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A convex primal formulation for convex hull pricing
Bowen Hua and Ross Baldick, University of Texas at Austin
Unit commitment and dispatch of generation in electricity markets involves the ISO sending target quantity instructions to each generator. Ideally, energy prices provide incentives for profit maximizing market participants to comply with efficient commitment and dispatch instructions in the short-term, and to develop new generation (based on anticipation of future energy prices) at the right place and time over the long-term. Various issues prevent this ideal, including the non-convexity of the underlying unit commitment problem, so that start-up and no-load costs of generation units may not be covered by sales of energy at locational marginal prices. To encourage generators to comply with commitment and dispatch, non-negative profit is guaranteed by the ISO, but this guarantee necessitates non-anonymous “uplift” payments. Because energy prices alone do not provide all the information about profitability in the market, incentives for new generation to enter the market are weakened. Moreover, the energy prices do not reveal to demand the full cost of providing energy, weakening incentives for demand response. Both issues arguably contribute to resource adequacy problems. Convex hull pricing is a uniform pricing scheme that minimizes the amount of uplift payments. The Lagrangian dual problem of the unit commitment and economic dispatch problem has been used to determine the convex hull prices. This approach is computationally expensive for market implementations. We propose a convex programming formulation for convex hull pricing that is polynomially solvable. We describe explicitly the convex hull of individual units’ feasible commitment and dispatch decisions. We show that our formulation leads to the exact convex hull prices when ramping constraints are not present, and that the exactness is preserved when ancillary services or any linear system-wide constraints (such as the transmission constraints) are considered. When ramping constraints are considered, an exponential number of valid inequalities are needed to describe the convex hulls. In this case, we use a tractable approximation. Numerical tests are conducted on several examples found in the literature. Although this work provides for convenient implementation of convex hull pricing, significant issues remain. For example, the incentives for generators to reveal truthful start-up and no-load information are not well understood under either the traditional uplift approach or convex hull pricing. Moreover, under convex hull prices, profit maximizing generators have incentives to deviate from the ISO-determined dispatch instructions.
