THÈSE Pour obtenir le grade de DOCTEUR DE LA COMMUNAUTÉ UNIVERSITÉ GRENOBLE ALPES Spécialité : GENIE ELECTRIQUE Arrêté ministériel : 25 mai 2016 Présentée par Felix KOETH Thèse dirigée par Nicolas RETIERE préparée au sein Laboratoire de Génie Electrique dans l’École Doctorale EEATS Etude des propriétés spectrales des graphes des réseaux élec- triques. Investigations of Spectral Graph Properties in Power Systems Thèse soutenue publiquement le 19 décembre 2019, devant le jury composé de : Monsieur Didier Georges Professeur des Universités, Grenoble-INP, Président Monsieur Ettore Bompard Professeur des Universités, Politecnico di Torino, Rapporteur Monsieur Philippe Jacquod Professeur des Universités, HES-SO Valais-Wallis, Rapporteur Monsieur Jose Luis Dominguez Charge de Recherche, IREC Barcelona, Examinateur Monsieur Jean-Guy Caputo Maitres de conférénce - INSA Rouen, Invitée Monsieur Nicolas Retiere Professeur des Universités, Université Grenoble Alpes, Directeur de thèse iii Contents General Introduction - Overview 1 Thesis Outline . .2 Résumé en français . .4 1 Geometry and Power Systems 11 1.1 The Role of Geometry . 11 1.2 Power Systems . 13 1.2.1 Transmission of Energy - Power Flow . 13 DC Power Flow . 15 1.2.2 Generation of Energy - Swing Equation . 16 1.2.3 Consumption of Energy - Load Models . 17 1.3 The Role of Geometry in Power System Stability . 18 Complexity and Spectral Solving . 20 2 Spectral Graph Theory 23 2.1 Fundamental Properties of Graphs . 23 2.1.1 Simple Graphs . 23 2.1.2 Networks and Graphs . 24 2.1.3 Paths and Connected Graphs . 25 2.1.4 Subgraphs and Connected Components . 25 2.1.5 Properties and Metrics of Graphs . 26 Vertex Degree . 26 Shortest Path and Diameter . 27 2.1.6 Examples for Graphs . 27 Basic Graphs . 28 Random Graphs . 28 2.1.7 Graph Matrices . 30 Adjacency Matrix . 30 Incidence Matrix . 30 Laplacian Matrix . 31 2.2 Spectral Properties of Graphs . 32 iv 2.2.1 The Laplacian Pseudo Inverse . 33 2.2.2 Algebraic Connectivity . 35 2.2.3 Nodal Domains . 35 2.2.4 Coherency . 37 2.3 Spectral Graph Theory on Power Systems . 38 2.3.1 Small Signal Stability . 38 2.3.2 Transient Stability . 39 2.3.3 Conclusion . 41 3 Spectral Analysis of power flow 43 3.1 Methodology and Theory . 43 3.2 Numerical Results . 45 3.2.1 Spectral Properties of the IEEE 118 Test System . 45 3.2.2 Powerflow Decomposition . 46 3.2.3 Flows in Line . 48 3.3 Conclusions . 54 4 Spectral Properties of Dynamical Power Systems 55 4.1 Modelling and Background . 55 4.1.1 Quadratic Eigenvalue Problem . 57 Overdamped Behaviour . 57 Companion Form . 58 4.1.2 Kron Reduction . 59 4.1.3 Condition Numbers . 59 4.1.4 Oscillatory Networks . 60 4.2 The Laplacian Spectrum and the QEP . 61 Dissipation Mode . 62 4.3 Experimental Investigations . 62 4.3.1 Time Response and Oscillations . 63 4.3.2 Eigenvectors . 64 4.3.3 Spectrum of the Reduced Power System Laplacian . 65 4.3.4 Parameter Influence . 66 Bifurcation Diagrams . 66 Randomly Perturbed Matrices . 67 4.3.5 Pseudo Spectra . 69 4.3.6 Condition Numbers . 71 4.3.7 Eigenvector Sensitivity . 71 4.3.8 Damped and Undamped Oscillations . 73 v 4.4 Proposal - Perturbations of the Undamped System . 75 4.4.1 Small Damping Perturbation . 