Điện tử công suât mạch MMC Thesis modular multilevel converters for railway supply

Luận văn, Điện tử công suất, đề tài tốt nghiệp, đồ án, thực tập tốt nghiệp, đề tài

Modular Multilevel Converters for

Railway Supply

Arif Haider

XR-EE-EME-2010:006

R OYAL I NSTITUTE OF T ECHNOLOGY

S CHOOL OF E LECTRICAL E NGINEERING

E LECTRICAL M ACHINES AND P OWER E LECTRONICS

Submitted to the School of Electrical Engineering in partial fulfillment of the

requirements for the degree of Master

Stockholm 2010

XR-EE-EME-2010:006

This document was prepared using Microsoft Word 2003

Sammanfattning

Normalt strưmfưrsưrjs tåg från en kontaktledning med växel- eller likspänning Kontakledningen fưrses med effekt på olika sätt t.ex med likriktare vid likspänningsmatning eller från transformatorer, roterande eller statiska omformare vid växelspänningsmatning Växelspänningssystem som använder 16 3 Hz behưver omriktare från trefas- till enfas-

spänning Sådana omriktare har framställts med hjälp av roterande omformare, omriktare eller spänningsstyva omriktare (Voltage Source Converter = VSC) kopplade back-back med gemensam likspänningssida Det har fưreslagits att man skulle använda en topologi känd som "modulär multinivå omriktare" (Modular Multilevel Converter = MMC), liknande den som används i kraftưverfưring med hưgspänd likstrưm (High Voltage Direct Current = HVDC), dock utrustad med fullbryggor i submodulerna De gemensamma skenorna, vilka motsvarar likspänningsuttagen i HVDC-omriktarna, kommer då att utgưra de två utgående växelspänningsterminalerna

cyklo-Ett 15 kV system baserat på MMC kan direkt producera den ưnskade växelspänningen och man eliminerar därigenom behovet av en stor och dyrbar lågfrekvenstransformator Dessutom reduceras fưrlusterna och de dynamiska prestanda fưrbättras Effekten som tillfưrs tågen pulserar till fưljd av den enfasiga matningen via kontaktledningen Kapaciteten i MMC-omriktarens interna energilager är tillräckligt stor fưr att reducera de (subharmoniska) pulsationerna i effekten som dras från det matande nätet

Strukturen fưr MMC-omriktaren med full-bryggor beskrivs i [1] och det påpekas att full fyrkvadrantdrift åstadkoms utan separat energitillfưrsel till de individuella likspänningskondensatorerna i submodulerna Alla submoduler har samma märkdata och systemet är därfưr enkelt skalbart Man noterar också att låginduktiva skensystem inte är nưdvändiga fưr att koppla ihop submodulernas kondensatorer; vanliga kablar kan användas

I denna rapport har en kontinuerlig modell (ộndligt antal submoduler, ộndlig switchfrekvens) av den MMC-baserade frekvensomriktaren fưr järnvägsmatning utvecklats i Matlab/Simulink I modellen levereras 15 kV växelspänning 16 3Hz från ett 50 Hz trefasnät

via en trasformator med huvudspäningen 15 kV Egenskaperna för omriktaren byggd med den nya teknologin introduceras baserat på en öppen styrstrategi Simuleringar och analytiska undersökningar visar omriktarens attraktiva egenskaper Kontinuerlig drift med aktiv last såväl som dynamiskt beteende vid laständringar, återmatning och reaktiva belsatningsförändringar demonstreras med simuleringsresultat Strömmarna på den matande sidan har försumbar harmonisk distortion och de bildar ett symmetriskt trfassystem vid stationär drift

Slutligen har en nollföljdskomponent adderats till spänningen (common mode voltage injection) för att maximera utnyttjningen av omriktarens huvudkrets

Abstract

Trains normally are powered from a catenary supplying AC or DC to the rolling stock Various means can power the catenary e.g rectifier stations for DC systems and transformers, rotating or static converters for AC systems For AC systems using 16 3Hz three-phase to single-phase converters are required Such converters have been implemented using rotating converters, cyclo-converters and Voltage Source Converters (VSC) connected back-back with a common DC link It has been proposed that the Modular Multilevel Converter (MMC) topology, similar to the one used in HVDC, can be used for the AC/AC conversion, however using sub-modules equipped with full-bridges The common bars for the phase legs, corresponding to the DC bars in the HVDC, then will constitute the single-phase AC side terminals

