Wide-band modeling of modular multilevel converters …scientiairanica.sharif.edu/article_3997_ed14c6971db50f6a550c5c... · This paper presents a model for a Modular Multilevel Converter

Scientia Iranica D (2016) 23(6), 2881{2890

Sharif University of TechnologyScientia Iranica

Transactions D: Computer Science & Engineering and Electrical Engineeringwww.scientiairanica.com

Invited Paper

Wide-band modeling of modular multilevel convertersusing extended-frequency dynamic phasors

S. Rajesvaran and S. Filizadeh�

Department of Electrical and Computer Engineering, University of Manitoba,75A Chancellor's Circle, Winnipeg, MB R3T 5V6,Canada.

Received 12 March 2016; received in revised form 20 April 2016; accepted 18 July 2016

KEYWORDSModular MultilevelConverter (MMC);Voltage-SourceConverter (VSC);High-Voltage DirectCurrent (HVDC);Electromagnetictransient (EMT);Dynamic phasors;Modeling.

Abstract. This paper presents a model for a Modular Multilevel Converter (MMC)using extended-frequency dynamic phasors. The model is based upon a series of harmonic-frequency representations that are obtained for the fundamental and dominant harmoniccomponents. Depending on the requirements of the study to be conducted and the desiredlevel of model accuracy, a low-order model (i.e., average-value) or an arbitrarily wide-band model may be constructed. The paper describes the principles of modeling usingextended-frequency dynamic phasors, and applies them to an MMC connected to a powersystem represented using a Thevenin equivalent. The model contains details of the MMC'scontrol system including its high-level control circuitry, as well as voltage balancing andsynchronization components. The developed model is then validated by comparing itagainst a fully-detailed electromagnetic transient (EMT) simulation model developed inPSCAD/EMTDC simulator. Comparisons are made to establish the accuracy of the model(both its low-order and wide-band variants) and to assess the computational advantages ito�ers compared to conventional EMT models.© 2016 Sharif University of Technology. All rights reserved.

1. Introduction

In High-Voltage Direct Current (HVDC) transmissionsystems, thyristor-based Line-Commutated Converters(LCC) have been used for decades. LCC-HVDC o�ersthe bene�ts of a mature technology and is availableat ratings reaching thousands of MWs per converter.LCC-HVDC systems, however, su�er from technicalchallenges such as large �ltering requirements, limitedability to operate into weak ac systems, and con-verter commutation failure. As such, a new breedof HVDC converters, which are based upon voltage-source converters (VSC-HVDC), have been pursued toalleviate these problems. VSCs utilize fully-controlled

*. Corresponding author. Tel.: +1-204-480-1401;E-mail addresses: [email protected] (S.Rajesvaran); shaahin.�[email protected] (S. Filizadeh)

switches such as Insulated-Gate Bipolar Transistors(IGBTs) instead of thyristors to avoid commutationfailure problems [1]. A class of VSC topologies, knownas Modular Multilevel Converters (MMCs), has beenintroduced [2], which o�er reduced harmonics (andhence reduced �ltering requirements), lower switchinglosses, and superior output quality to conventionaltwo-level and multilevel voltage-source converters. AnMMC is constructed using a cascade connection of alarge number of identical units known as submodules.Commonly used types include half-bridge and full-bridge submodules [1-4].

The bene�ts of MMC-based HVDC convertersare achieved at the expense of increased control andoperating complexity. MMC converters deploy a largenumber of submodule capacitors, whose voltages needto be tightly regulated and balanced for proper op-eration. Additionally, the generation of high-quality

2882 S. Rajesvaran and S. Filizadeh/Scientia Iranica, Transactions D: Computer Science & ... 23 (2016) 2881{2890

output voltages without incurring large switching lossesrequires use of advanced waveform synthesis tech-niques [5-7].

