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Abstract: This paper presents a new stochastic self-scheduling model for generation companies (GENCOs) in day-ahead

electricity markets including energy and reserves auctions. The proposed stochastic model takes into account both the

uncertainty of predicted market prices and forced outages of generating units. Also, financial risk of GENCOs is formulated

through well-known conditional value-at-risk (CVaR) index. The proposed self-scheduling model is tested on the IEEE 118-bus

test system and the obtained results are discussed.

Keywords: Electricity Market, Stochastic Self-Scheduling, Forced Outage, Uncertainty

NOMENCLATURE

A. Variables:

,SUu tU Start-up cost of unit u in period t ($)

,SDu tU Shut-down cost of unit u in period t ($)

, ,G

u t sP Power of unit u sold in the energy auction in period t and scenario s (MW)

, ,u t sSR Spinning reserve capacity sold by unit u in period t and scenario s (MW)

, ,u t sNSR Non-spinning reserve capacity sold by unit u in period t and scenario s (MW)

, ,Total

u t sP Total power sold in all auctions by unit u in period t and scenario s (MW)

1 Corresponding Author: Nima Amjady, Tel.: (+98)-(231)-(3525264); Fax: (+98)-(21)-(22668428); E-mail:

amjadi@semnan.ac.ir

Stochastic Self-Scheduling of Generation Companies in Day-

Ahead Multi-Auction Electricity Markets Considering

Uncertainty of Units and Electricity Market Prices

Behdad Vatani*, Student Member, IEEE, Nima Amjady*,1, Senior Member, IEEE, and

Hamidreza Zareipour†, Senior Member, IEEE

*Department of Electrical Engineering, Semnan University, Semnan, Iran

†Department of Electrical and Computer Engineering, Schulich School of Engineering, University of

Calgary, Calgary, Alberta, Canada

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,Gt s Market clearing price for energy in period t and scenario s ($/MW)

,SRt s Market clearing price for spinning reserve capacity in period t and scenario s ($/MW)

,NSRt s Market clearing price for non-spinning reserve capacity in period t and scenario s ($/MW)

, , ,A

u t s sY Non-negative auxiliary variable used for modeling non-decreasing limits, where A=[G,SR,NSR]

, , ,Au t s sX Free auxiliary variable used for modeling non-anticipativity limits, where A=[G,SR,NSR]

, ,At s sB Binary variable, which is zero if , ,

A At s t s and one otherwise, where A=[G,SR,NSR]

s Probability for scenario s of the set of generated scenarios

Rs Updated probability for scenario s of the set of reduced scenarios

Value at risk (VaR)

sv Additional variable corresponding to CVaR

,u tz Binary variable indicating acceptance status of unit u in the energy auction during period t (1/0 for accepted/not-

accepted)

,u tzn Binary variable indicating acceptance status of unit u for non-spinning reserve in period t (1/0 for accepted/not-

accepted)

,u tx Binary variable indicating start-up status of unit u for energy auction in period t

,u ty Binary variable indicating shut-down status of unit u for energy auction in period t

,u tI Binary variable indicating online status of unit u in period t (1/0 for online/offline)

, ,u t sw Binary variable representing state of unit u in period t and scenario s (1=Available, 0=Unavailable)

B. Functions:

, , (.)FTu t sF Fuel consumption function of unit u in period t and scenario s for fuel type FT

, , (.)ETu t sF Emission function of unit u in period t and scenario s for emission type ET

(.)Prob Probability function

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C. Constants:

uSU Start-up cost of unit u ($)

uSD Shut-down cost of unit u ($)

uUT Minimum up time of unit u

uDT Minimum down time of unit u

,Min

G uP Lower limits for power of unit u (MW)

,Max

G uP Upper limits for power of unit u (MW)

MaxuSR Maximum spinning reserve capacity sold by unit u (MW)

MaxuNSR Maximum non-spinning reserve capacity sold by unit u (MW)

uRU Ramp up limit of unit u

uRD Ramp down limit of unit u

uSUR Start-up ramp limit of unit u

uSDR Shut-down ramp limit of unit u

,FTu tSU Start-up fuel consumption for unit u in period t for fuel type FT

,FTu tSD Shut-down fuel consumption for unit u in period t for fuel type FT

,ETu tSU Start-up emission for unit u in period t for emission type FT

,ETu tSD Shut-down emission for unit u in period t for emission type FT

MaxFTF Fuel consumption limit for fuel type FT

MaxET System emission limit for emission type ET

FTt Fuel price in period t for fuel type FT ($/Mbtu)

Per unit confidence level

A non-negative weight factor that weighs conditional robust profit versus expected profit

uFOR Forced Outage Rate (FOR) of unit u connected

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U Number of generation units

T Number of time periods

RN Number of reduced scenarios obtained from the scenario reduction technique

1. INTRODUCTION

Optimal self-scheduling is a vital task for GENCOs in electricity markets. By self-scheduling, a GENCO tries to maximize its

profit from selling power and ancillary services in the competitive electricity market while meeting the prevailing equality and

inequality constraints of the generating units and power system such as minimum on/off duration, generation capacity limits,

ramping up/down limits of units and GENCO’s demand constraint [1]. However, self-scheduling of GENCOs is a complex

optimization problem due to, e.g., uncertain electricity market prices and mixed integer nature and nonlinearities of production

functions. Because of its importance and complexity, many studies have been performed in this area. Reference [2] presents a

bidding strategy for GENCOs to participate in multi-auction pool-based electricity market, comprising energy, AGC and reserve

auctions. The single period formulation of [2] is solved as a mixed-integer program. In [3], a framework to obtain optimal

bidding strategy of a price-taker GENCO under price uncertainty is presented. Probability density functions of next day energy

prices are predicted and using the obtained probabilistic information, a self-scheduling problem is formulated and solved to

derive bidding strategy of the GENCO. The research work of [4] addresses the trade-off between the maximum profit and

minimum risk. The two conflicting objectives are combined using a risk tolerance parameter. In [5], a bidding strategy to

optimize energy and reserve offer curves for a hydrothermal power producer under a deregulated daily electricity market is

presented. Their solution method is based on Lagrangian relaxation and dynamic programming. In [6], an offering strategy for

GENCOs is presented considering the day-ahead market in the first-stage and the adjustment and reserve markets in the second-

stage. The producer participates in electricity markets under residual demand uncertainty that is only considered in the first-

stage. In [7], a multimarket framework is presented for a producer to participate in day-ahead, automatic generation control

(AGC) and balancing markets. In order to solve the framework, a stochastic programming methodology is presented in [7]

considering that self-scheduling is known a priori. A heuristic algorithm is also used to reduce the number of scenarios. In [8], a

risk-constrained bidding strategy for a thermal electricity producer participating in day-ahead auctions of energy and ancillary

services is presented. Expected downside risk is used as financial risk measure in [8] included as a constraint in the model.

