Self-scheduling of a wind producer based on Information Gap Decision Theory

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Energy 81 (2015) 588e600

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Energy

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Self-scheduling of a wind producer based on Information GapDecision Theory

M. Moradi-Dalvand a, B. Mohammadi-Ivatloo b, *, N. Amjady c, H. Zareipour d,A. Mazhab-Jafari d

a Department of Electrical and Computer Engineering, Shahid Beheshti Uinversity (Abbaspour Technical College), Tehran, Iranb Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, Iranc Department of Electrical Engineering, Semnan University, Semnan, Irand Power and Energy Systems Group, Department of Electrical and Computer Engineering, University of Calgary, Alberta, Canada

a r t i c l e i n f o

Article history:Received 13 March 2014Received in revised form18 November 2014Accepted 1 January 2015Available online 2 February 2015

Keywords:IGDT (Information-Gap Decision Theory)Electricity marketsSelf-schedulingUncertaintyWind producer

* Corresponding author. Tel.: þ98 4113393744.E-mail addresses: [email protected]

[email protected], [email protected]@tavanir.org.ir (N. Amjady), h.zareipor@ucalga

http://dx.doi.org/10.1016/j.energy.2015.01.0020360-5442/© 2015 Elsevier Ltd. All rights reserved.

a b s t r a c t

In a competitive market where all producers must participate in the market, WPPs (wind power pro-ducers) face two sources of uncertainty: (i) future market prices, and (ii) their production capability incoming hours. In this paper a risk-constrained optimal self-scheduling method for a WPP considering theuncertainty associated with market prices and wind generation is proposed. IGDT (Information GapDecision Theory) is used to address theses uncertainties in WPP's self-scheduling. The proposed IGDT-based model is a bilevel programming approach, which is transformed to an equivalent single levelbilinear programming model that can be solved using available solvers. Numerical simulations anddiscussions are provided.

© 2015 Elsevier Ltd. All rights reserved.

1. Introduction

1.1. Background and motivation

Utilization of renewable energies for electricity production hassignificantly grown in the recent years. Global warming and en-ergy security are mentioned as two main drivers for this trend[1,2]. Using renewable energies for electricity production canresult in lower consumption of fossil fuels, decreasing CO2 emis-sions and therefore alleviating global warming. In addition, ad-vances in renewable electricity generation technologies canmitigate concerns about limited sources of fossil fuels. Alternativesupport schemes are used in various jurisdictions to promotegeneration of electricity from renewable energies, specifically,using wind power. The main three schemes are fixed feed-intariffs, feed-in premium and green certificates [3,4]. In feed-intariff scheme, WPPs (wind power producers) are paid with a

(M. Moradi-Dalvand),(B. Mohammadi-Ivatloo),

ry.ca (H. Zareipour).

fixed price for the total wind energy amount fed into the grid. Infeed-in premium scheme, WPP participates in electricity marketand receives market income and a fixed regulated premium for itsproduced energy. In green certificate scheme, some marketplayers are obligated to procure green electricity or its equivalentgreen certificate. In this scheme, income of a WPP comes fromparticipation in electricity market and selling its green certificates(which come from generating green electricity) to obligatedparties. WPPs that operate under feed-in premium and greencertificate support schemes need to participate in an electricitymarket. While electricity markets have different structures, theyshare the challenges of scheduling of WPPs who are supported byfeed-in premium and green certificate schemes. In particular, thechallenges include variability, and to some extent unpredict-ability, of the generated wind power and market price volatility. Indeterministic self-scheduling approaches, the uncertainty of windpower and electricity prices are ignored [5], which could lead toconsiderable profit losses. The economic effect of price forecastinginaccuracies on self-scheduling of thermal and hydro producer isstudied in Ref. [6]. In this paper, we apply IGDT (Information GapDecision Theory) to model the self-scheduling problem of a WPPin presence of these two sources of uncertainty.

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Nomenclature

ParametersblDt forecasted (expected) day-ahead energy marketclearing price for time interval t ($/MWh).blþt , bl�t expected excess/deficit generation imbalance price fortime interval t ($/MWh).

lPt the premium for renewable generation for timeinterval t ($/MWh).

B0 expected maximum profit based on the forecastedwind generations and market prices ($). B0 is alsoshown with rc in the paper.

Bc critical profit ($).s profit deviation factor.M sufficiently large constant.BBðþ;MaxÞ

t =Bðþ;MinÞt upper/lower bounds for Bþt .

Bð�;MaxÞt =Bð�;MinÞ

t upper/lower bounds for B�t .

Variablesq decision variables.PWt power generation of WPP for time interval t (MWh).u uncertain variables.lDt day-ahead energy market clearing price for time

interval t ($/MWh).lþt , l

�t excess/deficit generation imbalance price for time

interval t ($/MWh).PDt the offered power to day-ahead energymarket for time

interval t (MWh).

Dþt , D

�t excess/deficit generation in time interval t (MWh).

a horizon of relative variations of uncertain parametersaround their forecasted values.

a optimal robustness function value.Bþt the lowest profit of WPP when excess generation is

occurred.B�t the lowest profit of WPP when deficit generation is

occurred.

Binary variablesb a binary variable which is used to determine excess (1)

or deficit generation (0).

FunctionsRð:Þ system model in IGDT method.Uð:Þ fractional uncertainty model in IGDT method.að:Þ robustness function in IGDT method.fu(x,y*) upper level objective function in bi-level optimization

problem.gu(x,y*) upper level inequality constraints in bi-level

optimization problem.fl(x,y) lower level objective function in bi-level optimization

problem.gl(x,y) lower level inequality constraints in bi-level

optimization problem.hl(x,y) lower level equality constraints in bi-level

optimization problem.

