In this study, we consider a collusion model for competitive pool based electricity markets operated by an independent system operator (ISO), where it aims to prevent tacit collusion among generators. In order to determine the existence of tacit collusion in the market, we have employed the the game-theoretic bilevel optimization model proposed by [1]. This model represents the market clearing mechanism, where generators determine their bids in order to maximize their profit while the system operator allocates power and determine locational electricity prices. The resulting optimization problem is a bilevel multi-criteria problem with non-linear terms, which is already complex and difficult to solve. We provide reformulations and linearization methods to obtain equivalent problems, e.g., a mathematical problem with equilibrium constraints (MPEC), a mixed integer non-linear problem (MINLP) and a mixed integer problem (MIP). A simple 6-bus system is used to test the rate of detection for collusive states for the reformulations and the results are presented. We have found that MIP models and reformulations detect strong collusive states. This model can guide ISOs in identifying and preventing cases of generator collusion.