume market forwards .................................................................................................. 34

Table 2 Storage capacity limits, unit costs, injection/withdrawal rate...................................................................... 34

Table 3 Production costs and limits from Sources/Supplies..................................................................................... 35

Table 4 Demand requirement for each of the time period in MMBtu .................................................................... 37

Table 5 Spot market prices at each of the time period in $/MMBtu ....................................................................... 37

Table 6 Capacity Acquisition Strategy for Deterministic Approach, all units are stated in MMBtu, unless

otherwise indicated ........................................................................................................................................................... 38

Table 7 Overall Cost of the System using the Deterministic Approach, all units are stated in MMBtu unless


Table 8 Spot Market Trading Profits from Deterministic Approach, all units are stated in MMbtu unless


Table 9 Cost to Meet Demand from Deterministic Approach, all units are stated in MMbtu unless otherwise

indicated ............................................................................................................................................................................. 42

Table 10 Spot market prices in the three scenarios considered ................................................................................ 43

Table 11 Cost to Meet Demand for the Three Scenarios Considered Using Stochastic - Minimizing Expected

Cost Approach .................................................................................................................................................................. 44

Table 12 Capacity Acquisition Strategy for Stochastic Expected Cost Approach (Deterministic Demand), all

units are stated in MMBtu, unless otherwise indicated ............................................................................................... 45

Table 13 Overall System Cost for Stochastic Expected Cost Approach (Deterministic Demand), all units are

stated in MMBtu, unless otherwise indicated ............................................................................................................... 47

Table 14 Spot Market Transaction for Stochastic Expected Cost Approach (Deterministic Demand), all units

are stated in MMBtu, unless otherwise indicated ........................................................................................................ 50

Table 15 Cost to Meet Demand Stochastic Expected Cost Approach (Deterministic Demand), all units are


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Table 16 Stochastic Demand Values ............................................................................................................................. 54

Table 17 Capacity Acquisition Strategy for Stochastic Expected Cost Approach (Stochastic Demand), all units


Table 18 Overall System Cost for Stochastic Expected Cost Approach (Stochastic Demand), all units are


Table 19 Total Profit from Spot Market Trading Transactions for Stochastic Expected Cost Approach

(Stochastic Demand), all units are stated in MMBtu, unless otherwise indicated .................................................. 60

Table 20 Total Cost to Meet Demand for Stochastic Expected Cost Approach (Stochastic Demand), all units


Table 21 VaR and CVaR with various beta values for UB for forward type= 10.................................................. 65

Table 22 VaR, CVaR, Maximum number of forwards purchased with respect to varying UB for forward type

.............................................................................................................................................................................................. 65

Table 23 Expected loss and standard deviation with the expected cost and CVaR approaches ......................... 66

Table 24 Capacity Acquisition Strategy for Stochastic Minimizing CVaR Approach with =0.99, all units are


Table 25 Overall System Cost for Stochastic Minimizing CVaR Approach with =0.99, all units are stated in

MMBtu, unless otherwise indicated ............................................................................................................................... 69

Table 26 Total Profit from Spot Market Transactions for Stochastic Minimizing CVaR Approach with

=0.99, all units are stated in MMBtu, unless otherwise indicated ........................................................................... 72

Table 27 Total Cost to meet demand for Stochastic Minimizing CVaR Approach with =0.99, all units are


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CHAPTER 1 - INTRODUCTION

1.1 Natural Gas Market in North America

The countries in North America recognize the important role of a competitive natural gas market on

the economic, environmental and social welfare. In the U.S., there has been long belief in

competitive markets, based on private ownership of energy capital and resources, to ensure the

optimal supplies and consumption of natural gas.[2] In Canada, deregulation of the natural gas

industry began in 1984, mainly aimed for a more open market, allowing market prices to be

determined by supply and demand forces. Some of the significant structural changes in policy

included 1) The option for consumers to purchase from a supplier other than their Local Utility,

such as from a Marketer or a Producer; 2) Opening up pipeline capacities to third parties; and 3)

Lifting the 30-year reserve-to-production ratio policy, freeing up large quantities of gas in storages

for export opportunities. [1] Although many restrictions were removed, today, items such as

transport and storage fees, transmission and other areas where the market does not adequately serve

its policy objectives are still regulated or controlled by regulators such as the National Energy Board

(NEB). The deregulation of the natural gas market allowed new participants to compete and enabled

them with access to pipelines distribution capacities.

With more new participants in a deregulated and integrated North American market, there is a need

for new natural gas transaction and risk management system solutions to manage the growing price

risks from a more dynamic environment.

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1.2 Financial and Operations Risk Management

A participant in the Natural Gas market with a gas portfolio faces the challenges of managing both

financial and operational risks to meet various demand levels mean while optimizing their

profitability. Natural Gas prices are heavily correlated to oil prices and greatly influenced by weather

conditions. [1] As a result, fluctuations in oil prices driven by market supply and demand, and

geopolitical events around the globe greatly impact the market prices. In addition, further

uncertainties are introduced from unpredictable and uncontrollable weather conditions (Figure 1)

Since natural gas is typically used in space heating, in colder winters, the consumption of the

commodity is substantially higher than average. The increase in demand during these periods creates

the demand shock effect in which spot market prices become highly volatile, thus making

financial and operational planning difficult.

Market instruments, such as forward contracts, are available to allow market participants to mitigate

the risks of price volatility. A forward contract can be purchased from a seller for delivery of a

specified quantity of natural gas at a pre-agreed location, time and price in the future for a price at

Figure 1 Source: NYMEX, Impact of events on Natural Gas Prices

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the time when the contract is engaged. 1By entering a forward contract, gas price will be locked to

hedge against future uncertainties in the spot market. We will also consider the addition of another

instrument known as the swing option, which allows the flexibility of the contract holder to

receive volumes of natural gas within a predefined range at pre-agreed time periods. The swing

option will facilitate risk management and give the purchaser of the contract the volume flexibility to

vary their demand levels of the forward in the future.

Another type of contract is called a Swap contract. Similar to a forward contract, the swap locks in

the value of the commodity at a pre-agreed price. In a swap contract, the seller of the swap agrees

to pay the buyer for the increase in price of the underlying commodity above an agreed-upon value

(the price of the swap) at the time when the swap expires, and the buyer agrees to pay the seller for

any decreases below the agreed-upon value. [3] Thus, the seller of a swap is protected against any

decreases in the price of the commodity, and a buyer is protected against any price increases. In this

thesis, we develop an optimization model that can be used by the seller of the swap contract to

minimize the cost for meeting demand. The swing options are also considered, in which the buying

of the contract can withdrawal a range of volume of gas during a time period.

It is of an interest to the seller of the contract to determine the optimal forward contract acquisition

strategies to minimize the overall network cost to meet demand.

1 A forward contract differs from a future contract in that physically delivery is intended for the former, and not in

the latter. [1]

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1.3 Natural Gas Network and Portfolio

A Natural Gas Network consists of producing basins, storage areas, demand locations, hubs and

pipelines connecting them. In the portfolio problem, we will assume the provider of the case study

has its own sources of supply. Examples of such sourcing locations which contribute to the North

American natural gas supply are: the Western Canadian Sedimentary Basin (WCSB) in Alberta; the

Mid-continent in the central and mid-western U.S.; and the Gulf of Mexico region (Figure 3).

Natural Gas is extracted, produced and distributed from the basins to the markets and Local

Distribution Companies (LDCs), and eventually to the residential and commercial end-users.

The pricing and trading of the gas occur at market hubs. Hubs update both spot and future gas

prices on a daily basis, which serve as benchmark prices for trading transactions.2 Major hub

locations in North America are: AECO in Canada, and the Henry Hub in Louisiana U.S (Figure 2).

2 Units of price is stated in $/Bcf

Figure 2 Source: Petroleum Encyclopedia, Oil and Gas Journal,

Schematic Diagram of North American Market Hubs

Figure 3 Source: US Energy Policy, Schematic Diagram of North

American Producing Basins

Natural Gas Basins

in North America

North American

Market Centers and

Each hub sets the referencing price independently; therefore there are variations

The variation in prices is related to the cost of transport, as well as the supply and demand

conditions in the region. High volatility of prices on the market translates to potential arbitrage

opportunities in time, in which gas can be purchased for a lower price a

at a higher price in the future. Potential arbitrage opportunities also exist in

market prices of gas are set independently at these hubs, there are variations in prices between

locations. In such situation, profits can be made by purchasing gas at a lower price from one

location, transported to sell for a high price at another. In addition to physical trading, electronic

trading is also available. For the purpose of this analysis, only physical tradin

Connecting the producing basin, the end

the transport pipelines. Major pipeline systems in North America are the Trans

Canada and the PG&E Gas transm

the maximum flow of natural gas over a specified period of time for which

Figure 4 Source: CAPP, North American Transport Pipeline Network

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Each hub sets the referencing price independently; therefore there are variations between locations.

ion in prices is related to the cost of transport, as well as the supply and demand

High volatility of prices on the market translates to potential arbitrage

, in which gas can be purchased for a lower price at the present, stored and sold

at a higher price in the future. Potential arbitrage opportunities also exist in location. Since the spot


uation, profits can be made by purchasing gas at a lower price from one


trading is also available. For the purpose of this analysis, only physical trading will be considered.

the end-users (LDCs) natural gas markets, and storage facilities

Major pipeline systems in North America are the Trans-Canada Pipelines in

PG&E Gas transmission Northwest in the U.S. (Figure 4). The capacity represents

the maximum flow of natural gas over a specified period of time for which a pipeline system or

Source: CAPP, North American Transport Pipeline Network

North American Transport

Pipeline Network

between locations.

ion in prices is related to the cost of transport, as well as the supply and demand

High volatility of prices on the market translates to potential arbitrage

t the present, stored and sold

. Since the spot


uation, profits can be made by purchasing gas at a lower price from one


g will be considered.

