A Hybrid Stochastic and Robust Optimization Approach - IEEE Xplore

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 30, NO. 4, JULY 2015

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Contracting Strategies for Renewable Generators: A Hybrid Stochastic and Robust Optimization Approach Bruno Fanzeres, Student Member, IEEE, Alexandre Street, Member, IEEE, and Luiz Augusto Barroso, Senior Member, IEEE

Abstract—We present a new methodology to support an energy trading company (ETC) to devise contracting strategies under an optimal risk-averse renewable portfolio. The uncertainty in the generation of renewable energy sources is accounted for by exogenously simulated scenarios, as is customary in stochastic programming. However, we recognize that spot prices largely depend on unpredictable market conditions, making it difficult to capture its underlying stochastic process, which challenges the use of fundamental approaches for forecasting. Under such framework, industry practices make use of stress tests to validate portfolios. We then adapt the robust optimization approach to perform an endogenous stress test for the spot prices as a function of the buy-and-sell portfolio of contracts and renewable energy generation scenarios. The optimal contracting strategy is built through a bilevel optimization model that uses a hybrid approach, mixing stochastic and robust optimization. The proposed model is flexible to represent the traditional stochastic programming approach and to express the ETC's uncertainty aversion in the case where the price distribution cannot be precisely estimated. The effectiveness of the model is illustrated with examples from the Brazilian market, where the proposed approach is contrasted to its stochastic counterpart and both are benchmarked against observed market variables. Index Terms—Conditional value-at-risk, power system economics, robust optimization, stochastic optimization.

NOMENCLATURE Constants Sale price of the PPA ($/MWh). Price of the capacity payment contract with a renewable unit ($/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. Date of publication September 25, 2014; date of current version June 16, 2015. This work was supported in part by UTE Parna´ıba Geração de Energia S.A. through R&D project ANEEL PD-76250001/2013. 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 of one MW during the relevant period. One average MW during one year corresponds to 8760 MWh.

Firm energy certificate of the renewable unit (avgMW). Energy production of the renewable generator in period and scenario (MWh). Number of hours of period . Probability of the renewable production scenario . Opportunity cost of the money in percentage per sub-set of periods, e.g., per year. Opportunity cost of the money in percentage per period of the th subset of periods, e.g., per month. First period of the th subset of periods. Risk aversion parameter that combines the expected value and the CVaR of the th subset of periods. Risk aversion parameter that defines the confidence level of the CVaR risk measure of the th subset of periods. Reference or nominal spot-price scenario in period and scenario ($/MWh). Maximum positive deviation from the spot-price reference scenario in period and scenario ($/MWh). Maximum negative deviation from the spot-price reference scenario in period and scenario ($/MWh). Budget that control the conservatism level of the spot-price stress scenarios in each subset . Maximum return (%) of the spot-price stress scenario from period to . Minimum return (%) of the spot-price stress . scenario from period to Decision Variables that the ETC is willing to Percentage of supply through the PPA. Percentage of the unit that is brought to the ETC portfolio through a capacity contract.

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Auxiliary variable that achieves the -value-at-risk of the partial PV of the revenue in each sub-period at the optimal solution ($/MWh). Auxiliary variable that represents the left deviation of the revenue scenario from the value in each sub-period ($/MWh). Spot price in period and scenario

($/MWh).

Percentage of the positive deviation from the reference price in period and scenario . Percentage of the negative deviation from the reference 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 each sub-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 set is divided. Set of renewable units. Set of scenarios.

A

I. INTRODUCTION

MAJOR challenge for power generation companies (Gencos) in competitive electricity markets is to determine an optimal contracting strategy for its assets that maximizes their value. This strategy should take into account the company's risk profile and adequately treat all types of external risks, such as the uncertainties in energy production and the spot prices. Energy trading companies (ETC) play a key role in providing market penetration and diversifying risk of Gencos in order to serve consumers. To hedge against the volatility of spot prices, generators or ETC sign mid- and/or long-term energy supply contracts, which is a financial instrument that plays an important role in the power system reform of many countries worldwide. A supply contract involves the need to deliver a given volume of energy over a time horizon by a

seller against a fixed payment by the buyer. In most electricity markets, such contracts are only financial instruments (see [1] for a review of the financial instruments used in electricity markets 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, the plant will not generate but meet its contractual obligations by purchasing (cheaper) energy in the Wholesale Energy Market (WEM). Conversely, if spot price is high, the plant will produce its own energy, thus avoiding expensive purchases. In other words, in the case of thermoelectric generators, the physical production of the asset is a natural hedge and, except for the hazard of outages, mitigates production risks [3]. In the case of renewable generators—such as hydroelectric, wind and biomass plants—the probabilistic and seasonal nature of their physical production poses a joint volume and price risk: a situation can arise where in moments of high spot prices, the production of a renewable generator is low, which can result in high financial exposures in the short-term market if the plant is contracted with a consumer [5]–[7]. The situation is made worse for hydro plants in hydro-dominated countries because spot prices are negatively correlated with hydro production (the so-called hydrological risk, see [8]). The development of renewable electricity contracting strategies in competitive markets via ETC has been widely studied in different contexts: the ETC acts as a purchaser of renewable energy and creates a product that bundles the energy purchased in order to serve a firm load. This involves the definition of the optimal mix of sources to be bought by the ETC to mitigate the price and quantity risk (taking advantage of the complementarity usually existing among the energy production from the different renewable sources) as well as the definition of the optimal amount of load to be served through a co-optimization model [5]–[8]. The “standard” modeling approach of this ETC portfolio optimization problem considers the representation of uncertainty in renewable production and market prices by scenarios generated via Monte Carlo simulation, usually utilizing a production costing model (market equilibrium simulated by a fundamental approach) [9] or a statistical regression on past market prices [10]. One of the main difficulties in implementing such modeling is how to produce meaningful future paths of spot prices. The spot price formation is a very complex process, which results from the combination of market (hydrology, demand, supply, outages, fuel availability, etc.) and political (interference on prices, out of merit order dispatches, etc.) conditions [10]. Even in countries such as Chile, Peru and Brazil, where a production costing model determines the system dispatch and the spot prices (as short-run marginal costs) and the market participants use the same production costing models to predict spot prices, forecasts have proven to be inaccurate and very difficult [11]. When the complexity of the true underlying uncertainty processes requires the knowledge of input parameters whose dynamics are difficult to predict, practitioners generally treat this inaccuracy by means of stress tests, where stress “scenarios” are used to validate the performance of the optimal portfolios under adverse 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

