Projected solution of a pay-as-bid electricity market model

Abstract This paper aims to provide a detailed study of an electricity market model initially presented by Aussel et al. (2016). This model can be formulated as a multi-leader-common-follower game, in which N producers act as leaders and an independent system operator acts as follower. The revenue bid function on which each producer operates belongs to a subset of $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> , and this generates computational difficulties due to the nonsmoothness of the functions. It also means that the solution of the follower is not guaranteed to be unique. For this reason, Aussel et al. (2016) approximate the bid function of each leader with a quadratic function and introduce the concept of a projected solution. This paper proves the existence of a projected solution to the model when $$N=2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> and, under suitable assumptions, for $$N\ge 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> as well. Furthermore, thanks to the potential structure of the leaders’ game, the projected solution can be determined using KKT conditions. Some numerical tests witness the validity of the approach.