Harmonic Modeling of a Diode-Clamped Multilevel Voltage ...downloads.hindawi.com/journals/mpe/2018/3250761.pdfResearchArticle Harmonic Modeling of a Diode-Clamped Multilevel Voltage

Research ArticleHarmonic Modeling of a Diode-Clamped Multilevel VoltageSource Converter for Predicting Uncharacteristic Harmonics

Tian Hao Huang and Kuo Lung Lian

Department of Electrical Engineering National Taiwan University of Science and Technology Taiwan

Correspondence should be addressed to Kuo Lung Lian ryankuoliangmailcom

Received 15 October 2018 Accepted 5 November 2018 Published 5 December 2018

Academic Editor Vincenzo Vespri

Copyright copy 2018 Tian Hao Huang and Kuo Lung Lian This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

In response to the growing demand for medium- and high-power trends multilevel voltage source converters (VSCs) have beenattracting growing considerations One of the widely used VSCs are the diode-clamped multilevel VSC (DCM-VSC) As theseconverters proliferate their harmonic impact may become significant Nevertheless a harmonic model for the DCM-VSC iscurrently lacking in the literature In this paper the ABCD matrix mapping the input harmonics to the output harmonics ofDCM-VSC is derived As the matrix is formulated in the time-domain the output harmonics are exact and do not suffer fromharmonic truncation errors As the paper will demonstrate the derived ABCD matrix can be easily applied to a microgrid systemand users can easily predict all the uncharacteristic harmonics when a microgrid is subjected to various conditions of imbalanceIn addition to all the results being validated with those of PSCADEMTDC the computation time of the proposed method is incontrast much shorter

1 Introduction

Voltage source converters (VSCs) have been widely used inthe modern power system [1] They are the main intercon-nection devices for distributed generators (DGs) and energystorage systems in amicrogrid system [2] At the transmissionand distribution levels they are also the main building blocksof the flexible ac transmission system devices and custompower controllers respectively While many benefits may berealized from proliferation of VSCs in the power grid thesecome at the expense of generating harmonic distortion Itis therefore essential to obtain accurate harmonic modelsfor VSCs to predict their generated current and voltageharmonic under various conditions and to understand howthese harmonics will interact with the rest of the powersystem

Electromagnetic transient programs such as PSCADEMTDC can provide accurate harmonic analysis resultsHowever its disadvantage is that an initialization transientmust die out before steady-state is reached when Fourieranalysis can be performed [3]The simulation time step mustbe sufficiently small for the VSCs to capture the switching

dynamics and this leads to excessively long simulation timesAlternatively one can rely on harmonic power flow to obtainthe harmonic solutions However a frequency map (FM)between convertersrsquo input and output harmonics usuallyrequires deriving Hence the main objective of this paper isto derive the FM of a multilevel VSC

Depending on the number of the voltage levels VSCs canbe classified into two-level and multilevel (more than two-level) converters The multilevel VSCs have the advantagesof reduced voltage stress on power semiconductor switcheslower distortions of the input current and lower dvdt ascompared with their two-level counterparts Consequentlymultilevel VSCs have become attractive even for the low-and medium-power range such as in the distributed energyconversion application [4] Among many various types ofmultilevel VSCs diode-clamped multilevel VSCs (DCM-VSCs) are the most widely used for utility and high-powerindustry applications [5] ever since their first appearance in1981 [6] As these converters become more dispersed theirharmonic impact on the system could be significant Therehave been numerous publications [7ndash9] addressing harmonicmodeling of a two-level VSC Nevertheless a generalized

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 3250761 16 pageshttpsdoiorg10115520183250761

2 Mathematical Problems in Engineering

+

AC side

-

DC side

Figure 1 Equivalent external network of DCM-VSC

harmonic model of the DCM-VSC has not been developedyet Thus the focus of this paper is on the DCM-VSCs

