Power System Harmonic Analysis

, by Arrillaga, Jos; Smith, Bruce C.; Watson, Neville R.; Wood, Alan R.

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ISBN: 9780471975489 | 0471975486
Cover: Hardcover
Copyright: 10/7/1997

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Quality of power supply is now a major issue worldwide making harmonic analysis an essential element in power system planning and design. Power System Harmonic Analysis presents novel analytical and modelling tools for the assessment of components and systems, and their interactions at harmonic frequencies. The recent proliferation of power electronic equipment is a significant source of harmonic distortion and the authors present effective techniques to tackle this real engineering problem. Features include: * Introduction to the main harmonic modelling philosophies * Analysis of the behaviour of harmonic sources, stressing the interaction of ac/dc converters with the power system * Information showing the reader how to predict accurately the levels of voltage and current harmonics throughout the power system * Explanation of the techniques currently used for the prediction of harmonic content and the more advanced algorithms recently developed to determine both characteristic and uncharacteristic harmonic levels * Description of methods to facilitate accurate assessment of harmonic sources and precise harmonic flow analysis * Practical guidance on the prediction of unstable conditions and uncharacteristic harmonics Presenting effective techniques for the analysis and resolution of harmonic interactions, this valuable book will be an asset to engineers and researchers involved in the planning, design and operation of power systems. Power System Harmonic Analysis will also serve as a useful reference for postgraduate students following courses in power systems and power electronics disciplines.

Preface

1 Introduction

(6)

1.1 Power System Harmonics

(1)

1.2 The Main Harmonic Sources

(1)

1.3 Modelling Philosophies

(1)

1.4 Time Domain Simulation

(1)

1.5 Frequency Domain Simulation

(1)

1.6 Iterative Methods

(1)

1.7 References

(2)

2 Fourier Analysis

(26)

2.1 Introduction

(1)

2.2 Fourier Series and Coefficients

(3)

2.3 Simplifications Resulting from Waveform Symmetry

(3)

2.4 Complex Form of the Fourier Series

(2)

2.5 Convolution of Harmonic Phasors

(2)

2.6 The Fourier Transform

(2)

2.7 Sampled Time Functions

(1)

2.8 Discrete Fourier Transform

(4)

2.9 Fast Fourier Transform

(2)

2.10 Transfer Function Fourier Analysis

(5)

2.11 Summary

(1)

2.12 References

(2)

3 Transmission Systems

(64)

3.1 Introduction

(1)

3.2 Network Subdivision

(2)

3.3 Frame of Reference used in Three-Phase System Modelling

(2)

3.4 Evaluation of Transmission Line Parameters

(9)

3.4.1 Earth Impedance Matrix [Z(e)]

(2)

3.4.2 Geometrical Impedance Matrix [Z(g)] and Admittance Matrix [Y(g)]

(2)

3.4.3 Conductor Impedance Matrix [Z(c)]

(5)

3.5 Single Phase Equivalent of a Transmission Line

(6)

3.5.1 Equivalent PI Models

(6)

3.6 Multiconductor Transmission Line

(9)

3.6.1 Nominal PI Model

(4)

3.6.2 Mutually Coupled Three-Phase Lines

(2)

3.6.3 Consideration of Terminal Connections

(1)

3.6.4 Equivalent PI Model

(2)

3.7 Three-Phase Transformer Models

(4)

3.8 Line Compensating Plant

(2)

3.8.1 Shunt Elements

(2)

3.8.2 Series Elements

(1)

3.9 Underground and Submarine Cables

(4)

3.10 Examples of Application of the Models

(23)

3.10.1 Harmonic Flow in a Homogeneous Transmission Line

(4)

3.10.2 Harmonic Analysis of Transmission Line with Transpositions

(9)

3.10.3 Harmonic Analysis of Transmission Line with Var Compensation

(3)

3.10.4 Harmonic Analysis in a Hybrid HVdc Transmission Link

(7)

3.11 Summary

(1)

3.12 References

(3)

4 Direct Harmonic Solutions

(36)

4.1 Introduction

(1)

4.2 Nodal Harmonic Analysis

(3)

4.2.1 Incorporation of Harmonic Voltage Sources

100

(1)

4.3 Harmonic Impedances

101

(13)

4.3.1 Generator and Transformer Modelling

101

(1)

4.3.2 Distribution and Load System Modelling

102

(2)

