Multiple-Periods Locally-Facet-Based MIP Formulations for the Unit Commitment Problem

The thermal unit commitment (UC) problem has historically been formulated as a mixed integer quadratic programming (MIQP), which is difficult to solve efficiently, especially for large-scale systems. The tighter characteristic reduces the search space, therefore, as a natural consequence, significantly reduces the computational burden. In literatures, many tightened formulations for a single unit with parts … Read more

A Graph-based Decomposition Method for Convex Quadratic Optimization with Indicators

In this paper, we consider convex quadratic optimization problems with indicator variables when the matrix Q defining the quadratic term in the objective is sparse. We use a graphical representation of the support of Q, and show that if this graph is a path, then we can solve the associated problem in polynomial time. This … Read more

On the Formulation Dependence of Convex Hull Pricing

Convex hull pricing provides a potential solution for reducing out-of-market payments in wholesale electricity markets. This paper revisits the theoretical construct of convex hull pricing and explores its important but underappreciated formulation-dependence property. Namely, convex hull prices may change for different formulations of the same unit commitment problem. After a conceptual exposition of the property, … Read more

Convex Hull Results on Quadratic Programs with Non-Intersecting Constraints

Let F be a set defined by quadratic constraints. Understanding the structure of the closed convex hull cl(C(F)) := cl(conv{xx’ | x in F}) is crucial to solve quadratically constrained quadratic programs related to F. A set G with complicated structure can be constructed by intersecting simple sets. This paper discusses the relationship between cl(C(F)) … Read more

Conic Mixed-Binary Sets: Convex Hull Characterizations and Applications

We consider a general conic mixed-binary set where each homogeneous conic constraint involves an affine function of independent continuous variables and an epigraph variable associated with a nonnegative function, $f_j$, of common binary variables. Sets of this form naturally arise as substructures in a number of applications including mean-risk optimization, chance-constrained problems, portfolio optimization, lot-sizing … Read more

Polyhedral Analysis of Symmetric Multilinear Polynomials over Box Constraints

It is well-known that the convex and concave envelope of a multilinear polynomial over a box are polyhedral functions. Exponential-sized extended and projected formulations for these envelopes are also known. We consider the convexification question for multilinear polynomials that are symmetric with respect to permutations of variables. Such a permutation-invariant structure naturally implies a quadratic-sized … Read more

A Geometric View of SDP Exactness in QCQPs and its Applications

Let S denote a subset of Rn defined by quadratic equality and inequality constraints and let S denote its projected semidefinite program (SDP) relaxation. For example, take S and S to be the epigraph of a quadratically constrained quadratic program (QCQP) and the projected epigraph of its SDP relaxation respectively. In this paper, we suggest … Read more

A Unified Approach to Solve Convex Hull Pricing and Average Incremental Cost Pricing

This paper introduces a unified approach to solving convex hull pricing (CHP) and average incremental cost (AIC) pricing problems. By developing a convex hull and convex envelope formulation for individual resources, a CHP model that minimizes uplift can be solved by linear programming (LP) using relaxation of the binary terms of the security constrained unit … Read more

Enhancements of Extended Locational Marginal Pricing – Advancing Practical Implementation

Price formation is critical to efficient wholesale electricity markets that support reliable operation and efficient investment. The Midcontinent Independent System Operator (MISO) developed the Extended Locational Marginal Pricing (ELMP) with the goal of more completely reflecting resource costs and generally improving price formation to better incent market participation. MISO developed ELMP based on the mathematical … Read more

Convex Hull Representations for Bounded Products of Variables

It is well known that the convex hull of {(x,y,xy)}, where (x,y) is constrained to lie in a box, is given by the Reformulation-Linearization Technique (RLT) constraints. Belotti et al. (2010) and Miller et al. (2011) showed that if there are additional upper and/or lower bounds on the product z=xy, then the convex hull can … Read more