As VSCs may be subjected to various power qualitydisturbance the derived harmonic models of DCM-VSCsshould be general and able to account for various conditionssuch as unbalanced grid voltage [10 11] and unbalanced dcvoltages [12 13] The imbalance in the dc voltage comes from(i) the DGs with different voltage levels connected to thedc link of the converters [14 15] and (ii) the inherent dcvoltage imbalance of DCM-VSCs [16] It will be demonstratedin the paper that the two cases will lead to drasticallydifferent harmonic profiles The first dc voltage imbalancemay result in low-order even current harmonics at the ac side[14] which in turn may cause undesired dc component andunequal positive and negative peak values [17] However inthe second case no even harmonics are generated but someuncharacteristic odd harmonics are generated Moreover aswill be demonstrated in the paper the harmonic profileof converters will also alter significantly in a microgridenvironment or in the presence of neighboring convertersTherefore it is essential to have an accurate harmonic modelcapable of predicting uncharacteristic harmonics under vari-ous conditions

The current models for DCM-VSCs are inadequate forharmonic modeling Khajehoddin et al [4] proposed acurrent flow model for the DCM-VSCs They formulatedthe capacitor currents as well as the input dc current asa function of the switching functions and output currentsNevertheless they ignored the high-order harmonics of theswitching functions whichmay result in inaccurate side bandharmonic arising with converter modulation Generalizedaveraged models of DCM-VSCs have been proposed in [18]However all the high-order harmonics are effectively elimi-nated in averaging operations thereby producing inaccurateharmonic results Jayaram et al [19] derived a five-levelDCM-VSC in abc frame for which the switching functions areobtained using curve fitting However such amodel becomesvery complex when the number of level is high He et al[20] have modeled the DCM-VSC as a Norton equivalentcircuit However they have assumed that the capacitors arelarge enough so that the dc link voltage can be assumedto be constant Such models are only accurate in predictingcharacteristic harmonics and are not able to account for theuncharacteristic harmonics under unbalanced conditions

Based on the work of Larsen et al [21] the FM of an AC-DC converter is more conveniently modeled by means of anABCD matrix A DCM-VSC can be visualized in Figure 1The harmonic injections caused by external networks canbe modeled as ac voltage harmonics 119881119886119888 (in this papercapital letters represent harmonic domain variables whereas

lower-case letters represent time-domain variables) and dccurrent harmonics 119868119889119888 at the point of common couplingThe converter in turn produces ac current harmonics and dcvoltage harmonics 119881119889119888 via the ABCD matrix as stated in (1)

[119868119886119888119881119889119888] = [119860119861119862119863][119881119886119888119868119889119888] (1)

Therefore the main goal of this paper is to derive anaccurate ABCDmatrix for a DCM-VSC under various condi-tions The proposed method essentially consists of two stepsthe first is to derive the state-space equations of the DCM-VSC and the second is to treat the harmonics of interestas state variables and solve the entire augmented state-spaceequations piecewise with the periodicity constraint imposedThen the ABCDmatrix is obtained by mapping the resultingsubmatrix of the state transition matrix to the harmonicdomain by a transformation matrix The advantage of theproposed method is that

(i) calculation times are much shorter than the brute-force time-domain method as it directly bypasses thetransient state

(ii) different from the harmonic domain methods thereis no need to know how many harmonics need to beincluded to obtain the harmonics of interest

(iii) the predicted harmonics are exact and do not sufferfrom harmonic truncation errors [7]

(iv) it can be easily expanded to study a microgrid systemcontaining multiple numbers of DCM-VSCs

To validate the proposed model all the predicted harmonicsare verified with those obtained from PSCADEMTDC

The rest of the paper is organized as follows Section 2describes the state-space model of a DCM-VSC Section 3shows how the harmonic ABCD coupling matrix is derivedIn Section 4 various scenarios of imbalance for a DCM-VSC and DCM-based microgrid are studied and the pre-dicted results are verified with those of PSCADEMTDCdemonstrating the validity of the proposed method Finallya conclusion is given in Section 5

2 Generalized State-Space Representationof a DCM-VSC

Different from other conventional multilevel converters suchas flying capacitors or cascaded multilevel converters DCM-VSCs have forbidden states [22] in which the output voltageis different if the current direction is different under the sameswitching function state Figure 2 shows the three-level VSCwhere the upper switches are assumed to be complementarywith the lower switches in each leg

To illustrate the forbidden states Figure 3 is referred toand without loss of generality only phase A leg of the VSCis shown When 11990411 turns on and 11990412 turns off and if thecurrent flows as shown in Figure 3(a) V119886119900 = minusV1198882 Howeverwhen the current flows in the reverse direction the circuitbecomes Figure 3(b) and V119886119900 = V1198881 Consequently 11990411 = 1 and