Reference [9] studies the effect of unit failure on forward contracting in medium-term study horizon. The problem is solved by

using a two-stage stochastic programming approach. In [10], in order to provide the optimal involvement in a future electricity

market, a multistage scenario tree, generated by a heuristic procedure, for the whole year is presented. Reference [11] presents a

bi-level programming approach to derive the optimal offering strategy considering uncertainty in demand and rival behaviors. In

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their bi-level approach, the upper-level is devoted to the profit maximization of the strategic producer, while the lower-level

models the market clearing procedure and price calculation. In [12], a novel scenario reduction technique based on the forward

selection algorithm is presented. In their approach, the most common probability distance, called Kantorovich distance, is used.

Reference [13] addresses a weekly bidding strategy for producers based on three-stage stochastic programming approach. The

first stage includes commitment decisions and forward contracting, the second stage determines Monday pool offering and the

third one derives pool offers for the rest of days of the week. In [14], a self-scheduling strategy for a thermal GENCO in day-

ahead energy and reserve markets is presented. Their start-up modeling includes hot, warm and cold start-up types. The

formulation is solved as a mixed-integer linear program. In [15], offering strategy based on robust optimization for price taker

GENCOs participating in pool electricity markets is presented. Their technique considers price confidence intervals instead of

using price scenarios and solves a set of robust mixed-integer linear programming problems. In [16], the strategic behavior of a

producer over one year is optimized through a bi-level approach. The upper level maximizes the profit of the strategic GENCO

considering investment decisions subject to lower level problems corresponding to market clearing scenarios. In [17], a self-

scheduling model for thermal generating units during commissioning is presented. In the model, a producer can participate either

as a price-taker or price-maker in the day-ahead energy market. The problem is solved as a mixed-integer linear program.

In this paper, a new two-stage stochastic self-scheduling model for GENCOs participating in multi-auction electricity markets,

including energy and reserves auctions, is presented. The proposed model considers the uncertainty of both forecasted market

prices and availability of the generating units. Moreover, the model optimizes both commitment and dispatch variables of the

GENCO. At the same time, financial risk of GENCOs is appropriately modeled within the proposed approach. To the best of the

authors’ knowledge, such a model has not been presented in the previous research works. To better illustrate the new

contributions of the paper, the proposed model is briefly compared with some other recently published self-scheduling models.

Compared with [9] that models the uncertainty of both forced outage of units and predicted market prices, the proposed approach

can also consider reserves auctions in addition to energy auction and optimize commitment variables in addition to dispatch

variables. The self-scheduling methods of [5-8] consider reserve auctions in addition to energy auction. However, these methods

can only model the uncertainty of the market prices. Moreover, commitment decisions are not taken into account in [6,7]. On the

other hand, the proposed self-scheduling approach also models forced outage of generating units in addition to the uncertainty of

the market prices and includes commitment decisions as well as reserves’ auctions. Compared with [1,2,14], presenting multi-

auction non-stochastic self-scheduling models, the proposed approach presents a stochastic self-scheduling model.

In this paper, it is assumed that the GENCO is a price taker, which is well-known assumption in the area used in many other

research works, such as [3,4,9,13,14]. Also, in line with [3,4,18,19], we assume that the self-schedule of the GENCO is accepted

in the market. The rest of the paper is organized as follows. In Section 2, the proposed two-stage stochastic self-scheduling

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model is presented. In Section 3, the proposed model is applied to the IEEE 118-bus test system and the obtained results are

discussed. Section 4 concludes the paper.

2. PROPOSED TWO-STAGE STOCHASTIC SELF-SCHEDULING MODEL

The objective function (OF) of the proposed stochastic self-scheduling framework is a combination of expected profit (EP),

i.e. profit without considering risk, and conditional robust profit (CRP), which is profit with considering risk, as follows:

Maximize OF = EP + λ.CRP (1)

Low and high values of the weight factor λ causes that the OF focuses on the EP and CRP leading to more desirable objective

functions for risk-seeker and risk-averse GENCOs, respectively. CRP is mathematically defined as [20]:

CRP = EP – CVaR (2)

where CVaR (conditional value-at-risk) is an effective financial risk measure in portfolio optimization theory [21] used for risk

management. By definition, with respect to a specified probability level α, the value-at-risk (VaR) of a portfolio is the lowest

amount µ such that, with probability α the loss will not exceed µ or equivalently the profit will not be lower than robust profit

(RP), whereas the CVaR is defined as the conditional expected loss under the condition that it exceeds the amount µ. These

concepts are graphically shown in Fig. 1 [18]. By substituting (2) in (1), we have:

Maximize OF = (1+λ).EP – λ.CVaR (3)

The objective function of (3) can be imagined as a combination of the profit function and risk measure. By modeling EP and

CVaR in the proposed two-stage stochastic framework, the following formulation for OF is obtained:

, ,

1 1

1 ( )

U TSU SDu t u t

u t

Maximize OF U U

, , , , , ,

1 1 1

. ( ).

