A Stochastic Programming Approach to Natural Gas Portfolio and Transport Optimization

Transcription

1 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

2 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. 2

3 Acknowledgements This thesis is supervised by Professor R.H. Kwon who provided me with guidance in research and enlightenment through discussions. 3

4 Table of Contents Abstract... 2 Acknowledgements... 3 List of Figures... 5 List of Tables... 6 CHAPTER 1 - INTRODUCTION Natural Gas Market in North America Financial and Operations Risk Management Natural Gas Network and Portfolio Risk Measure Tools and Methodology Value-at-risk Conditional Value-at-risk (CVaR) CVaR Minimization Stochastic Programming CVaR Minimization Approach in Stochastic Programming CHAPTER 2 PROBLEM FORMULATION Deterministic Approach Stochastic Approach Minimizing Expected Cost Stochastic Approach Minimizing CVaR CHAPTER 3 COMPUTATIONAL RESULTS Deterministic Approach Expected Cost Approach Deterministic demand Stochastic demand CVaR Approach Conclusion APPENDIX : REFERENCES OPL format for Deterministic Model OPL Format for Stochastic Model with minimizing Expected Cost OPL format for stochastic model with minimizing CVaR

5 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 Figure 3 Source: US Energy Policy, Schematic Diagram of North American Producing Basins Figure 4 Source: CAPP, North American Transport Pipeline Network Figure 5 Schematic diagram of the Flow of gas between locations within a gas network Figure 6 Portfolio loss distribution of the Expected Cost Approach and CVaR Approach

6 List of Tables Table 1 Fixed and variable volume market forwards Table 2 Storage capacity limits, unit costs, injection/withdrawal rate Table 3 Production costs and limits from Sources/Supplies Table 4 Demand requirement for each of the time period in MMBtu Table 5 Spot market prices at each of the time period in $/MMBtu Table 6 Capacity Acquisition Strategy for Deterministic Approach, all units are stated in MMBtu, unless otherwise indicated Table 7 Overall Cost of the System using the Deterministic Approach, all units are stated in MMBtu unless otherwise indicated Table 8 Spot Market Trading Profits from Deterministic Approach, all units are stated in MMbtu unless otherwise indicated Table 9 Cost to Meet Demand from Deterministic Approach, all units are stated in MMbtu unless otherwise indicated Table 10 Spot market prices in the three scenarios considered Table 11 Cost to Meet Demand for the Three s Considered Using Stochastic - Minimizing Expected Cost Approach Table 12 Capacity Acquisition Strategy for Stochastic Expected Cost Approach (Deterministic Demand), all units are stated in MMBtu, unless otherwise indicated Table 13 Overall System Cost for Stochastic Expected Cost Approach (Deterministic Demand), all units are stated in MMBtu, unless otherwise indicated Table 14 Spot Market Transaction for Stochastic Expected Cost Approach (Deterministic Demand), all units are stated in MMBtu, unless otherwise indicated Table 15 Cost to Meet Demand Stochastic Expected Cost Approach (Deterministic Demand), all units are stated in MMBtu, unless otherwise indicated

7 Table 16 Stochastic Demand Values Table 17 Capacity Acquisition Strategy for Stochastic Expected Cost Approach (Stochastic Demand), all units are stated in MMBtu, unless otherwise indicated Table 18 Overall System Cost for Stochastic Expected Cost Approach (Stochastic Demand), all units are stated in MMBtu, unless otherwise indicated 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 Table 20 Total Cost to Meet Demand for Stochastic Expected Cost Approach (Stochastic Demand), all units are stated in MMBtu, unless otherwise indicated Table 21 VaR and CVaR with various beta values for UB for forward type= Table 22 VaR, CVaR, Maximum number of forwards purchased with respect to varying UB for forward type Table 23 Expected loss and standard deviation with the expected cost and CVaR approaches Table 24 Capacity Acquisition Strategy for Stochastic Minimizing CVaR Approach with β=0.99, all units are stated in MMBtu, unless otherwise indicated Table 25 Overall System Cost for Stochastic Minimizing CVaR Approach with β=0.99, all units are stated in MMBtu, unless otherwise indicated Table 26 Total Profit from Spot Market Transactions for Stochastic Minimizing CVaR Approach with β=0.99, all units are stated in MMBtu, unless otherwise indicated Table 27 Total Cost to meet demand for Stochastic Minimizing CVaR Approach with β=0.99, all units are stated in MMBtu, unless otherwise indicated