LDCs) natural gas markets, and storage facilities are

Canada Pipelines in

The capacity represents

a pipeline system or

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portion thereof is designed or contracted, not limited by existing service conditions. [9] For market

participants, pipeline capacity contracts can be acquired through auctions for long term periods of

typically 10 years or less. These capacities give rights to the contract holders to transport volumes of

gas along a fixed pipeline path between locations. A procurement strategy for capacity acquisition

must be developed to optimize the usage and reduce the overall system cost for distribution.

As the demand for natural gas is significantly higher in the winter months, excess production from

the summer is typically injected into storages and withdrawn during peak demands. For market

participants, storages can be used as a tool for risk management. Gas can be purchased during the

summer time when prices are relatively lower for storage and used for delivery in the winter when

market prices are higher and more volatile. Some limitations do exist for storage injection and

withdrawals as the cost, volume and time required to transport in and out of these area are

constrained.

As the complexity of the gas network increases, managing market risks to meet demands under

various uncertainties become non-trivial. To optimize portfolio performance, a strategy for resource

acquisition and dispatch must be developed in addition to purchasing of forward contracts.

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1.4 Risk Measure Tools and Methodology

1.4.1 Value-at-risk

Value-at-Risk (VaR) is a tool for measuring risks by determining the maximum decrease in the value

(loss) of the portfolio (in monetary value) given a confidence level x 100% over a specified period

of time. The loss, represented by (, ), is a function of the decision vector and the random vector . The loss can take either a positive value or a negative one. In the latter case, the negative loss would describe a gain. For each decision , (, ) takes on some random value having a distribution brought about by the set of uncertain parameters, . We will assume the vector follows a probability distribution denoted by (). The cumulative probability of the loss (, ) not exceeding a threshold value then takes on the form [4]

(x, ) = ()(,) . Where is the VaR values of the loss associated with the decision vector at a given probability level in (0, 1). [4]

(x) = min | (x, ) } Because the loss function of a portfolio is not normally distributed, but rather heavy-tailed. By

minimizing the value-at-risk, a set of optimal decisions is selected that does not favour large losses.

There exist undesirable mathematical probabilities associated with VaR in the application of

optimization such as the non-subadditivity and non-convexity properties. [4] In the case of sub-

additivity, the VaR associated with a combination of portfolios can be greater than the sum of the

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risks of the individual portfolios. That is ( + !) > + (!) . This is undesirable because portfolio diversification should reduce risk. The non-convexity property makes optimization

difficult when calculated using scenarios because the VaR as a function of can exhibit multiple local extrema.

1.4.2 Conditional Value-at-risk (CVaR)

Due to implications with using VaR in optimization, Conditional Value-at-Risk (CVaR), with better

mathematical properties is preferred and used in this thesis. The value of CVaR is by definition

greater than the value of VaR at a given confidence level . Thus, a low CVaR will also yield a low

VaR value.[4][5]

It is shown in [4][6] that the CVaR is defined by

(x) = (1 )&' (, )()(,)() . Thus CVaR represents the conditional expectation of the loss associated with vector x in the case

which the loss exceeds VaR or (x).

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1.5 CVaR Minimization

To utilize CVaR (x) defined by (x) = (1 )&' ( (, )()(,)() in an optimization problem, a single function is developed to characterize (x) and (x). The simplified function becomes[4]

F(x, ) = + (1 )&' ( (, )()(,) . It was shown in [4] that the F(x, ) is related to CVaR or (x) by the formula

(x) = min

F(x, ) = F(x, (x, )). F(x, ) is convex and continuously differentiable, thus CVaR can be easily determined by minimizing the function numerically. Furthermore, the VaR or , for which F(x, ) depend on, does not need to be calculated independently in order to derive (x). Instead VaR can be obtained as a by product.

F(x, ) can be approximated from a set of sample data ',+,, . that belongs to a probability space ().

F/(x, ) = + (q(1 ))1 123x, yk6 7q

k=1+

Although F/(x, ) is not differentiable with respect to , however, it still can be minimized through various methods, including linear programming. It was shown in [6] that the linear programming

representation of the problem takes on the form:

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min

F/(x, ) = + (q(1 ))1 1 z9qk=1 Subjected to the following constraints:

9 z9 3x, yk6 z9 0.

1.6 Stochastic Programming

Stochastic programming is a method for modeling optimization problems by taking into

consideration of the uncertainty of the problem data. Unlike in a deterministic model (Simple Linear

Programming model), in which the optimal solution is obtained by assuming the input parameters

are accurate; in a stochastic model, the optimal solution is obtained by applying random variations to

these input parameters to project the optimal decision that would be prepare for various

uncertainties. The optimal set of decisions will be one which maximizes the expected value of the

objective function for all possible outcomes. If a deterministic approach is adopted, in the case

where the projected data is inaccurate, the original set of optimal decisions obtained from this

approach would limit opportunities and yield undesired results.

The two-stage Stochastic Linear Programming model with integer recourse model is widely used

within Stochastic Programming. In the first-stage of the model, a set of decisions defined by linear

constraints are made. After a random event occurs, affecting the outcome of the first-stage decision,

a recourse action is taken in the second stage to enable corrections to the solution altered by the

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random event. The optimal solution will be obtained by choosing both the first-stage decisions and

the second-stage recourse decisions that will maximize the expected outcome.

The general two-stage stochastic model with recourse takes on the form [7]

?@A + BCD(, E)| = F, 0G where D(, E(H)) = IA |J = (H) L, 0}. Vector represents the first-stage decisions. The second-stage decisions or the corrective actions denoted by are determined for each scenario described by some random vector E(H), where H an element which determines the nature of the random event with some probability value of occurring. E(H) is formed from a combination of elements in IA , L, A, representing the objective function for the second-stage decisions, the resulting values of the first-stage decisions, and the sum of the values from the first-

stage and second-stage decisions, respectively. For our gas portfolio problem, only the Amatrix is considered random.

In our two-stage model, the first stage determines the volume of pipeline capacity contracts to

engage in and the number of forwards to purchase for each of the planning time periods. The

second stage decisions give the amount of gas to withdraw/inject from storage facilities, the amount

of gas to purchase/sell on the spot markets, and finally the transport of gas from storage, source and

the spot market to meet demand requirements, for each of the planning time periods. Discrete

scenarios are generated by considering the uncertainty of demands and gas spot prices. We will

employ stylized probabilities for the input parameters.

Arbitrage opportunity are also considered in the model, and occurs when there is a price mismatch

between either the forward prices and the spot market price, or two spot market prices at separate

locations, for a specific time. The number of forwards which can be purchased will be bounded by a

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limit defined by the user to avoid unrealistic situations where large quantities of forwards are

purchased in case of an attractive arbitrage opportunity.

1.7 CVaR Minimization Approach in Stochastic Programming

Combining the approximation function of Conditional Value-at-Risk in section 1.5 and the general

two-stage stochastic programming model in Section 1.6, the two-stage stochastic model minimizing

the CVaR with discrete probabilities for (H) is defined in [6] as minM,N,O,P DQRPS (T, ) = + (1 )&' 1 UVTV

.VW'

Subjected to the constraints

JV = L TV @A + IAV TV 0, V. The problem will determine the VaR, CVaR, the first stage decisions , and the second stage recourse decisions . The CVaR minimization approach in two-stage stochastic programs has been previously applied to

Replication of Electricity Custom Contract problem in [6].

CHAPTER 2 PROBLEM FORMULATION

2.1 Deterministic Approach

We will propose a deterministic linear programming m

Deterministic models for the application of gas portfolio and transport have been

in [10]. The strategy is to determine the

demand while optimizing total revenue of the portfolio. The strategy would consist of acquisition of

pipeline capacities, purchasing of mark

storages, buying from the spot market

Figure 5 Schematic diagram of the Flow of gas between locations within a gas network

We will assume that the market participant owns

natural gas and |X| storage spaces for storing gas.at || locations at time period Y for interest for the analysis. Assuming

fixed, therefore this revenue stream

We will also consider |Z[| different market forwards available each time periods in the future. The two types of forwards considered, fixed and variable volume

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EM FORMULATION

Approach

propose a deterministic linear programming model for our portfolio optimization problem.

Deterministic models for the application of gas portfolio and transport have been previously

determine the resource acquisition and dispatch volumes required to meet

revenue of the portfolio. The strategy would consist of acquisition of

pipeline capacities, purchasing of market forwards, generation from producing basins, utilization of

storages, buying from the spot market, at each of the future time periods.

