Neural network constraint satisfaction is crucial for safety-critical applications such as power system optimization, robotic path planning, and autonomous driving. However, existing constraint satisfaction methods face efficiency-applicability trade-offs, with hard constraint methods suffering from either high computational complexity or restrictive assumptions on constraint structures. The Sampling Kaczmarz-Motzkin (SKM) method is a randomized iterative algorithm for solving large-scale linear inequality systems with favorable convergence properties, but its argmax operations introduce non-differentiability, posing challenges for neural network applications. This work proposes the Trainable Sampling Kaczmarz-Motzkin Network (T-SKM-Net) framework and, for the first time, systematically integrates SKM-type methods into neural network constraint satisfaction. The framework transforms mixed constraint problems into pure inequality problems through null space transformation, employs SKM for iterative solving, and maps solutions back to the original constraint space, efficiently handling both equality and inequality constraints. We provide theoretical proof of post-processing effectiveness in expectation and end-to-end trainability guarantees based on unbiased gradient estimators, demonstrating that despite non-differentiable operations, the framework supports standard backpropagation. On the DCOPF case118 benchmark, our method achieves 4.27ms/item GPU serial forward inference with 0.0025% max optimality gap with post-processing mode and 5.25ms/item with 0.0008% max optimality gap with joint training mode, delivering over 25× speedup compared to the pandapower solver while maintaining zero constraint violations under given tolerance.

Power systems around the world are experiencing an energy revolution that substitutes fossil fuels with renewable energy. Such a transition poses two significant challenges: highly variable generators that add short-term and long-term difficulties for supply–demand balance, and a high proportion of convertor-based devices that may jeopardize power system security and stability. At the same time, machine learning techniques provide more opportunities to study the complex power system security and stability problems. This article summarizes the machine learning framework to embed security rules into power system operation optimization under high renewable energy penetration. First, we explore how high penetration renewable energy impacts power system security and stability. Then, we review how the complex security and stability boundary of power systems is modeled using various machine learning techniques. Finally, we show how the machine learning model is transformed into optimization constraints that can be embedded into the power system operation model. The framework is substantiated through case studies of practical power systems.