A Cooperative Game for the Realized Profit of an Aggregation of Renewable Energy Producers Pratyush Chakraborty, Enrique Baeyens, Pramod P. Khargonekar, and Kameshwar Poolla Abstract— The aggregation of renewable energy has signif- icant potential to mitigate undesirable characteristics such as intermittency and variability and thereby facilitate grid inte- gration. Using cooperative game theory, it has been shown that aggregation is also beneficial for renewable energy producers because they can increase their expected profit by making a coalition, bidding a joint contract that maximizes the expected profit and sharing the profit in a way that keeps the game stable. However, we show that the realized (as opposed to expected) profit of the coalition, using the contract that maximizes the expected profit, cannot be suitably distributed among its members. We propose an alternative coalition contract and prove that it allows for a satisfactory distribution of the realized profit among the coalition members keeping the game stable. We design a new payoff allocation that lies in the core of the game of the realized profit. Finally, we analyze the cost of stabilizing the game by evaluating the loss of expected profit that a coalition incurs by bidding the stabilizing contract. I. I NTRODUCTION Sustainable social and economic development requires affordable access to the energy resources necessary to meet basic human needs and to serve productive processes without jeopardizing wellbeing of future generations. However, most of the current energy driving global economies comes from the combustion of fossil fuel that accounts for 56.6% of the greenhouse gas emissions [1]. Renewable energy resources, particularly wind and solar, offer a promising potential to reduce greenhouse gas emissions by displacing fossil fuel sources for electricity production. The main challenge for large-scale integration of wind and solar generation into the electric grid lies in their inherent variability, uncertainty, and nondispatchability [2]–[4]. The aggregation of geographically disperse sources offers a solution to the variability of renewable energy. Wind and solar power are usually negatively correlated in geographi- cally near locations, while wind power tends to decorrelate as the distance increases [5]. Besides, aggregation provides economical benefit for the producers of renewable energy. The economic benefit of wind power aggregation was studied This work is supported in part by NSF grants ECCS-1129061 and CNS- 1239274 Pratyush Chakraborty is with the Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL, USA pchakraborty@ufl.edu Enrique Baeyens is with Instituto de las Tecnolog´ ıas Avanzadas de la Producci´ on, Universidad de Valladolid, Valladolid, Spain enrbae@eis.uva.es Pramod P. Khargonekar is with the Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL, USA ppk@ufl.edu Kameshwar Poolla is with the Department of Mechanical Engineering, University of California, Berkeley, CA, USA poolla@berkeley.edu in a two-settlement market setting in [6] using coopera- tive game theory. The expected profit of a coalition can always be improved if a set of independent wind power producers decide to make a joint bid for a given time interval. It was also proved that there always exists a fair payoff allocation of the expected profit that is satisfactory for every coalition member. However, the expected profit is a theoretical quantity and the actual realized profit that is obtained in any given run can be very different from the expected profit. It is possible to devise payoff allocation mechanisms of the realized profit in such a way that the payment that each member receives approaches almost surely a suitable payoff allocation as the number of time intervals increases. Nevertheless, there are two reasons that can hinder the formation or continued operation of stable coalitions. The first reason is that the coalition members need to share and agree upon statistical models of their production. Second, the realized profit could turn out to be very different from the expected value during many time intervals over a long period. In such a case, one or more renewable energy producers may become dissatisfied with the profit payoff allocation mechanism and decide to abandon the coalition. To deal with these issues, some researchers have proposed a competitive game approach [7], [8]. In this paper, we continue to develop our previous co- operative game framework in the setting of realized profit. We propose a new joint contract that ensures long-term stable coalitions because it provides a superadditive value for the cooperative game of the realized profit. Consequently, the realized profit obtained by a coalition bidding the new stabilizing contract in the two-settlement market is greater than that obtained by the coalition members if they act independently trying to improve their realized profits. The cooperative game is also balanced. Besides, the producers do not need to agree on a common statistical framework. Each of them only uses statistical information about its own production and need not share it with the rest of the coalition members. In addition, we are able to design a payoff allocation for the game of the realized profit under the stabilizing contract that does not require to solve any linear program. However, the new stabilizing contract has a drawback. Since it does not maximize the expected profit of the coalition, there is a loss as compared to maximum expected profit. We are able to quantify this loss. The remainder of this paper is organized as follows. A brief review of cooperative game theory, a summary of previous results about aggregation of renewable energy using cooperative games, its limitations and the formulation of 2016 IEEE 55th Conference on Decision and Control (CDC) ARIA Resort & Casino December 12-14, 2016, Las Vegas, USA 978-1-5090-1837-6/16/$31.00 ©2016 IEEE 5805