Similar to other high-power converter systems,many of the studies pertinent to the design, controllertuning, and operating characteristics of MMC-HVDCsystems rely on detailed computer simulations. In par-ticular, electromagnetic transient (EMT) simulatorsare prime simulation platforms for this purpose due totheir wide-band representation of transient phenomenain switching circuits. The complexity of MMC circuitryand the presence of a large number of switches posemajor computational di�culties for EMT simulations.It is often observed that simpli�cations need to be madein order to enable EMT simulation of MMC systemswithin reasonable computing time. MMC models withreduced computational intensity fall into two generalcategories of averaged (i.e., low-frequency) models,and low-intensity full-order (also known as detailedequivalent) models. Generally, MMC models aredeveloped with a pre-determined modeling outcome inmind. Low-order average-value models are inherentlyunable to capture transients of high frequencies, anddetailed models naturally include as much detail aspossible within the simulation bandwidth [8].

This paper proposes a new MMC model thatis based upon extended-frequency dynamic phasors.These mathematical artifacts are based upon decom-posing a (quasi-) periodic waveform into constituentharmonics, which are then included in the �nal model.By doing so, the user of the resulting model canjudiciously select the model's desired level of accuracyby including as many harmonic components as needed.This a�ords a high degree of control to the user toadjust the band-width of the model depending on therequirements of the study to be conducted. It isobvious to note that this modeling approach will alsoo�er exibility in its �nal computational complexity.

Modeling using dynamic phasors has been appliedto such complex systems as LCC-HVDC, VSC-HVDC,and wind energy systems [9-11]. A preliminary (andonly low-order) dynamic phasor model of an MMCsystem is presented in [12,13] wherein the eigenvaluesof the system are studied; the presentations in [12,13],however, do not include any validation of the model.Another phasor model of an MMC is presented in [14],which is developed in a rotating d-q frame. However,this model only includes the fundamental and thesecond harmonic corresponding to the MMC arm'scirculating current. As such, expansion of these modelsto include arbitrary higher-order harmonics, as is donein this paper, remains unresolved.

The extended-frequency dynamic phasor modelof an MMC in this paper includes its controls and isvalidated against a detailed EMT model developed inPSCAD/EMTDC electromagnetic transient simulation

program. The dynamic phasor model is developeddirectly in the original abc frame, which enables directinterfacing possibility with other system componentsmodeled in the abc domain (e.g., in the context of EMTsimulations). To facilitate the readability of the math-ematical development of the model, a Nomenclature isincluded at the end of the paper.

2. Mathematical principles of modeling usingdynamic phasors

Modeling using dynamic phasors is based upon de-composition of a quasi-periodic time-domain waveforminto its harmonics using Fourier analysis over a slidingwindow. As the window slides along the time axis, thetime evolution of the Fourier coe�cients is obtained.For reconstruction, all or a subset of harmonics may beselected. The former produces the original waveformand the latter gives an approximation of the originalwaveform. The approximation can be made moreaccurate if a larger subset of harmonics is selected forreconstruction, albeit at the expense of more computa-tions. The governing equations for decompositions ofa signal x(t) over its period T0 are given in Eqs. (1)and (2):

x(�) =+1X

k=�1hxik(t)ej!k� (1)

where, � 2 [t� T0; t]:

hxik(t) =1T0

Z t

t�T0

x(�)e�j!k�d�; (2)

where hxik(t) is the kth Fourier coe�cient (the kth-order dynamic phasor) of the signal and k denotesthe order of the harmonic. Decomposition into dy-namic phasors is a linear operation. In applying thistransformation to dynamical systems, the following twoproperties are often used [15]:

dhxik(t)dt

=�dx(t)dt

�k� j!khxik(t); (3)

hu(t)v(t)ik =+1Xl=�1

hu(t)ilhv(t)ik�1: (4)

Eqs. (2) to (4) are normally used to model a systemin the extended-frequency dynamic phasor domain.Then, Eq. (1) is used to reconstruct a waveformin the time-domain. As stated earlier, any desirednumber of extended-frequency dynamic phasors maybe calculated and any desired number may be used inreconstruction.

