Options based reserve procurement strategy for wind generators - Using binomial trees

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Options Based Reserve Procurement Strategy forWind Generators—Using Binomial Trees

Reza Ghaffari and Bala Venkatesh, Senior Member, IEEE

Abstract—Wind and solar PV are the most mature forms of re-newable energy and are integral to our clean energy strategy. Theirintermittency poses technical and economic challenges. Technicalchallenges are load balancing, frequency regulation, etc. Economicchallenges include providing least costing load balancing (reserves)services to these intermittent generators.This paper considers a future electricity market situation

wherein wind generators are required to forecast and bid to supplyenergy. The future electricity market treats wind generators sim-ilar to conventional generators penalizing for underproductionand pays poorly for overproduction.An intra-day ( 24 h) secondary market is proposed in this

paper where a wind generator and a reserve provider can bilat-erally trade in reserves. Reserves are traded in the market bypurchasing options to buy reserves at predetermined strike pricesby paying premiums. These reserves include call and put optionsto address underproduction and overproduction. A binomialtree approach for estimating possible deviation from the forecastvalue is used. A new optimization formulation is proposed thatuses binomial tree option pricing technique to determine optimalvalues of strike prices and premiums for call and put options. Twoexamples illustrate the benefits of the proposed idea.

Index Terms—Binomial tree, call option, energy option, ISO,market spot price, market volatility, put option, spinning reserve,strike price, wind volatility.

NOMENCLATURE

Stock Price Example: Financial Market

S Current price of stock.

uS, dS, pS Increasing coefficient, decreasingcoefficient, and probability of increasing inbinomial tree for stocks.Volatility of financial instrument.

rS (Interest rate 1) for one time period.

KSc, KSp Call and put option strike prices.

SC, SP Call and put option values for premiums.

Marginal Price: Electricity Marketm Forecasted market price at Tth hour.

um, dm, pm Increasing coefficient, decreasingcoefficient. and probability of increasingin a binomial tree for marginal price ofenergy.

Manuscript received December 28, 2011; revised April 25, 2012; acceptedJune 28, 2012. Paper no. TPWRS-01248-2011.The authors are with the Ryerson University, Toronto, ON M5B2K3, Canada

(e-mail: [email protected]).Digital Object Identifier 10.1109/TPWRS.2012.2210574

Volatility of reserve price for a specifiedperiod.

Kmc, Kmp Call and put option strike prices for reserve.

mC&mP Call and put premiums for reserve.

Forecasted EnergyE Forecasted wind energy for Tth hour.

uE, dE, pE Increasing coefficient, decreasingcoefficient and probability of increasing ina binomial tree for forecasted wind energy(E).Volatility of wind energy (E) for a specifiedperiod.

EC Expected energy to be under produced.

EP Expected energy to be over produced.

Common Variablest, T Time, number of steps time.

CCC Capacity cost for call option.

CCP Capacity cost for put option.

Profit/Loss Variables for Wind Gen and Reserve ProviderPenalty and discount factor imposed towind generation company for under/overproduction.

GWC, GWP Profit of wind generation company fromcall and put options.

GTC, GTP Profit of reserve provider from call and putoptions.

Actual Values at hour T (Example 2)mA Actual market price at Tth hour.

EA Actual value of wind energy available atTth hour.

N Number of points in the historical data.

I. INTRODUCTION

T HE demand for energy is on the rise worldwide. The en-vironmental costs associated with conventional energy re-

sources such as natural gas, coal, etc. are hefty. Renewable en-ergy resources provide tremendous environmental benefits anda way to mitigate green house gas effects. Many power systemsaround the world have started changing their energy generationportfolio to include significant amounts of renewable energy

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sources such as wind, solar, etc. Such transitions to integraterenewable energy resources are met with challenging complex-ities due to their characteristics. One such challenge is a mis-match between demand and supply, which is usually caused byuncertainty of production by renewable energy resources. Thischallenge needs to be addressed to make the transition smootherand economically feasible.Wind is a non-dispatchable source of energy and is plagued

by uncertainty in its availability. Decisions on the amount ofwind energy to be traded are therefore risky. Uncertainty in theavailability of wind energy constitutes a major obstacle in theintegration of wind generators into the conventional electricitymarket framework in economic terms. In response to this chal-lenge, three different approaches can or have been adopted byregulators [1]:1) Full Regulated—Wind generation is managed in the elec-tricity market as a negative demand, but wind power pro-ducers are paid a regulated tariff for their actual energy pro-duction. This way, wind generators are released from therisk stemming from uncertainty in their availability at theexpense of losing the opportunity of making a higher profitthrough competitive market participation.