not represent the true worst-case price scenario for the portfolio in hands. For example, different ETC portfolios may require different worst-case scenarios. One alternative to overcome this difficulty is to consider endogenously-defined stress-price scenarios that create the worst-case financial adversity for the ETC portfolio. In other words, they would be defined by an embedded optimization problem that ensures they really represent the worst possible realization of this uncertainty against the ETC portfolio revenue. Robust optimization (RO) has recently emerged [12]–[14] and proven to be a powerful tool for addressing decision-making problems involving uncertainty parameters whose true probability distribution is difficult to predict, which is exactly the case of our spot price modeling challenge. Recent applications have dealt with robust-based decisions in operations and scheduling [15]–[18], bidding strategies of generators in the short-term spot market [19] and in portfolios with wind and storage [20], demand response and consumption strategies [22], [23], integration 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 renewable energy portfolio optimization problem. We propose a model that combines in one single optimization problem the endogenous stress-analysis approach for the spot prices (via RO) with the stochastic approach for the representation of renewable production. A. Objective of This Work The objective of this work is to present a portfolio optimization model to determine the risk-adjusted optimal trading strategy for an ETC in a contract market backed by a renewable energy portfolio to serve a firm load. The two relevant uncertainties are the future spot prices and the production of the renewable energy sources, which are treated in our portfolio optimization model by robust and stochastic optimization, respectively. We use the RO approach with polyhedral uncertainty set to constrain spot-price stress scenarios within the agent's prior hypothesis for adverse conditions. To obtain the optimal portfolio, spot prices are endogenously determined by a set of second-level optimization problems, one for each exogenous scenario of renewable energy generation (assumed to follow a “well-behaved” stochastic process). Prices are allowed to deviate from a reference scenario toward a worst-case realization within a constrained uncertainty set. The modeling approach allows for the consideration of deterministic or stochastic reference scenarios for the spot prices, which can be obtained following standard time-series approaches or by means of equilibrium models. In other words, we consider that the spot-price scenarios may be misspecified and consist of only an approximation of the true underlying stochastic process whereas renewable production scenarios are produced via the classic Monte Carlo approach. The model proposed in this paper can be seen as an extension of the model proposed in [6] to account for the modeling uncertainty in the spot 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 “survive” even if the spot price reaches its cap or floor value, depending on what is the worst case for the portfolio (considering

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scenarios of renewable production), during (set by the decision maker) periods (hours/months) within a given time horizon (day/year). This technique proves to be rather flexible for incorporating other constraints and conceptually speaking it resembles—but is not equal to—the security criterion , which is widely used in operation problems, as shown in [15]. Our methodology is flexible to consider the standard case where the decision maker fully believes in his/her price scenarios. B. Contributions Regarding the Existing Literature The contribution of our work is to present an alternative methodology to devise the ETC's optimal contracting strategy via a hybrid robust-stochastic optimization model. More objectively, the proposed model aims at recognizing the well-defined stochastic dynamics of renewable energy production as well as the difficulty to forecast spot prices, thus ensuring the robustness of the results against unexpected price variations. The methodology is flexible to model spot-price stress scenarios, endogenously defined in the portfolio problem, as well as to follow a pure stochastic approach2 that considers exogenously-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 RO approach (see [12] for more details), and solved by commercial solvers [26]. We illustrate the methodology to devise a portfolio of renewable sources (wind and small hydros) in the Brazilian system, where the proposed approach is contrasted to its fully stochastic 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 presented in [17], which explores the modeling of an expanded polyhedral uncertainty set to account for different load patterns or security criteria in the unit commitment problem; [18], which combines the robust and stochastic approaches to account for both the worst-case and the expected dispatch cost under nodal injection uncertainty through a weighted objective function; and [20], which deals with portfolios with wind and storage in the short-term market. Differently from previous reported works, we consider as a motivation for our hybrid approach the fact that one of the uncertainty parameters, the spot price, is of difficult modeling via stochastic programming and the rest of the uncertainties are assumed to follow a known and “well-behaved” stochastic process. To the best of our knowledge, no work has been performed on applying RO to the definition of commercial strategies for portfolio of renewables in contract markets, which is the subject of our work. On the methodological side, [17] and [18] also provide hybrid robust-stochastic models. Our paper differs on the way this hybridization is done: via a concave function, which accounts for the well-known coherent measure of risk, namely the conditional value at risk [28] ([17] and [18] use a linear combination approach without risk constraints). Finally, it is worth mentioning that the problems addressed in [17] and [18] lie in the class of adjustable (two-stage) RO models [34], requiring more 2However, the challenges associated to the computation of a joint probability distribution of the uncertain factors are seen as an additional motivation to adopt the proposed stochastic-robust approach.

