Contracting Strategies for Renewable Generators: A Hybrid Stochastic and Robust Optimization...

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Contracting Strategies for Renewable Generators: AHybrid Stochastic and Robust Optimization Approach

Bruno Fanzeres, Student Member, IEEE, Alexandre Street, Member, IEEE, andLuiz Augusto Barroso, Senior Member, IEEE

Abstract—We present a new methodology to support an energytrading company (ETC) to devise contracting strategies underan optimal risk-averse renewable portfolio. The uncertainty inthe generation of renewable energy sources is accounted for byexogenously simulated scenarios, as is customary in stochastic pro-gramming. However, we recognize that spot prices largely dependon unpredictable market conditions, making it difficult to cap-ture its underlying stochastic process, which challenges the use offundamental approaches for forecasting. Under such framework,industry practicesmake use of stress tests to validate portfolios.Wethen adapt the robust optimization approach to performan endoge-nous stress test for the spot prices as a function of the buy-and-sellportfolio of contracts and renewable energy generation scenarios.The optimal contracting strategy is built through a bilevel opti-mization model that uses a hybrid approach, mixing stochastic androbust optimization. The proposedmodel is flexible to represent thetraditional stochastic programming approach and to express theETC’s uncertainty aversion in the case where the price distributioncannot be precisely estimated. The effectiveness of the model isillustrated with examples from the Brazilian market, where theproposed approach is contrasted to its stochastic counterpart andboth are benchmarked against observed market variables.

Index Terms—Conditional value-at-risk, power system eco-nomics, robust optimization, stochastic optimization.

NOMENCLATURE

Constants

Sale price of the PPA ($/MWh).

Price of the capacity payment contract with arenewable unit ($/MWh).

Maximum volume that can be sold in the PPA.Defined by the consumer willingness to contract(avgMW).1

Manuscript received December 25, 2013; revised May 02, 2014 and July 10,2014; accepted August 06, 2014. This work was supported in part by UTEParnaı́ba Geração de Energia S.A. through R&D project ANEEL PD-7625-0001/2013. Paper no. TPWRS-01630-2013.B. Fanzeres and A. Street are with the Electrical Engineering Department,

Pontifical Catholic University of Rio de Janeiro (PUC-Rio), Rio de Janeiro, RJ,Brazil (e-mail: [email protected]; [email protected]).L. A. Barroso is with PSR Consulting, Rio de Janeiro, RJ, Brazil (e-mail:

[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/TPWRS.2014.2346988

1An average MW is equivalent to the continuous production/consumption ofone MW during the relevant period. One average MW during one year corre-sponds to 8760 MWh.

Firm energy certificate of the renewable unit(avgMW).

Energy production of the renewable generator inperiod and scenario (MWh).

Number of hours of period .

Probability of the renewable productionscenario .

Opportunity cost of the money in percentage persub-set of periods, e.g., per year.

Opportunity cost of the money in percentageper period of the th subset of periods, e.g., permonth.

First period of the th subset of periods.

Risk aversion parameter that combines theexpected value and the CVaR of the th subsetof periods.

Risk aversion parameter that defines theconfidence level of the CVaR risk measure of theth subset of periods.

Reference or nominal spot-price scenario inperiod and scenario ($/MWh).

Maximum positive deviation from the spot-pricereference scenario in period and scenario($/MWh).

Maximum negative deviation from the spot-pricereference scenario in period and scenario($/MWh).

Budget that control the conservatism level of thespot-price stress scenarios in each subset .

Maximum return (%) of the spot-price stressscenario from period to .

Minimum return (%) of the spot-price stressscenario from period to .

Decision Variables

Percentage of that the ETC is willing tosupply through the PPA.

Percentage of the unit that is brought to the ETCportfolio through a capacity contract.

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2 IEEE TRANSACTIONS ON POWER SYSTEMS

Auxiliary variable that achieves the-value-at-risk of the partial PV of therevenue in each sub-period at the optimalsolution ($/MWh).

Auxiliary variable that represents the leftdeviation of the revenue scenario from the valuein each sub-period ($/MWh).

Spot price in period and scenario ($/MWh).

Percentage of the positive deviation from thereference price in period and scenario .

Percentage of the negative deviation from thereference price in period and scenario .

Dual Variables

Dual variable of the spot price envelope constraint(3) in each period and scenario .

Dual variable of the budget constraint (4) in eachsub-period and scenario .

Dual variable of the maximum return constraint(5) in each period and scenario .

Dual variable of the minimum return constraint(6) in each period and scenario .

Dual variable of the upper bound constraint(7) in each period and scenario .

Dual variable of the upper bound constraint(8) in each period and scenario .

Sets

Set of periods of the supply contract.

Subset of periods. .

Set of the number of sub-periods in which the setis divided.

Set of renewable units.

Set of scenarios.

I. INTRODUCTION

A MAJOR challenge for power generation companies(Gencos) in competitive electricity markets is to de-

termine an optimal contracting strategy for its assets thatmaximizes their value. This strategy should take into accountthe company’s risk profile and adequately treat all types ofexternal risks, such as the uncertainties in energy productionand the spot prices. Energy trading companies (ETC) play akey role in providing market penetration and diversifying riskof Gencos in order to serve consumers. To hedge against thevolatility of spot prices, generators or ETC sign mid- and/orlong-term energy supply contracts, which is a financial instru-ment that plays an important role in the power system reform ofmany countries worldwide. A supply contract involves the needto deliver a given volume of energy over a time horizon by a

seller against a fixed payment by the buyer. In most electricitymarkets, such contracts are only financial instruments (see [1]for a review of the financial instruments used in electricitymarkets and [2]–[4] for application examples).Bilateral contracts provide an adequate hedge against spot

prices in the case of thermal plants: if the spot price is low, theplant will not generate but meet its contractual obligations bypurchasing (cheaper) energy in the Wholesale Energy Market(WEM). Conversely, if spot price is high, the plant will pro-duce its own energy, thus avoiding expensive purchases. In otherwords, in the case of thermoelectric generators, the physicalproduction of the asset is a natural hedge and, except for thehazard of outages, mitigates production risks [3]. In the case ofrenewable generators—such as hydroelectric, wind and biomassplants—the probabilistic and seasonal nature of their physicalproduction poses a joint volume and price risk: a situation canarise where in moments of high spot prices, the production ofa renewable generator is low, which can result in high financialexposures in the short-termmarket if the plant is contracted witha consumer [5]–[7]. The situation is madeworse for hydro plantsin hydro-dominated countries because spot prices are negativelycorrelated with hydro production (the so-called hydrologicalrisk, see [8]).The development of renewable electricity contracting strate-