2) Partially Regulated—Wind generators compete in all thetrading floors available in the electricity market, includingthe balancing market, in which they must face the costof their energy output deviations. For each producedMWh of energy, they are paid the price resulting fromthe market-clearing process plus a subsidy intended tostrengthen the competitiveness of wind producers in theelectricity market.

3) Full Competition—According to a pure competitivemarket, wind producers must bear the burden of themarket as any other market participant. This situation isachieved by eliminating the subsidies mentioned before inapproaches 1 and 2. In order for a wind power producerto compete in such a strictly competitive market, a fewmethods have existed. One of them is based on the com-bined use of wind power and energy storage technologiessuch as pumped-storage facilities, compressed air facili-ties, etc. However, based on [1], plants including storagefacilities have no clear incentive to associate with windfarms. Another relevant area of possibilities is the use offinancial options in an intra-day trading framework as atool for wind producers to hedge against wind generationuncertainty [2]. The latter (financial options in an intra-daytrading framework) is the main subject of this paper. Noclear financial instruments have been available so far tohedge against the risk of production uncertainty [1].

References[3]–[6] study how to take wind energy and con-vert it to different form of storable energy. Methods range frompumped storage facilities to underground compressed air facil-ities.This paper considers an electricity market where there is no

subsidy or preference is given to wind generators—a possiblefuture electricity market scenario. It relates to the third scenariodescribed above.The economic difficulty arising from the mismatch between

actual supply and committed supply of energy from renewable

Fig. 1. Schematic of intra-day secondary market for reserve trade.

resources such as wind generators is a very big challenge. In apure competitive market model, the production facility is penal-ized if it under produces and is given a fraction of the energy’smarket value if it over produces. The issue of under productioncan be resolved by developing a secondary market scheme thatallows such facilities to buy the energy reserve from thermalplants or energy storage system as back up energy. A similarmodel for over production can also be created. Thus a new sec-ondary market formulation in an intra-day framework is pro-posed in this paper. In this secondary market, a bilateral trade isestablished between wind generators and reserve provider suchas thermal generators or storage units.Fig. 1 shows a simple grid connection including renewable

and non-renewable sources of energy. In this figure, the dashlines represents the financial transactions between parties wherewind generation companies purchase the rights (either buy orsell of energy) from thermal or storage companies which is ca-pable of supplying reserves.Energy is a commodity and can be traded in the market. An

“option” is a trading concept that is used to reduce the risk asso-ciated with buying or selling a commodity. An option holder hasthe right to buy or sell a commodity at a future time for a pre-determined price (strike price). There are several option tradingmodels available that fall under two major categories: 1) a call(the right to buy) option, and 2) a put (the right to sell) option.In the proposed model in this paper the call option is exercisedin the case of underproduction, while the put option is exercisedin the case of over production.Next section explains the financial markets of stocks for

describing options and their pricing mechanism using binomialtree. Thereafter, the next sections describe a binomial treestrategy to determine forecast errors in wind energy. Section IVdevelops necessary theory for pricing of call and put optionsincluding strike prices and premiums. Thereafter Section Vdevelops the complete optimization formulations for opti-mizing the bilateral trade market for option pricing of reserves.Thereafter results of a sample study are provided.


