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Abstract—This paper proposes a mathematical model to manage natural gas supply and energy portfolio of a generation company. The model incorporates financial risks associated with the decision-making process of buying and selling both natural gas and electricity while keeping the interaction between the two markets. Using stochastic programming framework, the problem formulation considers uncertainties associated with electricity prices and natural gas consumption, which results in a large scale mixed integer linear programming problem. The financial risks are measured by the conditional-value-at-risk (CVaR) index. A simplified test system is presented and later solved using Xpress-IVE student edition. Value of stochastic solution is calculated, which provides the value of the stochastic model.

Index Terms—Conditional Value at Risk, Energy Portfolio, Natural Gas Supply portfolio, Risk management.

I. NOMENCLATURE Most of the nomenclature follow from [23] and [24].

A. Indices i Hour j Day

Scenarios Set of blocks of contracts that are active during hour i,

day j.

B. Decision Variables Energy generated by a company at hour i, day j, scenario

(MW-h). Energy sold to the pool at hour i, day j, scenario

(MW-h). Power sold through forward contract c block b (MW). Power bought through forward contract c block b (MW). Amount of natural gas delivered from based load natural

gas contract (MBTU). Amount of natural gas delivered from intra-day natural

gas contract at hour i, day j (MBTU).

This work is supported by Singapore Ministry of Education–Academic

Research Fund, Grant No. R-263-000-487-112.

Amount of natural gas delivered from swing natural gas contract at hour i, day j (MBTU). Amount of natural gas injected into natural gas storage facility at hour i, day j (MBTU). Amount of natural gas withdrawn from natural gas storage facility at hour i, day j (MBTU). Amount of natural gas bought from spot market to satisfy electricity production obligation at hour i, day j (MBTU). Amount of natural gas sold to spot market at hour i, day j (MBTU).

Binary variable, equal to 1 if energy forward contract is signed to buy energy, 0 otherwise.

Binary variable, equal to 1 if energy forward contract is signed to sell energy, 0 otherwise.

C. Parameters OC Operation cost of a generation company ($/MW-h) HR Heat rate of a generation company (MBTU/MW-h)

Time duration of intra-day natural gas (hour) Number of hour of a forward contract c, block b

Maximum power that can be sold through block b of a forward contract c (MW).

Maximum power that can be bought through block b of a forward contract c (MW).

MP Maximum capacity of a generation company (MW). mP Minimum capacity of a generation company (MW).

Available power at hour i, day j (MW), unrestricted. Energy price sold through forward contract c block b

($/MW-h). Energy price bought through forward contract c block

b ($/MW-h). Price of base load natural gas ($/MBTU). Price of intra-day natural gas ($/MBTU). Price of swing natural gas at day j ($/MBTU).

Price of natural gas in spot market at day j, scenario ($/MBTU).

Storage facility charge for natural gas ($/MBTU). MG Maximum natural gas in storage facility at the end of

period (MBTU).

An Optimization Model for Risk Management in Natural Gas Supply and Energy Portfolio of a

Generation Company

Usama Asif Panida Jirutitijaroen Department of Electrical & Computer Engineering Department of Electrical & Computer Engineering National University of Singapore National University of Singapore Singapore Singapore usamaasif03@gmail.com elejp@nus.edu.sg

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mG Minimum natural gas in storage facility at the end of period (MBTU). Γ Electricity pool price at hour i, day j, scenario ($/MW-h).

Forecasted natural gas consumption at hour i, day j, scenario (MBTU).

Probability of scenario (MBTU). Confidence level for risk measure Weighting factor for risk measure

II. INTRODUCTION OMPETITION in deregulated electricity market of Singapore has motivated generation companies to use

more efficient ways of producing electricity. As of today, about 80% of electricity in Singapore is generated from natural gas [1]. Optimizing short-term natural gas supply portfolio involves decisions regarding timing, quantity and price of buying and selling natural gas while optimizing electricity production portfolio involves decisions regarding timing, quantity and price of buying and selling electricity. A generation company needs to make a decision of day-to-day fuel mix supply and also of make-or-buy electricity in order to maximize a profit and minimize financial risks at the same time. Thus, a model that intends to incorporate all the financial risks threatening the profits of a generation company must not only take into consideration both the natural gas and electricity market but also the unavoidable interaction between the two.