M. Moradi-Dalvand et al. / Energy 81 (2015) 588e600 589

1.2. Literature review

Despite the environmental profits, the variability associatedwith wind power poses operational and economical challenges. Tomitigate the impacts of wind power uncertainty in power systemoperation, various approaches have been proposed in the literature.Scenario-based stochastic optimization modeling has been pro-posed in Refs. [7e11]. In these papers, a presumed probabilitydistribution function is considered for wind generation and somescenarios are generated as the representation of wind generation.In Ref. [7], a two-stage stochastic programming formulation ispresented for scheduling thermal generation units and responsiveloads in a power system with high penetration of wind power. Aparticle swarm optimization based approach is proposed for dy-namic economic dispatch which uncertainties of wind and load areaddressed through scenarios in Ref. [8]. A risk-constrained sto-chastic programming method for participation of aWPP in the day-ahead market is presented in Ref. [9]. The considered risk measureis the CVaR (conditional value-at-risk) and the uncertainties areday-ahead and imbalance prices as well as wind generation. In Ref.[10] unit commitment problem with wind uncertainty is modeled.The normal distribution is used to model the wind power fore-casting error. CVaR (conditional value at risk), as a prevalent risk-measure, has been utilized for managing wind producers' risk inRefs. [12,13]. Chance constrained method is used to model winduncertainty in Ref. [14]. In Ref. [15], wind generation is modeled as anegative load. Uncertainty of aggregated load and wind generationin each bus of the system is modeled using interval optimization.Interval optimization and scenario-based approaches have beencompared in Ref. [16]. In interval optimization approaches, there isno need to presume a probability distribution for uncertainty.However, they require a confidence interval, which should be

selected carefully. In comparison with the scenario-based sto-chastic optimization approach, interval optimization has lowercomputational burden and is more sensitive to confidence interval[16]. Fuzzy modeling approach is applied to model available windpower generation in Refs. [17,18]. Robust Optimization is applied tomodel wind uncertainty in Ref. [19]. The objective of the proposedprogram in Ref. [19] is minimizing the cost of supplying electricityunder worst wind power output scenario. Information Gap Deci-sion Theory is introduced in Ref. [20] as a decision making methodunder uncertainty. IGDT method is a non-probabilistic and non-possibilistic (non-fuzzy) method. This method has been appliedto different decision making processes in the literature. IGDTmethod has been used in Ref. [21] for modeling life cycle engi-neering design problems. A review or recent risk managementtechniques applied to energy systems is provided in Ref. [22]. Inrecent years, the IGDT method has been implemented in operationproblems of power systems [23e34]. The previous IGDT-basedoptimization approaches used in power systems cope successfullywith the uncertainties of electricity market prices and loads andprovide robust decisions. The IGDT method has been used todetermine robust participation of a large consumer into the day-ahead and subsequent adjustment markets in Ref. [23]. The un-certain parameters of [23] are day-ahead and adjustment marketprices. In Ref. [24], IGDT-based risk-constrained electricity pro-curement of a large consumer from different resources, i.e., pool, itsgeneration facilities and bilateral contracts, has been modeled. Theuncertain parameter in Ref. [24] is the pool price. IGDT method hasbeen used in Ref. [25] for modeling market price uncertainty inshort term scheduling of a thermal GenCo. A robust decisionmaking tool using IGDT method for load procurement of a DNO(Distribution Network Owner) has been proposed in Ref. [26]. Poolmarket electricity price and load are considered as uncertain

M. Moradi-Dalvand et al. / Energy 81 (2015) 588e600590

parameters in Ref. [26]. IGDT method is implemented in Ref. [27]for bidding strategy problem of GenCos considering the bilateralcontacts. This method is extended to solve the scheduling problemof large electricity utilities in Ref. [28]. Demand response programand price uncertainties are modeled in Ref. [29] using IGDTmethod. A multi-stage transmission expansion planing in presenceof wind farms is obtained using IGDT method in Ref. [30]. In Ref.[31], the uncertainties of capital cost of transmission lines as well asdemand are modeled in transmission expansion planning problemusing IGDT method. An IGDT-based midterm, i.e. one month tothree years, energy procurement method is presented in Ref. [32].The problem of GenCo's portfolio management is modeled in Refs.[33,34]. However, to the best of the authors' knowledge, noresearchwork has concurrentlymodeled the uncertainty sources ofwind power generation and balancing electricity market prices forday-ahead self-scheduling of a WPP in the IGDT framework, whichis specific to this paper.

1.3. Assumptions and contributions

In the present paper, we consider a market structure whereWPPs must participate in the market, and thus, submit their offersto the system operator. It is assumed that a feed-in premiummechanism is in place to promote wind development. In feed-inpremium schemes, WPPs participate in electricity markets andreceive a fixed regulated premium for their produced energy, inaddition to market prices [3,4]. A feed-in premium mechanism isconsidered a trade-off solution because it exposes WPPs to marketprofits without imposing new considerable risks on them [3]. Thefeed-in premium support scheme is used in Spain and Denmark[3,35]. In this paper, we used IGDT for addressing wind generationand price uncertainties in WPP scheduling in order to achieveschedule with maximum tolerable horizon of uncertainty for acritical benefit. Unlike other uncertainty modeling approaches, theobjective of the IGDT method is maximizing the tolerable horizonof uncertainty while satisfying a predetermined objective. Thehorizon of uncertainty defines the relation betweenwhat is known,i.e., predicted values of uncertain parameters, and what couldhappen in reality. Another difference of IGDT method with otherrisk management approaches is that IGDT method guarantees acertain predetermined performance for objective function, pro-vided that the realized forecast error falls into the maximized ho-rizon of uncertainty. In comparison with stochastic programmingand chance-constrained based methods [12e14,36], there is noneed to know the probability distribution function of wind poweruncertainty in the proposed IGDT-based method. Unlike fuzzyuncertainty modeling methods [17,18], in IGDT approach, there isno need to know or assign membership functions to uncertainparameters. Wind generation uncertainty modeling in the pro-posed method is similar to that of the Interval [15,16] and Robust[19] optimization approaches in the sense that in these methods anuncertainty band is defined. This band, referred to as confidenceinterval in Robust and Interval optimization methods, is called thehorizon of uncertainty or info-gap uncertainty in IGDT method.However, in Interval and Robust optimization methods, the lowerand upper level of this band should be determined as input pa-rameters, whereas there is no need to determine these bounds inIGDT. In robust optimization methods, the robustness region orhorizon of uncertainty is fixed before solving the problem [37]. Thatregion is optimized in the solution procedure of IGDT method suchthat the solution is robust for a maximized horizon of uncertainty.In other words, the uncertainty band in other methods is defined asinput of the optimization procedure, and the goal of the decisionmaker is maximizing the objective function. However, in IGDTmethod, the profit deviation factor is defined as a parameter and

the objective of the decision maker is maximizing the horizon ofuncertainty while a critical profit is satisfied.

