Statistical Foundations of Generative AI for Optimal Control Problems in Power Systems: Comprehensive Review and Future Directions

With the rapid advancement of deep learning, generative artificial intelligence (Gen-AI) has emerged as a powerful tool, unlocking new prospects in the power systems sector. Despite the evident success of these methods and the rapid growth of this field in the power systems community, there is still a pressing need for a deeper understanding of how different evaluation metrics relate to the underlying statistical structure of the models. Another related important question is what tools can be used to quantify the different uncertainties, which are inherent in these problems, and stem not only from the physical system but also from the nature of the generative model itself. This paper attempts to address these challenges and provides a comprehensive review of existing evaluation metrics for generative models applied in various power system tasks. We analyze how these metrics align with the statistical properties of the models and explore their strengths and limitations. We also examine different sources of uncertainty, distinguishing between uncertainties inherent to the learning model, those arising from measurement errors, and other sources. Our general aim is to promote a better understanding of generative models as they are being applied in power systems to support this fascinating growing trend.