GHAFFARI AND VENKATESH: OPTIONS BASED RESERVE PROCUREMENT STRATEGY FOR WIND GENERATORS 3

II. BINOMIAL TREE FOR OPTION PRICING

Option pricing of electricity has been a research subject. Sincethe financial markets of electric power systems differs from tra-ditional financial markets in certain important aspects, pricingand trading in “electricity options” is challenging. Reference [7]considers the pricing of electricity swing options that hedge theelectricity price risk andalsopartly the risks in theoptionowner’sload pattern. Also [8] tries to use Black and Scholes formula-tion for electricity option pricing. Since there are no reliable op-tion prices available, the most dependable way to analyze optionpricing on electricity contracts is to estimate models for the un-derlying assets and, from these, derive the corresponding optionprices [8]. “Given thatmany of the existing options on electricitycontracts are, in fact, options on electricity forwards rather thanon the actual spot price, this involves modeling both electricityspot and forwardprices” [8]. In [9] theoptionpriceof spinning re-serve is studiedusing theBlackandScholes formula.Howevernostudy about using Binomial option pricing has been found in theelectricitymarket.Also the studiesmentionedabove consider thetransactionbetweenthedemandandsupplyandnoneof thosewasrelated to the transactions between generation companies. Fur-ther, importantly—alloptionsdealwithenergy(commodity)—notwith ancillary services such as reserves on an intra-day tradetimeframe.The binomial option pricingmodel was first proposed by Cox,

Ross, and Rubinstein in [10] and later was further discussed in[11]. Essentially, the model uses a “discrete-time” model of thechanging price over time of the underlying financial instrument.The Binomial options pricingmodel approach is widely used be-cause it is able to handle a variety of situations for which othermodels cannot easily be applied. This is because this model isbased on an underlying instrument over a period of time ratherthan a single point. As a consequence, it is used to value Amer-ican options that are exercisable at any time in a given intervalas well as Bermudan options that are exercisable at a specifictime. Relatively simple, this model is easily implementable bycomputers. Although computationally slower than theBlack-Sc-holes formula, it is more accurate, particularly for longer-datedoptions on securities with dividend payments [12]. In some liter-atures like [13] the concept of trinomial tree instead of binomialtree in option pricing was investigated however in this paper wefocus on the concept of binomial tree. Binomial trees are used ingeneral when the evolution follows Gaussian distribution whichis the case for predicted near-term loads and hence price. Fig. 2shows a typical binomial option tree. S is the initial price of thestock or starting node. Each price node can move either upwardor downward in the next time step (t). The upward coefficient isuS and the downward coefficient is dS. The variable pS is theprobability of stock price moving upward. In these equations, rSis one plus the interest rate for the time period .The variable volatility of financial instrument for a speci-fied period before expiration and T is the number of steps. Thisprobability can be calculated using [10]

(1)

(2)

(3)

Fig. 2. Binomial tree for stock price ( time steps).

In general, one may write

(4)

The probability of reaching the jth node in the final Tth timestep is identified in the binomial tree. This value is

The jth node of the Tth time step shows a possible value ofthe stock to be: . Accordingly, the deviation fromthe future stock price (or strike price) of KS at the jth node inthe final Tth time step is: .

A. Call Option

When a party wishes to buy, then it would investigate a “calloption”. A call option, often simply labelled a “call”, is a fi-nancial contract between two parties, the buyer and the seller ofthis type of option. The buyer of the call option has the right, butnot the obligation to buy an agreed quantity of a certain com-modity (the underlying) from the seller of the option at a certaintime (the expiration date) for a certain price (the strike price).The seller (or “writer”) is obligated to sell the commodity if thebuyer decides to exercise the right to buy. The buyer pays a fee(called a premium or options price) for this right. In some casesthe premiummay become zero (premium neutral). The probablevalue for the call option at the jth node of the Tth time step canbe calculated using the following equation considering a pos-sible strike price of KSc for the call option:

(5)

From the above, it is obvious that certain nodes in the finaltime step have no value.Now considering the entire final time step , the total

call option value can be computed as using (5):

(6)



This value of SC reflects the premium, the buyer is expectedto pay at time step to purchase the “call option” (rightto buy the stock) at the end of time period T at the call optionstrike price of KSc.