Unlike the other uncertainty modeling approaches used foroptimization problems, which optimize the objective function, theobjective of the IGDT method is maximizing the tolerable horizonof uncertainty while satisfying a predetermined objective. Thehorizon of uncertainty defines the relation betweenwhat is known,i.e., predicted value of uncertain parameter, and what could happenin reality. Another difference of IGDT method with other riskmanagement tools is that IGDT method guarantees a certain pre-determined performance for objective function, provided that therealized forecast error falls into the maximized horizon of uncer-tainty. IGDT method does not require neither PDF (probabilitydistribution function) of the uncertain parameter, which is used inmost of the probabilistic methods, nor the membership functionswhich are required in fuzzy approaches. In comparison with sto-chastic programming and chance-constrained based methods[12e14,36], there is no need to know the probability distributionfunction of wind power uncertainty in the proposed IGDT-basedmethod. Unlike fuzzy uncertainty modeling methods [17,18], inIGDT approach, there is no need to know or assign membershipfunctions to uncertain parameters. Wind generation uncertaintymodeling in the proposed method is similar to that of the Interval[15,16] and Robust [19] optimization approaches in the sense thatin these methods an uncertainty band is defined. This band,referred to as confidence interval in Robust and Interval optimi-zation methods, is called the horizon of uncertainty or info-gapuncertainty in IGDT method. However, in Interval and Robustoptimization methods, the lower and upper level of this bandshould be determined as input parameters, whereas there is noneed to determine these bounds in IGDT. In other words, in robustoptimization methods the robustness region or horizon of uncer-tainty is fixed before solving the problem while is optimized in thesolution procedure of IGDT method such that the solution is robustfor a maximized horizon of uncertainty. Briefly, in the robustoptimization methods the uncertainty band is defined as input ofthe optimization procedure and the goal of the decision maker ismaximizing the objective functionwhereas in the IGDTmethod theprofit deviation factor is defined as a parameter and the objective ofthe decision maker is maximizing the horizon of uncertainty whilea critical profit is satisfied. Rather, this band is maximized in theoptimization process such that the solution is robust for a maxi-mized horizon of uncertainty.

The main contributions of this paper can be summarized asfollows:

1. We propose a new non-probabilistic and non-possibilisticmethod for handling wind power and market price un-certainties in IGDT framework.

2. The proposed WPP self-scheduling IGDT-based model leads to abi-level programming problem. Since the lower level program isnon-convex, KKT (KarusheKuhneTucker) conditions can not beused for casting the program into a single level program. Wepropose a method to cast this bi-level model to a single-levelprogramming problem which can be solved using commercialsolvers.

1.4. Paper organization

The remainder of this paper is organized as follows. Section 2provides background of the implemented IGDT method for riskmanagement. Section 3 describes the structure of the consideredmarket, problem formulation without and with considering un-certainties, proposed IGDT-based robust scheduling formulation


and conversion of the bilevel IGDT-based formulation to anequivalent single level problem. Section 4 provides simulation re-sults for an illustrative case and a test system. Finally concludingremarks are provided in Section 5.

2. Information Gap Decision Theory

The IGDT model can be described using three elements, namelya system model, an uncertainty model and a performancerequirement. The input/output structure of the studied system isdescribed in the system model, i.e., R(q,u), considering a decisionvariable q and the uncertain parameter u [20]. In scheduling theoperation of a wind power producer R(q,u) can be its total profit.There are different methods for representing uncertainty modelusing IGDT [20]. The more common uncertainty model in IGDT isfractional uncertainty model which can be represented as follows.

Uða; buÞ ¼ �u :

��u� bubu�� a

�; a � 0 (1)

where, bu shows the forecasted values of the uncertain parametersand a is the horizon of uncertainty. The forecasted values of theuncertain parameters are input while the horizon of uncertainty isa variable which is determined in the decision making process.Note that since the horizon of uncertainty is a variable, the lowerand upper bounds of the uncertainty horizonwill be determined inthe solution procedure. The structure of fractional uncertaintymodel is shown in Fig. 1. This uncertainty model implies that thelength of horizon of uncertainty is proportional to the forecastedvalue of the uncertain parameter.

Although different performance functions can be defined inIGDT context, robustness function is considered in this paperbecause of its capability for modeling worst case scenarios. Therobustness function models the immunity of the decision againstthe unfavorable deviations of the uncertain parameter from theforecasted value. The minimum demanded or desired reward valueof the decision making problem is expressed using robustnessfunction, which can be defined as follows:

aðq; rcÞ ¼ maxa

�a :

�minimum requirementrc is always satisfied

��(2)

This immunity function ensures that reward, i.e., the profits inthis paper, will not be less than a critical value rc, provided theuncertain parameters fall within the region of uncertainty. TheIGDT robustness function aðq; rcÞ represents maximum info-gapuncertainty that decision variable q can tolerate with the perfor-mance not being worse than rc. The IGDT model is a bilevel pro-gram. A general form of bilevel program can be stated as follows[38]:

maxx

f uðx; y�Þ (3)

s.t.

guðx; y�Þ � 0 (4)

Fig. 1. Simple illustration of fractional uncertainty model.

y� ¼ arg�miny

f lðx; yÞ (5)

s.t.

glðx; yÞ � 0 (6)

hlðx; yÞ ¼ 0o

(7)

where the superscripts u and l indicate upper and lower levels ofthe bilevel problem, respectively. The upper level program isformed by (3) and (4) and (5)e(7) form the lower level program.