For a 15 kV system the MMC directly can produce the requested AC voltage, thereby eliminating the bulky and costly low-frequency transformer Moreover the losses are reduced and the dynamical performance can be improved The power to the rolling stock pulsates due

to the single-phase connection to the catenary The MMC has sufficient internal energy storage capability that can minimize these pulsations (sub-harmonics) in the power drawn from the grid

The structure of the MMC with full-bridges is described in [1] and it is pointed out that full four quadrant operation can be achieved without separate energy supply to the DC-capacitors All sub-modules have the same ratings and the system is easy to scale It is also pointed out that no low-inductive busbars are needed to connect the capacitors; ordinary cables can be used

In this work a continuous model (infinte number of sub-modules, infinite switching frequency) of the AC/AC MMC-based railway supply is developed in Matlab/Simulink It provides single-phase AC 15 kV / 16 3Hz and it is fed from the 50 Hz network through a transformer with the line-line voltage 15 kV The basic characteristics of this new converter technology will be introduced based on an open-loop control strategy Simulations and analytical investigations underline the attractiveness of this converter Steady state operation with active load as well as dynamic behaviour of the converter at load changes, regeneration

and reactive load changes has been demonstrated with simulation results Negligible harmonic distortion appears in the line-side currents, which form a symmetrical three-phase system in steady state

Finally common mode voltage injection has been applied in order to maximize the utilisation converter hardware

Keywords: Modular Multilevel Converter, Traction, Power transmission, MMC, Future

Technology , Electric Railway, Electric Traction, Electrification

Acknowledgements

Thanks to Allah the most merciful the most compassionate My deepest gratitude goes to my supervisors, Prof Hans-Peter Nee and Prof Lennart Ängquist It is an honor and a pleasure for me to have Prof Hans-Peter Nee as my supervisor and examiner It is also a privilege for

me to be a student of Prof Lennart Ängquist I am grateful for his generosity to share with

me his deep understanding and knowledge to develop the model Without his guidance, this project cannot reach the same level as it is today I am grateful to Hongbo Jiang (Banverket) for many inspiring discussions

At KTH, I would like to thank all the colleagues in the Electrical Machines and Power Electronics department In particular, I would like to thank PhD students Antonios Antonopoulos and Noman Ahmed for guidance and support

