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Diss. ETH No. 15269

Dynamic Modeling of Line and Capacitor

Commutated Converters forHVDC Power Transmission

A dissertation submitted to theSWISS FEDERAL INSTITUTE OF TECHNOLOGY

ZURICH

for the degree of DOCTOR OF TECHNICAL SCIENCES

presented byWOLFGANG HAMMER

Dipl.-Ing. (RWTH Aachen)born December 13, 1972

in Leverkusen, Germany

accepted on the recommendation of Prof. Dr. G oran Andersson, examiner

Prof. Dr. Ani Gole, co-examiner

2003


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Preface

This thesis presents the result of my research done during the yearsof 20002003 at the Power Systems Laboratory of the Swiss FederalInstitute of Technology (ETH) Zurich.

First of all I wish to thank Prof. Dr. G oran Andersson for his supervisionof my work and for the freedom of research that he granted me. I amparticularly indebted for his consent to the foundation of a spin-off company with the purpose of bringing to market a simulation softwarewhich was developed partly in the course of this thesis.

Special thanks go to Prof. Dr. Ani Gole for willing in to co-referee thisthesis.

I would also like to thank my colleagues at the Power System Labo-ratory both for the valuable discussions and for the welcome diversionfrom the research work. The coffee-breaks and the excursions will stayin my memory. Jost Allmeling, colleague and co-founder of the spin-off company, deserves special mention for relieving me of my companyduties during the nal stages of this thesis.

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Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Kurzfassung . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1 Introduction 13

2 Energy transmission by direct current 17

2.1 Line commutated converter . . . . . . . . . . . . . . . . 17

2.2 Capacitor commutated converter . . . . . . . . . . . . . 192.3 Voltage-sourced converter . . . . . . . . . . . . . . . . . 20

3 Modeling techniques for power electronics devices 23

3.1 Piecewise LTI models . . . . . . . . . . . . . . . . . . . 233.1.1 Generic system matrix . . . . . . . . . . . . . . . 253.1.2 State dependencies . . . . . . . . . . . . . . . . . 28

3.1.3 Undetermined switch variables . . . . . . . . . . 313.1.4 State inconsistencies . . . . . . . . . . . . . . . . 343.1.5 Implementation . . . . . . . . . . . . . . . . . . . 38

3.2 State-space averaging . . . . . . . . . . . . . . . . . . . 393.2.1 Local average . . . . . . . . . . . . . . . . . . . . 393.2.2 Local -component . . . . . . . . . . . . . . . . . 393.2.3 Generalized averaging . . . . . . . . . . . . . . . 403.2.4 Phasor dynamics models . . . . . . . . . . . . . . 41

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Contents 7

6 Case studies 103

6.1 HVDC control scheme . . . . . . . . . . . . . . . . . . . 1046.1.1 Voltage dependent current order limit . . . . . . 1046.1.2 Current controller . . . . . . . . . . . . . . . . . 1046.1.3 Gamma controller . . . . . . . . . . . . . . . . . 1046.1.4 Phase locked loop . . . . . . . . . . . . . . . . . 1066.1.5 Firing pulse generator . . . . . . . . . . . . . . . 106

6.2 LCC case study . . . . . . . . . . . . . . . . . . . . . . . 1086.3 CCC case study . . . . . . . . . . . . . . . . . . . . . . . 112

6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 1137 Summary and Conclusions 117

A Analytical solution of the CCC commutation equations119

Bibliography 123


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Abstract

Energy transmission by means of direct current is an attractive alter-native to ac transmission for a number of special applications. One of these applications, i. e. the use of fast HVDC controls to stabilize atransmission system, gains more and more importance as todays powersystems are operated closer to their stability limits.

Therefore, there is an increasing need to understand the dynamic in-teractions between HVDC converters and the ac systems. Convenientmodels are needed that will both facilitate control design and allow fastand accurate transient simulations. However, the analysis of HVDCconverter systems is challenging because of their hybrid nature, as theyincorporate both continuous-time dynamics and discrete events.

In this thesis, dynamic models are developed for HVDC converters basedon turn-on devices. A new dynamic modeling approach is presentedwhich takes into account the dynamics of a variable direct current. Thisis in contrast to the standard quasi-static approach, which assumes aconstant current and therefore lacks accuracy when transient phenom-ena are of interest. The investigations are restricted to phasor-basedmodels for use with transient stability programs. The major contribu-tion of this thesis is a novel dynamic model for the capacitor commu-tated converter which considers the inuences not only of a variabledirect current but also of unbalanced capacitor voltages.

Case studies demonstrate the impact of the dynamic models on thestudy of control interactions between an HVDC converter with its con-trols and an ac system. The proposed dynamic CCC model is shown tobe valid in a frequency range up to the fundamental ac frequency.

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Kurzfassung

Hochspannungs-Gleichstrom- Ubertragung ist fur gewisse Anwendungeneine attraktive Alternative zur Drehstrom- Ubertragung, die in Energie-versorgungsnetzen ublicherweise verwendet wird. Aufgrund der immerhoheren Auslastung heutiger Ubertragungsnetze gewinnt eine solcheAnwendung, die Verwendung der schnellen HG U-Regelung zur Bek amp-fung von Stabilitatsproblemen, mehr und mehr an Bedeutung.

Vor diesem Hintergrund wird es immer wichtiger, das dynamische Wech-selspiel zwischen der HG U und dem Drehstromnetz genau zu verstehen.

Es besteht ein wachsender Bedaf nach ad aquaten Modellen, die sowohlfur die Auslegung von Reglersystemen verwendet werden k onnen, alsauch schnelle und genaue transiente Simulationen erm oglichen. Auf-grund ihres hybriden Charakters stellen leistungselektronische Ger atewie die HGU-Umrichter allerdings eine besondere Herausforderung dar,da sie kontinuierliche Vorg ange und diskrete Ereignisse vereinen.

In dieser Arbeit werden dynamische Modelle f ur HG U-Umrichter basie-rend auf Einschaltelementen hergeleitet. Dazu wird ein neuartiger,dynamischer Modellierungsansatz verwendet, der den Einuss eines ver-anderlichen Gleichstromes ber ucksichtigt. Dies steht im Gegensatz zumherk ommlichen quasi-station aren Ansatz, bei dem der Gleichstrom alskonstant angenommen wird. Auf einer solchen Annahme beruhendeModelle sind daher zwangslaug ungenau, wenn transiente Vorg angebetrachtet werden. Die Untersuchungen beschr anken sich auf Zeiger-basierte Modelle, wie sie in Simulatoren fur transiente Stabilit atspro-bleme verwendet werden. Der Hauptbeitrag dieser Arbeit ist ein neuesModell fur den so genannten Capacitor Commutated Converter , das

nicht nur die Auswirkungen eines variablen Gleichstromes beschreibtsondern auch den von unsymmetrisch geladenen Kondensatoren.

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12 Kurzfassung

Vergleichsrechnungen demonstrieren den Einuss, den die Verwendungdynamischer Modelle auf die Untersuchung von Wechselwirkungen zwi-schen einer HG U-Regelung und dem Drehstromnetz hat. Das hiervorgeschlagene dynamische Model des Capacitor Commutated Converter erweist sich als g ultig uber einen weiten Frequenzbereich bis hinauf zurNetzfrequenz.


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Chapter 1

Introduction

While the very rst practical applications of electricity were based ondirect current, this technology was quickly replaced by three-phase al-ternating current because of various advantages. Alternating currentcan be transformed easily between different voltage levels used for gen-eration, transmission, distribution, and use. In particular, the use of transformers made long-distance, high-voltage power transmission pos-sible. Circuit breakers for alternating current can take advantage of the natural current zeros that occur twice per cycle. The ac inductionmotor is cheap and robust and serves the majority of industrial and res-idential purposes. In contrast, the commutators of dc machines requiremaintenance and impose voltage, speed and size limitations [1].

Still, in spite of the principal use of alternating current in power systems,there are some applications for which direct current is the better if notonly choice, even taking into account the cost of the equipment that isnecessary to convert between ac and dc [6], [7]:

Bulk power transmission on long overhead linesAt very long distances, ac overhead lines consume large amounts of reactive power which furthermore are dependent on the amountof active power transfered. In addition to the additional lossescaused by the reactive current, this also gives rise to stability

problems. DC lines, on the other hand, do not consume reactivepower.

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14 Chapter 1. Introduction

Power transmission via cableDue to its construction, a cable has a much higher capacitanceper unit length than an overhead line. This means that even for acable of moderate length (50 km), the reactive current can utilizea major part of the total current capability when transmittingac power. In the case of dc transmission, apart from the initialcharging, the cable draws no capacitive current in steady state.

Transmission between unsynchronized ac systemsAC power transmission is physically only possible between syn-chronized systems. When two systems operate at different fre-quencies (such as 50 Hz and 60 Hz), or even when they operate atthe same nominal frequency but have divergent frequency controlregimes, the only practical way to transmit power between themis by means of a dc connection.