The objective of this paper is to build a model for a generation company that incorporates the financial risks in the decision-making process of buying and selling in both natural gas and electricity markets while keeping in mind the interaction between the two. The optimization problem will consider uncertainties associated with the electricity price. The fluctuation of the electricity demand throughout a day is represented by the demand of natural gas to produce the same amount of electricity. The financial risks will be measured by the conditional-value-at-risk (CVaR) index. This risk index is used to model the risk preferences of the generation company and study the effect of changes in risk preferences on the optimal results.

A. Literature Review Recent research on the risk management in electricity and

natural gas market is presented. Reference [5], [6], and [7] gives an overview of risk management in electricity markets. A comprehensive work [8] proposed different methods available for market risk analysis. Statistical characterization of risk due to forward contracts in electricity markets is given in [9]. A stochastic programming framework to manage hydropower resources considering both forward contracts and spot markets is presented in [10]. Even though this framework is not directly related to the use of natural gas as a mean of producing electricity, it serves as a good example of formulating the problem of our interest. References [11], [12] and [13] are related to using stochastic programming to make

decisions in electricity markets. Cost minimization framework to optimize natural gas

portfolio is discussed in [14]-[17]. This framework does not take into account all the practical constraints. Ignoring the interaction with the electricity markets, this framework minimally considers the associated financial risks faced by a generation company in today’s deregulated electricity market. Reference [18] mentioned how the risk neutral assumption leads to riskier decisions while using the cost minimization framework. Theoretical framework to optimize natural gas buying is presented in [19] but it neither takes into account practical constraints nor does it consider the interaction with the electricity market. Modeling techniques such as mean-reverting process and diffusion process to model the natural gas market are provided in [20] and [21]. Detailed review of natural gas price volatility is given in [22].

Recently, significant amount of work has been done to incorporate the financial risk associated with the natural gas buying procedure and electricity production procedure but the practicality of these works is limited as both the natural gas and electricity market has either been separately treated or complete interaction has been ignored. Thus the motivation for this work is the complete interaction between the two markets. Reference [23] provides comprehensive risk cost trade-off framework to optimize natural gas portfolio for electric utility companies while [24] provides stochastic programming framework to solve the problem of a power producer facing the possibility of signing forward contracts. These two papers separately provide comprehensive view of natural gas and electricity market. Our aim in this work is to propose an integration of the two markets presented in [23] and [24].

This paper intends to combine the effects of both natural gas and electricity markets into the formulation to present a complete picture of interaction between the two markets. The CVaR is proposed as a risk index to manage risk faced by a generation company. The application of CVaR will also help us study how changes in the risk preference of the generation company affect the optimal results and thus how this framework can be integrated into company-wide risk management.

The problem is formulated in the next section. A simple test system is presented and studied in section IV. Value of stochastic solution is discussed in section V to justify the use of stochastic programming. Concluding remarks are given in section VI.

III. PROBLEM FORMULATION In this section, an interaction between natural gas and

electricity market is first presented in section A. General stochastic programming framework is explained in section B. The decision making process involving with time period is given in section C. An energy portfolio model is proposed in section D.

C

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A. An Electricity Generation Company We consider a generation company with

using natural gas as the main source of fuelbetween a natural gas and electricity marketthe supply and demand of the two coimbalance of supply and demand of one of thleads to the imbalance of supply and demandthis paper, we consider that natural gas and etraded in following two markets.

1. A future market represented by the forw2. A pool market or a spot market. The price of natural gas or electricity in a

is fixed as one of the terms of the contract wspot market is highly volatile. The fixed prcontract may protect the generation comvolatility of returns but it is less profitable. Dbuy or sell i.e. trade in a spot market or contract and if sign a future contract which fsign, how much to sell or buy are imcontributing to the risk faced by a generatiomodel presented in this paper aims to answerThe representation of the interaction betweecompany and the two markets under considthe natural gas market and electricity market1.