B. Put Option

When a party wishes to sell, it investigates “put option”. Wemay build the same tree for put option or simply labelled a “put”.It is obvious that the put option has zero value if stock marketprice at a certain node is more than strike price.A put is a financial contract between two parties. The buyer

of the put option has the right, but not the obligation to sell anagreed quantity of a certain commodity (the underlying) to theseller of the option at a certain time (the expiration date) for acertain price (the strike price). The probable value for the calloption at the jth node of the Tth time step can be calculated usingthe following equation considering a possible put option strikeprice of KSp:

(7)

From the above, it is obvious that certain nodes in the finaltime step have no value. Now considering all the ofthe final time step , the total put option value can becomputed as follows using (7):

(8)

This value SP reflects the premium, the seller is expected topay at time step to purchase the “put option” (right tosell the stock) at the end of time period T where the buyer isobligated to buy at the put option strike price of KSp.In summary, given a stock price “S” and volatility , one

may create a binomial tree to predict changes in the stock price.The attributes of this binomial tree can be computed using (4).Collating information from the end nodes at the Tth time step,for given strike prices of KSc and KSp for call and put options,respectively, one may determine premium values now at time

to be SC and SP, respectively.We use this analogy in the next section to determine potential

amount of wind energy that a wind generator may have in excessor be in deficit at short-term (a few hours) commitment in theday-ahead electricity market.

III. PROPOSED MODEL FOR NEAR-TERMWIND GENERATION FORECAST ERROR

Wind Energy output from a wind turbine is forecasted in thenear term and such forecasts have large errors. In this paper, wepropose the use of binomial trees to model potential variationof wind energy output from the forecasted value. This help iscomputing amount of purchase or sale in call or put optionsrespectively.

Fig. 3. Binomial tree for wind energy ( time steps).

Fig. 3 shows the evolution of a wind forecast Binomial tree.is the present forecast of wind energy at Tth hour. Here en-

ergy is the variable just like stock market price in the previousexample. The volatility is it is used to calculate the up-ward coefficient uE, the downward coefficient dE and the prob-ability of moving upward pE. The relations of between these aredetermined as before using (4):

(9)

Call option deals with underproduction. Hence we expect fu-ture values are lower than E. Hence we compare the energyvalues of the nodes in the Tth hour with E. If the future valueturns out to be lesser than E, the call option will be realized asfollows using (6):

(10)It is noted that the optimal amount of energy procured by

the wind company, as we will discuss later, considering reservecapacity price, may be lesser than this amount. Accordingly

(10.1)

The put option deals with overproduction. Hence we expectfuture values are higher than E. If the future value turns out tobe higher than E, then, the put option will be realized. In generalfor T number of time steps similar to (8):

(11)It is noted that the optimal amount of energy sold by the wind

company, as we will discuss later, considering reserve capacityprice, may be lesser than this amount. Accordingly

(11.1)

The mismatch between supply and demand for electricitygenerated from a renewable resource such as a wind generatorcould be costly. The production facility is penalized if it under



Fig. 4. Binomial tree for spot price of energy ( time steps).

produces. It is given a fraction of the energy’s market value forenergy (spot price) if it over produces. The mismatch betweenforecasted renewable energy and real output, which is usuallygoverned by uncertainty of production, needs to be addressedto make the transition from conventional to renewable energysources faster and economically feasible in a purely competi-tive market setting. The next section proposes an option basedstrategy to procure reserves. It is structured using binomial treeand stock market theory in Section I.

IV. PROPOSED MODEL FOR RESERVE AND OPTION PRICING

The issue of under production can be resolved by developinga market scheme that allows such facilities to buy the spinningreserve from other energy sources. A similar model for overproduction can also be used. Energy is a commodity and canbe traded in the market. An “option” is a trading concept thatis used to reduce the risk associated with buying or selling acommodity. The purpose of this research is to find proper fi-nancial tools in order to create a channel between renewablecompanies and other sources of reserves to reduce the effect offorecast uncertainty. In this direction, consider that the presentforecasted value of spot price for the Tth hour is “m”. Consid-ering volatility in energy price equaling , one may computethe binomial tree values as follows using (4):

(12)