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complex solution methodologies (see [35]) than the single-stage 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 describes the modeling approach proposed for the representation of the uncertainties. Section III presents the bilevel formulation for the hybrid robust-stochastic renewable portfolio selection model, and Section IV shows its single-level-equivalent counterpart. Section V presents two case studies. Section VI concludes this work and discusses extensions and future research. II. UNCERTAINTY MODELING APPROACH Two main risk factors are considered in the ETC's future revenue: , which is a random variable that expresses the amount of renewable energy produced by the generating unit in period (MWh), and , which is the random variable that represents the spot price in period ($/MWh). Throughout this work, random variables are assumed to be discrete in a finite support following stochastic programming standards. Both uncertainty factors are characterized by a set of possible scenarios and probabilities, , as customary in stochastic optimization [31]. A. Electricity Contracts and the Energy Trading Company Standard financial power purchase agreements (PPA) and capacity payment contracts are considered in this work. A PPA is a bilateral supply contract agreement in which the buyer pays a fixed price for a given fixed amount of energy to the seller party [6]–[8]. In capacity payment contracts with renewables, the generator (selling counterpart) transfers a percentage of its future, therefore uncertain, production to the buyer in exchange of a fixed payment. The fixed payment of the capacity contracts can be parameterized in two terms: price and quantity. The latter is based on the contracted percentage multiplied by the firm energy certificates (FEC) of the unit [5]–[8]. The FEC of a renewable unit is issued by the system regulator based on a long-term quantile (generally set to the median or 10%) of the observed generation. The role of the FEC in the Brazilian regulatory framework is to limit the total amount of energy each unit can sell through contracts (see [11] for further discussion on this subject). Therefore, a capacity payment contract with a unit acts as if the buyer “rents” percent of the power plant, which includes both the generation and FEC. In the case of renewables, for simplicity purposes, we assume that the variable generating cost is zero; however, such cost could be easily included in the model, as shown in [6]. Differently from the short-term market, the financial contract market does not impose any real energy exchange, all transactions are purely financial. The contract market operator runs a market clearing after the determination of the spot prices, which occurs in the short-term market, by clearing the differences between all volumes produced (consumed) and contracted of each agent at this price. In this framework, ETCs play an important role in the market acting as risk managers, creating commercial opportunities, and mitigating risks. Fig. 1 illustrates the proposed contractual scheme for the ETC. At the left

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

and right-hand-side of this figure, the ETC buys two capacity contracts with two complementary renewables and therefore, receives the right of commercialize the bought generation. This provides the ETC with a positive settlement in the contract market clearing regarding the purchased proportion of the generation of each unit (first term of the market settlement payment—at the top of the figure), , in exchange of a fixed payment, . Then, at the bottom of the figure, the ETC sells a standard flat PPA for a consumer, e.g., industries or large commercial entities, ensuring the amount to be delivered. Due to this agreement, the ETC receives a fixed income from the consumer, , in exchange of a negative settlement in market clearing regarding the sold amount (second term of the market 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 income due to the PPA sell; 2) a fixed expenditure due to the capacity payments for all renewable generating units—we assume a set of units; and 3) a random term that accounts for the net contract market settlement by the spot price between the renewable generation, acquired by means of the capacity contracts, and the total amount sold through the PPA. Expression (1) illustrates 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 to back sales in the forward market. For simplicity and didactic purposes, it is assumed that the ETC has no existing portfolio, which could be easily included in the model by an additional constant term in (1). Note that in this contracting scheme, the ETC bears all of the price and quantity risk because the capacity contracts do not guarantee a fixed energy delivery to fulfill the bilateral PPA with the consumer. If, on the one hand, the fixed revenue, due to the first term of expression (1), increases as the ETC sells more energy through the PPA, on the other hand, the