S. Rajesvaran and S. Filizadeh/Scientia Iranica, Transactions D: Computer Science & ... 23 (2016) 2881{2890 2883

3. MMC modeling using extended-frequencydynamic phasors

Figure 1 shows the circuit diagram of a simple MMCconnected to an ac system. The Kirchho�'s voltagelaw is applied to the upper- and lower-arm loops andalso to the upper- and lower-arm submodules to obtainthe following four state equations for the arm inductorcurrents and submodule capacitor voltages as shown inEq. (5):

Ldiujdt

=V udc � riudc�vucj �Riuj � (Ls+Ltf )d(iuj � jlj)

dt

�Rs(juj � jlj)� Vsj ;

Ldiljdt

=V ldc�rildcvlcj �Rij�(Ls+Ltf )d(iuj � jlj)

dt

+Rs(iuj � jlj) + Vsj ;

Cmdvuav�jdt

=nuj iujN

;

Cmdvlav�jdt

=nljiljN

; (5)

where j = a; b; c, and superscripts u and l denote theupper and the lower arms, respectively. Other variablesare de�ned in the Nomenclature at the end of the paperand are marked in Figure 1 as well. Here, vucj , vlcj ,iudc, and ildc in the equations shown in Eq. (5) can besubstituted as follows:

vucj = nuj vuav�j ; and vlcj = nljv

lav�j ;

iudc = iua + iub + iuc ; and ildc = ila + ilb + ilc:

To avoid complexity, the summation and the di�erenceof the upper and lower inductor currents and capacitorvoltages are considered. The resulting state equationsare given in Eq. (6):

didjdt

=��

R+ 2RsL+ 2Ltf + 2Ls

�idj

+��riddc � (nsjvdav�j + nsjvdav�j)=2� 2Vsj

L+ 2Ltf + 2Ls

�;

disjdt

=��RL

�isj

+�Vdc � risdc � (nsjvsav�j + ndjvdav�j)=2

L

�;

dvdav�jdt

=

nsjidj + ndj isj

2NCm

!;

dvdav�jdt

=

nsjisj + ndj idj

2NCm

!j = a; b; c; (6)

where superscripts s and d denote the summation anddi�erence, respectively.

In Eq. (6), other than the state variable, theonly unknown is nj , which is the switching signal forthe MMC. Generating switching signal is explained inSection 4.4. From the state equations in Eq. (6), thecorresponding dynamic phasor equations are obtainedusing the basic operations given in Eqs. (2), (3), and(4). The resulting dynamic phasor equations are givenin Eq. (7), as shown in Box I, wherein j denotes aphase and k denotes a harmonic order: The di�erencebetween the upper- and the lower-arm currents directly

Figure 1. Schematic diagram of an MMC connected to an ac system.


dhidj ikdt

= �j!khidj ik ��

R+ 2RsL+ 2Ltf + 2Ls

�hidj ik

+��rhirdcik � +1P

l=�1hnsjilhvdac�jik�l +

+1Pl=�1

hndj ilhvsav�jik�l!=2� 2hVsjik

L+ 2Ltf +2 Ls

�;

dhisjikdt

= �j!khisjik ��RL

�hisjik +

� hVdcik � rhisdcik � � +1Pl=�1

hnsjilhvsav�jik�l ++1Pl=�1

hndj ilhvdav�jik�l�=2

L

�;

dhvdav�jikdt

= �j!khvdav�jik +� +1Pl=�1

hnsjilhidj ik�l ++1Pl=�1

hndj ilhisjik�l2NCm

�;

dhvsav�jikdt

= �j!khvsav�jik +� +1Pl=�1

hnsjilhisjik�l ++1Pl=�1

hndj ilhidj ik�l2NCm

�: (7)

Box I

gives the line current. Note that hidj i and hvdav�ji onlycontain odd harmonics; likewise, hisji and hvsav�jionlycontain even harmonics.

