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Digital Object Identifier: 10.1109/TSG.2013.2271283

IEEE TRANSACTIONS ON SMART GRID, VOL. 4, NO. 4, DECEMBER 2013 2191

Risk Measure Based Robust Bidding Strategy forArbitrage Using a Wind Farm and Energy StorageAnupam A. Thatte, Student Member, IEEE, Le Xie, Member, IEEE, Daniel E. Viassolo, Member, IEEE, and

Sunita Singh, Member, IEEE

Abstract—This paper proposes the use of a risk measure basedrobust optimization bidding strategy for dispatching a windfarm in combination with energy storage. Through coordinationwith energy storage devices, variable wind generators can beutilized as dispatchable energy producers in the deregulatedelectricity market. The total profit from sale of electricity can beincreased by exploiting arbitrage opportunities available due tothe inter-temporal variation of electricity prices in the day aheadmarket. A case study is presented to show that as the forecasterror in electricity price increases, the robust optimization basedbidding strategy has an increasing probability of yielding bettereconomic performance than a deterministic optimization basedbidding strategy. The uncertainty set for robust optimization isselected based on the coherent risk measure conditional valueat risk (CVaR). Uncertainties in electricity price forecasting andwind power forecasting are considered. The resulting robustoptimization based bidding strategy is evaluated using MonteCarlo simulation for different choices of uncertainty sets.

Index Terms—Bidding strategy, electricity market, energystorage, risk measure, robust optimization, uncertainty set, windpower.

I. INTRODUCTION

T HE INSTALLED capacity of renewable generationresources such as wind and solar in power systems

is increasing in many countries. The variability and limitedpredictability of these renewable resources poses challenges topower system operations. Due to these characteristics, it is dif-ficult for system operators to dispatch wind generators as theydispatch conventional generators. Storage technologies canhelp in firming the production of wind generation and providebenefits to the system over different time scales. For example,storage would enable wind generators to be dispatched, aswell as allow them to participate in ancillary services such asfrequency regulation [1].

Manuscript received September 04, 2012; revised January 22, 2013, May 27,2013; accepted June 06, 2013. Date of publication September 12, 2013; dateof current version November 25, 2013. This work was supported in part byVestas Technology R&D and in part by State Key Laboratory of Control andSimulation of Power System and Generation Equipments, Tsinghua University,China. Paper no. TSG-00549-2012.A. A. Thatte and L. Xie are with the Department of Electrical and Computer

Engineering, Texas A&M University, College Station, TX 77843 USA (e-mail:[email protected]; [email protected]).D. E. Viassolo is with Halliburton, Houston, TX 77072 USA (e-mail:

[email protected]).S. Singh is with Bechtel, Houston, TX 77056 USA. (e-mail: svo.

[email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TSG.2013.2271283

Utility scale storage such as batteries and flywheels are nowbeing deployed in power systems [2]. An important revenuestream for energy storage is energy arbitrage in short-termelectricity markets, such as the day-ahead electricity market.Therefore, the combination of wind farms and energy storagecan be dispatched using optimization based methods thatmaximize the economic returns. The self-scheduling problemof the combination of pumped storage and hydro plants in amarket environment has been investigated [3], [4]. Castron-uovo and Lopes [5] formulated the operation strategy for awind generator combined with hydro pumped storage facilityas a linear deterministic optimization problem. The value ofcombining wind farms with energy storage for energy arbitragein short-term electricity markets has also been analyzed [6]. Inaddition to the uncertainty in electricity price forecasts, windfarms also experience difficulty in accurately predicting theirproduction in a day-ahead market time-frame.Many researchers have proposed using the stochastic pro-

gramming approach to deal with uncertainty in generator deci-sion making [7]–[9]. However, the stochastic programming ap-proach is computationally challenging due to the large numberof scenarios that have to be considered. Additionally, stochasticprogramming also requires knowledge of the probability distri-bution of uncertain variables, which may not be available. Morerecently Robust Optimization (RO) has received attention asan alternate approach to deal with uncertainty in optimizationproblems. The RO approach has been applied across a varietyof domains including portfolio optimization, supply chain man-agement, network flows, circuit design, wireless networks, andmodel parameter estimation [10].The main feature of the RO approach is that it uses a non-