net 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, expression (1) can reach negative values and significantly jeopardize the company financial performance. Therefore, according to expression (1), in defining the optimal sell and buy percentages , the ETC must take into account the tradeoff between the fixed payment, composed of the first and second terms, and the price and quantity risk introduced by the third term. Note that the ETC plays the role of a risk manager in the proposed model and may therefore embody a variety of other different commercial roles (such as an electricity retailer or a Genco). For example, the model devised in this paper can be used by a Genco devising its optimal investment strategy in a portfolio of renewable technologies to mitigate the price and quantity risk and then supplying a consumer through a longterm flat PPA. Under this setting, the capacity payment in (1) can represent the annualized investment cost and interests expenses of the units owned by the company. In this context, the fixed capacity payment term, , in (1) could be replaced by a simpler fixed payment term, , representing those expenses in each period of the cash flow time horizon. Then, expression (1) can represent the net revenue of the generation company selling a long-term PPA as a function of the percentages of the renewable units in which the company is willing to invest. It is worth mentioning that this business format, in which an ETC is formed to represent a given Genco in the contract market, is largely adopted in countries where the financial contract markets are representative such as in Brazil. B. Renewables Generation Scenarios The modeling approach adopted to generate the scenarios of the renewable production is based on simulation procedures of periodic stochastic processes via, e.g., a Monte Carlo procedure. For the model proposed here, the generation scenarios for each plant are considered as input data, and, therefore, exogenous to the model. For example, in [5], a Monte Carlo procedure is proposed to simulate renewable production scenarios based on a vector-autoregressive stochastic process with variance law. C. Polyhedral Uncertainty Set (PUS) for Spot Prices Spot price scenarios are usually obtained by utilizing a production costing model (a fundamentalist approach) [9] or through a statistical regression on past market prices [10]. On the one hand, the fundamentalist approach takes into account ex-ante hypothesis on market uncertainties, such as fuel prices, availability, supply expansion scenario, hydrology, etc. Statistical models, on the other hand, are based on the assumption that past realizations explain future prices. Both approaches can be easily challenged: in the first, any deviation of the assumptions affects the estimated probability distribution of prices, whereas the second approach is not suitable for markets with technological developments, in which the supply mix changes significantly over time and does not make the historical record a good proxy for the future. Our proposal considers the price uncertainty by means of endogenously generated scenarios following the RO approach. Such scenarios are defined within

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

a polyhedral uncertainty set, which can be used to robustify the portfolio by means of the use of worst-case analyses. To allow the ETC to express its inter-temporal risk-preference, 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 generation scenario , there are subsets of partial spot price time series, 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) is a copy of except for the last term, which is where disregarded to account for expressions (5) and (6). Lagrangian multipliers (or dual variables) are shown after each constraint to ease the understanding of the robust counterpart that will be presented next. Expression (3) defines the envelope for the spot-price time series. Expression (4) constrains the number of periods within that the stress scenarios can deviate from the reference. Expressions (5) and (6) constrain the maximum and minimum returns to and , respectively. Finally, expressions (7) and (8) sets the bounds for and . Fig. 2 depicts the arrangement of the PUS'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 approach, where a set of exogenous scenarios are used to characterize the uncertainties, following previous reported works [5]–[8]. By making the reference scenarios meet the simulated ones and , the grey areas of Fig. 2 reduce to the exogenous scenario and the pure stochastic approach is met. Moreover, the same rationale applies to consider an exogenously defined static-stress scenario. However, under the proposed framework, by increasing the value of from zero

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to some positive value, the decision maker admits a deviation, e.g., of periods within each sub-period , from the reference scenario that, in this work, is explored as an endogenous stress scenario, because it is going to be chosen to create the worst-case adversity. In this framework, the proposed approach has the advantage of not requiring the specification of the entire stress-scenario path. Instead, the definition of the stress scenario is parameterized by intuitive parameters, such as the maximum and minimum price values, the maximum and minimum returns between periods, and the number of periods in which the price can deviate from the reference. Depending on the agent modeling choice and available information, the reference scenarios can be either obtained by a degenerated distribution, where only one trajectory of price is considered and repeated for all reference scenarios in expression (3), or by a non-degenerated distribution, where different price trajectories are simulated by a given methodology and accounted for in (3). Notwithstanding, it is important to say that in the stress analysis, the decision maker is not willing to reproduce nor to capture the dynamic of the true underlying spot-price process, but rather guard against the occurrence of unpredictable adverse conditions. This is a step beyond in the subject of stress analysis largely used in industry practices, which generally relies on exogenous (static) stress-scenarios. Additionally, such an approach can be combined with the standard pure-stochastic approach in many different ways, e.g., constraining the loss under the stress scenarios and optimizing for the scenarios obtained by a stochastic model. For didactic purposes, we consider the case in which the decision maker maximizes the portfolio value under the stress scenarios. For each renewable generation scenario and portfolio decision-vector, , the present value . of expression (1) is minimized for each subset of periods Thus, for a given , the stress scenario for the ETC outcome within each is obtained by the following deterministic optimization problem:

(9) The main idea behind expression (9) is that for each simulated renewable generation scenario and portfolio decision-vector , the stress scenario for the spot prices is endogenously obtained for the entire contract horizon by means of sub-level optimization problems, as defined in (9). The spot-price scenarios are defined to penalize as much as possible the revenue present value of the ETC, considering a set of constraints for the spot behavior to control the degree of stress produced. A partial present value accounts for the money value over the time within each subset of periods by means of a discount rate . In this setting, means the worst-case present value of the 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 with for some arbitrary scenario and portfolio . different values for

Note that controls the conservatism of the solution in each sub-level optimization problem. For example, in a scenario , a small induces a low conservative solution because only few sub-periods can decouple from the reference and penalize the ETC's revenue. Hence, the larger the value of , the more freedom the sub-problems have to attack the portfolio, which would lead to more conservative solutions. Fig. 3 illustrates a stress spot-price scenario for a set of three subsets of periods (3 years on a monthly basis). Therefore, each subset of periods is built to represent the months of each of the three years as follows: , and , where . For illustrative purposes, a hypothetical renewable generation scenario and 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 line represents the stress scenario. In the first and third year, the worst-case spot scenario is obtained by decreasing the value of the energy during the sub-periods of the highest production surplus. In contrast, in the second year, the worst-case spot scenario is obtained by increasing the energy price during the two periods in which the renewable portfolio faces its worst production. This simple example shows the different structures for the stress scenario that resemble the well-known security criterion used in power system operation [15]. Under such interpretation, the optimized portfolio is set to withstand a spot-price spike, or deviations from the expected price pattern, in up to periods, e.g., months, in a subset of the contract horizon, e.g., a year. Finally, it is important to emphasize that the definition of the subset of periods is arbitrary in the proposed framework and constitute a modeling choice of the decision maker. This would allow for the decision maker to make use of the proposed uncertainty modeling approach for long-term strategies that would last for many accounting periods. III. HYBRID ROBUST AND STOCHASTIC MODEL The objective of the model is to determine the optimal renewable portfolio to back up a bilateral PPA sale in the forward market. Within this objective, the model determines the decision


