Optimization-Based Exploration of the Feasible Power Flow Space for Rapid Data Collection

TL;DR

This work tackles the challenge of sampling the non-convex feasible space of optimal power flow (OPF) by proposing an optimization-based data-collection framework that sequentially solves OPF with nonlinear objectives aimed at maximizing dispersion among solutions. It builds an exhaustive rejection-sampling reference and uses the Hausdorff distance to quantify how well each objective covers the feasible space, validating the approach on five IEEE/PGLib test systems with large-scale data generation. The key finding is that nonlinear objectives that apply a logarithmic distance and incorporate multiple variables, particularly both power and voltage terms (e.g., $f_{36}$, $f_{38}$), most effectively reveal remote regions of the space; exponential objectives perform poorly. The results provide concrete guidance for data collection in learning-based OPF studies and come with public code to enable replication and extension to larger systems or distribution grids.

Abstract

This paper provides a systematic investigation into the various nonlinear objective functions which can be used to explore the feasible space associated with the optimal power flow problem. A total of 40 nonlinear objective functions are tested, and their results are compared to the data generated by a novel exhaustive rejection sampling routine. The Hausdorff distance, which is a min-max set dissimilarity metric, is then used to assess how well each nonlinear objective function performed (i.e., how well the tested objective functions were able to explore the non-convex power flow space). Exhaustive test results were collected from five PGLib test-cases and systematically analyzed.

Figures (4)

• Figure 1: Depicted is the Hausdorff distance (green arrow) associated with points collected in some 2-dimensional feasible space $f(x,y)$. The orange points exhaustively approximate the space, while the blue points are the sequential solutions of a given nonlinear objective function. The green arrow connects the orange point (farthest to the right) whose minimum distance to an iterative solution is maximized.
• Figure 2: Progression of the Hausdorff distance on the 14-bus system, as more iterations of Alg. \ref{['algo:iterative']} are performed, when just voltage $\bm v$ is measured in (\ref{['eq: distance_edit']}). $f_{36}$ and $f_{38}$ are highlighted, and diverging functions are excluded.
• Figure 3: Exhaustive data is plotted as blue dots and function sampling is plotted as red dots. This figure portrays the difference between good (left) and bad (right) nonlinear objective functions by comparing the active power generation at buses 1 and 2 in the IEEE 14-bus system. In both cases, there is some clustering on the edges. However, this is much more severe for $f_{32}$.
• Figure 4: Progression of the Hausdorff distance on the 14-bus system, as more iterations of Alg. \ref{['algo:iterative']} are performed, when just power $\bm p$ is measured in (\ref{['eq: distance_edit']}). This figure clearly shows two different groups of functions. One group gets stuck at a value of 0.31, and a second group overcomes this first limit and reaches a lower value of 0.05. The second group samples the feasible space much more effectively. $f_{36}$ and $f_{38}$ are highlighted for posterior analysis.