probabilistic approach to deal with the uncertainty. Uncertaintyis addressed by constructing an uncertainty set and the solutionsobtained are robust to all realizations of uncertain data withinthe defined uncertainty set. This definition of uncertainty leadsto a more tractable problem. The question that arises in this re-gard is as to the selection of these uncertainty sets. One methodthat has been suggested to determine the uncertainty set is touse risk measures commonly used in finance industry [11]. Infinancial portfolio optimization the future values of the assetsare uncertain, similarly in the generator scheduling problem themarket clearing price of electricity in the day-ahead market isuncertain at the time of generator bidding. The uncertainty setcan be determined based on a coherent risk measure such as con-ditional value at risk (CVaR) [12]. Consequently a robust opti-mization bidding strategy can be obtained based on the risk pref-erence of the wind farm operator. Robust optimization solves for

1949-3053 © 2013 IEEE

2192 IEEE TRANSACTIONS ON SMART GRID, VOL. 4, NO. 4, DECEMBER 2013

the worst-case, consequently it will yield conservative results ifthe forecast errors are low. However, since the robust approachyields solutions that are immunized to all realizations of uncer-tain data within the uncertainty set, it maybe a suitable approachwhen forecast errors are high.The main contributions of this paper are:• to propose the use of risk measure based robust optimiza-tion bidding strategy, for energy arbitrage in the day-aheadelectricity market, using the combination of a wind farmand energy storage.

• to verify through a case study that the robust approach hasa higher probability of yielding better economic returnscompared to a deterministic optimization approach, for ahigh forecast error in day-ahead electricity market clearingprice.

The rest of this paper is organized as follows. Section IIpresents the background on robust optimization, risk measuresused in finance industry, incorporating decision maker’s riskaversion and the risk measure based construction of uncer-tainty sets. Section III presents the formulation of the robustoptimization based bidding strategy. Further, the decisionmaking process for selecting the optimum bidding strategy,from the point of view of the wind farm operator, is presented.In Section IV, the economic performance of the robust opti-mization approach is compared to a deterministic optimizationapproach, using Monte Carlo simulation, for increasing forecasterror in day-ahead electricity market price. Also, a case studyis presented to illustrate the risk measure based robust biddingstrategy for an energy arbitrage application using the combi-nation of a wind farm and a generic energy storage. This isfollowed by concluding remarks and future work in Section V.

II. BACKGROUND

A. Robust Optimization

Robust optimization is a relatively recent technique for op-timization under uncertainty. RO offers a non-probabilistic ap-proach to deal with uncertainty through the use of an uncertaintyset. The optimization solves for the worst case of uncertaintywithin the uncertainty set, hence the solution is feasible for allrealizations of uncertain variables within the given uncertaintyset. The other advantage is that for many classes of optimizationproblems the RO formulation is tractable [10].Over the past few years researchers have proposed robust

optimization approaches for power system applications [13].Due to higher penetration of generation from variable renewablesources such as wind and solar, as well as increasing demandside participation enabled by smart grid technologies, unit com-mitment in deregulated electricity markets has become morechallenging. Bertsimas et al. [14] propose a two-stage adaptiverobust optimizationmodel for the security constrained unit com-mitment problem in the presence of nodal net injection uncer-tainty. The method used is based on Benders’ decompositionand the level of conservatism of the solution is controlled byan uncertainty budget. Similarly [15], [16] and [17] also applyrobust optimization for the unit commitment problem, with theuncertainty set determined by an uncertainty budget. Jiang et

al. [18] propose a method to provide a robust unit commitmentschedule for thermal generators in the day-ahead market withwind power fluctuations. Zheng et al. [19] propose applying ro-bust optimization to economic dispatch in addition to the unitcommitment problem.The uncertainty set can be based on some historical infor-

mation about the values of the uncertain parameters. One waythat has been suggested is based on the correspondence of theuncertainty set with risk measures commonly used in finance[11], [12]. In this paper we apply the risk measure approach todetermine the uncertainty set for robust optimization based bidscheduling for a wind farm combined with energy storage.