Schematic diagram of the Flow of gas between locations within a gas network

assume that the market participant owns |\| producing basins, or sources that storage spaces for storing gas. ]^_ volumes of gas must be delivered for demand

for Y = 1, L`, where L` denotes the number of time periods ofAssuming the price charged on a unit of gas used to fulfill the demands is

this revenue stream will be omitted in the objective function.

different market forwards available to deliver specific volumes of gas at

The two types of forwards considered, fixed and variable volume

odel for our portfolio optimization problem.

previously studied

volumes required to meet

revenue of the portfolio. The strategy would consist of acquisition of

et forwards, generation from producing basins, utilization of

Schematic diagram of the Flow of gas between locations within a gas network

asins, or sources that can supply

volumes of gas must be delivered for demand

denotes the number of time periods of

price charged on a unit of gas used to fulfill the demands is

ver specific volumes of gas at

The two types of forwards considered, fixed and variable volume

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forwards are denoted as abcMd^ = 1, eZ[bcMdê and afPgcPhid = eZ[bcMdê + 1, , |Z[| , respectively. Where eZ[bcMdê represented the number of fixed volume forwards and eZ[fPgcPhide represents the number of variable volume contracts. A forward can be purchased at price `[fj_ at Hub , for 1, , lm}. The forward will deliver |[fj_| volume of natural gas during time period Y. In addition to the forward contracts, spot market purchases can also be made at each hubs for the

price of `nj_ . per unit of gas. Finally, pipeline capacities must be acquired at a volume of oPh for pipeline (, F) for the duration of analysis. The cost per unit of capacity will assumed to be CCab.

The following parameters and decision variables are used in the deterministic model.

Parameters

t Time period Y 0, , L`} (a,b) Pipeline between locations a and b. (, F) 1, , Z} h, g Hub location 1, , lm}, p 1, , lm} i Storage location = 1, , qm} v Set of market forwards, fixed and variable volume a 1, , Z[} e Source location \ 1, , Bm} d Demand location 1, , ]m} CSI Unit cost for gas injection from storage

CSW Unit cost for gas withdrawal from storage

SR Maximum rate of storage injection or withdrawal

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CE Unit cost for gas extraction from source

CC Unit cost of capacity for pipeline

SUB Maximum allowable storage volume of storage

PUB Maximum allowable pipeline volume which can be purchased for pipeline

EUB Maximum allowable extraction volume for source

LBvt Minimum allowable volume of Forwardvht that can be drawn at time t

UBvt Maximum allowable volume of Forwardvht that can be drawn at time t

PMht Price of gas on the spot market at Hubh at time t

PFvth Price of gas forward associated with Forwardvht at Hubh at time t

Ddt Gas demand at location d at time t

MBS Minimum benchmark strategy

Decision variable

STit Volume in Storageit at time t

RMTht Volume that sold at Hubh at time t

BMTht Volume that purchased at Hubh at time t

VFvht Volume that will be delivered by Forwardvht at Hubh for time t

ETet Volume extracted from Sourceet at time t

FDvhdt Amount of Forwardvht used to meet demand at location d, at time t

FMvhgt Amount of Forwardvht used to sell at Hubg at time Y FSvhit Amount of Forwardvht used to put in Storageit at time t

SDidt Amount of Storageit used to meet demand at location d, at time t

SMiht Amount of Storageit used to sell at Hubh at time t

EDedt Amount of Sourceet used to meet demand at location d, at time t

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EMeht Amount of Sourceet used to sell at Hubh at time t

ESeit Amount of Sourceet used to put in Storagei at time t

BMDhdt Amount of gas purchased on the spot market at Hubh for demand at location d, at time t

BMShit Amount of gas purchased in the spot market at Hubh for Storagei, at time t

BMMhgt Amount of gas purchased in the spot market at Hubh for Hubg, at time t

FLabt Amount of flow through in pipeline (a,b) at time t

Cab Amount of capacity bought for pipeline (, F) NFPvht Number of Forwardvht purchased at Hubh, at time t

The objective function minimizes the overall system costs, which includes cost associated with

pipeline capacity acquisition, buying in the spot market, cost of producing at sources, cost of market

forwards, cost of storage injection and withdrawals, subtracted by revenues from selling on the spot

market.

n=> r 1 @s@=Y @tXYXuPv wcwdv + 1 1 txyx @tXYXbzg{Pg^v_c|d + 1 1 (s}x@X=>p @tXYX x\a\>}\X)j~hv_c|d+ 1 1 (=>\@Y=t> @tXYX + y=Yxy @tXYX)_zgPudv_c|d + 1 1 Xt}x@\ X}ss @tXYXvz~gdv_c|d

Where:

1 = 1 oo(oPh)(P,h)

1 1 = 1 1 1 `[fj_[fj_

fW'

jW'A

_W'

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1 1 ( ) = 1 1 `x=@$`}xtxxF=Yxp\ + `}xtx]\ + `}xtxqYtxp\j~hv_c|d qtxt\@Y=t>([txyxtxqYtxp\ + qt}x@\txqYtxp\v_zgPudv_c|d+ `}xtxqYtxp$ + otXYtqYtxp\(qYtxp\tx]\ + qtxt

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_c qLc_' = 1 1 [qfjc_ jW'

RfW' + 1 Bqdc_

dW' + 1 !nqjc_

jW' 1 q]c^_

^W' + 1 qncj_

jW' + qLc_

Storage limits

Amount stored at a particular storage is less than or equal to the maximum capacity at the storage

_c qLc_ q! Amount withdrawn from storage during time t is less than or equal to the amount in storage at the

beginning of time t _c 1 q]c^_^W' + 1 qncj_

jW' qLc_

The rate at which volume is injected or withdrawn from a particular storage is bounded by the

maximum rate SR

_c 1 q]c^_^W' + 1 qncj_

jW' q _c 1 1 [qfjc_

jW'R

fW' + 1 Bqdc_

dW' + 1 !nqjc_

jW' q

Spot market purchase balance

Amount of gas purchased at a spot market is equal to the sum of the total volume used to satisfy

demand requirements, for delivery to storage, and for location arbitrage strategy

_j !nL j_ = 1 !n]j^_^W' + 1 !nqjc_

cW' + 1 !nnuj_

uW'

Spot market sell balance

Amount of gas purchased at a spot market is equal to the sum of the total volume delivered from

forwards, production, storage, and location arbitrage strategy

_j nL j_ = 1 qncj_cW' 1 Bndj_

dW' + 1 1 [nfuj_

fW'

uW' + 1 !nnuj_

uW'

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Forward balance

Total forward is equal to the amounts put in storage, used for demand, and sold on the spot market.

_jf [ fj_ = 1 [nfju_uW' + 1 [qfjc_

cW' + 1 []fj ^ _

^W'

Forward limits

The volume delivered by a particular forward is bounded by the maximum number and minimum

number of forwards which can be purchased

_fj m! f_ 1 Z[ f`ju_uW' [fj_ !f_ 1 Z[ f`ju_

uW'

Source balance

The total amount delivered from production is equal to the sum of the amount sold on the spot

market, delivered to storage, and used to satisfy demand

_d BL d_ = 1 Bndj_jW' + 1 Bqdc_

vW' + 1 B]d^_

^W'

Source limits

The total amount drawn from a source is less than the maximum production capacity

_d BLd_ B! Capacity

Amount flown through the pipeline is less than the amount of capacity bought

_(P,h)[mPh_ oPh Amount of capacity bought is less than or equal to the maximum capacity which can be acquired for

the pipeline

_(P,h) oPh `! Demand Balance

27

The demand requirement is satisfied from volumes delivered by forwards, spot purchases,

production and storages

_^ ]^_ = 1 1 []fj ^ _ fW'

jW' + 1 q]c^_

cW' + 1 B]d^_

dW' + 1 !n]j^_

jW'

Minimum Benchmark Strategy

The minimum benchmark strategy can be a set of contracts which can be obtained conveniently on

the market to satisfy demand requirements. Establishing a minimum benchmark strategy will

eliminate undesired decisions in which the cost to meet demand from the optimal solution exceed

the minimum benchmark strategy. 1 1 1 1 `[fj_[]fj^_fW'

jW' + oqJ 1 q]c^_

cW' + oB 1 B]d^_

dW'

A_W' + 1 `nj_!n]j^_

jW'

^W' n!q

28

2.2 Stochastic Approach Minimizing Expected Cost

In the stochastic approach with the objective function of minimizing the expected system cost, we

introduce a set of scenario dependent parameters. We will assume the spot market prices and

demands are uncertain. The uncertainty is factored in by representing possible outcomes in discrete

scenarios; each scenario is given a probability of occurrence. The second stage decisions or the

recourse actions will differ for each scenario to determine the optimal solution when a particular

scenario occurs.