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vector through a risk-averse trading strategy, which is driven by the maximization of the certainty equivalent of the ETC's future cash flow. We use the left-tail conditional value-at-risk (CVaR) [27] to account for the risk in a risk-adjusted objective function composed of a convex combination between the CVaR and the expected value [28]. The portfolio model is a hybrid approach between robust and stochastic optimization and can be stated as follows:

(10) (11) (12) Fig. 4. Overview of the stochastic max-min optimization approach.

(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 set of worst-case revenue scenarios obtained as the objective function of second-level minimization problems. Therefore, the proposed 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, and play the role of risk-averse parameters for the ETC. The first term, weighted by , recovers the left-tail -CVaR of the 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 pracgenerally ranges from 0.90 to 0.99. The tical applications, second term, multiplied by the complementary weight , accounts for the expected value of the net income in the objective function. For further details on the economic interpretation of the adopted risk-metric, we refer to Appendix A. Expression (11) is part of the CVaR assessment, and the net revenue of the 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 riskpreference for different subset of periods. Furthermore, in this model, the CVaR measures the risk of the ETC under a given level of modeling uncertainty defined by the PUS. Thus, both the CVaR and the PUS are two combined ways to express the agent's risk-aversion. However, the role of the former is to assess the risk through the scenarios, while the latter inputs a level of uncertainty in those scenarios to consider the fact that the model that generates spot price scenarios is only an approximation 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 optimum amounts to be contracted by the ETC. Given the portfolio selected by the first-level problem, for each simulated renewable generation scenario, the worst-case revenue stream is assessed for all sub-periods by means of the spot-price stress scenarios

following expression (9). Then, the objective function is evaluated according to (10). IV. SINGLE-LEVEL EQUIVALENT MODEL Problem (10)–(14) is a bilevel optimization problem and cannot be directly solved by commercial optimization solvers. Based on the RO approach presented in [12], an efficient maximization-equivalent formulation can be provided for (10)–(14) using the following: 1) derive the dual objective function of the linear program defined by the right-hand-side of (9). It constitutes a lower bound for the worst-case settlement term, for all dual feasible solutions; 2) derive the dual feasible constraints of the linear program defined by the right-hand side of (9); 3) replace in (10) and (11) by the dual objective function found in 1) and add in (10)–(14), the dual feasible constraints derived in 2). For an interested reader, we refer to [12] for further details on the aforementioned transformation. The equivalent single-level model for problem (10)–(14) is the following:

(15)

(16) (17) (18)

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(19)

(20)

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

(21) (22) (23) (24) (25) In (15), the dual variables of the second level problem defined by (9) are optimization variables. Expressions (16) and (19)–(25) follow steps 1) and 2), respectively. For expository purposes, expression (16) is used to facilitate the model understanding. 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 programming models with recourse [31]. In this paper the extensive form of the stochastic program was solved (no decomposition used) but we highlight our problem structure is suitable for decomposition methods, such as the Benders approach [32]. Under such decomposition framework, the problem structure could be explored to decompose the sub-problem and solve it by sub-periods and scenarios. V. CASE STUDY In this section, two case studies illustrate the applicability of the proposed methodology with realistic data from the Brazilian power sector. In the first case study, we motivate the usage and analyze the results of the portfolio model (15)–(25) as a strategic tool to define the optimal medium-term renewable portfolio to back up a one-year flat PPA. In the second case study, we consider a case where an ETC needs to define its optimal participation in a given set of renewable projects to supply a long-term PPA. In this setting, we assume a 10-year PPA, as typically required by financial institutions to provide competitive interest rates 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 back the PPA sale: a run-of-river small hydro (SH) plant with 30 MW of installed capacity and 17.4 avgMW of FEC, and a wind power plant (WPP) with 54.6 MW of installed capacity and 27.12 avgMW of FEC. Furthermore, we also assume a deterministic spot price reference and maximum/minimum deviation parameters throughout both case studies by dropping its scenario index: and . We make

