Restricted power domination and fault-tolerant power domination on grids

AbstractThe power domination problem is to find a minimum placement of phase measurement units (PMUs) for observing the whole electric power system, which is closely related to the classical domination problem in graphs. For a graph G=(V,E), the power domination number of G is the minimum cardinality of a set S⊆V such that PMUs placed on every vertex of S results in all of V being observed. A vertex with a PMU observes itself and all its neighbors, and if an observed vertex with degree d>1 has only one unobserved neighbor, then the unobserved neighbor becomes observed. Although the power domination problem has been proved to be NP-complete even when restricted to some special classes of graphs, Dorfling and Henning in [M. Dorfling, M.A. Henning, A note on power domination in grid graphs, Discrete Applied Mathematics 154 (2006) 1023–1027] showed that it is easy to determine the power domination number of an n×m grid. Their proof provides an algorithm for giving a minimum placement of PMUs. In this paper, we consider the situation in which PMUs may only be placed within a restricted subset of V. Then, we present algorithms to solve this restricted type of power domination on grids under the conditions that consecutive rows or columns form a forbidden zone. Moreover, we also deal with the fault-tolerant measurement placement in the designed scheme and provide approximation algorithms when the number of faulty PMUs does not exceed 3

Introduction to Robust Power Domination

Sensors called phasor measurement units (PMUs) are used to monitor the electric power network. The power domination problem seeks to minimize the number of PMUs needed to monitor the network. We extend the power domination problem and consider the minimum number of sensors and appropriate placement to ensure monitoring when $k$ sensors are allowed to fail with multiple sensors allowed to be placed in one location. That is, what is the minimum multiset of the vertices, $S$, such that for every $F\subseteq S$ with $|F|=k$, $S\setminus F$ is a power dominating set. Such a set of PMUs is called a $k$-robust power domination set. This paper generalizes the work done by Pai, Chang and Wang in 2010 on vertex-fault-tolerant power domination, which did not allow for multiple sensors to be placed at the same vertex. We provide general bounds and determine the $k$-robust power domination number of some graph families.Comment: 19 pages, 2 figure

Improved fault-tolerant PMU placement using algebraic connectivity of graphs

Due to perpetual and innovative technological advancements, the need for reliable and stable power generation and transmission has been increasing dramatically over the years. Smart grids use advanced technologies to provide self-monitoring, self-checking and self-healing power networks, including smart metering devices capable of providing accurate measurements of the network\u27s power components. Among the most important metering devices in this context are Phasor Measurement Units (PMUs) . PMUs are metering devices that provide synchronized measurements of voltage, current and phase angle differences using signals from the GPS satellites. However, due to the high cost of such advanced metering devices, studies were performed to determine the minimum number of PMUs required and their strategic placements in the power networks to provide full system observability. In this thesis, we consider fault-tolerant PMU placement aiming to minimize the number of PMUs while maintaining system observability under various contingencies. Conventionally, the optimal number of PMUs in a system is determined based on the system\u27s connectivity matrix under no contingency. This thesis considers fault- tolerant PMU placement under single and double branch failures. We propose algebraic connectivity, or Fiedler value, to identify the worst- case branch failures in terms of connectivity degradation. The proposed PMU placement accounts for this worst-case and covers a large percentage of other single and double branch failures. Furthermore, we propose the usage of Fiedler vector to provide a PMU placement that would ensure that the system remains fully observable during system partitioning into separate sub-systems. The resulting placements are compared with those obtained without considering connectivity degradation or system partitioning in terms of the percentages of observable systems during any single and double branch failures. The proposed PMU placements have increased percentages of fully observable systems in the event of any single or double branch failures compared to non—contingency based placement, with a reasonable increase in number of PMUs, and for some placement approaches no increase in PMUs is needed for providing a higher percentage of fully observable systems

Improved fault-tolerant PMU placement using algebraic connectivity of graphs

Due to perpetual and innovative technological advancements, the need for reliable and stable power generation and transmission has been increasing dramatically over the years. Smart grids use advanced technologies to provide self-monitoring, self-checking and self-healing power networks, including smart metering devices capable of providing accurate measurements of the network’s power components. Among the most important metering devices in this context are “Phasor Measurement Units (PMUs)â€. PMUs are metering devices that provide synchronized measurements of voltage, current and phase angle differences using signals from the GPS satellites. However, due to the high cost of such advanced metering devices, studies were performed to determine the minimum number of PMUs required and their strategic placements in the power networks to provide full system observability. In this thesis, we consider fault-tolerant PMU placement aiming to minimize the number of PMUs while maintaining system observability under various contingencies. Conventionally, the optimal number of PMUs in a system is determined based on the system’s connectivity matrix under no contingency. This thesis considers fault- tolerant PMU placement under single and double branch failures. We propose algebraic connectivity, or Fiedler value, to identify the worst- case branch failures in terms of connectivity degradation. The proposed PMU placement accounts for this worst-case and covers a large percentage of other single and double branch failures. Furthermore, we propose the usage of Fiedler vector to provide a PMU placement that would ensure that the system remains fully observable during system partitioning into separate sub-systems. The resulting placements are compared with those obtained without considering connectivity degradation or system partitioning in terms of the percentages of observable systems during any single and double branch failures. The proposed PMU placements have increased percentages of fully observable systems in the event of any single or double branch failures compared to non—contingency based placement, with a reasonable increase in number of PMUs, and for some placement approaches no increase in PMUs is needed for providing a higher percentage of fully observable systems