8 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. 8

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

10 the time when the contract is engaged. 1 By 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] 10

11 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). Natural Gas Basins in North America North American Market Centers and Figure 3 Source: US Energy Policy, Schematic Diagram of North American Producing Basins Figure 2 Source: Petroleum Encyclopedia, Oil and Gas Journal, Schematic Diagram of North American Market Hubs 2 Units of price is stated in $/Bcf 11

12 Each hub sets the referencing price independently; therefore there are variations between locations. The variation ion 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 aatt the present, stored and sold at a higher price in the future. Potential arbitrage opportunities also exist in location.. Since the spot market prices of gas are set independently at these hubs, there are variations in prices between locations. In such situation, uation, 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 tradingg will be considered. Connecting the producing basin, the end end-users (LDCs) LDCs) natural gas markets, and storage facilities are the transport pipelines. Major pipeline systems in North America are the Trans Trans-Canada Canada Pipelines in Canada and the PG&E Gas transm 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 North American Transport Pipeline Network Figure 4 Source: CAPP, North American Transport Pipeline Network 12

13 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. 13

14 1.4 Risk Measure Tools and Methodology 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 β 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] Ψ(x, α) = (, ) ( ). Where α is the VaR values of the loss associated with the decision vector at a given probability level in (0, 1). [4] α (x) = min α Ψ(x, α) β} 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 subadditivity, the VaR associated with a combination of portfolios can be greater than the sum of the 14

15 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 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 β. 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 β) (, ) ( ) (, ) ( ). Thus CVaR represents the conditional expectation of the loss associated with vector x in the case which the loss exceeds VaR or α (x). 15

16 1.5 CVaR Minimization To utilize CVaR (x) defined by (x) = (1 β) (, ) ( ) (, ) ( ) in an optimization problem, a single function is developed to characterize (x) and α (x). The simplified function becomes[4] F (x, ) = α + (1 β) (, ) (, ) ( ). It was shown in [4] that the F β (x, ) is related to CVaR or β (x) by the formula β (x) = min F β (x, ) = F β (x, (x, β)). α F β (x, ) is convex and continuously differentiable, thus CVaR can be easily determined by minimizing the function numerically. Furthermore, the VaR or, for which F β (x, ) depend on, does not need to be calculated independently in order to derive β (x). Instead VaR can be obtained as a by product. F β (x, ) can be approximated from a set of sample data,, that belongs to a probability space ( ). q + F β (x, ) = α + (q(1 β)) 1 x, y k α k=1 Although F β (x, ) is not differentiable with respect to α, 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: 16

17 min α F β (x, ) = α + (q(1 β)) 1 z q k=1 Subjected to the following constraints: z x, y k α z 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 17

18 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] + (, ) =, 0 where (, ( )) = = h( ), 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 ( ), where an element which determines the nature of the random event with some probability value of occurring. ( ) is formed from a combination of elements in,, h, 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 firststage and second-stage decisions, respectively. For our gas portfolio problem, only the h matrix 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 18

19 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 min (, ) = + (1 ),,, Subjected to the constraints = h + 0,. 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]. 19

20 CHAPTER 2 PROBLEM FORMULATION 2.1 Deterministic Approach We will propose a deterministic linear programming model for our portfolio optimization problem. Deterministic models for the application of gas portfolio and transport have been previously studied in [10]. The strategy is to determine the resource acquisition and dispatch volumes required to meet demand while optimizing total 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. Figure 5 Schematic diagram of the Flow of gas between locations within a gas network We will assume that the market participant owns producing basins, or sources that can supply natural gas and storage spaces for storing gas. volumes of gas must be delivered for demand at locations at time period for = 1,, where denotes the number of time periods of interest for the analysis. Assuming the price charged on a unit of gas used to fulfill the demands is fixed, therefore this revenue stream will be omitted in the objective function. We will also consider different market forwards available to deliver specific volumes of gas at each time periods in the future. The two types of forwards considered, fixed and variable volume 20

