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Thesis Harmonic

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  • Dynamic Analysis of Harmonicsin Electrical Systems

    Erik Mllerstedt

    Lund 2000

  • Till Anna

    Department of Automatic ControlLund Institute of TechnologyBox 118SE-221 00 LUNDSweden

    ISSN 02805316ISRN LUTFD2/TFRT--1060--SE

    c&2000 by Erik Mllerstedt. All rights reserved.Printed in Sweden by Bloms i Lund Tryckeri AB.Lund 2000

  • Contents

    Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . 71. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

    1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 Contributions of the Thesis . . . . . . . . . . . . . . . . . 101.3 How to read the Thesis . . . . . . . . . . . . . . . . . . . 13

    2. Power System Stability . . . . . . . . . . . . . . . . . . . . . 142.1 Power System Stability Analysis . . . . . . . . . . . . . . 142.2 Harmonic Analysis . . . . . . . . . . . . . . . . . . . . . . 16

    3. Linear Time Periodic Systems . . . . . . . . . . . . . . . . . 203.1 Frequency Separation of LTP Systems . . . . . . . . . . 223.2 LTP System Analysis . . . . . . . . . . . . . . . . . . . . 233.3 Transformation to Time Invariant Representations . . . 263.4 An LTP Model of a Sampled-Data System . . . . . . . . 30

    4. The Harmonic Transfer Function . . . . . . . . . . . . . . . 334.1 The Harmonic Transfer Function . . . . . . . . . . . . . 334.2 Structure of the HTF . . . . . . . . . . . . . . . . . . . . 344.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.4 A Lifting Interpretation . . . . . . . . . . . . . . . . . . . 444.5 LTP System Analysis using HTFs . . . . . . . . . . . . . 464.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 51

    5. Convergence and Computational Issues . . . . . . . . . . 525.1 Roll-off for LTP Systems . . . . . . . . . . . . . . . . . . . 535.2 Roll-off and Power System Modeling . . . . . . . . . . . 58

    6. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . 637. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64I. Harmonic Analysis of Distribution Networks . . . . . . . 69II. A Harmonic Transfer Function Model for a Diode Con-

    verter Train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85III. Out of Control Because of Harmonics An Analysis of

    the Harmonic Response of an Inverter Locomotive . . . 101IV. Robust Control of Power Converters . . . . . . . . . . . . 127

    5

  • Contents

    6

  • Acknowledgments

    In Cambridge you have to state that your dissertation does not includesanything which is the outcome of work done in collaboration. Guess I amlucky that this thesis is to be defended in Lund. There are many people,without whom this thesis would never have been in your hand.

    First I would like to thank my supervisor Bo Bernhardsson, who has aunique blend of theoretical knowledge and practical know-how. He is alsoa very pedagogical inspirer and is extremely helpful in times of short dead-lines. My second supervisor Anders Rantzer has also played an importantrole in the writing of this thesis. The project was initiated by Sven ErikMattsson but, unfortunately, he left the department after I had finishedmy licentiate thesis.

    One of the persons who have meant the most for my time at the de-partment is Henrik Olsson. He was the one who convinced me to start myPhD studies when he was a most inspiring teaching assistant in the coursein adaptive control. He also saw the need for our methods in the anal-ysis of converter locomotives, and established invaluable contacts withDaimler-Chrysler, Adtranz, and ABB Corp. Research.

    Many thanks goes to my PAL Johan Eker for helping me out with allsorts of computer problems, and for filling my ears with such lovely music.

    I would also like to thank all colleagues at the department for sixfantastic years. Especially I would like to mention Karl Johan strm,Bjrn Wittenmark, and Per Hagander, for making every possible effortto make all of us to get on well. Not to mention the fifth floor babes.What would the world be like without you? I would also like to thank LeifAndersson for excellent computer support.

    Many thanks to Henrik Sandberg for long-desired collaborations, andfor giving me the possibility to leave everything half-finished, and to An-drey Ghulchak for many valuable comments on the manuscript.

    During the years of this project I have always appreciated the help Ihave received from the people at IEA. Especially I would like to thankOlof Samuelsson, who always have time for discussions and has readnumerous manuscripts during the years.

    I would like to thank Alec Stothert for a very nice and interesting stayat ABB Corp. Research in Baden-Dttwil, and Markus Meyer at Adtranz,Zrich, for good collaborations.

