A Convex Primal Formulation for Convex Hull Pricing

Abstract

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.

Impacts of Price Formation Efforts Considering High Renewable Penetration Levels and System Resource Adequacy Targets

“…The resulting prices and payments only minimize the combination of make-whole payments and lost opportunity costs (i.e., they do not minimize make-whole payments alone) and are not guaranteed to be revenue adequate. After originally implementing an approximation to Convex Hull Pricing, Midcontinent ISO (MISO) has updated the formulation to include better approximations (Wang et al 2013;Hua and Baldick 2017). Implementation of full Convex Hull Pricing is immensely computationally difficult (Schiro et al 2016;Zheng et al 2018), which is why ISOs have approximated the method for implementation.…”

Hybrid Automatic Lung Segmentation on Chest CT Scans

“…Because there are many hollow parts in the extracted lung region, it is impossible to directly extract the outer contour of the lung by the contour extraction method [30], therefore the lung region obtained by the Otsu and PSCCL can be treated as a point set. We use the CH [31] to find the extreme points from the disordered point set. The principle is to calculate the direction of the intersection of the two vectors by Graham scan [32] and find all the vertexes along the boundary of the convex hull.…”

Correction

“…We can observe that our extended formulation not only describes an integral polytope with respect to variables ðy, u, a, b, c, hÞ due to Theorem 1, but also provides an optimal objective converging to that of the deterministic single-UC problem (1) when f ðÁÞ is a general convex function, as the number of line segments increases, due to the compactness of the feasible region and bounded objective value for the single-UC problem. Furthermore, since f ðÁÞ is a quadratic function in general, an alternative way to obtain an optimal solution that is binary with respect to variables ðy, u, a, b, c, hÞ under the quadratic function f ðÁÞ is to utilize the convex envelope of f t ðx t , y t Þ as described in [3] (Theorem 3) and the integral polytope proved in our Theorem 1, because Theorem 1 provides an explicit description of the convðX g Þ defined in [3]. Thus, following [3], the convex envelope of our original formulation (9) in [2] when w s tk is a quadratic function (e.g., w s tk ðq s tk , b tk Þ ¼ a s tk ðq s tk Þ 2 þ b s tk q s tk þ c s tk b tk ) can be described as…”

“…Furthermore, since f ðÁÞ is a quadratic function in general, an alternative way to obtain an optimal solution that is binary with respect to variables ðy, u, a, b, c, hÞ under the quadratic function f ðÁÞ is to utilize the convex envelope of f t ðx t , y t Þ as described in [3] (Theorem 3) and the integral polytope proved in our Theorem 1, because Theorem 1 provides an explicit description of the convðX g Þ defined in [3]. Thus, following [3], the convex envelope of our original formulation (9) in [2] when w s tk is a quadratic function (e.g., w s tk ðq s tk , b tk Þ ¼ a s tk ðq s tk Þ 2 þ b s tk q s tk þ c s tk b tk ) can be described as…”

“…An independent work compared with [3] is available in [1]. Both of them investigate extended formulations of the UC problem from the perspective of nonlinear programming, while our work in [2] focuses on the extended formulations with piecewise linear convex objective functions in the form of linear program.…”