Scenario independent parameters

t Time period Y 0, , L`} s Scenario X 1, , Lq} (a,b) Pipeline between locations a and b. (, F) 1, , Z} h, g Hub location 1, , lm}, p 1, , lm} i Storage location = 1, , qm} v Set of market forwards, fixed and variable volume a 1, , Z[} e Source location \ 1, , Bm} d Demand location 1, , ]m} CSI Unit cost for gas injection from storage

CSW Unit cost for gas withdrawal from storage

SR Maximum rate of storage injection or withdrawal

CE Unit cost for gas extraction from source

CC Unit cost of capacity for pipeline

SUB Maximum allowable storage volume of storage

29

PUB Maximum allowable pipeline volume which can be purchased for pipeline

EUB Maximum allowable extraction volume for source

LBvt Minimum allowable volume of Forwardvht that can be drawn at time t

UBvt Maximum allowable volume of Forwardvht that can be drawn at time t

PFvth Price of gas forward associated with Forwardvht at Hubh at time t

MBS Minimum benchmark strategy

Scenario dependent parameters

PMhts Price of gas on the spot market at Hubh at time t, in scenario s

Probs Probability of occurrence associated with scenario s

Ddts Gas demand at location d at time t, scenario s

Scenario independent decision variable

VFvht Volume that will be delivered by Forwardvht at Hubh for time t,

Cab Amount of capacity bought for pipeline (, F) Decision variable

STits Volume in Storageit at time t, scenario s

RMThts Volume that sold at Hubh at time t, scenario s

BMThts Volume that purchased at Hubh at time t, scenario s

ETets Volume extracted from Sourceet at time t, scenario s

FDvhdts Amount of Forwardvht used to meet demand at location d, at time t, scenario s

FMvhgts Amount of Forwardvht used to sell at Hubg at time t, scenario s

FSvhits Amount of Forwardvht used to put in Storageit at time t, scenario s

SDidts Amount of Storageit used to meet demand at location d, at time t, scenario s

SMihts Amount of Storageit used to sell at Hubh at time t, scenario s

30

EDedts Amount of Sourceet used to meet demand at location d, at time t, scenario s

EMehts Amount of Sourceet used to sell at Hubh at time t, scenario s

ESeits Amount of Sourceet used to put in Storagei at time t, scenario s

BMDhdts Amount of gas purchased on the spot market at Hubh for demand at location d, at time t, scenario s

BMShits Amount of gas purchased in the spot market at Hubh for Storagei, at time t, scenario s

BMMhgts Amount of gas purchased in the spot market at Hubh for Hubg, at time t, scenario s

FLabts Amount of flow through in pipeline (a,b) at time t, scenario s

NFPvhts Number of Forwardvht purchased at Hubh, at time t, scenario s

n=> r 1 oo(oPh)(P,h) + 1 1 1 `[fj_[fj_

fW'

jW'A

_W'+ 1 `nj_v 1 !nnju_vuW' + 1 !n]j^_v

^W'

+ 1 !nqjc_vcW' 1 [nfuj_v

uW' 1 Bndj_vdW' 1 qncj_v

cW'

1 !nnuj_vuW'

jW'+ 1 oq 1 1 [qfjc_v

jW'

fW'+ 1 Bqdc_vdW' + 1 !nqjc_v

jW'

+ oqJ 1 q]c^_v^W' + 1 qncj_v

jW'cW'

+ 1 oB 1 B]d^_v^W' + 1 Bndj_v

jW'+ 1 Bqdc_vcW'

dW'

_W+ 1 `xtFvAvW' 1 1 `nj_v 1 !nnju_v

uW'

+ 1 !n]j^_v^W' + 1 !nqjc_v

cW' 1 [nfuj_vuW' 1 Bndj_v

dW'

1 qncj_vcW'

jW'A

_W' 1 !nnuj_vuW' + 1 1 oq 1 1 [qfjc_v

jW'

fW'

+ 1 Bqdc_vdW' + 1 !nqjc_v

jW' + oqJ 1 q]c^_v^W' + 1 qncj_v

jW'

cW'A

_W'+ 1 1 oB 1 B]d^_v^W' + 1 Bndj_v

jW'

+ 1 Bqdc_vcW'

dW'A

_W'

Subject to the following constraints

v_c qLc_'v = 1 1 [qfjc_vjjW'R

fW' + 1 Bqdc_v

dW' + 1 !nqjc_v

jjW' 1 q]c^_v

^W' + 1 qncj_v

jW' + qLc_v v_c qLc_v q! v_c 1 q]c^_v^W' + 1 qncj_v

jW' qLc_v

31

v_c 1 q]c^_v^W' + 1 qncj_v

jW' q v_c 1 1 [qfjc_vjjW'

RfW' + 1 Bqdc_v

dW' + 1 !nqjc_v

jjW' q

v_j !nL j_v = 1 !n]j^_v^W' + 1 !nqjc_v

cW' + 1 !nnuj_v

uW'

v_j nL j_v = 1 qncj_vcW' 1 Bndj_v

dW' + 1 1 [nfuj_v

fW'

uW' + 1 !nnuj_v

uW'

v_jf [ fj_ = 1 [nfju_vuW' + 1 [qfjc_v

cW' + 1 []fj^_v

^W'

_fj m!jf_ 1 Z[ f`ju_uW' [fj_ !f_ 1 Z[ f`ju_

uW'

v_d BL d_v = 1 Bndj_vjW' + 1 Bqdc_v

vW' + 1 B]d^_v

^W'

v_d BLd_v B! v_(P,h)[mPh_v oPh v_(P,h) oPh `! v_^ ]^_ = 1 1 []fj^_vfW'

jjW' + 1 q]c^_v

cW' + 1 B]d^_v

dW' + 1 !n]j^_v

jW'

1 `xtFvAvW' 1 1 1 1 `[fj_[]fj^_v

fW'

jW' + oqJ 1 q]c^_v

cW' + oB 1 B]d^_v

dW'A

_W' + 1 `nj_v!n]j^_v

jW'

^W' n!q

32

2.3 Stochastic Approach Minimizing CVaR

In this section, we formulate a new two-stage stochastic model that will allows us to reduce greater

risk for the portfolio by minimizing the expected loss with the risk measure Conditional Value-at-

Risk. With the newly defined objective function discussed in section 1.7 and the set of constraints

developed in section 2.2 the new linear programming problem becomes:

n=> + 11 1 sxtFvmtvA

vW'

Subject to the following constraints

v_c qLc_'v = 1 1 [qfjc_vjjW'R

fW' + 1 Bqdc_v

dW' + 1 !nqjc_v

jjW' 1 q]c^_v

^W' + 1 qncj_v

jW' + qLc_v v_c qLc_v q! v_c 1 q]c^_v^W' + 1 qncj_v

jW' qLc_v

v_c 1 q]c^_v^W' + 1 qncj_v

jW' q v_c 1 1 [qfjc_vjjW'

RfW' + 1 Bqdc_v

dW' + 1 !nqjc_v

jjW' q

v_j !nL j_v = 1 !n]j^_v^W' + 1 !nqjc_v

cW' + 1 !nnuj_v

uW'

v_j nL j_v = 1 qncj_vcW' 1 Bndj_v

dW' + 1 1 [nfuj_v

fW'

uW' + 1 !nnuj_v

uW'

v_jf [ fj_ = 1 [nfju_vuW' + 1 [qfjc_v

cW' + 1 []fj^_v

^W'

_fj m!jf_ 1 Z[ f`ju_uW' [fj_ !f_ 1 Z[ f`ju_

uW'

33

v_d BL d_v = 1 Bndj_vjW' + 1 Bqdc_v

vW' + 1 B]d^_v

^W' v_d BLd_v B! v_(P,h)[mPh_v oPh v_(P,h) oPh `! v_^ ]^_ = 1 1 []fj^_vfW'

jjW' + 1 q]c^_v

cW' + 1 B]d^_v

dW' + 1 !n]j^_v

jW'

1 `xtFvAvW' 1 1 1 1 `[fj_[]fj^_v

fW'

jW' + oqJ 1 q]c^_v

cW' + oB 1 B]d^_v

dW'A

_W' + 1 `nj_v!n]j^_v

jW'

^W' n!q

v mtv 1 oo(oPh)(P,h) + 1 1 1 `[fj_[fj_

fW'

jW'A

_W'+ 1 1 `nj_v 1 !nnju_vuW' + 1 !n]j^_v

^W' + 1 !nqjc_v

cW' 1 [nfuj_v

uW' 1 Bndj_v

dW'

jW'

A_W

1 qncj_vcW' 1 !nnuj_v

uW' + 1 1 oq 1 1 [qfjc_v

jW'

fW' + 1 Bqdc_v

dW' + 1 !nqjc_v

jW'

cW'A

_W+ oqJ 1 q]c^_v^W' + 1 qncj_v

jW' + 1 1 oB 1 B]d^_v

^W' + 1 Bndj_v

jW' + 1 Bqdc_v

cW'

dW'

A_W

34

CHAPTER 3 COMPUTATIONAL RESULTS

Numerical examples are provided for different cases using the deterministic, stochastic - minimizing

expected cost, and stochastic minimizing CVaR approaches as discussed in the previous chapter.

We will conduct our analysis for four time periods or a total of four months. The natural gas

network will be composed of three separate storage areas with maximum storage capacities, injection

and withdrawal rates and unit costs listed in (Table 2); two supply or source locations with their

respective production limits presented in (Table 3). Finally, the pipeline capacities will be limited to a

maximum of 10,000,000 MMBtu available for purchase in contract with an average unit cost of

$0.86/MMBtu per capacity purchased for each pipeline. There also exist six different market

forward types available for delivery at different time periods as listed in (Table 1). Forwards can be

delivered to any of the market hubs for redistribution and we set the maximum number of forwards

which can be purchased of each type for delivery at each hub to 10 units (ie. 10 forwards/forward

type/delivery hub).