    I also want to mention my dear friends who looked after Albin duringmy first stumbling weeks on paternity leave; Kerstin, Loffe, Johan, Mor,Matti, Ingrid, Johan, Linda, Amalia, and Ingela.

    7

  • My PhD studies have been financed within the Elektra project by Sven-ska Elfretagens Forsknings- och Utvecklings- Elforsk- AB, Statens En-ergimyndighet, and the Swedish National Board for Industrial and Tech-nical Development DNUTEKE.

    Finally, thanks to all my friends and family who have encouraged meduring all years. Most precious of all, however, has been, and will alwaysbe, my wife Anna, whose patience and support seem to be never-ending,and Hugo and Albin. I guess I am the luckiest man on earth!

    Erik

    8

  • 1Introduction

    1.1 Motivation

    Economical and environmental reasons force a more efficient use of thepower networks. Improved network operation has been made possible withthe introduction of active power electronic components for power condi-tioning, protection, and conversion. Characteristic for power electronics isthat they have switching dynamics, and that they are highly dependenton control. The switching nature of power electronics results in increasedharmonic injection into the grid.

    The common approach to steady state stability analysis and control de-sign, however, assumes sinusoidal conditions. This assumption is basedon conditions in traditional power systems, where the amount of harmon-ics is neglectable. Even though harmonics exist, they are believed not toaffect the stability of the system. Hence, harmonics are not consideredin stability analysis. With an increasing amount of switching componentsattached to the grid, this picture has to change. Harmonics can lead to un-predicted interaction between components, which in worst case can leadto instability. To guarantee stable network operation, the dynamic aspectsof the harmonics have to be considered. A problem is that such analysisis complicated, and as power systems are very large and complex, one hasto rely on simplified analysis and simulations.

    This thesis describes a method to combine the continuous dynamicsof the power system with the switching power electronics. The idea isto linearize the system around the periodic steady-state operating point,which normally can be derived by assuming a sinusoidal grid voltage. Theresult is a linear time periodic DLTPE model, which captures the couplingbetween different frequencies that arise due to the periodic dynamics. Afrequency domain framework is developed, and it is shown that many

    9

  • Chapter 1. Introduction

    results developed for common linear time invariant DLTIE systems canbe generalized to time periodic systems. Hence, many existing results onrobustness analysis, robust controller design and model reduction can bedirectly applied.

    1.2 Contributions of the Thesis

    A frequency domain framework for modeling and analysis of electricalsystems with a mix of continuous and discrete dynamics is developed. Thisis an infinite dimensional transfer function matrix H DsE which describesthe input-output relation for an LTP system in frequency domain2666666664

    ...

    YDs j 0EYDsE

    YDs j 0E...

    3777777775

    2666666664

    ......

    ...

    H1,1DsE H1,0DsE H1,1DsE H0,1DsE H0,0DsE H0,1DsE H1,1DsE H1,0DsE H1,1DsE

    ......

    .... . .

    3777777775

    2666666664

    ...

    UDs j 0EUDsE

    UDs j 0E...

    3777777775.

    With the HTF, analysis of LTP systems can be performed with methodsdeveloped for multi-input multi-output LTI systems.

    The HTF approach is shown to be closely related to methods for LTPsystems analysis of digital control systems. This means that many strongresults developed for the analysis of the inter-sample behavior of sampled-data system also can be applied to power system analysis.

    The HTF is an infinite dimensional operator, which for computationshas to be approximated by a finite dimensional operator. This may lead toconvergence problems. The concept of roll-off for LTP systems is definedand can be used to justify the use of truncated HTFs. Relations to powersystem modeling and the concept of computational causality are discussed.

    The thesis contains four papers with different power system applica-tions. Below, the contents of the papers is briefly summarized. Referencesto related publications are also given. The work in Papers II, II, and IV,has been done in collaboration with people from the industry.

    10

  • 1.2 Contributions of the Thesis

    Paper I

    Mllerstedt, E. and B. Bernhardsson: Harmonic analysis of distributionnetworks. Submitted Nov. 1999 to IEEE Trans. on Power Systems.

    ContributionsThe paper presents a method to analyze the periodic steady-state solutionof power distribution networks. Components with nonlinear and switchingdynamics are replaced by Harmonic Norton Equivalents which describethe steady-state coupling between different harmonics. It is