4. Dynamic phasor modeling of MMCcontrollers

In the considered MMC test system, the real powerand the RMS terminal voltage are controlled via closed-loop control systems shown in Figure 2. As shown, theerrors between the reference and the actual variablesof interest are fed through Proportional-Integral (PI)controllers that eventually generate commands for theconverter's modulation index and phase shift. Theconverter's output voltage phase-shift is relative toa reference sinewave that is achieved using a Phase-Locked Loop (PLL) unit connected to the Point ofCommon Coupling (PCC). The controller inputs arethe reference values of real power and RMS terminalvoltage; the controller outputs are also (quasi-)dc

Figure 2. Schematic diagram of the MMC controlsystem: (a) Real power controller; and (b) Terminalvoltage controller.

quantities. As such, in their dynamic phasor model,inclusion of harmonics is not necessary.

4.1. PI controller modeling4.1.1. Real power controllerFrom Figure 2(a), the state equations for the real powercontroller can be obtained as follows:dydt

= Pref � Pmeas;

� = kp2dydt

+ ki2y: (8)

Real power has primarily only a dc component. There-fore, only one value of � is needed (i.e., no harmonic-order model for the PI controller is developed); Pref isthe ordered power and the actual power is determinedas follows:

Pmeas = Vta:ia + Vtb:ib + Vtc:ic:

hPmeasik =1Xl=0

hVtailhiaik�l +1Xl=0

hVtbilhibik�l

+1Xl=0

hVtcilhicik�l;

hPmeasi0 =1Xl=0

hVtailhiai�l+hVtbilhibi�l+hVtcilhici�l;(9)


4.1.2. Terminal RMS voltage controller modelThe following state equations of the voltage controllerare developed for the control block diagram shown inFigure 2(b):

dzdt

= Vref � Vmeas;

m = kp3dzdt

+ ki3z: (10)

The terminal voltage is a staircase waveform obtainedusing stacking of an appropriate number of submod-ules; due to its quarter-cycle symmetry, it containsonly odd harmonics. The controller input, Vmeas, is theline-to-line RMS value. The real and imaginary valuesof the terminal voltage for each harmonic are known,and the basic RMS equation and the correspondingdynamic phasor equation are given as follows:

Vmeas =r

32:Vtapeak;

hVmeasik =r

32:2qRe (hVtaik)2 + Im (hVtail)2: (11)

The dynamic phasor form of Eq. (10) is shown inEq. (12). In Eq. (12), Vmeas represents the magnitudeof kth harmonic that is a dc value. Therefore, theoutput modulation indices for each harmonic are alsodc values:

hdzdtik = hVref ik � hVmeasik;

hmik = kp3hdzdt ik + ki3hzik: (12)

Using Eq. (12) for each odd harmonic, the modulationindex values corresponding to each harmonic can beobtained. By using the modulation index values in theswitching signal, the terminal voltage can be regulated.To �nd the reference values for each harmonic, thenearest level control (explained in Section 4.3) is used.

4.2. Modeling of the phase locked loopA PLL is a feedback control system that rapidly tracksthe phase angle of a given sinusoidal voltage. Adynamic phasor model of the PLL is developed in [9]and its block diagram is shown in Figure 3. This modelis directly used in this study as well. Here, !0 is thenominal system frequency (i.e., 2� 60 rad/s in a 60-Hzsystem).

From Figure 3, the following state equations areobtained for the PLL:dxdt

= \Vt � �;d�dt

= kp1 (\Vt � �) + ki1x: (13)

Figure 3. Dynamic phasor model of a Phase LockedLoop (PLL).

For conversion to the dynamic phasor domain, onlythe k = 0 component of the PLL state equationsis considered. The output � is used to generate theswitching signals for the converter, as explained inSection 4.4.

