1 2 3 4 5 6 7 8 91011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556576061

Abstract— Renewable sources of energy such as wind, have

been focus of many studies in recent years. The clean and

economic nature of renewable resources makes them very

appealing candidates for the future’s energy planning, however

the uncertain nature of them makes the integration process

complicated. One method of dealing with this uncertainty is to

use new storage technologies and the other way is to procure

reserve dedicated to these renewable sources of energy. This

paper suggests a method for wind producers to buy reserve from

other non-renewable sources to mitigate the uncertainty. This

paper investigates the usage of Black and Scholes mathematical

model not only for pricing the options but also for estimating the

amount of possible errors in wind energy forecast for a future

time span. As far as the authors are aware of, such a novel

approach has not yet been investigated. This method utilizes the

concept of historic volatility in finance and introduces the historic

volatility of wind energy which can be used for wind energy

forecast error estimation. Further to the authors’ previous

research on the use of “binomial tree” as both financial and

energy forecast model, this research applies the Black and

Scholes model and reports on the advantages of the latter.

I. NOMENCLATURE

Stock market :

t Time in future, current time t = 0 hours TH Time period for historical data λs Current value of stock price

σs Volatility per hour of financial instrument

rs Interest rate Cs, Ps Call and put option premiums for stock at

time t Ks Strike price of stock option at time t N(x0) Cumulative distribution function of the

standard normal distribution at point x0. d1s, d2s Interim parameters of Black and Scholes

formula for stock market Call option ( Under-production):

OCiλE Locational marginal price of energy at bus

result of optimization for call option QCwλE

Quoted price of energy by wth wind generator for call option

QCkλE

Quoted price of energy by kth reserve provider for call option

iKc Optimal call option strike price of energy at ith bus

iCm Optimal call option premium at ith bus

1 R. Ghaffari and B. Venkatesh are with Ryerson University, Toronto, ON,

M5B2K3, Canada (e-mail: bala@ryerson.ca).

wEU Expected energy underproduction for wth wind generator

wEU Contracted energy to be purchased for wth wind generator

KEC Contracted energy to be provided by the reserve provider for kth reserve provider

KEC

Max. capacity of kth reserve provider

TAWC Total avoidable loss for wind generators in call option

TERC Total extra revenue for reserve providers in call option

Put option (Over-production): OPiλE Locational marginal price of energy at bus

result of optimization for put option QPwλE

Quoted price of energy by wth wind generator for put option

QPkλE

Quoted price of energy by kth reserve provider for put option

iKp Optimal put option strike price of energy at ith bus

iPm Optimal put option premium at ith bus

wEO Expected energy overproduction for wth wind generator

wEO Contracted energy to be sold by wth wind generator

KEP Contracted extra energy to be purchased by the reserve provider for kth reserve provider

KEP

Max. negative capacity of kth reserve provider

TAWP Total avoidable loss for wind generators in put option

TERP Total extra revenue for reserve providers in put option

General terms:

ww d2E,d1E Interim parameters of Black and Scholes formula for energy forecast at wth wind generator

ii d2λ,d1λ Interim parameters of Black and Scholes formula for energy price at ith bus

wEF

Forecasted energy at wth wind generator

λAi

Actual energy price at ith bus in Example-2

iλE Estimated price of energy by ISO at ith bus

wσE Volatility of wind energy for wth wind generator

iσλ Volatility per hour of energy price at ith bus

NB Total bus number NW Total wind connected bus number

Wind Energy Forecast Error Estimation

Using Black & Scholes Mathematical Model

Reza Ghaffari and Bala Venkatesh, Senior Member, IEEE1

CCECE 2014 1569883975

1

978-1-4799-3010-9/14/$31.00 ©2014 IEEE CCECE 2014 Toronto, Canada

1 2 3 4 5 6 7 8 91011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556576061

NK Total number of reserve connected buses

γ0 Penalty or discount factor imposed to wind generator

δ)(V,Pi Total power injected to ith bus

δ)(V,Q i Total reactive power injected to ith bus

ii QG,PG

Active and reactive power of the generator connected to ith bus

ii QD,PD

Active and reactive power of the load connected to ith bus

δ)(V,Sl

Flow of the lth line

PC OBJ,OBJ Objective functions for call and put options

II. INTRODUCTION

IND Energy has been focus of many studies in recent

years. The clean and economic nature of renewable

resources makes them very appealing candidates for

the future’s energy planning, however their uncertain nature

makes the integration process complicated. Demand and

supply in a power system must always remain in balance on

any instant or the frequency of the system will transgress

beyond the acceptable operation range. Each power system

always maintains certain amount of reserve to deal with

contingency situations such as sudden loss of a generator

,however a wind generation farm always inject un-predictable

amount of energy to the system which is impossible to be

forecasted accurately. One method of dealing with this

uncertainty is to use new storage technologies and the other

way is to procure reserve dedicated to these renewable sources

of energy [1, 2].