Fig. 1. An interaction between natural gas market and egeneration company [5]

Fig. 1 shows four possible transactigeneration company and electricity marketselling and buying electricity in the future eand selling and buying electricity in the market. A generation company can buy the elfuture contract to sell in the spot marketbecause of the high volatility of spot price more profitable. A generation company can athe produced electricity in the future market tbut it would bring fewer returns as comparedspot market. Reference [24] covers only transactions and excludes the possibility of bfrom the spot market. If the price of buyingthe spot market is less than generating egeneration company may decide to buy elecmarket rather than producing it itself. Moremodel assumes that the cost of production constant. This assumption may not be alwproduction cost depends on the cost of natproduce electricity. This brings our attransactions between a generation company

EE

generating units l. The interaction t is dependent on ommodities. The hese commodities d of the other. In electricity can be

ward contract.

forward contract while the price in rice of the future mpany from the Deciding when to

to sign a future future contract to mportant factors on company. The r these questions. en the generation deration namely, t is given in Fig.

electricity market of a

ions between a t. These include electricity market

electricity spot lectricity through . This is riskier but likely to be

also decide to sell to reduce the risk d to selling in the three of these

buying electricity g electricity from lectricity then a

ctricity from spot eover, the above

of electricity is ways valid as the tural gas used to ttention to the

y and the natural

gas market. Fig. 1 also shows three possib

generation company and natural gbuying in the spot market and buyiDifferent contracts have differenconditions. Natural Gas products different groups as follow. 1. Base Load Gas (BLG): These

generation company at a fixedthroughput the period of contra

2. Intra-Day Gas (IDG): These cgeneration company at a fixedperiod of contract at a fixed certain pre-decided time of thelike BLG).

3. Swing Gas (SWG): These conthe gas supply is only on demelectricity from the spot markthat the supply is guaranteed anpremium.

4. Storage Facility (STF): This is company to store the natural gto injection and withdrawal oStorage can be used to store exto be used for electricity prodreached its full capacity, a gensell excess gas to the spot markthis work that without having the generation company does nthe spot market to earn the prcompany policy.

B. A Stochastic Programming FramA stochastic programming frame

this problem due to uncertainties inand the natural gas consumption. delay the decision to buy electricitspot market till the spot prices are sign the future contracts has to beoperational period. Therefore, weusing the recourse model. The amoelectricity or natural gas from the variables. The recourse model takesplanning for one scenario can be affmaterialization of another scenarirepresents the expected profit.

The aim of the recourse modedecision that is well positioned scenarios. The optimal value of firpossible for the second stage decisioutcomes of the scenarios and tunfavorable outcomes.

C. Decision Making Process The focus of this paper is a short

and electricity production portfolioday-to-day operations of a generatio

TENCON 2009

le transactions between a as market are selling and ing from forward contract. nt pricing and supplying

can be divided into four

contracts provide gas to a d flow rate 24 hours a day act at a fixed price. contracts provide gas to a d flow rate throughput the

price but only during a e day (not 24 hour supply

ntracts are most flexible as mand. It is just like buying ket however, difference is nd thus the price charged is

the ability of a generation as. There are costs related of gas from this storage. cess gas or the gas waiting

duction. If the storage has neration company needs to ket. It has been assumed in reached the storage limit,

not deliberately sell gas in rofit. This depends on the

mework ework is used to formulate n the electricity pool price In this problem, we can ty or natural gas from the known but the decision to

e made at the start of the e formulate this problem ounts of buying or selling spot market are recourse

s into account the fact that ffected by the possibility of io. Thus, it meaningfully

el is to give a first stage after considering all the

rst stage decision makes it on to exploit the favorable to be less vulnerable to

t-term natural gas portfolio o, as they directly impact on company. The duration

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of the portfolio considered here is one month. Forward contract decisions are made at the beginning of the month. While, decisions related to spot market are made throughout the month. At the beginning of each month, the optimization problem is solved to decide which contract to sign to buy electricity in the future market, which contract to sign to sell electricity in the future market and which contract to sign to buy natural gas in the future market. The future contracts are considered to be monthly contracts in this case. It should be noted that the trading time-span can be extended beyond one month; however, it greatly increases the problem size.