gies in competitive markets via ETC has been widely studiedin different contexts: the ETC acts as a purchaser of renewableenergy and creates a product that bundles the energy purchasedin order to serve a firm load. This involves the definition of theoptimal mix of sources to be bought by the ETC to mitigatethe price and quantity risk (taking advantage of the comple-mentarity usually existing among the energy production fromthe different renewable sources) as well as the definition of theoptimal amount of load to be served through a co-optimizationmodel [5]–[8].The “standard” modeling approach of this ETC portfolio op-

timization problem considers the representation of uncertaintyin renewable production and market prices by scenarios gen-erated via Monte Carlo simulation, usually utilizing a produc-tion costing model (market equilibrium simulated by a funda-mental approach) [9] or a statistical regression on past marketprices [10]. One of the main difficulties in implementing suchmodeling is how to produce meaningful future paths of spotprices. The spot price formation is a very complex process,which results from the combination of market (hydrology, de-mand, supply, outages, fuel availability, etc.) and political (in-terference on prices, out of merit order dispatches, etc.) con-ditions [10]. Even in countries such as Chile, Peru and Brazil,where a production costing model determines the system dis-patch and the spot prices (as short-run marginal costs) and themarket participants use the same production costing models topredict spot prices, forecasts have proven to be inaccurate andvery difficult [11].When the complexity of the true underlying uncertainty pro-

cesses requires the knowledge of input parameters whose dy-namics are difficult to predict, practitioners generally treat thisinaccuracy by means of stress tests, where stress “scenarios” areused to validate the performance of the optimal portfolios underadverse conditions. However, these scenarios are usually static,generally exogenously determined by an expert, and often do


FANZERES et al.: CONTRACTING STRATEGIES FOR RENEWABLE GENERATORS: A HYBRID STOCHASTIC AND ROBUST OPTIMIZATION APPROACH 3

not represent the true worst-case price scenario for the portfolioin hands. For example, different ETC portfolios may require dif-ferent worst-case scenarios. One alternative to overcome thisdifficulty is to consider endogenously-defined stress-price sce-narios that create the worst-case financial adversity for the ETCportfolio. In other words, they would be defined by an em-bedded optimization problem that ensures they really representthe worst possible realization of this uncertainty against the ETCportfolio revenue.Robust optimization (RO) has recently emerged [12]–[14]

and proven to be a powerful tool for addressing decision-makingproblems involving uncertainty parameters whose true proba-bility distribution is difficult to predict, which is exactly the caseof our spot price modeling challenge. Recent applications havedealt with robust-based decisions in operations and scheduling[15]–[18], bidding strategies of generators in the short-term spotmarket [19] and in portfolios with wind and storage [20], de-mand response and consumption strategies [22], [23], integra-tion of plug-in hybrid electric vehicles [24], just to name a few.This work leverages on robust and stochastic optimization

to represent different uncertainties in different ways in a re-newable energy portfolio optimization problem. We propose amodel that combines in one single optimization problem the en-dogenous stress-analysis approach for the spot prices (via RO)with the stochastic approach for the representation of renewableproduction.

A. Objective of This Work

The objective of this work is to present a portfolio opti-mization model to determine the risk-adjusted optimal tradingstrategy for an ETC in a contract market backed by a renew-able energy portfolio to serve a firm load. The two relevantuncertainties are the future spot prices and the production of therenewable energy sources, which are treated in our portfoliooptimization model by robust and stochastic optimization,respectively. We use the RO approach with polyhedral uncer-tainty set to constrain spot-price stress scenarios within theagent’s prior hypothesis for adverse conditions. To obtain theoptimal portfolio, spot prices are endogenously determinedby a set of second-level optimization problems, one for eachexogenous scenario of renewable energy generation (assumedto follow a “well-behaved” stochastic process). Prices are al-lowed to deviate from a reference scenario toward a worst-caserealization within a constrained uncertainty set.The modeling approach allows for the consideration of

deterministic or stochastic reference scenarios for the spotprices, which can be obtained following standard time-seriesapproaches or by means of equilibrium models. In other words,we consider that the spot-price scenarios may be misspecifiedand consist of only an approximation of the true underlyingstochastic process whereas renewable production scenarios areproduced via the classic Monte Carlo approach. The modelproposed in this paper can be seen as an extension of the modelproposed in [6] to account for the modeling uncertainty in thespot prices by means of an embedded stress analysis.An example of a practical result that this model can provide

is the optimal contracting strategy such that the ETC may “sur-vive” even if the spot price reaches its cap or floor value, de-pending on what is the worst case for the portfolio (considering

scenarios of renewable production), during (set by the deci-sion maker) periods (hours/months) within a given time horizon(day/year). This technique proves to be rather flexible for incor-porating other constraints and conceptually speaking it resem-bles—but is not equal to—the security criterion , whichis widely used in operation problems, as shown in [15]. Ourmethodology is flexible to consider the standard case where thedecision maker fully believes in his/her price scenarios.

B. Contributions Regarding the Existing Literature

The contribution of our work is to present an alternativemethodology to devise the ETC’s optimal contracting strategyvia a hybrid robust-stochastic optimization model. More objec-tively, the proposed model aims at recognizing the well-definedstochastic dynamics of renewable energy production as wellas the difficulty to forecast spot prices, thus ensuring the ro-bustness of the results against unexpected price variations. Themethodology is flexible to model spot-price stress scenarios,endogenously defined in the portfolio problem, as well asto follow a pure stochastic approach2 that considers exoge-nously-defined scenarios to characterize all uncertainty factors.To solve the problem, the proposed model is rewritten as a

maximization program, following the standard single-stage ROapproach (see [12] for more details), and solved by commercialsolvers [26]. We illustrate the methodology to devise a portfolioof renewable sources (wind and small hydros) in the Braziliansystem, where the proposed approach is contrasted to its fullystochastic counterpart and both are benchmarked against actual(observed) market variables.There are many papers in the literature dealing with RO