RN U TR FT FT Gs t u t s u t s u t s

s u t FT

F P w

, , , ,

1 1

1 .( )

U TSU SD

u t s u t u t

u t

w U U

, , , , , , , , ,

1 1

( . . . )

U TG G SR NSRt s u t s t s u t s t s u t s

u t

P SR NSR

1

1.

1

RNRs s

s

(4)

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The first four rows of (4) model the part (1+λ).EP of (3) such that the first row indicates the first stage of the proposed two-

stage stochastic framework representing the start-up and shut-down costs. As the objective function of the self-scheduling model

is profit, which should be maximized, the costs are appeared as negative terms in it. The next three rows of (4) represent the

second stage of the stochastic model including fuel cost, impact of forced outage of generating units on the start-up and shut-

down costs, and revenues obtained from the sold energy, spinning and non-spinning reserves. If a unit becomes unavailable in a

scenario (wu,t,s=0), its fuel cost is eliminated and its start-up and shut-down costs calculated as deterministic variables in the first

stage are removed from that scenario. To remove the effect of these costs (appeared as the negative terms in the first stage) from

the scenario in which the unit is not available, they are added as positive terms to that scenario. For instance, assume that a shut-

down for a unit in hour 9 is planned in the day-ahead market. The cost of this decision is considered in the first stage of the

proposed two-stage stochastic model, i.e. the first row of (4). However, this unit fails in hour 7 in a scenario. Consequently, in

that scenario, the unit is not available in hour 9 so that the planned shut-down is run for it. Hence, the planned shut-down cost in

the first stage should be removed from the second stage for that scenario, which is implemented in the third row of (4). Also, its

revenue becomes zero due to the constraints (7)-(12), which result in , , , , , , 0Gu t s u t s u t sP SR NSR . The proposed formulation

focuses on day-ahead electricity markets, wherein GENCOs sell their generation in energy auction and capacity in spinning and

non-spinning reserves’ auctions, such as the day-ahead electricity markets of New York Independent System Operator (NYISO)

and Electric Reliability Council of Texas (ERCOT) [22,23]. Therefore, only the power sold in energy auction ( , ,G

u t sP ) is

considered in the fuel cost.

The part λ.CVaR of (3) is mathematically modeled in the last row of (4) and the following constraints (5) and (6):

, , , , , ,

1 1 1

. ( ).

RN U TR FT FT G

s k t u t k u t k u t k

k u t FT

F P w

, , , ,

1 1

1 .( )

U TSU SD

u t k u t u t

u t

w U U

, , , , , , , , ,

1 1

( . . . )

U TG G SR NSRt k u t k t k u t k t k u t k

u t

P SR NSR

, , , , , ,

1 1

. ( ).

U TFT FT Gt u t s u t s u t s

u t FT

F P w

, , , ,

1 1

1 .( )

U TSU SD

u t s u t u t

u t

w U U

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, , , , , , , , , ,

1 1

( . . . )

U TG SR NSRt s G u t s t s u t s t s u t s

u t

P SR NSR

Rs N (5)

0 Rs s N (6)

Proof of this formulation for CVaR can be found in the Appendix.

Equations (7)-(12) represent power generation and spinning/non-spinning reserve capacity limits considering acceptance status

of the unit in the energy and non-spinning reserve auctions, online status of the unit and availability status of the unit:

, , , , , , , , , ,. . . . , ,Min G Max RG u u t u t s u t s G u u t u t sP z w P P z w u U t T s N (7)

, , , , ,0 . . , ,Max Ru t s u u t u t sSR SR z w u U t T s N (8)

, , , , ,0 . . , ,Max Ru t s u u t u t sNSR NSR zn w u U t T s N (9)

, , , ,u t u t u tz zn I u U t T (10)

, , , , , , , , , ,Total G Ru t s u t s u t s u t sP P SR NSR u U t T s N (11)

, , , , , , , , , ,. . . . , ,Min Total Max RG u u t u t s u t s G u u t u t sP I w P P I w u U t T s N (12)

Note that wu,t,s variables are specified values for each scenario. Thus, wu,t,s multiplied by zu,t in (7) and (8), znu,t in (9) and Iu,t in

(12) does not make the problem non-linear. Each unit cannot simultaneously participate in the spinning and non-spinning

reserve’s auctions as the unit can be synchronized or not. To mathematically implement this limit, the binary variable ,u tI is

defined to allow only one of zu,t and znu,t becomes 1 based on (10). For instance, if unit u in period t is accepted in the energy

auction, then zu,t=1 and znu,t= 0; if it is accepted in the non-spinning reserve auction, then zu,t= 0 and znu,t=1; if it is neither

accepted in the energy auction nor non-spinning reserve auction, then zu,t=znu,t=0. Constraints (11) and (12) limit total capacity of

the unit considering its online status Iu,t.

Start-up and shut-down costs associated with energy market are modeled in (13) and (14):

, ,0 . ,SUu t u u tU SU x u U t T (13)

, ,0 . ,SDu t u u tU SD y u U t T (14)

In (13) and (14), the start-up and shut-down costs associated with deploying the accepted non-spinning reserve capacity of a unit

is not considered as the ISO (and not a GENCO) decides about deploying this capacity in the real-time environment based on the

load forecast error, failure of the other units, etc. In other words, this information (such as the status of units of the other

GENCOs in the next day) is not available for a GENCO when designing its bidding strategy for day-ahead market.

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Minimum up and down time limits for commitment of a unit in the energy market are modeled through the constraints (15)

and (16), respectively. Also, (17) implements the logical relation among the start-up and shut-down indicators and commitment

variables:

, ,

1

,

u

t

u n u t

n t UT

x z u U t T

(15)

, ,

1

1 ,

u

t

u t u n

n t DT

z y u U t T

(16)

, 1 , , , 0 ,u t u t u t u tz z x y u U t T (17)

In (15), if xu,n=1, t-UTu+1≤ n ≤ t, it makes zu,t=1. In (16), if yu,n=1, t-DTu+1≤ n ≤ t, it makes zu,t= 0. For the initial hours of the

current day, the back-shift operator of (15) and (16) can extend to the previous day such that n = 0 indicates hour 24 of the

previous day, n = –1 indicates hour 23 of the previous day and so on. According to [24], the 3-binary variable formulation

(including start-up and shut-down indicators and commitment variables) is more effective and faster than the 1-binary variable

formulation (only including commitment variables) presented in [25]. Moreover, the 3-binary variable formulation presented in

this paper has fewer constraints and is more effective than that presented in [24].