Mathematical Problems in Engineering 3

Table 1 Legitimate switching states of phase A

11990411 11990412 V1198861199000 0 V11986210 1 01 1 minusV1198622

Table 2 Switching status of 1199041198861199041198861 1199041198862 1199041198863119904119886 997888rarr 1198901198861 1 0 0

119904119886 997888rarr 1198901198862 0 1 0119904119886 997888rarr 1198901198863 0 0 1

11990412 = 0 constitute a forbidden state because the output voltageis differentwhen the current direction is different In contrastwhen 11990411 = 0 11990412 = 1 the circuit remains that shown inFigure 4 regardless of the direction of current flow Such as astate is a legitimate state Table 1 lists all the possible legitimateswitching states for phase A

To have a better representation of all these states Figure 2is replaced with Figure 5 where all the bidirectional switchesare replaced with single-pole triple-throw switches [23] 119904119886in Figure 5 can be represented by three switching states 11990411988611199041198862 and 1199041198863 When 119904119886 is connected to 1198901198861 in Figure 5 1199041198861=11199041198862=0 and 1199041198863=0 Table 2 summarizes the switching status for119904119886 Similar rules apply to 119904119887 and 119904119888

Based on Table 2 one can derive a generalized m-level(m is assumed to be odd) three-phase four-wire model (seeFigure 6) of DCM-VSC with parameters such as inductanceand resistance for each phase that may not necessarily bethe same In addition without loss of generality the stimuliare assumed to take the form of 119906119899119890119895(119899120596119905+120579119899) where 119899 is thenumber of harmonics and 120579119899 the phase angle at the n-thharmonic

119889119889119905 [

119909119906] = [

119865 1198660 Ω][

119909119906] (2)

where

119909 = [119894119886 119894119887 119894119888 V1198621 1198881198622 sdot sdot sdot V119862(119898minus1)]119879119906 = [V119904119886 V119904119887 V119904119888 119894119871]119879

(3)

Submatrices 119865 119866 and Ω are stated in the AppendixMoreover the derivation of a simple three-level DCM-VSCcan also be found in [24]

3 Derivation of the ABCD Matrix forHarmonic Evaluation

Equation (2) is a time-domain state-space representation Tostep into the harmonic domain one can treat the harmonicsof interest as harmonic states and augment them to (2) byrecasting a Fourier integral into a differential equation [7] asstated in (4)

119889119909ℎ119889119905 = 119895119899120596119909ℎ + 1119879119890minus119895119899120596119879119909 (119905) (4)

where 119909ℎ is the harmonic state 119879 is the period and 119909ℎ(0) =0If one wants to know the harmonics for the three-phase

current and all the 119898 minus 1 capacitor voltages the overallequation is as stated in (5)

119889119889119905 [[[

119909119906119909ℎ]]]= [[

[119909119906119909ℎ]]]

(5)

where

= [[[119865 119866 00 Ω 0119863 0 119869

]]]

119863 =[[[[[[[[

119863111986321198632+119898

]]]]]]]]

1198631 =

[[[[[[[[[[[[[[[[[[[[[[[[[

119890119895119899120596119879119879 0 sdot sdot sdot 0

119890119895(119899minus1)120596119879119879 0 sdot sdot sdot 0 1119879 0 sdot sdot sdot 0

119890minus119895(119899minus1)120596119879119879 0 sdot sdot sdot 0119890minus119895119899120596119879119879 0 sdot sdot sdot 0

]]]]]]]]]]]]]]]]]]]]]]]]](2lowast119899+1)times(2+119898)

1198632+119898 =

[[[[[[[[[[[[[[[[[[[[[[[[[

0 sdot sdot sdot 0 119890119895119899120596119879119879

0 sdot sdot sdot 0 119890119895(119899minus1)120596119879119879

0 sdot sdot sdot 0 1119879 0 sdot sdot sdot 0 119890minus119895(119899minus1)1205961198791198790 sdot sdot sdot 0 119890minus119895119899120596119879

119879

]]]]]]]]]]]]]]]]]]]]]]]]](2lowast119899+1)times(2+119898)