models. We believe our work is more related to the ones pre-sented in [17], which explores the modeling of an expandedpolyhedral uncertainty set to account for different load patternsor security criteria in the unit commitment problem; [18], whichcombines the robust and stochastic approaches to account forboth the worst-case and the expected dispatch cost under nodalinjection uncertainty through a weighted objective function;and [20], which deals with portfolios with wind and storagein the short-term market. Differently from previous reportedworks, we consider as a motivation for our hybrid approach thefact that one of the uncertainty parameters, the spot price, is ofdifficult modeling via stochastic programming and the rest ofthe uncertainties are assumed to follow a known and “well-be-haved” stochastic process. To the best of our knowledge, nowork has been performed on applying RO to the definition ofcommercial strategies for portfolio of renewables in contractmarkets, which is the subject of our work.On the methodological side, [17] and [18] also provide hy-

brid robust-stochastic models. Our paper differs on the way thishybridization is done: via a concave function, which accountsfor the well-known coherent measure of risk, namely the con-ditional value at risk [28] ([17] and [18] use a linear combina-tion approach without risk constraints). Finally, it is worth men-tioning that the problems addressed in [17] and [18] lie in theclass of adjustable (two-stage) RO models [34], requiring morecomplex solutionmethodologies (see [35]) than the single-stage

2However, the challenges associated to the computation of a joint probabilitydistribution of the uncertain factors are seen as an additional motivation to adoptthe proposed stochastic-robust approach.



RO approach [12] employed in the proposed model. Therefore,our methodology requires only a linear programming solver[26].

C. Organization of This Work

The rest of this paper is organized as follows. Section II de-scribes the modeling approach proposed for the representationof the uncertainties. Section III presents the bilevel formulationfor the hybrid robust-stochastic renewable portfolio selectionmodel, and Section IV shows its single-level-equivalent coun-terpart. Section V presents two case studies. Section VI con-cludes this work and discusses extensions and future research.

II. UNCERTAINTY MODELING APPROACH

Two main risk factors are considered in the ETC’s future rev-enue: , which is a random variable that expresses the amountof renewable energy produced by the generating unit in pe-riod (MWh), and , which is the random variable that repre-sents the spot price in period ($/MWh). Throughout this work,random variables are assumed to be discrete in a finite supportfollowing stochastic programming standards. Both uncertaintyfactors are characterized by a set of possible sce-narios and probabilities, , as customary instochastic optimization [31].

A. Electricity Contracts and the Energy Trading Company

Standard financial power purchase agreements (PPA) and ca-pacity payment contracts are considered in this work. A PPA isa bilateral supply contract agreement in which the buyer paysa fixed price for a given fixed amount of energy to the sellerparty [6]–[8]. In capacity payment contracts with renewables,the generator (selling counterpart) transfers a percentage ofits future, therefore uncertain, production to the buyer in ex-change of a fixed payment. The fixed payment of the capacitycontracts can be parameterized in two terms: price and quantity.The latter is based on the contracted percentage multiplied bythe firm energy certificates (FEC) of the unit [5]–[8]. The FECof a renewable unit is issued by the system regulator based on along-term quantile (generally set to the median or 10%) of theobserved generation. The role of the FEC in the Brazilian regu-latory framework is to limit the total amount of energy each unitcan sell through contracts (see [11] for further discussion on thissubject). Therefore, a capacity payment contract with a unit actsas if the buyer “rents” percent of the power plant, which in-cludes both the generation and FEC. In the case of renewables,for simplicity purposes, we assume that the variable generatingcost is zero; however, such cost could be easily included in themodel, as shown in [6].Differently from the short-term market, the financial con-

tract market does not impose any real energy exchange, alltransactions are purely financial. The contract market operatorruns a market clearing after the determination of the spotprices, which occurs in the short-term market, by clearing thedifferences between all volumes produced (consumed) andcontracted of each agent at this price. In this framework, ETCsplay an important role in the market acting as risk managers,creating commercial opportunities, and mitigating risks. Fig. 1illustrates the proposed contractual scheme for the ETC. Atthe left and right-hand-side of this figure, the ETC buys two

Fig. 1. ETC contractual scheme. Continuous lines for energy rights and dottedlines for financial income/payments.

capacity contracts with two complementary renewables andtherefore, receives the right of commercialize the bought gen-eration. This provides the ETC with a positive settlement in thecontract market clearing regarding the purchased proportion ofthe generation of each unit (first term of the market settlementpayment—at the top of the figure), , inexchange of a fixed payment, .Then, at the bottom of the figure, the ETC sells a standardflat PPA for a consumer, e.g., industries or large commercialentities, ensuring the amount to be delivered. Due to this agree-ment, the ETC receives a fixed income from the consumer,

, in exchange of a negative settlement inmarket clearing regarding the sold amount (second term of themarket settlement—at the top of the figure), .Under the proposed contracting scheme, the stochastic cash

flow of the ETC can be divided into three parts: 1) a fixed in-come due to the PPA sell; 2) a fixed expenditure due to the ca-pacity payments for all renewable generating units—we assumea set of units; and 3) a random term that accounts for the netcontract market settlement by the spot price between the renew-able generation, acquired by means of the capacity contracts,and the total amount sold through the PPA. Expression (1) il-lustrates the net revenue stream of an ETC for a given scenario

under the proposed commercial model:

(1)

According to (1), the ETC purchases generation capacity toback sales in the forward market. For simplicity and didacticpurposes, it is assumed that the ETC has no existing portfolio,which could be easily included in the model by an additionalconstant term in (1). Note that in this contracting scheme, theETC bears all of the price and quantity risk because the capacitycontracts do not guarantee a fixed energy delivery to fulfill thebilateral PPA with the consumer. If, on the one hand, the fixedrevenue, due to the first term of expression (1), increases as theETC sells more energy through the PPA, on the other hand, thenet revenue due to the settlement in the spot market, third term



of (1), decreases. This materializes the price and quantity threat,which is related with scenarios with low renewable production,below the amount sold, and high spot price. In such cases, ex-pression (1) can reach negative values and significantly jeopar-dize the company financial performance. Therefore, accordingto expression (1), in defining the optimal sell and buy percent-ages , the ETC must take into account thetradeoff between the fixed payment, composed of the first andsecond terms, and the price and quantity risk introduced by thethird term.Note that the ETC plays the role of a risk manager in the