21 forwards are denoted as = 1, and = + 1,,, respectively. Where represented the number of fixed volume forwards and represents the number of variable volume contracts. A forward can be purchased at price at Hub h, for h 1,, }. The forward will deliver volume of natural gas during time period. In addition to the forward contracts, spot market purchases can also be made at each hubs for the price of. per unit of gas. Finally, pipeline capacities must be acquired at a volume of for pipeline (, ) for the duration of analysis. The cost per unit of capacity will assumed to be CC ab. The following parameters and decision variables are used in the deterministic model. Parameters t (a,b) Time period 0,, } Pipeline between locations a and b. (, ) 1,, } h, g Hub location h 1,, }, 1,, } i v e d CSI CSW SR Storage location 1,, } Set of market forwards, fixed and variable volume 1,, } Source location 1,, } Demand location 1,, } Unit cost for gas injection from storage Unit cost for gas withdrawal from storage Maximum rate of storage injection or withdrawal 21

22 CE CC SUB PUB EUB LB vt UB vt PM ht PF vth D dt MBS Unit cost for gas extraction from source Unit cost of capacity for pipeline Maximum allowable storage volume of storage Maximum allowable pipeline volume which can be purchased for pipeline Maximum allowable extraction volume for source Minimum allowable volume of Forward vht that can be drawn at time t Maximum allowable volume of Forward vht that can be drawn at time t Price of gas on the spot market at Hub h at time t Price of gas forward associated with Forward vht at Hub h at time t Gas demand at location d at time t Minimum benchmark strategy Decision variable ST it RMT ht BMT ht VF vht ET et FD vhdt FM vhgt FS vhit SD idt SM iht ED edt Volume in Storage it at time t Volume that sold at Hub h at time t Volume that purchased at Hub h at time t Volume that will be delivered by Forward vht at Hub h for time t Volume extracted from Source et at time t Amount of Forward vht used to meet demand at location d, at time t Amount of Forward vht used to sell at Hub g at time Amount of Forward vht used to put in Storage it at time t Amount of Storage it used to meet demand at location d, at time t Amount of Storage it used to sell at Hub h at time t Amount of Source et used to meet demand at location d, at time t 22

23 EM eht ES eit BMD hdt BMS hit BMM hgt FL abt C ab NFP vht Amount of Source et used to sell at Hub h at time t Amount of Source et used to put in Storage i at time t Amount of gas purchased on the spot market at Hub h for demand at location d, at time t Amount of gas purchased in the spot market at Hub h for Storage i, at time t Amount of gas purchased in the spot market at Hub h for Hub g, at time t Amount of flow through in pipeline (a,b) at time t Amount of capacity bought for pipeline (, ) Number of Forward vht purchased at Hub h, 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. + + (h ) + ( + h ) + Where: = ( ) (, ) = 23

24 ( ) = ( + + ) = + + ( + ) = ( + + ) + ( + ) = = + + Constraints Storage balance The amount of natural gas in storage is equal to the amount injected and withdrawn. Gas stored is equivalent to the amount delivered from forwards, spot purchase, and production; gas withdrawn is equivalent to the amount used to satisfy demand requirement and amount sold on the spot market 24

25 = Storage limits Amount stored at a particular storage is less than or equal to the maximum capacity at the storage Amount withdrawn from storage during time t is less than or equal to the amount in storage at the beginning of time t + The rate at which volume is injected or withdrawn from a particular storage is bounded by the maximum rate SR 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 = + + 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 =

26 Forward balance Total forward is equal to the amounts put in storage, used for demand, and sold on the spot market. = + + Forward limits The volume delivered by a particular forward is bounded by the maximum number and minimum number of forwards which can be purchased 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 = + + Source limits The total amount drawn from a source is less than the maximum production capacity Capacity Amount flown through the pipeline is less than the amount of capacity bought (, ) Amount of capacity bought is less than or equal to the maximum capacity which can be acquired for the pipeline Demand Balance (, ) 26

27 The demand requirement is satisfied from volumes delivered by forwards, spot purchases, production and storages = 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