Pulse propagation, graph cover, and packet forwarding

We study distributed systems, with a particular focus on graph problems and fault tolerance. Fault-tolerance in a microprocessor or even System-on-Chip can be improved by using a fault-tolerant pulse propagation design. The existing design TRIX achieves this goal by being a distributed system consisting of very simple nodes. We show that even in the typical mode of operation without faults, TRIX performs significantly better than a regular wire or clock tree: Statistical evaluation of our simulated experiments show that we achieve a skew with standard deviation of O(log log H), where H is the height of the TRIX grid. The distance-r generalization of classic graph problems can give us insights on how distance affects hardness of a problem. For the distance-r dominating set problem, we present both an algorithmic upper and unconditional lower bound for any graph class with certain high-girth and sparseness criteria. In particular, our algorithm achieves a O(r·f(r))-approximation in time O(r), where f is the expansion function, which correlates with density. For constant r, this implies a constant approximation factor, in constant time. We also show that no algorithm can achieve a (2r + 1 − δ)-approximation for any δ > 0 in time O(r), not even on the class of cycles of girth at least 5r. Furthermore, we extend the algorithm to related graph cover problems and even to a different execution model. Furthermore, we investigate the problem of packet forwarding, which addresses the question of how and when best to forward packets in a distributed system. These packets are injected by an adversary. We build on the existing algorithm OED to handle more than a single destination. In particular, we show that buffers of size O(log n) are sufficient for this algorithm, in contrast to O(n) for the naive approach.Wir untersuchen verteilte Systeme, mit besonderem Augenmerk auf Graphenprobleme und Fehlertoleranz. Fehlertoleranz auf einem System-on-Chip (SoC) kann durch eine fehlertolerante Puls- Weiterleitung verbessert werden. Das bestehende Puls-Weiterleitungs-System TRIX toleriert Fehler indem es ein verteiltes System ist das nur aus sehr einfachen Knoten besteht. Wir zeigen dass selbst im typischen, fehlerfreien Fall TRIX sich weitaus besser verhält als man naiverweise erwarten würde: Statistische Analysen unserer simulierten Experimente zeigen, dass der Verzögerungs-Unterschied eine Standardabweichung von lediglich O(log logH) erreicht, wobei H die Höhe des TRIX-Netzes ist. Das Generalisieren einiger klassischer Graphen-Probleme auf Distanz r kann uns neue Erkenntnisse bescheren über den Zusammenhang zwischen Distanz und Komplexität eines Problems. Für das Problem der dominierenden Mengen auf Distanz r zeigen wir sowohl eine algorithmische obere Schranke als auch eine bedingungsfreie untere Schranke für jede Klasse von Graphen, die bestimmte Eigenschaften an Umfang und Dichte erfüllt. Konkret erreicht unser Algorithmus in Zeit O(r) eine Annäherungsgüte von O(r · f(r)). Für konstante r bedeutet das, dass der Algorithmus in konstanter Zeit eine Annäherung konstanter Güte erreicht. Weiterhin zeigen wir, dass kein Algorithmus in Zeit O(r) eine Annäherungsgüte besser als 2r + 1 erreichen kann, nicht einmal in der Klasse der Kreis-Graphen von Umfang mindestens 5r. Weiterhin haben wir das Paketweiterleitungs-Problem untersucht, welches sich mit der Frage beschäftigt, wann genau Pakete in einem verteilten System idealerweise weitergeleitetwerden sollten. Die Paketewerden dabei von einem Gegenspieler eingefügt. Wir bauen auf dem existierenden Algorithmus OED auf, um mehr als ein Paket-Ziel beliefern zu können. Dadurch zeigen wir, dass Paket-Speicher der Größe O(log n) für dieses Problem ausreichen, im Gegensatz zu den Paket-Speichern der Größe O(n) die für einen naiven Ansatz nötig wären

Metric-locating-dominating sets of graphs for constructing related subsets of vertices

© 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/A dominating set S of a graph is a metric-locating-dominating set if each vertex of the graph is uniquely distinguished by its distances from the elements of S , and the minimum cardinality of such a set is called the metric-location-domination number. In this paper, we undertake a study that, in general graphs and specific families, relates metric-locating-dominating sets to other special sets: resolving sets, dominating sets, locating-dominating sets and doubly resolving sets. We first characterize the extremal trees of the bounds that naturally involve metric-location-domination number, metric dimension and domination number. Then, we prove that there is no polynomial upper bound on the location-domination number in terms of the metric-location-domination number, thus extending a result of Henning and Oellermann. Finally, we show different methods to transform metric-locating-dominating sets into locating-dominating sets and doubly resolving sets. Our methods produce new bounds on the minimum cardinalities of all those sets, some of them concerning parameters that have not been related so farPeer ReviewedPostprint (author's final draft

Optimal bounds on codes for location in circulant graphs

Identifying and locating-dominating codes have been studied widely in circulant graphs of type Cn(1,2,3,...,r) over the recent years. In 2013, Ghebleh and Niepel studied locating-dominating and identifying codes in the circulant graphs Cn(1,d) for d=3 and proposed as an open question the case of d>3. In this paper we study identifying, locating-dominating and self-identifying codes in the graphs Cn(1,d), Cn(1,d-1,d) and Cn(1,d-1,d,d+1). We give a new method to study lower bounds for these three codes in the circulant graphs using suitable grids. Moreover, we show that these bounds are attained for infinitely many parameters n and d. In addition, new approaches are provided which give the exact values for the optimal self-identifying codes in Cn(1,3) and Cn(1,4)