proposed model and may therefore embody a variety of otherdifferent commercial roles (such as an electricity retailer or aGenco). For example, the model devised in this paper can beused by a Genco devising its optimal investment strategy in aportfolio of renewable technologies to mitigate the price andquantity risk and then supplying a consumer through a long-term flat PPA. Under this setting, the capacity payment in (1)can represent the annualized investment cost and interests ex-penses of the units owned by the company. In this context, thefixed capacity payment term, , in (1) could be re-placed by a simpler fixed payment term, , representing thoseexpenses in each period of the cash flow time horizon. Then, ex-pression (1) can represent the net revenue of the generation com-pany selling a long-term PPA as a function of the percentages ofthe renewable units in which the company is willing to invest. Itis worth mentioning that this business format, in which an ETCis formed to represent a given Genco in the contract market, islargely adopted in countries where the financial contract mar-kets are representative such as in Brazil.

B. Renewables Generation Scenarios

The modeling approach adopted to generate the scenarios ofthe renewable production is based on simulation procedures ofperiodic stochastic processes via, e.g., aMonte Carlo procedure.For the model proposed here, the generation scenarios for eachplant are considered as input data, and, therefore, exogenousto the model. For example, in [5], a Monte Carlo procedure isproposed to simulate renewable production scenarios based ona vector-autoregressive stochastic process with variance law.

C. Polyhedral Uncertainty Set (PUS) for Spot Prices

Spot price scenarios are usually obtained by utilizing aproduction costing model (a fundamentalist approach) [9] orthrough a statistical regression on past market prices [10]. Onthe one hand, the fundamentalist approach takes into accountex-ante hypothesis on market uncertainties, such as fuel prices,availability, supply expansion scenario, hydrology, etc. Statis-tical models, on the other hand, are based on the assumptionthat past realizations explain future prices. Both approaches canbe easily challenged: in the first, any deviation of the assump-tions affects the estimated probability distribution of prices,whereas the second approach is not suitable for markets withtechnological developments, in which the supply mix changessignificantly over time and does not make the historical recorda good proxy for the future. Our proposal considers the priceuncertainty by means of endogenously generated scenariosfollowing the RO approach. Such scenarios are defined within

Fig. 2. Uncertainty characterization: spot price polyhedral uncertainty sets.

a polyhedral uncertainty set, which can be used to robustify theportfolio by means of the use of worst-case analyses.To allow the ETC to express its inter-temporal risk-prefer-

ence, the set of periods, , is partitioned into subsets of periods,, e.g., months of each year, , in the time horizon. In

our approach, we consider that for each renewable generationscenario , there are subsets of partial spot price timeseries, each one of them related to a given subset of periods .In this work, we use the following polyhedral uncertainty set(PUS) to represent those sets:

(2)

(3)

(4)

(5)

(6)

(7)

(8)

where is a copy of except for the last term, which isdisregarded to account for expressions (5) and (6). Lagrangianmultipliers (or dual variables) are shown after each constraintto ease the understanding of the robust counterpart that will bepresented next.Expression (3) defines the envelope for the spot-price time se-

ries. Expression (4) constrains the number of periods withinthat the stress scenarios can deviate from the reference. Expres-sions (5) and (6) constrain the maximum and minimum returnsto and , respectively. Finally, expressions (7) and (8) setsthe bounds for and . Fig. 2 depicts the arrangement of thePUS’s over time and for different renewable energy scenarios.It is important to mention that the PUS (2)–(8) is flexible

enough to be particularized to the pure stochastic modeling ap-proach, where a set of exogenous scenarios are used to char-acterize the uncertainties, following previous reported works[5]–[8]. By making the reference scenarios meet the simulatedones and , the grey areas of Fig. 2 reduceto the exogenous scenario and the pure stochastic approach ismet. Moreover, the same rationale applies to consider an ex-ogenously defined static-stress scenario. However, under theproposed framework, by increasing the value of from zero



to some positive value, the decision maker admits a deviation,e.g., of periods within each sub-period , from the refer-ence scenario that, in this work, is explored as an endogenousstress scenario, because it is going to be chosen to create theworst-case adversity.In this framework, the proposed approach has the advantage

of not requiring the specification of the entire stress-scenariopath. Instead, the definition of the stress scenario is parameter-ized by intuitive parameters, such as the maximum and min-imum price values, the maximum and minimum returns be-tween periods, and the number of periods in which the price candeviate from the reference. Depending on the agent modelingchoice and available information, the reference scenarioscan be either obtained by a degenerated distribution, where onlyone trajectory of price is considered and repeated for all refer-ence scenarios in expression (3), or by a non-degenerated dis-tribution, where different price trajectories are simulated by agiven methodology and accounted for in (3). Notwithstanding,it is important to say that in the stress analysis, the decisionmaker is not willing to reproduce nor to capture the dynamic ofthe true underlying spot-price process, but rather guard againstthe occurrence of unpredictable adverse conditions. This is astep beyond in the subject of stress analysis largely used in in-dustry practices, which generally relies on exogenous (static)stress-scenarios. Additionally, such an approach can be com-bined with the standard pure-stochastic approach in many dif-ferent ways, e.g., constraining the loss under the stress scenariosand optimizing for the scenarios obtained by a stochastic model.For didactic purposes, we consider the case in which the de-

cision maker maximizes the portfolio value under the stress sce-narios. For each renewable generation scenario and portfolio de-cision-vector, , the present valueof expression (1) is minimized for each subset of periods .Thus, for a given , the stress scenario for the ETC outcomewithin each is obtained by the following deterministicoptimization problem:

(9)

Themain idea behind expression (9) is that for each simulatedrenewable generation scenario and portfolio decision-vector, the stress scenario for the spot prices is endogenously ob-tained for the entire contract horizon by means of sub-leveloptimization problems, as defined in (9). The spot-price sce-narios are defined to penalize as much as possible the revenuepresent value of the ETC, considering a set of constraints for thespot behavior to control the degree of stress produced. A par-tial present value accounts for the money value over the timewithin each subset of periods by means of a discount rate .In this setting, means the worst-case present value ofthe ETC revenue for a given scenario at the first period of ,accounted for by .

