Daniel K. Molzahn

Daniel K. Molzahn, Matthew Niemerg, Dhagash Mehta et al.

The power flow equations model the steady-state relationship between the power injections and voltage phasors in an electric power system. By separating the real and imaginary components of the voltage phasors, the power flow equations can be formulated as a system of quadratic polynomials. Only the real solutions to these polynomial equations are physically meaningful. This paper focuses on the maximum number of real solutions to the power flow equations. An upper bound on the number of real power flow solutions commonly used in the literature is the maximum number of complex solutions. There exist two- and three-bus systems for which all complex solutions are real. It is an open question whether this is also the case for larger systems. This paper investigates four-bus systems using techniques from numerical algebraic geometry and conjectures a negative answer to this question. In particular, this paper studies lossless, four-bus systems composed of PV buses connected by lines with arbitrary susceptances. Computing the Galois group, which is degenerate, enables conversion of the problem of counting the number of real solutions to the power flow equations into counting the number of positive roots of a univariate sextic polynomial. From this analysis, it is conjectured that the system has at most 16 real solutions, which is strictly less than the maximum number of complex solutions, namely 20. We also provide explicit parameter values where this system has 16 real solutions so that the conjectured upper bound is achievable.

Paprapee Buason, Sidhant Misra, Daniel K. Molzahn

Accurate modeling of power flow behavior is essential for a wide range of power system applications, yet the nonlinear and nonconvex structure of the underlying equations often limits their direct use in large-scale optimization problems. As a result, linear models are frequently adopted to improve computational tractability, though these simplifications can introduce excessive approximation error or lead to constraint violations. This paper presents a linear approximation framework, referred to as Conservative Bias Linear Approximations (CBLA), that systematically incorporates conservativeness into the approximation process. Rather than solely minimizing local linearization error, CBLA constructs linear constraints that bound the nonlinear functions of interest over a defined operating region while reducing overall approximation bias. The proposed approach maintains the simplicity of linear formulations and allows the approximation to be shaped through user-defined loss functions tailored to specific system quantities. Numerical studies demonstrate that CBLA provides more reliable and accurate approximations than conventional linearization techniques, and its integration into a unit commitment formulation results in improved feasibility and reduced operating costs.

Rosemarie Santa Gonzalez, Ryan Piansky, Sue M Bae et al.

The integration of artificial intelligence (AI) and optimization hold substantial promise for improving the efficiency, reliability, and resilience of engineered systems. Due to the networked nature of many engineered systems, ethically deploying methodologies at this intersection poses challenges that are distinct from other AI settings, thus motivating the development of ethical guidelines tailored to AI-enabled optimization. This paper highlights the need to go beyond fairness-driven algorithms to systematically address ethical decisions spanning the stages of modeling, data curation, results analysis, and implementation of optimization-based decision support tools. Accordingly, this paper identifies ethical considerations required when deploying algorithms at the intersection of AI and optimization via case studies in power systems as well as supply chain and logistics. Rather than providing a prescriptive set of rules, this paper aims to foster reflection and awareness among researchers and encourage consideration of ethical implications at every step of the decision-making process.

Sihan Zeng, Alyssa Kody, Youngdae Kim et al.

With the increasing penetration of distributed energy resources, distributed optimization algorithms have attracted significant attention for power systems applications due to their potential for superior scalability, privacy, and robustness to a single point-of-failure. The Alternating Direction Method of Multipliers (ADMM) is a popular distributed optimization algorithm; however, its convergence performance is highly dependent on the selection of penalty parameters, which are usually chosen heuristically. In this work, we use reinforcement learning (RL) to develop an adaptive penalty parameter selection policy for the AC optimal power flow (ACOPF) problem solved via ADMM with the goal of minimizing the number of iterations until convergence. We train our RL policy using deep Q-learning, and show that this policy can result in significantly accelerated convergence (up to a 59% reduction in the number of iterations compared to existing, curvature-informed penalty parameter selection methods). Furthermore, we show that our RL policy demonstrates promise for generalizability, performing well under unseen loading schemes as well as under unseen losses of lines and generators (up to a 50% reduction in iterations). This work thus provides a proof-of-concept for using RL for parameter selection in ADMM for power systems applications.

Alyssa Kody, Samuel Chevalier, Spyros Chatzivasileiadis et al.

Nonlinear power flow constraints render a variety of power system optimization problems computationally intractable. Emerging research shows, however, that the nonlinear AC power flow equations can be successfully modeled using Neural Networks (NNs). These NNs can be exactly transformed into Mixed Integer Linear Programs (MILPs) and embedded inside challenging optimization problems, thus replacing nonlinearities that are intractable for many applications with tractable piecewise linear approximations. Such approaches, though, suffer from an explosion of the number of binary variables needed to represent the NN. Accordingly, this paper develops a technique for training an "optimally compact" NN, i.e., one that can represent the power flow equations with a sufficiently high degree of accuracy while still maintaining a tractable number of binary variables. We show that the resulting NN model is more expressive than both the DC and linearized power flow approximations when embedded inside of a challenging optimization problem (i.e., the AC unit commitment problem).