Abstract-A numerical method based on a relaxation algorithm and the Nikaido-Isoda function is presented for the calculation of Nash-Cournot equilibria in electricity markets. Nash equilibrium is attained through a relaxation procedure applied to an objective function, the Nikaido-Isoda function, which is derived from the existing profit maximization functions calculated by the generating companies. We also show how to use the relaxation algorithm to compute, and enforce, a coupled constraint equilibrium, which occurs if regulatory, generation, and distribution (and more) restrictions are placed on the companies and entire markets. Moreover, we use the relaxation algorithm to compute players' payoffs under several player configurations. This is needed for the solution of our game under cooperative game theory concepts, such as the bilateral Shapley value and the kernel. We show that the existence of both depends critically on demand price elasticity. The numerical method converges to a unique solution under rather specific but plausible concavity conditions. A case study from the IEEE 30-bus system, and a three-bus bilateral market example with a dc model of the transmission line constraints are presented and discussed.

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We study a trade-off between economic and environmental benefits using a two-stage optimal control setting where the player can switch to a cleaner technology that is environmentally 'efficient' but economically less productive. We provide an analytical characterization of the solution paths for the case where the considered utility functions are increasing and strictly concave with respect to consumption and decreasing linearly with respect to the pollution stock. We establish that in this context, an isolated player will either immediately start using the cleaner technology or for ever continue applying the old 'dirty' technology. In a two-player dynamic game (between two neighboring countries) where the pollution results from a sum of two consumptions, we prove existence of a Nash (open-loop) equilibrium, in which each player chooses the technology selfishly, i.e., without considering the choice made by the other player. A Stackelberg game solution displays the same properties. Under cooperation, the country reluctant to adopt the clean technology under autarky will adopt the cleaner technology provided it benefits from some 'transfer' from the more environment-friendly partner. ‡ All formulae derivations are available upon request from the authors. § Notice that, given the law of motion of the stock of pollution, and P(0) 0, we have necessarily P(t) 0, ∀t 0. Hence P(T )>0. ¶ We have not tried nonlinear natural decay schemes.

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