This paper investigates the usage of Black and Scholes

mathematical model not only for pricing the options but also

for estimating the amount of possible errors in wind energy

forecast in a future horizon. When it comes to procuring the

reserve from different reserve providers in the system, the

feasibility of delivery and security of the network cannot be

taken for granted. Each transmission line of the system must

not be overloaded as a result of reserve procurement. This

research integrates the network security constraints into the

whole framework of reserve trades via our proposed option

market. Section III gives a brief in Option Pricing theory.

Section IV discusses the theory of Black and Scholes

formulation. Section V proposes a new methodology to

forecast possible under or over productions of wind energy

using Black and Scholes mathematical model. Sections VI and

VII provide the optimization formulation. At the end, results

of the proposed method implemented in a 6-bus test system

are reported and analyzed in section VIII .

III. OPTION PRICING THEORY

Option is a financial tool which gives the option buyers the right to buy a commodity at a certain time; however it doesn’t create any obligation to the option buyers (unlike futures).There are two types of options: call option and put option. Call option gives the buyer the right to buy a stock or commodity for a certain price at a certain time. Put option gives the buyer the right to sell a stock or commodity for a

certain price at a certain time. The aforementioned price is called strike or exercise price and the aforementioned certain time is called expiration or maturity time or date. A European

option can only be exercised on maturity time. An American

option can be exercised at any time before the expiry. Option buyers pay “premium” or “option price” to obtain these rights. Options have a major difference from futures and that is the lack of obligation of exercising [3].

There are several different models for option pricing such as binomial tree model and Black and Scholes model [4]. The binomial tree model in finance is thoroughly discussed in [5], and [6]. The Black-Scholes model can also be used when the limiting distribution is assumed to be a lognormal distribution which means that the price movement is continuous. In [7], modeling and evaluating electricity options markets with intelligent agents are discussed. [8] explains the usage of options for transaction with pump storages. Also [9] discuss the usage of the Black and Scholes model for electricity option pricing. In [10] the option pricing is used to procure spinning reserve and [11] discusses the electricity swing options that hedge the electricity price risk.

IV. BLACK AND SCHOLES MODEL

The Black-Scholes model was first published in [4] by three

economists Fischer Black, Myron Scholes and Robert Merton.

The Black-Scholes model is used to calculate the price of

European options for a particular commodity, assuming no

dividends paid during the option's lifetime. The binomial

model is a discrete-time model evaluating the stock price

movements, in each time interval between price movements.

As the time interval becomes smaller, the limiting distribution

can have two forms. If as time interval approaches the value of

zero and price changes approaches zero too, the limiting

distribution can be assumed to be a normal or lognormal

distribution. If as time interval approaches the value of zero

and price changes remain significant, the limiting distribution

cannot be assumed to be normal but is assumed as a Poisson

distribution. The Black-Scholes model can be used when the

limiting distribution is considered as a lognormal distribution

which means that the price movement is continuous.

The value of a call option premium on a stock in the Black-

Scholes model can be calculated as a function of the following

variables at hour t:

( ) ( )d2s.NKs.ed1sλs.NCs rs.t−−= (1)

With:

ts

t/2)s(rs s/Ks)ln(d1s

2

σ

σλ ⋅++= (2)

tσsd1s(t)d2s −= (3) N(x0) is the cumulative distribution function of the standard normal distribution at point x0:

= ∫

−

−x0

inf

.dxx

e2π

1N(x0)

221

(4)

The value of put option premium can be calculated using: trseKss Cs Ps ⋅−⋅+−= λ (5)

Volatility for a stock can be computed using a set of hourly

W

2

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historical values recorded for TH+1 hours as below. Consider

a set of stock price values λs, the volatility per hour may be

computed as below:

∑ ∑= =

−⋅=

TH

1t

2TH

1tTH1

1)-λs(-t

λs(-t)ln

1)-λs(-t

λs(-t)ln

1-TH

1 σs

(6)

In theory Black and Scholes model is only correct if the short term interest rate, rs is constant. In practice the formula is usually used with the interest rate, rs being set as equal to the risk free interest rate on an investment that lasts for time T [12]. When the stock price becomes very large, a call option is almost certain to be exercised and then becomes very similar

to a forward contract because when λs becomes very large,

d1s and d2s become very large too and therefore N(d1s) and N(d2s) are both close to one. Also when the stock price becomes very large, the put option would be zero i.e. the put option has no value. When the stock price becomes very small, d1s and d2s become very large negative values. N(d1s) and N(d2s) are both close to zero and call option price would be close to zero as well i.e. the call option has no value.