3. Problem formulation

3.1. Market structure

A pool with day-ahead and balancing (real time) markets isconsidered in this paper. The balancing market clearing procedureconsidered in this paper is taken from Ref. [39], which is inspiredfrom Iberian Peninsula electricity market in Spain. It is supposedthat all market players must trade all generated or demanded en-ergies in power pool. Market players must participate in day-aheadmarket through offering for electricity generation and bidding forconsumption. Market participants should schedule themselvesbased on the cleared day-ahead market. If a player deviates fromthe scheduled amount (awarded power), the deviation is clearedusing balancing market mechanism. In this mechanism if a pro-ducer generates more than the awarded power, its excess genera-tion is cleared with positive imbalance price. Positive imbalanceprice is lower or equal to the cleared day-ahead energy marketprice. If a producer generates less than the scheduled value, itsdeficit generation is cleared with negative imbalance price which isgreater or equal to the cleared day-ahead energy market price.Positive (negative) imbalance price is equal to day-ahead clearingmarket price when real time total system generation is lower(greater) than demand. Therefore one of the positive and negativeimbalance prices is equal to the day-ahead clearing price. In such amarket, generators should avoid unscheduled generation whichleads to their economic loss.

3.2. WPP self scheduling without considering uncertainty

In a feed-in premium support scheme, WPP should participatein electricity market like other producers. Thus, WPP should offerto the day-ahead market. It is supposed that the WPP is a pricetaker player and therefore, its decision has no effect on the day-ahead market clearing prices. The WPP needs to determine theoptimal quantity to offer to the market. Thus, the WPP wouldforecast its generation for each interval of the day-ahead marketscheduling horizon. In this stage, we assume that the WPP canforecast its day-ahead power production with no error. In thiscondition, the WPP self scheduling problem is presented in(8)e(12).

B0 ¼ maxPDt ;D

þt ;D

�t

Xt

�lPtbPWt þ blDt PDt þ blþt Dþ

t � bl�t D�t

�(8)

s.t.:

bPWt ¼ PDt þ Dþ

t � D�t ct (9)

0 � PDt � PMax ct (10)


0 � Dþt � bPW

t ct (11)

0 � D�t � PMax ct (12)

The overall structure and idea of this model comes from Ref.[39]. The objective function of WPP, which is presented in (8), ismaximizing total expected profit of the WPP. The income of theWPP comes from the feed-in premium, the day-ahead market po-wer sale, excess generation in balancing markets. The only cost isthe deficit generation in the balancingmarket. The power balancingfor a WPP is presented in (9). The produced power of a WPP equalsto its day-ahead offered power plus positive deviation minusnegative deviations for each time interval. The offered power ofWPP should be positive and lower than its generation capacity, asindicated in (10). Excess generation, i.e., Dþ

t , is a positive variable

which its maximumvalue is expected power generation, i.e., bPWt , as

shown in (11). The maximum value of Dþt occurs when WPP offers

zero power production to day-aheadmarket. Deficit generation, i.e.,D�t , is a positive variable and its maximum value is the capacity of

WPP, i.e., PMax, as shown in (12). The maximum value of D�t occurs

when WPP offers its maximum capacity to day-ahead market andno wind power generation is realized. Bearing in mind that lþt � lDt

and l�t � lDt , when WPP has a completely perfect forecast for itswind power production, the optimal solution has no dependencieson positive and negative imbalance prices and PDt ¼ PWt ,Dþ

t ¼ 0 andD�t ¼ 0. It should be noted that this problem may have more than

one optimal solution. For example suppose lþt ¼ lDt for a certainhour. In this condition each positive PDt and Dþ

t which satisfiesbPWt ¼ PDt þ Dþ

t , gives optimal values with equivalent objectivevalue. However PDt ¼ PWt , Dþ

t ¼ 0 and D�t ¼ 0 is an optimal solution

in all situations.

3.3. The proposed IGDT-based WPP self scheduling with wind andprice uncertainty

The uncertain parameters are wind power generation (PWt ), day-ahead market price (lDt ) and balancing market prices (lþt and l�t ).Thereforewe can represent uncertain parameters for time interval tby ut ¼ fPWt ; lDt ; l

þt ; l

�t g. Uncertainty set can be defined for all

planning horizon intervals as u ¼ fu1;u2;…;uTg. The decisionvariables are the amount of powers to be offered to the day-aheadmarket, i.e., PDt ; t2f1;2;…; Tg. IGDT is used to handle the uncer-tainty of day-ahead and balancing market prices as well as windgenerations at operation time intervals. It is assumed that theforecasted values of all uncertain parameters are available. Theforecasted values of uncertain parameters are denoted by bPw

forwind power generation, blD

for day-ahead energy market and blþ

and bl�for positive and negative balancing market prices, respec-

tively. The set of forecasted values of uncertain parameters can be

shown as but ¼ fbPWt ; blDt ; blþt ; bl�t g. bu is defined similar to u. The de-

cision variables are power and excess/deficit generations which canbe shown as q ¼ fPDt ;Dþ

t ;D�t g. A fractional information-gap un-

certainty model is used for uncertainty modeling of wind powergeneration and day-ahead clearing prices. A fractional information-gap uncertainty model with only upper (lower) bound is used fordeficit (excess) imbalance prices. Excess (deficit) imbalance pricesshould be lower (more) than day-ahead clearing price, which ismodeled with fractional information-gap uncertainty model (1).Decision making under uncertainty is different from deterministicscheduling procedure. Robust self-scheduling is a decision which

immunes WPP under a range of uncertainties and gains a pre-determined profit for it.

3.4. Robust self-scheduling

The general form of the robustness function in IGDT basedmethod was presented in (2). The robust self-scheduling of a WPPcan be modeled using (13)e(27).

a ¼ maxa;PD

t

a (13)

s.t.:

B� � Bc ¼ ð1� sÞB0 (14)

0 � PDt � PMax; ct (15)

B� ¼(

minPWt ;lDt ;l

þt ;l

�t ;D

þt ;D

�t

Xt

lPt P

Wt þ lDt P

Dt

þlþt Dþt � l�t D

�t

!(16)

s.t.:

PWt ¼ PDt þ Dþt � D�

t ; ct (17)

0 � Dþt � PWt ; ct (18)

0 � D�t � PMax; ct (19)

0 � PWt � PMax; ct (20)

0 � lþt � lDt ; ct (21)

lDt � l�t ; ct (22)

Dþt D

�t ¼ 0; ct (23)

ð1� aÞbPWt � PWt � ð1þ aÞbPW

t ; ct (24)