In addition to these more traditional applications concerning rather thestationary operation of networks, there is a fourth aspect of HVDCtransmission which gains more and more importance with the growingglobal interconnection of transmission systems:

Parallel ac and dc transmissionIn interconnected ac systems technical problems like transient in-stability or power oscillations can occur when an interconnectionbetween two parts is relatively weak. In such a case, the fastcontrols of an additional HVDC link in parallel to the ac inter-connection can be used to stabilize the system [6].

With todays power systems being operated closer to their stability lim-its, and particularly in view of this last aspect, there is an increasing

need to understand the dynamic interactions between HVDC convertersand the ac system. Convenient models are needed that will both facil-itate control design and allow fast and accurate transient simulations.However, the analysis of HVDC converter systems is challenging be-cause of their hybrid nature, as they incorporate both continuous-timedynamics (associated with the voltages and currents of capacitors andinductors) and discrete events (due to the switching of the valves).

Transient stability programs typically use power electronics device mod-

els that are based on quasi-static approximations. Such models rely onthe assumption that the ac system is in sinusoidal steady-state and,


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15

in case of an HVDC current source converter, that the direct currentis ripple-free and constant. Due to these assumptions, the models arestrictly valid only in the steady state and lack accuracy when transientstability or other dynamic phenomena are of interest [8].

In order to characterize the dynamic behavior of hybrid systems, so-called sampled-data models have been developed for various power elec-tronics devices including the line commutated HVDC converter [9].Such models nd application in the eigenvalue analysis of power sys-tem dynamics. While these models are quite accurate and also offer ananalytical basis for controller design, they have also some disadvantages:the model derivation is relatively complicated; due to the linearizationsinvolved the models are only valid for small perturbations around a xedoperating point; and they do not interface well with the continuous-timetransient stability programs.

Another new approach for modeling power electronics devices is thephasor dynamics method. With this method the periodic currents andvoltages associated with a device are described in terms of time-varyingFourier coefficients. By restricting the focus to the fundamental fre-quency components a natural dynamic extension of the usual quasi-static phasor equations can be obtained. This method has been suc-

cessfully demonstrated on two different FACTS devices [10].The phasor dynamics method cannot be applied directly to HVDC con-verters. However, some of its key ideas will be used in this thesis toderive models of the line commutated converter and the capacitor com-mutated converter, which capture the dynamic behavior of these devicesand can be easily interfaced with transient stability programs.

The following chapters in this thesis are organized as follows:

Chapter 2 gives a brief overview of the historical development of HVDC transmission including the relatively young application of voltage-sourced converters.

Chapter 3 reviews three different modeling techniques for powerelectronics devices: the sampled-data method and the phasor dy-namics method mentioned above, and piecewise LTI models thatcan be used for detailed time-domain simulations.

In chapters 4 and 5, dynamic models for the line commutatedconverter and the capacitor commutated converter are developedstarting from standard quasi-static models.


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16 Chapter 1. Introduction

Chapter 6 presents two case studies that demonstrate the impactof the dynamic models on the study of control interactions betweenan HVDC converter with its controls and an ac system.

Chapter 7 summarizes the contributions of this thesis and givessome suggestions for future work.


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Chapter 2

Energy transmission bydirect current

Energy transmission by means of high voltage direct current has anover 100 year old history. The rst system was designed by a French

engineer, Rene Thury, at the end of the 19th century [1]. In this systemthe direct voltage was generated at the sending end of a transmissionline by a number of dc generators connected in series, which were drivenby prime movers. At the receiving end, a comparable number of dcmotors drove low-voltage dc or ac generators. Due to the limitations of dc machines the maximum circuit voltage achieved was 125 kV and themaximum power transfered was 19.3 MW. The last line was operateduntil 1937.

2.1 Line commutated converter

At around this time, the rst experimental HVDC converters basedon controllable switches were installed. Initially, these switches weremercury-vapor lled tubes, devices which can conduct current only inone direction (which is why these devices are also often called valves ).By applying an appropriate voltage to a control electrode, the valve

can be prevented from starting to conduct. However, once a valve con-ducts, conduction does not cease until the current reverses. The rst

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18 Chapter 2. Energy transmission by direct current

I d

Figure 2.1: Line commutated converter circuit

commercial HVDC transmission system built with mercury arc valveswas a link between the Swedish mainland and the island of Gotland.The system began service in 1954 and transmitted 20 MW at 100 kVover a distance of 96 km through a single-conductor cable, with returnpath through the sea and earth. Later systems had ratings as high as1854 MW at 463 kV such as Bipole 1 of the Nelson River HVDCtransmission system in Manitoba, Canada, which was installed between1970 and 1977 and transmits dc power over a distance of 900 km [11].About 1955, a new solid-state switching device called thyristor was de-veloped [12]. This device can be crudely described as a diode withan added control electrode and thus has (ideally) the same switchingcharacteristics as the mercury arc valve while being cheaper and morecompact. The thyristor was soon capable of handling the high voltagesand powers required for HVDC transmission and began to replace themercury arc valves. The rst HVDC project with the new thyristor

valves was again Gotland, commissioned in 1970. The voltage of theserst converters was only 50 kV (6 pulse monopole), but since then thehighest ratings have grown to 600 kV, 6300 MW (Itaipu, built 19841987, two 12 pulse bipoles) [13].The basic module of a conventional HVDC converter using mercury-arcor thyristor valves is the three-phase, full-wave bridge circuit shown inFig. 2.1, also known as a Graetz-bridge. Compared with several possiblealternative congurations, this circuit provides the best utilization of

the converter transformer and the lowest voltage across non-conductingvalves [1], [2].


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2.2. Capacitor commutated converter 19

The direct voltage of the converter can be varied continuously betweena negative and a positive peak value by controlling the instants at whichthe valves start conducting. This is discussed in detail in chapter 4. Theproperties of the valves impose several limitations on the operation of the converter [14]:

Since a valve can only be turned on when there is a positive voltageacross it, the converter can only draw an inductive current fromthe ac network.

A valve cannot be turned off actively. Instead, the current throughone valve has to be brought to zero by turning on another valve(a process known as commutation). The converter has to be op-erated in such a way that the voltage across the valve that justturned off remains negative for a sufficiently large time after thecommutation.

The voltages required for the turning on and off of the valves haveto be provided by the ac network (hence: line commutated con-verter). In networks with a low short circuit ratio, even small net-work disturbances can lead to commutation failures. This problemis aggravated by the ac shunt capacitors which are usually neededto compensate the reactive power consumed by the converter.

2.2 Capacitor commutated converter

These problems are mitigated by the capacitor commutated convertershown in Fig. 2.2. This circuit also consists of a three-phase, full-wavebridge, but in addition it has series capacitors on the ac side. Thedc current ows through each capacitor in either forward or reversedirection during the conduction period of the upper or lower group valvein the corresponding phase. This charges the capacitors in such a waythat their voltages aid in the commutation process. Details are givenin chapter 5. In principle, the capacitor commutated converter couldthus be operated well into the capacitive region. However, the capacitorvoltages also add to the valve peak voltages, and hence it is practicalto limit the capacitor size to a value that allows operation with a near

unity power factor while the valve stress does not exceed approximately110% of that of the conventional converter [15]. In comparison with


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20 Chapter 2. Energy transmission by direct current

I d

Figure 2.2: Capacitor commutated converter circuit

the conventional converter, the capacitor commutated converter has thefollowing prominent advantages:

Because of the very low reactive power consumption, the need forlarge shunt compensators is eliminated. The additional commutation voltage provided by the capacitorsgreatly reduces the risk of commutation failures even in weak ac

systems.

Although references to this conguration can be found as early as 1954[16], there was no interest in its commercial use until the 1990s becausethe control and protection requirements were considered to be exces-sively complex. The rst commercial application, taken into full servicein June 2000, is a 1100 MW asynchronous back-to-back link betweenArgentina and Brazil [14].

2.3 Voltage-sourced converter

A third type of converter with completely different properties becamepossible with the advent of the IGBT (insulated gate bipolar transistor),a new semiconductor device with turn-off as well as turn-on capabilities.The basic circuit is shown in Fig. 2.3.

Voltage-sourced converters (VSC) operate with a smooth dc voltageprovided by a storage capacitor. By turning on either the upper or the


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2.3. Voltage-sourced converter 21

U d

Figure 2.3: Voltage-sourced converter circuit

lower IGBT in one leg of the converter it is possible to impose a positiveor a negative voltage on the corresponding ac phase. The fast switchingcapability of the IGBT allows to create a pulse width modulated acvoltage, the phase of which can be controlled freely and the amplitudewithin the limits given by the dc voltage. Thus, the converter canoperate in all four quadrants of the P-Q plane.

Since the commutation does not depend on the ac network voltage, theconverter can be connected with an extremely weak, or altogether blackac network. Another advantage is that the ac output voltage of theconverter can be changed extremely quickly. However, the voltage andpower ratings of IGBTs are as yet far below those of thyristors andso applications with voltage-sourced converters are limited to low andmedium power.