Forward Type

($/MMBtu) Upperbound (MMBtu)

Lowerbound (MMBtu) Time 1 Time 2 Time 3

F1fixed 7.50 8.40 10.40 5000 5000

F2fixed 7.40 8.70 8.90 8500 8500

F3fixed 7.30 8.50 8.40 7300 7300

F1variable 7.80 8.90 8.60 7400 7000

F2variable 7.90 9.10 9.40 3200 3000

F3variable 8.00 8.10 6.70 6500 5000

Table 1 Fixed and variable volume market forwards

Location Maximum Storage

Capacities

Injection/Withdrawal Rates (MMBtu/Time

Period)

Unit Cost of Injection/Withdrawal

($/Mmbtu)

Storage 1 1000000 10000 0.05

Storage 2 1000000 10000 0.05

Storage 3 1000000 10000 0.05

Table 2 Storage capacity limits, unit costs, injection/withdrawal rate

35

Location Time 0 Time 1 Time 2 Time 3

Production Limits (MMBtu)

Source 1 10000 10000 10000 10000

Source2 15000 15000 15000 15000

Unit cost of Production ($/MMBtu)

Source 1 5.50 5.50 5.50 5.50

Source2 5.50 5.50 5.50 5.50

Table 3 Production costs and limits from Sources/Supplies

3.1 Deterministic Approach

We will consider five separate locations where demand must be met in the volumes listed in( Table

4). We will assume that the spot market prices of natural gas at each of the hubs at each time period

will take on the values listed in (Table 5).

It is determined that the minimum cost benchmark to meet demands between time periods 1 to 3

contains 3 F3 variable (5000 MMBtu), 1 F3 variable (6100 MMBtu), 3 F3 fixed forwards for time 1;

3 F1fixed, 9 F3 variable (6500 MMBtu), 1 F3 variable (6000 MMBtu) forwards for time 2; and 9 F3

Variable (6500 MMBtu), 1 F3 Variable (5900 MMBtu), 2 F3 fixed, 1 F2 variable (3000 MMBtu)

forwards for time 3. The total cost of the minimum bench mark is $1,514,440.

Assuming all demands and Spot Market prices will follow the values as predicted, the optimal

strategy to the deterministic problem is shown in Table 6, Table 7,Table 8 and Table 9. The overall

system cost takes on a negative value, meaning arbitrage opportunities are taken. That is when in

excess of meeting the demand requirement; natural gas is also sold on the spot market to generate

revenue. The optimal strategy involves purchasing the maximum number of forwards with a

relatively large price difference between the forward price and spot market price; purchasing on the

spot market at the hub which has the lowest price for selling at the hub location with the highest

36

price during the time period; and producing at the maximum capacity at all two sources. The

delivery of the forwards are generally set at the hub locations where they are sold on the spot market

to reduce/eliminate pipeline acquisition costs. For example, during time period 2, 10 forwards of F1

fixed are delivered to both hubs 1 and 3 for spot market selling without redistribution. This results

in a revenue generation of $0.03 per unit and $0.14 per unit for hubs 1 and 3, respectively without

additional transportation cost considerations. The maximum volume purchased on the spot market

for arbitrage opportunity in location is dictated by the maximum pipeline capacity. For example,

during time period 1, a total of 20,000,000 MMBtu of gas is purchased at hub 2 for the unit price of

$4.26 to sell at Hubs 1 and 3 at 10,000,000 MMBtu each for a price difference of $3.19. A spot

market location arbitrage strategy is also dependent on the relative pipeline acquisition costs and the

availability of the capacity in the case when a more favorable opportunity also exists (such as in the

case for time period 3, when forwards F3 fixed and F3 variable are favored over the Hub 3 spot

market price for selling at Hub 1). Both sources are set to produce at their maximum capacities as

the cost of production is considerably lower than spot and forwards prices. As the storage

withdrawal and injection rates are restricted, the optimal strategy also includes selling on the spot

market from the reserves during time periods 0 (Now) and 3 when the spot prices are the highest,

and storing gas from purchases off of the spot market during time period 1 when the market price is

the lowest. Demand requirements are satisfied by a combination of volumes from spot market

purchases and forwards, determined by the lower cost strategy out of the two options. Using

volumes from storages are not favored as the withdrawal rate is highly limited.

The calculated total cost to meet demand for time periods 1, 2, and 3 is $1,375,650 compared to our

minimum benchmark strategy of $1,514,440. This results in a total saving of $138,790.00 using the

deterministic model. This saving value can be used as a benchmark for setting price strategies on the

gas volumes delivered to the demand customers.

37

Location (MMBtu)

Time 1 Time 2 Time 3

Demand 1 9600 12000 16000

Demand 2 6400 14000 15000

Demand 3 8000 17000 16500

Demand 4 6500 18000 15500

Demand 5 7500 18500 19000

Table 4 Demand requirement for each of the time period in MMBtu

Location ($/MMBtu)

Time 0 Time 1 Time 2 Time 3

Spot Price Hub1 9.00 7.45 8.43 9.45

Spot Price Hub2 9.10 4.26 8.34 7.64

Spot Price Hub3 8.90 7.45 8.54 8.45

Table 5 Spot market prices at each of the time period in $/MMBtu

38

Capacity Acquisition Strategy

Destination Locations

Storage 1 Storage 2 Storage 3 Hub 1 Hub 2 Hub 3 Demand 1 Demand2 Demand 3 Demand 4 Demand 5 D

e

p

a

r

t

u

r

e

L

o

c

a

t

i

o

n

s

Storage 1 0 0 0 10000 0 0 0 0 0 0 0

Storage 2 0 0 0 10000 0 0 0 0 0 0 0

Storage 3 0 0 0 10000 0 0 0 0 0 0 0

Hub 1 0 0 0 0 0 0 0 0 0 0 0

Hub 2 10000 10000 10000 10000000 0 10000000 16000 15000 17000 18000 19000

Hub 3 0 0 0 10000000 0 0 0 0 0 0 0

Source 1 0 0 0 10000 0 0 0 0 0 0 0

Source 2 0 0 0 15000 0 0 0 0 0 0 0

Total Capacity Acquired 30170000

Total Cost of Capacity $ 25,946,200.00

Table 6 Capacity Acquisition Strategy for Deterministic Approach, all units are stated in MMBtu, unless otherwise indicated

39

Spot Price ($/MMBtu)

Total Purchased on

the Spot Market

Total Sold on Spot Market

F1 fixed

F2 fixed

F3 fixed

F4 variable

F5 variable

F6 variable

Total in Storage Total Production

from Source (MMBtu)

Total Cost ($) Total Revenue

($) Net Cost ($)

Time 0

Hub 1 $9.00 0 10040000 Storage 1 5000 Source 1 10000 55,000.00 90,360,000.00 -90,305,000.00 Hub 2 $9.10 0 0 Storage 2 5000 Source 2 15000 82,500.00 0.00 82,500.00 Hub 3 $8.90 10000000 0 Storage 3 5000 89,000,000.00 0.00 89,000,000.00

Time 1

Hub 1 $7.45 0 10256000 0 85000 73000 0 0 0 Storage 1 0 Source 1 10000 1,217,150.00 76,407,200.00 -75,190,050.00 Hub 2 $4.26 20068000 0 0 0 0 0 0 0 Storage 2 0 Source 2 15000 85,572,430.00 0.00 85,572,430.00 Hub 3 $7.45 0 10085000 0 85000 73000 0 0 0 Storage 3 0 1,162,150.00 75,133,250.00 -73,971,100.00

Time 2

Hub 1 $8.43 0 10140000 5000

0 0 0 0 0 65000 Storage 1 10000 Source 1 10000 1,002,000.00 85,480,200.00 -84,478,200.00 Hub 2 $8.34 20018500 4000 0 0 0 0 0 65000 Storage 2 10000 Source 2 15000 167,563,790.00 33,360.00 167,530,430.00

Hub 3 $8.54 0 10188000 5000

0 0 73000 0 0 65000 Storage 3 10000 1,567,500.00 87,005,520.00 -85,438,020.00

Time 3

Hub 1 $9.45 0 20384000 0 85000 73000 74000 32000 65000 Storage 1 10000 Source 1 10000 2,797,900.00 192,628,800.00 -189,831,400.00 Hub 2 $7.64 20017000 0 0 0 0 0 0 65000 Storage 2 10000 Source 2 15000 153,448,380.00 0.00 153,447,880.00 Hub 3 $8.45 9862000 10000000 0 0 73000 0 0 65000 Storage 3 10000 84,383,100.00 84,500,000.00 -117,400.00

Total System Cost (Revenue) -103,696,430.00

Table 7 Overall Cost of the System using the Deterministic Approach, all units are stated in MMBtu unless otherwise indicated

40

Spot Market Profits

Spot Market Hub Location for Selling


Spot Purchase for Arbitrage

F1 fixed F2 fixed F3 fixed F4

variable F5

variable F6

variable

Total Storage for Spot Market

Selling

Total Production towards Spot Market

Selling Net Profit ($)