76 4.4.2 Companion Form Method . 77 Validation of Eigenvalue Perturbation . 78 4.4.3 Eigenvector Perturbations . 79 4.5 Conclusions . 80 5 Localisation of Modes 83 5.1 Background, Definitions and Simple Examples . 83 5.1.1 Fork . 84 5.1.2 High Impedance Line . 85 5.1.3 Exact Localisation . 86 5.1.4 Approximate Localisation . 88 5.2 Localisation Bounds . 88 5.2.1 Localisation of Modes . 89 5.2.2 Localisation in Subgraphs . 91 5.3 Empirical Studies of Landscape Properties . 94 5.3.1 Simple Systems . 94 Fork Graph . 94 High Impedance Line . 95 5.3.2 Landscape Statistics of the IEEE 118 Test System . 96 5.4 Localisation Candidates . 98 5.4.1 Node removal - High Frequencies . 98 5.4.2 Node Removal - Low Frequencies . 101 5.4.3 Mode Prediction . 104 5.4.4 Bounds on the Relative Complement Subgraph . 105 5.5 Localisation and Resonance . 106 5.5.1 Small Subgraphs . 107 5.5.2 Statistical Properties . 108 5.6 Conclusion . 109 6 Conclusions and Outlook 113 6.1 Conclusions . 113 6.2 Outlook . 114 6.2.1 Localisation Bounds . 114 6.2.2 Spectral Properties of Dynamical Power Systems . 115 6.2.3 Nodal Domains and Localisation Bounds . 116 6.2.4 Eigenvectors from Eigenvalues . 117 vi 6.2.5 Application to realistic power system cases . 118 6.2.6 Beyond AC systems . 118 A Additional Theoretical Results 121 A.1 State Space Formalism . 121 A.2 Solution to the Laplace Equation on Graphs . 122 A.3 Improved Landscape Bounds . 123 A.4 Proposal - from the GEP to the Eigenvalue Problem . 124 B Additional Data and Figures 127 B.1 Additional Information for Chapter 4 . 127 B.1.1 Dynamical properties of the IEEE 300 test case . 127 B.1.2 Angles Between Eigenvector Examples . 129 B.2 Additional Information for Chapter 5 . 129 B.2.1 Localisation in the IEEE 300 Test Case . 129 B.2.2 Example Graphs showing Localisation . 130 Fork . 130 High Impedance Line . 131 Bibliography 133 Summaries 145 Summary in English . 145 Summary in French . 146 Acknowledgements 147 Abstract 149 vii List of Figures 1 Flux par ligne pour la deuxième plus petite et la plus grande valeur propre. .5 2 Oscillations du modèle de système électrique linéarisé. .6 3 Spectres du cas d’essai IEEE 118 avec inertie modifiée aléatoirement M et paramètre d’amortissement D....................7 4 Nœuds restants après la suppression des nœuds en utilisant les paysages.8 1.1 Nine possible networks with four identical resistors with resistance R. 11 1.2 Schematic drawing of the start delta transformation. 12 1.3 Example transmission system in Germany . 14 1.4 Overview over time and length scales related to power system oper- ation and design . 21 2.1 Conditioning numbers for the IEEE 118 test case . 24 2.2 Example for a random graph with different subgraphs . 26 2.3 Vertex degree and diameter of the Florentine families graph. 27 2.4 eigenvector components of the subgraph with the largest x....... 28 2.5 Examples for basic graphs. 28 2.6 Creating process of random graphs . 29 2.7 Influence of the random parameter p on random graphs. 30 2.8 Creating process of random graphs . 36 2.9 Two example graphs which exhibit coherency in the Fiedler eigenvector 38 3.1 Ordered eigenvalues and Fiedler vector drawn on a graph plots for the IEEE 118 test system . ..