Smart Grid using Big Data Analytics

Preface xv Acknowledgments xix Some Notation xxi 1 Introduction 1 1.1 Big Data: Basic Concepts 1 1.2 Data Mining with Big Data 9 1.3 A Mathematical Introduction to Big Data 13 1.4 A Mathematical Theory of Big Data 28 1.5 Smart Grid 34 1.6 Big Data and Smart Grid 36 1.7 Reading Guide 37 Bibliographical Remarks 39 Part I Fundamentals of Big Data 41 2 The Mathematical Foundations of Big Data Systems 43 2.1 Big Data Analytics 44 2.2 Big Data: Sense, Collect, Store, and Analyze 45 2.3 Intelligent Algorithms 48 2.4 Signal Processing for Smart Grid 48 2.5 Monitoring and Optimization for Power Grids 48 2.6 Distributed Sensing and Measurement for Power Grids 49 2.7 Real-time Analysis of Streaming Data 50 2.8 Salient Features of Big Data 51 2.9 Big Data for Quantum Systems 54 2.10 Big Data for Financial Systems 55 2.11 Big Data for Atmospheric Systems 73 2.12 Big Data for Sensing Networks 74 2.13 Big Data forWireless Networks 75 2.14 Big Data for Transportation 78 Bibliographical Remarks 78 3 Large Random Matrices: An Introduction 79 3.1 Modeling of Large Dimensional Data as Random Matrices 79 3.2 A Brief of Random MatrixTheory 81 3.3 Change Point of Views: From Vectors to Measures 85 3.4 The Stieltjes Transform of Measures 86 3.5 A Fundamental Result: The Marchenko-Pastur Equation 88 3.6 Linear Eigenvalue Statistics and Limit Laws 89 3.7 Central LimitTheorem for Linear Eigenvalue Statistics 99 3.8 Central LimitTheorem for Random Matrix S.1T 101 3.9 Independence for Random Matrices 103 3.10 Matrix-Valued Gaussian Distribution 110 3.11 Matrix-ValuedWishart Distribution 112 3.12 Moment Method 112 3.13 Stieltjes Transform Method 113 3.14 Concentration of the Spectral Measure for Large Random Matrices 114 3.15 Future Directions 117 Bibliographical Remarks 117 4 Linear Spectral Statistics of the Sample Covariance Matrix 121 4.1 Linear Spectral Statistics 121 4.2 Generalized Marchenko-Pastur Distributions 122 4.3 Estimation of Spectral Density Functions 127 4.4 Limiting Spectral Distribution of Time Series 146 Bibliographical Remarks 154 5 Large Hermitian Random Matrices and Free Random Variables 155 5.1 Large Economic/Financial Systems 156 5.2 Matrix-Valued Probability 157 5.3 Wishart-Levy Free Stable Random Matrices 166 5.4 Basic Concepts for Free Random Variables 168 5.5 The Analytical Spectrum of theWishart-Levy Random Matrix 172 5.6 Basic Properties of the Stieltjes Transform 176 5.7 Basic Theorems for the Stieltjes Transform 179 5.8 Free Probability for Hermitian Random Matrices 185 5.9 Random Vandermonde Matrix 196 5.10 Non-Asymptotic Analysis of State Estimation 200 Bibliographical Remarks 201 6 Large Non-Hermitian Random Matrices and Quatartenionic Free Probability Theory 203 6.1 Quatartenionic Free ProbabilityTheory 204 6.2 R-diagonalMatrices 209 6.3 The Sum of Non-Hermitian Random Matrices 216 6.4 The Product of Non-Hermitian Random Matrices 220 6.5 Singular Value Equivalent Models 226 6.6 The Power of the Non-Hermitian Random Matrix 234 6.7 Power Series of Large Non-Hermitian Random Matrices 239 6.8 Products of Random Ginibre Matrices 246 6.9 Products of Rectangular Gaussian Random Matrices 249 6.10 Product of ComplexWishart Matrices 252 6.11 Spectral Relations between Products and Powers 254 6.12 Products of Finite-Size I.I.D. Gaussian Random Matrices 258 6.13 Lyapunov Exponents for Products of Complex Gaussian Random Matrices 260 6.14 Euclidean Random Matrices 264 6.15 Random Matrices with Independent Entries and the Circular Law 273 6.16 The Circular Law and Outliers 275 6.17 Random SVD, Single Ring Law, and Outliers 285 6.18 The Elliptic Law and Outliers 295 Bibliographical Remarks 305 7 The Mathematical Foundations of Data Collection 307 7.1 Architectures and Applications for Big Data 307 7.2 Covariance Matrix Estimation 