Rubido Energy Transmission and Synchronization in Complex Networks

Mathematical Principles
1. Auflage 2016
ISBN: 978-3-319-22216-5
Verlag: Springer Nature Switzerland
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 130 Seiten

Reihe: Springer Theses

This work tackles the problems of understanding how energy is transmitted and distributed in power-grids as well as in determining how robust this transmission and distribution is when modifications to the grid or power occur. The most important outcome is the derivation of explicit relationships between the structure of the grid, the optimal transmission and distribution of energy, and the grid's collective behavior (namely, the synchronous generation of power). These relationships are extremely relevant for the design of resilient power-grid models. To allow the reader to apply these results to other complex systems, the thesis includes a review of relevant aspects of network theory, spectral theory, and novel analytical calculations to predict the existence and stability of periodic collective behavior in complex networks of phase oscillators, which constitute a paradigmatic model for many complex systems.

Nicolas Rubido received his Ph.D. in 2014 from the University of Aberdeen, which he owes to the 2011 the Scottish Universities Physics Alliance studentship prize. He received his B.Sc. (2008) and M.Sc. (2010) degrees from the Universidad de la República, Uruguay, where he is now an Adjunct Professor. There, he was awarded a research initiation scholarship (2009) to study turbulence phenomena and a post-graduate scholarship (2010) to research into synchronization phenomena in gregarious fireflies, which resulted in various publications. In general, his research focuses on understanding the structure and behaviours of different complex systems from a mathematical and data-analysis perspective.

Autoren/Hrsg.

Rubido, Nicolás

Inhaltsverzeichnis

1;Supervisor's Foreword;7
2;Abstract;9
3;Acknowledgments;10
4;Contents;12
5;Notations;14
5.1;Observations on Notations;14
5.2;Notations for Methods;14
5.3;Notation for Transmission of Energy;15
5.4;Notation for Synchronisation;16
6;1 Introduction;17
6.1;1.1 Background;17
6.1.1;1.1.1 Complexity: A Shift from Reductionism;17
6.1.2;1.1.2 Complex Systems: Structure and Behaviour;20
6.2;1.2 Motivation;22
6.2.1;1.2.1 Transmission of Energy;22
6.2.2;1.2.2 Synchronisation;23
6.3;1.3 Summary of Results;24
6.4;References;27
7;2 Complex Networks;29
7.1;2.1 Networks;29
7.1.1;2.1.1 Notions and Definitions;30
7.1.2;2.1.2 Matrix Representations;33
7.1.3;2.1.3 Network Characterisation Methods;36
7.1.4;2.1.4 Archetypal Networks;38
7.2;2.2 The Laplacian Matrix;42
7.2.1;2.2.1 General Properties;42
7.2.2;2.2.2 Eigenvalues and Eigenvectors;44
7.2.3;2.2.3 Spectral Theorems;47
7.2.4;2.2.4 Spectral Properties of Laplacian Matrices;49
7.3;2.3 Resistance Distance;51
7.3.1;2.3.1 Laplacian Matrix Pseudo-inverse;52
7.3.2;2.3.2 Circulant Networks Spectral Characteristics;54
7.3.3;2.3.3 Equivalent Resistance, Its Bounds, and the Kirchhoff Index;56
7.4;References;59
8;3 Transmission of Energy;60
8.1;3.1 Flow Networks;61
8.1.1;3.1.1 AC/DC Flow Networks;61
8.1.2;3.1.2 DC Conservative Flow-Network Solutions;64
8.1.3;3.1.3 Configurational Edge-Capacity;67
8.1.4;3.1.4 Steady-State Cascade of Failures;69
8.1.5;3.1.5 Configurational Edge Capacity Margins;73
8.1.6;3.1.6 Community Detection via Flow Networks;76
8.2;3.2 Power Networks;78
8.2.1;3.2.1 Swing Equations: Power-Grid Model;78
8.2.2;3.2.2 Variational Equations;83
8.2.3;3.2.3 Stability Criteria for Networks of Generators;85
8.2.4;3.2.4 Equilibrium Solutions for Networks of Generators;89
8.2.5;3.2.5 Stability of Equilibrium Solutions in Circulant Networks;92
8.2.6;3.2.6 Energy Manifold for Networks of Generators;94
8.3;References;96
9;4 Synchronisation;99
9.1;4.1 Phase Oscillators;100
9.1.1;4.1.1 Model and Frameworks;101
9.1.2;4.1.2 Collective Behaviour: Conditions for Existence;103
9.1.3;4.1.3 Phase Master Stability Function;108
9.1.4;4.1.4 Fixed-Points Explicit Solutions;111
9.2;References;115
10;5 General Conclusions;117
11; Appendix;119
12; Curriculum Vitae;122