This binomial tree for spot price of energy is shown in Fig. 4.Let the call option strike price for purchase of energy at the Tthhour be Kmc. Using the binomial tree, the differences betweenvalues at end nodes at the Tth hour and call option strike priceKmc will determine call option premium value. They are deter-mined using (6) for call option as follows:

(13)

Let the put option strike price for sale of energy at the Tthhour be Kmp. Using the binomial tree, the differences betweenvalues at end nodes at the Tth hour and put option strike priceKmp will determine put option premium value. They are deter-mined using (8) for put options as follows:

(14)

Because the financial transactions take place only a few hoursahead of real energy delivery, the interest rate factor which isan important factor in finance is not considerable here and isassumed to be equal to one. The value of Kmc and Kmp are thestrike prices for call and put options respectively if reserves areused at the Tth hour. The values of mC and mP are premiumsto be paid by the wind generator to the reserve provider at hour

to obtain the right for call and put options. This valueshall be determined through optimization process described inthe next section.

V. PROPOSED OPTIMIZATION FORMULATION

The players in the market, wind generator and reserveprovider, must determine possible shortage or excess of energyin the Tth hour (10) and (11), determine optimal values of strikeprices Kmc and Kmp and corresponding option prices (mC andmP). These optimal values should maximize the profits of windgenerator and reserve provider.

A. Objectives

This section formulates objectives of the optimizationproblem. There are two possible scenarios, under and overproduction. These are dealt with hereunder to derive terms forobjectives.1) Underproduction: This case is true when the wind gen-

erator would underproduce than the committed energy E at theTth hour.The wind generator would potentially buy a call option con-

tract from the reserve provider at a premium of mC and elect touse it in the case of actual shortfall at the strike price of Kmc.Through this contract, both wind generator and reserve providerwould gain. The premise of the gain stems from the fact that inthe case of underproduction and if the wind generator did nothave a contract with the reserve provider, it would have to buy itfrom the ISO (independent system operator) at a rate .The factor represents the additional costs (penalty) incurredby the ISO to arrange for this energy delivery at short notice.For a shortfall of EC then this would total to: .On the other hand if the wind generator procures this directlythrough a reserve provider, it would cost the wind generator:

. Therefore the wind generator profits fromthe avoidable loss equaling

(15)

EC is the optimum amount of energy in call option contractwhich will be found through optimization and it has a maximum



and minimum given in (10.1). CCC is the capacity cost for calloption and is a function of market price and the amount of ca-pacity to be purchased. In other words it is the price for eachMWh of reserve capacity and can be formulated as

(16)

The coefficient of may change in different times and mar-kets. We use MWh. It should be mentioned that thestrike price, Kmc is the price of actual energy (if the call optionexercised) and is different from the cost of reserve capacity.At the same time, the thermal generators would gain from

selling this energy to the wind generator as they get to sell ECamount of energy at a call option strike price of Kmc and alsoreceive a premium of mC instead of selling at the market rate of“m”. Therefore the profit for reserve provider is

(17)

We also suggest a simple sharing formula that is based uponwind energy volatility and volume of energy being traded. Byusing this sharing formula, we can make sure that neither windgenerator or reserve service provider can earnmore than A timesof the other one’s profit. It can be demonstrated as follows:

(18)

(19)

The following attributes were used to define the value of Aas follows:1) When wind energy volatility is zero, this profit ratio mustnot be a real number (no contract will exist).

2) With increasing volatility, the profits should move fromwind generator to the reserve service provider.

3) At infinite volatility, no real contract exists.2) Overproduction: This case is true when the wind gener-

ator would overproduce than the committed energy E at the Tthhour.The wind generator would potentially buy a put option con-

tract from the reserve provider at a premium of mP and elect toexercise this contract at the strike price of Kmp in the case ofactual excess production. Through this contract, both wind gen-erator and reserve provider would gain. The premise of the gainstems from the fact that in the case of overproduction and if thewind generator did not have a contract with the reserve provider,it would have to sell it from the independent system operator(ISO) at a rate . The factor represents the additionalcosts incurred by the ISO to arrange for this energy absorptionat short notice. For an excess of EP then this would total to:

. On the other hand if the wind generator sellsthis directly to a reserve provider, it would get the wind gen-erator: . Therefore the wind generator profitsfrom the avoidable loss equaling

(20)

EP is the optimal amount of energy in put option contract whichis found through optimization and it has a maximum and min-imum given in (11.1). CCP is the capacity cost for put option

and is a function of market price and the amount of capacity.In other words it is the price for each MWh of negative reservecapacity and can be formulated as

(21)

The coefficient of MWh is used in this paper andmay change in different times and markets. It should be men-tioned that the strike price, Kmp is the price of actual energy (ifthe put option exercised) and is different from the cost of nega-tive reserve capacity.At the same time, the reserve provider would gain from

buying this energy from the wind generator as they get to sellEP amount of energy at a put option strike price of Kmp andalso receive a premium of mP instead of selling at the marketrate of “m”. This would be profit for the reserve provider wouldbe

(22)

Just as in (17), a simple sharing formula that is based uponvolatility and volume of energy

(23)

(24)

B. Complete Formulation

The objective of this proposed formulation is to maximizethe avoidable costs for the wind generator both for call andput options caused by underproduction and overproduction sce-narios respectively. Therefore adding profits from avoided costsof (15) and (18):

(25)

Subject to:Energy that might be required:

(26)

(26.1)

(27)

(27.1)

Reserve Provider Profits:

(28)

(29)



TABLE IDATA FOR EXAMPLE 1

Profit sharing Constraints:

(30)

(31)

(32)

Call and Put Option Premium Costs:

(33)

(34)

VI. SIMULATION RESULTS

Example #1: The proposed method is programmed to op-timally price and procure reserves considering wind volatilityvalues from 0.6 to 1.0 in steps of 0.1 andmarket volatility valuesfrom 0.1 to 0.3 in steps of 0.05. Table I provides details of dataused in this problem.On optimizing by solving the proposed formulation pre-

sented in (25)–(34), the solution provides optimal values ofstrike prices, amount of energy to be purchased or sold andpremiums for call and put options for a given wind energyvolatility and market price volatility . Figs. 5 and 6provide details of variations in the strike prices with changingwind and market volatility. As the volatility increases, largeruncertainty results in corresponding larger possible deviationfrom forecasted wind power output. It is clearly seen that as thewind volatility (uncertainty) increases and for smaller marketprice volatility values, the call option strike price increases andbecomes more favorable to the reserve provider (Fig. 5).At the same breadth, a smaller market price volatility and

higher wind volatility results in lesser put option strike prices(Fig. 6).The formulation of (25)–(34) when solved maximizes sum of

avoidable costs for wind generator for both cases: when thereis (1) underproduction (call option) and (2) overproduction(put option) for a given wind/market volatility. The graph inFig. 7 depicts these optimal values for various wind and marketvolatility values. The descending shape in avoidable costs forlarger market volatility values is due to the larger premiumsfor call and put options. Figs. 8 and 9 report the variations ofpremiums in this example for call and put options respectively.

Fig. 5. Wind and market volatility versus call option strike prices.

Fig. 6. Wind and market volatility versus put option strike prices.

Fig. 7. Optimal values of avoidable costs of wind generator for various windand market volatility values.

Example 2 (Realistic Data): In this example, actual windenergy produced by Port Alma wind farm in Ontario, Canadais used for analysis. It is assumed that 5 separate call and putcontracts are needed for 5 consecutive dates of March 1, 2012to March 5, 2012. All contracts are needed for 13:00 hours ofeach day and must be made 4 hours ahead at 9:00 hours. Fig. 10shows recorded data of wind energy injected to the grid for12 hours prior to the contract time (9:00 hours). Fig. 11 shows



Fig. 8. Optimal values of call option premiums.

Fig. 9. Optimal values of put option premiums.