308 7.3 Spectral Estimators for Large Random Matrices 312 7.4 Asymptotic Framework for Matrix Reconstruction 319 7.5 Optimum Shrinkage 329 7.6 A Shrinkage Approach to Large-Scale Covariance Matrix Estimation 331 7.7 Eigenvectors of Large Sample Covariance Matrix Ensembles 338 7.8 A General Class of Random Matrices 351 Bibliographical Remarks 359 8 Matrix Hypothesis Testing using Large RandomMatrices 361 8.1 Motivating Examples 362 8.2 Hypothesis Test of Two Alternative Random Matrices 363 8.3 Eigenvalue Bounds for Expectation and Variance 364 8.4 Concentration of Empirical Distribution Functions 369 8.5 Random Quadratic Forms 381 8.6 Log-Determinant of Random Matrices 382 8.7 General MANOVA Matrices 383 8.8 Finite Rank Perturbations of Large Random Matrices 386 8.9 Hypothesis Tests for High-Dimensional Datasets 391 8.9.1 Motivation for Likelihood Ratio Test (LRT) and Covariance Matrix Tests 392 8.10 Roy's Largest Root Test 428 8.11 Optimal Tests of Hypotheses for Large Random Matrices 431 8.12 Matrix Elliptically Contoured Distributions 444 8.13 Hypothesis Testing for Matrix Elliptically Contoured Distributions 446 Bibliographical Remarks 452 Part II Smart Grid 455 9 Applications and Requirements of Smart Grid 457 9.1 History 457 9.2 Concepts and Vision 458 9.3 Today's Electric Grid 459 9.4 Future Smart Electrical Energy System 464 10 Technical Challenges for Smart Grid 471 Bibliographical Remarks 483 11 Big Data for Smart Grid 485 11.1 Power in Numbers: Big Data and Grid Infrastructure 485 11.2 Energy's Internet:The Convergence of Big Data and the Cloud 486 11.3 Edge Analytics: Consumers, Electric Vehicles, and Distributed Generation 486 11.4 CrosscuttingThemes: Big Data 486 11.5 Cloud Computing for Smart Grid 488 11.6 Data Storage, Data Access and Data Analysis 488 11.7 The State-of-the-Art Processing Techniques of Big Data 488 11.8 Big Data Meets the Smart Electrical Grid 488 11.9 4Vs of Big Data: Volume, Variety, Value and Velocity 489 11.10 Cloud Computing for Big Data 490 11.11 Big Data for Smart Grid 490 11.12 Information Platforms for Smart Grid 491 Bibliographical Remarks 491 12 Grid Monitoring and State Estimation 493 12.1 Phase Measurement Unit 493 12.2 Optimal PMU Placement 495 12.3 State Estimation 495 12.4 Basics of State Estimation 495 12.5 Evolution of State Estimation 496 12.6 Static State Estimation 497 12.7 Forecasting-Aided State Estimation 500 12.8 Phasor Measurement Units 501 12.9 Distributed System State Estimation 502 12.10 Event-Triggered Approaches to State Estimation 502 12.11 Bad Data Detection 502 12.12 Improved Bad Data Detection 504 12.13 Cyber-Attacks 504 12.14 Line Outage Detection 504 Bibliographical Remarks 504 13 False Data Injection Attacks against State Estimation 505 13.1 State Estimation 505 13.2 False Data Injection Attacks 507 13.3 MMSE State Estimation and Generalized Likelihood Ratio Test 508 13.4 Sparse Recovery from Nonlinear Measurements 512 13.5 Real-Time Intrusion Detection 515 Bibliographical Remarks 515 14 Demand Response 517 14.1 Why Engage Demand? 517 14.2 Optimal Real-time Pricing Algorithms 520 14.3 Transportation Electrification and Vehicle-to-Grid Applications 522 14.4 Grid Storage 522 Bibliographical Remarks 523 Part III Communications and Sensing 525 15 Big Data for Communications 527 15.1 5G and Big Data 527 15.2 5GWireless Communication Networks 527 15.3 Massive Multiple Input, Multiple Output 528 15.4 Free Probability for the Capacity of the Massive MIMO Channel 537 15.5 Spectral Sensing for Cognitive Radio 539 Bibliographical Remarks 539 16 Big Data for Sensing 541 16.1 Distributed Detection and Estimation 541 16.2 Euclidean Random Matrix 547 16.3 Decentralized Computing 548 Appendix A: Some Basic Results on Free Probability 551 Appendix B: Matrix-Valued Random Variables 557 References 567 Index 601