Time 0

Hub 1 Hub 1 9.00 0 Storage 1 5000 Source 1 10000

Hub 2 9.10 0

Storage 2 5000 Source 2 15000

Hub 3 8.90 10000000 Storage 3 5000

1000000

134250

87500 1,221,750.00

Hub 2 Hub 1 9.00 0


Hub 2 9.10 0


Hub 3 8.90 0 Storage 3 0

0

0

0 0.00

Hub 3 Hub 1 9.00 0


Hub 2 9.10 0


Hub 3 8.90 0 Storage 3 0 0 0 0 0.00

Time 1

Hub 1 Hub 1 7.45 0 0 85000 73000 0 0 0 Storage 1 0 Source 1 10000

Hub 2 4.26 10000000 0 0 0 0 0 0 Storage 2 0 Source 2 15000

Hub 3 7.45 0 0 0 73000 0 0 0 Storage 3 0 31900000 0 4250 21900 0 0 0 0 48750 31,974,900.00



Hub 3 7.45 0 0 0 0 0 0 0 Storage 3 0

0 0 0 0 0 0 0 0 0 0.00



Hub 3 7.45 0 0 85000 0 0 0 0 Storage 3 0

31900000 0 4250 0 0 0 0 0 0 31,904,250.00

Time 2 Hub 1 Hub 1 8.43 0 50000 0 0 0 0 65000 Storage 1 0 Source 1 10000


Hub 3 8.54 0 0 0 0 0 0 0 Storage 3 0 900000 1500 0 0 0 0 21450 0 73250 996,200.00



Hub 3 8.54 0 0 0 0 0 0 0 Storage 3 0 0 0 0 0 0 0 960 0 0 960.00



Hub 3 8.54 50000 0 73000 0 0 65000 Storage 3 0

2000000 7000 0 2920 0 0 28600 0 0 2,038,520.00

Time 3 Hub 1 Hub 1 9.45 0 85000 73000 74000 32000 65000 Storage 1 10000 Source 1 10000


Hub 3 8.45 9862000 0 0 73000 0 0 65000 Storage 3 10000

27962000 0 46750 153300 62900 1600 357500 282000 98750 28,964,800.00

41



Hub 3 8.45 0 0 0 0 0 0 0 Storage 3 0

0 0 0 0 0 0 0 0 0 0.00



Hub 3 8.45 0 0 0 0 0 0 0 Storage 3 0

8100000 0 0 0 0 0 0 0 0 8,100,000.00

Total Net Profit 105,201,380.00

Table 8 Spot Market Trading Profits from Deterministic Approach, all units are stated in MMbtu unless otherwise indicated

42

Cost to Meet Demand

Time Period Demand Location


Spot Purchase

for Demand

F1fixed F2

fixed F3

fixed F4

variable F5

variable F6

variable Cost to Meet Demand ($)

Time 1 Demand 1 Hub 1 7.45 0 0 0 0 0 0 0 0.00

Hub 2 4.26 9600 0 0 0 0 0 0 40,896.00

Hub 3 7.45 0 0 0 0 0 0 0 0.00 40,896.00 Demand 2 Hub 1 7.45 0 0 0 0 0 0 0 0.00

Hub 2 4.26 6400 0 0 0 0 0 0 27,264.00

Hub 3 7.45 0 0 0 0 0 0 0 0.00 27,264.00 Demand 3 Hub 1 7.45 0 0 0 0 0 0 0 0.00

Hub 2 4.26 8000 0 0 0 0 0 0 34,080.00

Hub 3 7.45 0 0 0 0 0 0 0 0.00 34,080.00 Demand 4 Hub 1 7.45 0 0 0 0 0 0 0 0.00

Hub 2 4.26 6500 0 0 0 0 0 0 27,690.00

Hub 3 7.45 0 0 0 0 0 0 0 0.00 27,690.00 Demand 5 Hub 1 7.45 0 0 0 0 0 0 0 0.00

Hub 2 4.26 7500 0 0 0 0 0 0 31,950.00

Hub 3 7.45 0 0 0 0 0 0 0 0.00 31,950.00 Total Cost to Meet Demand During Time 1 161,880.00 Time 2 Demand 1 Hub 1 8.43 0 0 0 0 0 0 0 0.00

Hub 2 8.34 0 0 0 0 0 0 12000 97,200.00

Hub 3 8.54 0 0 0 0 0 0 0 0.00 97,200.00 Demand 2 Hub 1 8.43 0 0 0 0 0 0 0 0.00

Hub 2 8.34 0 0 0 0 0 0 14000 113,400.00

Hub 3 8.54 0 0 0 0 0 0 0 0.00 113,400.00 Demand 3 Hub 1 8.43 0 0 0 0 0 0 0 0.00

Hub 2 8.34 0 0 0 0 0 0 17000 137,700.00

Hub 3 8.54 0 0 0 0 0 0 0 0.00 137,700.00 Demand 4 Hub 1 8.43 0 0 0 0 0 0 0 0.00

Hub 2 8.34 0 0 0 0 0 0 18000 145,800.00

Hub 3 8.54 0 0 0 0 0 0 0 0.00 145,800.00 Demand 5 Hub 1 8.43 0 0 0 0 0 0 0 0.00

Hub 2 8.34 18500 0 0 0 0 0 0 154,290.00

Hub 3 8.54 0 0 0 0 0 0 0 0.00 154,290.00 Total Cost to Meet Demand During Time 2 648,390.00 Time 3 Demand 1 Hub 1 9.45 0 0 0 0 0 0 0 0.00

Hub 2 7.64 1500 0 0 0 0 0 14500 108,610.00

Hub 3 8.45 0 0 0 0 0 0 0 0.00 108,610.00 Demand 2 Hub 1 9.45 0 0 0 0 0 0 0 0.00

Hub 2 7.64 0 0 0 0 0 0 15000 100,500.00

Hub 3 8.45 0 0 0 0 0 0 0 0.00 100,500.00 Demand 3 Hub 1 9.45 0 0 0 0 0 0 0 0.00

Hub 2 7.64 0 0 0 0 0 0 16500 110,550.00

Hub 3 8.45 0 0 0 0 0 0 0 0.00 110,550.00 Demand 4 Hub 1 9.45 0 0 0 0 0 0 0 0.00

Hub 2 7.64 15500 0 0 0 0 0 0 118,420.00

Hub 3 8.45 0 0 0 0 0 0 0 0.00 118,420.00 Demand 5 Hub 1 9.45 0 0 0 0 0 0 0 0.00

Hub 2 7.64 0 0 0 0 0 0 19000 127,300.00

Hub 3 8.45 0 0 0 0 0 0 0 0.00 127,300.00 Total Cost to Meet Demand During Time 2 565,380.00

The Total Cost to Meet Demand 1,375,650.00 Table 9 Cost to Meet Demand from Deterministic Approach, all units are stated in MMbtu unless otherwise indicated

43

3.2 Expected Cost Approach

We now consider the case which uncertainty exists with the spot market prices at each of the time

periods. We will assume that the spot market prices are stochastic and take on the values as listed in

Table 10. with the probabilities of occurrence at 35%, 45% and 20% for scenarios 1, 2, and 3,

respectively.

Scenario Probability Location ($/MMBtu)

Time 1 Time 2 Time 3

Scenario 1 0.35 Spot Price Hub1 7.45 8.43 9.45

Spot Price Hub2 4.26 8.34 7.64


Scenario 2 0.45 Spot Price Hub1 9 9.1 11.32

Spot Price Hub2 6 9 8.95


Scenario 3 0.2 Spot Price Hub1 7 8.2 9.11

Spot Price Hub2 5 8.2 7.5

Spot Price Hub3 7 8.4 8.21

Table 10 Spot market prices in the three scenarios considered

3.2.1 Deterministic demand

As the demand requirements; forward options; cost of the minimum benchmark; storage,

production and capacity prices and limitations remain the same as the deterministic model, the

optimal replication strategy is shown in Table 12, Table 13, Table 14, and Table 15. The spot market

prices for scenario 1 are the same as the conditions used in the deterministic model. Scenarios 2 and

3, presents the cases for which the spot price are expected to be relatively higher and lower,

respectively. Under scenario 2, when the spot prices are relatively higher for all time periods, it is

favorable to purchase more forwards to sell on the spot market. Purchasing more forwards would

produce undesirable outcomes for some instances in scenarios 3 (For example, Time 1, spot markets

1 and 3), when the spot market prices drop below the original prices of the forwards. However,

since the optimal strategy is based on the expected value, therefore, because of a higher probability

of occurrence for scenarios 1 and 2, the overall strategy would remain as taking advantage of the

arbitrage opportunity by purchasing more forwards. If each scenario is calculated independently

using the deterministic model, the volumes of forwards purchased to sell on the spot market would

wholly depend on the relative prices of the two. With the stochastic approach, a strategy which

optimizes the expected results is determined. Such strategy would reduce the risks of uncertainty by

taking into consideration of different possible outcomes. The uncertainty of the prices does not

44

negatively impact the type of strategies associated with location arbitrage, and selling on the spot

market from both production and storages, because recourse decisions are determined for each time

period under each scenario to take into consideration of the negative effects of undesired outcomes.

However the magnitude of the arbitrage opportunity is affected by the uncertainty of prices.

Similar to the deterministic model, in the stochastic approach, demand requirements are satisfied by

a combination of volumes from spot market purchases and forwards, determined by the lower cost

strategy out of the two options. In scenario 2, when the spot market prices are high, more demand

volumes are met through forward contracts. While in scenario 3, more demand volumes are met

through spot market purchases.

