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An algorithm that allows a market participant to maximize its individual welfare in electricity spot markets is presented. The use of the algorithm in determining market equilibrium points, called Nash equilibria, is demonstrated. The start of the algorithm is a spot market model that uses the optimal power flow (OPF), with a full representation of the transmission system and inclusion of consumer bidding. The algorithm utilizes price and dispatch sensitivities, available from the Hessian matrix and gradient of the OPF, to help determine an optimal change in an individual's bid. The algorithm is shown to be successful in determining local welfare maxima, and the prospects for scaling the algorithm up to realistically sized systems are very good. Nash equilibria are investigated assuming all participants attempt to maximize their individual welfare. This is done by iteratively solving the individual welfare maximization algorithm until all individuals stop modifying their bids.