Power System Harmonic Analysis

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

• ISBN13: 9780471975489
• ISBN10: 0471975486
• Edition: 1st
• Format: Hardcover
• Copyright: 1997-10-07
• Publisher: Wiley

Summary

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.

Author Biography

Jos Arrillaga is an experienced author, now an Emeritus Professor from the Department of Electrical and Computer Engineering at the University of Canterbury, New Zealand. He has written 10 books, including five for Wiley on the topic of electrical power systems, such as Power System Harmonics, Second Edition, Computer Modelling of Electrical Power Systems, Second Edition and High Voltage Direct Current Transmission. He has also written over 350 journal and conference papers. During the course of his career he has supervised around 50 PhD and 60 MSc/ME theses, most of them on the subject of high voltage direct current transmission, and he has also participated and convened several working groups. In 1997 he was awarded the Uno Lamm medal for outstanding contributions to HVDC transmission and he was in the New Years Honours list as a member of the New Zealand order of Merit.

Neville Watson is a professor with the Department of Electrical and Computer Engineering at the University of Canterbury. His research interests include power systems, power flow and harmonics. He is a senior member of IEEE and member of Institution of Professional Engineers New Zealand.

Table of Contents

Preface	xi
1 Introduction	1	(6)
1.1 Power System Harmonics	1	(1)
1.2 The Main Harmonic Sources	1	(1)
1.3 Modelling Philosophies	2	(1)
1.4 Time Domain Simulation	3	(1)
1.5 Frequency Domain Simulation	3	(1)
1.6 Iterative Methods	4	(1)
1.7 References	5	(2)
2 Fourier Analysis	7	(26)
2.1 Introduction	7	(1)
2.2 Fourier Series and Coefficients	7	(3)
2.3 Simplifications Resulting from Waveform Symmetry	10	(3)
2.4 Complex Form of the Fourier Series	13	(2)
2.5 Convolution of Harmonic Phasors	15	(2)
2.6 The Fourier Transform	17	(2)
2.7 Sampled Time Functions	19	(1)
2.8 Discrete Fourier Transform	20	(4)
2.9 Fast Fourier Transform	24	(2)
2.10 Transfer Function Fourier Analysis	26	(5)
2.11 Summary	31	(1)
2.12 References	31	(2)
3 Transmission Systems	33	(64)
3.1 Introduction	33	(1)
3.2 Network Subdivision	33	(2)
3.3 Frame of Reference used in Three-Phase System Modelling	35	(2)
3.4 Evaluation of Transmission Line Parameters	37	(9)
3.4.1 Earth Impedance Matrix [Z(e)]	37	(2)
3.4.2 Geometrical Impedance Matrix [Z(g)] and Admittance Matrix [Y(g)]	39	(2)
3.4.3 Conductor Impedance Matrix [Z(c)]	41	(5)
3.5 Single Phase Equivalent of a Transmission Line	46	(6)
3.5.1 Equivalent PI Models	46	(6)
3.6 Multiconductor Transmission Line	52	(9)
3.6.1 Nominal PI Model	52	(4)
3.6.2 Mutually Coupled Three-Phase Lines	56	(2)
3.6.3 Consideration of Terminal Connections	58	(1)
3.6.4 Equivalent PI Model	59	(2)
3.7 Three-Phase Transformer Models	61	(4)
3.8 Line Compensating Plant	65	(2)
3.8.1 Shunt Elements	65	(2)
3.8.2 Series Elements	67	(1)
3.9 Underground and Submarine Cables	67	(4)
3.10 Examples of Application of the Models	71	(23)
3.10.1 Harmonic Flow in a Homogeneous Transmission Line	71	(4)
3.10.2 Harmonic Analysis of Transmission Line with Transpositions	75	(9)
3.10.3 Harmonic Analysis of Transmission Line with Var Compensation	84	(3)
3.10.4 Harmonic Analysis in a Hybrid HVdc Transmission Link	87	(7)
3.11 Summary	94	(1)
3.12 References	94	(3)
4 Direct Harmonic Solutions	97	(36)
4.1 Introduction	97	(1)
4.2 Nodal Harmonic Analysis	98	(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