Machine Learning and Convex Optimization for Secure Power System Operation.

Abstract

Growing concerns about climate change drive a world-wide decarbonization of energy and transportation systems. The share of intermittent and distributed renewable generation and the usage of electric vehicles increase, elevating uncertainty and complexity for power system operation. To ensure secure power system operation, these developments necessitate the explicit representation of uncertainty in operational tools such as the non-convex AC optimal power flow (AC-OPF). At the same time, the increased uncertainty requires system operators to identify potential failure risks and take corrective actions to ensure system security closer to real-time, leading to computational challenges. Machine learning including neural networks has significant potential to accelerate and enhance operational tools such as system security assessment. However, the lack of high-quality datasets and the black-box nature of neural networks present barriers for its successful application in practice. This thesis proposes convex optimization methods for secure power system operation which explicitly represent uncertainty, and methods to remove these barriers for machine learning in safety-critical power system applications. To account for uncertainty associated with intermittent generation and stochastic loads, this thesis introduces the first convex formulation of a chance-constrained AC-OPF problem which can obtain AC-feasible solutions with near-global optimality guarantees. This formulation extends to power system security criteria and interconnected AC and HVDC grids. Chance constraints define a maximum allowable probability of constraint violation and tractable formulations are derived for Gaussian and robust uncertainty sets, assuming piecewise affine approximations of the system state as a function of the uncertainty realizations. Semidefinite relaxations of the non-convex AC-OPF problems allow to obtain near-global optimality guarantees. Two metrics are proposed to characterize inexact convex relaxations and an in-depth investigation of these for a large number of power networks demonstrates that solutions to inexact convex relaxations can exhibit substantial distances to AC-feasibility. To obtain AC-feasible solutions for the proposed chance-constrained AC-OPF formulations, a penalization method and systematic procedures to choose penalty weights are introduced. Decomposition techniques including Benders decomposition and iterative solution algorithms address scalability. Using realistic wind forecast data and a range of case studies, the proposed formulation ensures compliance with the chance constraints and obtains tight near-global optimality guarantees, while existing methods relying on the DC-OPF approximation lead to constraint violations. Machine learning including neural networks shows promising performance for a range of power system applications, e.g., by performing security assessment at a fraction of the time required by conventional approaches. However, its successful application requires large-scale datasets with sufficiently balanced classes of secure and insecure operating points and a detailed description of the security boundary. These datasets are challenging to obtain as historical data of abnormal operating conditions is highly limited and the problem dimensionality is substantial. To efficiently create datasets with these properties, this thesis proposes modular and parallelizable algorithms. Infeasibility certificates using convex relaxations are proposed to a-priori classify large parts of the possible operating region as insecure. Then, directed walks using sensitivity measures traverse the remaining unclassified operating region, and characterize the security boundary in detail. For a range of test cases, simulation results show drastic reductions in the initial unclassified operating regions, balanced datasets are created, and illustrative data-driven applications are evaluated. Besides the lack of high-quality datasets, the black-box nature of neural networks presents a major barrier towards their adoption for safety-critical power system applications. To remove this barrier, this thesis introduces, for the first time, a framework to obtain formal guarantees of neural network behavior in power systems, leveraging mixed-integer linear reformulations of trained neural networks. First, formal guarantees are obtained using security classifiers as a guiding example. These guarantees take the form of certificates for continuous input regions which guarantee that the neural network predicts the same classification. Using this framework, the neural network robustness is evaluated by identifying adversarial examples and neural networks are systematically re-trained to improve robustness. Second, formal guarantees are obtained for neural networks trained to predict solutions to DC-OPF problems. Solving mixed-integer linear programs allows to obtain worst-case guarantees for neural network predictions related to physical constraint violations, distances from predicted to optimal decision variables, and sub-optimality. The proposed methodology is demonstrated for a range of case studies, and has the potential to build the missing trust of system operators in neural networks, unlocking their potential for safety-critical applications.