Power Systems Signal Processing for Smart Grids -- Contents -- About the Authors -- Preface -- Accompanying Websites -- Acknowledgments -- 1 Introduction -- 1.1 Introduction -- 1.2 The Future Grid -- 1.3 Motivation and Objectives -- 1.4 Signal Processing Framework -- 1.5 Conclusions -- References -- 2 Power Systems and Signal Processing -- 2.1 Introduction -- 2.2 Dynamic Overvoltage -- 2.2.1 Sustained Overvoltage -- 2.2.2 Lightning Surge -- 2.2.3 Switching Surges -- 2.2.4 Switching of Capacitor Banks -- 2.3 Fault Current and DC Component -- 2.4 Voltage Sags and Voltage Swells -- 2.5 Voltage Fluctuations -- 2.6 Voltage and Current Imbalance -- 2.7 Harmonics and Interharmonics -- 2.8 Inrush Current in Power Transformers -- 2.9 Over-Excitation of Transformers -- 2.10 Transients in Instrument Transformers -- 2.10.1 Current Transformer (CT) Saturation (Protection Services) -- 2.10.2 Capacitive Voltage Transformer (CVT) Transients -- 2.11 Ferroresonance -- 2.12 Frequency Variation -- 2.13 Other Kinds of Phenomena and their Signals -- 2.14 Conclusions -- References -- 3 Transducers and Acquisition Systems -- 3.1 Introduction -- 3.2 Voltage Transformers (VTs) -- 3.3 Capacitor Voltage Transformers -- 3.4 Current Transformers -- 3.5 Non-Conventional Transducers -- 3.5.1 Resistive Voltage Divider -- 3.5.2 Optical Voltage Transducer -- 3.5.3 Rogowski Coil -- 3.5.4 Optical Current Transducer -- 3.6 Analog-to-Digital Conversion Processing -- 3.6.1 Supervision and Control -- 3.6.2 Protection -- 3.6.3 Power Quality -- 3.7 Mathematical Model for Noise -- 3.8 Sampling and the Anti-Aliasing Filtering -- 3.9 Sampling Rate for Power System Application -- 3.10 Smart-Grid Context and Conclusions -- References -- 4 Discrete Transforms -- 4.1 Introduction -- 4.2 Representation of Periodic Signals using Fourier Series -- 4.2.1 Computation of Series Coefficients.

4.2.2 The Exponential Fourier Series -- 4.2.3 Relationship between the Exponential and Trigonometric Coefficients -- 4.2.4 Harmonics in Power Systems -- 4.2.5 Proprieties of a Fourier Series -- 4.3 A Fourier Transform -- 4.3.1 Introduction and Examples -- 4.3.2 Fourier Transform Properties -- 4.4 The Sampling Theorem -- 4.5 The Discrete-Time Fourier Transform -- 4.5.1 DTFT Pairs -- 4.5.2 Properties of DTFT -- 4.6 The Discrete Fourier Transform (DFT) -- 4.6.1 Sampling the Fourier Transform -- 4.6.2 Discrete Fourier Transform Theorems -- 4.7 Recursive DFT -- 4.8 Filtering Interpretation of DFT -- 4.8.1 Frequency Response of DFT Filter -- 4.8.2 Asynchronous Sampling -- 4.9 The z-Transform -- 4.9.1 Rational z-Transforms -- 4.9.2 Stability of Rational Transfer Function -- 4.9.3 Some Common z-Transform Pairs -- 4.9.4 z-Transform Properties -- 4.10 Conclusions -- References -- 5 Basic Power Systems Signal Processing -- 5.1 Introduction -- 5.2 Linear and Time-Invariant Systems -- 5.2.1 Frequency Response of LTI System -- 5.2.2 Linear Phase FIR Filter -- 5.3 Basic Digital System and Power System Applications -- 5.3.1 Moving Average Systems: Application -- 5.3.2 RMS Estimation -- 5.3.3 Trapezoidal Integration and Bilinear Transform -- 5.3.4 Differentiators Filters: Application -- 5.3.5 Simple Differentiator -- 5.4 Parametric Filters in Power System Applications -- 5.4.1 Filter Specification -- 5.4.2 First-Order Low-Pass Filter -- 5.4.3 First-Order High-Pass Filter -- 5.4.4 Bandstop IIR Digital Filter (The Notch Filter) -- 5.4.5 Total Harmonic Distortion in Time Domain (THD) -- 5.4.6 Signal Decomposition using a Notch Filter -- 5.5 Parametric Notch FIR Filters -- 5.6 Filter Design using MATLAB® (FIR and IIR) -- 5.7 Sine and Cosine FIR Filters -- 5.8 Smart-Grid Context and Conclusions -- References -- 6 Multirate Systems and Sampling Alterations.