Many thanks to my parents and siblings for their love and support

Arif Haider

Stockholm, Sweden

November 2010

Sammanfattning ii

Abstract iv

Acknowledgements vii

Chapter 1 1

Introduction 1

1.1 Background 1

1.2 Project objectives and outline of the thesis 4

Chapter 2 6

Converter Technologies and Topologies 6

2.1 Converter Technologies 6

2.1.1 Line-commutated Current source Converters (LCC) 6

2.2 Some Multilevel VSC Topologies 9

2.2.1 Diode Clamped or Neutral point clamped Voltage Source Converter (NPC) 9

2.2.2 Capacitor Clamped or Flying Capacitor Voltage Source Converter (FC) 10

2.2.3 Modular Multilevel Cascaded Bridge converter (MMC) 12

2.3 Summary 14

Chapter 3 16

Modular Multilevel Converter (MMC) Model for Railway Supply 16

3.2 Topology and Description of the Model 16

3.2.1 Single Phase Circuit of the converter 18

3.2.2 Three Phase Circuit of the converter 19

3.2.3 Capacitor Voltages and currents in the arms 22

3.2.4 Calculation of Energy in Capacitor Banks 23

3.3 Converter Function 26

3.3.1 Continuous Model modulation 27

3.3.2 Dynamics of total capacitor voltages in the arms 28

3.4 Summary 29

Chapter 4 31

Modeling and Simulation Results 31

4.1 Converter Circuit Specification and Optimization of Parameters 31

4.2 Simulation Results Analysis 34

4.2.1 Steady state behaviour of the Converter 34

4.2.1.1 Resistive load: 35

4.2.2 Dynamic behaviour of the converter 37

4.2.2.1 Regeneration 37

4.2.2.2 Load Changes 40

4.2.2.3 Reactive load changes on the single-phase side 43

4.2.2.4 Reactive load changes on the three-phase side 48

4.2.2.5 Change of Stored Energy in the Converter Capacitors: 51

4.2.3 Third Harmonic Injection at three-phase AC side 52

4.3 Summary 55

5.2 Future work 59

References 61

A Equations to Estimate the Phase Quantities 63

B List of Symbols and Abbreviations 66

C Matlab 69

D Simulink Model 71

The first small scale electric locomotive was demonstrated by Siemens in 1876, only 50 years

after the first train ride for transporting people in 1825 The future development started with

electric railway connections for industrial purposes The first non-industrial electric railway

connection in Sweden was built in 1890 as part of Roslagsbanan

The DC-powered railways had a low voltage and were therefore not suitable for longer

distances The supply could not easily be changed to an AC-system because it was difficult to

make single-phase AC-machines suitable for locomotives A three-phase system was not an

attractive solution since it would require three connections, all of which are on high potential

A single-phase machine suitable for locomotives was developed by Westinghouse in 1902 It

was however only suitable for low frequencies Initially 15 or 25 Hz were used and later on

3

16 Hz was introduced

In Sweden the railway is supplied from the national 50 Hz electrical network Initially this

was done by connecting two synchronous machines with the pole pair ratio 3:1 Later on

converters based on power electronics were developed The first one in Sweden was a

cycloconverter and was installed in 1972 This was followed by self-commutated converters

with intermediate DC-connection Today the self-commutated converters have fully replaced

the cycloconverter in new installations in Sweden Although the converters have a high

efficiency and are easy to synchronize, they have some drawbacks compared to the rotary machines The synchronous machines do not require filters and they have an inherent transient smoothing capability due to the large rotor mass The larger machine based stations

in Sweden are still in operation and there are no plans to replace them in the next few decades The total number of converter stations in Sweden is 28 both rotary and static converters Minimum number of converter units per station is two while the maximum is five The minimum available rated converter power is 2.4 MVA and the maximum is 10 MVA A new MMC from Siemens with power 24 MVA is currently in the installation phase

Today, the well known two-level voltage source converter (2L-VSC) is dominating for low voltage industrial and traction applications It is not possible to connect such converters with only a single power semiconductor in each arm to medium or high voltage networks To overcome this problem new families of multilevel converters has emerged [4]

One solution is a Medium Voltage and High Voltage multilevel converter, also called Pulsed Step Inverters (PSI), which was proposed in [5] The main circuit is shown in Fig.1.1 together with typical output waveforms obtained using PWM in each stage (upper) or fundamental frequency switching of each stage (lower) The main characteristic of this technique is the series connection of step inverter stages

Fig 1.1: Main circuit (left) and possible output waveforms (right) of PSI

(Source: ref [5])

An alternative solution is the three-level neutral point clamped voltage source converter VSC or NPC) These converters have technical advantages like simple power part, a low component count and simple protection The high switching losses and a poor harmonic spectrum make them less favorable [9]

(3L-Emerging technology offers a new solution called the Modular Multilevel Converter (MMC) [1-2] MMC comprises strings of sub-modules, each one containing capacitors and power semiconductor switches The output waveform is synthesized by controlling the switches to insert or bypass a selected number of capacitors to create a stair-case waveform that approximates the sinusoidal wave [4-8] It can be used for for conversion from three-phase

AC to single-phase AC for traction This versatile traction converter concept is favorable due

to its modularity, its superior control characteristics as well as its good adaptability to various traction drives Thus, this converter topology might even have the potential to replace the conventional transformer of future AC-fed traction vehicles The main technical and economical aspects are given in [10] as

“Modularity:

- Scalable to different power- and voltage levels

- Independent of the state of the art of fast developing power devices

Multilevel waveform:

- Expandable to any number of voltage steps

- Low total harmonic distortion

- Dynamic division of voltage to the power devices

High availability:

- Use of approved devices

- Redundant operation

Failure management:

- Fail safe operation on device failures

- Avoidance of mechanical destruction (high current magnetic forces and arcing)

Investment and life cycle cost:

- Standard components

- Modular construction ”

1.2 Project objectives and outline of the thesis

The project that is reported in this thesis deals with the proposed MMC three-phase to phase converter mentioned above The objectives of the project are:

single-1 Develop a continuous model (i.e a model having infinite output voltage resolution and infinite switching frequency) of a 10 MVA AC/AC MMC that produces 16 3 Hz output with voltage 15 kV (nom), 16.5 kV (max)

2 To select the optimal input voltage on the three-phase input side Determine adequate parameters for the components in the model

3 Try to design an open loop control structure for the converter

4 To investigate steady state operation with active load as well as dynamic behaviour of the converter at load changes, regeneration and reactive load changes

5 To investigate the benefit of injecting third harmonic voltage in the three-phase AC side modulator in order to fully utilize the converter hardware

The project is conducted by both theoretical analysis and time simulations The outline of the thesis is:

Chapter 2 A short introduction of various converter technologies is given Different types of

multilevel converter topologies are briefly explained Their basic structure, advantages and disadvantages are given

Chapter 3 The model of the MMC, which is developed for the traction power supply, is

presented The structure and function aspects of the converter are described

Chapter 4 Optimization and selection of parameters for the converter model is given The

dynamic response of the model at different conditions is given Simulation results are compiled and analyzed

Chapter 2

Converter Technologies and Topologies

This chapter presents different converter topologies In section 2.1, a brief introduction to two major technologies, i.e., Line Commutated Converters (LCC) and forced-commutated Voltage Source Converters (VSC) are described In section 2.2, three different types of multilevel converter topologies are discussed: Diode Clamped or Neutral point clamped (NPC), Capacitor Clamped or Flying Capacitor Voltage Source Converter (FC), and Modular Multilevel Cascaded Bridge converter (MMC)

2.1 Converter Technologies

There are two main types of converter technologies that are used in power transmission and industrial applications Both of them have their advantages and disadvantages which make one better over the other in certain circumstances LCC and VSC are described briefly below

2.1.1 Line-commutated Current source Converters (LCC)

The LCC has been around for more than eighty years Originally it was implemented using mercury-arc valves However, during the 1960'ies the thyristor was invented and it replaced the old valves

The LCC technology has been used for railway supply in the form of cycloconverters to transform three-phase 50 Hz power into single-phase 16 3 Hz as shown in Fig.2.1 This converter simply comprises two anti-parallel six-pulse thyristor bridges The converter can operate in four quadrants (rectifier mode and inversion mode) Both the positive and negative converter can supply voltage at either polarity but the positive converter only supplies positive current and the negative converter only negative current The low-frequency output

instantaneous current direction, controlling its firing angle so that the desired instantaneous voltage is obtained

Ua Ub Uc

Fig 2.1 Three-phase to single-phase Cycloconverter

Two major problems related to the cycloconverter are the need for reactive power and the risk of commutation failures They also need a large amount of filters and they occupy huge space as shown in the Fig.2.2

Fig 2.2 LCC station (Courtesy ABB)

2.1.2 Forced-commutated Voltage Source Converters (VSC)

VSC technology was introduced about forty years ago Today this technology uses Insulated Gate Bipolar Transistors (IGBTs) or Integrated Gate Commutated Thyristor (IGCT) as switches These switches can be turned ON and turned OFF freely while in the LCC the voltage polarity must be reversed to turn OFF the switches

In VSC the polarity of the converter bridge voltage is constant and the power to the converter bridge can flow in both directions depending on the current direction In the LCC, on the other hand, the current direction is given so that the power flow direction is determined by the bridge's voltage polarity Moreover, the VSC does not need reactive power for its operation but rather it can provide reactive power to the system Due to their advantages, the VSC has been used since decades in industry, for transmission and distribution, traction etc The basic VSC is shown in Fig 2.3 (Upper single line diagram, lower real picture)