119869 = diag (1198691 1198692 sdot sdot sdot 1198692+119898)1198691 = 1198692 = sdot sdot sdot = 1198692+119898 = diag (minus119895119899120596minus 119895 (119899 minus 1) 120596 sdot sdot sdot 0 sdot sdot sdot 119895 (119899 minus 1) 120596 119895119899120596)

(6)


+

-

+

-

sa

sb

sc

ao

bo

co

a

b

c

C

C

11 21 31

12 22 32

31

32

21

22

11

12

Figure 2 Three-phase three-level DCM-VSC

+

-

+

-

+

-

+

-

ao ao

C

C

C

C

Figure 3 DCM-VSC forbidden states (a) Left current out of converter (b) Right current into the converter

+

-

+

-

ao

C

C

(a)

+

-

+

-

ao

C

C

(b)

Figure 4 DCM-VSC legitimate states (a) current out of converter (b) current into the converter

+

-

+

-

C

C

sa

sb

sc

a

b

c

aa

bb

cc

a

b

c

Figure 5 The equivalent single-pole triple-throw model for a three-level DCM-VSC


Figure 6 The equivalent single-pole m-throw model for a three-phase four-wire m-level DCM-VSC

Evaluating (5) over 119879 one gets (7)[[[

119909 (119879)119906 (119879)119909ℎ (119879)

]]]= Φ[[

[

119909 (0)119906 (0)119909ℎ (0)

]]]

(7)

where

Φ = 119890119899119904119908 (119879minus119905119899119904119908 ) sdot sdot sdot 1198902(1199052minus1199051)1198901(1199051minus0) (8)

119899119904119908 is the number of switching timesAs proved in [7] Φ in general takes the following form

Φ = [[[

120572 120573 00 120574 0120589 120578 120585

]]]

(9)

Imposing the periodicity constraints

119909 (119905 + 119879) = 119909 (119905) (10)

where 119905 is generally set to 0 and the values of 119909ℎ(119879) areessentially the harmonics of interests [7] In other words theoutput harmonics such asAC current harmonics 119868119886119887119888 andDCvoltage harmonics119881119889119888 are equal to119909ℎ(119879) and can be obtainedanalytically as stated in (11)

[119868119886119887119888119881119889119888] = [119872][V119904119886119887119888 (0)119894119889119888 (0) ] (11)

where

[119872] = 120589 [(119868 minus 120572)minus1 120573] + 120578 (12)

Thematrix119872 in (11) relates the steady-state initial conditionsof the ac voltages and dc currents (ie V119904119886119887119888(0) and 119894119889119888(0)) to

the ac current harmonics 119868119886119888 and dc voltage harmonics 119881119889119888Because the input is the time-domain variable one needs tofind the relation between input steady-state initial conditions(V119904119886119887119888(0) and 119894119889119888(0))) and input harmonics (ie119881119904119886119887119888 and 119868119889119888)As the stimuli are in the form of 119906119899119890119895(119899120596119905+120579119899)

[119881119904119886119887119888119868119886119887119888 ] = [V119904119886119887119888 (0)119894119889119888 (0) ] (13)

Hence

[119868119886119887119888119881119889119888] = [119860119861119862119863]119886119887119888 [119881119904119886119887119888119868119889119888 ] (14)

where [119860119861119862119863]119886119887119888 = [119872] Alternatively one can also formu-late (14) in terms of the sequential components (zero positiveand negative sequence components are designated as 0 1 and2 respectively) Since

119868119886119887119888 = 119875 sdot 119868012 (15)

119881119904119886119887119888 = 119875 sdot 119881119904012 (16)

where

119875 = [[[[

1 1 11 1198862 1198861 119886 1198862

]]]]

119886 = 1ang120∘(17)

Substituting (15) into (16) and converting harmonic phasecurrents 119868119886119887119888 into harmonic sequential currents 119868012 oneobtains

[119875 00 119868] [

119868012119881119889119888] = [119860119861119862119863]119886119887119888 [

119875 00 119868] [

119881119886012119868119889119888 ] (18)


[119868012119881119889119888] = [119860119861119862119863]012 [119881012119868119889119888 ] (19)

where [119860119861119862119863]012 = [ 119875 00 119868 ]minus1 [119860119861119862119863]119886119887119888 [ 119875 0