ð1� aÞblDt � lDt � ð1þ aÞblDt ; ct (25)

ð1� aÞblþt � lþt ; ct (26)

l�t � ð1þ aÞbl�t ; cto

(27)

It should benoted that obtaining theminimumrequirement is notstraightforward in the WPP scheduling. Hence, we need to solve anoptimization problem to obtain the minimum requirement. Thisminimum requirement is dependent to the uncertainty horizon,which is the solution of another optimization problem. Due to thiscross-relation between the solutions of two optimizationproblems, abi-level optimizationmodel should be implemented. The upper levelprogram is formulated in (13)e(15) and is used to determinemaximum possible horizon of uncertainty which guarantee a pre-determined profit. The lower level program is presented (16)e(27),which isused todetermine the lowestprofit fora certain scheduleanduncertainty horizon. Theuncertaintyhorizona and the offeredpowerPDt is determined in theupper level. Although time series of electricityprice and wind power usually have different volatility levels andforecast errors, we consider the same uncertainty horizon for both ofthem for the sake of simplicity. Themain reason of this simplification


is tobetterdescribe theunderlying ideaand illustrate theapplicabilityof IGDT method for solving WPP self-scheduling problem. The pro-posed model can be extended to consider different uncertainty ho-rizons, e.g. two uncertainty horizons for wind power generation andelectricity market price here. In this manner, the IGDT-based self-scheduling model will be a multi-objective optimization problem,which its objectives are different uncertainty horizons. This problemcan be solved by adding a multi-objective solution method to theproposed IGDT approach. For instance, ε-constraint method [40]converts the multi-objective optimization problem to a set of singleobjective problems, which each one can be solved by the IGDTapproach. In the lower level problem, the values of uncertain pa-rameters in uncertainty horizon which result in minimum profit aredetermined. In this problem, the objective of theWPP is maximizingthe horizon of uncertainties while the total profit is greater than apredeterminedvalueBc. The criticalprofit, i.e.Bc, is a percentageof theexpected profit when scheduling is done based on the forecastedvalues without considering uncertainties, i.e. B0, as described in Sec-tion 3.2. In other words, Bc¼ (1� s)B0. Equation (15) is equal to (10).Equations (17)e(19) are equal to (9), (11) and (12), respectively,whichare described in Section 3.2. Although PWt , lDt , l

þt and l�t are uncertain

parameters, they follow certain logics which are shown in (20)e(22).The realizedWPP generation is lower than its capacitywhich is givenin (20). As stated in Section 3.1, excess (deficit) generation imbalanceprice is a positive valuewhich is equal or lower (equal or greater) thanday-ahead clearing price for different hours which are presented in(21)and (22). It isobvious thatexcessanddeficit generationcannotbeoccurred simultaneously which is implied in (23). Equations(24)e(27)model the informationgapmodel.A lower (upper)bound isconsidered for lþt (l�t ) and its upper (lower) bound is limited to lDtbased on (21) and (22).

3.5. Equivalent single level program of robust self-schedulingproblem

The robust self-scheduling problem, described in Section 3.4, is abilevel program, which should be converted to a single level prob-lem for solving it with commercial solvers. Awidely used procedureis deriving the first order necessary optimality conditions, i.e., KKT(KarusheKuhneTucker) conditions, of lower level program andrepresenting the lower level program by these constraints in theupper level [41,42]. In Ref. [41] a bilevel programming model fortransmission expansion planning is presented in which the upperlevel program models the investment phase while the lower levelprogram represents market clearing. A bilevel approach to solvemid-term decision making of a retailer is presented in Ref. [42]. Theupper level program determines the selling prices whereas thelower level programmodels the competitions between retailers andclient demand supplied by the decisionmaker supplier. Note that inthese papers the lower level program is convex. However, since thelower level program is non-convex because of constraints (23), thisprocedure can not be used for this problem. In the following wepropose a method to convert it to a single level program withoutderiving KKT conditions. The goal of lower level program is deter-mining the values of uncertain parameters which lead to minimumprofit for constant values of uncertainty horizon (a) and offeredpower (PDt ). The explanation of the proposed procedure is presentedhere for one hour for the sake of simplicity. However, this explana-tioncanbeextended foranyoperationhorizon, oneday in thispaper,easily. For a fixed a and PDt two conditions can occur:

1. Excess generation

This condition occurs when the produced power in real time isgreater than the offered power in the day-ahead market, i.e.,

PWt > PDt . In this condition, the profit of the WPP presented in (16)can be rewritten as follows:

Bþt ¼ minPWt ;lDt ;l

þt

lPt P

Wt þ lDt P

Dt

þlþt�PWt � PDt

�! (28)

where, Bþt is the lowest profit of WPP when excess generation isoccurred. It should be noted that for fixed value of PDt , the WPP'sprofit in (28) monotonically decreases by decreasing wind power,day-ahead market price and excess generation imbalance price.Thus, for a given uncertainty horizon, the minimum profit occurs atthe minimum wind power generation, minimum day-ahead mar-ket price and minimum excess generation imbalance price asshown in (29)e(31).

PWt ¼ ð1� aÞbPWt (29)

lDt ¼ ð1� aÞblDt (30)

lþt ¼ ð1� aÞblþt (31)

2. Deficit generation

This condition occurs when the produced power in real time islower than the offered power in the day-aheadmarket, i.e., PWt < PDt .In this condition, the profit of the WPP presented in (16) can berewritten as follows:

B�t ¼ minPWt ;lDt ;l

�t

lPt P

Wt þ lDt P

Dt

�l�t�PDt � PWt

�! (32)

where, B�t is the lowest profit of WPP when deficit generation isoccurred. It should be noted that for fixed value of PDt , the WPP'sprofit in (32) monotonically decreases by decreasing wind powerand day-ahead market price and increasing deficit generationimbalance price. Thus, for a given uncertainty horizon, the mini-mum profit occurs at the minimum wind power generation, min-imum day-ahead market price and maximum deficit generationimbalance price as shown in (33)e(35).



l�t ¼ ð1þ aÞbl�t (35)

When PDt � ð1� aÞbPWt (PDt � ð1þ aÞbPW

t ), we can be sure thatthe excess (deficit) generation condition occurs. However, when

ð1� aÞbPWt � PDt � ð1þ aÞbPW

t , both conditions can be occurred. Inboth excess and deficit conditions decreasing the wind generationresults in lower profit. Since the objective of the lower level pro-gram is minimizing the profit, the minimum generation is occurredwhenwind generation get its minimum possible value. Considering

that the optimal wind generation in all cases is ð1� aÞbPWt , the PWt

can be replaced with it in the model including both excess anddeficit conditions.