Ramp up/down limits considering the effect of start-up/shut-down ramps can be modeled as follows:

, , , 1, , 1 ,. . , ,G G Ru t s u t s u u t u u tP P RU z SUR x u U t T s N (18)

, 1, , , , ,. . , ,G G Ru t s u t s u u t u u tP P RD z SDR y u U t T s N (19)

Non-decreasing constraints should be considered to avoid from unreasonable bids in the scenarios. For instance, a scenario

with higher energy price than another scenario should not have lower energy bid with respect to it. These constraints for energy

and reserves’ markets can be represented as follows:

, , , , , , , , , , , , ,( ).( 1) .( )G G G G Gu t s u t s u t s u t s u t s s t s t sP P w w Y , Ru U t T s s N (20)

, , , , , , , , , , , , ,( ).( 1) .( )SR SR SRu t s u t s u t s u t s u t s s t s t sSR SR w w Y , Ru U t T s s N (21)

, , , , , , , , , , , , ,( ).( 1) .( )NSR NSR NSRu t s u t s u t s u t s u t s s t s t sNSR NSR w w Y , Ru U t T s s N (22)

In (20)-(22), if wu,t,s , wu,t,s′ or both of them become zero, the left-hand-side of these constraints becomes 0. In these cases, the

non-negative auxiliary variables , , ,G

u t s sY , , , ,SR

u t s sY and , , ,NSR

u t s sY are forced to be 0. On the other hand, if wu,t,s = 1 and wu,t,s′ =1, then

, ,G Gt s t s will lead to , , , ,

G Gu t s u t sP P and similarly for the reserves. It is noted that (20)-(22) are different with respect to the non-

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decreasing constraints presented in [8] and [13], as the uncertainty source of units’ unavailability is also considered in (20)-(22),

despite the non-decreasing constraints of [8] and [13].

Non-anticipativity limits enforce the following bundle constraints on the scenarios:

, , , , , , , , ,

1

. ,

tG G G G R

u t s u t s u t s s n s s

n

P P X B u U t T s s N

(23)

, , , , , , , , ,

1

,

tSR SR R

u t s u t s u t s s n s s

n

SR SR X B u U t T s s N

(24)

, , , , , , , , ,

1

,

tNSR NSR R

u t s u t s u t s s n s s

n

NSR NSR X B u U t T s s N

(25)

Based on (23), if the realization of the uncertain energy price variables from hour 1 to t is the same in two scenarios s and s',

the decisions made for these two scenarios (energy bids) in hour t should be the same to prevent from obtaining unreasonable

scenarios. This constraint is implemented for the reserves’ markets through (24) and (25). An advantage of the proposed non-

anticipativity limits (23)-(25) is considering the history of scenarios, as each multi-period scenario has time coupling constraints,

to correctly identify identical ones and bundle the decisions made for these scenarios. Moreover, the binary variables , ,Gn s sB ,

, ,SRn s sB and , ,

NSRn s sB are determined for the scenarios and so do not increase the dimensions of the solution space as well as the

computation burden of the proposed optimization model.

Fuel and emission limits are presented in (26) and (27), respectively. However, for practical purposes, the quadratic functions

, , (.)FTu t sF and , , (.)E

u t sF can be accurately approximated by a set of piecewise linearized blocks, commonly used as the linear model

in the literature [25]. This approximation is also adopted in this paper.

, , , , , , , , , ,

1

( ) . . .

TFT G FT FT Max

u t s u t s u t u t u t u t u t s FT

u FT t

F P SU x SD y w F

Rs N (26)

, , , , , , , , , ,

1

( ) . . .

TET G ET ET Max

u t s u t s u t u t u t u t u t s

u ET t

F P SU x SD y w ET

Rs N (27)

The decision variables of the proposed stochastic self-scheduling framework as an optimization model are as follows:

First stage decision variables:

, , , ,, , , ,u t u t u t u tz x y zn u U t T (28)

Second stage decision variables:

, , , , , ,, , , ,G Ru t s u t s u t sP SR NSR u U t T s N (29)

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Also, deterministic outcome (DO) of the stochastic self-scheduling framework can be defined as follows:

, , ,, , , ,, , , , , , , ,Gu t u t u tu t u t u t u tDO z x y zn P SR NSR u U t T (30)

where ,Gu tP , ,u tSR , and ,u tNSR represent expected value of , ,

Gu t sP , , ,u t sSR , and , ,u t sNSR over the scenarios Rs N , respectively.

The only remaining part of the proposed stochastic self-scheduling model is generation of the scenarios. Two uncertainty

sources are modeled in the proposed stochastic framework and so each scenario should include realization of the random

variables corresponding to these uncertainty sources. The first uncertainty source is related to price forecast error, which is

modeled by the scenario generation approach presented in [26] and [27] for modeling the uncertainty of wind power forecast and

load forecast, respectively. The first step of this approach includes a statistical analysis as well as a quantization technique for

constructing the relative frequency distribution of the forecast error. Then, the probabilities of the forecast errors, occupying the

interval [0,1], are arranged on a roulette wheel. Afterward, each scenario is generated by using Lattice Monte Carlo Simulation

(LMCS), which uniformly distributes the scenarios throughout the uncertainty space. Using this approach, a number of 24-h

scenarios are generated for each of energy, spinning and non-spinning reserve auctions.

The second uncertainty source is related to the unavailability of generating units of the GENCO modeled by an approach

based on the two-state Markov chain model. To generate a scenario by this approach, random values in the range of [0,1] by the

number of units are generated. For instance, each scenario generated by this approach for IEEE 118-bus test system with 54 units

includes 54 random values. If each of these random numbers is less than FOR of the corresponding unit, that unit is considered

unavailable in the scenario and vice versa. For each unavailable unit of the scenario, a discrete random number in the range of

[1,24] is generated to determine its forced outage hour.