V. WIND ENERGY FORECAST ERROR EXPECTATION

Predicted near-term natural wind energy forecast error by a specific wind farm in some earlier researches has been assumed to have a Gaussian distribution with a constant variance that assumes the modeling errors are uncorrelated and normally distributed [13], [14] and [15] although Weibull [16] and Beta [17] distributions have also been used. In this part we discuss about the similar natures forecasts of wind energy and stock price. They both fluctuates in time and we can assume that for the time step approaching zero the wind energy prediction errors become smaller and the limiting probabilistic distribution can be assumed to be a normal distribution. As an analogy to finance where the stock price at any time may be lower or higher than the strike price of an option associated with the stock, wind energy injected by a generator into the connected power system may also be under produced or over produced with respect to the forecasted wind energy value at a certain time in the near future. Now instead of stock and strike prices in (1) we replace them by the

amount of forecasted wind energy, wEF at wth wind generator

at a certain time, t in the near future. Using (1) to (3) and knowing that there is no interest rate factor in this case, the expected energy over produced can be calculated as:

)N(d2EEF)N(d1EEF(t)EO wwwww ⋅−⋅= (7)

With:

tσE2

1

tσE

/2).t(σd1E w

w

2

w == wE (8)

tσEd1Ed2E www −= (9)

At zero interest rate, the expected energy under-produced is analogous to a put option and therefore can be calculated using (5):

ww EOEU = (10)

Fig. 1. Variation of forecast error percentage with respect to time for constant

forecast of 100 MWh and volatility =0.15

Fig. 2. Variation of wd1E , wd2E , )N(d1Ew and )N(d2Ew with respect to

time for constant forecast of 100 MWh and volatility =0.15

Similar to (6), the wind energy volatility ( wσE ) can be

computed from historic values of wind energy. As the time

approaches infinity, the expected energy under produced or

over produced will reach the limit of wEF which means that

the forecast error reaches 100% as shown in Fig. 1. Fig. 2

shows the variation of wd1E , wd2E , )N(d1Ew and )N(d2Ew

with respect to time. At t=0, )N(d1Ew , )N(d2Ew are both

equal to 0.5 in standard normal distribution and as t

approaches to infinity, )N(d1Ew approaches to one and

)N(d2Ew approaches to zero.

VI. PROPOSED MODEL FOR RESERVE OPTION PRICING

When underproduction happens, wind plants must procure

their energy deficits from other reserve providers. In the case

of overproduction wind plants need to sell their energy excess

at the best prices possible. It is assumed that wind plants with

under-production will face a penalty factor of γ0 . similarly it

is also assumed that wind plants with over-production will

face a discount factor of γ0 . The factor γ0 represents the

additional costs (penalties) imposed by the ISO in order to

arrange for these alternative energy deliveries at short notice.

Let us assume that iλE are the optimal energy price of call and

put options at ith bus of the system and iKc and iKp are the

optimal strike prices for call and put options at aforementioned

bus (not necessarily equal). By using (1) to (3) and by

assuming zero interest rate for the option contract and by

considering different price volatility for different buses, the

equation used to compute premium is:

).N(d2λKc).N(d1λλECm iiiii −= (11)

With:

0

50

100

0 50 100 150 200 250 300

-3

-2

-1

0

1

2

3

0 50 100 150 200 250 300t

t

)%/E(EO ww

wd1E

wd2E

)N(d1E w

)N(d2E w

3

1 2 3 4 5 6 7 8 91011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556576061

tσλ

/2).tλ (σ)/KcE ln(λd1λ

i

2iii

i

+= (12)

tσλd1λd2λ iii −= (13)

And by using (5) the equation for put premium is:

iiiiiii KpλE-).N(d2λKp).N(d1λλEPm +−= (14) With:

tσλ

/2).tλ (σ)/KpE ln(λd1λ

i

2iii

i

+= (15)

tσλd1λd2λ iii −= (16)

After running the optimization discussed in the next section these parameters will be found.

