The transition to sustainable energy demands a fundamental redesign of the electrical grid, especially at the distribution level. With the increasing deployment of distributed energy resources, the development of reliable models, accurate state estimation methods, and robust control systems have become crucial to overcome the challenges posed by the sparse and dynamic nature of these systems, and maintain grid stability and efficiency.

The first part of the thesis addresses the complexities of Distribution Grid (DG) identification, where frequent topology changes and low signal-to-noise ratios complicate the accurate estimation of grid parameters. Traditional power flow methods, although well-established at the transmission level, require significant adaptation for DGs due to issues such as zero current injections and the sparse connectivity of nodes. This work introduces a Bayesian framework that leverages prior knowledge about grid topology and line parameters. In order to integrate it into the estimation process despite the challenges posed by non-smooth and non-convex prior likelihood functions, a novel Alternating Iteratively-Reweighted Least Squares (AIRLS) algorithm is proposed, demonstrating robust convergence properties and applicability beyond DG identification.

The second part of the thesis shifts the focus to the operational challenges arising from measurement and process noise, which are often non-Gaussian and difficult to model. Classical state estimation and control techniques are typically inadequate in the face of these complex noise distributions, leading to conservative and suboptimal performance. To address this, the thesis explores the use of Distributionally Robust Optimization (DRO) methods, which optimize system performance under the worst-case noise distributions that resemble empirical data. These methods are particularly promising due to their data-driven nature and strong mathematical guarantees. However, significant theoretical work is required to obtain practical control algorithms that can be applied in the real world. The work provides new theoretical insights into DRO, offering tractable solutions for real-time state estimation and infinite-horizon linear control design problems that ensure robustness without undue conservatism.

Through these contributions, the thesis advances the state of the art in DG identification, state estimation, and control under uncertainty, offering practical methods with strong theoretical underpinnings that can be applied to the ongoing automation and optimization of power systems.