0 119868 ] and 119868 is theidentity matrix

4 Case Studies

To validate the uncharacteristic harmonics predicted by theproposed method five case studies are investigated In thefirst case the characteristic harmonics produced by a three-level DCM-VSC under a balanced ac grid are first evaluatedThen the uncharacteristic harmonics due to the negativesequence ac voltage injection are studied The second case isto investigate the uncharacteristic harmonics of a three-levelDCM-VSC with unequal dc voltage levels caused by externalDGs The third case is to show that the proposed model canreproduce the inherent unbalanced dc voltage of a five-levelDCM-VSC In the fourth case the proposed model is shownto be able to predict the uncharacteristic harmonics arisingfrom an imbalance in system parameters The converterparameters are extracted from [25] and listed in Table 3Note that for Cases 1 to 3 119877119873 and 119871119873 are assumed to beinfinity Thus a three-phase three-wire DCM-VSC system isinvestigated In contrast 119877119873 and 119871119873 for Case 4 are finite anda three-phase four-wire DCM-VSC system is investigatedFinally in case 5 a microgrid consisting of the converterspresented in Cases 2 3 and 4 is modeled using the proposedmethod

The modulation strategy for multilevel converters can beclassified into three types specifically carrier-based pulsewidth modulation (PWM) space vector modulation andselective harmonic elimination Among them the carrier-based modulation strategy is widely used in industry [26]The carrier-based PWM can be further classified into phase-shifted PWM(PS-PWM) and level-shifted PWM(LS-PWM)PS-PWM is rarely used for DCM-VSC as it has no modularstructure and the carrier cannot be associated with a partic-ular cell or treated independently [27] The LS-PWM on theother hand can be easily adapted to a DCM There are threetypes of LS-PWMs [28]mdash(i) in-phase disposition (IPD)where all carrier are in phase (ii) alternative phase oppositedisposition (APOD) where all carriers are alternatively inopposite disposition and (iii) phase opposite disposition(POD) where all the positive carriers are in phase with eachother and in opposite phase of the negative carriers Forthe case studies the IPD-LS-PWM is used because of itsbetter power quality [28]The switching times are obtained bysolving the transcendental equations of the modulating andcarrier signals

41 Case 1 Balanced and Unbalanced AC Grid (NegativeSequence Voltage Injection) The three-level DCM-VSC isfirst connected to a balanced AC grid Figure 7 shows themaximum difference (119868119889) of the ac currents obtained bythe proposed method and those by PSCADEMTDC undervarious time steps As the figure shows the simulation step119905119904119905119890119901 needs to reduce to 01120583119904 in order for PSCADEMTDCmodel to reach the same level of accuracy as the proposed

50 20 011510 005 001PSCADEMTDC solution time step (uS)

01

1

10

40

）＞

(A)

Figure 7 |119868119889| versus solution time step

120

100

80

60

40

20

0

minus20

minus40

minus60

Iabc

(Am

pere

)

0 0002 0004 0006 0008 001 0012 0014 0016 0018

time (seconds)

Proposed Method(phase A)PSCADEMTDC (phase A)Proposed Method(phase B)PSCADEMTDC (phase B)Proposed Method(phase C)PSCADEMTDC (phase C)

Figure 8 Balanced steady-state AC current waveforms

method Figure 8 shows the resulting balanced three-phasecurrents for both PSCADEMTDC (119905119904119905119890119901 = 01120583119904) and theproposed method Nevertheless the calculation time forPSCADEMTDC with time step of 01120583119904 is about 85 timeslonger than the proposedmethod Figure 9 shows the currentharmonic spectra in terms of its positive and negativesequence components It clearly shows that under balancedcondition the DCM-VSC only produces characteristic har-monics

It is to be noted that input stimuli in (2) can be easilyextended to other stimuli such as the cosine function bymeans of Eulerrsquos formula which is stated in (20)

cos (120596119905 + 120579) = 119890119895(120596119905+120579) + 119890minus119895(120596119905+120579)2 (20)

The following as an example are the step-by-step pro-cedures for calculating the positive sequence of the funda-mental current and dc voltage harmonic under the balancedcondition