VII. OPTIMIZATION FORMULATION

The first step of this model is to calculate the expected under or over production of wind energy at the time of forecast, and to determine the optimal strike and option prices at each node (bus) of the system. In the case of under production, it is assumed that all of wind generators participating in an option market have under production at the maturity time. These wind generators can use the amount of reserve they purchased in the option market to avoid penalties imposed by ISO. They have already paid call option prices (premiums) to one or more reserve providers located at different buses to enjoy this privilege.

In the case of over production, it is assumed that all of wind generators participating in an option market have over production at the maturity time. These wind generators can use the amount of negative reserve they purchased in the option market to avoid discounted prices imposed by ISO. By assuming all having under production or all having over production at the same time the model points out the maximum option prices.

A. Call Option

The objective function is to maximize the benefit of the call option market participants at Nash market equilibrium to determine EUw and ECk:

∑∑==

⋅−⋅=NK

1k

QCkk

NW

1w

QCwwC λEECλEEUOBJ

(17)

Subject to:

ww EUEU0 ≤≤ (18)

kk ECEC0 ≤≤ (19)

The following constraints ensure the network feasibility which includes active and reactive power balance at each bus and also transmission lines’ limits:

iiwik

ii PDhour 1

EU

hour 1

ECPGδ)(V,P −+−= ∈∈ (20)

iii QDQGδ)(V,Q −= (21)

lll δ) S(V,SS ≤≤−

(22)

iii VVV ≤≤ (23)

The lambda multiplier corresponding to the ith bus in the

equality (20) is OCiλE .

B. Marginal Prices in Call Option

The value of the Lagrangian multiplier associated with the real power balance equation (20) at each bus must be equal to the total amount of money payable at that bus.

iiOCi CmKcλE +=

(24)

After determining OCiλE by optimizing (17)-(23), solving (11)

and (24), one can determine ii Cm and Kc .

If the solution benefits both reserve service providers and wind generators, the following must be true:

iOCii λEγ0)(1λEλE ⋅+≤≤

(25)

C. Put Option

The objective function is to maximize the benefit of the put option market participants at Nash market equilibrium to determine EPk and EOw:

∑∑==

⋅−⋅=NW

1w

QPww

QPk

NK

1k

kP λEEOλEEPOBJ

(26)

Subject to:

ww EOEO0 ≤≤ (27)

kk EPEP0 ≤≤ (28)

iiwik

ii PDhour 1

EU

hour 1

ECPGδ)(V,P −+−= ∈∈ (29)

iii QDQGδ)(V,Q −= (30)

lll δ) S(V,SS ≤≤−

(31)

iii VVV ≤≤ (32)

The lambda multiplier corresponding to the ith bus in the

equality (29) is OPiλE .

D. Marginal Prices in Put Option

At the ith bus, the total payable money should be equal to strike price less the option price.

iiOPi PmKpλE −=

(33)

By solving (14) and (33), we can obtain the value of Kpi and Pmi. For a solution benefitting both wind generators and reserve providers, the following must be true:

iOPii λEλEγ0).λE-(1 ≤≤

(34)

VIII. SIMULATION RESULTS

The proposed method is tested and demonstrated in a 6-bus

test system. The results are discussed below.

Example: IEEE 6-bus test system

Using IEEE 6-bus test system, we assume that there are two wind farms connected to buses 4 and 5 (W1 and W2) and there are two reserve providers connected to buses 2 and 3 (G1 and G2). The single-line diagram is shown in Fig.3. Each transmission line has the maximum capacity of 250 MVA and two 220 + j20 MVA loads are connected to buses 5 and 6. G1 and G2 are scheduled to supply 200 MWh each before participating in any option contract. Also the penalty/discount

4

1 2 3 4 5 6 7 8 91011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556576061