4.3. Nearest level control and reference voltageharmonics

The MMC control system determines the fundamentalvalue of Vref and from that, a staircase voltage wave-form closely resembling a sinewave with such magni-tude must be generated. To generate the staircasewaveform, a modulation technique is needed. Thereare a number of high-frequency PWM techniques thatresult in high switching losses. Therefore, in this paper,a low-frequency modulation technique known as theNearest Level Control (NLC) is used [7]. For thismethod, the nominal value of the submodule capacitorvoltage is needed, which can be calculated as follows:

Vcap =VdcN: (14)

To obtain the staircase waveform, the following formu-lae are used:

For even values of N :

Vout =

(nVcap; Vref <

�n+ 1

2

�Vcap

(n+ 1)Vcap; Vref >�n+ 1

2

�Vcap

where 0 � n � (N2 ).

For odd values of N :

Vout =

(�n� 1

2

�Vcap; Vref < nVcap�

n+ 12

�Vcap; Vref > nVcap

where:

0 � n ��N � 1

2

�: (15)

For the odd and even numbers of submodulesper arm, the staircase waveform is di�erent. However,the �nal number of the levels in the output staircasewaveform is N + 1.

Figure 4 shows the staircase waveform for a halfcycle when the number of submodules is 8. Therefore,


Figure 4. Staircase waveform (positive half-cycle shown)for N = 8.

the number of levels seen in the output waveform is9 for both positive and negative half cycles. Theswitching instants between levels correspond to thenearest level angles. By using Eq. (2) for each value ofk, the kth Fourier coe�cient can be calculated, whichcan then be converted to line-to-line RMS and given asthe reference voltage for each harmonic in the voltagecontroller.

4.4. Generation of switching signalsTo generate the switching signals for the upper- andlower-arms, the modulation index and the phase anglefrom the controller and PLL angle are required. Theupper- and lower-arm switching signals for phase à'are given in Eq. (16):

nua =N2� N

2

1Xk=1

hmik sin(k!t+ k� + �k);

nla =N2

+N2

1Xk=1

hmik sin(k!t+ k� + �k); (16)

where k is an odd number. For phases `b' and `c', phaseshifts of 120� and 240� need to be added, respectively.Since the di�erence and the sum of the switchingsignals are used in Eqs. (6) and (7), the sum anddi�erence of the switching signals in Eq. (16) need tobe obtained as follows:

nda = �N1Xk=1

hmik sin(k!t+ k� + �k)

nsa = N: (17)

As can be seen from Eq. (17), the sum of the switchingsignals is always a constant and it is equal to thenumber of submodules per arm of the MMC. Theonly time-variant part is the di�erence of the switchingsignals. Therefore, using the basic form in Eq. (2), thedynamic phasor representation of ndj can be found asgiven in Eq. (18):

nda =�N�hmi1 sin(!t+ � + �1) + hmi3 sin(3!t

+ 3� + �3) + hmi5 sin(5!t+ 5� + �5) + :::�

hndaik = �Nhmik2j

ej(k�+�k) =jNhmik

2ej(k�+�k): (18)

Similarly, the switching signals corresponding to othertwo phases can be obtained. For odd values of k, thekth harmonic of switching signal can be obtained.

5. Model validation

A detailed switching model of the MMC test systemis developed using PSCAD/EMTDC electromagnetictransient simulation program. EMT simulators suchas PSCAD/EMTDC are widely used to study fasttransients in power systems with embedded power elec-tronics. Due to their detailed component models andaccurate solution methods, EMT simulation results aregenerally accepted to be close representations of thereal phenomena, and as such it is common to use themas benchmarks for validation of other simulators. Theparameters of the simulated circuit are given in Table 1.

To show the details of the output staircase wave-form, the number of submodules per arm (N) isselected as 5. In addition, to minimize the burden onmodeling, large capacitors (5000 �F) are selected sothat the e�ect of circulation current becomes minimal.

Table 1. Power system parameters.