B. Risk Measures

In the field of finance the portfolio allocation problem is anoptimization problem where the uncertain coefficients are thefuture asset returns. Risk measures are used to quantify the like-lihood and size of potential losses. A risk measure is effectivelya mapping from a set of random variables (e.g., portfolio re-turns) to the set of real numbers. The aim of the portfolio opti-mization is to find the minimum risk portfolio in the set of fea-sible portfolios. Analogous to this, the bid scheduling problemfor the combination of energy from wind farm and storage de-vice can also be framed as an optimization problem. The aim isto maximize the profit from sale of electricity under uncertain-ties in electricity price forecasting and wind power forecasting.1) Value at Risk (VaR): VaR gives the monetary risk associ-

ated with the bid schedule of the combination of the wind farmand energy storage, due to uncertainties in the forecasts. VaRis computed as the maximum profit over a target time horizonsuch that the probability of the profit being less than or equal tothis value is less than or equal to [20].Given a confidence level and the normally dis-

tributed random variable, profit, Value at Risk for the operatingday is defined as

(1)

The main disadvantages of VaR are that it does not capturetail cases, and VaR is not coherent.2) Coherent Risk Measures: A risk measure is coherent

if it satisfies the following four conditions [21].1) Sub-additivity:2) Homogeneity: For any3) Monotonicity: if4) Translational invariance: , for anyconstant c

Convexity, defined asfollows from above conditions. The main

consequence of coherency is the preservation of convexity,which in turn implies computational tractability of optimization[12], [22].3) Conditional Value at Risk (CVaR): CVaR is defined as the

conditional expectation of the profit, given that the profit is lessthan or equal to the VaR value. Thus given a confidence level

,

(2)

THATTE et al.: RISK MEASURE BASED ROBUST BIDDING STRATEGY FOR ARBITRAGE USING A WIND FARM AND ENERGY STORAGE 2193

Therefore CVaR is the expected value of the worstcases of profit. Compared to VaR, CVaR is a conservative riskmeasure since it captures the tail of the probability distributionof profit. The main advantage of CVaR is that it is coherent [23].This addresses the motivation of using the CVaR risk measurein this paper. CVaR being a coherent risk measure preserves theconvexity of the robust counterpart to the linear optimizationproblem with uncertain data. Therefore, the resulting robust op-timization problem is tractable.

C. Decision Maker’s Risk Aversion

CVaR captures the tail of the bidding profit scenarios as spec-ified by a confidence level . Therefore for a given confidencelevel CVaR can be used as a performance measure to comparedifferent bidding strategies. By changing the confidence levelthe conservatism level of this performance measure can be ad-justed. For instance if we take % the CVaR measureis reduced to the worst case scenario only. Whereas if we take

% then CVaR gives the mean for all profit scenarios.Thus the confidence level can be used to represent the deci-sion maker’s risk aversion. A more risk averse decision makermay choose a larger value of , while a risk tolerant decisionmaker may choose a smaller value of .

D. Risk Measures and Uncertainty Sets

Expressing the decision maker’s risk preference as a coherentrisk measure, allows us to formulate the optimization problemwith uncertain data as a robust optimization problem with aconvex uncertainty set. Example 3.2 in Bertsimas and Brown[11] specifies the link between CVaR risk measure and polyhe-dral uncertainty sets, as follows.Given samples of data i.e., , the uncertainty

set for the uncertain vector corresponding to is

(3)

If we assume the probability distribution of data samplesas and also take , then for some

this has the interpretation of the convex hull of all j-pointaverages of matrix .Without loss of generality this result can also be used to

find the uncertainty set for cost coefficients of the optimizationproblem [11]. Thus we can find the uncertainty set for pricebased on wind farm operator’s choice of and the historicaldata of prices represented by the vectors . Thechoice of affects the size of the uncertainty set. Namely alarger yields a larger uncertainty set, while a smaller value ofyields a smaller uncertainty set.

III. FORMULATION

The bid scheduling for the combination of a wind farm andenergy storage device is formulated as a robust linear optimiza-tion problem which aims to maximize the total profit from saleof electricity in the day ahead market [24]. The inputs are theforecasts of electricity prices and wind farm power production,

Fig. 1. Schematic of wind farm and energy storage.