factor imposed by ISO for under/over production is γ0=0.5. The original ISO estimated market prices are given in Table I. Also quoted prices are shown in Table II. The forecasted wind energy at each wind farms is assumed to be 200 MWh .Option contracts are taking place 4-hour ahead of option exercise time. Each reserve provider is ready to provide maximum amount of 70 MWh reserve. The market price volatility for all nodes is assumed to be 10% per hour. 6-Bus test system line data can be found in Table III. Fig. 4 shows how strike and premium call options change with increasing of wind energy volatility. It is obvious that higher wind energy volatility increases the uncertainty at the time of contract and the prices for majority of the buses increase. Scheduled energy for wind under/over production at buses 4 and 5 (wind connected buses) can be found in Fig. 5. It is obvious that higher wind energy volatility increases the uncertainty at the time of contract and higher value of required reserve for wind generators is required. The amount of optimal reserve provided by reserve suppliers at buses 2 and 3(G1 and G2) for the under production case are shown in Fig. 6. The negative reserve for overproduction is shown in Fig. 7 and it is provided by only G2 resulting of quoted prices. Now let’s observe the effect of transmission lines’ constraints in this test system. Let’s assume that the maximum capacity of line between buses 2 and 5 is changed from 90 MVA to 50 MVA for a wind volatility value of 0.15. As it is shown in Fig 8 , the tighter security constraint of the line means that W2 connected to bus 5 can purchase lower amount of energy to compensate for the under production however W1 connected to bus 4 is not limited. Variation of optimal scheduled reserve with respect to security constraint variation of line 2-5 is shown in Fig. 8. It is obvious that the lower amount of allowable flow on this line will affect the scheduled reserve amounts. Variation of optimal call strike prices with respect to security constraint variation of line 2-5 is shown in Fig. 9. The sensitivity of prices to the security constraint of line 2-5 is obvious in this graph. Price of majority of buses increases as a result of tighter security constraint. The following parameters can help us to measure the economic benefits of the model; however they must not be confused with objective function:

)Cm.(KcEUγ0).λE.(1EUTAWCNW

wi1w

iiw

NW

wi1w

iw ∑∑∈

=∈

=

+−+=

(35)

∑∈=

−+=NK

ki1k

iiik )λECm.(KcECTERC (36)

∑∑∈

=∈

=

⋅−⋅−−⋅=NW

wi1w

iw

NW

wi1w

iiw λEγ0)(1EO)Pm(KpEOTAWP

(37)

( )[ ]∑∈=

−−⋅=NK

ki1k

iiik PmKpλEEPTERP (38)

TABLE I

ISO ESTIMATED NODAL PRICES ($/MWh)

Bus

Number 1

GS

Bus

Number 2

G1

Bus

Number 3

G2

Bus

Number 4

W1

Bus

Number 5

W2

28.45 31.79 29.46 27.23 33.80

Fig. 3. Single-Line diagram of the 6-Bus test system.

TABLE II

QUOTED PRICES OF THE EXAMPLE ($/MWh)

Bus

Number

1

Gs

Bus

Number

2

G1

Bus

Number

3

G2

Bus

Number

4

W1

Bus

Number

5

W2

Under production Ν/Α 34.96 35.35 38.12 43.94

Over production Ν/Α 23.84 27.98 14.97 20.28

TABLE III

6-BUS TEST SYSTEM LINE DATA

From

Bus

Number

To Bus

Number

Resistance

(p.u.)

Inductance

(p.u.)

Half line

Admittance

(p.u.)

Transformer1 1 6 0.123 0.218 -

Transformer2 1 4 0.080 0.270 -

Line1 4 6 0.097 0.207 0

Line2 5 2 0.102 0.240 0

Line3 2 3 0.123 1.250 0

Line4 5 6 0.100 0.240 0

Line5 4 3 0.100 0.240 0

Fig. 4. Call option strike prices and premiums at all buses for under-production with respect to Wind Volatility ($/MWh)

Fig. 5. Wind Energy under or over-production,

0

10

20

30

40

50

0

10

20

30

40

50

iKC

0.05 0.10 0.15 0.20

1 3

6 2 5

4

W2

G2 Gs

G1

W1

Cmi

MWh

0.25

5 4 3 2 Bus

Bus 4 5 W1 W2

0.05 0.10 0.15 0.20 0.25

wσE

wσE

5

1 2 3 4 5 6 7 8 91011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556576061

Fig. 6. Optimal amounts of positive reserve (call option) for wind under production

Fig. 7. Optimal amounts of negative reserve (put option) for wind over production

Fig. 8. Purchased reserve by Wind Generators (W1 and W2) and optimal reserve purchased from reserve providers (G1 and G2) with respect to security

constraint of line 2-5 for under production (call option)

Fig. 9. Optimal call strike prices with respect to security constraint of line 2-5

TABLE IV

RESULTS OF THE CASE STUDY

B&S

Volatility

of Wind

Energy

σEw

TAWC

($)

TERC

($)

TAWP

($)

TERP

($)

OBJC

+

OBJP

($)

0.05 179.55 49.35 208.54 24.06 287.52

0.10 326.36 100.13 396.92 47.51 548.36

0.15 436.51 151.92 562.41 70.32 781.14

0.20 505.24 205.07 706.12 92.42 984.22

0.25 514.29 334.29 824.91 113.75 1152.59

IX. CONCLUSION

This paper proposes and investigates the usage of Black and

Scholes mathematical model not only for pricing options but also for estimating the amount of possible errors in wind energy forecast in a future time span. This method utilizes the concept of historic volatility in finance and introduces the historic volatility of wind energy which can be used for wind energy forecast error estimation. In both cases of call and put options, wind generators and reserve providers would benefit by getting better financial returns than the default case of having no option trading. The amount of revenue they may gain depends on the reserve capacity, network topology and the wind energy volatility level.

