Mathematical programming formulations for the alternating current optimal power flow problem

Abstract

Power flow refers to the injection of power on the lines of an electrical grid, so that all the injections at the nodes form a consistent flow within the network. Optimality, in this setting, is usually intended as the minimization of the cost of generating power. Current can either be direct or alternating: while the former yields approximate linear programming formulations, the latter yields formulations of a much more interesting sort: namely, nonconvex nonlinear programs in complex numbers. In this technical survey, we derive formulation variants and relaxations of the alternating current optimal power flow problem.

Applications of optimization models for electricity distribution networks

“…In this review, we start from the assumption that mathematical optimization is an appropriate framework to automate the process of decision making. We note that a number of surveys and reviews have been published in this space, including Dubey et al (2020), G. Wang and Hijazi (2018), Bienstock et al (2020) and Molzahn and Hiskens (2019). To distinguish ourselves, in this review we choose to focus on:…”

“…In this review, we start from the assumption that mathematical optimization is an appropriate framework to automate the process of decision making. We note that a number of surveys and reviews have been published in this space, including Dubey et al (2020), G. Wang and Hijazi (2018), Bienstock et al (2020) and Molzahn and Hiskens (2019). To distinguish ourselves, in this review we choose to focus on: Distribution networks, with careful consideration of the structure of three‐phase (European style) low‐voltage (LV) grids; Model‐based approaches representing the steady‐state physics and limits of the network in the context of non‐negligible phase unbalance ; Implementation of the optimization models in software codes, with considerations on the state‐of‐the‐art, and reproducibility of the results. …”

Certifying Global Optimality of AC-OPF Solutions via the CS-TSSOS Hierarchy

“…The CS-TSSOS hierarchy is implemented in the Julia package TSSOS 4 . In TSSOS, the minimal initial relaxation step is accessible via the commands cs tssos first and cs tssos higher!…”

“…Because of the inequality S ij S * ij ≤ (s u ij ) 2 , the resulting optimization problem contains a quartic constraint. To implement Shor's relaxation for QCQP, we then relax this quartic constraint to a quadratic constraint using the trick described in [4,Sec. 5.3].…”

“…Background on the AC-OPF problem. The AC-OPF is a fundamental problem in power system, which has been extensively studied in recent years; see the survey [4] and references therein. One can formulate the AC-OPF problem as a POP either with real variables [7] or with complex variables [11].…”