The expected cost to meet demand is $1,447,077.00, with an expected savings of $67,363.00. With

the uncertainty of the spot market prices factored in our model, the cost to meet demand for each

scenario is still less than the minimum benchmark cost. All decisions determined by the stochastic

model would heavily rely on correct scenarios and expected values used.

Scenario Total Cost to Meet Demand

Minimum Benchmark Cost

1 $1,429,270.00 $1,514,440.00

2 $1,472,650.00 $1,514,440.00

3 $1,420,700.00 $1,514,440.00

Table 11 Cost to Meet Demand for the Three Scenarios Considered Using Stochastic - Minimizing Expected Cost Approach

45

Capacity Acquisition Strategy

Destination Locations

Storage 1 Storage 2 Storage 3 Hub 1 Hub 2 Hub 3 Demand 1 Demand2 Demand 3 Demand 4 Demand 5

D

e

p

a

r

t

u

r

e

L

o

c

a

t

i

o

n

s

Storage 1 0 0 0 10000 0 0 0 0 0 0 0

Storage 2 0 0 0 10000 0 0 0 0 0 0 0

Storage 3 0 0 0 10000 0 0 0 0 0 0 0

Hub 1 0 0 0 0 0 0 0 0 0 0 0

Hub 2 10000 10000 10000 10000000 0 10000000 16000 15000 17000 18000 19000

Hub 3 0 0 0 10000000 0 0 0 0 0 0 0

Source 1 0 0 0 10000 0 0 0 0 0 0 0

Source 2 0 0 0 15000 0 0 0 0 0 0 0

Total Capacity Acquired 30170000

Total Cost of Capacity $ 25,946,200.00 Table 12 Capacity Acquisition Strategy for Stochastic Expected Cost Approach (Deterministic Demand), all units are stated in MMBtu, unless otherwise indicated

46

Scenario Spot Price ($/MMBtu)

Total Purchased

on the Spot

Market

Total Sold on Spot

Market

F1 fixed

F2 fixed

F3 fixed

F4 variable

F5 variable

F6 variable

Total in Storage Total

Production from Source

Total Cost ($)

Total Revenue

($) Net Cost ($)

Time 0

Hub 1 9.00 0 10040000 Storage 1 5000 Source 1 10000 55000 90360000 -90,305,000.00 Hub 2 9.10 0 0 Storage 2 5000 Source 2 15000 82500 0 82,500.00 Hub 3 8.90 10000000 0 Storage 3 5000 89000000 0 89,000,000.00

System Cost During Time 0 (Now) -1,222,500.00 Time 1

Scenario 1 (0.35)

Hub 1 7.45 0 10478000 50000 85000 73000 74000 32000 65000 Storage 1 0 Source 1 10000 2,942,150.00 78061100 -75,118,950.00 Hub 2 4.26 20068000 0 0 0 0 0 0 0 Storage 2 0 Source 2 15000 85,572,430.00 0 85,572,430.00 Hub 3 7.45 0 10208000 50000 85000 73000 74000 0 0 Storage 3 0 2,114,350.00 76049600 -73,935,250.00

-63,481,770.00

Scenario 2 (0.45)

Hub 1 9.00 0 20404000 50000 85000 73000 74000 32000 65000 Storage 1 0 Source 1 10000 2,942,150.00 183636000 -180,693,850.00 Hub 2 6.00 20068000 0 0 0 0 0 0 0 Storage 2 0 Source 2 15000 120,490,750.00 0 120,490,750.00 Hub 3 8.50 9926000 10208000 50000 85000 73000 74000 0 0 Storage 3 0 86,485,350.00 86768000 -282,650.00

-60,485,750.00

Scenario 3 (0.20)


-39,322,750.00 System Cost DuringTime 1 -57,301,757.00

Time 2

Scenario 1 (0.35)


-2,366,000.00

Scenario 2 (0.45)

Hub 1 9.10 0 20223000 50000 0 73000 0 0 65000 Storage 1 10000 Source 1 10000 1,622,500.00 184029300 -182,406,800.00 Hub 2 9.00 10024500 123000 50000 0 73000 0 0 65000 Storage 2 10000 Source 2 15000 91,870,500.00 1107000 90,763,500.00 Hub 3 8.90 9950000 138000 50000 0 73000 0 0 65000 Storage 3 10000 90,122,500.00 1228200 88,894,300.00

-2,749,000.00

Scenario 3 (0.20)

Hub 1 8.20 0 213000 50000 0 73000 0 0 65000 Storage 1 10000 Source 1 10000 1,622,500.00 1746600 -124,100.00 Hub 2 8.20 10047500 156000 50000 0 73000 0 0 65000 Storage 2 10000 Source 2 15000 84,039,500.00 1279200 82,760,300.00 Hub 3 8.40 0 10188000 50000 0 73000 0 0 65000 Storage 3 10000 1,567,500.00 85579200 -84,011,700.00

-1,375,500.00 System Cost DuringTime 2 -2,340,250.00

Time 3

Scenario 1 (0.35)

Hub 1 9.45 0 20384000 0 85000 73000 74000 32000 65000 Storage 1 10000 Source 1 10000 2,797,900.00 192628800 -189,830,900.00 Hub 2 7.64 20066000 49000 0 0 0 0 0 65000 Storage 2 10000 Source 2 15000 153,822,740.00 374360 153,448,380.00 Hub 3 8.45 9926000 10138000 0 0 73000 74000 0 65000 Storage 3 10000 85,560,300.00 85666100 -105,800.00

-36,488,320.00

Scenario 2 (0.45)

Hub 1 11.32 0 20384000 0 85000 73000 74000 32000 65000 Storage 1 10000 Source 1 10000 2,797,900.00 230746880 -227,948,980.00 Hub 2 8.95 10017000 0 0 0 0 0 0 65000 Storage 2 10000 Source 2 15000 90,170,650.00 0 90,170,650.00 Hub 3 8.95 10000000 212000 0 0 73000 74000 0 65000 Storage 3 10000 91,186,600.00 1897400 89,289,200.00

-48,489,130.00

Scenario 3 (0.20)

Hub 1 9.11 0 20384000 0 85000 73000 74000 32000 65000 Storage 1 10000 Source 1 10000 2,797,900.00 185698240 -182,900,340.00 Hub 2 7.50 20035500 18500 0 0 0 0 0 65000 Storage 2 10000 Source 2 15000 150,784,750.00 138750 150,646,000.00

47

Hub 3 8.21 9853000 10065000 0 0 73000 74000 0 65000 Storage 3 10000 82,578,730.00 82633650 -54,920.00 -32,309,260.00

System Cost During Time 3 -41,052,872.50

Total System Cost -101,917,379.50 Table 13 Overall System Cost for Stochastic Expected Cost Approach (Deterministic Demand), all units are stated in MMBtu, unless otherwise indicated

48

Spot Market Profits

Time Period

Scenario

Spot Market Hub

Location for Selling


Spot Purchase

for Arbitrage

F1 fixed

F2 fixed

F3 fixed

F4 variable

F5 variable

F6 variable

Total Storage for Spot Market Selling

Total Production towards Spot Market

Selling Net Profit ($)

Time 0

Hub 1 Hub 1 9.00 Storage 1 5000 Source 1 10000

Hub 2 9.10


Hub 3 8.90 10000000 Storage 3 5000

1000000 134250 87500 1221750

Hub 2 Hub 1 9.00

Storage 1

Source 1

Hub 2 9.10

Storage 2

Source 2

Hub 3 8.90

Storage 3

0 0 0 0.00

Hub 3 Hub 1 9.00

Storage 1

Source 1

Hub 2 9.10

Storage 2

Source 2

Hub 3 8.90 Storage 3

0

0

0 0.00

Time 1

Scenario 1 (0.35)

Hub 1 Hub 1 7.45

50000 85000 73000 74000 32000 65000 Storage 1

Source 1 10000

Hub 2 4.26 10000000 Storage 2

Source 2 15000

Hub 3 7.45

74000 Storage 3

31900000 -2500 4250 10950 -51800 -14400 -35750 0 48750 31,859,500.00

Hub 2 Hub 1 7.45

Storage 1

Source 1

Hub 2 4.26

Storage 2

Source 2

Hub 3 7.45

Storage 3

0 0 0 0 0 0 0 0 0 0.00

Hub 3 Hub 1 7.45

Storage 1

Source 1

Hub 2 4.26 10000000 Storage 2

Source 2

Hub 3 7.45 50000 85000 73000

Storage 3

31900000 -2500 4250 10950 0 0 0 0 0 31,912,700.00

Scenario 2 (0.45)

Hub 1 Hub 1 9.00 50000 85000 73000 74000 32000 65000 Storage 1

Source 1 10000

Hub 2 6.00 10000000 Storage 2

Source 2 15000

Hub 3 8.50 9926000

74000 Storage 3

34963000 75000 136000 124100 177600 35200 65000 0 87500 35,663,400.00

Hub 2 Hub 1 9.00

Storage 1

Source 1

Hub 2 6.00

Storage 2

Source 2

Hub 3 8.50

Storage 3

0 0 0 0 0 0 0 0 0 0.00

Hub 3 Hub 1 9.00

Storage 1

Source 1

Hub 2 6.00 10000000 Storage 2

Source 2

Hub 3 8.50 50000 85000 73000

Storage 3

25000000 50000 93500 87600 0 0 0 0 0 25,231,100.00

Scenario 3 (0.20)

