  Strategic bidding problems in electricity markets are widely studied in power
systems, often by formulating complex bi-level optimization problems that are
hard to solve. The state-of-the-art approach to solve such problems is to
reformulate them as mixed-integer linear programs (MILPs). However, the
computational time of such MILP reformulations grows dramatically, once the
network size increases, scheduling horizon increases, or randomness is taken
into consideration. In this paper, we take a fundamentally different approach
and propose effective and customized convex programming tools to solve the
strategic bidding problem for producers in nodal electricity markets. Our
approach is inspired by the Schmudgen's Positivstellensatz Theorem in
semi-algebraic geometry; but then we go through several steps based upon both
convex optimization and mixed-integer programming that results in obtaining
close to optimal bidding solutions, as evidenced by several numerical case
studies, besides having a huge advantage on reducing computation time. While
the computation time of the state-of-the-art MILP approach grows exponentially
when we increase the scheduling horizon or the number of random scenarios, the
computation time of our approach increases rather linearly.
