Computational Methods for the Design and Operations of Electric Power Systems: Towards Resiliency and Security
State estimation is an important tool for continuously monitoring the power system and aims to recover the underlying system voltage phasors, given supervisory control and data acquisition (SCADA) measurements and a model that encompasses the system topology and specifications. To ensure an accurate state estimation, it is essential to have the capability of detecting bad data. Assuming that the network parameters are known and the measurement devices are correctly calibrated, the main source of bad data is topological errors in the model. In this dissertation, we propose a methodology for robust power system state estimation (PSSE) modeled by AC power flow equations when there exists a small number of topological errors. The developed technique utilizes the availability of a large number of SCADA measurements and minimizes the L-1 norm of nonconvex residuals augmented by a nonlinear, but convex, regularizer. Representing the power network by a graph, we first study the properties of the solution obtained from the proposed NLAV estimator and demonstrate that, under mild conditions, this solution identifies a small subgraph of the network that contains the topological errors in the model used for the state estimation problem. Then, we introduce a method that can efficiently detect the topological errors by searching over the identified subgraph. In addition, we develop a theoretical upper bound on the state estimation error to guarantee the accuracy of the proposed state estimation technique.
The power flow equations are nonlinear, and may admit multiple solutions. In the past, the conventional wisdom was to assume that the solution becomes unique by restricting it to "realistic'' or "physically realizable'' values. However, various examples in the literature show that multiple solutions may persist even after restricting either voltage magnitudes or phase angle differences to "physically realizable'' values. This dissertation establishes sufficient conditions for the uniqueness of AC power flow solutions via the monotonic relationship between real power flow and the phase angle difference. More specifically, we prove that the P-Theta power flow problem has at most one solution for any acyclic or GSP graph. In addition, for arbitrary cyclic power networks, we show that multiple distinct solutions cannot exist under the assumption that angle differences across the lines are bounded by some limit related to the maximal girth of the network. We also introduce a series-parallel operator and show that this operator obtains a reduced and easier-to-analyze model for the power system without changing the uniqueness of power flow solutions.
In the next part of this dissertation, the above work is extended and we establish general necessary and sufficient conditions for the uniqueness of P-Theta power flow solutions in an AC power system using properties of the monotone regime and the power network topology. We show that the necessary and sufficient conditions can lead to tighter sufficient conditions for the uniqueness in several special cases. Our results are based on the previously introduced notion of maximal girth and a new notion of maximal eye. Moreover, we develop a series-parallel reduction method and search-based algorithms for computing the maximal eye and maximal girth, which are necessary for the uniqueness analysis. Reduction to a single line using the proposed reduction method is guaranteed for 2-vertex-connected Series-Parallel graphs.
In the final part of this dissertation, we present a methodology based on homotopy to find the globally optimal solutions of nonconvex optimization problems. Optimal power flow (OPF) is a fundamental problem in power systems analysis for determining the steady-state operating point of a power network that minimizes the generation cost. In anticipation of component failures, such as transmission line or generator outages, it is also important to find optimal corrective actions for the power flow distribution over the network. The problem of finding these post-contingency solutions to the OPF problem is challenging due to the nonconvexity of the power flow equations and the large number of contingency cases in practice. In this paper, we introduce a homotopy method to solve for the post-contingency actions, which involves a series of intermediate optimization problems that gradually transform the original OPF problem into each contingency-OPF problem. We show that given a global solution to the original OPF problem, a global solution to the contingency problem can be obtained using this homotopy method, under some conditions. With simulations on Polish and other European networks, we demonstrate that the effectiveness of the proposed homotopy method is dependent on the choice of the homotopy path and that homotopy yields an improved solution in many cases.