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. independent parameters t s (a,b) Time period 0,, } 1,, } Pipeline between locations a and b. (, ) 1,, } h, g Hub location h 1,, }, 1,, } i v e d CSI CSW SR CE CC SUB Storage location 1,, } Set of market forwards, fixed and variable volume 1,, } Source location 1,, } Demand location 1,, } Unit cost for gas injection from storage Unit cost for gas withdrawal from storage Maximum rate of storage injection or withdrawal Unit cost for gas extraction from source Unit cost of capacity for pipeline Maximum allowable storage volume of storage 28

29 PUB EUB LB vt UB vt PF vth MBS Maximum allowable pipeline volume which can be purchased for pipeline Maximum allowable extraction volume for source Minimum allowable volume of Forward vht that can be drawn at time t Maximum allowable volume of Forward vht that can be drawn at time t Price of gas forward associated with Forward vht at Hub h at time t Minimum benchmark strategy dependent parameters PM hts Prob s D dts Price of gas on the spot market at Hub h at time t, in scenario s Probability of occurrence associated with scenario s Gas demand at location d at time t, scenario s independent decision variable VF vht Volume that will be delivered by Forward vht at Hub h for time t, C ab Amount of capacity bought for pipeline (, ) Decision variable ST its RMT hts BMT hts ET ets FD vhdts FM vhgts FS vhits SD idts SM ihts Volume in Storage it at time t, scenario s Volume that sold at Hub h at time t, scenario s Volume that purchased at Hub h at time t, scenario s Volume extracted from Source et at time t, scenario s Amount of Forward vht used to meet demand at location d, at time t, scenario s Amount of Forward vht used to sell at Hub g at time t, scenario s Amount of Forward vht used to put in Storage it at time t, scenario s Amount of Storage it used to meet demand at location d, at time t, scenario s Amount of Storage it used to sell at Hub h at time t, scenario s 29

30 ED edts EM ehts ES eits BMD hdts BMS hits BMM hgts FL abts NFP vhts Amount of Source et used to meet demand at location d, at time t, scenario s Amount of Source et used to sell at Hub h at time t, scenario s Amount of Source et used to put in Storage i at time t, scenario s Amount of gas purchased on the spot market at Hub h for demand at location d, at time t, scenario s Amount of gas purchased in the spot market at Hub h for Storage i, at time t, scenario s Amount of gas purchased in the spot market at Hub h for Hub g, at time t, scenario s Amount of flow through in pipeline (a,b) at time t, scenario s Number of Forward vht purchased at Hub h, at time t, scenario s ( ) + (, ) Subject to the following constraints =

31 + + + = + + = + + = + + = + + (, ) (, ) =

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: Subject to the following constraints = = + + = + + =

33 = + + (, ) (, ) = ( ) + (, )

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) Time 1 Time 2 Time 3 Upperbound (MMBtu) Lowerbound (MMBtu) F1fixed F2fixed F3fixed F1variable F2variable F3variable 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 Storage Storage Table 2 Storage capacity limits, unit costs, injection/withdrawal rate 34

35 Location Time 0 Time 1 Time 2 Time 3 Production Limits (MMBtu) Source Source Unit cost of Production ($/MMBtu) Source Source 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 35

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, 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. 36

37 Location (MMBtu) Time 1 Time 2 Time 3 Demand Demand Demand Demand Demand Table 4 Demand requirement for each of the time period in MMBtu Location ($/MMBtu) Time 0 Time 1 Time 2 Time 3 Spot Price Hub Spot Price Hub Spot Price Hub Table 5 Spot market prices at each of the time period in $/MMBtu 37

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 Storage Storage Departure Locations Storage Hub Hub Hub Source Source Table 6 Capacity Acquisition Strategy for Deterministic Approach, all units are stated in MMBtu, unless otherwise indicated Total Capacity Acquired Total Cost of Capacity $ 25,946,