D. Energy Portfolio Model This section describes a mathematical formulation of the

monthly energy portfolio of a generation company. 1) Objective Function

The objective function (1) of this problem is to maximize the profit and risk measure. = Total Revenue – Total Cost Risk measure (1)

Total revenue (2) consists of the expected revenue from selling electricity to the pool market and the revenue from selling electricity through future contract. Total Revenue Γ

(2) Total cost (3) comprises of the cost of buying electricity

through future contract, the operational cost for running the generation units, and the cost of natural gas, bought through several contracts accordingly. Total Cost

Natural Gas Cost

(3)

The natural gas cost (4) is from based load contract, intra-day contract, swing contract, storage facility cost including injection, , and withdrawal, . The last term correspond to gas bought from spot market, , to satisfy generation obligation and excess gas sold to spot market, . Natural Gas Cost 30 · 24 ·30

(4)

Risk measure used in this analysis is Conditional Value-at-Risk (CVaR). It is calculated from the expected profit not exceeding Value-at-Risk (VaR). VaR is a maximum profit such that probability of the profit being lower than or equal to this value is lower than or equal to 1 , a confidence level. In general, CVaR and VaR is calculated from (5) and (6) respectively. Profit | Profit (5)max | Profit 1 (6)

Using intermediate variable, , CVaR can be found [24] by maximizing (7). This intermediate variable will be zero if a scenario has a profit greater than VaR else it will equal to the difference of VaR and the profit of the scenario . 11

(7)

s.t. Profit 0, 0, (8)

CVaR in (7) is then multiplied by a risk factor, , to represent tradeoff between risk and profit. High weighing factor (closer to 1) implies a conservative portfolio while low weighing factor (closer to 0) characterizes risk taker behavior. Adjustment of this weighing factor will help modeling the risk preferences of a generation company. 2) Problem Constraints

Constraint (9) expresses that electricity sold in pool is equal to electricity bought, in future market minus electricity sold in future market, plus energy generated,

and available power carried forward from the previous planning period,

= , , , , , (9)

Maximum and minimum production capacity of a generation company is described by (10). , , ,

(10)

Constraints (11) and (12) limit the energy bought and sold through each block of each future contract. , ,

(11), , (12)

Constraint (13) and (14) limit the maximum power bought and sold through each contract. , (13) , (14)

Constraint (15) ensures that the generation company cannot buy or sell energy through the same contract. 1,

(15)

Storage facility constraint for natural gas is expressed by (16). , ,

(16)

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Constraint (17) states that gas supply and consumption must be balanced at all time. , , , (17)

The last constraint (18) deals with conversion of natural gas consumption to electricity production through a heat rate, HR, of a generation company. , , ,

(18)

All natural gas and energy decision variables are greater than or equal to zero. The problem is simplified in the next section to make it implementable using Xpress-IVE student editions.

IV. A TEST SYSTEM The resulting formulation in the previous section is a large

scale mixed-integer linear programming problem. The problem size depends not only on the number of contracts but also on the number of scenarios. To simplify the problem, the time span is reduced from a month to six hours. The problem size is given in Table I. This simplified form of the proposed framework is solved using Xpress-IVE student edition.

