Analytical approximations and decision-making techniques for power systems under uncertainty

The integration of distributed energy resources and increasing adoption of electric vehicles continue to drive uncertainty in power systems to an unprecedented level. In view of reduced applicability of traditional analysis and decision-making methods, this dissertation aims to address the need for new approaches, by attempting to solve three different sets of problems. This dissertation first proposes an analytical power flow approximation and develops a closed-form power flow framework. The thesis proposes a novel framework using Gaussian process regression to learn node voltage as a closed-form function of effective bus load or injection vector. The proposed approximation is valid over a subspace of load, where the `subspace' is used to describe a hypercube within which uncertain loads/injections lie. The approximation framework explains the system's behavior under uncertainty via Gaussian process hyper-parameter based interpretability. The proposed method achieves low mean absolute error of order $10^{-5}$ (per unit) in voltage magnitude and $10^{-4}$ (rad.) in angle when tested on 33-Bus and 56-Bus systems. Also, the proposed framework is used to develop an optimal steady-state voltage control framework using a linear voltage-power relationship. Simulation results on the 69-Bus system have shown the independence of approximation error with system loading and uncertainty levels. Therefore, control mechanisms developed based on the proposed analytical approximation are suitable under uncertainty. The framework also addresses the challenges posed by independent entity integration (e.g., peer-to-peer energy trading), as independent operational decisions of prosumers affect the entire network. The proposed probabilistic feasibility set construction method-- for privacy-preserving feasibility assessment-- is tractable and handles non-parametric injection uncertainties by applying the worst-case performance analysis theorem over the proposed closed-form power flow approximation. The developed method's ability to capture the variations in probabilistic feasibility set for storage connected at a node in a 33-Bus network is shown via numerical simulations under different levels of total renewable penetration and uncertainty. The second problem set deals with power systems' decision-making problems under non-parametric uncertainties of injections. Fundamentally, this part of dissertation solves three optimal power flow problems under uncertainty where probability distributions (and/or distribution parameters) of random injections are not known i.e., non-parametric uncertainties are present. Firstly, this thesis presents a novel approach for performing uncertainty quantification through an interpretable and non-parametric probabilistic optimal power flow formulation using Gaussian process learning. The proposed method is benchmarked-- under non-parametric uncertainty settings-- against Monte-Carlo simulations and state-of-art methods on 14-Bus, 30-Bus, and 118-Bus systems and is shown to provide accurate solutions with significantly less elapsed time. Additionally, the thesis introduces a novel state-aware stochastic optimal power flow problem formulation, targeting a day-ahead base solution that minimizes generation cost while also reducing deviations in generation and voltage set-points during real-time operation. The Gaussian process learning is used to obtain a non-parametric (distributionally robust) affine policy for real-time generation and voltage set-point changes, and simulations on a 14-Bus system are used to demonstrate distributional robustness. The proposed formulation is able to reduce the expectation of voltage and generation deviation by more than 60\% with an additional day-ahead scheduling cost of only 4.68\%. The dissertation also presents a novel joint chance-constrained optimal power flow formulation for non-parametric uncertainties. This work proposes a novel approach, which uses maximum mean discrepancy penalization to convert probabilistic constrain into a deterministic objective penalty-- achieving the same complexity as a deterministic optimal power flow problem. Simulation results and benchmarking against existing approaches on 24-Bus, 30-Bus, and 57-Bus systems validate the proposed method's non-parametric nature, show the ability to obtain a probabilistically feasible solution, and indicate better computational performance. The third research direction focuses on challenges of small-signal stability due to low inertia sources (renewable generation) integration and in power systems' operation. In this thesis, a novel probabilistic robust small-signal stability framework is developed, based on Gaussian process learning. The proposed framework defines a robust stability certificate for a state subspace with a given probability. The critical eigenvalue of the system matrix is obtained as a function of state variables, within a state subspace, with probabilistic guarantees on errors. The proposed certification-- with the subspace-based and confidence-based search mechanisms-- constitutes a holistic framework for assessing stability under non-parametric uncertainties without state-specific input-to-output approximations. The results on a Three-machine, Nine-bus WSCC system show that the proposed certificate can find a probabilistically robust stable state subspace, with a given probability. The thesis also approaches small-signal stability from the network structure-preserving differential algebraic equation formulation point of view, using a logarithmic norm. A sufficient condition for stability using bilinear matrix inequality and its inner approximation as linear matrix inequality is proposed. The work also provides a necessary base to develop tractable construction techniques for robust stability regions for power system operations. Furthermore, using the novel sufficient stability condition, a convexified small-signal stability constrained optimal power flow is presented that does not rely on eigenvalue analysis. The developed formulation is based on semi-definite programming and uses an objective penalization strategy for feasible solution recovery, making the method computationally efficient. An effectiveness study on WECC 9-Bus, New England 39-Bus, and 118-Bus test systems shows that the proposed method can achieve a stable equilibrium point without inflicting a high stability-induced additional cost. Benchmarking with the current non-convex optimization approaches shows improved computational performance as well. Overall, this thesis presents a comprehensive approach to address the challenges in analysis and operational decision-making of the power system under uncertainty, without relying on probability distribution information of uncertain injections.Doctor of Philosoph