use of the methodology presented in [5] to jointly simulate a set of 2000 scenarios for the renewable energy generation based on the historical data available for the SH and WPP. In Fig. 5, the average and extreme quantiles (5% and 95%) of the simulated scenarios for the renewable production are depicted for the year of 2012. For reproducibility purposes, [29] makes available 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 opportunity of avgMW by R$/MWh and that both renewable units agree to sell 100% of their FEC by R$/MWh-of-FEC. In this first case study, . Thus, . The risk-aversion parameters are set to and the effect of varying parameter for 1, 2, and 3 is studied. For comparison purposes, a pure stochastic (PS) model (similar to the one proposed in [6]) is used to provide a base-case solution to be compared with the proposed hybrid stochastic-robust (HSR) model. The PS model considers the spot price scenarios as exogenous variables produced by a least cost dispatch model based on the SDDP methodology [9]. As mentioned in Section II-C, the PS model can be obtained from the HSR model by making the reference spot-price scenarios equal to the simulated ones and . Official system data from December 2011 were used to produce the set of spot price scenarios for January 2012 to December 2012. In this case-study, the methodology used provides paired scenarios that account for the correlation between both processes (see [5] for further details).3 Fig. 6 shows the statistics for the set of simulated spot price scenarios. The reference scenario used to run the robust portfolio model (15)–(25) was set to the simulated average, shown in Fig. 6, to capture the conditional information of the spot prices throughout the contract horizon. Nevertheless, the maximum positive and negative deviations from the reference scenario were chosen to allow the stress scenario to reach the price cap (730 R$/MWh) and floor (12.1 R$/MWh), respectively, in any period. To constrain the worst-case spot-price scenarios, the maximum and minimum returns were obtained from the historical 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 scenarios is case-dependent and out of the scope of this paper (they are consider as input data in our model).


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

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

TABLE I ROBUST PORTFOLIO MODEL INPUT DATA FOR CASE STUDY I

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

TABLE II OPTIMAL CONTRACTING OF EACH CANDIDATE OPTION IN THE STOCHASTIC AND HYBRID STOCHASTIC-ROBUST CASE (AVGMW)

and renewable production for both plants in each month and scenario). We have solved (15)–(25), and the optimal amount sold and contracted of each technology is shown in Table II. Each problem was solved in less than one minute using a Dell Inspiron 15R Special Edition Laptop. We observed that when the spot price is treated as an exogenous variable (second column of Table II), the optimal strategy for the trader is to back its sales solely on the wind generator. One way to interpret this result is to observe that the wind production pattern is similar to the spot price one, i.e., high spot price seasons coincide, in general, with high wind power production (see Figs. 5 and 6 for better comparison). In contrast, hydro production presents almost full complementarity with 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 supporting it by a mixed portfolio that is composed of both wind and small hydro. This result is due to the stress-scenarios for

the spot prices that penalize portfolios with high exposure to the spot market, which occurs whenever the portfolio generation is below 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 generation observed in 2012. In this year, spot prices lie within the 5th and 95th quantiles. However, the observed price for 2012 can be considered as atypical (when compared to the historical profile) since we have experienced an unusual spot-price dynamic (path through the year) with two price peaks, in April and November (see Fig. 8). As a conclusion, the role of the price dynamic is also considered when building the optimal HSR portfolio, because in (9), the stress-scenario is conceived to attack the portfolio with the worst-case path within a subset of periods. Therefore, the robust approach also guards the portfolio against the occurrence of 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 generation profile throughout the months. The effect of such mitigation is shown in Fig. 8, where the impact of the first semester spot-price disturbance (April and May) is attenuated under the robust solutions. Moreover, the excess of capacity contracted by the HSR model 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 to not 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,

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

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

the solution obtained with both models were evaluated under the observed generation and prices for 2012 and for two other representative years: 2010 (a typical year with respect to spot price realization—lower values at the beginning of the year and high values at the end) and for 2008 (an unusual year—a price spike 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 outcome for this year realized as “expected” in accordance with the pattern of the simulated scenarios. However, when the spot price realization presents a different pattern with respect to the price scenarios simulated by the stochastic model, the hybrid stochastic-robust model outperforms the pure stochastic one. We highlight that the HSR model outperforms the PS one when the spot price increases in the last months of the year as well, indicating that the robust counterpart was also able to capture the effects of “standard” and “normal” pattern of prices. Finally, in Fig. 9, convergence analysis is performed by analyzing the convergence of the optimal-primal variables (decision vector as a function of the number of scenarios . , are shown since In this figure, only the buying variables, is the same (equal to 10 avgMW) the optimal solution of for all cases. Note that the optimal solutions stabilize when the number of scenarios meets 1100. This suggests that convergence is achieved. This fact provides strong evidences that the solution obtained, which make use of the 2000 scenarios, converged and the results presented are accurate with respect to the asymptotic probability distribution. B. Case Study II: Long-Term Portfolio Strategy In this second case study, we apply the HSR model to the same units (wind and SH) considered before. However, we now consider an ETC deciding its willingness to invest in

TABLE IV ROBUST PORTFOLIO MODEL INPUT DATA FOR CASE STUDY II. ALL MONTHLY DATA ARE THE SAME FOR ALL YEARS

both units. We consider that the ETC has the same price and quantity selling opportunity described in the previous case study but for a 10-year PPA horizon, from January 2016 to is the set December 2025. In this case, of years and is the collection of sets containing the months of each year , i.e., ; and . In this case study, the risk-aversion parameters are set to be constant: , and for 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 structural data inputs. The proposed approach provides the decision maker with a tool consistent with well-known stress analysis that is able to consider the information available for the stochastic nature of the renewables generation. The model developed in [5] was used to simulate the joint wind and SH production scenarios throughout the contract horizon, and the data presented in Table IV were used to constrain the related spot-price stress scenarios, in which the same sequence of monthly data (all data, including the renewable are used for each years and scenario production for both plants in each month , 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 a and 2. 0.05 step-basis. Fig. 10 shows the curve for The vertical axis represents the present value (PV) of the expected value of the 10-year revenue and the horizontal axis stands for risk. The latter is evaluated as the difference between the PV of the expected value and the PV of the CVaR % , i.e., the second and first terms of expression (10), respectively (see Appendix B for further details about the development of a risk metric in the multi-period setting). and , the risk is higher than For the expected return, indicating that the PV of the CVaR % is , the investment negative. This fact suggests that under decision in renewables requires high risk aversion to mitigate


