Encoding frequency stability constraints in the operation problem is challenging due to its complex dynamics. Recently, data-driven approaches have been proposed to learn the stability criteria offline with the trained model embedded as a constraint of online optimization. However, random sampling of stationary operation points is less efficient in generating balanced stable and unstable samples. Meanwhile, the performance of such a model is strongly dependent on the quality of the training dataset. Observing this research gap, we propose a gradient-based data generation method via forward-mode automatic differentiation. In this method, the original dynamic system is augmented with new states that represent the dynamic of sensitivities of the original states, which can be solved by invoking any ODE solver for a single time. To compensate for the contradiction between the gradient of various frequency stability criteria, gradient surgery is proposed by projecting the gradient on the normal plane of the other. In the end, we demonstrate the superior performance of the proposed sampling algorithm, compared with the unrolling differentiation and finite difference. All codes are available at https://github.com/xuwkk/frequency_sample_ad.

Deterministic frequency deviations (DFDs) critically affect power grid frequency quality and power system stability. A better understanding of these events is urgently needed as frequency deviations have been growing in the European grid in recent years. DFDs are partially explained by the rapid adjustment of power generation following the intervals of electricity trading, but this intuitive picture fails especially before and around noonday. In this article, we provide a detailed analysis of DFDs and their relation to external features using methods from explainable Artificial Intelligence. We establish a machine learning model that well describes the daily cycle of DFDs and elucidate key interdependencies using SHapley Additive exPlanations (SHAP). Thereby, we identify solar ramps as critical to explain patterns in the Rate of Change of Frequency (RoCoF).