TABLE INOMENCLATURE

whereas the outputs are the hourly power injection profiles ofthe wind farm and the energy storage. The bids comprisingof the hourly power injection totals of the wind farm and en-ergy storage device are submitted to the market. Upon marketclearing the system operator sends the dispatch signal com-prising of successful injection bids, to the wind farm operator(Fig. 1). The wind farm and energy storage inject power tothe grid to match the dispatch commands. The bidding strategyleverages the storage device by exploiting the arbitrage oppor-tunities, available due to inter-temporal price variations. The ab-breviation is used for Danish Kroner. The nomenclatureused is given in Table I.

A. Robust Optimization Bidding Strategy

The objective function and constraints for the robust opti-mization bidding strategy are as follows.

(4)


s.t.

(5)

(6)

(7)

(8)

(9)

(10)

where is the uncertain electricity price variable, is the un-certain available wind power, is the uncertainty set for elec-tricity price, and is the uncertainty set for available windpower.The objective function (4) consists of revenue from elec-

tricity sales from both the wind farm and storage, marginal costof wind generation, degradation costs associated with chargingand discharging of storage device, and energy storage operationcosts. By minimizing the negative of the total profit, (4) effec-tively maximizes the total profit. The inter-temporal rampingconstraint for the wind farm is given by (5). The upper limiton the wind farm’s power injection to the grid is specified by(6). The energy, charging and discharging power limits of thestorage device are specified by (7)–(9). The dynamic equationfor the storage device is (10). In the implementation code, inorder to maintain tractability, this equality constraint is con-verted into two inequality constraints with small tolerances. Theterm is the firming power provided by storageto compensate for wind power forecast errors. It is assumed thatthe storage device can only be charged using the wind powerproduction and not by the grid (Fig. 1). This assumption is usedsince we are interested in analyzing the impact of storage on theutilization of wind resource. The power loss in storage chargingand discharging is a function of and , the discharging andcharging efficiencies of the storage device respectively. Theroundtrip efficiency of the storage device can be taken as theproduct of these two values, i.e., .The problem (4)–(10) is of the form

(11)

where is the vector of decision variables, and are vectors ofuncertain data. The uncertain problem (11) can be reformulatedas

(12)

When the uncertainty sets are polyhedral they can be repre-sented by matrix inequalities. Thus (12) can be written in themin-max form as

(13)

Taking the dual of the inner maximization subproblem we get

(14)

Thus the original problem is transformed into a tractable linearprogramming formulation.

B. Decision Making Process for Wind Farm Operator

In this section the decision making process from the perspec-tive of a price taking wind farm operator for participating inthe day ahead electricity market is discussed. We propose an of-fline decision to choose the optimization model based on MonteCarlo simulation which makes use of historical data on the un-certain variables, followed by an online decision which uses thechosen optimization model and the forecasts of wind power andelectricity price to yield the bidding strategy. The flowchart forthis decision making process is shown in Fig. 2.1) We assume that historical data on wind power forecasterror and electricity price forecast error is available to thedecision maker. This data can be used to estimate the typeof probability distribution in forecast error and the worstcase of uncertainty that may be experienced by the windfarm operator.

2) Next the performance of the optimization based biddingfor multiple values of the parameter are comparedusing Monte Carlo simulation. The considered optimiza-tion models may range from deterministic %to worst-case robust % . Based on the profitsobtained in the Monte Carlo runs CVAR can be estimatedfor each model in order to compare their performance.

3) The decision maker selects the optimization model basedon both the relative performance as obtained from MonteCarlo simulations as well as the decision maker’s confi-dence in the forecast. The decision maker may use histor-ical data on forecast error to decide the confidence level in agiven forecasting method. If the decision maker has a highlevel of confidence in the forecast it may not make sense touse a more robust approach since it would be too conserva-tive. However, if the decision maker believes that the priceforecast error is likely to be high, choosing a more robustapproach, i.e., a higher value of may be appropriate. Arisk averse decision maker may choose a more robust ap-proach even when the price forecast error is anticipated tobe medium, in order to minimize potential loss of revenue


Fig. 2. Flowchart for wind farm operator decision making.

in case the actual price forecast error is higher than antic-ipated. The uncertainty set associated with the particularchoice of can be obtained as described in Section II-D.