4.3.3 Induction Motor Model

104

(3)

4.3.4 Detail of System Representation

107

(2)

4.3.5 System Impedances

109

(5)

4.3.6 Existing Non-linearities

114

(1)

4.4 Computer Implementation

114

(14)

4.4.1 Structure of the Algorithm

114

(2)

4.4.2 Data Programs

116

(10)

4.4.3 Application Programs

126

(1)

4.4.4 Post Processing

127

(1)

4.5 Summary

128

(2)

4.6 References

130

(3)

5 AC-DC Conversion -- Frequency Domain

133

(40)

5.1 Introduction

133

(1)

5.2 Characteristic Converter Harmonics

133

(11)

5.2.1 Effect of Transformer Connection

137

(1)

5.2.2 Twelve-pulse Related Harmonics

138

(1)

5.2.3 Higher Pulse Configurations

139

(1)

5.2.4 Insufficient Smoothing Reactance

140

(1)

5.2.5 Effect of Transformer and System Impedance

141

(3)

5.3 Frequency Domain Model

144

(13)

5.3.1 Commutation Analysis

147

(3)

5.3.2 Control Transfer Functions

150

(1)

5.3.3 Transfer of Waveform Distortion

151

(5)

5.3.4 Discussion

156

(1)

5.4 The Converter Frequency Dependent Equivalent

157

(12)

5.4.1 Frequency Dependent Impedance

160

(4)

5.4.2 Converter DC Side Impedances

164

(2)

5.4.3 Converter AC Side Positive Sequence Impedances

166

(1)

5.4.4 Converter AC Side Negative Sequence Impedances

166

(1)

5.4.5 Simplified Converter Impedances

167

(1)

5.4.6 Example of Application of the Impedance Models

168

(1)

5.5 Summary

169

(2)

5.6 References

171

(2)

6 Harmonic Instabilities

173

(20)

6.1 Introduction

173

(1)

6.2 Composite Resonance -- A Circuit Approach

174

(6)

6.2.1 The Effect of Firing Angle Control on Converter Impedance

175

(1)

6.2.2 Test Case

176

(3)

6.2.3 Discussion

179

(1)

6.3 Transformer Core Related Harmonic Instability in AC-DC Systems

180

(10)

6.3.1 AC-DC Frequency Interactions

180

(2)

6.3.2 Instability Mechanism

182

(1)

6.3.3 Instability Analysis

183

(4)

6.3.4 Dynamic Verification

187

(1)

6.3.5 Characteristics of the Instability

188

(1)

6.3.6 Control of the Instability

189

(1)

6.4 Summary

190

(1)

6.5 References

191

(2)

7 Machine Non-linearities -- Harmonic Domain

193

(30)

7.1 Introduction

193

(1)

7.2 Synchronous Machine

193

(14)

7.2.1 The Frequency Conversion Process

194

(1)

7.2.2 Harmonic Model in dq Axes

195

(1)

7.2.3 Two-phase Transformation dq to XXX

196

(2)

7.2.4 Admittance Matrix [Y(XXX)]

198

(1)

7.2.5 Admittance Matrix [Y(abc)]

199

(1)

7.2.6 Illustration of Harmonic Impedances

200

(2)

7.2.7 Model Validation

202

(1)

7.2.8 Accounting for Saturation

202

(3)

7.2.9 Norton Equivalent

205

(1)

7.2.10 Case Studies

206

(1)

7.3 Transformers

207

(14)

7.3.1 Representation of the Magnetisation Characteristics

208

(1)

7.3.2 Norton Equivalent of the Magnetic Non-Linearity

209

(2)

7.3.3 Generalisation of the Norton Equivalent

211

(5)

7.3.4 Full Harmonic Electromagnetic Representation

216

(1)

7.3.5 Case Study

216

(5)

7.4 Summary

221

(1)

7.5 References

221

(2)

8 AC-DC Conversion -- Harmonic Domain

223

(18)

8.1 Introduction

223

(1)

8.2 The Commutation Process

224

(3)

8.2.1 Star Connection Analysis

224

(2)

8.2.2 Delta Connection Analysis

226

(1)

8.3 The Valve Firing Process

227

(2)

8.4 DC-Side Voltage

229

(5)

8.4.1 Star Connection Voltage Samples

229

(1)

8.4.2 Delta Connection Voltage Samples

230

(2)

