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In modeling battery energy storage systems (BESS) in power systems, binary variables are used to represent the complementary nature of charging and discharging. A conventional approach for these BESS optimization problems is to relax binary variables and convert the problem into a linear program. However, such linear programming relaxation models can yield unrealistic fractional solutions, such as simultaneous charging and discharging. In this paper, we develop a regularized mixed-integer programming (MIP) model for the optimal power flow (OPF) problem with BESS. We prove that, under mild conditions, the proposed regularized model admits a zero integrality gap with its linear programming relaxation; hence, it can be solved efficiently. By studying the properties of the regularized MIP model, we show that its optimal solution is also near optimal to the original OPF problem with BESS, thereby providing a valid and tight upper bound for the OPF problem with BESS. The use of the regularized MIP model allows us to solve a trilevel [Formula: see text]-[Formula: see text]-[Formula: see text] network contingency problem, which is otherwise intractable to solve. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: N. Jiang (as a graduate student at the Georgia Institute of Technology) and W. Xie were supported in part by the National Science Foundation [Grant 2246414] and the Office of Naval Research [Grant N00014-24-1-2066]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0771 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0771 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .
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Hidden Integrality and Semirandom Robustness of SDP Relaxation for Sub-Gaussian Mixture Model
We consider the problem of estimating the discrete clustering structures under the sub-Gaussian mixture model. Our main results establish a hidden integrality property of a semidefinite programming (SDP) relaxation for this problem: while the optimal solution to the SDP is not integer-valued in general, its estimation error can be upper bounded by that of an idealized integer program. The error of the integer program, and hence that of the SDP, are further shown to decay exponentially in the signal-to-noise ratio. In addition, we show that the SDP relaxation is robust under the semirandom setting in which an adversary can modify the data generated from the mixture model. In particular, we generalize the hidden integrality property to the semirandom model and thereby show that SDP achieves the optimal error bound in this setting. These results together highlight the “global-to-local” mechanism that drives the performance of the SDP relaxation. To the best of our knowledge, our result is the first exponentially decaying error bound for convex relaxations of mixture models. A corollary of our results shows that in certain regimes, the SDP solutions are in fact integral and exact. More generally, our results establish sufficient conditions for the SDP to correctly recover the cluster memberships of [Formula: see text] fraction of the points for any [Formula: see text]. As a special case, we show that under the [Formula: see text]-dimensional stochastic ball model, SDP achieves nontrivial (sometimes exact) recovery when the center separation is as small as [Formula: see text], which improves upon previous exact recovery results that require constant separation.
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- PAR ID:
- 10327714
- Date Published:
- Journal Name:
- Mathematics of Operations Research
- ISSN:
- 0364-765X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1287/moor.2021.1216