Fig. 3. Stress scenario for the spot price obtained for three different years withdifferent values for for some arbitrary scenario and portfolio .

Note that controls the conservatism of the solution in eachsub-level optimization problem. For example, in a scenario ,a small induces a low conservative solution because onlyfew sub-periods can decouple from the reference and penalizethe ETC’s revenue. Hence, the larger the value of , the morefreedom the sub-problems have to attack the portfolio, whichwould lead to more conservative solutions. Fig. 3 illustrates astress spot-price scenario for a set of three subsets of periods(3 years on a monthly basis). Therefore, each subset of periodsis built to represent the months of each of the three years

as follows: ,and , where . For illus-trative purposes, a hypothetical renewable generation scenarioand a decision vector were chosen to calculate the energy

settlement (third term of expression (9)). Moreover,and were set to illustrate the response of the

worst-case stress model.In Fig. 3 the bars represent the energy settled by the spot price

in expression (9), , the dotted

line represents the spot price reference, and the continuous linerepresents the stress scenario. In the first and third year, theworst-case spot scenario is obtained by decreasing the value ofthe energy during the sub-periods of the highest production sur-plus. In contrast, in the second year, the worst-case spot sce-nario is obtained by increasing the energy price during the twoperiods in which the renewable portfolio faces its worst pro-duction. This simple example shows the different structures forthe stress scenario that resemble the well-known securitycriterion used in power system operation [15]. Under such inter-pretation, the optimized portfolio is set to withstand a spot-pricespike, or deviations from the expected price pattern, in up toperiods, e.g., months, in a subset of the contract horizon, e.g., ayear.Finally, it is important to emphasize that the definition of the

subset of periods is arbitrary in the proposed framework andconstitute a modeling choice of the decision maker. This wouldallow for the decision maker to make use of the proposed un-certainty modeling approach for long-term strategies that wouldlast for many accounting periods.

III. HYBRID ROBUST AND STOCHASTIC MODEL

The objective of the model is to determine the optimal re-newable portfolio to back up a bilateral PPA sale in the forwardmarket.Within this objective, the model determines the decision



vector through a risk-averse trading strategy, which is drivenby the maximization of the certainty equivalent of the ETC’sfuture cash flow. We use the left-tail conditional value-at-risk(CVaR) [27] to account for the risk in a risk-adjusted objectivefunction composed of a convex combination between the CVaRand the expected value [28]. The portfolio model is a hybrid ap-proach between robust and stochastic optimization and can bestated as follows:

(10)

(11)

(12)

(13)

(14)

Problem (10)–(14) is a particular case of a bilevel problem,in which the first-level problem maximizes a concave function,namely the CVaR combined with the expected value, of a setof worst-case revenue scenarios obtained as the objective func-tion of second-level minimization problems. Therefore, the pro-posed model is a sort of max-min optimization problem.In expression (10), the risk-averse certainty equivalent of the

ETC is maximized. In such an expression, andplay the role of risk-averse parameters for the ETC.

The first term, weighted by , recovers the left-tail -CVaR ofthe th subset of periods of net income. Roughly speaking, the-CVaR can be understood as the average of the %

worst-case scenarios of the total net income within . In prac-tical applications, generally ranges from 0.90 to 0.99. Thesecond term, multiplied by the complementary weight ,accounts for the expected value of the net income in the objec-tive function. For further details on the economic interpretationof the adopted risk-metric, we refer to Appendix A. Expres-sion (11) is part of the CVaR assessment, and the net revenue ofthe ETC is evaluated under the worst-case spot-price scenario(stress scenario) for each subset of periods. Expression (12)constrains PPA sales to the acquired amount of FECs, and (13)and (14) impose bounds on decision variables.Within this model, the ETC is allowed to express its risk-

preference for different subset of periods. Furthermore, in thismodel, the CVaR measures the risk of the ETC under a givenlevel of modeling uncertainty defined by the PUS. Thus, boththe CVaR and the PUS are two combined ways to express theagent’s risk-aversion. However, the role of the former is to as-sess the risk through the scenarios, while the latter inputs a levelof uncertainty in those scenarios to consider the fact that themodel that generates spot price scenarios is only an approxima-tion of the real one.In Fig. 4, an outlook of the proposed model is depicted. In

the first level, a unique portfolio vector is defined with the op-timum amounts to be contracted by the ETC. Given the portfolioselected by the first-level problem, for each simulated renewablegeneration scenario, the worst-case revenue stream is assessedfor all sub-periods by means of the spot-price stress scenarios

Fig. 4. Overview of the stochastic max-min optimization approach.

following expression (9). Then, the objective function is evalu-ated according to (10).

IV. SINGLE-LEVEL EQUIVALENT MODEL

Problem (10)–(14) is a bilevel optimization problem andcannot be directly solved by commercial optimization solvers.Based on the RO approach presented in [12], an efficient maxi-mization-equivalent formulation can be provided for (10)–(14)using the following:1) derive the dual objective function of the linear program de-fined by the right-hand-side of (9). It constitutes a lowerbound for the worst-case settlement term, for all dual fea-sible solutions;

2) derive the dual feasible constraints of the linear programdefined by the right-hand side of (9);

3) replace in (10) and (11) by the dual objective func-tion found in 1) and add in (10)–(14), the dual feasible con-straints derived in 2).

For an interested reader, we refer to [12] for further details onthe aforementioned transformation. The equivalent single-levelmodel for problem (10)–(14) is the following:

(15)

(16)

(17)

(18)



(19)

(20)

(21)

(22)

(23)

(24)

(25)

In (15), the dual variables of the second level problemdefined by (9) are optimization variables. Expressions (16) and(19)–(25) follow steps 1) and 2), respectively. For expositorypurposes, expression (16) is used to facilitate the model un-derstanding. Therefore, step 3) is performed in this expression.Finally, in (18), expressions (12)–(14) are repeated. Model(15)–(25) is the implementable version of problem (10)–(14),which belongs to the class of two-stage stochastic linear pro-grammingmodels with recourse [31]. In this paper the extensiveform of the stochastic program was solved (no decompositionused) but we highlight our problem structure is suitable fordecomposition methods, such as the Benders approach [32].Under such decomposition framework, the problem structurecould be explored to decompose the sub-problem and solve itby sub-periods and scenarios.