Fig. 10. Efficient frontier curve for 0.5 to 0.99 on a 0.05 step-basis.

and 2 varying the parameter

TABLE V OPTIMAL CONTRACTING OF EACH CANDIDATE OPTION FOR THE 10-YEAR CASE STUDY FOR DIFFERENT RISK LEVELS OF

from

AND 2 IN (AVGMW)

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Fig. 11. Sensitivity analysis of the optimal solution of the HSR model with and 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 the results are accurate with respect to the asymptotical probability distribution. VI. CONCLUSION

the solvency risk (chance of a negative PV). Table V depicts the portfolio decisions associated with each point of the efficient frontier curve. Observe that in Table V, as the risk-averse parameter grows, the excess of energy bought by the ETC also grows, representing a hedge against the price and quantity risk. These results follow the same idea explained in Case Study I. As the parameter grows, price spikes can attack the portfolio in different periods. In this context, the optimal contracting model aims to create a flatter generation profile, bringing more hydro generation to complement the portfolio, thus reducing the exposure to the short-term market. In general, the efficient frontier varies with respect to the conservativeness level of the robust counterpart, i.e., with . In this sense, to create a long-term portfolio, the decision maker should well-adjust its risk preference to the parameters of the model proposed because the solution obtained is highly dependent on these parameters. For example, a 0.05 variation on creates a different portfolio and the relationship between risk and expected revenue. At last, in Fig. 11, we analyze the convergence of the primal on the number of scenarios considvariables solution ered in the optimization problem for the case of . Again, only the buying variables, , are shown since the opis the same (equal to 10 avgMW) for all timal solution of cases.

This work established a new methodology to determine the risk-constrained optimal portfolio of renewable sources to support a firm selling in the forward market. Such a model assesses the optimal amount to trade in contracts by the ETC, given their respective specifications (prices, starting dates, durations, etc.), which is robust with respect to unexpected variations in spot prices but that also considers the stochastic nature of the production of renewable assets in a risk-constrained setting. This approach provides an alternative to current models based on the simulation of prices. Additionally, it incorporates a generalization of stress analysis to account for the modeling uncertainty in the optimal renewable portfolio model by means of endogenously defined stress scenarios. Such an approach provides the decision maker with an intuitive way to control the level of stress produced by the stress scenarios following the idea of well-known deterministic security criteria such as the . We illustrated the methodology to devise a portfolio of renewable sources (wind and small hydros) in the Brazilian system, where the approach of this paper was contrasted with its pure-stochastic counterpart and both are benchmarked against actual (observed) market variables. Ongoing research related to this paper includes a definition of a stochastic conservatism parameter correlated with the generation of the plants, the use of stochastic spot reference prices to account for ambiguity aversion. The inclusion of new financial products in the model—such as call options—is also a relevant topic of research due to the nonlinearities introduced on the model.

APPENDIX A RISK-METRIC'S ECONOMIC INTERPRETATION Expression (10) stands for a composite objective function that accounts for both the CVaR and the expected value of the ETC's cash flow. The objective function in (10) can be understood as

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the ETC's certainty equivalent (CE), since, in the optimal solution, the value of the objective function is the amount of money at the beginning of the study horizon with which the ETC is indifferent, with respect to the objective function of (10), with the optimal portfolio (we refer to [28] for further details on CVaR and CE). This is due to the fact that both the CVaR and the expected value applied to a “deterministic” number, which is equivalent to a degenerated random variable with all scenario assuming 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 the intertemporal preference of the agent? If the agent agrees that his preference functional of one period is the convex combination between the CVaR and the expected value, following previous reported works (see [5]–[7], [28], and references therein), in the multi-period setting one has two possibilities: 1) apply this composite measure to the present value of the random cash flow or 2) apply such measure to the random revenue of each period and then assess the present value of the resultant deterministic cash flow [33]. Although not typical, such composite measure has 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 preserving the temporal correlation of the cash flow stream, on the other hand, it is unable to capture the risk-preference between periods. For instance, consider two stochastic cash flows, namely C1 and C2, of two years and two scenarios each with 50%–50% of probability: in the first cash flow, C1, both scenarios are equal to zero $ for both years; in the second one, C2, scenario one values Billion $ in the first year and 1.1 Billion $ in the second year, and the second scenario is built with the negative values of the first scenario. Assuming , and typical risk-averse parameters, such as and , approach 1) is indifferent between C1 and C2, and, clearly, approach 2) prefers C1 rather than C2.4 One can argue that the hell-and-heaven dynamic present in C2 makes C1 preferable than C2 in practical applications because (i) companies do not have access to unlimited leverage to withstand hell and achieve heaven if the first scenarios is observed and (ii) the fruits of heaven would be shared among investors before hell takes place in the second year if the second scenario materializes. This simple example illustrates that a practical approach should lie in between of approaches 1) and 2). Moreover, it should reflect the company's financial access to external leverage and the time window with which the company performance is evaluated. In this work, we propose a metric that generalizes the aforementioned approaches. By selecting the subsets of periods, , both approaches can be obtained as well as intermediate approaches with trimestral or semiannual accounting periods. APPENDIX B RISK METRIC IN THE MULTI-PERIOD SETTING Expression (10) stands for a convex combination between the PV of the CVaR of each subset of periods and the PV of the 4This happens because, under the second approach, the CVaR assumes the value of the worst-case scenario in each period.