Hub 1 Hub 1 7.00 50000 85000 73000 74000 32000 65000 Storage 1

Source 1 10000

Hub 2 5.00 10000000 Storage 2

Source 2 15000

Hub 3 7.00

50000

74000 Storage 3

20000000 -50000 -34000 -21900 -118400 -28800 -65000 0 37500 19,719,400.00

Hub 2 Hub 1 7.00

Storage 1

Source 1

Hub 2 5.00 Storage 2

Source 2

49

Hub 3 7.00

Storage 3

0 0 0 0 0 0 0 0 0 0.00

Hub 3 Hub 1 7.00

Storage 1

Source 1

Hub 2 5.00 10000000 Storage 2

Source 2

Hub 3 7.00

85000 73000

Storage 3

20000000 0 -34000 -21900 0 0 0

0

0 19,944,100.00

Time 2

Scenario 1 (0.35)

Hub 1 Hub 1 8.43 50000 73000 65000 Storage 1

Source 1 10000

Hub 2 8.34 10000000

Storage 2

Source 2 15000

Hub 3 8.54

Storage 3

900000 1500 0 -5110 0 0 21450 0 73250 991,090.00

Hub 2 Hub 1 8.43

Storage 1

Source 1

Hub 2 8.34

38000 73000 32500 Storage 2

Source 2

Hub 3 8.54

Storage 3

0 -2280 0 -11680 0 0 7800 0 0 -6,160.00

Hub 3 Hub 1 8.43

Storage 1

Source 1

Hub 2 8.34 10000000

Storage 2

Source 2

Hub 3 8.54 50000 73000 65000 Storage 3

2000000 7000 0 2920 0 0 28600 0 0 2,038,520.00

Scenario 2 (0.45)

Hub 1 Hub 1 9.10 50000 73000 65000 Storage 1

Source 1 10000

Hub 2 9.00 10000000 Storage 2

Source 2 15000

Hub 3 8.90 9950000 50000 0 Storage 3 10000

2990000 70000 0 43800 0 0 65000 90500 90000 3,349,300.00

Hub 2 Hub 1 9.10

Storage 1

Source 1

Hub 2 9.00 50000 73000

Storage 2

Source 2

Hub 3 8.90

Storage 3

0 30000 0 36500 0 0 0 0 0 66,500.00

Hub 3 Hub 1 9.10

Storage 1

Source 1

Hub 2 9.00

Storage 2

Source 2

Hub 3 8.90

73000 65000 Storage 3

0 0 0 29200 0 0 52000 0 0 81,200.00

Scenario 3 (0.20)

Hub 1 Hub 1 8.20 50000 73000 65000 Storage 1

Source 1 10000

Hub 2 8.20

Storage 2

Source 2 15000

Hub 3 8.40

Storage 3

0 -10000 0 -21900 0 0 6500 0 67500 42,100.00

Hub 2 Hub 1 8.20

Storage 1

Source 1

Hub 2 8.20

36000 73000 47000 Storage 2

Source 2

Hub 3 8.40

Storage 3

0 -7200 0 -21900 0 0 4700 0 0 -24,400.00

Hub 3 Hub 1 8.20

Storage 1

Source 1

Hub 2 8.20 10000000

Storage 2

Source 2

Hub 3 8.40 50000 73000 65000 Storage 3

2000000 0 0 -7300 0 0 19500

0

0 2,012,200.00

Time 3

Scenario 1 (0.35)

Hub 1 Hub 1 9.45 85000 73000 74000 32000 65000 Storage 1 10000 Source 1 10000

Hub 2 7.64 10000000 Storage 2 10000 Source 2 15000

Hub 3 8.45 9926000 74000

Storage 3 10000

28026000 0 46750 76650 125800 1600 178750 282000 98750 28,836,300.00

Hub 2 Hub 1 9.45

Storage 1

Source 1

Hub 2 7.64 49000 Storage 2

Source 2

Hub 3 8.45

Storage 3

50

0 0 0 0 0 0 46060 0 0 46,060.00

Hub 3 Hub 1 9.45

Storage 1

Source 1

Hub 2 7.64 10000000

Storage 2

Source 2

Hub 3 8.45 73000

65000 Storage 3

8100000 0 0 3650 0 0 113750 0 0 8,217,400.00

Scenario 2 (0.45)


Hub 2 8.95 10000000


Hub 3 8.95 10000000

Storage 3 10000

47400000 0 205700 213160 201280 61440 300300 338100 145500 48,865,480.00

Hub 2 Hub 1 11.32

Storage 1

Source 1

Hub 2 8.95

Storage 2

Source 2

Hub 3 8.95

Storage 3

0 0 0 0 0 0 0 0 0 0.00

Hub 3 Hub 1 11.32

Storage 1

Source 1

Hub 2 8.95

Storage 2

Source 2

Hub 3 8.95 73000 74000 65000 Storage 3

0 0 0 40150 25900 0 146250 0 0 212,300.00

Scenario 3 (0.20)


Hub 2 7.50 10000000 Storage 2 10000 Source 2 15000

Hub 3 8.21 9853000 73000 74000

Storage 3 10000

24967700 0 17850 103660 75480 -9280 156650 271800 90250 25,674,110.00

Hub 2 Hub 1 9.11

Storage 1

Source 1

Hub 2 7.50

18500 Storage 2

Source 2

Hub 3 8.21

Storage 3

0 0 0 0 0 0 14800 0 0 14,800.00

Hub 3 Hub 1 9.11

Storage 1

Source 1

Hub 2 7.50 10000000 Storage 2

Source 2

Hub 3 8.21

65000 Storage 3

7100000 0 0 0 0 0 98150 0 0 7,198,150.00

Total Profit from Selling on the Spot Market 103,562,411.50 Table 14 Spot Market Transaction for Stochastic Expected Cost Approach (Deterministic Demand), all units are stated in MMBtu, unless otherwise indicated

51

Cost to meet Demand

Time Period

Scenario Demand Location


Spot Purchase

for Demand

F1fixed F2

fixed F3

fixed F4

variable F5

variable F6

variable Cost to Meet Demand ($)

Time 1

Scenario 1

Demand 1 Hub 1 7.45 0 0.00

Hub 2 4.26 9600 40,896.00

Hub 3 7.45 0 0.00

40,896.00

Demand 2 Hub 1 7.45 0 0.00

Hub 2 4.26 6400 27,264.00

Hub 3 7.45 0 0.00

27,264.00

Demand 3 Hub 1 7.45 0 0.00

Hub 2 4.26 8000 34,080.00

Hub 3 7.45 0 0.00

34,080.00

Demand 4 Hub 1 7.45 0 0.00

Hub 2 4.26 6500 27,690.00

Hub 3 7.45 0 0.00

27,690.00

Demand 5 Hub 1 7.45 0 0.00

Hub 2 4.26 7500 31,950.00

Hub 3 7.45 0 0.00

31,950.00

Total Cost to Meet Demand during Time 1 in Scenario 1 161,880.00

Scenario 2

Demand 1 Hub 1 9.00 0.00

Hub 2 6.00 9600 57,600.00

Hub 3 8.50 0.00

57,600.00

Demand 2 Hub 1 9.00 0.00

Hub 2 6.00 6400 38,400.00

Hub 3 8.50 0.00

38,400.00

Demand 3 Hub 1 9.00 0.00

Hub 2 6.00 8000 48,000.00

Hub 3 8.50 0.00

48,000.00

Demand 4 Hub 1 9.00 0.00

Hub 2 6.00 6500 39,000.00

Hub 3 8.50 0.00

39,000.00

Demand 5 Hub 1 9.00 0.00

Hub 2 6.00 7500 45,000.00

Hub 3 8.50 0.00

45,000.00

Total Cost to Meet Demand during ime 1 in Scenario 2 228,000.00

Scenario 3

Demand 1 Hub 1 7.00 0.00

Hub 2 5.00 9600 48,000.00

Hub 3 7.00 0.00

48,000.00

Demand 2 Hub 1 7.00 0.00

Hub 2 5.00 6400 32,000.00

Hub 3 7.00 0.00

32,000.00

Demand 3 Hub 1 7.00 0.00

Hub 2 5.00 8000 40,000.00

Hub 3 7.00 0.00

40,000.00

Demand 4 Hub 1 7.00 0.00

Hub 2 5.00 6500 32,500.00

Hub 3 7.00 0.00

32,500.00

Demand 5 Hub 1 7.00 0.00

Hub 2 5.00 7500 37,500.00

Hub 3 7.00 0.00

37,500.00

Total Cost to Meet Demand duringTime 1 in Scenario 3 190,000.00

52

Total Expected Cost to Meet Demand at Time 1 197,258.00

Time 2

Scenario 1

Demand 1 Hub 1 8.43 0.00

Hub 2 8.34 12000

100,800.00

Hub 3 8.54

0.00 100,800.00 Demand 2 Hub 1 8.43

0.00

Hub 2 8.34

14000 113,400.00

Hub 3 8.54

0.00

113,400.00 Demand 3 H

A Stochastic Programming Approach to Natural Gas Portfolio and Transpofrt Optimisation BSc

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