As the first and second uncertainty sources are independent, the scenarios generated by the first and second approaches can be

combined as follows:

s = (Probability of the realized price values) × (Probability of the realized states for generating units)

, , , , , , ,1 1 1 1 1

( ). ( ). ( ) 1 1T T T U T

G SR NSRt s t s t s u t s u u t s u

t t t u t

Prob Prob Prob w FOR w FOR

(31)

where s represents probability of each scenario generated for the whole problem considering its both uncertainty sources.

Modeling these two uncertainty sources for large-scale GENCOs may lead to a large number of scenarios increasing the

computation burden of the proposed model. To remedy this problem, scenario reduction technique based on Kantorovich

distance, provided by SCENRED2 tool in GAMS software package [28], is used in the proposed stochastic self-scheduling

approach. This scenario reduction technique eliminates similar and less probable scenarios and retains a minimum subset of the

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most effective scenarios, which are used in the proposed stochastic self-scheduling model. Also, the probability of each reduced

scenario Rs is obtained from s values by the scenario reduction technique considering the merged scenarios. For the

mathematical details of this technique, the interested reader can refer to [29,30].

3. NUMERICAL RESULTS

The proposed stochastic self-scheduling framework is tested on the well-known IEEE 118-bus test system, which its data is

obtained from [31,32]. To generate price scenarios, real price data of energy, spinning and non-spinning reserves’ auctions for

year 2011 and the last 50 days of year 2010 of the NYISO electricity market is used [22]. The price forecast strategy presented in

[33] is used in this research work. This strategy is trained by the data of the last 50 days of year 2010 and then predicts the hourly

price values for the first day of year 2011. Afterward, the sliding window of the 50 days’ historical data proceeds by one day and

the hourly prices of the second day are forecasted. By repeating this cycle, price forecasts and forecast errors for the whole year

2011 are produced, used as the input data for the first uncertainty modeling approach. The second approach is fed by the FOR

data of the units [32].

The proposed scenario generation approach generates 1000 scenario trees by the length of 24 hours. Then, among the 1000

scenario trees, 20 most probable and dissimilar ones are selected by SCENRED2 tool. Also, α=0.95 is considered in the

numerical experiments of the paper [20]. Outline of the numerical experiments is as follows. At first, power sold in the energy

auction and reverses capacity sold in each scenario are illustrated. After that, the effect of the uncertainty sources on the outcome

of the self-scheduling strategy, including profit and risk of the producer, is evaluated. As well, the offer curves in different hours

are depicted. Then, by varying the weight factor λ, results of the model for different risk averse and risk seeker GENCOs are

presented. Finally, an after-the-fact analysis is presented to validate the results of the proposed stochastic self-scheduling

strategy.

In Table 1, power sold in the energy auction, spinning and non-spinning reserve capacity sold in each scenario are presented.

This Table illustrates that the results change from one scenario to another due to variation of the realized prices and units’

availability in the scenarios. In this Table, λ=1 is considered.

The impact of unavailability of generating units on expected profit (EP) is shown in Fig. 2. This Figure depicts that as FOR

increases, leading to more unreliable units of the GENCO, the expected profit monotonically decreases. In Fig. 2, λ=1 is

considered. To evaluate the effect of the other uncertainty source, accuracy of the price forecasts should be considered,

commonly measured by mean absolute percentage error (MAPEP). Obtained results from the proposed strategy with different

MAPEP values are shown in Table 2. In this numerical experiment, λ=1 and FOR of units is considered to be zero to evaluate the

effect of price forecast uncertainty. In Table 2, actual profit is obtained from Profits, defined in (A8) of the Appendix, such that

13

its decision variables are set based on DO in (30), and its uncertain variables ,Gt s , ,

SRt s and ,

NSRt s are set according to the actual

prices. Also, financial loss indicates difference between the values reported in the second and third columns (EP – Actual Profit).

By increasing MAPE in Table 2, leading to generation of more inaccurate price scenarios, EP of the GENCO increases.

However, this increase is unrealistic and actual profit of the GENCO really decreases, as shown in Table 2, due to incorrect

decisions made by inaccurate price forecasts. This numerical experiment shows that EP may not be a good measure of a

GENCO’s revenue when the price forecasts are inaccurate. Higher price forecast errors also lead to increase of financial loss of

the GENCO as shown in Table 2. At the same time, higher MAPE results in larger differences between price scenarios and

actual price or higher dispersion of price scenarios, which in turn lead to higher dispersion of profits of the scenarios. Thus,

probability density function of the profit becomes wider and so CVaR increases (Fig. 1), which can be seen from Table 2.

In Fig. 3, 4, 5 and 6, the energy offer curves in the four hours of 2, 9, 18 and 22 for λ=0, 1 and 10 are shown. According to

these figures, by increasing offered energy price, offered power for energy auction increases. Moreover, by increasing λ, the

power offered in the energy auction decreases due to adopting a more conservative bidding strategy by the GENCO. Similar

results have been obtained for the other hours and for the reserves’ auctions.

In the next numerical experiment, reported in Table 3, the effect of changing the weight factor λ is evaluated. For all cases of

this Table, FOR of units are kept constant as their original values and 20 price scenarios corresponding to 1×MAPEP are

considered. In Table 3, by increasing λ, the self-scheduling model more focuses on CRP in (1), depicting a risk-averse GENCO,

and so EP slightly decreases, while CRP increases. Consequently, CVaR decreases based on (2) leading to a more conservative

bidding strategy.