Fig. 10. Recorded data of wind energy injected to the grid at Port Alma for 12hours prior to the contract time (9:00 hours).

recorded nodal prices of energy at Port Alma for 12 hours priorto the contract time (9:00 hours). This data is found at the Inde-pendent Electricity System Operator (IESO) archive [14].The last 12 hours data is used to calculate both energy and

market historical volatility values for each contract. Historicalvolatility is a statistical calculation that tells option tradershow rapid price movements have been over a given period.The most common method of calculating historical volatility iscalled the standard deviation. Standard deviation measures thedispersion of a set of data points from its average. The more

Fig. 11. Recorded data of market nodal prices at Port Alma for 12 hours priorto the contract time (9:00 hours).

disperse (spread out) the data is, the higher the deviation. Thisdeviation is referred as historical volatility.The historical data in this paper is assumed to have a log-

normal distribution. For example if market price at 1:00 hoursis 19.86 $/MWh and at 2:00 hours is 20.82 $/MWh, the loga-rithmic change is . After calculatinglogarithmic change at each hour, it is possible to calculate thehistorical volatility. In general if is the number of pointsin the historical data and is stock price at end of nth intervaland T is number of time steps from contract to maturity, we canwrite expressions for average value SA and volatility as fol-lows [15]:

(35)

(36)

In this example, and . The same concept is used tocalculate wind energy historical volatility. The calculated priceand energy volatility values are shown in Table III. The fore-casted values and the actual values (what really happened atthat time) of wind energy and market price for 13:00 hours foreach day is shown in Table IV. On optimizing by solving theproposed formulation presented in (25)–(34), the solution pro-vides optimal values of strike prices, amount of energy to bepurchased or sold and premiums for call and put options for eachday (March 1 to March 5, 2012) at 13:00 hours. The penalty anddiscount factor is assumed to be similar to Example 1.Fig. 12 shows the optimal call and put option strike prices

for each day and Fig. 13 shows the forecasted profit of windgenerator and reserve service provider for each day at 13:00hours.Fig. 14 shows the actual revenue of wind generator with

and without contract using the actual wind energy amountsand market prices that occurred at 13:00 hours of each day(Table IV). It is obvious that on March 1 and 2 at 13:00 hours,wind generator over produced and on March 3 to 5 at 13:00hours, wind generator under produced. In order to calculate theactual revenue of wind generator, we must compare the total



Fig. 12. Daily variation of strike prices and market price at 13:00 hours duringMarch 1, 2012 to March 5, 2012.

Fig. 13. Daily variation of call and put option forecasted profits of both entitiesat 13:00 hours during March 1, 2012 to March 5, 2012.

energy available and the total energy contracted for each day at13:00 hours. Taking mA is the actual value of market price at13:00 hours and EA is the actual value of wind energy availableat 13:00 hours, two cases each might exist for under and overproduction scenarios. These cases are outlined in Table II.As shown in Table IV, in March 1st and 2nd there were over

production with the condition of case 1 of Table II. On March 3and 5 there was under production with the condition of Case 2of Table II and on March 4 there was under production with thecondition of Case 1 of Table II. The results show us that in thefive-day period of study and only for 13:00 hours of each day,the wind generator can save the total amount of $850.5 by usingthe proposed method.

VII. CONCLUSION

This paper is targeted to address challenges faced by windgenerators due to their production uncertainties in a future elec-tricity market that is devoid of market subsidies to wind en-ergy and develop an appropriate mitigation strategy.

Fig. 14. Actual revenues for wind generator with and without option contract.

TABLE IIREVENUE OF WIND GENERATOR IN DIFFERENT SCENARIOS

TABLE IIIPORT ALMA ENERGY AND PRICE VOLATILITY VALUES

Towards this end, an intra-day secondary market forprocuring reserves by intermittent renewable generators toovercome forecast uncertainties and market participation com-mitment deviations is proposed. Electricity market spot pricesand wind energy outputs forecasts are propagated throughbinomial trees characterized by volatility to determine possibledeviations in committed energy to the market. Call option isused to procure energy from the reserve provider in case of



TABLE IVPORT ALMA ACTUAL AND FORECASTED DATA

underproduction by wind generator. In the event of overpro-duction by wind generator, put option is used to sell excessproduction to the reserve provider.In both cases of call (underproduction) and put (overproduc-

tion) options, both wind generator and reserve provider benefitby getting a better economic return than that offered by the ISOthat applies a penalty for wind generator and market rate for re-serve provider.Option pricing strategies are used to relate volatility in en-

ergy prices to call and put option strike prices and associatedpremium costs. These relations are used formulate an optimiza-tion challenge that optimizes strike prices, premium costs andamount of reserve to be purchased or sold for call and put op-tions such that profits of wind generator and reserve serviceprovider are maximized.Two illustrative examples demonstrate the benefit of creation

of this intra-day secondary market. The results demonstrate howthewind generator’s eroded profits in the face of increasing fore-cast uncertainties can be restored by taking optimal mitigatingactions—procuring reserve options through the proposed sec-ondary market.

