Optimal Power Flow With Regression-Based Small-Signal Stability Constraints

In modern power systems, with a growing integration of inverter-based resources, small-signal stability is becoming increasingly important. However, current practice in operation and planning studies is based on solving optimization problems, mainly based on Optimal Power Flow calculation (OPF), considering only static stability constraints. The small-signal stability of such optimal solutions is typically addressed in separate, subsequent studies. Avoid costly preventive actions or post-oscillation corrective actions require a tool that can compute optimal and stable solutions within a unified framework. To address this need, this paper proposes a Small-signal Stability Constrained Optimal Power Flow (SSC-OPF). To alleviate the computational burden, leveraging Machine Learning to formulate the stability constraints is an emerging and effective solution. In this work regression functions are employed, quantifying the oscillation damping via a numerical stability indicator (the Damping Index (DI)). Through qualitative and quantitative analyses, Multivariate Adaptive Regression Splines emerges as a suitable regression technique, offering a continuous and derivable expression. This choice bypasses the need for integer variables and the formulation of challenging Mixed-Integer Linear Programming optimization, making SSC-OPF implementation more feasible with the use of the Interior-point algorithm. In this study, the SSC-OPF is implemented in two configurations. In one scenario, the optimization control variables include only the generators’ power dispatch. In the second scenario, the possibility of tuning converters control parameters is also considered. Both implementations are tested and compared on a 9-bus power system, with two objective functions exemplifying the use of the SSC-OPF for power system operation and planning. In all test cases, the regression-based constraint exhibits high accuracy in determining the DI, with an
$R^{2}$
higher than 0.96. This accuracy ensures that none of the SSC-OPF solutions are misclassified in terms of stability. When optimization relies solely on generator dispatch, the SSC-OPF might fail to find stable solutions due to system limitations, like generator constraints. However, incorporating controller tuning overcomes this issue, achieving optimal and stable solutions across all test cases. From a computational perspective, although the inclusion of the small-signal constraint increases the computational burden, it remains within the same order of magnitude as the number of iterations required to solve a conventional OPF.