V. CASE STUDY

In this section, two case studies illustrate the applicability ofthe proposed methodology with realistic data from the Brazilianpower sector. In the first case study, we motivate the usage andanalyze the results of the portfolio model (15)–(25) as a strategictool to define the optimal medium-term renewable portfolio toback up a one-year flat PPA. In the second case study, we con-sider a case where an ETC needs to define its optimal participa-tion in a given set of renewable projects to supply a long-termPPA. In this setting, we assume a 10-year PPA, as typically re-quired by financial institutions to provide competitive interestrates for project financing.For expository purposes, both case studies assume

% is set to the equivalent monthly rate)and the opportunity of contracting two renewable units to backthe PPA sale: a run-of-river small hydro (SH) plant with 30MW of installed capacity and 17.4 avgMW of FEC, and awind power plant (WPP) with 54.6 MW of installed capacityand 27.12 avgMW of FEC. Furthermore, we also assume adeterministic spot price reference and maximum/minimumdeviation parameters throughout both case studies by droppingits scenario index: and . We make

Fig. 5. Monthly statistics for the generation profile of complementary renew-able generation units in the Brazilian power system: SH and WP.

use of the methodology presented in [5] to jointly simulatea set of 2000 scenarios for the renewable energy generationbased on the historical data available for the SH and WPP. InFig. 5, the average and extreme quantiles (5% and 95%) of thesimulated scenarios for the renewable production are depictedfor the year of 2012. For reproducibility purposes, [29] makesavailable all data used in both case studies.

A. Case Study I: Medium-Term Portfolio Strategy

In this case study, we assume that the ETC has a selling op-portunity of avgMW by R$/MWh andthat both renewable units agree to sell 100% of their FEC by

R$/MWh-of-FEC. In this first case study, .Thus, . The risk-aversion parame-ters are set to and the effect of varyingparameter for 1, 2, and 3 is studied. For comparison pur-poses, a pure stochastic (PS) model (similar to the one pro-posed in [6]) is used to provide a base-case solution to be com-pared with the proposed hybrid stochastic-robust (HSR) model.The PS model considers the spot price scenarios as exogenousvariables produced by a least cost dispatch model based on theSDDP methodology [9]. As mentioned in Section II-C, the PSmodel can be obtained from the HSR model by making thereference spot-price scenarios equal to the simulated ones and

. Official system data from December 2011 were usedto produce the set of spot price scenarios for January 2012 toDecember 2012. In this case-study, the methodology used pro-vides paired scenarios that account for the correlation betweenboth processes (see [5] for further details).3

Fig. 6 shows the statistics for the set of simulated spot pricescenarios. The reference scenario used to run the robust port-folio model (15)–(25) was set to the simulated average, shownin Fig. 6, to capture the conditional information of the spotprices throughout the contract horizon. Nevertheless, the max-imum positive and negative deviations from the reference sce-nario were chosen to allow the stress scenario to reach the pricecap (730 R$/MWh) and floor (12.1 R$/MWh), respectively, inany period. To constrain the worst-case spot-price scenarios, themaximum and minimum returns were obtained from the histor-ical data for each month and are presented in Table I (see also[29] for all data used in this case study, including the spot price

3The definition of how to derive generation and spot-price reference scenariosis case-dependent and out of the scope of this paper (they are consider as inputdata in our model).



Fig. 6. Average and extreme quantiles (5% and 95%) for the simulated spotprices: from January 2012 to December 2012 on a monthly basis.

TABLE IROBUST PORTFOLIO MODEL INPUT DATA FOR CASE STUDY I

TABLE IIOPTIMAL CONTRACTING OF EACH CANDIDATE OPTION IN THE STOCHASTIC

AND HYBRID STOCHASTIC-ROBUST CASE (AVGMW)

and renewable production for both plants in each month andscenario).We have solved (15)–(25), and the optimal amount sold and

contracted of each technology is shown inTable II. Each problemwas solved in less than one minute using a Dell Inspiron 15RSpecial Edition Laptop. We observed that when the spot priceis treated as an exogenous variable (second column of Table II),the optimal strategy for the trader is to back its sales solely onthe wind generator. One way to interpret this result is to observethat the wind production pattern is similar to the spot price one,i.e., high spot price seasons coincide, in general, with high windpower production (see Figs. 5 and 6 for better comparison). Incontrast, hydro production presents almost full complementaritywith these two variables, thus adding no value to the portfolio.When the model (15)–(25) is utilized with and 2,

the optimal strategy is also to sell the total of the PPA, but sup-porting it by a mixed portfolio that is composed of both windand small hydro. This result is due to the stress-scenarios for

Fig. 7. Generation profile of the portfolios found with the stochastic and hybridstochastic-robust models for the observed data (renewables generation) duringthe contract horizon: months of 2012.

Fig. 8. Spot prices and revenue profile of the portfolios found with the sto-chastic and robust models for the observed data (renewables generation andspot prices) during the contract horizon: months of 2012.

the spot prices that penalize portfolios with high exposure to thespot market, which occurs whenever the portfolio generation isbelow the amount sold. This situation can be observed in Fig. 7,which shows an out-of-sample comparison (in energetic terms)for the first three portfolios shown in Table II with actual gener-ation observed in 2012. In this year, spot prices lie within the 5thand 95th quantiles. However, the observed price for 2012 can beconsidered as atypical (when compared to the historical profile)since we have experienced an unusual spot-price dynamic (paththrough the year) with two price peaks, in April and November(see Fig. 8). As a conclusion, the role of the price dynamic is alsoconsidered when building the optimal HSR portfolio, because in(9), the stress-scenario is conceived to attack the portfolio withthe worst-case path within a subset of periods. Therefore, the ro-bust approach also guards the portfolio against the occurrenceof unusual price dynamics.It is possible to see that the portfolios found with the robust

model mitigate purchases in the spot market by raising the gen-eration profile throughout the months. The effect of such miti-gation is shown in Fig. 8, where the impact of the first semesterspot-price disturbance (April and May) is attenuated under therobust solutions. Moreover, the excess of capacity contracted bythe HSRmodel provides an extra benefit during the second peak(November) not fully obtained by the PS approach. For ,the stress created to the portfolio is so high that it is optimal tonot set up the business.Nevertheless, it is important to note that it is not always the

case that the HSR model outperforms the PS one. In Table III,



TABLE IIIBACK TEST IN THE YEARLY REVENUE OF THE ETC (MMR$)