To correctly evaluate the effectiveness of a stochastic framework, its performance should be assessed in a sufficiently long

run. However, such a long run of the stochastic self-scheduling model cannot be executed on the IEEE 118-bus test system, due

to the absence of real operational data. Even a real producer may not accept a self-scheduling strategy that its long run

performance cannot be guaranteed. To remedy this problem, an after-the-fact analysis is proposed in this paper to simulate a long

run so that the efficiency of the stochastic self-scheduling model can be better evaluated. This analysis can be described as the

following step by step procedure:

Step 1: A number of trial scenarios (NTS) are produced by the suggested scenario generation approaches, separate from the 20

scenarios used for executing the stochastic framework.

Step 2: Profit of each trial scenario ts (1 ≤ ts ≤ NTS), i.e. Profitts, is obtained from (A8) such that its decision variables are set

based on DO in (30) and its uncertain variables ,Gt ts , ,

SRt ts and ,

NSRt ts , t T as well as , ,u t tsw , ,u U t T are set according to

14

the generated values for trial scenario ts. In other words, Profitts evaluates outcome of the stochastic model for one realization of

the uncertain variables.

Step 3: The aggregated profit (AP) of the trial scenarios, as a measure of the after-the-fact performance, is computed as follows:

1

1

. ts

NTS

ts

ts

NTS

ts

ts

A

Profi

P

t

(32)

where ts is the probability of the trial scenario ts. AP for NTS = 10,000, 20,000 and 50,000 is reported in Table 4. Close values

of AP and expected profit (EP), obtained by the stochastic model, validate its performance for long run.

Step 4: As a measure of convergence for the after-the-fact analysis, the variation coefficient β [34] is computed as below:

.

NTS

NTS AP

(33)

where NTS is the standard deviation of the profit values for the NTS trial scenarios. Very low values of β, much lower than 1%,

indicate good convergence of the proposed after-the-fact analysis for this test case.

The proposed stochastic self-scheduling framework is a mixed integer linear programming (MILP) optimization model.

Linearity is an important advantage of the proposed framework, which significantly decreases its computation time. To better

illustrate this advantage, the size of the problem for IEEE 118-bus test case as well as computation time of the proposed

approach are presented in Table 5. All simulations of this paper are performed using CPLEX 12.4 solver [35] within GAMS

software package on the personal computer 2.2 GHz Intel Core i7 with 8GB of RAM. From Table 5 it is seen that despite the

large size of the optimization problem for IEEE 118-bus test system, computation time of the proposed approach is very low,

about 5.4 minutes, on the simple hardware set, which is completely acceptable within a day-ahead decision making framework.

4. CONCLUSION

In this paper, a new stochastic self-scheduling strategy and scenario generation techniques to model the uncertainty sources of

unavailability of units and forecasted market prices for GENCOs participating in energy and reserves’ auctions is presented. It

has been shown that considering both the uncertainty sources affects the modeling approach of different technical and financial

constraints of GENCOs. The suggested self-scheduling strategy can be adapted to different risk-averse and risk-seeker

producers. It has been demonstrated that more unreliable units can lead to decrease of the expected profit of the GENCO. Also,

inaccurate price forecasts not only decreases the actual profit of the producer, but also can be misleading for it. The performance

of a stochastic model against the uncertainty sources can be correctly evaluated based on its long run results as the short-term

results may be volatile. In the absence of real operational data, to simulate long run evaluation of the self-scheduling model, an

15

after-the-fact analysis is proposed in this paper. This analysis shows that the aggregated results obtained from the model for a

large number of realized scenarios, representing different realizations of the uncertain variables, are close to its expected results

based on a limited number of scenarios, illustrating robustness of the proposed self-scheduling strategy.

5. APPENDIX

Based on the definition of CVaR, given in section 2, it can be mathematically modeled as follows [21]:

, ,

,

, ( ) , ( )

1( )

h r h r

h r

h r p d h r p d

CVaR = =p d

(A1)

where the vectors r and θ represent the portfolio decisions and uncertain variables, respectively; p(θ) indicates probability

density function of θ; h(r,θ) is financial loss of the portfolio. For our model, r includes the first and second stage decision

variables presented in (28) and (29) and θ is as follows:

, , , , ,, , , , ,G SR NSR Ru t s t s t s t sw u U t T s N

(A2)

In [20], it has been shown that (A1) for discrete cases can be formulated as below:

1

1.

1

RNRs s

s

CVaR +

(A3)

( , ) 0, Rs s sv h r v s N (A4)

where s indicates realization of the uncertain variables for scenario s. The formulation of (A3) for CVaR is appeared in the last

row of (4). To obtain the constraints (5) and (6), at first, the financial loss for discrete cases is written as follows:

( , )s sh r EP Profit (A5)

where Profits represents the profit obtained in scenario s. In other words, the financial loss of scenario s, i.e. h(r,θs), is the amount

that Profits is less than the expected profit or the part of the expected profit that is not realized. By substituting (A5) in (A4), the

following relation is obtained:

0, Rs s sv EP Profit v s N (A6)

The first four rows of (4) represent (1+λ).EP. By removing (1+λ) from this formulation, EP can be obtained as follows:

, ,

1 1

( )

U TSU SDu t u t

u t

EP U U

, , , , , ,

1 1 1

. ( ).

RN U TR FT FT Gk t u t k u t k u t k

k u t FT

F P w

16

, , , ,

1 1

1 .( )

U TSU SD

u t k u t u t

u t

w U U

, , , , , , , , ,

1 1

( . . . )

U TG G SR NSRt k u t k t k u t k t k u t k

u t

P SR NSR

(A7)

Also, Profits can be written as below:

, ,

1 1

( )

U TSU SD

s u t u t

u t

Profit U U

, , , , , ,

1 1

. ( ).

U TFT FT Gt u t s u t s u t s

u t FT

F P w

, , , ,

1 1

1 .( )

U TSU SD

u t s u t u t

u t

w U U

, , , , , , , , , ,

1 1

( . . . )

U TG SR NSR Rt s G u t s t s u t s t s u t s

u t

P SR NSR s N

(A8)

Profits includes start-up and shut-down costs (first row), fuel cost (second row), impact of forced outage of generating units on

the start-up and shut-down costs (third row), and revenues obtained from the sold energy, spinning and non-spinning reserves

(fourth row) as described in section 2. However, Profits represents the costs and revenues realized in scenario s and does not have

the expectation operator of EP. By substituting (A7) and (A8) in (A6) and eliminating the common deterministic terms, i.e. the

first rows of (A7) and (A8), the constraints (5) and (6) are obtained, which completes the proof.