Parameters Value

AC system rated voltage 290 kVShort circuit ratio 4.0 \80�

Rated power 500 MWRated dc voltage, internal resistance 500 kV, 0.001Converter transformer turns ratio 290:290Converter transformer impedance 0.05 pu, 700 MVAArm inductor 1 mHSubmodule capacitor 5000 �FNominal voltage of submodule capacitor 100 kV


The parameters of the PI controllers are given inTable 2. The values of proportional (kp) and integralgain (ki) are selected based upon trial and error.

In the following sections, all PSCAD/EMTDCresults are obtained with a 50 �s time-step and alldynamic phasor results are obtained with a 100 �stime-step.

5.1. Model validation for the fundamentalfrequency

In Figure 5, the comparison of results from the dy-namic phasor and EMT models is shown only for thefundamental component. Although the EMT modelincludes all harmonics, the fundamental componentsof the shown waveforms are extracted using a standardFFT block. The dynamic phasor model in this set ofsimulations only includes the fundamental component.The results clearly show that the waveforms are wellmatched.

5.2. Model validation including harmonicsFigure 6 shows the comparison of the EMT results andthe dynamic phasor representation of harmonics up tothe 17th harmonic. If more harmonics are added tothe dynamic phasor model, it will produce much moreaccurate results that will eventually converge moreclosely at the EMT traces. However, even with the

Table 2. Controller parameters.

Controllers kp ki

PLL 50 900

Real power controller 0.04 0.25

Terminal voltage controller 0.005 50

Figure 5. Comparison of steady-state results(fundamental only).

Figure 6. Comparison of steady-state results withinclusion of harmonics.

harmonics up to 17th, the waveforms show a very closesimilarity with the EMT simulation results.

5.3. Validation with a step change in terminalvoltage reference

To test the model accuracy further, a step decrease isapplied to the terminal voltage reference from 1 p.u.to 0.75 p.u. at 2 s. The smooth terminal voltagetransition obtained through the voltage controller indynamic phasor model is shown in Figure 7.

Figure 8 shows the comparison of fundamentalresults from the EMT model and the dynamic phasormodel for the step change in voltage.

It can be seen that results from both modelsclosely match at transient (around 2 s) and in steadystate. The slow delay observed in the current waveformcan be the e�ect of the voltage controller.

Figure 9 shows the comparison of waveforms withthe same step change in the voltage reference withinclusion of the harmonics. The EMT model hasall the harmonics and the dynamic phasor model hasharmonics up to the 17th. For the terminal voltagechange from 1 p.u to 0.75 p.u., the levels in output

Figure 7. Response to a step change in the terminalvoltage reference.


Figure 8. Comparison of the responses to a step changein the voltage reference (fundamental only).

Figure 9. Comparison of the responses to a step changein the voltage reference (including harmonics).

voltage changed from 6 to 4. The line current increasedto maintain constant power ow at 500 MW.

5.4. Model validation with a step change inthe real power reference

A step change in the real power order from 300 MWto 500 MW is applied to both models at 2 s, andthe RMS power variation obtained from the dynamicphasor model is shown in Figure 10.

Figure 11 shows the variation in the phase à'voltage and current (fundamental component only)waveforms in response to the power order change. Itis observed that the dynamic phasor model closelyfollows the step change compared to the EMT model.Note that similar to the previous case, the funda-mental component of the EMT solution is obtained

Figure 10. Response to a step change in terminal realpower.

Figure 11. Comparison of the responses to a step changein the power reference (fundamental only).

using a standard FFT block in PSCAD/EMTDC fromthe original response waveform including all harmon-ics.

Figure 12 shows the phase à' voltage and currentwith inclusion of harmonics for a step change in thepower order. It can be observed that the power orderchange does not a�ect the ac voltage in a noticeableway and the change is only observed in the linecurrent.