In order to properly include both excess and deficit conditionsin the formulation, a binary variable (bt) can be used for each

Fig. 2. The steps of the proposed self-scheduling method.


hour to detect excess (bt¼ 1) or deficit (bt¼ 0) generation. Thesingle level robust self-scheduling program can be written as(36)e(47).

a ¼ maxa;PD

t

a (36)

s.t.:

B� � Bc ¼ ð1� sÞB0 (37)

0 � PDt � PMax ct (38)

Bþt ¼ lPt PWt þ lDt P

Dt þ lþt

�PWt � PDt

�(39)

B�t ¼ lPt PWt þ lDt P

Dt � l�t

�PDt � PWt

�(40)



lþt ¼ ð1� aÞblþt (43)

l�t ¼ ð1þ aÞbl�t (44)

PDt � PWt þ ð1� btÞM ct (45)

PDt � PWt � btM ct (46)

B� ¼Xt

Bþt bt þ B�t ð1� btÞ (47)

where M is a sufficiently large constant M� PMax. Since excess anddeficit generation result in different objective functions, i.e., (39) forexcess generation and (40) for deficit generation, (45) and (46) aredefined to detect excess and deficit condition. Bearing in mind thatbt is a binary variable, the WPP profit is determined in (47) as thesummation of excess and deficit objective times their occurrencecondition, i.e., bt ¼ 1 for excess generation and bt ¼ 0 for deficitgeneration. It should be mentioned that one of the terms is alwayszero depending on the value of bt in (47). For bt¼ 1, the second termwill be zero and the first term will be zero for bt ¼ 0. Hence, (47)calculate the total benefit considering the excess or deficit condi-tion. Equation (47) can be written in the form of (48)e(53) tosimplify the problem by removing multiplication of binary andcontinuous variables. Using this linearization technique is validhere, since the continuous variables, i.e. Bþt and B�t , are boundedvariables [43].

B� ¼Xt

pþt þ p�

t (48)

Bðþ;MinÞt bt � pþ

t � Bðþ;MaxÞt bt ; ct (49)

Bþt � Bðþ;MaxÞt ð1� btÞ � pþ

t ; ct (50)

pþt � Bþt � Bðþ;MinÞ

t ð1� btÞ; ct (51)

Bð�;MinÞt ð1� btÞ � p�

t � Bð�;MaxÞt ð1� btÞ; ct (52)

B�t � Bð�;MaxÞt bt � p�

t � B�t � Bð�;MinÞt bt ; ct (53)

where Bðþ;MaxÞt , Bðþ;MinÞ

t , Bð�;MaxÞt , Bð�;MinÞ

t are upper and lowerbounds for Bþt and B�t , respectively.

The steps of the proposed self-scheduling method are shown inFig. 2. At the first step, the forecasted values, i.e., bPW

t , blDt , blþt , bl�t , areused to determine the optimal deterministic solution. The onlyoutput of this step that will be used in the next step is themaximum expected profit based on the forecasted values, i.e., B0.The values of B0 and s are used to determine the critical profit, i.e.,Bc¼ (1� s)B0, which is theminimum demanded or desired profit ofthe WPP. The solution provides the bidding strategy of the WPP forthe day-ahead electricity market considering the uncertaintyhorizon.

The only input parameter, which is needed in addition to theforecasted values of uncertain variables, is the profit deviationfactor, i.e., s. No PDFs (probability distribution functions), fuzzymembership functions or confidence intervals of the uncertainvariables are needed for the proposed methodology. Note that theforecasted values of the uncertain variables are available to WPPs,and have become part of their operation. The profit deviation factors indicates the level of risk-averseness of theWPP; the higher the s,the more risk-averse is the WPP. Higher values of the profit devi-ation factor leads to a more robust self-scheduling strategy and theobtained critical profit will be valid for a wider range of realizationsof the uncertain variables. However, the cost of this robustness is alower critical profit. Thus, a WPP can adjust the robustness level ofits bidding strategy based on the minimum tolerable profit, whichis an important advantage of the proposed approach. Also, theproposed method requires less information compared to the othernon-deterministic bidding strategies, such as stochastic program-ming, fuzzy and robust optimization approaches. Thus, the pro-posed IGDT-based bidding strategy is straightforward to implementin practice. After implementing the proposed approach, s acts as asetting for it. By changing s, the WPP can obtain the best trade-offbetween the acceptable uncertainty ranges and critical profit basedon its preferences.

4. Simulation results and discussions

The problem is modeled as a Mixed Integer Nonlinear Programand has been solved using DICOPT [44] solver under GAMS (GeneralAlgebraic Mathematical System) environment [45]. All simulationshave been done using a typical PC systemwith 4 GB RAM and 2 GHzCPU speed. The lower and upper bounds of the variables are pre-sented in Table 1. Moreover, the considered parameters for bothcases are listed in Table 2.

Table 1Bounds of the variables and parameters.

Parameter Value

M 1000Bðþ;MaxÞt 50,000

Bðþ;MinÞt 50,000

Bð�;MaxÞt 0

Bð�;MinÞt 0

PMax 100

Table 3Model statistics.

Block of variables 10 Single variables 194Block of equations 11 Single equations 218Non zero elements 674 Discrete variable 24

Table 4Optimal robustness function value and schedule for case I considering wind


4.1. An illustrative one hour scheduling example

In this section, the proposed model is applied to a typical windfarm aggregator with 100MW capacity for a single time period. It issupposed that the forecasted wind generation is 50 MWh, thepremium is 10 $/MWh, forecasted day ahead market price is 50 $/MWh and forecasted positive and negative imbalance prices are45 $/MWh and 55 $/MWh, respectively. Excess and deficit gener-ation imbalance prices in this paper are considered 0.9 and 1.1times of the day-ahead market price, respectively. These values areinspired from the current Iranian electricity market regulations[36].