New topologies for VSC might make them even more attractive in the future But VSC also exhibits certain drawbacks; specifically the losses caused by the high switching frequency

that is required to reduce AC filter size The VSC also produces unwanted EMI at high di

dt in the arm currents

2.2 Some Multilevel VSC Topologies

Different topologies for multilevel VSC are presented here Every topology has its advantages and drawbacks MMC is becoming popular because of its advantages over the rest of the converter topologies

2.2.1 Diode Clamped or Neutral point clamped Voltage

Source Converter (NPC)

The three-level Diode Clamped VSC, also called neutral-point-clamped (NPC) converter, is shown in Fig 2.4 This converter has two clamping diodes in each phase which distinguish it from the two-level converter This converter contains twelve unidirectional active switches Diodes are connected in anti-parallel with the switches and six neutral point clamp diodes across the legs The AC terminal of each phase of the VSC can be switched to three different voltage levels, that is, the positive, the negative and the neutral mid-point The concept may

be expanded to include more voltage levels However, with an even number of voltage levels, the neutral point is not accessible [9]

i dc

u dc

L L L

The major advantages and disadvantages of NPC voltage source converter are given as

• Voltage balance across the DC bus capacitors at high number of levels is a significant challenge

The three-level NPC is a simple way to extend the existing voltage and power range of two- level converters because all the switches are operated at commutation voltage which is half of the DC link voltage Currently it is the most wide spread technology in the medium voltage range

2.2.2 Capacitor Clamped or Flying Capacitor Voltage Source Converter (FC)

The three-level Flying Capacitor or Capacitor Clamped VSC is shown in Fig 2.5 The structure of the Flying-Capacitor Multilevel converter is similar to that of the Diode-Clamped converter Capacitors are used in the FC instead of clamping diodes in the NPC The size of the voltage step in the output voltage waveform is the voltage increment corresponding to half the DC voltage

Fig 2.5 Three-level flying capacitor voltage-source converter

The Flying Capacitor VSC has phase redundancies which make it favorable over NPC that has only line to line redundancies These redundancies allow a choice of charging and discharging of specific capacitors and can be incorporated in the control system for balancing the voltages across the various levels [13] Static VAR generation is one of the most common applications for this topology

The major advantages and disadvantages of the FC are given as

Advantages:

• The DC link capacitor voltages can be balanced due to phase redundancies

• The large number of capacitors enables the converter to ride through short duration power outages and deep voltage sags

• Harmonic content will be low due to the increase of the number of output voltage levels

Disadvantages:

• To start up the converter, pre-charging of all the capacitors is a complex procedure

• The large numbers of capacitors are expensive

• The capacitors are bulky, which complicates the mechanical design Packaging becomes more difficult with high number of voltage levels

• The converter control is complicated

For industrial medium voltage drive, four-level FC are currently used At low and medium switching frequency the high expense of the flying capacitor is the main concern of this VSC topology

2.2.3 Modular Multilevel Cascaded Bridge converter (MMC)

The recently proposed Modular Multilevel Converter (MMC) concept [1-2,10] attracts significant interest for high-voltage converter applications due to its advantages and less complexity over conventional topologies Fig 2.6a) shows the MMC topology for HVDC In this case, three-phase AC power is converted into DC The sub-modules in this case contains half-bridges as shown in Fig 2.6b) One major feature of the MMC is that no common capacitor is connected between the converter's common bars Contrary to the VSC concepts

in the preceding sections, which all have a big common DC link capacitor, in the MMC, the

DC capacitors have been distributed in the sub-modules [12]

a) Modular Multilevel Converter MMC

b) Half Bridge Sub Module

c) Full Bridge Sub Module

Fig 2.6 (a) Modular Multilevel Converter MMC, (b) Half Bridge Sub Module (c) Full Bridge Sub

Module

In the railway supply, both sides of the converter are connected to AC terminals The converter is fed from three-phase AC voltage and the output is single-phase AC voltage as depicted in Fig 2.6 a) In this case the sub-modules must be full bridges, shown in Fig 2.6.(c) The reason is that the diodes in the half-bridges would clamp the output voltage between the common bars, making it impossible to reduce it below the line-line voltage amplitude of the input three-phase AC voltage Sub-modules with full-bridges do not impose this limitiation, but permits that the voltage between the common bars may vary freely