4) Once the optimization model is chosen the decision makercan use the dashboard tool to determine the optimum bid-ding strategy. The uncertainty set for wind power can beselected based on probabilistic forecasts. The dashboarduses the forecasts of the day-ahead electricity prices andwind power production for the operating day as inputs inthe optimization.

Thus the optimization based bidding strategy for the combina-tion of wind farm and energy storage can be obtained.

C. Dashboard Tool for Bidding Strategy Selection

In order to facilitate bidding strategy selection a software toolwith a dashboard GUI will be made available to the decisionmaker (Fig. 3). This software tool allows the decision makerto use historical data in an offline study to estimate the typeof probability distribution of the forecast error as well as theworst case of uncertainty that the wind farm operator may ex-perience. The tool then compares the performance of the opti-mization model for different values of the parameter , basedon Monte Carlo simulation, and determines the uncertainty set.Thus the uncertainty set is selected through an offline process.Next in the online phase the optimization routine is run to deter-mine the bidding strategy for the wind farm and energy storagedevice for the day-ahead electricity market. The inputs are thewind power forecast and electricity price forecast for the oper-ating day.

IV. CASE STUDY

In this section a case study is presented in which the eco-nomic performance of the robust optimization based bidding

Fig. 3. Dashboard for bidding strategy selection.

TABLE IIWIND FARM AND STORAGE DEVICE PARAMETERS

strategy is evaluated. For simplicity we assume that all thehourly bids submitted by wind farm and energy storage combi-nation to the market operator are successful. The characteristicsof the wind farm and a generic energy storage device arepresented in Table II.

A. Performance of Robust Optimization Bidding Strategy

The performance of the robust optimization bidding strategyrelative to the deterministic approach is analyzed for differentlevels of electricity price forecast error. The deterministicmodel, which is presented in [24], uses the point forecasts ofthe inputs in the optimization to find the bid schedule. Boththe robust and deterministic optimization based strategies areobtained for a set of price forecasts with error measured inMean Absolute Error (MAE). MAE is the unweighted averageof the absolute values of the forecast errors.

To evaluate the performance of each bidding strategy MonteCarlo simulation is used. A number of scenarios of actual elec-tricity price (1000 scenarios) are considered. The result we areinterested in is the number of scenarios where the profit fromrobust optimization (RO) based bidding strategy is greater thanthat from the deterministic optimization (DO) based biddingstrategy. The results for several days are analyzed (Fig. 4). Inall the cases for an increase in price forecast error there is a cor-responding increase in the probability of getting better results


Fig. 4. Relative performance of robust optimization based bidding vs. priceforecast error.

using robust optimization compared to using deterministic op-timization.It has been observed that day-aheadwind power forecast error

as a percentage of installed capacity has MAE in the range of15%–25% for a single wind farm [25], [26]. Wind farms thatwant to bid into day-ahead market have a serious problem ofdealing with the uncertainty due to high wind power produc-tion forecast errors. Further the electricity price forecast for theday ahead time horizon can have an error of up to around 15%[27]. The robust optimization approach can be used to managethe uncertainty due to high day-ahead forecast error, and obtainbetter economic performance compared to the deterministic op-timization approach.Bertsimas et al. [28], [29] address the issue of the perfor-

mance guarantee of the robust optimization model versus thebudget of uncertainty. Based on their approach we consider theproblem (12) and assume that each of the elements of thevector of uncertain cost coefficients belongs to a symmetric in-terval centered at the point forecast value withmaximum deviation . The parameter is defined by

, and is called the budget of uncertaintyof the cost coefficients. is the upper bound of the aggregatescaled deviations of the actual values of the coefficients fromtheir point forecast values, and thus can be used to represent theaccuracy of forecasting. The key result from [28] can be summa-rized as follows. For the robust optimization problem let bean optimal solution and the optimal objective function value,then if is chosen to be ,where is the number of uncertain variables (here

hours since we consider one day) and is the cumula-tive distribution function of the standard Gaussian random vari-able. Thus the actual value of the profit from bidding will exceedthe predicted value with probability at least equal to , andthis value is the performance guarantee. Fig. 5 shows the theo-retical performance guarantee of the robust optimization model,under above assumptions on uncertainty in coefficients, for dif-ferent values of the budget of uncertainty [29].

Fig. 5. Performance guarantee of robust optimization model.