REFERENCES[1] A. J. Conejo, M. Carrion, and J. M. Morales, “Decision making under

uncertainty in electricity markets,” in 2010 Int. Series in OperationsResearch and Management Science.

[2] K. Hedman and G. Sheble, “Comparing hedging methods for windpower: Using pumped storage hydro units vs. options purchasing,”in Proc. Int. Conf. Probabilistic Methods Applied to Power Systems,PMAPS’06, Stockholm, Sweden.

[3] R. B. Schainker, “Executive overview: Energy storage options for asustainable energy future,” in Proc. IEEE Power Engineering SocietyGeneral Meeting, Jun. 6-10, 2004, vol. 2, pp. 2309–2314.

[4] B. Roberts and J. McDowall, “Commercial success in power storage,”IEEE Power Energy Mag., vol. 3, no. 2, pp. 24–30, Mar.–Apr. 2005.

[5] E. D. Castronuovo and J. A. P. Lopes, “On the optimization of the dailyoperation of a wind-hydro power plant,” IEEE Trans. Power Syst., vol.19, no. 3, pp. 1599–1606, Aug. 2004.

[6] W. Leonhard and E. M. Grobe, “Sustainable electrical energy supplywith wind and pumped storage—A realistic long term strategy orutopia?,” in Proc. IEEE Power Engineering Society General Meeting,Jun. 6-10, 2004, vol. 2, pp. 1221–1225.

[7] G. C. Pflug and N. Broussev, “Electricity swing options: Behaviouralmodels and pricing,”Eur. J. Oper. Res., vol. 197, pp. 1041–1050, 2009.

[8] E. Hjalmarsson, “Does the Black-Scholes formula work for electricitymarkets? A nonparametric approach,” in Working Papers in Eco-nomics, Jul. 2003, no. 101.

[9] M. Rashidinejad, Y. H. Song, and M. H. Javidi, “Option pricing ofspinning reserve in a deregulated electricity market,” in Proc. 2000Symp. Nuclear Power Systems, Lyon, France, Oct. 18–19, 2000.

[10] J. Cox, S. Ross, and M. Rubinstein, “Option pricing: A simplified ap-proach,” J. Financial Econ., vol. 7, pp. 229–263, Sep. 1979.

[11] M. Rubinstein, “Implied binomial trees,” J. Finance, Jul. 1994, Presi-dential Address to the American Finance Association.

[12] R. J Rendleman, Jr. and B. J. Bratter, “Two-state option pricing,” J.Finance 24, pp. 1093–1110, 1979.

[13] S.-I. Liu, “Trinomial tree option pricing via threshold-Garch model,”Int. J. Res. Rev. Appl. Sci., May 2011.

[14] [Online]. Available: http://www.ieso.ca.[15] J. C. Hull, Fundamentals of Futures and Option Markets, 4th ed.

Englewood Cliffs, NJ: Prentice Hall, 2002, p. 237.

RezaGhaffari is currently pursuing the Ph.D. degree in the Department of Elec-trical and Computer Engineering of Ryerson University, Toronto, ON, Canada.His research area includes power system analysis and optimization, unit com-

mitment, and energy markets.

Bala Venkatesh (M’95–SM’08) is an Associate Professor in the Departmentof Electrical and Computer Engineering, Ryerson University, Toronto, ON,Canada. His research area includes power system analysis and optimization,unit commitment, and energy markets. He is also the Director of the Center forUrban Energy at Ryerson University.

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Options based reserve procurement strategy for wind generators - Using binomial trees

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