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In certain electricity markets, because of non-convexities that arise from their operating characteristics, generators that follow the independent system operator's (ISO's) decisions may fail to recover their cost through sales of energy at locational marginal prices. The ISO makes discriminatory side payments to incentivize the compliance of generators. Convex hull pricing is a uniform pricing scheme that minimizes these side payments. The Lagrangian dual problem of the unit commitment problem has been solved in the dual space to determine convex hull prices. However, this approach is computationally expensive. We propose a polynomially-solvable primal formulation for the Lagrangian dual problem. This formulation explicitly describes for each generating unit the convex hull of its feasible set and the convex envelope of its cost function. We cast our formulation as a second-order cone program when the cost functions are quadratic, and a linear program when the cost functions are piecewise linear. A 96period 76-unit transmission-constrained example is solved in less than fifteen seconds on a personal computer.

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This paper proposes an efficient method for evaluating composite system reliability via subset simulation. The central idea is that a small failure probability can be expressed as a product of larger conditional probabilities, thereby turning the problem of simulating a rare failure event into several conditional simulations of more frequent intermediate failure events. In existing methods, system states are simply assessed in a binary secure/failure manner. To fit into the context of subset simulation, the adequacy of system states is parametrized with a metric based on linear programming, thus allowing for an adaptive choice of intermediate failure events. Samples conditional on these events are generated by Markov chain Monte Carlo simulation. The proposed method requires no prior information before imulation. Different models for renewable energy sources can also be accommodated. Numerical tests show that this method is significantly more efficient than standard Monte Carlo simulation, especially for simulating rare failure events.