B. Results of Robust Optimization Bidding Strategy

The robust optimization given in Section III yields the powerinjection profile for the wind farm and storage device for eachhour of the entire day. Based on these injection profiles the bids,which are the hour-by-hour total power injection of the windfarm and energy storage combination, are obtained. The bids forthe day-ahead electricity market have to be submitted 12 hoursbefore the beginning of the operating day. Electricity price datafrom Nordpool for West Denmark is used for the simulations. Itis assumed that during actual operation any excess wind genera-tion is sold in the hour-ahead market, whereas any deficit has tobe purchased from the hour-ahead market at the clearing pricefor that hour. The total profit from electricity sales is calculatedbased on the actual values of market clearing price. This set-tlement is done after the end of the operating day. The robustoptimization problem is implemented and solved in MATLABusing linprog solver and the YALMIP toolbox [30].In order to analyze the performance of the optimization based

bidding strategy, Monte Carlo simulation method is used fora what-if type analysis based on assumptions about forecasterrors. The Cauchy distribution is considered to be a reason-able model for the distribution of wind power forecast error[31] and electricity price forecast error [32]. Based on histor-ical data consisting of 3 months of hourly forecasts and actualvalues of market clearing price and wind power, we also findthat using Cauchy distribution to model the errors is a reason-able assumption (Figs. 6 and 7). Therefore for both the windfarm power production and the electricity price an error is gen-erated at random for each hour of the day by sampling a Cauchydistribution within bounds defined by 90% confidence interval.For each hour of the day the realization of the actual value ofthe input quantity (i.e., wind farm power production and elec-tricity price) is obtained by subtracting the error from the fore-cast value. In this manner scenarios of actual windfarm power production and electricity price are generated for thegiven day using random sampling. For each scenario based onthe bids obtained from the robust optimization the profit for theoperating day can be calculated. These 1000 values are assumed


Fig. 6. Histogram of normalized price forecast error.

Fig. 7. Histogram of normalized wind power forecast error.

to be independent identically distributed (i.i.d.) observations ofthe profit. Based on the order statistics, the VaR and the CVaRcan be estimated from these observations [33].In this case study the robust bidding strategy is analyzed for

one operating day. The forecasts and actual data for the elec-tricity price and wind power production are provided by Vestas.Fig. 8 shows the forecast and actual hourly electricity pricesfor the day-ahead market, as well as the actual prices in thehour-ahead market for the given day. The error between fore-cast and actual day-ahead price is high in hours 17 and 21.Table III shows the results of the evaluation of the robust bid-ding strategy using Monte Carlo simulation, for different uncer-tainty sets based on wind farm operator’s choice of parameter. Table IV shows the 95% CVaR values for different uncer-tainty sets and forecast errors in prices measured inMAE%. Therobust bidding strategy for the combination of the wind farmand storage corresponding to % and % isshown in Fig. 9. In hours 3–5 when the forecast price is low, partof the wind energy is used to charge the storage device. In hour11 when the forecast price is high the stored energy is injectedinto the grid. Therefore the storage device can be used to take

Fig. 8. Electricity prices for given day.

Fig. 9. Results of robust optimization for %.

TABLE IIIRESULTS OF MONTE CARLO SIMULATION FOR PRICE UNCERTAINTY SETS

advantage of arbitrage opportunities that result from temporalvariations in the electricity price.The simulation is conducted on a laptop with Intel Core 2 Duo

2.2 GHz CPU with 4 GB of RAM. It takes 13.3 seconds to gen-erate the Monte Carlo scenarios. For each value of the robustoptimization and evaluation takes on average 7.2 seconds.In order to determine uncertainty sets for wind power pro-

duction around the deterministic forecast we use probabilisticforecasts. Percentiles of a probabilistic forecast are usually de-fined such that the probability of wind power production being


TABLE IVCVAR FOR COMBINATIONS OF AND

Fig. 10. Wind deterministic and percentile forecast.

