Optimal Topology Design for Disturbance Minimization in Power Grids

The transient response of power grids to external disturbances influences their stable operation. This paper studies the effect of topology in linear time-invariant dynamics of different power grids. For a variety of objective functions, a unified framework based on $H_2$ norm is presented to analyze the robustness to ambient fluctuations. Such objectives include loss reduction, weighted consensus of phase angle deviations, oscillations in nodal frequency, and other graphical metrics. The framework is then used to study the problem of optimal topology design for robust control goals of different grids. For radial grids, the problem is shown as equivalent to the hard "optimum communication spanning tree" problem in graph theory and a combinatorial topology construction is presented with bounded approximation gap. Extended to loopy (meshed) grids, a greedy topology design algorithm is discussed. The performance of the topology design algorithms under multiple control objectives are presented on both loopy and radial test grids. Overall, this paper analyzes topology design algorithms on a broad class of control problems in power grid by exploring their combinatorial and graphical properties.Comment: 6 pages, 3 figures, a version of this work will appear in ACC 201

Bicriteria Network Design Problems

We study a general class of bicriteria network design problems. A generic problem in this class is as follows: Given an undirected graph and two minimization objectives (under different cost functions), with a budget specified on the first, find a <subgraph \from a given subgraph-class that minimizes the second objective subject to the budget on the first. We consider three different criteria - the total edge cost, the diameter and the maximum degree of the network. Here, we present the first polynomial-time approximation algorithms for a large class of bicriteria network design problems for the above mentioned criteria. The following general types of results are presented. First, we develop a framework for bicriteria problems and their approximations. Second, when the two criteria are the same %(note that the cost functions continue to be different) we present a ``black box'' parametric search technique. This black box takes in as input an (approximation) algorithm for the unicriterion situation and generates an approximation algorithm for the bicriteria case with only a constant factor loss in the performance guarantee. Third, when the two criteria are the diameter and the total edge costs we use a cluster-based approach to devise a approximation algorithms --- the solutions output violate both the criteria by a logarithmic factor. Finally, for the class of treewidth-bounded graphs, we provide pseudopolynomial-time algorithms for a number of bicriteria problems using dynamic programming. We show how these pseudopolynomial-time algorithms can be converted to fully polynomial-time approximation schemes using a scaling technique.Comment: 24 pages 1 figur

Optimising a nonlinear utility function in multi-objective integer programming

In this paper we develop an algorithm to optimise a nonlinear utility function of multiple objectives over the integer efficient set. Our approach is based on identifying and updating bounds on the individual objectives as well as the optimal utility value. This is done using already known solutions, linear programming relaxations, utility function inversion, and integer programming. We develop a general optimisation algorithm for use with k objectives, and we illustrate our approach using a tri-objective integer programming problem.Comment: 11 pages, 2 tables; v3: minor revisions, to appear in Journal of Global Optimizatio

An O(1)-Approximation for Minimum Spanning Tree Interdiction

Network interdiction problems are a natural way to study the sensitivity of a network optimization problem with respect to the removal of a limited set of edges or vertices. One of the oldest and best-studied interdiction problems is minimum spanning tree (MST) interdiction. Here, an undirected multigraph with nonnegative edge weights and positive interdiction costs on its edges is given, together with a positive budget B. The goal is to find a subset of edges R, whose total interdiction cost does not exceed B, such that removing R leads to a graph where the weight of an MST is as large as possible. Frederickson and Solis-Oba (SODA 1996) presented an O(log m)-approximation for MST interdiction, where m is the number of edges. Since then, no further progress has been made regarding approximations, and the question whether MST interdiction admits an O(1)-approximation remained open. We answer this question in the affirmative, by presenting a 14-approximation that overcomes two main hurdles that hindered further progress so far. Moreover, based on a well-known 2-approximation for the metric traveling salesman problem (TSP), we show that our O(1)-approximation for MST interdiction implies an O(1)-approximation for a natural interdiction version of metric TSP