5.5. Analysis of the simulation errorIt is straightforward to note that a dynamic phasorbased model can be made more accurate if a largernumber of harmonics are included in the model. Thissection shows how doing so a�ects the accuracy ofa simulated waveform. As an example, the dynamicphasor-based representation of the terminal voltagewaveform with an increasing number of odd harmonicis compared against the actual staircase waveform.The RMS error between the dynamic phasor voltagewaveforms and the actual staircase waveform is plottedin Figure 13 as a function of the number of harmoniccomponents included.

It is observed from Figure 13 that inclusion of


Figure 12. Comparison of the responses to a step changein the power reference (including harmonics).

Figure 13. Model's error as a function of harmonicsincluded.

every odd harmonic (up to the 17th as shown) leads toa reduction in RMS error. In general, the observationmade in Figure 13 holds true for models developedusing dynamic phasors. The model accuracy improveswhen a larger number of constituent harmonics areincluded; however, it must also be noted that mostpower systems dampen high-frequency harmonics dueto their inductive nature. Therefore, as the harmonicfrequencies increase, their expected contribution isexpected to decrease. As such, inclusion of high-frequency harmonics may not have marked contribu-tions to the accuracy of the results beyond a certainpoint. This is also evident in Figure 13, as the RMSerror appears to reach a at level beyond the 13thharmonic.

Note that in practical MMC con�gurations, amuch larger number of submodules are normally used.Therefore, the harmonic contents of the actual voltagewaveform will be much reduced compared to the caseconsidered in this paper. As such, it is expected that forsimulations of practical MMC systems, a much smallernumber of harmonics need to be included to achievehigh levels of accuracy.

6. Conclusion

An extended-frequency dynamic phasor model of anMMC was presented and validated against a detailedEMT model in PSCAD/EMTDC. The fundamental-frequency results of the dynamic phasor model closelymatched the fundamental frequency response of EMTsimulations. By adding more harmonics to the dynamicphasor model, further accuracy was achieved. It wasalso shown that a dynamic phasor model, even withconsideration of an extended number of harmonics, isexecutable with a much larger time step than requiredin an EMT simulation, thereby o�ering signi�cantcomputational advantages.

Acknowledgements

This research was funded by the Natural Sciences andEngineering Research Council (NSERC) of Canada,MITACS Accelerate, and Manitoba Hydro.

Nomenclature

ij Arm currentsvav�j Submodule average voltagesnj Switching functionsidc DC currentsvcj Sum of submodule voltages in the armVdc; r DC link voltage and DC source internal

resistance! = 2�f Angular frequencyN Number of submodules per armCm Submodule capacitanceLtf Converter transformer inductanceR;L Arm resistance and arm inductanceRs; Ls AC network Th�evenin resistance and

inductanceVsj AC network source voltage

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Biographies

Shailajah Rajesvaran graduated with a BSc (Hons.)from the University of Peradeniya in Peradeniya, SriLanka, in 2012. Since Jan. 2014, she has been pursuingan MSc degree in the Department of Electrical andComputer Engineering at the University of Manitoba.Prior to joining the University of Manitoba, she wasan Electrical Design Engineer with Hairu EngineeringConsultancy Company (Pvt.) Ltd in Sri Lanka.

Ms. Rajesvaran's research interests are in powerelectronic applications in power systems, and modellingand simulation of advanced power converters.

Shaahin Filizadeh received his BSc and MSc degreesin Electrical Engineering from Sharif University ofTechnology in Tehran, Iran, in 1996 and 1998, respec-tively. He obtained his PhD from the University ofManitoba, Canada, in 2004. He is currently a profes-sor with the Department of Electrical and ComputerEngineering at the University of Manitoba, where he isalso Associate Head (graduate programs).

His areas of interest include electromagnetic tran-sient simulation, nonlinear optimization, and powerelectronic applications in power systems and vehiclepropulsion. Dr. Filizadeh is a senior member of theIEEE and a registered professional engineer in theProvince of Manitoba.

Documents

Wide-band modeling of modular multilevel converters …scientiairanica.sharif.edu/article_3997_ed14c6971db50f6a550c5c... · This paper presents a model for a Modular Multilevel Converter