Stability, control, and optimization of nonlinear dynamical systems with applications in electric power networks

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Gholami, Amin
Sun, X. Andy
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Electric power systems in recent years have witnessed an increasing adoption of renewable energy sources as well as restructuring of distribution systems into multiple microgrids. These trends, together with an ever-growing electricity demand, are making power networks operate closer to their stability margins, thereby raising numerous challenges for power system operators. In this thesis, we focus on two major challenges: How to efficiently assess and certify the stability of power systems; and how to optimize the operation of multiple microgrids while maintaining their stability. In the first part of the thesis, we focus on the first question, and study one of the most fundamental models of power systems, namely the swing equation model. We develop sufficient conditions under which the equilibrium points of swing equations are asymptotically stable. We also discuss the connection between the stability of equilibrium points and the network structure. This for example reveals an analog of Braess’s Paradox in power system stability, showing that adding power lines to the system may decrease the stability margin. Based on the developed theories, we also introduce several distributed control schemes for maintaining the stability of the system. Since swing equations belong to a more general class of second-order ordinary differential equations (ODEs) which are the cornerstone of studying many other physical and engineering systems, a considerable part of this thesis is devoted to the study of this general class of ODEs, where we investigate the impact of damping as a system parameter on the stability, hyperbolicity, and bifurcation in such systems. In the second part of the thesis, we address the second question and provide a computationally efficient method for optimizing multi-microgrid operation while ensuring its stability. Our goal is to maintain the frequency stability of multi-microgrid networks under an islanding event and to achieve optimal load shedding and network topology control with AC power flow constraints. Attaining this goal requires solving a challenging optimization problem with stability constraints. To cope with this challenge, we develop a strong mixed-integer second-order cone programming (MISOCP)-based reformulation and a cutting plane algorithm for scalable computation of the problem. The optimization frameworks and stability certificates developed in this thesis can be used as powerful decision support tools for power system operators.
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