The rst commercial project was once more commissioned on Gotlandand taken into service in November 1999 [17]. A power of 50 MW is

transferred through two underground cables of 70 km length at a voltagelevel of 80 kV from the south of the island to the north. A similarinstallation (3 60 MW, 80 kV) was commissioned and brought intooperation in 2000 to connect the grids of Queensland and New SouthWales, Australia [18]. Depending on the manufacturer, voltage-sourcedconverter based HVDC systems are called HVDC Light or HVDC plus

[19].

The dynamics of the VSC are well understood because it has been used

already for a long time, e. g. in adjustable speed drives. It is thereforenot treated in this thesis.


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Chapter 3

Modeling techniques forpower electronics devices

In this chapter different ways of modeling power electronics devices willbe reviewed. The rst section deals with state-space models that can

be used for simulations in the time-domain. A new method is presentedfor the easy computation of the state-space matrices.

The second section discusses a technique called state-space averagingwhich can be used to derive phasor dynamics models for power elec-tronics devices. The purpose of these models is to extend the standardphasor model, which in principle is valid only for pure sinusoidal condi-tions in steady state, to be valid over a larger frequency range.

In the third section a brief review of the theory of sampled-data models

will be given. These models are particularly useful to study the small-signal stability of a system at a specic operating point.

3.1 Piecewise LTI models

State-space models are increasingly used in power electronics applica-tions. One reason for this is their focus on variables that are central

to describing the dynamic evolution of a system. Another reason isthat the same formalism can be applied to both continuous-time and

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3.1. Piecewise LTI models 25

value [21]. The described formulation procedure is quite complex. Fur-thermore, approximating an impulse with a large value may again leadto numerical problems.

In this section, a new method is proposed based on simple matrix oper-ations which addresses the problem of impulses and a few others whichare described below.

3.1.1 Generic system matrix

The rst step is setting up a generic system matrix containing the cir-

cuit equations for all possible combinations of switch positions. Thealgorithm comprises the following steps [22]:

1. For a circuit with e arbitrary elements and n nodes, nd thee (n 1) independent loop equations and n 1 independentnode equations.

2. Eliminate the dependent variables, e. g. by applying Ohms law.

The voltages and currents of switch elements are left undeter-mined.

3. Order the independent variables as follows:

x y s x u T (3.3)

wherex = [ v C i L ] (state derivatives)

y = [ y vm y i m ] (output variables)s = [ v S i S ] (switch variables)x = [ v C i L ] (state variables)u = [ v src i src ] (input variables)

The variables on the left hand side are unknown, those on theright hand side known variables.

4. Calculate M gen , the reduced row echelon form ( RREF ) of theequation system. This matrix has the following general block


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variables:yvD = vD yi D = iD yi L = iL (3.10)

In the matrix notation this reads as

iL vC yvD yi D yi L vT iT vD iD iL vC vsrc

0 0 0 0 0 1 0 1 0 0 0 1L 0 0 0 0 0 0 1 0 R 1 00 C 0 0 0 0 0 0 0 1 G 00 0 0 0 0 0 1 0 1 1 0 00 0 1 0 0 0 0 1 0 0 0 00 0 0 1 0 0 0 0 1 0 0 00 0 0 0 1 0 0 0 0 1 0 0

(3.11)

The generic system matrix M gen is the RREF of (3.11):


1 0 0 0 0 0 0 1L 0 RL 1L 00 1 0 0 0 0 0 0 0 1C

GC 0

0 0 1 0 0 0 0 1 0 0 0 00 0 0 1 0 0 0 0 1 0 0 00 0 0 0 1 0 0 0 0 1 0 00 0 0 0 0 1 0 1 0 0 0 10 0 0 0 0 0 1 0 1 1 0 0

(3.12)

Now consider that the the state-space matrices shall be calculated forthe case where the transistor conducts and the diode blocks. Therefore

L RiLiT

vT

vsrc C

iD

vD vC G

Figure 3.1: Buck converter


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A

A

A

va

vb

vc

L a

Lb

L c

ia

ib

ic

Figure 3.2: Example circuit for state dependencies

how to resolve the dependency and eliminate the dependent states. Con-sider the three-phase circuit in Fig. 3.2. This circuit can be described

by two loop equations and one node equation:La iL a Lb iL b = va vb (3.15)Lb iL b Lc iL c = vb vc (3.16)

0 = iL a + iL b + iL c (3.17)

The three line currents are chosen as output variables:

yi a = ia yi b = ib yi c = ic (3.18)

The matrix notation is

iL a iL b iL c yi a yi b yi c iL a iL b iL c va vb vc

La Lb 0 0 0 0 0 0 0 1 1 00 Lb Lc 0 0 0 0 0 0 0 1 10 0 0 0 0 0 1 1 1 0 0 00 0 0 1 0 0 1 0 0 0 0 00 0 0 0 1 0 0 1 0 0 0 00 0 0 0 0 1 0 0 1 0 0 0

(3.19)

Evidently the equation system is under-determined because the left sub-matrix has a rank of 5 whereas there are six unknown variables. Onthe other hand, the third row represents an equation that involves onlyknown variables. In fact this row, containing only state variables, in-

dicates a dependence between these states. As a consequence, M gencannot be calculated.


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32 Chapter 3. Modeling techniques

This gives for the generic matrix M gen :

vD 1 iD 1 vD 2 iD 2 vsrc

1 0 1 0 10 1 0 1 0

(3.25)

Now consider both diodes to be blocking, i. e. the currents iD 1 and iD 2are zero, and the voltages vD 1 and vD 2 are unknown. With the columnsof iD 1 and iD 2 set to zero, the matrix M {0,0} is

vD 1 iD 1 vD 2 iD 2 vsrc

1 0 1 0 10 0 0 0 0

(3.26)

and it is clear that the diode voltages can not be determined because theequation system is under-determined. In this situation it is not possibleever to decide whether the diodes should turn on or not!

This problem would of course not arise if the switches were modeledwith a large off resistance. In that case the open switches would forman ohmic voltage divider which evenly distributes the voltage vsrc amongthe diodes. The solution for ideal switches is then to introduce a virtual innite off-resistance so that

voff = R ioff (3.27)where ioff denotes a virtual innitesimal off-current.

From Eq. (3.24) we know that any current that ows through one diodewill also ow through the other, be it virtual or not. Multiplying thisequation by R

gives

R iD 1 ,off R iD 2 ,off = 0 vD 1 ,off vD 2 ,off = 0 (3.28)

This equation is appended to M {0,0}vD 1 iD 1 vD 2 iD 2 vsrc

1 0 1 0 11 0 1 0 0

(3.29)


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which thus has a rank of 2. The RREF of M {0,0} can now be calculatedasvD 1 iD 1 vD 2 iD 2 vsrc

1 0 0 0 1

2

0 0 1 0 12(3.30)

and yields the correct result vD 1 = vD 2 = vsrc / 2.

The currents through parallel closed switches can be determined in asimilar manner: For closed switches a virtual innite on-conductanceG is dened so that

ion = G von (3.31)where von denotes a virtual innitesimal on-voltage.If a circuit has two switches connected in parallel, the loop equation forthe switch voltages is:

vS 1 vS 2 = 0 (3.32)When both switches are closed these voltages will become virtual on-voltages. Multiplying the loop equation by G and using Eq. (3.31)gives

iS 1 ,on

iS 2 ,on

= 0

so that both switches will carry the same current.

General rule

If, after applying the switch conditions, the system matrix is under-determined and contains rows with all entries zero:

1. Reorder the columns of the generic matrix M gen , so that thevirtual switch variables ( von and ioff ) appear in the rightmostcolumns.

2. Calculate the RREF M of the reordered generic matrix. Thebottom row(s) of M now contain loop or node equations involvingonly the virtual switch variables.

3. Append these equations to M and on the way replace ioff withvoff and von with ion . (This corresponds to multiplying every nodeequation with R and every loop equation with G.)


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3.1.4 State inconsistencies

It was pointed out in the introduction of this section that switchingmay introduce impulses into the circuit because a capacitor voltage or aninductor current are forced to change abruptly. It is crucial to accuratelydetect such impulses in order to determine the correct topology afterswitching because they might cause other switches (e. g. diodes) tochange their state.

Rather than trying to approximate an innite impulse with a largevalue the method developed here splits the circuit response into twoparts: a non-impulsive component and an impulsive one. While theimpulsive component still is innite, its weight is nite and hence canbe represented on a computer. This approach is described in [23] forthe modied nodal analysis.