6. REFERENCES

[1] Arroyo, J. M., and Conejo, A. J., ‘Optimal response of a thermal unit to an electricity spot market’, IEEE Trans. Power

Syst., 2000, 15, (3), pp. 1098-1104

[2] Arroyo, J. M., and Conejo, A. J., ‘Optimal Response of a Power Generator to Energy, AGC, and Reserve Pool-Based

Markets’, IEEE Trans. Power Syst., 2002, 17, (2), pp. 404-410

[3] Conejo, A. J., Nogales, F. J., and Arroyo, J. M., ‘Price-Taker Bidding Strategy Under Price Uncertainty’, IEEE Trans.

Power Syst., 2002, 17, (4), pp. 1081-1088

[4] Conejo, A. J., Nogales, F. J., Arroyo, J. M., and Bertrand, R. G., ‘Risk-Constrained Self-Scheduling of a Thermal Power

Producer’, IEEE Trans. Power Syst., 2004, 19, (3), pp. 1569-1574

[5] Ni, E., Luh, P. B., and Rourke, S., ‘Optimal integrated generation bidding and scheduling with risk management under a

deregulated power market’, IEEE Trans. Power Syst., 2004, 19, (1), pp. 600–609

17

[6] Baillo, A., Ventosa, M., Rivier, M., and Ramos, A., ‘Optimal offering strategies for generation companies operating in

electricity spot markets’, IEEE Trans. Power Syst., 2004, 19, (2), pp. 745–753

[7] Plazas, M. A., Conejo, A. J., and Prieto, F. J., ‘Multimarket Optimal Bidding for a Power Producer’, IEEE Trans. Power

Syst., 2005, 20, (4), pp. 2041-2050

[8] Li, T., Shahidehpour, M., and Li, Z., ‘Risk-Constrained Bidding Strategy With Stochastic Unit Commitment’, IEEE Trans.

Power Syst., 2007, 22, (1), pp. 449-458

[9] Pineda, S., Conejo, A. J., and Carrion, M., ‘Impact of Unit Failure on Forward Contracting’, IEEE Trans. Power Syst., 2008,

23, (4), pp. 1768–1775

[10] Conejo, A. J., Bertrand, R. G., Carrion, M., Caballero, A., and de Andrés, A., ‘Optimal involvement in futures markets of a

power producer’, IEEE Trans. Power Syst., 2008, 23, (2), pp. 703–711

[11] Ruiz, C., and Conejo, A. J., ‘Pool strategy of a producer with endogenous formation of locational marginal prices’, IEEE

Trans. Power Syst., 2009, 24, (4), pp. 15-21

[12] Morales, J. M., Pineda, S., Conejo, A. J., and Carrion, M., ‘Scenario Reduction for Futures Market Trading in Electricity

Markets’, IEEE Trans. Power Syst., 2009, 24, (2), pp. 878–888

[13] Garces, L. P., and Conejo, A. J., ‘Weekly Self-Scheduling, Forward Contracting, and Offering Strategy for a Producer’,

IEEE Trans. Power Syst., 2010, 25, (2), pp. 657-666

[14] Simoglou, C. K., Biskas, P. N., and Bakirtzis, A. G., ‘Optimal Self-Scheduling of a Thermal Producer in Short-Term

Electricity Markets by MILP’, IEEE Trans. Power Syst., 2010, 25, (4), pp. 1965-1977

[15] Baringo, L., and Conejo, A. J., ‘Offering Strategy Via Robust Optimization’, IEEE Trans. Power Syst., 2011, 26, (3), pp.

1418-1425

[16] Kazempour, S. J., Conejo, A. J., and Ruiz, C., ‘Strategic Generation Investment Using a Complementarity Approach’, IEEE

Trans. Power Syst., 2011, 26, (2), pp. 940-948

[17] Simoglou, C. K., Biskas, P. N., and Bakirtzis, A. G., ‘Optimal Self-Scheduling of Thermal Units During Commissioning’,

IEEE Trans. Power Syst., 2012, 27, (1), pp. 181-188

[18] Jabr, R. A., ‘Robust Self-Scheduling Under Price Uncertainty Using Conditional Value-at Risk’, IEEE Trans. Power Syst.,

2005, 20, (4), pp. 1852-1858

[19] Jabr, R. A., ‘Self-scheduling under ellipsoidal price uncertainty: conic-optimisation approach’, IET Gener. Transm. Distrib.,

2007, 1, (1), pp. 23-29

[20] Amjady, N., Vahidinasab, V., ‘Security-Constrained Self-Scheduling of Generation Companies in Day-Ahead Electricity

Markets Considering Financial Risk’, Energy Conversion and Management, 2013, 65, pp. 164–172

18

[21] Rockafellar, R. T., Uryasev, S., ‘Optimization of conditional value-at-risk’, J. Risk, 2000, 2, (3), pp. 21–41

[22] New York Independent System Operator, NYISO, Available: http://www.nyiso.com, accessed December 2012

[23] The Electric Reliability Council of Texas, ERCOT, Available: http://www.ercot.com, accessed December 2012

[24] Ostrowski, J., Anjos, M. F., and Vannelli, A., ‘Tight Mixed Integer Linear Programming Formulations for the Unit

Commitment Problem’, IEEE Trans. Power Syst., 2012, 27, (1), pp. 39-46

[25] Carrion, M. and Arroyo, J. M., ‘A computationally efficient mixed-integer linear formulation for the thermal unit

commitment problem’, IEEE Trans. Power Syst., 2006, 21, (3), pp. 1371–1378

[26] Amjady, N., Rashidi, A. A., and Zareipour, H., ‘Stochastic security-constrained joint market clearing for energy and

reserves auctions considering uncertainties of wind power producers and unreliable equipment’, Euro. Trans. Electr. Power.

