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Power producers have to take decisions concerning their bidding strategy in the presence of uncertain prices. The paper addresses the problem of generator self-scheduling when locational marginal prices are subject to ellipsoidal uncertainty. A robust counterpart of the uncertain self-scheduling problem is formulated as a second-order cone program and solved using a commercially available interior-point package. The safety parameter, which trades between profit and risk, is determined based on a prespecified chance that the realised loss would be greater than the corresponding value at risk. Simulation results are presented on the IEEE 30-bus test system. A Monte Carlo simulation is used to compare the stability properties of the schedules obtained.