Further research is suggested by using different wind energy forecast error mathematical models such as Gaussian, Beta, Cauchy, Weibull and etc. with comparing the results for the same test system and optimization model. The authors of this paper have submitted their research in Gaussian and Cauchy wind energy error models to other Journals which are under review at this time.

X. REFERENCES

[1] R. Ghaffari and B. Venkatesh, Options based Reserve procurement strategy for wind generators- Using binomial trees, IEEE transactions

on power systems, accepted 2012. [2] A. J. Conejo, M. Carrion, J. M. Morales, Decision making under

uncertainty in electricity markets, 2010 International series in

operations research and management science. [3] [Online]. Available: www.investopedia.com [4] Fischer Black, Myron Scholes, The Pricing of Options and Corporate

Liabilities, Political Economy, vol. 81, issue 3 May. 1973, pp. 637-654. [5] J. Cox, S. Ross and M. Rubinstein, Option pricing: A simplified

approach, Financial Economics, vol. 7, Sep. 1979, pp. 229-263. [6] M. Rubinstein, implied binomial trees. Finance, Jul. 1994, Presidential

Address to the American Finance Association. [7] Lane, D.W. ; Richter, C.W., Jr. ; Sheble, G.B. “Modeling and evaluating

electricity options markets with intelligent agents” Int. Conf. on Electric

Utility Deregulation and Restructuring and Power Technologies2000,

pp203-208 [8] Hedman, K.W., Sheble, G.B., “Comparing Hedging Methods for Wind

Power: Using Pumped Storage Hydro Units vs. Options Purchasing” Int. Conf. on Probabilistic Methods Applied to Power Systems 2006,

pp1-6

[9] E. Hjalmarsson, “Does the Black-Scholes formula work for electricity markets? A nonparametric approach,” Working Papers in Economics no

101, July 2003 [10] M. Rashidinejad, Y.H. Song, M.H. Javidi, “Option Pricing of Spinning

Reserve in a Deregulated Electricity Market” 2000 Symposium on

Nuclear Power Systems,18-19 October, Lyon, France.

[11] Georg C. Pflug, Nikola Broussev, “Electricity swing options: Behavioural models and pricing,” European Journal of Operational

Research, 197 (2009) 1041–1050

[12] J. C. Hull, Fundamentals of Futures and Option Markets, 4th ed., Prentice Hall, New Jersey 2002, p. 237.

[13] Wang, J. H., Shahidehpour, M., & Li, Z. Y. (2008). Security-constrained unit commitment with volatile wind power generation. IEEE

Transaction on Power Systems, 23(3), 1319-1327. [14] Ortega-Vazquez, M. and Kirschen, D., Estimating Spinning Reserve

Requirements in Systems With Significant Wind Power Generation Penetration, IEEE Transactions on Power Systems, 2009

[15] Doherty, R. and O'Malley, M., A New Approach to Quantify Reserve Demand in Systems with Significant Installed Wind Capacity, IEEE

Transactions on Power Systems, 2005

[16] K. Dietrich, et al., "Stochastic unit commitment considering uncertain wind production in an isolated system," in 4th Conference on Energy

Economics and Technology, Dresden, Germany, 2009. [17] H. Bludzuweit, et al., "Statistical Analysis of Wind Power Forecast

Error," IEEE Transactions on Power Systems, vol. 23, pp. 983 - 991, August 2008.

0

20

40

60

80

0

20

40

60

80

100

0

20

40

60

80

151617181920212223

MVA095S2 =− MVA075S2 =− MVA055S2 =−

5 4 3 2 Bus

MWh

Bus 2 3 G 1 G2

0.05 0.10 0.15 0.20 0.25

MVA095S2 =− MVA075S2 =− MVA055S2 =−

MWh

Bus 2 3 G1 G2

MWh

3 Bus

0.05 0.10 0.15 0.20 0.25 wσE

wσE

Bus 4 5 W1 W2

$/MWh K ic

6