expected value. In the optimal value, the objective function can be written as follows:

(26) This expression can be rearranged in order to achieve the following bi-criterion penalization form:

(27) In (27), the first term accounts for the expected value of the project and the second term for a multi-period measure of risk (negative deviation from the expected value), following the agent's chosen approach to measure risk in the multi-period setting (see Appendix A for further details). If the risk-aversion parameters are constant for all , (27) is equivalent to the PV of the expected cash flow penalized by the following multi-period measure of risk:

(28) It is important to notice that (27) can be seen as the Lagrange dual function of the risk constrained version of this problem where only the first term of this expression is maximized subject to a constraint on the maximum value of expression (28). Therefore, by solving model (15)–(25) for different values of between 0 and 1, one can draw an efficient frontier by plotting on the horizontal axis the risk measures associated with each value of and the corresponding expected value terms on the vertical axis. ACKNOWLEDGMENT The authors would like to thank FICO (Xpress-MP developer) for the academic partnership program with the Electrical Engineering Department of Pontifical Catholic University of Rio de Janeiro, Brazil (PUC-Rio). The authors would also like thank the LAMPS researchers for the daily exchanges and their insightful considerations. REFERENCES [1] S. Hunt, Making Competition Work in Electricity. New York, NY, USA: Wiley Finance, 2002, vol. I. [2] N. Yu, L. Tesfatsion, and C. Liu, “Financial bilateral contract negotiation in wholesale electricity markets using Nash bargaining theory,” IEEE Trans. Power Syst., vol. 27, no. 1, pp. 251–267, Feb. 2012. [3] C. Vásquez, M. Rivier, and I. J. Pérez-Arriaga, “A market approach to long-term security of supply,” IEEE Trans. Power Syst., vol. 17, no. 3, pp. 349–357, 2002, Aug.. [4] L. A. Barroso, J. Rosenblatt, A. Guimaraes, B. Bezerra, and M. V. Pereira, “Auctions of contracts and energy call options to ensure supply adequacy in the second stage of the Brazilian power sector reform,” in Proc. IEEE PES General Meeting 2006, Montreal, QC, Canada, Jun. 2006, pp. 1–8.


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[28] A. Street, “On the conditional value-at-risk probability-dependent utility function,” Theory and Decision, vol. 68, no. 1–2, 2010. [29] Dataset used in the case studies [Online]. Available: http://goo.gl/ hpy4CT [30] R. Green, “Competition in generation: The economic foundations,” Proc. IEEE, vol. 88, no. 2, pp. 128–139, Feb. 2000. [31] J. R. Birge and F. Louveaux, Introduction to Stochastic Programming, 2nd ed. New York, NY, USA: Springer, 2011. [32] D. Bertsimas and J. Tsitsiklis, Introduction to Linear Optimization. Nashua, NH, USA: Athena Scientific, 1997. [33] D. Luenberger, Investment Science. Oxford, U.K.: Oxford Univ. Press, 1998. [34] A. Ben-Tal, A. Goryashko, E. Guslitzer, and A. Nemirovsk, “Adjustable robust solutions of uncertain linear programs,” Math. Program., vol. 99, no. 2, pp. 351–376, Mar. 2004. [35] B. Zeng and L. Zhao, “Solving two-stage robust optimization problems using a column-and-constraint generation method.,” Oper. Res. Lett., vol. 41, no. 5, pp. 457–461, Sep. 2013. [36] H. Föllmer and A. Schied, Stochastic Finance: An Introduction in Discrete Time. New York, NY, USA: De Gruyter, 2011, vol. III. [37] L. Baringo and A. J. Conejo, “Risk-constrained multi-stage wind power investment,” IEEE Trans. Power Syst., vol. 28, no. 1, pp. 401–411, Feb. 2013. [38] G. Pflug, “Some remarks on the value-at-risk and the conditional valueat-risk,” Probab. Constrain. Optimiz.: Methodol. Applicat., vol. 49, no. 1, pp. 272–281, 2000. [39] B. Rudloff, A. Street, and D. M. Valladão, “Time consistency and risk averse dynamic decisions models: Definition, interpretation and practical consequences,” Eur. J. Oper. Res., vol. 234, no. 3, pp. 743–750, May 2014.

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

Alexandre Street (S’06–M’10) received the M.Sc. degree and the D.Sc. degree in electrical engineering (operations research) from the Pontifical Catholic University of Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil. From 2003 to 2007, he participated in several projects related to strategic bidding in the Brazilian energy auctions and market regulation at the Power System Research Consulting (PSR), Rio de Janeiro, Brazil. From August 2006 to March 2007, 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 optimization 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 in mathematics 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 systems economics focusing on hydrothermal systems. He has been a lecturer in Latin America, Europe, and the United States/Canada. Dr. Barroso was the recipient of the 2010 IEEE PES Outstanding Young Engineer Award and is currently an Associate Editor of the IEEE TRANSACTIONS ON POWER SYSTEMS and the IEEE TRANSACTIONS ON SMART GRID.