Polynomial time algorithms and extended formulations for unit commitment problems

Correction

“…In this notice, we correct imprecise statements in the article above [2]. In Proposition 1, we claimed that formulation (9) produces optimal solutions that are binary with respect to a, b, c and h for general convex function w s tk ðq s tk , b tk Þ: This statement is not precise.…”

“…The updated integral formulations with the precise statements have been available at https://arxiv.org/abs/1906.07862 (i.e., formulation (2) in Section II and corresponding proof in Section III).…”

“…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. Finally, the authors thank Yanan Yu, Tiziano Bacci, Antonio Frangioni, and Claudio Gentile for their sincere suggestions.…”

Data-Driven Stochastic Scheduling for Energy Integrated Systems

“…For instance, more than 33% of the total electricity consumption served by the California Independent System Operator (CAISO) are provided from renewable energy resources [1]. Nevertheless, due to the usually intermittent nature of wind power, the high penetration of such a renewable energy source fluctuates, leading to a fluctuation in power generation, which involves sophisticated power system operation and creates difficulties for the system to schedule power generation [2]. An example is the duck curve [3], a well-known new pattern of the CAISO demand curve.…”