IEEE TRANSACTIONS ON POWER SYSTEMS 1 A Hybrid MILP and Benders Decomposition Approach to Find the Nucleolus Quota Allocation for a Renewable Energy Portfolio Lucas Freire, Student Member, IEEE, Alexandre Street, Member, IEEE, Delberis A. Lima, Member, IEEE, and Luiz Augusto Barroso, Senior Member, IEEE Abstract—Portfolios of renewable electricity sources are inter- esting risk-management mechanisms for trading in electricity con- tract markets. When formed by players belonging to different com- panies, their stability relies on the way the beneｿt generated by the optimal portfolio is allocated. The challenge of ｿnding a fair and efｿcient allocation can be mathematically formulated in terms of ｿnding the Core of a cooperative game, which in turn is stated as an optimization problem with a set of constraints that exponentially grows with the number of participants, quickly becoming compu- tationally intractable. Moreover, the right-hand-side of each con- straint relies on a given coalition value, which in our case is ob- tained by a two-stage stochastic optimization model. This paper presents an efｿcient methodology based on mixed —linear pro- gramming and Benders decomposition to ｿnd the Nucleolus share of large-scale renewable portfolios. Case studies are presented with data from the Brazilian power system. Index Terms—Benders decomposition, cooperative game, nucle- olus, risk-aversion, renewable energy pool. NOMENCLATURE This section lists the main notation used throughout the paper. Additional symbols with subscripts “ ” and “ ” are used to indicate the value of a speciｿc variable at iterations and , respectively. Functions Worst-case gain as a function of the coalition vector . Stochastic net revenue of a pool of renewable generators in period as a function of the contract amount . Stochastic net revenue in both contract and spot markets in period as a function of the contract amount and coalition vector . Characteristic function . It assigns to each coalition vector a real number that measures the collective value of its optimal selling strategy. Manuscript received June 11, 2014; revised October 12, 2014; accepted November 10, 2014. This work was supported in part by UTE Parnaiba Geração de Energia S.A. through R&D project ANEEL PD-7625-0001/2013. Paper no. TPWRS-00794-2014. L. Freire, A. Street, and D. A. Lima are with the Electrical Engineering De- partment, PUC-Rio, Rio de Janeiro, Brazil (e-mail: lucasfreire1@gmail.com; street@ele.puc-rio.br; delberis@ele.puc-rio.br). L. A. Barroso is with PSR, Rio de Janeiro, Brazil (e-mail: luiz@psr-inc.com). Digital Object Identiｿer 10.1109/TPWRS.2014.2374532 Constants Unitary production cost of unit in period in /MWh. Total amount of FEC issued by the regulator to generator in average-MW. Amount of energy produced by the generating unit , in period , and scenario in MWh. Number of hours in period . The risk-free opportunity cost of money between two periods (%). Lower bound for the worst-case gain in the th iteration of the Benders algorithm. Number of players in the grand coalition or renewable pool. Probability of scenario . Price of the ｿnancial contract in /MWh. Upper bound for the worst-case gain in the th iteration of the Benders algorithm. Left-tail percentile of the future stochastic net revenue probability distribution. Tolerance level for the difference between Upper and Lower bounds of the worst case gain. Risk-aversion parameter for the risk measure function. Spot price at period and scenario in /MWh. Random Variables Amount of energy produced by the generating unit during period in MWh. Spot price at period in /MWh. Decision Variables Amount of energy sold in ｿnancial contracts in average-MW. -th element of the binary vector that deｿnes a given coalition. It assumes value 1 if player belongs to such coalition and 0 otherwise. Percentage of the future stochastic net revenue of the pool allocated to player . Auxiliary variable that accesses the -Value-at-Risk of the probability distribution function of the revenue. 0885-8950 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.