### Abstract

Increasing penetration of volatile wind-based generation into the fuel mix is leading to growing supply-side volatility. As a consequence, the reliability of the power grid continues to be a source of much concern, particularly since the impact of supply-side risk exposure, arising from aggressive bidding,1 is not felt by risk-seeking generation firms; instead, the system operator is largely responsible for managing shortfalls in the real-time market. We propose an alternate design in which the cost of such risk is transferred to firms responsible for imposing such risk. The resulting strategic problem can be cast as a two-period generalized stochastic Nash game with shared strategy sets. A subset of equilibria is given by a solution to a related stochastic variational inequality, that is shown to be both monotone and solvable. Computing solutions of this variational problem is challenging since the size of the problem grows with the cardinality of the sample space, network size and the number of participating firms. Consequently, direct schemes are inadvisable for most practical problems. Instead, we present a distributed regularized primal-dual scheme and a dual projection scheme where both primal and dual iterates are computed separately. Rates of convergence estimates are provided and error bounds are developed for inexact extensions of the dual scheme. Unlike projection schemes for deterministic problems, here the projection step requires the solution of a possibly massive stochastic programme. By utilizing cutting plane methods, we ensure that the complexity of the projection scheme scales slowly with the size of the sample space. We conclude with a study of a 53-node electricity network that allows for deriving insights regarding market design and operation, particularly for accommodating firms with uncertain generation assets.

Original language | English (US) |
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Pages (from-to) | 1095-1138 |

Number of pages | 44 |

Journal | Optimization Methods and Software |

Volume | 28 |

Issue number | 5 |

DOIs | |

State | Published - Oct 1 2013 |

### Keywords

- Nash games
- cutting plane methods
- projected gradient schemes
- stochastic programming
- variational inequalities

### ASJC Scopus subject areas

- Software
- Control and Optimization
- Applied Mathematics

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## Cite this

*Optimization Methods and Software*,

*28*(5), 1095-1138. https://doi.org/10.1080/10556788.2012.676756