The method is again illustrated with the help of the buck converter inFig. 3.1. Consider that the transistor has been conducting (and hencethe diode is blocking), and the current through the inductor L has builtup to a certain positive value. Now the transistor is switched off. It isclear that the current through the inductor cannot abruptly be broughtto zero. Hence, an alternative path has to be found which is achieved

by the turning-on of the free-wheeling diode.However, with the method developed here so far, a computer programcannot decide that the diode needs to be turned on as will be demon-strated now. Applying the conditions for both switches turned off tothe generic matrix yields


1 0 0 0 0 0 0 1L 0 RL 1L 00 1 0 0 0 0 0 0 0

1C

GC 0

0 0 1 0 0 0 0 1 0 0 0 00 0 0 1 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 1 0 00 0 0 0 0 1 0 1 0 0 0 10 0 0 0 0 0 0 0 0 1 0 0

(3.33)

The system is under-determined, and the last row indicates that thestate variable iL has become a dependent variable. According to the


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rule given in subsection 3.1.2, this dependence is resolved by adding theequation iL = 0 to the system, which gives for M {0,0}:


1 0 0 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0 GC 00 0 1 0 0 0 0 0 0 0 1 00 0 0 1 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 1 10 0 0 0 0 0 0 1 0 0

1 0

0 0 0 0 0 0 0 0 0 1 0 0(3.34)

Looking at the third row, one sees that the diode voltage is given byyvD = vC and so, as vC is generally positive, the diode will not turnon!What the matrix notation is lacking so far is a possibility to representthe impulsive voltage that the innite diL /dt will produce across the

inductor when the current is interrupted. At the instant of the discon-tinuity, the inductor can be modeled as a voltage source with

vL = L iL (t t0) (3.35)where t0 denotes the time of the discontinuity and iL is the differ-ence between the inductor currents before and after the discontinuity,iL (t+0 ) iL (t0 ). Since iL (t+0 ) = 0 we have

iL =

iL (t0 ) (3.36)

Therefore, at the instant of the discontinuity, the loop equation (3.7) isreplaced by

vD = Ri L + vC Li L (t0 ) (t t0) (3.37)Divided by L and written in the matrix notation this reads as

iL vC yvD yi D yi L vT iT vD iD iL vC vsrc iL

0 0 0 0 0 0 0 1

L 0 R

L 1

L 0 1(3.38)


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37/128


right hand side and ll it with the identity matrix:

x y s x u x

I x 0 X X X I x

0 I y X X X 0

0 0 X X X 0

(3.41)

The identity matrix I x on the right hand side effectively adds an im-pulsive voltage source in series with every inductor, and an impulsivecurrent source in parallel with every capacitor.

Since the rst N (x ) rows already are in the reduced row echelon form,the entries in the right-most columns will normally not propagate intothe output equations below. Hence, the impulsive sources will normallyremain inactive when the switch conditions for a specic topology areapplied. In this case the RREF of the matrix M for a specic combi-nation of switch positions has the following form:

x y s x u x

I x 0 0 A B I x

0 I y 0 C D 0

0 0 X X X 0

(3.42)

However, if due to the switch conditions one or more state dependenciesoccur and are resolved by applying the rule given in subsection 3.1.2,the identity matrix I

x on the left hand side is destroyed. During the

subsequent calculation of the RREF , the entries in the rst rows willtherefore be reordered and inuence the rows below. The RREF of M then has the following form:

x y s x u x

I x 0 0 A B X

0 I y 0 C D C ()

0 0 X X X X

(3.43)


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The evaluation of the output variables is now split in two parts:

y = C x + D u

y () = C () x(3.44)

Whenever an entry of y () is non-zero, it will overrule the respectiveentry of y .

3.1.5 Implementation

The methods described here have been implemented in a computer pro-gram PLECS [22], an extension to the software package MATLAB/Simu-link. The program accepts circuits that consist of passive elements ( R,L, C ), independent voltage and current sources, voltage and currentmeters, and switches both externally controlled (e. g. ideal switches)and internally controlled (e. g. diodes) or both (e. g. thyristors, GTOsand circuit breakers).

At the beginning of a simulation, the program sets up the generic systemmatrix, determines the initial topology and calculates the correspondingstate-space matrices. During a simulation, the state derivatives arecalculated and passed on to the solver engine of Simulink. At everytime step, the state of each switch is determined from external controlsignals or from internal circuit variables such as the current trough orthe voltage across a diode. Whenever the state of a switch changes, anew set of state-space matrices is computed and afterwards all switchesare checked again to determine whether subsequent switching actionsare necessary (as in the last example of the buck converter). Once astable circuit topology is obtained, the simulation continues.

PLECS has been used for all time-domain simulations in the followingchapters.


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3.2. State-space averaging 39

3.2 State-space averaging

In many cases when studying power electronics circuits the interest lies

not so much in the instantaneous values of voltages and currents butrather in their average values [3]. Obvious examples are the dc voltageand current of an HVDC converter which are normally ltered in orderto remove the ripple, before they are fed into the various controllers.This motivates the study of models that directly describe the local av-erage behavior of circuit variables even during transient conditions.

3.2.1 Local average

The local (or moving) average of a variable x is dened by

x(t) = 1T

t

tT x( ) d (3.45)

for a xed T . An appropriate choice of T is required in order to obtainuseful results. In general, it is a multiple of the shortest regular switch-ing interval associated with the operation of the circuit being analyzed.In the special case where x is periodic, and T is chosen to equal the pe-riod, x is just the usual average. This special case is very important inpower electronics, because the steady-state waveforms in typical circuitsare indeed periodic.

3.2.2 Local -component

As a generalization of Eq. (3.45) the averaging operation can be done us-ing a weighing function other than a constant unity factor. Specically,the local -component of x at time t can be dened as

x (t) = 1T

T

tT x( )ej d (3.46)

In this notation the local average x(t) of Eq. (3.45) is the local 0-component x0(t). The choices of and T are usually interrelated. For

instance, with = 2/T , x

in steady-state is the complex amplitudeof the fundamental frequency component for the periodic waveform x.


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3.2. State-space averaging 41

3.2.4 Phasor dynamics models

Applied to power systems applications, the generalized averaging method

has been used to extend the standard phasor model, which in principleis valid only for pure sinusoidal conditions. An attractive feature of the resulting phasor dynamics models is that the frequency range, overwhich it should be valid, can be chosen by retaining the appropriateFourier coefficients.

In [8] the method is used to develop a phasor dynamics model for aThyristor-Controlled Series Capacitor (TCSC). It is interesting to notethat although the model uses only the fundamental coefficients it is capable of taking into account the effects of higher harmonics veryaccurately.

This is achieved by using the following approximation. The steady-state waveform xs of a variable corresponding to a certain mode of operation(in the cited example the series capacitor voltage waveform for a specicconduction angle) is decomposed into its (constant) Fourier coefficientsxs k , which are normalized with respect to the fundamental coefficient.

For the evaluation of the term f (x, u ) 1 as in Eq. (3.52), the dynamic waveform x is then approximately reconstructed from only the dynamicfundamental coefficient x 1 (t) and the static steady-state coefficientsxs k as

x( ) k

xs k x 1 (t)ejk s (3.53)

This approximation turns out to be an excellent improvement over usingonly the fundamental coefficient, while it is substantially simpler thanthe full inclusion of higher harmonics.

A similar approach will be used in chapter 5 where a steady-state rela-tionship is used as an approximation for the dynamic relationship be-tween a dynamic phasor and the short-time average of a space-vector.


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3.3 Sampled-data modeling

The majority of power electronics circuits operate cyclically. When

studying such circuits it can be useful to work with models that in-volve quantities sampled once per cycle. A general approach to obtainsuch models for arbitrary circuits is described in [25] and reviewed herebriey.

3.3.1 Large-signal sampled data description

In one cycle k of a cyclically operating, piecewise linear system, the state

vector x (t) is governed by a succession of LTI state-space equations of the form

x = A i x (t) + B i u (t) tk + T k,i 1 < t tk + T k,i (3.54)where tk is the starting time of the k-th cycle and the T k,i the are relativetransition times in the k-th cycle at which the switch conguration (andthus the state-space matrices) changes.

For a given x (tk ) the evolution of the system is completely determined

by the input waveforms and the transition times. These input wave-forms and transition times are in turn governed by a set of independentcontrolling parameters pk . The dependence of the transition times onthese parameters and the state vector at the beginning of a cycle canbe expressed by a set of constraint equations:

c (x (tk ), pk , T k ) = 0 (3.55)

where T k is a vector of all T k,i . These equations can describe bothexternally controlled transitions, such as the turning-on of a thyristor,or transitions caused by internal circuit variables, such as the turning-off of a thyristor.

On integrating Eq. (3.54) over the interval from tk to tk+1 one obtainsa sampled-data description of the form

x (tk+1 ) = f (x (tk ), pk , T k ) (3.56)

The function f is often called advance map , as it advances the state

vector from one cycle to the next, or Poincare map [26], a mathematicaltool used for the study of the nonlinear system dynamics.


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3.3. Sampled-data modeling 43

3.3.2 Perturbations about a nominal steady state

If a system described by Eq. (3.54) is in cyclic steady state, it followsthat

x = f (x , p , T ) (3.57)

with x , p and T denoting constant steady-state values of the sampledstate vector, parameters and transition times.