(1) The input stimuli for phases A B and C are (V1199042)119890minus119895(120596119905+120579) (V1199042)119890119895(120596119905+120579) (V1199042)119890minus119895(120596119905+120579minus21205873) (V1199042)119890119895(120596119905+120579minus21205873) and (V1199042)119890minus119895(120596119905+120579+21205873) (V1199042)119890119895(120596119905+120579+21205873) Note that V119904 is the magnitude ofthe source voltage


Table 3 The parameter list of the three-level DCM-VSC

Parameter Value Parameter Value119877119886119877119887119877119888 01Ω firing angle 120575 0∘119871119886 119871119887119871119888 0004119867 modulation index119898119886 0833119891 60 Hz modulation factor119898119891 1511986211198622 1000 120583119865 source voltage 380 119881

0 5 10 15 20 25 30 35 40Harmonics Number

0

20

40

60

Proposed MethodPSCADEMTDC

） 1(A

)

(a)


0

5

10


） 2(A

)

(b)

Figure 9 (a) The harmonic spectra of ac currents of Figure 8 in terms of positive sequence components (b) the harmonic spectra of accurrents of Figure 8 in terms of negative sequence components

(2) Substitute the parameter values listed in Table 3 intothe form similar to (5)

(3) Evaluate (7) to obtain Φ(4) Extract the submatrices 119860 119861 119862 and 119863 which are as

follows

119860= [[[16490 + 00489119895 16709 minus 02624119895 minus07822 minus 14525119895 minus10626 + 15416119895 minus08668 + 14036119895 minus06083 minus 12792119895minus07822 minus 14525119895 minus06083 minus 12792119895 minus08668 + 14036119895 16709 + 02624119895 16490 + 00489119895 minus1626 + 15416119895minus08668 + 14036119895 minus10626 + 15416119895 16490 + 00489119895 minus06083 minus 12792119895 minus07822 minus 14525119895 16709 minus 02624119895

]]]

(21)

119861 = [[[01887 + 00148119895minus01072 + 01560119895minus00815 minus 01708119895

]]]

(22)

119862 = [58125 minus 03137119895 58125 + 03137119895 minus3178 minus 48769119895 minus3178 + 48769119895 minus26345 + 51907119895 minus26345 minus 5190711989558125 minus 03137119895 58125 + 03137119895 minus3178 minus 48769119895 minus3178 + 48769119895 minus26345 + 51907119895 minus26345 minus 51907119895] (23)

119863 = [00] (24)

(5) The ac current and dc voltage harmonics can beobtained by evaluating (14) Note that the ac currentharmonics need to be multiplied by two to obtain thecorrect phasor forms

[[[[[[[[[[

11986811198861198681119887119868111988811988101198881

11988101198882

]]]]]]]]]]

=[[[[[[[[[

2 0 0 0 00 2 0 0 00 0 2 0 00 0 0 1 00 0 0 0 1

]]]]]]]]]

[[[[[[[[[

2276 minus 034j1109 + 1988j1167 minus 1954j

2920329203

]]]]]]]]]

=[[[[[[[[[[

minus4553 minus 068j2218 + 3977j2335 minus 3909j

2920329203

]]]]]]]]]]

(25)

where 1198681119886 1198681119887 and 1198681119888 denote the fundamental currentsof phases A B and C respectively and1198810

1198881 and11988101198882 are

the two dc voltages across 1198621 and 1198622



0

20

40

60


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)

Figure 10 Unbalanced ac grid case (a) the harmonic spectra of ac currents of in terms of positive sequence components (b) the harmonicspectra of ac currents of in terms of negative sequence components


0

20

40

60


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)

Figure 11 Case 2 (a) the harmonic spectra of ac currents in terms of positive sequence components (b) the harmonic spectra of ac currentsin terms of negative sequence components

(6) Evaluate (19) and the fundamental positive (11986811 )negative (11986812) and zero sequence (11986810 ) currents and thetwo dc voltages are stated as (26)

[[[[[[[[[[

11986810119868111198681211988101198881

11988101198882

]]]]]]]]]]

=[[[[[[[[[

04553 minus 068j

02920329203

]]]]]]]]]

(26)

Next a 5 negative sequence of the fundamental voltageis added to the AC grid and the voltage takes the followingform