4.1.1. WPP self scheduling without considering uncertaintyIn this part, perfect forecasted values are considered for wind

generation and market prices. The optimal schedule of WPP whenthe scheduling is done without considering uncertainties is50 MWh. The expected profit of WPP in this condition is10� 50 þ 50� 50¼ 3000$.

4.1.2. WPP self scheduling with uncertainty in wind powergeneration

In this section it is supposed that themarket prices forecasting isperfect and WPP wants to determine the horizon of uncertaintywhich guarantees achievement of a predetermined profit. Simula-tions are done for a range of s between 0 and 0.7. The optimalrobustness function value a and optimal schedule PD are shown inTable 4 for different profit deviation factors. As depicted in Fig. 3,optimal robustness function value varies linearly with respect tocritical profit. The model statistics is presented in Table 3.

4.1.3. WPP self scheduling with uncertainty in market pricesIn this section, it is supposed that the wind generation fore-

casting is perfect and WPP wants to determine the maximum ho-rizon of uncertainty of market prices whichWPP can tolerate and atleast obtain the profit of Bc. The optimal robustness function value aand optimal schedule PD are shown in Table 5 for different profitdeviation factors. Since wind power forecast is considered

Table 2Simulation data.

Data Illustrative Day-ahead

(Subsection 4.1) (Subsection 4.2)

Wind capacity (MW) 100 100Premium ($/MWh) 10 10Expected wind generation (MWh) 50 Fig. 8Expected day-ahead market

price ($/MWh)50 Fig. 9

Expected balancing marketexcess generation price ($/MWh)

45 0.9 * expected day-aheadmarket prices

Expected balancing marketdeficit generation price ($/MWh)

55 1.1 * expected day-aheadmarket prices

s 0 to 0.7 0.25

completely perfect, the optimal schedule (PDt ) becomes equal to itfor different profit deviation factors (s). As illustrated in Fig. 3,optimal robustness function value varies linearly with respect toprofit deviation factor similar to self-scheduling considering winduncertainty. Also as shown in Fig. 3, for a typical profit deviationfactor wider range of robustness can be achieved for self-scheduling under price uncertainty in comparison with wind un-certainty. The reason of this robustness is the premium. Under priceuncertainty, we assumed that the wind generation forecast is per-fect and since the premium is a predetermined fixed value, pre-mium payment does not vary with variation inmarket prices whichresults in wider range of robustness in comparison with winduncertainty.

4.1.4. WPP self scheduling with uncertainties in wind powergeneration and market prices

In this section both wind power generation and market pricesare considered as source of uncertainty. The optimal robustnessfunction value a and optimal schedule PD are shown in Table 6 fordifferent profit deviation factors. For a typical profit deviation fac-tor, for example s¼ 0.3, the maximumvalue of uncertainty horizonwhich the critical profit, $2100 here, is guaranteed to be obtained is0.178 or 17.8% and the optimal schedule is 41.0977MW. In this case,the critical profit could not be guaranteed if forecast errors be morethan the maximum allowable value, 17.8% here. If the realized windgeneration and market prices fall into this band, the profit will begreater than or equal to $2100. The obtained profit for differentschedules when the worst case in the obtained robust horizon fors¼ 0.15, s¼ 0.3 and s¼ 0.45 are shown in Figs. 4e6, respectively. Itcan be observed from these figures that the maximum benefit oc-curs for lower scheduled power by increasing the value of s. Worstcase here refers to the lowest values of wind generation, day-aheadmarket price and excess generation imbalance price and highestvalue of deficit generation imbalance price. It can be observed thatthe critical profit of $2100 could not be achieved for schedules otherthan 41.0977 MW. It is clear that robustness imposes cost. Supposeforecasting of uncertain parameters is done with no error. In thiscondition, a risk neutral WPP gains more than a risk-averse WPP.

uncertainty.

Profit deviationfactor (s)

Criticalprofit ($) (Bc)

Optimum robustnessfunction value (a)

Optimal schedule(MWh) (PD)

0 3000 0 500.05 2850 0.05 47.50.1 2700 0.1 450.15 2550 0.15 42.50.2 2400 0.2 400.25 2250 0.25 37.50.3 2100 0.3 350.35 1950 0.35 32.50.4 1800 0.4 300.45 1650 0.45 27.50.5 1500 0.5 250.55 1350 0.55 22.50.6 1200 0.6 200.65 1050 0.65 17.50.7 900 0.7 15

Fig. 3. Optimum robustness function value (a) versus critical profit (Bc) for case I.


The difference between these two profits is called RC (RobustnessCost). Robustness cost as a function of robustness function value isshown in Fig. 7. As seen in this figure, robustness cost increaseswith robustness function value. In other words, more robustschedule leads to more profit loss when forecasted values arerealized.

4.2. Day-ahead self-scheduling based on the proposed method

In this section, the uncertainties associated with wind genera-tion, day-ahead market price and excess/deficit imbalance pricesare considered. Similar to illustrative example, aWPPwith 100MWcapacity is considered here. Also the premium is 10 $/MWh for allhours. Wind generation forecasts and day-ahead market prices areshown in Figs. 8 and 9, respectively. The profit of WPP withouttaking uncertainty into account is $51,669.1. The simulation resultsfor the arbitrary profit deviation factor of 0.25 are presented. Thecorresponding critical profit for s ¼ 0.25 is $38,751.825. Simula-tions take less than one second to be solved. The optimal robust-ness function value is 0.147 or 14.7%. It means if the uncertainparameters varies up to 14.7%, then earning the critical profit of$38,751.825 is guaranteed. 14.7% is the maximum possible level ofrobustness which can be achievedwith profit deviation factor equal

Table 5Optimal robustness function value and schedule for case I considering priceuncertainty.





