Construction of power flow feasibility sets

We develop a new approach for construction of convex analytically simple regions where the AC power flow equations are guaranteed to have a feasible solutions. Construction of these regions is based on efficient semidefinite programming techniques accelerated via sparsity exploiting algorithms. Resulting regions have a simple geometric shape in the space of power injections (polytope or ellipsoid) and can be efficiently used for assessment of system security in the presence of uncertainty. Efficiency and tightness of the approach is validated on a number of test networks

Robust stability assessment for future power systems

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mechanical Engineering, 2018.Cataloged from PDF version of thesis. "Due to the condition of the original material, there are unavoidable flaws in this reproduction. Some pages in the original document contain text that is illegible"--Disclaimer Notice page.Includes bibliographical references (pages 119-128).Loss of stability in electrical power systems may eventually lead to blackouts which, despite being rare, are extremely costly. However, ensuring system stability is a non-trivial task for several reasons. First, power grids, by nature, are complex nonlinear dynamical systems, so assessing and maintaining system stability is challenging mainly due to the co-existence of multiple equilibria and the lack of global stability. Second, the systems are subject to various sources of uncertainties. For example, the renewable energy injections may vary depending on the weather conditions. Unfortunately, existing security assessment may not be sufficient to verify system stability in the presence of such uncertainties. This thesis focuses on new scalable approaches for robust stability assessment applicable to three main types of stability, i.e., long-term voltage, transient, and small-signal stability. In the first part of this thesis, I develop a novel computationally tractable technique for constructing Optimal Power Flow (OPF) feasibility (convex) subsets. For any inner point of the subset, the power flow problem is guaranteed to have a feasible solution which satisfies all the operational constraints considered in the corresponding OPF. This inner approximation technique is developed based on Brouwer's fixed point theorem as the existence of a solution can be verified through a self-mapping condition. The self-mapping condition along with other operational constraints are incorporated in an optimization problem to find the largest feasible subsets. Such an optimization problem is nonlinear, but any feasible solution will correspond to a valid OPF feasibility estimation. Simulation results tested on several IEEE test cases up to 300 buses show that the estimation covers a substantial fraction of the true feasible set. Next, I introduce another inner approximation technique for estimating an attraction domain of a post-fault equilibrium based on contraction analysis. In particular, I construct a contraction region where the initial conditions are "forgotten", i.e., all trajectories starting from inside this region will exponentially converge to each other. An attraction basin is constructed by inscribing the largest ball in the contraction region. To verify contraction of a Differential-Algebraic Equation (DAE) system, I also show that one can rely on the analysis of extended virtual systems which are reducible to the original one. Moreover, the Jacobians of the synthetic systems can always be expressed in a linear form of state variables because any polynomial system has a quadratic representation. This makes the synthetic system analysis more appropriate for contraction region estimation in a large scale. In the final part of the thesis, I focus on small-signal stability assessment under load dynamic uncertainties. After introducing a generic impedance-based load model which can capture the uncertainty, I propose a new robust small signal (RSS) stability criterion. Semidefinite programming is used to find a structured Lyapunov matrix, and if it exists, the system is provably RSS stable. An important application of the criterion is to characterize operating regions which are safe from Hopf bifurcations. The robust stability assessment techniques developed in this thesis primarily address the needs of a system operator in electrical power systems. The results, however, can be naturally extended to other nonlinear dynamical systems that arise in different fields such as biology, biomedicine, economics, neuron networks, and optimization. As the robust assessment is based on sufficient conditions for stability, there is still room for development on reducing the inevitable conservatism. For example, for OPF feasibility region estimation, an important open question considers what tighter bounds on the nonlinear residual terms one can use instead of box type bounds. Also, for attraction basin problem, finding the optimal norms and metrics which result in the largest contraction domain is an interesting potential research question.by Hung Dinh Nguyen.Ph. D