TABLE I PROBLEM SIZE OF A TEST SYSTEM

Continuous Variable 103 Stochastic Variable 42

Binary Variable 2 Constraints 169

Data used in this application is taken from [23] and [24]

which is mainly obtained from Electricity Reliability Council of Texas (ERCOT). Two electricity forward contracts are considered each with one buying and one selling block of 160MW. Five scenarios are considered each with different probability of having different electricity pool prices and natural gas demand. The consumption of natural gas is taken from historical data given in [23]. A generator company has an operational cost of 51$/MW-h and maximum capacity of 300MW. The heat rate for such generation unit is found to be 10 MBTU/MW-h from NYE Thermodynamics Corporation Canada [27]. All four types of natural gas products mentioned above are taken into account and the cost of each product are taken from [23] and given in Table II.

TABLE II GAS PRODUCT COST

Gas Product Cost ($/MBTU)

Base Load Gas 6 Intra-Day Gas 6.1

Swing Gas 5.5 Storage Facility 0.12

For the given data, the expected profit is found to be

$431,151 with $188,881 being Value-at-Risk when risk weighing factor, W, is taken to be 1. Value of stochastic solution is calculated and the meaningful of stochastic formulation is given in the following section.

V. VALUE OF STOCHASTIC SOLUTION In the stochastic framework, there is a measure that help us

understand how good the stochastic solution is when compared with the deterministic one [28]. This measure is called “Value of the Stochastic Solution” (VSS). It is a difference of the two values. The first one is value of the objective function computed using the stochastic model. In order to compute the second value, the stochastic variables are replaced by their expected values and the first stage decisions from the solution of this deterministic model are used to find the individual objective values for each of the different scenarios. The average value of these objective functions is the second value needed. The percentage difference between the two values indicates whether it is worth the effort to use stochastic framework to compute the model.

For the test system, $278,446 is the profit obtained by solving the stochastic model while $242,629 is the value obtained from the optimizer of the deterministic model. The VSS for this application is calculated below. 278446 242629242629 100% 14.7%

It can be concluded from this percentage that it is worth to formulate the problem using stochastic programming since the percentage is rather small. Thus, the effort spend on stochastic modeling and the complications of computing the volatility in uncertain data elements is important to optimize the natural gas supply and electricity production portfolio for the generation company.

VI. CONCLUSION Electricity and natural gas forward contracts allow the

generation company to hedge against the risk of profit volatility. This volatility comes from the uncertain electricity prices in the pool market and also from the uncertain natural gas consumption which is driven by the electricity demand. We propose a formulation for one-month production planning with financial risks threatening the profits of the generation company. A comprehensive study of both the natural gas and electricity market is done to incorporate the interaction between these two markets into the proposed risk-constrained stochastic programming framework with recourse. The resulting problem is a large scale mixed-integer programming problem.

The Conditional-Value-at-Risk index helps to incorporate the risk preferences of the generation company in the proposed model. This flexible and comprehensive planning tool allows the company to also study the effect of risk preferences on the optimized results. The “Value for Stochastic Solution” is then calculated to appreciate the trade-off and benefit of stochastic framework. The VSS percentage value is small enough to justify the stochastic framework instead of the deterministic approach.

For future work, a solution algorithm for the large-scale model will be proposed. The application of L-shaped algorithm is one of the tools that will be studied. The basic

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idea of this algorithm is to decompose the problem and approximate the second stage objective function by a piecewise linear function. This can help reduce the problem size which makes the problem solvable in a timely manner. This framework can also be extended to optimize the mid-term and long-term natural gas supply and electricity production portfolio.

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VIII. BIOGRAPHIES

Usama Asif is currently working with Unilever Asia Pvt LTD since June 2009. He received his B. Eng. Degree from Department of Electrical and Computer Engineering, National University of Singapore. He can be reached at usamaasif03@gmail.com Panida Jirutitijaroen (S’05, M’08) is currently an Assistant Professor at Department of Electrical and Computer Engineering, National University of Singapore since January 2008. She received the B.Eng. degree (Hon.) from Chulalongkorn University, Bangkok, Thailand, in 2002, and the Ph.D. degree in electrical engineering at Texas A&M University in August 2007. She was a post doctoral researcher at Texas A&M University from September to December 2007. Her research interests are power system reliability and optimization.

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