Starting from such a steady-state solution, the dynamics of small per-turbations from the steady state can be investigated. For this purposeit is convenient to use a notation such as the following

x

k = x

(tk ) x

pk = pk pT k = T k T

(3.58)

From Eqs. (3.55) to (3.57) it then follows that

x k+1 = f (x + x k , p + pk , T + T k ) x (3.59)with

c(x + x k , p + pk , T + T k ) = 0 (3.60)

Carrying out multi-variable Taylor series expansions in these equations,retaining only linear terms, and combining the two resulting equationsin order to eliminate T k gives

x k+1 F x k + G u k (3.61)with

F = f x f T c T

1 c x

G = f

p f T

c T

1 c p

(3.62)

This constitutes an approximate LTI sampled-data model for pertur-bations of the system from a cyclic steady state. One property of theJacobian matrix F is that if all its eigenvalues have magnitudes smaller

than one, the cyclic steady state of the system is asymptotically stablewithout further control action.


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Chapter 4

Line commutatedconverter models

In this chapter the conventional line-commutated converter is analyzed.Its topology is the three-phase, full-wave bridge circuit shown in Fig. 4.1.

The following idealizations form the basic assumptions of the analysis:

The ac voltage is stiff and may be represented by an ideal sinu-soidal source in series with a lossless inductance. The direct current is ripple-free and (in steady-state operation)constant. The valves are ideal switches with zero on-resistance and inniteoff-resistance. They change instantaneously between these two

states.

The valves are red at equal intervals of one-sixth cycle (60 ).The instantaneous line-to-neutral voltages of the ac sources are takenas

ea = E m cos(t + 60 )eb = E m cos(t

60) (4.1)

ec = E m cos(t 180)45


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46 Chapter 4. LCC models

ea

eb

ec

ia

ib

ic

L c

u did

Figure 4.1: Equivalent circuit for the line commutated converter. Thevalves are numbered in the order of their ring

The corresponding line-to-line voltages are then

eac = ea ec = 3 E m cos(t + 30 )eba = eb ea = 3 E m cos(t 90)ecb = ec eb = 3 E m cos(t + 150 )

(4.2)

4.1 Static model

The development of the equations governing the steady-state operationof the converter is described in standard literature (e.g. [1], [2], [4]). Itis repeated briey for reference in this section.

4.1.1 Analysis without commutation overlap

Fig. 4.2 shows the typical wave-forms of the converter if the ac induc-tance Lc is neglected. In the top graph the ac line-to-neutral voltagesare drawn in thin lines and, in heavy lines, the potentials of the positiveand negative dc terminals with respect to ac neutral. The middle graphshows the ac line-to-line voltages and, in a heavy line, the instantaneousdirect voltage ud . The bottom graph shows the constant dc current and,in a heavy line, the ac line current ia .

At any given instant, one valve of the upper commutation group andone of the lower row are conducting. Therefore, the instantaneous direct


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4.1. Static model 47

a)

b)

c)ia id

ea eb ec

eab eac ebc eba eca ecb

120

0

60

60

120

180

240

t

t

t

t

A

Figure 4.2: Voltage and current waveforms without overlap: (a) posi-tive and negative dc terminal potentials, (b) instantaneousdirect voltage, (c) phase a line current.

voltage at any time equals one of the six line-to-line voltages. The in-stant at which the direct voltage changes to another line-to-line voltage

is controlled via the ring angle .

Average direct voltage

It is assumed that the valves are red at equal intervals. Hence, udconsists of six identical segments of 60 width each, and so the averagedirect voltage can be found by averaging the direct voltage over any 60 interval.

Considering the interval indicated by the shaded area A in Fig. 4.2 the


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average direct voltage is given by

U d =

3

A =

3

+60

(eb ec) d(t)=

3

+60

3 E m cos (t 30) d(t)=

3 3

E m

+60

cos(t 30) d(t)= U di 0 sin(t 30) +60

= U di 0 sin( + 30 ) sin( 30)= U di 0 2 sin 30 cos = U di 0 cos (4.3)

where

U di 0 = 3 3

E m (4.4)

is the so-called ideal no-load direct voltage .

The ignition angle may range from 0 to 180. Outside of this range,the commutation voltage at the ring instant is negative and the com-mutation fails.

AC current magnitude and phase

Since the direct current I d is constant by assumption and each valveconducts for a period of 120 , the ac currents are rectangular blockswith magnitude I d and width 120 as shown in Fig. 4.2. The phasedisplacement changes with the ring angle .

The peak value of the fundamental frequency component of the alter-nating line current can be determined by Fourier analysis:

I (1) = 2 3 I d (4.5)


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By assumption, the converter is lossless and therefore the ac activepower must equal the dc power:

32E m I

(1)

cos = U d I d= U di 0I d cos (4.6)

where denotes the angle by which the fundamental frequency compo-nent of the line current lags the line-to-neutral voltage.

Inserting Eqs. (4.4) and (4.5) gives:

32 E m

2 3 I d cos =

3 3 E m I d cos

cos = cos (4.7)

That is, the ring angle shifts the fundamental frequency componentof the ac current by an angle = with respect to the ac line-to-neutralvoltage.

4.1.2 Analysis including commutation overlap

Because of the inductance Lc present at the ac-terminals the phase cur-rents cannot change instantaneously. Therefore, the dc current requiresa nite time to transfer from one phase to another. This phenomenonis called commutation overlap . Its effect on the voltage and currentwaveforms of the converter circuit is shown in Fig. 4.3. The angle cor-responding to the time needed for commutation is denoted by . The

angle = + is called the extinction angle .Fig. 4.4 shows the equivalent converter circuit during commutation fromvalve 1 to valve 3. At the beginning of commutation (i.e. when valve 3 isred), the phase current ia equals the direct current while ib is still zero.The voltage eb ea then drives a current through the loop containingvalves 1 and 3. The commutation ends when ia has decreased to zeroand ib has taken over the whole dc current:

At t = : ia = I d and ib = 0At t = : ia = 0 and ib = I d (4.8)


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a)

b)

c)

ia

ia

ibib

ib id

ea eb ec


e a + e b2

A

120

0

60

60

120

180

240

t

t

t

t

Figure 4.3: Voltage and current waveforms showing the effect of overlap:(a) positive and negative dc terminal potentials, (b) instan-taneous direct voltage, (c) phase a and b line currents.

The mesh equation for the above described loop is:

eb ea = Lcdibdt Lc

diadt

(4.9)

The sum of ia and ib during commutation equals the direct current;therefore,

diadt

+ dibdt

= did

dt = 0

diadt

= dibdt (4.10)


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ea

eb

ec

ia

ib

ic

L c

u did

Figure 4.4: Equivalent LCC circuit during commutation from valve 1to valve 3 (non-conducting valves not shown)

Inserting Eqs. (4.2) and (4.10) into (4.9) yields

3 E m sin(t) = 2 Lc dibdt dib

dt =

3 E m2Lc

sin(t) (4.11)

Integration over the duration of the commutation gives

dibdt

dt = 3 E m2Lc

(cos cos )

and nally, inserting the boundary conditions (4.8) into the left-handside:

I d = 3 E m2Lc

(cos

cos ) (4.12)

From this equation the extinction angle (and ultimately the overlapangle ) can easily be determined for a given ring angle . It alsoallows the calculation of the ideal maximum ring angle max for whichthe commutation will succeed in a converter with ideal valves. Since

cos 1for any angle :

I d 3 E m2Lc (cos + 1)


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cos max = 2Lc 3 E m I d 1 (4.13)


During commutation the two impedances in the commutation loop actas a voltage divider that sets the potential of the positive converterterminal to the average of the two line voltages. It is only after thecommutation that the terminal potential recovers to the voltage of theon-going phase.

The consequence is that the voltage/angle-area A derived in Eq. (4.3) isdecreased by an area A as shown in Fig. 4.3. This results in a voltagedrop U d of the average direct voltage.

A =

eb ea + eb

2 d(t) =

eb ea

2 d(t)

= 3 E m

2

sin(t) d(t) = 3 E m

2cos(t)

= 3 E m2 (cos cos ) U d =

3

A

= 3 3 E m

2 (cos cos ) =

U di 02

(cos cos ) (4.14)

Comparison of Eqs. (4.12) and (4.14) shows that the voltage drop isdirectly proportional to the dc current:

U d = 3

LcI d (4.15)

The total average direct voltage is thus

U d = U di 0 cos U d= U di 0 cos RcI d (4.16)

withRc = 3

Lc (4.17)


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Rc is called the equivalent commutation resistance . It accounts for thevoltage drop due to commutation. However, it is not a real ohmicresistance and thus consumes no active power.