0 3000 0 500.05 2850 0.06 500.1 2700 0.12 500.15 2550 0.18 500.2 2400 0.24 500.25 2250 0.3 500.3 2100 0.36 500.35 1950 0.42 500.4 1800 0.48 500.45 1650 0.54 500.5 1500 0.6 500.55 1350 0.66 500.6 1200 0.72 500.65 1050 0.78 500.7 900 0.84 50

to 0.25. The optimal schedule for s¼ 0.25 and the forecasted valuesshown in Figs. 8 and 9 is illustrated in Fig. 10.

In the engineering decision making problems, the decision aremade by methods which generally have simplifications to someextent. In order to check the efficiency of the decisions, somemethods with more details are used. Monte Carlo method is anextremely used for decision test phase. In the context of the paper'sproblem, the decisions are the submitted power quantity to theday-ahead market. In order to test them, it can be considered somerealizations of uncertain variables, i.e. day-ahead and balancingmarket prices andwind generation. This work is called aftere theefact analysis since this test is only possible when the uncertainvariables are realized. Indeed, the after e the e fact analysis is aparticular Monte Carlo method which is adopted for the self-scheduling problem. This procedure is similar to out of sampleassessment in stochastic programming [39]. Also, it performed inRef. [25] in order to test the proposed IGDT-based self-schedulingmethod. In addition, it should be noted that in the case of thecomparison of deterministic method with the proposed one,similar approach is taken. However, instead of generation so manyscenarios we consider only one scenario for more explanation.

It is supposed that the realized wind generation and day-aheadprices are (1e14%) ¼ 86% of their forecasted values. Since thevariation of the uncertain variables is lower than the optimal

Table 6Optimal robustness function value and schedule for case I considering wind andprice uncertainties.





0 3000 0 500.05 2850 0.0276 48.6190.1 2700 0.056 47.20150.15 2550 0.0851 45.74450.2 2400 0.1151 44.24430.25 2250 0.1461 42.6970.3 2100 0.178 41.09770.35 1950 0.2112 39.4410.4 1800 0.2456 37.720.45 1650 0.2815 35.92680.5 1500 0.319 34.05120.55 1350 0.3584 32.0810.6 1200 0.4 300.65 1050 0.4443 27.78720.7 900 0.4917 25.4138

Fig. 4. Profit from different schedules when the worst case in the uncertainty horizon with s ¼ 0.15 is realized.



Fig. 7. The robustness cost versus robustness function value (a).


Fig. 8. Forecasted and simulated after e the e fact wind generations for one day.

Fig. 9. Forecasted and simulated after e the e fact day-ahead prices for one day.

Fig. 10. Optimal daily schedule with s ¼ 0.25.


robustness function value, i.e., 14.7%, the after e the e fact profitshould be greater than critical profit, i.e., $38,751.825. The after ethe e fact profit with the proposed self-scheduling method is$39,221.523 and with deterministic self-scheduling is $38,230.135.Note that the after e the e fact profit of the proposed IGDT-basedapproach is higher than critical profit and higher than determin-istic self-scheduling after e the e fact profit. To validate the resultthree cases have been considered here.

Case I: In this case, day-ahead market prices and wind genera-tions are generated randomly greater than their forecastedvalues but lower than the upper limit of the robustness region,i.e, (1þ a) times the forecasted value.Case II: Day-ahead market prices and wind generations aregenerated randomly lower than their forecasted values butgreater than the lower limit of the robustness region, i.e. (1� a)times the forecasted value.Case III: Day-ahead market prices and wind generations aregenerated randomly in the robustness region.

Typical simulated after e the e fact wind generations and day-ahead market prices for the three cases are shown in Figs. 8 and 9,respectively. Figs. 8 and 9 illustrate that each scenario includes real-izedvalues for theuncertain variables of the hourlymarket prices and

Table 7Analysis of WPP's profit using simulated after e the e fact prices and wind gener-ations for one day.

Scenariotype

Min ($) Mean ($) Max ($) Standarddeviation ($)

Coefficientof Variation

I 53697.72 55732.96 57556.07 591.81 0.00034II 42360.44 44072.99 45626.15 513.71 0.00037III 46426.66 49713.01 53597.01 1082.90 0.00069


wind generations in the scheduling day. For each case,1000 scenariosare generated. The occurrence probability of the system excess anddeficit generation for each hour and for all scenarios is consideredequal. The mean, standard deviation, minimum, maximum and co-efficientofvariation foraftere thee factprofits for the three cases areshown in Table 7. The coefficient of variation, which is defined asStandard Deviation=ðMean�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNumber of Samples

pÞ, is used as a

measure of convergence [46e48]. The coefficient of variation is lowerthan 0.01 for different cases, which shows the good convergence ofthe simulation. The robust scheduling guarantees that after e the e

factprofit shouldbegreater than thecriticalprofit, i.e., $38,751.825.Ascanbeobserved fromTable7, aftere thee fact profits aregreater thanthe critical profit for all cases. It can also be observed from this tablethat the actual obtained profit depends on the profile of the realizedwind generation andmarket prices. If the forecasts be underestimatefor all hours, similar to Case I, the obtained profit will be greater.

5. Conclusion and further works

In this paper, a new IGDT-based formulation for scheduling of arisk-averse wind power producer considering wind and marketprice uncertainties is proposed. The consideredWPP participates inthe day-ahead energy market and imbalance market. The proposedIGDT-based robust scheduling model resulted in a bilevel optimi-zation problem. A non-KKT-based approach is proposed for con-verting the bilevel model to equivalent single level problem. Theresulting single-level problem was in non-linear form. A lineari-zation technique used to convert it to a linear problem. The prob-lem can be solved in less than a second, which confirms thecomputational efficiency of the method. The proposed robustscheduling guarantees obtaining a minimum level of critical profitprovided that the realized wind power production and marketprices fall into the maximized robustness region. The effectivenessof the proposed algorithm is evaluated using simulations on anillustrative example and a test system. After-the-fact analysis iscarried out to validate the effectiveness of the proposed method. Itis observed that the guaranteed profit is obtainable in differentsimulated scenarios. In this work, self-scheduling of a price-takerWPP is modeled using IGDT method. This model can be improvedfor price-makerWPPs as a future work. Moreover, correlation is notconsidered in this work. However, it can be modeled in the IGDTframework using ellipsoid-bound info-gap model as a future work.

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Self-scheduling of a wind producer based on Information Gap Decision Theory

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