119881119904119886 = 119881119904119890119895120596119905 + 005119881119904119890119895120596119905119881119904119887 = 119881119904119890119895120596119905minus120∘ + 005119881119904119890119895120596119905+120∘119881119904119888 = 119881119904119890119895120596119905+120∘ + 005119881119904119890119895120596119905minus120∘

(27)

Figure 10 shows the harmonic spectra of this unbalancedcase in terms of positive and negative sequence componentsIn this instance the uncharacteristic harmonics such as thenegative sequence fundamental and third harmonics areaccurately predicted by the proposed method

42 Case 2 Unequal DC Voltage due to DGs with DifferentVoltage Levels In this case we try to simulate the casewhere the split dc link capacitors of the DCM-VSC can becontrolled independently A commonly seen situation is tohave a different PV panel connected to each capacitor witheach panel being controlled independently to achieve its ownpoint of maximum power [14] Such a case leads to unequaldc voltage level for different capacitors with respect to theneutral point N To simulate such a case two DC sourceswith V1198631198621 = 350 V connected in parallel with 1198621 and V1198631198621 =230 V connected in parallel with 1198622 are connected to the dclinkThe resulting current spectra shown in Figure 11 featureundesired negative sequence even-order current harmonicssuch as the second and 14-th harmonics The values of theseharmonics are accurately predicted by the proposed methodwhen compared with those of PSCADEMTDC

43 Case 3 Inherent Unbalanced Capacitor Voltage for A5-Level DCM-VSC Different from the previous case thiscase study investigates a five-level DCM-VSCThe parametersettings are the same as those in Table 3 As stated in [1629] when the level number is greater than 3 an inherentdc voltage imbalance becomes apparent In addition asdemonstrated in [22] if the condition

119898119886 cos (120575) gt 055 (28)


0 0002 0004 0006 0008 001 0012 0014 0016 0018time (seconds)

120

140

160

180

200

220

240Vc

(vol

tage

)

Proposed Method (Vc1)Proposed Method (Vc2)


(a)

0 0002 0004 0006 0008 001 0012 0014 0016 0018time (seconds)

120

140

160

180

200

220

240

Vc (v

olta

ge)

PSCADEMTDC (Vc1)PSCADEMTDC (Vc2)


(b)

Figure 12 Case 3 (a) the capacitor voltage waveforms obtained by the proposed method (b) the capacitor voltage waveforms obtained bythe PSCADEMTDC

0

0

100

1

Har

mon

ic M

agni

tude

(V)

2

200

3 4

Harmonics Number

5 6 7 8 9


６1

６2

６3

６4

Figure 13 Harmonic spectra of each capacitor in the five-levelDCM-VSC

is satisfied no balancing strategy exists to balance thecapacitor voltages Hence for the chosen119898119886 and 120575 in Table 3voltage imbalance among capacitors is expected

Figures 12(a) and 12(b) show the DC voltage waveformsfor each capacitor (in this paper the capacitor labels fol-low those indicated in Figure 6) obtained by the proposedmethod and PSCADEMTDC Their corresponding har-monic spectra are in great agreement as shown in Figure 13Figure 14 shows the current waveforms of the five-levelDCM-VSC Figure 15 shows the corresponding current spec-tra Compared with Figure 9 one can see that the positivesequence 7-th 13-th 19-th and 25-th harmonics are muchgreater for the 5-level DCM-VSC Moreover for the negativesequence the value of the 5-th harmonic is about 5 timeslarger than that of 11-th harmonic for the 5-level casewhereas the 11-th harmonic for the 3-level converter is 3 times

30

25

20

15

10

5

0

minus5

minus10

minus15

Iabc

(Am

pere

)

0 0002 0004 0006 0008 001 0012 0014 0016 0018

Proposed Method(phase A)

PSCADEMTDC (phase A)

Proposed Method(phase B)

PSCADEMTDC (phase B)

Proposed Method(phase C)

PSCADEMTDC (phase C)

time (seconds)

Figure 14 Steady-state AC current waveforms of five-level DCM-VSC

higher than that of the 5-th harmonic as seen in Figure 9Nevertheless different from Case 2 there is no even currentharmonic generated at the ac side The reason is that withrespect to N there exists a symmetry between the upper andlower capacitor voltages In other words |V1198881| = |V1198884| and|V1198882| = |V1198883| Consequently such a symmetry with respect toNwill not generally lead to any even current harmonics at theac side