With Eq. (4.14) the average direct voltage could also be written as

U d = U di 0 cos U di 0

2 (cos cos )

= U di 0cos + cos

2 (4.18)


Due to the overlap the ac currents are no longer rectangular blocks.Instead, their shape is that of a deformed trapezoidal as can be seen inFig. 4.3. Still, Eq. (4.5) is a good approximation for the fundamentalfrequency component of the ac current:

I (1) 2 3

I d (4.19)

This approximation has a maximum error of 4.3% at = 60 and only

1.1% for 30 (the normal operating range) [1].Comparing again ac active power and dc power and using Eq. (4.18)gives

32

E m I (1) cos = U d I d

= U di 0I dcos + cos

2 (4.20)

and after substituting from Eqs. (4.4) and (4.19):

32

E m2 3

I d cos

3 3

E m I d cos + cos

2

cos cos + cos

2 (4.21)

With Eq. (4.18) another expression for the power factor cos is

cos U dU di 0 (4.22)


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55/128

4.2. Dynamic model 55


In the dynamic model the ac current is described by the dynamic phasor i, the fundamental component of which is given by

i (1) = 2 3

ej (arg( e ) ) (4.27)

where e denotes the mains voltage phasor.


Duration of overlap

As discussed in the previous section the duration of the commutationoverlap due to the inductance Lc is determined by the time it takesfor the on-going phase to take over the dc current. The commutationprocess with a non-constant dc current is shown in Fig. 4.5.

Hence, the conditions at the beginning and end of the commutation are

At t = : ia = id(t0 ) 2 I d and ib = 0

At t = : ia = 0 and ib = id (t0 ) + 2

I d(4.28)

The reference time t0 is chosen as the mid-point between the beginningand end of the commutation so that the average dc current during

ia

ibid

id (t 0 ) 2 I d

id (t 0 ) + 2 I d

t

Figure 4.5: Currents ia and ib during commutation showing the effectof a non-constant dc current


56/128


commutation is equal to id (t0):

t0 = +

2

The mesh equation for the commutation loop is again

eb ea = Lcdibdt Lc

diadt

(4.29)

Using the node equationia + ib = id

one nds that diadt

= did

dt dibdt

= I d dibdt

(4.30)

Substituting for di adt from (4.30) in (4.29) yields

3 E m sin(t) = 2 Lc dibdt Lc I d dib

dt =

3 E m2Lc

sin(t) +I d

2 (4.31)

The integral over the duration of the commutation is

dibdt

dt = 3 E m2Lc

(cos cos ) +

I d2

(4.32)

Inserting the boundary conditions (4.28) gives

id (t0) + 2

I d = 3 E m2Lc

(cos cos ) + 2

I d

id (t0) = 3 E m2Lc

(cos cos ) (4.33)

The conclusion of this result is that the duration of a commutation is

not inuenced by the derivative of the dc current. It only depends onthe average dc current during commutation.


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The instantaneous direct voltage during commutation is

ud = eb + ea2 12Lc I d ec Lc I d (4.34)This can be written as

ud = ud0 ud, ud, I d (4.35)where

ud0 = eb ec ud, =

eb ea2

12

Lc I d

ud, I d = 2 Lc I d

ud, is the additional drop of the direct voltage due to commutation.The average over a 60 interval is

U d, = 3

ud, d(t)= 3

eb ea2 12 Lc I d d(t)=

3 3 E m2

(cos cos ) 32

Lc I d (4.36)

Combining Eqs. (4.33) and (4.36) shows that the voltage drop due tooverlap depends on both the derivative of the dc current and its averagemagnitude during commutation:

U d, = 3 Lcid 32 Lc I d (4.37)Here the average direct current during commutation id (t0 ) has beenreplaced by the momentary direct current of the averaged model. Thetotal average direct voltage is thus

U d = U di 0 cos U d, U d, I d= U di 0 cos

3

Lcid 32

Lc I d 2Lc I d= U di 0 cos Rc id Lc I d (4.38)


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with

Rc = 3

Lc

Lc = 2 32 Lc (4.39)

Rc is again the equivalent commutation resistance introduced in theprevious section. The denition of Lc agrees with the notion of a meancommutation inductance as seen from the dc side, which is proposedin [28], albeit without further explanations. During non-commutationintervals it is 2 Lc and during commutation 32 Lc , thus:

Lc = 3

2Lc

3 +

32

Lc

= 3

23

2

Lc

= 2 32

Lc

Except for the term Lc I d , which is zero in steady-state, Eq. (4.38) isidentical to Eq. (4.16). The dynamic model is thus a generalization of the static model.


Using Eqs. (4.19) and (4.21), the fundamental ac current phasor is givenby

i (1) = 2 3 ej (arg( e )arccos( cos

+cos 2 )) (4.40)

4.3 Model validation

In order to validate the static and dynamic converter models and toassess their accuracy, their behavior is compared with detailed time-

domain simulations of a test circuit. The circuit used is a rectier withohmic-inductive load as shown in Fig. 4.6.


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4.3. Model validation 59

ea

eb

ec

ia

ib

ic

L c

u d

id

R load

L load

U load

Figure 4.6: Test circuit: converter with RL-load

4.3.1 Model equations

Both static and dynamic model are real rst-order state-space modelswith the direct current as the state variable.

For both models the overlap angle is given by

= arccos cos 2Lcid

3E m and the state equation is

ddt

id = 1Ld

3 3E m

cos U load (R load + Rc)id

withRc =

3

Lc

The only difference between the two models is the total dc inductanceLd . For the static model it is

Ld, stat = Lload

and for the dynamic model

Ld, dyn = L load + Lc = L load + 2 32 Lc


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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

0.75

1

1.25

i d [ p . u . ]

time [s]

Figure 4.7: Simulation of a step change of the ring angle from = 20to = 10. Thin continuous line: detailed simulation;heavy continuous line: dynamic model; dashed line: stan-dard model.

4.3.2 Qualitative comparison

Figure 4.7 shows the direct current response of the converter to a stepchange of the ring angle from = 20 to = 10. The converter isoperating in steady-state for one cycle before the step is applied. The

instantaneous direct current of the detailed simulation is drawn with athin continuous line. A heavy continuous line is used for the dynamicmodel, and a heavy dashed line for the standard model.

In steady-state both models yield the same direct current as expectedand show good accordance with the detailed simulation. However, theincrease of the direct current after the step-change of the ring angle aspredicted by the static model is too steep because the additional volt-age drop in the commutation reactances due to di/dt is not taken into

account. The dynamic model, on the other hand, follows the detailedsimulation very closely.

4.3.3 Quantitative comparison

The accuracy of the models is now assessed by modulating the mag-nitude of the mains voltage and the ring angle with varying frequen-cies. Special attention has to be given to the switching nature of the

converter with its periodicity of / 3. In order to avoid interferencesonly such modulation frequencies are chosen that are integer divisors of


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4.4. Conclusions 61

300 Hz. The magnitude and phase angle of the resulting modulationin the direct current are measured and compared with a detailed time-domain simulation. The parameters for the base case have been takenfrom [9] (with the commutation inductance adjusted to a fundamentalfrequency of 50 Hz): E m = 348.5717 kV, I d = 2 kA, Lc = 86 .5 mH,R load = 5 , L load = 0 .8288 H, U load = 495 kV. The converter operatesat a ring angle = 15.

The simulation results are summarized in Fig. 4.8. The plots show therelative amplitude error (top) and the absolute phase error (bottom) of the ac component on the direct current yielded by the two models ascompared with the detailed simulation. The plots on the left containthe results for a modulation of the mains voltage and those one the rightfor a modulation of the ring angle.

As expected, both models have the same small error at very low mod-ulation frequencies f 0.1 Hz. For higher frequencies the amplitudeerror of the static model steadily increases. The parameter variationsshow that for larger commutation reactances the error becomes largeras suggested by Eq. (4.38). On the other hand, the amplitude error of the dynamic remains very small, and its magnitude never exceeds 5 %.It is also less sensitive to variations of Lc .

An interesting detail can be seen in the phase error plot for the modu-lation of the ring angle: both models exhibit a phase error that growswith the modulation frequency. A closer inspection reveals that thiserror in fact is the sum of the phase error for the modulation of themains voltage and a term proportional to the modulation frequency.Further simulations showed that the slope of this additional phase erroris dependent on the commutation inductance Lc as well as the operatingpoint ring angle. A detailed study of this phenomenon has to be leftto future work.

4.4 Conclusions

In this chapter a dynamic modeling approach for the line commutatedconverter was presented, which takes into account the dynamics of an

averaged direct current. The resulting dynamic model is shown to bean extension of the conventional quasi-static model. This extension


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0

0

1 0

1 0

2 0

2 0

0 0

5 5 1 0 1 5

0 . 0 5 0 .

1 0 . 2

0 . 3

m o d u l a t i o n f r e q u e n c y

[ H z ]

m o d u l a t i o n f r e q u e n c y

[ H z ]

p h a s e e r r o r [ ] r e l . a m p l i t u d e e r r o r [ p . u . ]

M o d u l a t i o n o f

E m

M o d u l a t i o n o f

Figure 4.8: Simulation results for the quantitative comparison withparameter variations of the commutation inductance Lc( : 86.5 mH, : 43.25 mH, : 173 mH). Continuousline: dynamic model; dashed line: standard model.