6.1 Introduction -- 6.2 Basic Blocks for Sampling Rate Alteration -- 6.2.1 Frequency Domain Interpretation -- 6.2.2 Up-Sampling in Frequency Domain -- 6.2.3 Down-Sampling in Frequency Domain -- 6.3 The Interpolator -- 6.3.1 The Input-Output Relation for the Interpolator -- 6.3.2 Multirate System as a Time-Varying System and Nobles Identities -- 6.4 The Decimator -- 6.4.1 Introduction -- 6.4.2 The Input-Output Relation for the Decimator -- 6.5 Fractional Sampling Rate Alteration -- 6.5.1 Resampling Using MATLAB® -- 6.6 Real-Time Sampling Rate Alteration -- 6.6.1 Spline Interpolation -- 6.6.2 Cubic B-Spline Interpolation -- 6.7 Conclusions -- References -- 7 Estimation of Electrical Parameters -- 7.1 Introduction -- 7.2 Estimation Theory -- 7.3 Least-Squares Estimator (LSE) -- 7.3.1 Linear Least-Squares -- 7.4 Frequency Estimation -- 7.4.1 Frequency Estimation Based on Zero Crossing (IEC61000-4-30) -- 7.4.2 Short-Term Frequency Estimator Based on Zero Crossing -- 7.4.3 Frequency Estimation Based on Phasor Rotation -- 7.4.4 Varying the DFT Window Size -- 7.4.5 Frequency Estimation Based on LSE -- 7.4.6 IIR Notch Filter -- 7.4.7 Small Coefficient and/or Small Arithmetic Errors -- 7.5 Phasor Estimation -- 7.5.1 Introduction -- 7.5.2 The PLL Structure -- 7.5.3 Kalman Filter Estimation -- 7.5.4 Example of Phasor Estimation using Kalman Filter -- 7.6 Phasor Estimation in Presence of DC Component -- 7.6.1 Mathematical Model for the Signal in Presence of DC Decaying -- 7.6.2 Mimic Method -- 7.6.3 Least-Squares Estimator -- 7.6.4 Improved DTFT Estimation Method -- 7.7 Conclusions -- References -- 8 Spectral Estimation -- 8.1 Introduction -- 8.2 Spectrum Estimation -- 8.2.1 Understanding Spectral Leakage -- 8.2.2 Interpolation in Frequency Domain: Single-Tone Signal -- 8.3 Windows -- 8.3.1 Frequency-Domain Windowing.