Fig. 9. Sensitivity analysis of the optimal solution of the PS model and HSRmodel with and 2 on the number of scenarios of renewable generationconsidered in the model for the Case Study I.

the solution obtained with both models were evaluated underthe observed generation and prices for 2012 and for two otherrepresentative years: 2010 (a typical year with respect to spotprice realization—lower values at the beginning of the year andhigh values at the end) and for 2008 (an unusual year—a pricespike in January and low values at the end of the year).As shown in Table III, the PS model has the highest yearly

revenue in the typical year, mainly because the spot price out-come for this year realized as “expected” in accordance withthe pattern of the simulated scenarios. However, when the spotprice realization presents a different pattern with respect to theprice scenarios simulated by the stochastic model, the hybridstochastic-robust model outperforms the pure stochastic one.We highlight that the HSR model outperforms the PS one whenthe spot price increases in the last months of the year as well,indicating that the robust counterpart was also able to capturethe effects of “standard” and “normal” pattern of prices.Finally, in Fig. 9, convergence analysis is performed by an-

alyzing the convergence of the optimal-primal variables (deci-sion vector as a function of the number of scenarios .

In this figure, only the buying variables, , are shown sincethe optimal solution of is the same (equal to 10 avgMW)for all cases.Note that the optimal solutions stabilize when the number

of scenarios meets 1100. This suggests that convergence isachieved. This fact provides strong evidences that the solutionobtained, which make use of the 2000 scenarios, converged andthe results presented are accurate with respect to the asymptoticprobability distribution.

B. Case Study II: Long-Term Portfolio Strategy

In this second case study, we apply the HSR model to thesame units (wind and SH) considered before. However, we

TABLE IVROBUST PORTFOLIO MODEL INPUT DATA FOR CASE STUDY II.

ALL MONTHLY DATA ARE THE SAME FOR ALL YEARS

now consider an ETC deciding its willingness to invest inboth units. We consider that the ETC has the same price andquantity selling opportunity described in the previous casestudy but for a 10-year PPA horizon, from January 2016 toDecember 2025. In this case, is the setof years and is the collection of sets containingthe months of each year , i.e.,

; and. In this case study, the risk-aver-

sion parameters are set to be constant: , andfor all . Moreover, remains fixed and equal to

0.95 throughout the case study.Under a time horizon of 10 years, the specification of any

price model would rely on unpredictable economic and struc-tural data inputs. The proposed approach provides the decisionmaker with a tool consistent with well-known stress analysisthat is able to consider the information available for the sto-chastic nature of the renewables generation. The model devel-oped in [5] was used to simulate the joint wind and SH produc-tion scenarios throughout the contract horizon, and the data pre-sented in Table IV were used to constrain the related spot-pricestress scenarios, in which the same sequence of monthly dataare used for each years (all data, including the renewableproduction for both plants in each month and scenario

, are also available in [29]).To evaluate the results, an efficient frontier curve is created

by varying the risk-aversion parameter from 0.5 to 0.99 on a0.05 step-basis. Fig. 10 shows the curve for and 2.The vertical axis represents the present value (PV) of the

expected value of the 10-year revenue and the horizontal axisstands for risk. The latter is evaluated as the difference betweenthe PV of the expected value and the PV of the CVaR %, i.e.,the second and first terms of expression (10), respectively (seeAppendix B for further details about the development of a riskmetric in the multi-period setting).For and , the risk is higher than

the expected return, indicating that the PV of the CVaR % isnegative. This fact suggests that under , the investmentdecision in renewables requires high risk aversion to mitigatethe solvency risk (chance of a negative PV). Table V depicts the



Fig. 10. Efficient frontier curve for and 2 varying the parameter from0.5 to 0.99 on a 0.05 step-basis.

TABLE VOPTIMAL CONTRACTING OF EACH CANDIDATE OPTION FOR AND 2 INTHE 10-YEAR CASE STUDY FOR DIFFERENT RISK LEVELS OF (AVGMW)

portfolio decisions associated with each point of the efficientfrontier curve.Observe that in Table V, as the risk-averse parameter grows,

the excess of energy bought by the ETC also grows, representinga hedge against the price and quantity risk. These results followthe same idea explained in Case Study I. As the parametergrows, price spikes can attack the portfolio in different periods.In this context, the optimal contracting model aims to createa flatter generation profile, bringing more hydro generation tocomplement the portfolio, thus reducing the exposure to theshort-term market.In general, the efficient frontier varies with respect to the con-

servativeness level of the robust counterpart, i.e., with . Inthis sense, to create a long-term portfolio, the decision makershould well-adjust its risk preference to the parameters of themodel proposed because the solution obtained is highly depen-dent on these parameters. For example, a 0.05 variation oncreates a different portfolio and the relationship between riskand expected revenue.At last, in Fig. 11, we analyze the convergence of the primal

variables solution on the number of scenarios consid-ered in the optimization problem for the case of .Again, only the buying variables, , are shown since the op-timal solution of is the same (equal to 10 avgMW) for allcases.

Fig. 11. Sensitivity analysis of the optimal solution of the HSR model withand 2 on the number of scenarios of renewable generation considered

in the model for the Case Study II.

Note that again, the convergence is achieved and therefore theresults are accurate with respect to the asymptotical probabilitydistribution.

VI. CONCLUSION

This work established a new methodology to determine therisk-constrained optimal portfolio of renewable sources to sup-port a firm selling in the forward market. Such a model assessesthe optimal amount to trade in contracts by the ETC, given theirrespective specifications (prices, starting dates, durations, etc.),which is robust with respect to unexpected variations in spotprices but that also considers the stochastic nature of the pro-duction of renewable assets in a risk-constrained setting. Thisapproach provides an alternative to current models based onthe simulation of prices. Additionally, it incorporates a gener-alization of stress analysis to account for the modeling uncer-tainty in the optimal renewable portfolio model by means ofendogenously defined stress scenarios. Such an approach pro-vides the decision maker with an intuitive way to control thelevel of stress produced by the stress scenarios following theidea of well-known deterministic security criteria such as the

. We illustrated the methodology to devise a portfolioof renewable sources (wind and small hydros) in the Braziliansystem, where the approach of this paper was contrasted with itspure-stochastic counterpart and both are benchmarked againstactual (observed) market variables.Ongoing research related to this paper includes a definition

of a stochastic conservatism parameter correlated with the gen-eration of the plants, the use of stochastic spot reference pricesto account for ambiguity aversion. The inclusion of new finan-cial products in the model—such as call options—is also a rel-evant topic of research due to the nonlinearities introduced onthe model.