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4.4. Conclusions 63

comprises an averaged commutation reactance added to the dc sidewhich is very easy to implement.

The neglect of the direct current dynamics in the static model leads

to errors when studying the dynamic response of the converter. Theseerrors are sensitive to the ratio between the commutation reactanceand the smoothing reactance on the dc side. Therefore, the proposeddynamic model is expected to have a particular impact for the study of systems with small dc reactances such as back-to-back HVDC links.


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Chapter 5

Capacitor commutatedconverter models

In this chapter the capacitor commutated converter is analyzed. Itstopology is shown in Fig. 5.1. The same basic assumptions as in the

previous chapter apply:

The ac voltage is stiff and may be represented by an ideal sinu-soidal source in series with a lossless inductance. The direct current is ripple-free and (in steady-state operation)constant. The valves are ideal switches with zero on-resistance and inniteoff-resistance. They change instantaneously between these two

states.

The valves are red at equal intervals of one-sixth cycle (60 ).The instantaneous line-to-neutral voltages of the ac sources are takenas

ea = E m cos(t + 60 )eb = E m cos(t

60) (5.1)

ec = E m cos(t 180)65


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66 Chapter 5. CCC models

ea

eb

ec

ia

ib

ic

L c

C

u Ca

uCb

uCc

u d

id

Figure 5.1: Equivalent circuit for the capacitor commutated converter.

The vales are numbered in the order of their ring

The corresponding line-to-line voltages are then

eac = ea ec = 3 E m cos(t + 30 )eba = eb ea = 3 E m cos(t 90)ecb = ec eb = 3 E m cos(t + 150 )

(5.2)

As in the previous chapter, the converter is rst analyzed for the staticcase, i. e. a constant direct current I d and balanced capacitor voltagesuC . In a second step, the static model is then extended by dynamicterms which take into account a varying direct current and the effect of capacitor unbalance.

5.1 Static model


The typical wave-forms of the converter are given in Fig. 5.2. For aconstant dc current I d and equidistant ring pulses, the ac currents arerectangular blocks as shown in sub-plot (d) for phase a.

Since the capacitors are charged and discharged by the ac currents, thecapacitor voltages have a trapezoidal shape as can be seen in sub-plot

(c). During the 120 interval in which the current I d ows through acapacitor, its voltage decreases from its positive peak to its negative


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peak (note the sign conventions in Fig. 5.1). The peak value U C canthus be calculated as

C 2U C = 2

3I d

U C = 3C

I d (5.3)

Each capacitor gives a contribution to the dc voltage as long as thecorresponding phase is connected to the dc side, that is, during the in-tervals in which its voltage changes. The negative anks of the capacitorvoltages appear at the positive dc terminal, the positive anks at thenegative dc terminal. Hence, the total contribution of the capacitors tothe dc voltage, ud,C , has the saw-tooth shape shown in sub-plot (b).

The instantaneous dc voltage nally consists of 60 segments of theline-to-line voltages plus the contribution of the capacitors. This isillustrated in sub-plot (a).


The average direct voltage is found by averaging the direct voltage overany 60 interval. In the interval [ , + 60], the instantaneous directvoltage is

ud = eb ec + ud,C = eb ec + uCb uCc (5.4)

It can easily be seen in Fig. 5.2 that, as ud,C changes linearly from

+U C to

U C , its average over 60 is zero. Therefore

U d = 3

+60

(eb ec) d(t)= U di 0 cos (5.5)

with

U di 0 = 3 3

E m (5.6)

as already shown in Eq. (4.3) for the line commutated converter.


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a)

b)

c)

d)ia id

u d

ud,C

u CauCa uCb uCc


120

60 0 60 120 180 240

t

t

t

t

t

Figure 5.2: Voltage and current waveforms without overlap: (a) instan-taneous direct voltage, (b) total contribution of the capaci-tors to the direct voltage, (c) capacitor voltages, (d) phase aline current.


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The line current wave-forms are identical to those of the line commu-tated converter. Hence, the peak value of the fundamental frequencycomponent is

I (1) = 2 3

I d (5.7)

and the angle by which the fundamental frequency component of theline current lags the line-to-neutral voltage is

= (5.8)

Operating range

With the average direct voltage and the line current being identical,the major difference between the line commutated converter and thecapacitor commutated converter is that the CCC can be operated atring angles smaller than 0 or larger than 180 because the capacitorsprovide the additional commutation voltage 2 U C . This is illustrated inFig. 5.3.

The minimum (or maximum) ring angle that can theoretically beachieved is determined by

3E m sin + 2 U C > 0 sin( min / max ) =

2I d3 3CE m (5.9)


Fig. 5.4 shows the equivalent converter circuit during commutation fromvalve 1 to valve 3. The mesh equation for the commutation loop is

eb ea + uCb uCa = Lcdibdt Lc

diadt

(5.10)

Using Eq. (5.2) and taking into account that the direct current is con-stant this can be written as

3 E m sin(t) + uCb uCa = 2Lc dibdt (5.11)


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(a)

(b)ea + u Ca

ea + uCa

eb + uCb

eb + uCb

ec + uCc

ec + uCc

ea

ea

eb

eb

ec

ec

u +

u +

u

u

120

60 0 60 120 180 240

t

t

t

Figure 5.3: Extended operating range of the capacitor commutated con-verter for (a) < 0 and (b) > 180. Line voltages atthe ac sources are drawn in dotted lines, line voltages at theconverter terminals in solid lines. Heavy lines mark the po-tentials of the positive and negative dc terminal with respectto ac neutral.

Under normal operating conditions, the capacitors in phase a and b arecharged in such a way as to accelerate the commutation. Unfortunately,the capacitor voltages themselves depend again on the current ib:

C duCa

dt = ia = ib I d

C duCb

dt = ib

An analytical solution is presented in [29] and summarized in appendix A.However, the approach used there is not suitable for an extension into

the dynamic operation range. A much simpler solution which is nonethe-less quite accurate will therefore be developed here based partly on [5].


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Duration of overlap

The voltage and current wave-forms of the converter circuit includingcommutation overlap are shown in Fig. 5.5 and a detailed view of theline currents and capacitor voltages is given in Fig. 5.6.

It can be seen that the anks of the line currents during commutationare nearly linear and can thus approximately be written as

ib(t) t

I d for t (5.12)

This approximation is in general justied due to the additional com-mutation voltage provided by the capacitors. However, for ring angles

close to 0 or 180, i. e. when the natural commutation voltage changesmost rapidly, the commutation current will be notably nonlinear, par-ticularly if the capacitor peak voltage is small in comparison with themains voltage.

Assuming a linear commutation current, the capacitor voltage is aquadratic function of time and so the total change in the voltage of one capacitor during the commutation is

U C = 1

2 uC

t2

= 1

2 I dC

2

= 2C

I d (5.13)

ea

eb

ec

ia

ib

ic

L c

C

u Ca

uCb

uCc

u d

id

Figure 5.4: Equivalent CCC circuit during commutation from valve 1to valve 3 (non-conducting valves not shown)


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a)

b)

c)

d)

ia

ia

ibib

ib id

u d

ud,C

u CauCa uCb uCc


e ac + e bc2

120

60 0 60 120 180 240

t

t

t

t

t

Figure 5.5: Voltage and current waveforms showing the effect of overlap:(a) instantaneous direct voltage, (b) total contribution of the capacitors to the direct voltage, (c) capacitor voltages,(d) phase a and b line currents.


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a)

b)ia ibI d

U C U C U C

U C U C + U C uCa

uCb

t

t

Figure 5.6: Detailed view of the waveforms during commutation over-lap: (a) capacitor voltages, (b) line currents.

During one half-cycle, a capacitor participates in two commutations,

and in the (2 / 3) interval in between it is charged (or discharged) bythe constant current I d . Hence the total excursion of capacitor voltagefrom peak to peak is

2U C = 2 U C +23

I dC

= C

I d +23

I dC

= 23C I d

This givesU C =

3C

I d (5.14)

for the capacitor peak voltage. A comparison with Eq. (5.3) shows thatthe peak voltage is independent of the commutation overlap.