The output voltage from the MMC is adapted towards a sinuoidal waveform by instantaneously varying the number of inserted sub-modules At each step only one sub-module is switched Therefore the average switching frequency for each semiconductor device becomes quite low Thus the switching losses are greatly reduced although the harmonic content in the output voltage is very low

The MMC can easily be scaled up by inserting additional sub-modules in each arm, which makes it attractive for high-voltage applications An additional benefit of using full-bridge sub-modules is that the control system can instantaneously open the connection between the input and output sides in order to prevent heavy current at faults in either network Therefore fault recovery can be fast as the DC capacitors do not necessarily discharge during faults [14]

The major advantages and disadvantages of MMC for railway supply are

Advantages:

• Low harmonics at low switching frequency

• Protection is simplified by use of full-bridge sub-modules as described above

• The AC to AC conversion is done in one step In other converters, first the phase AC voltage is converted into DC and then the DC voltage is inverted into single-phase AC

to its modularity and less complex structure makes it favorable over the rest of NPC and FC topologies MMC would be most probably used as major voltage source converter at medium and high voltage for industry applications

Chapter 3

Modular Multilevel Converter (MMC)

Model for Railway Supply

This chapter describes the MMC model developed for railway supply In Section 3.1 a brief introduction to the railway system in Sweden is given Section 3.2 explains the topology and description of the model Section 3.3 describes functioning of the converter and this chapter

is summarized in Section 3.4

3.2 Topology and Description of the Model

A variety of main circuit configurations of cascaded full-bridge converters offers power conversion systems for various applications One such application is to create a single-phase voltage source to feed an electric railway from a three-phase source Modular multilevel topology is used for the circuit configuration Fig 3.1 shows the main circuit

Each leg has an upper arm and a lower arm All the upper arms are connected together to get one terminal of the single-phase system Similarly the lower arms are joined to the other terminal The upper and lower arms of each phase comprise N sub-modules Each arm has an inductor L and a resistor R The three-phase supply is assumed to be provided from stiff voltage sources The neutral point is grounded through a resistor Rz, which has high reistance

in order to prevent zero sequence current from passing through the three-phase AC voltage source The midpoint of the single-phase side is also grounded The ground potential also forms the reference for the voltages on the three-phase AC side Each three-phase quantity is represented by a space vector plus a zero-sequence component The phase quantities can be interpreted as

2 3 2 3

The converter is a combination of two circuits, namely the three-phase circuit and the phase circuit The power is transmitted from three-phase side to single-phase side The arms act as the interface between the three-phase and single-phase circuits and they shall insert the instantaneous voltage deviations between these systems In the model continuous controllable voltage sources are used In reality the voltages will be controlled by insertion or bypassing

single-of the sub-modules in the MMC obtained by modulation technique Pulse Width Modulation (PWM) technique is generally used The three-phase and single-phase circuits are described

in the following

3.2.1 Single Phase Circuit of the converter

The equivalent circuit of the single-phase network of the converter is shown in Fig 3.3 The midpoint of the single-phase side is grounded (in order to simplify the analysis)

Fig 3.31 phase network of the converter

Half of the single-phase voltage appear on both sides of the grounded point represented as

USU and USL The desired emfs inside the converter behind the impedance on the single-phase side have the same amplitude and phase and are given as

( ) ( )

ˆcos2ˆcos2

s SU

s SL

s CLz CUz

e

Currents i SU( ) ( )t ,i SL t flow when the single-phase side is connected to a load The currents are sinusoidal if the load is linear Ideally these currents are identical so that no zero-

sequence current flows through the three-phase AC side voltage sources Then all

zero-sequence current circulating through the single-phase circuit can only pass as zero zero-sequence currents through the arms of the converter The arm symmetry then splits the circulating zero-

sequence current into three equal parts In this case we have

ˆˆ

3.2.2 Three Phase Circuit of the converter

The zero-sequence components in the converter arms do not create any voltage differences between the three-phase AC side terminals Thus the zero-sequence voltages do not influence the current flow in the 3-phase AC circuit in any way However equation (6) shows that an average energy exchange between the single-phase network and the converter arms is at hand This cannot go on in steady state as then the capacitors in the arms will either accumulate infinite energy or be emptied A balance must be established by delivering energy from the three-phase AC side such that the capacitor energies in the arms remain constant The equivalent circuit for the three-phase part of the converter is constituted by an AC side

internal emf connected to the external voltage source through the resistive-inductive impedance (R/2, L/2) shown in Fig 3.4

