Orbitopal fixing for the full (sub-)orbitope and application to the Unit Commitment Problem

It is common knowledge that symmetries arising in integer programs could impair the solution process, in particular when symmetric solutions lead to an excessively large branch and bound (B&B) search tree. Techniques like isomorphic pruning [11], orbital branching [16] and orbitopal fixing [8] have been shown to be essential to solve very symmetric instances from … Read more

On the complexity of the Unit Commitment Problem

This article analyzes how the Unit Commitment Problem (UCP) complexity evolves with respect to the number n of units and T of time periods.A classical reduction from the knapsack problem shows that the UCP is NP-hard in the ordinary sense even for T=1. The UCP is proved to be strongly NP-hard.When either a unitary cost … Read more

The Min-up/Min-down Unit Commitment polytope

The Min-up/min-down Unit Commitment Problem (MUCP) is to find a minimum-cost production plan on a discrete time horizon for a set of fossil-fuel units for electricity production. At each time period, the total production has to meet a forecasted demand. Each unit must satisfy minimum up-time and down-time constraints besides featuring production and start-up costs. … Read more

Nash Equilibrium in a Pay-as-bid Electricity Market: Part 1 – Existence and Characterisation

We consider a model of a pay-as-bid electricity market based on a multi-leader-common-follower approach where the producers as leaders are at the upper level and the regulator as a common follower is at the lower level. We fully characterise Nash equilibria for this model by describing necessary and sufficient conditions for their existence as well … Read more

Nash Equilibrium in a Pay-as-bid Electricity Market: Part 2 – Best Response of a Producer

We consider a multi-leader-common-follower model of a pay-as-bid electricity market in which the producers provide the regulator with either linear or quadratic bids. We prove that for a given producer only linear bids can maximise his profit. Such linear bids are referred as the “best response” of the given producer. They are obtained assuming the … Read more