Fig. 11. Results of robust optimization for % (price) and 40%–60%(wind).

less than the value given by the percentile forecast is per-cent [34]. Fig. 10 shows the deterministic forecast, actual windpower production, and the probabilistic forecasts for 40 and 60percentiles for the given day. Pairs of probabilistic forecasts aretaken as the lower and upper bounds for the uncertainty set forwind power. Table V shows the results of the Monte Carlo sim-ulation, with different uncertainty sets for wind power, based onpairs of probabilistic forecasts. Thus the optimal robust bidding

TABLE VRESULTS OF MONTE CARLO SIMULATION FOR WIND UNCERTAINTY SETS

strategy is obtained which considers uncertainty in both elec-tricity price and wind power (Fig. 11).Finally we present a comparison of the robust optimization

approach to the stochastic optimization approach. For thestochastic model we solve the expected value problem usingthe sample average approximation method [35]. This involvessolving a large deterministic problem using the Monte Carlomethod. With samples the stochastic approachyields a mean profit of DKK 51 164 and takes 714.2 s to solve.Whereas for % the robust approach yields a mean profitof DKK 50 775 and takes 20.4 s to solve. The problem sizeof the stochastic approach increases linearly with and thecomputation time would be very large for a large sample size,whereas, the robust approach is comparable to the deterministicin computational effort required.

V. CONCLUSION

This paper presents a robust optimization approach, with un-certainty set based on risk measure used in finance industry, toobtain optimum bidding strategy for a wind farm and energystorage in a day-ahead electricity market. Robust optimizationbased strategy is seen to have increasing probability of yieldingbetter economic performance than the deterministic approachas the forecast error in electricity price increases. This is im-portant because wind producers who bid into day-ahead elec-tricity markets have to deal with uncertainty due to large fore-cast errors. The combination of wind power and energy storageleads to better utilization of the uncertain wind resource and in-creased economic performance through use of energy arbitrage.The economic performance of the bidding strategy is evaluatedusing Monte Carlo simulation by making suitable assumptionsabout the probability distribution of the electricity price andwind power forecast errors.Future work could include the integration of robust optimiza-

tion with a model predictive approach for hour-ahead and real-time markets. In order to improve the utilization of the renew-able resource other applications of the wind and energy storagecombination, including ancillary services such as frequency reg-ulation could also be considered.

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Anupam A. Thatte (S’12) received the B.E. degree in electrical engineeringfrom Pune University, India, in 2004 and the M.S. degree in electrical and com-puter engineering from Carnegie Mellon University, Pittsburgh, PA, USA, in2005. He is currently a student in the Ph.D. program at Texas A&M University,College Station, TX, USA.His industry experience includes an internship with ABB in Raleigh, NC,

USA. His research interests include modeling and control of power systems,renewable energy, and smart grids.

Le Xie (S’05–M’10) received the B.E. degree in electrical engineering fromTsinghua University, Beijing, China in 2004, the M.Sc. degree in engineeringsciences from Harvard University, Cambridge, MA, USA, in June 2005, and thePh.D. degree from the Department of Electrical and Computer Engineering atCarnegie Mellon University, Pittsburgh, PA, USA, in 2009.He is an Assistant Professor in the Department of Electrical and Computer

Engineering at Texas A&M University, College Station, TX, USA, where he isaffiliated with the Electric Power and Power Electronics Group. His research in-terests include modeling and control of large-scale complex systems, smart gridapplications in support of renewable energy integration, and electricity markets.

Daniel E. Viassolo received the Ph.D. degree from Purdue University, WestLafayette, IN, USA, in 2000.He is with Halliburton, in Houston, TX, USA. His expertise lies in the areas

of control systems and optimization, with a vast experience in applications toenergy systems such as power generation, wind turbines, combined-cycle powerplants, gas turbines. He holds 10 U.S. patents, and he has published 25+ papersin conference proceedings and journals of IEEE, ASME, AIAA, etc. Beforejoining Halliburton, he was with Vestas Wind Systems in Houston, TX, USA.

Sunita Singh (S’09–M’10) received her B.E. and M.Tech. degree from India.After working for 17 years in India in state utility, manufacturing industry, andteaching engineering courses in university, she finished her Ph.D. in 2009 atArizona State University, Tempe, AZ, USA, in the area of electromechanicalstate estimation of power systems.She worked with Pacific Northwest National Lab on smart grid initiatives

until 2010. Currently she is working with Bechtel, Houston, TX, USA, prior towhich she was with Vestas Wind Systems, Houston, TX, USA.

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