2012, DOI: 10.1002/etep.670

[27] Nejati, M., Amjady, N., and Zareipour, H., ‘A New Stochastic Search Technique Combined With Scenario Approach for

Dynamic State Estimation of Power Systems’, IEEE Trans. Power Syst., 2012, 27, (4), pp. 2093-2105

[28] Rosenthal, R. E., ‘GAMS: A User’s Guide’, Washington, DC: GAMS Development Corporation, Jan. 2012.

[29] Gröwe-Kuska, N., Heitsch, H., and Römisch, W., ‘Scenario reduction and scenario tree construction for power management

problems’, in Proc. IEEE Bologna Power Tech Conf., Bologna, Italy, Jun. 2003

[30] SCENRED2 Solver Documentation, Available: http://www.gams.com/dd/docs/solvers/scenred2.pdf, accessed December

2012

[31] IEEE 118-bus test system, Available: http://motor.ece.iit.edu/data/PBUCData.pdf, accessed December 2012

[32] IEEE 118-bus test system, Available: http://motor.ece.iit.edu/data/118bus_abreu.xls, accessed December 2012

[33] Amjady, N. and Keynia, F., ‘Day-Ahead Price Forecasting of Electricity Markets by Mutual Information Technique and

Cascaded Neuro-Evolutionary Algorithm’, IEEE Trans. Power Syst., 2009, 24, (1), pp. 306–318

[34] Leite Da Silva, A. M. , Da Fonseca Manso, L. A., De Oliveira Mello, J. C., and Billinton, R., ‘Pseudo-chronological

simulation for composite reliability analysis with time varying loads’, IEEE Trans. Power Syst., 2000, 15, (1), pp. 73–80

[35] The ILOG CPLEX, 2012, Available: http://www.ilog.com/ products/cplex, accessed December 2012

19

Fig. 1) Representation of VaR, CVaR, RP and CRP concepts

20

0 0.02 0.04 0.06 0.08 0.10 0.121.8

1.9

2

2.1

2.2

2.3

2.4x 10

6

FOR

Ex

pecte

d P

rofi

t ($

)

Fig. 2) Expected profit for different values of FOR of units

21

3000 3250 3500 3750 4000 4250 4500 4750 5000 5250 5500 5750 6000 6250

12

14

16

18

20

22

24

26

28

Offered power for energy auction (MW)

Off

ere

d e

nerg

y p

rice (

$)

Lambda=0 Lambda=1 Lambda=10

Fig. 3) Energy offer curves for hour 2

22

3000 3250 3500 3750 4000 4250 4500 4750 5000 5250 5500 5750 6000 6250 6500

12

14

16

18

20

22

24

26

28

Offered power for energy auction (MW)

Off

ere

d e

nerg

y p

rice (

$)

Lambda=0 Lambda=1 Lambda=10

Fig. 4) Energy offer curves for hour 9

23

5400 5600 5800 6000 6200 6400 6600 6800 7000

20

22

24

26

28

30

32

34

36

38

40

42

Offered power for energy auction (MW)

Off

ered

en

erg

y p

rice

($

)

Lambda=0 Lambda=1 Lambda=10

Fig. 5) Energy offer curves for hour 18

24

5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000

22

24

26

28

30

32

34

36

38

40

Offered power for energy auction (MW)

Off

ered

en

erg

y p

rice

($

)

Lambda=0 Lambda=1 Lambda=10

Fig. 6) Energy offer curves for hour 22

25

Table 1) Power sold in the energy auction, spinning and non-spinning reserve capacity sold for each scenario

Scenario

No.

Power sold in the

energy auction

(MW)

Spinning reserve

capacity sold

(MW)

Non-spinning reserve

capacity sold

(MW)

1 149174 5430 7610

2 140379 5863 8000

3 143053 5452 7760

4 145741 5894 7700

5 139918 5817 7900

6 148144 5429 7850

7 140357 5343 8000

8 148769 5412 7730

9 146230 5437 7230

10 140103 5425 8000

11 143752 5927 8000

12 143610 5814 7850

13 144677 5837 8000

14 147707 5967 8000

15 150124 5527 7720

16 148618 5482 7850

17 143647 5449 7790

18 146156 5487 7720

19 141661 5716 7520

20 152712 5937 7580

26

Table 2) Obtained results with different MAPE values

MAPE EP

($×106)

Actual profit

($×106)

Financial

loss

($×106)

CVaR

($×106)

1.5×MAPEP 2.516562 1.998383 0.518179 0.120174

1.25×MAPEP 2.430319 2.002390 0.427929 0.099525

1×MAPEP 2.345399 2.006619 0.338780 0.079671

0.75×MAPEP 2.261475 2.010495 0.250980 0.059834

0.5×MAPEP 2.178559 2.014733 0.163826 0.039336

27

Table 3) EP, CVaR and CRP for different values of λ

λ EP

($×106)

CRP

($×106)

CVaR

($×106)

λ=0 2.242812 2.026808 0.216004

λ=0.25 2.242467 2.093223 0.149244

λ=0.5 2.241016 2.096915 0.144101

λ=1 2.238345 2.100158 0.138187

λ=2.5 2.235123 2.101703 0.133420

λ=5 2.232532 2.102336 0.130196

λ=10 2.229892 2.102678 0.127214

λ=25 2.226929 2.102850 0.124079

λ=100 2.226783 2.102854 0.123929

28

Table 4) AP results of the trial scenarios for the proposed stochastic approach (λ=1)

NTS AP

($×106)

EP

($×106)

β

(×10-4)

10000 2.230260 2.238345 6.383

20000 2.238758 2.238345 4.530

50000 2.237754 2.238345 2.861

29

Table 5) Size of the optimization problem and computation time of the proposed approach for IEEE 118-bus test case

Constraints 3,399,623

Continuous variables 3,137,778

Binary variables 6,696

CPU time (sec) 323