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 $ Storage Source , ,360, ,305, Hub 2 $ Storage Source , , Hub 3 $ Storage ,000, ,000, Time 1 Hub 1 $ Storage 1 0 Source ,217, ,407, ,190, Hub 2 $ Storage 2 0 Source ,572, ,572, Hub 3 $ Storage 3 0 1,162, ,133, ,971, Time 2 Hub 1 $ Storage Source ,002, ,480, ,478, Hub 2 $ Storage Source ,563, , ,530, Hub 3 $ Storage ,567, ,005, ,438, Time 3 Hub 1 $ Storage Source ,797, ,628, ,831, Hub 2 $ Storage Source ,448, ,447, Hub 3 $ Storage ,383, ,500, , Table 7 Overall Cost of the System using the Deterministic Approach, all units are stated in MMBtu unless otherwise indicated Total System Cost (Revenue) -103,696,

40 Spot Market Profits Spot Market Hub Location for Selling Spot Price ($/MMBtu) Spot Purchase for Arbitrage F1 fixed F2 fixed F3 fixed F4 variable F5 variable F6 variable Total Storage for Spot Market Selling Total Production towards Spot Market Selling Net Profit ($) Time 0 Hub 1 Hub Storage Source Hub Storage Source Hub Storage ,221, Hub 2 Hub Storage 1 0 Source 1 0 Hub Storage 2 0 Source 2 0 Hub Storage Hub 3 Hub Storage 1 0 Source 1 0 Hub Storage 2 0 Source 2 0 Hub Storage Time 1 Hub 1 Hub Storage 1 0 Source Hub Storage 2 0 Source Hub Storage ,974, Hub 2 Hub Storage 1 0 Source 1 0 Hub Storage 2 0 Source 2 0 Hub Storage Hub 3 Hub Storage 1 0 Source 1 0 Hub Storage 2 0 Source 2 0 Hub Storage ,904, Time 2 Hub 1 Hub Storage 1 0 Source Hub Storage 2 0 Source Hub Storage , Hub 2 Hub Storage 1 0 Source 1 0 Hub Storage 2 0 Source 2 0 Hub Storage Hub 3 Hub Storage 1 0 Source 1 0 Hub Storage 2 0 Source 2 0 Hub Storage ,038, Time 3 Hub 1 Hub Storage Source Hub Storage Source Hub Storage ,964,

41 Hub 2 Hub Storage 1 0 Source 1 0 Hub Storage 2 0 Source 2 0 Hub Storage Hub 3 Hub Storage 1 0 Source 1 0 Hub Storage 2 0 Source 2 0 Hub Storage ,100, Total Net Profit 105,201, Table 8 Spot Market Trading Profits from Deterministic Approach, all units are stated in MMbtu unless otherwise indicated 41

42 Cost to Meet Demand Time Period Time 1 Time 2 Time 3 Demand Location Spot Price ($/MMBtu) Spot Purchase for Demand F1fixed F2 fixed F3 fixed F4 variable F5 variable F6 variable Cost to Meet Demand ($) Demand 1 Hub Hub , Hub , Demand 2 Hub Hub , Hub , Demand 3 Hub Hub , Hub , Demand 4 Hub Hub , Hub , Demand 5 Hub Hub , Hub , Total Cost to Meet Demand During Time 1 161, Demand 1 Hub Hub , Hub , Demand 2 Hub Hub , Hub , Demand 3 Hub Hub , Hub , Demand 4 Hub Hub , Hub , Demand 5 Hub Hub , Hub , Total Cost to Meet Demand During Time 2 648, Demand 1 Hub Hub , Hub , Demand 2 Hub Hub , Hub , Demand 3 Hub Hub , Hub , Demand 4 Hub Hub , Hub , Demand 5 Hub Hub , Hub , Total Cost to Meet Demand During Time 2 565, The Total Cost to Meet Demand 1,375, Table 9 Cost to Meet Demand from Deterministic Approach, all units are stated in MMbtu unless otherwise indicated 42

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. Probability Location ($/MMBtu) Time 1 Time 2 Time Spot Price Hub Spot Price Hub Spot Price Hub Spot Price Hub Spot Price Hub Spot Price Hub Spot Price Hub Spot Price Hub Spot Price Hub Table 10 Spot market prices in the three scenarios considered 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. s 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 43

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, 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. Total Cost to Meet Demand Minimum Benchmark Cost 1 $1,429, $1,514, $1,472, $1,514, $1,420, $1,514, Table 11 Cost to Meet Demand for the Three s Considered Using Stochastic - Minimizing Expected Cost Approach 44