APPENDIX ARISK-METRIC’S ECONOMIC INTERPRETATION

Expression (10) stands for a composite objective function thataccounts for both the CVaR and the expected value of the ETC’scash flow. The objective function in (10) can be understood asthe ETC’s certainty equivalent (CE), since, in the optimal solu-tion, the value of the objective function is the amount of money



at the beginning of the study horizon with which the ETC is in-different, with respect to the objective function of (10), with theoptimal portfolio (we refer to [28] for further details on CVaRand CE). This is due to the fact that both the CVaR and theexpected value applied to a “deterministic” number, which isequivalent to a degenerated random variable with all scenarioassuming the same value, return such value.However, an additional difficulty rises from the multi-period

setting. And it is related to the question of how to consider theintertemporal preference of the agent? If the agent agrees thathis preference functional of one period is the convex combina-tion between the CVaR and the expected value, following pre-vious reported works (see [5]–[7], [28], and references therein),in themulti-period setting one has two possibilities: 1) apply thiscomposite measure to the present value of the random cash flowor 2) apply such measure to the random revenue of each periodand then assess the present value of the resultant deterministiccash flow [33]. Although not typical, such composite measurehas gained a lot of attention in many areas, such as finance [36]and energy [37], due to its conceptual virtues (coherence [38]and time consistency [39]) and computational tractability [27].If on the one hand, approach 1) has the advantage of pre-

serving the temporal correlation of the cash flow stream, onthe other hand, it is unable to capture the risk-preference be-tween periods. For instance, consider two stochastic cash flows,namely C1 and C2, of two years and two scenarios each with50%–50% of probability: in the first cash flow, C1, both sce-narios are equal to zero $ for both years; in the second one,C2, scenario one values Billion $ in the first year and1.1 Billion $ in the second year, and the second scenario isbuilt with the negative values of the first scenario. Assuming

, and typical risk-averse parameters, such asand , approach 1) is indifferent between C1 and C2,and, clearly, approach 2) prefers C1 rather than C2.4 One canargue that the hell-and-heaven dynamic present in C2 makesC1 preferable than C2 in practical applications because (i) com-panies do not have access to unlimited leverage to withstandhell and achieve heaven if the first scenarios is observed and(ii) the fruits of heaven would be shared among investors be-fore hell takes place in the second year if the second scenariomaterializes. This simple example illustrates that a practical ap-proach should lie in between of approaches 1) and 2). More-over, it should reflect the company’s financial access to externalleverage and the time window with which the company perfor-mance is evaluated.In this work, we propose a metric that generalizes the afore-

mentioned approaches. By selecting the subsets of periods,, both approaches can be obtained as well as inter-

mediate approaches with trimestral or semiannual accountingperiods.

APPENDIX BRISK METRIC IN THE MULTI-PERIOD SETTING

Expression (10) stands for a convex combination between thePV of the CVaR of each subset of periods and the PV of theexpected value. In the optimal value, the objective function canbe written as follows:

4This happens because, under the second approach, the CVaR assumes thevalue of the worst-case scenario in each period.

(26)

This expression can be rearranged in order to achieve the fol-lowing bi-criterion penalization form:

(27)

In (27), the first term accounts for the expected value of theproject and the second term for a multi-period measure of risk(negative deviation from the expected value), following theagent’s chosen approach to measure risk in the multi-periodsetting (see Appendix A for further details). If the risk-aversionparameters are constant for all , (27) is equivalent tothe PV of the expected cash flow penalized by the followingmulti-period measure of risk:

(28)

It is important to notice that (27) can be seen as the Lagrangedual function of the risk constrained version of this problemwhere only the first term of this expression is maximized sub-ject to a constraint on the maximum value of expression (28).Therefore, by solving model (15)–(25) for different values ofbetween 0 and 1, one can draw an efficient frontier by plottingon the horizontal axis the risk measures associated with eachvalue of and the corresponding expected value terms on thevertical axis.

ACKNOWLEDGMENT

The authors would like to thank FICO (Xpress-MP devel-oper) for the academic partnership program with the ElectricalEngineering Department of Pontifical Catholic University ofRio de Janeiro, Brazil (PUC-Rio). The authors would also likethank the LAMPS researchers for the daily exchanges and theirinsightful considerations.

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Bruno Fanzeres (S’11) received the B.Sc. degree in electrical and industrial en-gineering and the M.Sc. degree in operations research from Pontifical CatholicUniversity of Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil. He is currentlypursuing the D.Sc. degree in operations research at the same University.His research interests include operations, planning, and power system eco-

nomics.

Alexandre Street (S’06–M’10) received the M.Sc. degree and the D.Sc. degreein electrical engineering (operations research) from the Pontifical Catholic Uni-versity of Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil.From 2003 to 2007, he participated in several projects related to strategic bid-

ding in the Brazilian energy auctions and market regulation at the Power SystemResearch Consulting (PSR), Rio de Janeiro, Brazil. FromAugust 2006 toMarch2007, he was a visiting researcher at the Universidad de Castilla-La Mancha,Ciudad Real, Spain. In the beginning of 2008, he joined PUC-Rio to teach op-timization in the Electrical Engineering Department as an Assistant Professor.His research interests include power system economics, optimization methods,and decision making under uncertainty.

Luiz Augusto Barroso (S’00–M’06–SM’07) received the B.Sc. degree inmathematics and the Ph.D. degree in operations research from COPPE/UFRJ,Rio de Janeiro, Brazil.He is a technical director at Power System Research Consulting (PSR), where

he has been providing consulting services and researching on power systemseconomics focusing on hydrothermal systems. He has been a lecturer in LatinAmerica, Europe, and the United States/Canada.Dr. Barroso was the recipient of the 2010 IEEE PES Outstanding Young En-

gineer Award and is currently an Associate Editor of the IEEE TRANSACTIONSON POWER SYSTEMS and the IEEE TRANSACTIONS ON SMART GRID.

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