Impact of Transmission Network Topology on Electrical Power Systems

Abstract

Power system reliability is a crucial component in the development of sustainable infrastructure. Because of the intricate interactions among power system components, it is often difficult to make general inferences on how the transmission network topology impacts performance of the grid in different scenarios. This complexity poses significant challenges for researches in the modeling, control, and management of power systems. In this work, we develop a theory that aims to address this challenge from both the fast-timescale and steady state aspects of power grids. Our analysis builds upon the transmission network Laplacian matrix, and reveals new properties of this well-studied concept in spectral graph theory that are specifically tailored to the power system context. A common theme of this work is the representation of certain physical quantities in terms of graphical structures, which allows us to establish algebraic results on power grid performance using purely topological information. This view is particularly powerful and often leads to surprisingly simple characterizations of complicated system behaviors. Depending on the timescale of the underlying problem, our results can be roughly categorized into the study of frequency regulation and the study of cascading failures. Fast-timescale: Frequency Regulation. We first study how the transmission network impacts power system robustness against disturbances in transient phase. Towards this goal, we develop a framework based on the Laplacian spectrum that captures the interplay among network topology, system inertia, and generator/load damping. This framework shows that the impact of network topology in frequency regulation can be quantified through the network Laplacian eigenvalues, and that such eigenvalues fully determine the grid robustness against low frequency perturbations. Moreover, we can explicitly decompose the frequency signal along scaled Laplacian eigenvectors when damping-inertia ratios are uniform across the buses. The insights revealed by this framework explain why load-side participation in frequency regulation not only makes the system respond faster, but also helps lower the system nadir after a disturbance, providing useful guidelines in the controller design. We simulate an improved controller reverse engineered from our results on the IEEE 39-bus New England interconnection system, and illustrate its robustness against high frequency oscillations compared to both the conventional droop control and a recent controller design. We then switch to a more combinatorial problem that seeks to characterize the controllability and observability of the power system in frequency regulation if only a subset of buses are equipped with controllers/sensors. Our results show that the controllability/observability of the system depends on two orthogonal conditions: (a) intrinsic structure of the system graph, and (b) algebraic coverage of buses with controllers/sensors. Condition (a) encodes information on graph symmetry and is shown to hold for almost all practical systems. Condition (b) captures how buses interact with each other through the network and can be verified using the eigenvectors of the graph Laplacian matrix. Based on this characterization, the optimal placement of controllers and sensors in the network can be formulated as a set cover problem. We demonstrate how our results identify the critical buses in real systems using a simulation in the IEEE 39-bus New England interconnection test system. In particular, for this testbed a single well chosen bus is capable of providing full controllability and observability. Steady State: Cascading Failures. Cascading failures in power systems exhibit non-monotonic, non-local propagation patterns which make the analysis and mitigation of failures difficult. By studying the transmission network Laplacian matrix, we reveal two useful structures that make the analysis of this complex evolution more tractable: (a) In contrast to the lack of monotonicity in the physical system, there is a rich collection of monotonicity we can explore in the spectrum of the Laplacian matrix. This allows us to systematically design topological measures that are monotonic over the cascading event. (b) Power redistribution patterns are closely related to the distribution of different types of trees in the power network topology. Such graphical interpretation captures the Kirchhoff s Law in a precise way and naturally suggests that we can eliminate long-distance propagation of system disturbances by forming a tree-partition. We then show that the tree-partition of transmission networks provides a precise analytical characterization of line failure localizability. Specifically, when a non-bridge line is tripped, the impact of this failure only propagates within well-defined components, which we refer to as cells, of the tree-partition defined by the bridges. In contrast, when a bridge line is tripped, the impact of this failure propagates globally across the network, affecting the power flow on all remaining transmission lines. This characterization suggests that it is possible to improve the system robustness by switching off certain transmission lines, so as to create more, smaller components in the tree-partition; thus spatially localizing line failures and making the grid less vulnerable to large-scale outages. We illustrate this approach using the IEEE 118-bus test system and demonstrate that switching off a negligible portion of transmission lines allows the impact of line failures to be significantly more localized without substantial changes in line congestion. Unified Controller on Tree-partitions. Combining our results from both the fast-timescale and steady state behaviors of power grids, we propose a distributed control strategy that offers strong guarantees in both the mitigation and localization of cascading failures in power systems. This control strategy leverages a new controller design known as Unified Controller (UC) from frequency regulation literature, and revolves around the powerful properties that emerge when the management areas that UC operates over form a tree-partition. After an initial failure, the proposed strategy always prevents successive failures from happening, and regulates the system to the desired steady state where the impact of initial failures are localized as much as possible. For extreme failures that cannot be localized, the proposed framework has a configurable design that progressively involves and coordinates across more control areas for failure mitigation and, as a last resort, imposes minimal load shedding. We compare the proposed control framework with the classical Automatic Generation Control (AGC) on the IEEE 118-bus test system. Simulation results show that our novel control greatly improves the system robustness in terms of the N-1 security standard, and localizes the impact of initial failures in majority of the load profiles that are examined. Moreover, the proposed framework incurs significantly less load loss, if any, compared to AGC, in all of our case studies.