Fig 3.4 3 phase network of the converter

It should be understood that any zero-sequence-free set of voltages can be inserted in the arms with out disturbing the status of the single-phase circuit The connected AC side voltage

is three-phase symmetric and given as

Similar power is fed into other arms of the converter The total average power from the single-phase zero-sequence component according to equation (6) and from the three-phase component according to equation (11) must be equal in steady state Thus

ˆ ˆ

s s s

v v

Fig 3.5 Block diagram of MMC

3.2.3 Capacitor Voltages and currents in the arms

The current i V can be given as

ˆ

ˆ

2ˆ

3

π

− Only equations for phase ‘a’ will be discussed here The equations remain the same for the other two phases with the corresponding phase shift

The ideal, desired instantaneous current in the capacitor arms is the sum of one third of the single-phase current and half of the three-phase current of the corresponding phase The instantaneous currents in the lower and upper arm of phase ‘a’ are given as

Similarly for the other two phases

3.2.4 Calculation of Energy in Capacitor Banks

To calculate the instantaneous energy in the capacitor bank, first the power across the capacitor bank in the arm is calculated The derivative of the energy stored in the capacitor bank is the power across that arm Integration of the derivatives thus gives the instantaneous energy in the capacitor banks in the given arm Calculation for phase ‘a’ is only shown The equations are similar for phase ‘b’ and ‘c’ with corresponding phase shift For the lower arms the negative product of the voltage and the current yields the instantaneous power fed into the capacitor given as

12ˆˆ

4ˆˆ

8ˆˆ

4ˆˆ

8ˆˆ

8ˆˆ

8ˆˆ

{ } { }0

The capacitor energy in the upper arm of phase 'a' , , can be freely selected and is given

CUa arm Cref

so that the total capacitor voltage in the upper arm can be estimated as

2 CUa

CUa

arm

W u

C

Equations (21) and (22) are used in the simulation model to calculate the energies across the arms Some terms are added to these equations later on when common mode voltage is injected on the three-phase AC side

3.3 Converter Function

The function of the converter is to provide stable single-phase AC voltage from the phase AC voltage, to keep the three-phase side line currents symmetric and to transfer the required power to the load from the three-phase AC network to the single-phase AC network

three-or, in case of regeneration, from the single-phase AC network to the three-phase AC network The block diagram of Fig 3.6 shows the simple function of the model Input power to the converter is the power provided by the three-phase grid The output power from the converter depends on the load The currents i V and i s are measured and given to the three-phase and single-phase phasor estimators respectively The estimated values are then given to the open loop controller, which calculates the energies in the arm capacitors using the energy reference value From the obtained energies the total capacitor voltage in each arm is determines according to equation (24) Finally the insertion index for each arm is obtained as the quotient between the desired inserted voltage in the arm, given by equation (15), and the estimated total voltage for that particular arm[15] The modulation method used for the continuous model is described in the following

ref

W∑

ref arm

3.3.1 Continuous Model modulation

One phase arm is shown in the Fig 3.7 Let N be the number of modules per arm for the converter In general, each arm is controlled by an insertion index n in the range -1 < n < 1

which is a function of time t means that all N modules in the arm are bypassed and

means that all N modules in the arm are inserted, either in the positive or negative direction, depending on the sign of n(t) Let each capacitor in the sub-module have the capacitance C Then the capacitance of the arm is given as

N

= (25) and the effective capacitance of the arm is given as

( )

arm x

C C

n t

= (26)

where the subscript x denotes the number of the arm

Fig 3.7 phase arm

In the simplest modulation approach, the direct modulation, it is assumed that the total capacitor voltage in all arms are constant at a given DC reference value The inserted arm voltages shall match the instantaneous voltage difference between the connected three-phase AC side voltages and the desired single-phase voltage Thus the modulation index n is calculated as

Cref

u∑