The voltage/time areas shaded in dark grey in Fig. 5.6 are cubic func-


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tions of time. For symmetry reasons, both are identical and equal to

A = 1

6 uC t3

= 16 I

d

C 3

= 2

62C I d =

2

2U C (5.15)

and so the total voltage/time area shaded in light grey is

(u

Cb u

Ca) dt =

2

U

C 2

2

2U

C

= (2 )

U C (5.16)

or, in terms of I d ,

(uCb uCa ) dt = (2 )

32C I d (5.17)

Thus, integrating Eq. (5.11) over the interval of overlap and insertingthe boundary conditions ib(t = ) = 0 and ib(t = ) = I d yields

3E m2LcI d

(cos cos( + )) +

62LcC (2 ) = 1 (5.18)

This equation is transcendental and needs to be solved for numerically.However, it can be further simplied by approximating cos( + ) witha second-order Taylor series:

cos( + ) cos sin 2

2 cos (5.19)

Eq. (5.18) then reduces to a quadratic equation in with the solution

4X + Y 2 Y 2X (5.20)


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where

X = 3E m2L

cI

d

cos 2

162L

cC

(5.21)

Y = 3E m2LcI d

sin + 2

62LcC (5.22)


During the commutation from valve 1 to valve 3, the contribution of

the capacitors to the direct voltage is given by

ud,C = uCa + uCb

2 uCc (5.23)

For symmetry reasons the average of ud,C taken over the commuta-tion interval is again zero. Therefore, Eq. (4.18) derived for the linecommutated converter including overlap can be applied here

U d = U di 0 cos + cos 2 (5.24)

It should be noted, however, that the average direct voltage of the CCCwill nonetheless differ from that of the LCC because the capacitorsinuence the overlap angle and, consequently, the extinction angle = + .


Likewise, as the shape of the line currents is the same as with the LCCincluding overlap, Eqs. (4.19) and (4.21) are also applicable:

I (1) 2 3

I d (5.25)

cos cos + cos

2 (5.26)


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5.2 Dynamic model

As in the analysis of the dynamic behavior of the LCC in the previous

chapter, the direct current will now be assumed as ripple free but slowlyvarying at the constant rate I d :

id (t) = id (t0 ) + I d (t t0) I d const. (5.27)The rst, immediate consequence of this is again an average voltagedrop U d, I d across the commutation reactances. The analysis of thisvoltage drop is identical to that of the LCC. The voltage drop can thusbe modeled by adding an averaged commutation reactance on the dc

side as given in Eq. (4.39):

Lc = 2 32

Lc

A second consequence of a non-constant direct current is that the peakvalue of the capacitor voltages is no longer directly proportional to thedirect current. For arbitrary line currents the individual voltages inthe capacitors are given by the following state-space equation (note the

signing conventions in Fig. 5.1):

C duC dt

= i (5.28)

The corresponding differential equation for the fundamental phasors is

C du (1)C

dt = i (1) jC u

(1)C (5.29)

In steady-state operation a relation between the magnitude of the fun-damental capacitor voltage phasor |u

(1)C |and the capacitor peak voltage

U C can be determined by means of Fourier analysis:

U C

|u(1)C |

= 2

6 3 (5.30)

The line current phasor has a magnitude of |i(1)

| = 2 3

id , and so,by setting the left hand side of Eq. (5.29) to zero, the steady-state


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magnitude of u (1)C is found to be

|u(1)C |=

2 3C

id (5.31)

Inserting this equation into Eq. (5.30) yields:

U C = 3C

id (5.32)

which shows that the phasor-based approach is consistent with the staticmodel.

Finally, as a third consequence of a non-constant direct current, thecapacitor voltages can become unbalanced , i. e. the voltages in different

phases have different peak values [30]. As a result the individual contri-butions of the capacitors to the direct voltage do not necessarily canceleach other out on the average as was the case in the static analysis.Hence, there will be another contribution to the average direct voltage, U d,C , caused by capacitor unbalance.

This effect is illustrated in Fig. 5.7, where the direct current in aninverter is ramped up from 0 .5 p.u. to 1.0 p.u. within half a cycle by atemporary increase of the rectier voltage. The inverter is operated at aconstant ring angle . The quantities shown are (from top to bottom):the individual capacitor voltages uC , the angle of the capacitor voltagephasor arg( u C ), the total instantaneous contribution of the capacitorsto the direct voltage ud,C , and the direct current id of the inverter.

During the rst cycle, the inverter operates in steady-state. Thus, thecapacitors are balanced and the average of ud,C is zero. After therst cycle, as soon as the direct current starts to increase, the capaci-tors become unbalanced. Consequently, the average contribution of thecapacitors to the direct voltage becomes non-zero. This in turn inu-

ences the direct current which begins to oscillate. It can be seen thatthese oscillations are naturally damped so that after a few cycles a newsteady-state is reached.

A comparison between Fig. 5.7 (b) and (c) indicates that there is a cor-relation between the phasor angle of the capacitor voltages and theiraverage contribution to the direct voltage. It will be shown in the fol-lowing analysis that the average contribution in fact can be described bya function of both the magnitude of the fundamental capacitor voltagephasor u (1)

C and its angle with respect to the fundamental line current

phasor i (1) .


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(a)

(b)

(c)

(d)

u d

, C [ p

. u ]

a r g ( u C

)

u C

[ p . u . ]

i d [ p

. u . ]

0

10

20

30

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

0

0

0

1

1

0.5

1

1

1

time [s]

Figure 5.7: Simulation of a transient current increase from 0 .5 p.u. to1.0 p.u. in an inverter operating at constant : (a) capac-itor voltages, (b) capacitor voltage phasor angle, (c) con-tribution of the capacitors to the direct voltage, (d) directcurrent. Voltages are referred to the capacitor peak voltageU C at nominal current.


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Average direct voltage inuence of capacitor unbalance

Consider the interval in which valves 5 and 6 are conducting, i. e. phasec is connected to the positive and phase b to the negative dc terminal of the converter. The corresponding equivalent circuit is shown in Fig. 5.8.

In this interval the line currents are

ia = 0ib =

id (5.33)

ic = id

and so the space-vector of the line current is given by

i = 2

3ia

ib2

ic2

+ j 3 (ib ic)

= 2

30 +

id2

id2

+ j 3 (id id )

= j 2

3 id (5.34)This means that regardless of whether the direct current is constantor not the short-time average of the line current space-vector only

ea

eb

ec

ia

ib

ic

L c

C

u Ca

uCb

uCc

ud

id

Figure 5.8: Equivalent circuit during conduction of valves 5 and 6. Non-conducting valves not shown.


80/128


has a negative imaginary part, i.e.:

arg( i ) = 90 (5.35)The space-vector of the capacitor voltage is given by

u C = 23

uCa uCb

2 uCc

2+

j 3 (uCb uCc ) (5.36)

Since phase c is connected to the positive and phase b to the negativedc terminal of the converter, the contribution of the capacitors to thedc voltage is

ud,C = uCc

uCb (5.37)

A comparison with Eq. (5.36) shows that ud,C is proportional to theimaginary part of the space-vector u C :

ud,C = 3 ( u C ) (5.38)Thus, the average contribution of the capacitors to the direct voltage inthe conduction interval of valves 5 and 6 is proportional to the imaginarypart of the short-time average of the space-vector in that interval:

U d,C = 3

+60

ud,C d(t)=

3

+60

3 ( u C ) d(t)= 3 3

+60

u C d(t)= 3 u C (5.39)

Considering the angle between the line current and capacitor voltagespace-vectors (and their averages), U d,C can also be interpreted asthe projection of u C onto i

U d,C = 3 | u C | cos arg( u C ) arg( i ) (5.40)


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which has the advantage of being independent from the considered con-duction interval. This is illustrated in gures 5.9 to 5.12.

The top plots in these gures show the instantaneous values of thecapacitor voltages, their contribution to the direct voltage and the phasecurrents. The coordinate system at the bottom represents the space-vector plane and shows the loci of the instantaneous space-vectors of capacitor voltage and line current as well as their short-time averages.The shaded areas highlight the respective conduction intervals.

Fig. 5.9 presents the steady-state operation with balanced capacitorvoltages. Due to the trapezoidal shape of the voltage waveforms thespace-vector u C follows the edge of a regular hexagon. The line current

space-vector i , on the other hand, only assumes six discrete values owingto the block-shaped current waveforms. During the conduction intervalof valves 5 and 6, u C can be seen to cover an area which is symmetricwith respect to the real axis. Accordingly, its average, u C , is purelyreal and hence encloses a 90 angle with i which is purely imaginary asshown above.

In gures 5.10 to 5.12 the capacitor voltages have been unbalancedby adding different dc offsets in the individual phases. As a result

the waveform of their contribution to the direct voltage ceases to bea regular sawtooth: individual teeth are shifted up or down, and sotheir average is non-zero. With regard to the voltage space-vector, theunbalance causes a shift so that the center of the hexagon lies outsidethe origin of the coordinate system. Due to this shift the angle between u C and i is larger than 90 in some intervals, and smaller in others. Itcan be seen in the gures that these are the intervals in which ud,C has a negative resp. positive average.

In order to evaluate Eq. (5.40), it is necessary to express the unknown average space-vectors u C and i in terms of the known fundamentalphasors u (1)C and i

(1) . Since the capacitor voltages and the line currentshave the same fundamental frequency and due to their / 3-periodicity,the angle between their fundamental phasors is the same as the anglebetween the average space-vectors:

arg( u C ) arg( i ) = arg( u(1)C ) arg( i (1) ) (5.41)

This only leaves the magnitude of | u C | to be determi

ccc and csc

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