8.4.3 Convolution of the Samples

232

(2)

8.5 Phase Currents on the Converter Side

234

(1)

8.6 Phase Currents on the System Side

234

(6)

8.7 Summary

240

(1)

8.8 References

240

(1)

9 Iterative Harmonic Analysis

241

(42)

9.1 Introduction

241

(1)

9.2 Fixed Point Iteration Techniques

241

(1)

9.3 The Method of Norton Equivalents

242

(4)

9.4 ABCD Parameters Model

246

(1)

9.5 Newton's Method

246

(25)

9.5.1 Functional Description of the Twelve Pulse Converter

248

(2)

9.5.2 Composition of Mismatch Functions

250

(3)

9.5.3 Solution Algorithm

253

(6)

9.5.4 Computer Implementation

259

(6)

9.5.5 Validation and Performance

265

(6)

9.6 Diagonalizing Transforms

271

(7)

9.7 Integrated Converter and Load Flow Solution

278

(1)

9.8 Summary

279

(2)

9.9 References

281

(2)

10 Converter Harmonic Impedances

283

(28)

10.1 Introduction

283

(1)

10.2 Calculation of the Converter Impedance

284

(20)

10.2.1 Perturbation Analysis

284

(4)

10.2.2 The Lattice Tensor

288

(6)

10.2.3 Derivation of the Converter Impedance by Kron Reduction

294

(6)

10.2.4 Sparse Implementation of the Kron Reduction

300

(4)

10.3 Variation of the Converter Impedance

304

(3)

10.4 Summary

307

(2)

10.5 References

309

(2)

Appendix I Efficient Derivation of Impedance Loci

311

(6)

I.1 Adaptive Sampling Scheme

311

(1)

I.2 Winding Angle Criterion

311

(6)

Appendix II Pulse Position Modulation Analysis

317

(10)

II.1 The PPM Spectrum

317

(1)

II.2 Contribution of Commutation Duration to DC Voltage

318

(2)

II.3 Contribution of Commutation Duration to AC Current

320

(2)

II.4 Contribution of Commutation Period Variation to AC Current

322

(3)

II.5 Reference

325

(2)

Appendix III Pulse Duration Modulation Analysis

327

(4)

III.1 The PDM spectrum

327

(2)

III.2 Firing Angle Modulation Applied to the Ideal Transfer Function

329

(1)

III.3 Reference

330

(1)

Appendix IV Derivation of the Jacobian

331

(22)

IV.1 Voltage Mismatch Partial Derivatives

331

(9)

IV.1.1 With Respect to AC Phase Voltage Variation

332

(3)

IV.1.2 With Respect to DC Ripple Current Variation

335

(2)

IV.1.3 With Respect to End of Commutation Variation

337

(2)

IV.1.4 With Respect to Firing Angle Variation

339

(1)

IV.2 Direct Current Partial Derivatives

340

(5)

IV.2.1 With Respect to AC Phase Voltage Variation

340

(2)

IV.2.2 With Respect to Direct Current Ripple Variation

342

(2)

IV.2.3 With Respect to End of Commutation Variation

344

(1)

IV.2.4 With Respect to Firing Angle Variation

345

(1)

IV.3 End of Commutation Mismatch Partial Derivatives

345

(3)

IV.3.1 With Respect to AC Phase Voltage Variation

346

(1)

IV.3.2 With Respect to Direct Current Ripple Variation

347

(1)

IV.3.3 With Respect to End of Commutation Variation

347

(1)

IV.3.4 With Respect to Firing Instant Variation

348

(1)

IV.4 Firing Instant Mismatch Equation Partial Derivatives

348

(1)

IV.5 Average Delay Angle Partial Derivatives

349

(4)

IV.5.1 With Respect to AC Phase Voltage Variation

349

(1)

IV.5.2 With Respect to DC Ripple Current Variation

350

(1)

IV.5.3 With Respect to End of Commutation Variation

350

(1)

IV.5.4 With Respect to Firing Angle Variation

351

(2)

Appendix V The Impedance Tensor

353

(8)

V.1 Impedance Derivation

353

(3)

V.2 Phase Dependent Impedance

356

(5)

Appendix VI Test Systems

361

(4)

VI.1 CIGRE Benchmark

361

(4)

Index

365

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Power System Harmonic Analysis

Summary

Author Biography

Table of Contents

Supplemental Materials