8.4 Interpolation in Frequency Domain: Multitone Signal -- 8.5 Interharmonics -- 8.5.1 Typical Interhamonic Sources -- 8.5.2 The IEC Standard 61000-4-7 -- 8.6 Interharmonic Detection and Estimation Based on IEC Standard -- 8.7 Parametric Methods for Spectral Estimation -- 8.7.1 Prony Method -- 8.7.2 Signal and Noise Subspace Techniques -- 8.8 Conclusions -- References -- 9 Time-Frequency Signal Decomposition -- 9.1 Introduction -- 9.2 Short-Time Fourier Transform -- 9.2.1 Filter Banks Interpretation -- 9.2.2 Choosing the Window: Uncertainty Principle -- 9.2.3 The Time-Frequency Grid -- 9.3 Sliding Window DFT -- 9.3.1 Sliding Window DFT: Modified Structure -- 9.3.2 Power System Application -- 9.4 Filter Banks -- 9.4.1 Two-Channel Quadrature-Mirror Filter Bank -- 9.4.2 An Alias-Free Realization -- 9.4.3 A PR Condition -- 9.4.4 Finding the Filters from P(z) -- 9.4.5 General Filter Banks -- 9.4.6 Harmonic Decomposition Using PR Filter Banks -- 9.4.7 The Sampling Frequency -- 9.4.8 Extracting Even Harmonics -- 9.4.9 The Synthesis Filter Banks -- 9.5 Wavelet -- 9.5.1 Continuous Wavelet Transform -- 9.5.2 The Inverse Continuous Wavelet Transform -- 9.5.3 Discrete Wavelet Transform (DWT) -- 9.5.4 The Inverse Discrete Wavelet Transform -- 9.5.5 Discrete-Time Wavelet Transform -- 9.5.6 Design Issues in Wavelet Transform -- 9.5.7 Power System Application of Wavelet Transform -- 9.5.8 Real-Time Wavelet Implementation -- 9.6 Conclusions -- References -- 10 Pattern Recognition -- 10.1 Introduction -- 10.2 The Basics of Pattern Recognition -- 10.2.1 Datasets -- 10.2.2 Supervised and Unsupervised Learning -- 10.3 Bayes Decision Theory -- 10.4 Feature Extraction on the Power Signal -- 10.4.1 Effective Value (RMS) -- 10.4.2 Discrete Fourier Transform -- 10.4.3 Wavelet Transform -- 10.4.4 Cumulants of Higher-Order Statistics -- 10.4.5 Principal Component Analysis.

10.4.6 Normalization -- 10.4.7 Feature Selection -- 10.5 Classifiers -- 10.5.1 Minimum Distance Classifiers -- 10.5.2 Nearest Neighbor Classifier -- 10.5.3 The Perceptron -- 10.5.4 Least-Squares Methods -- 10.5.5 Multilayer Perceptron -- 10.5.6 Support Vector Machines -- 10.6 System Evaluation -- 10.6.1 Estimation of the Classification Error Probability -- 10.6.2 Limited-Size Dataset -- 10.7 Pattern Recognition Examples in Power Systems -- 10.7.1 Power Quality Disturbance Classification -- 10.7.2 Load Forecasting in Electric Power Systems -- 10.7.3 Power System Security Assessment -- 10.8 Conclusions -- References -- 11 Detection -- 11.1 Introduction -- 11.2 Why Signal Detection for Electric Power Systems? -- 11.3 Detection Theory Basics -- 11.3.1 Detection on the Bayesian Framework -- 11.3.2 Newman-Pearson Criterion -- 11.3.3 Receiving Operating Characteristics -- 11.3.4 Deterministic Signal Detection in White Gaussian Noise -- 11.3.5 Deterministic Signals with Unknown Parameters -- 11.4 Detection of Disturbances in Power Systems -- 11.4.1 The Power System Signal -- 11.4.2 Optimal Detection -- 11.4.3 Feature Extraction -- 11.4.4 Commonly Used Detection Algorithms -- 11.5 Examples -- 11.5.1 Transmission Lines Protection -- 11.5.2 Detection Algorithms Based on Estimation -- 11.5.3 Saturation Detection in Current Transformers -- 11.6 Smart-Grid Context and Conclusions -- References -- 12 Wavelets Applied to Power Fluctuations -- 12.1 Introduction -- 12.2 Basic Theory -- 12.3 Application of Wavelets for Time-Varying Generation and Load Profiles -- 12.3.1 Fluctuation Analyses with FFT -- 12.3.2 Methodology -- 12.3.3 Load Fluctuations -- 12.3.4 Wind Farm Generation Fluctuations -- 12.3.5 Smart Microgrid -- 12.4 Conclusions -- References -- 13 Time-Varying Harmonic and Asymmetry Unbalances -- 13.1 Introduction -- 13.2 Sequence Component Computation.

13.3 Time-Varying Unbalance and Harmonic Frequencies.