44 Case 4 Unbalanced Parameters of a Three-Phase Four-Wire DCM-VSC In this case we investigate a three-phasefour-wire DCM-VSC system 119877119873 and 119871119873 are set equal to119877119887 and 119871119887 respectively The inductance of phase A 119871119886 isincreased by 20 whereas other parameters remain fixedFigure 16 shows the portion of [119860119861119862119863]012 which relates thepositive sequence input voltage harmonics 1198811199041 to the zero



0

2

4

6

8


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)

Figure 15 Case 3 (a) the current harmonic spectra of five-level DCM-VSC in terms of positive sequence components (b) the currentharmonic spectra of five-level DCM-VSC in terms of negative sequence components

Output harmonic Input harmonic

030

002

20 30

abs(

Out

putI

nput

)

10

004

200 10

006

minus10 minus10 0minus20 minus20minus30 minus30

Number (６1)number (）0)

Figure 16 The ABCDmatrix relating 1198811199041 to 1198680

sequence output current harmonics 1198680 Figure 17 shows theac current harmonic spectra in terms of the positive negativeand zero sequence components The figure clearly shows thenegative sequence second harmonic due to the unbalancedinductance values In addition it also clearly indicates thatthe derived ABCD matrix can accurately predict the zerosequence current harmonic at various frequency and theirvalues are in strong agreement with those obtained byPSCADEMTDC

45 Case 5 A Multiconverter Unbalanced Microgrid SystemFigure 18 shows a microgrid system consisting of multipleVSCs which is similar to the Aichimicrogrid system in Japan[30] With the aid of the proposed ABCD matrix the overallmicrogrid formulation can be derived

From Figure 18 one can see the following

119881119901119888119888 = 119881119904 minus 119885119904119868119904 (29)

119868119904 = 119868V1199041198881 + 119868V1199041198882 + 119868V1199041198883 + sdot sdot sdot + 119868V119904119888119899 (30)

Also each converter can be expressed as

[119868V119904119888119896119881119889119888119896] = [119860119896 119861119896119862119896 119863119896

][119881119901119888119888119868119889119888119896] (31)

where 119896 = 1 2 3 sdot sdot sdot 119899

Combining (29) (30) and (31) one can get (32)

[[[[[[[[[[

11986811990411988111988911988811198811198891198882119881119889119888119899

]]]]]]]]]]

= [119872]minus1 [119873][[[[[[[[[[

11988111990411986811988911988811198681198891198882119868119889119888119899

]]]]]]]]]]

(32)

where

119872 =[[[[[[[[[[

119868 + (1198601 + 1198602 + sdot sdot sdot + 119860119899) 119885119904 0 0 sdot sdot sdot 01198621119885119904 119868 0 sdot sdot sdot 01198622119885119904 0 119868 sdot sdot sdot 0 d 0

119862119899119885119904 0 0 sdot sdot sdot 119868

]]]]]]]]]]

119873 =[[[[[[[[[[

1198601 + 1198602 + sdot sdot sdot + 119860119899 1198611 1198612 sdot sdot sdot 1198611198991198621 1198631 0 sdot sdot sdot 01198622 0 1198632 sdot sdot sdot 0 d 0119862119899 0 0 sdot sdot sdot 119863119899

]]]]]]]]]]

(33)



0

20

40

60


） 1(A

)

(a)



0

5

10

） 2(A

)

(b)



0

05

1

15

2） 0

(A)

(c)

Figure 17 Case 4 (a) the current harmonic spectra of in terms of positive sequence components (b) the current harmonic spectra of interms of negative sequence components (c) the current harmonic spectra of in terms of zero sequence components

1

2

1

2

1

2

VSC1

VSC2

VSCn

1

2

1

2

Figure 18 A multiconverter microgrid system

Figure 19 shows a multi-DCM-VSC microgrid systemAssume that the ac grid is unbalanced and takes the formof (27) VSC1 in Figure 19 is a three-level DCM-VSC havingunequal DC voltage due to DGs with different voltage levelas described in Case 2 VSC2 is a 5-level DCM-VSC withinherent unbalanced capacitor voltage as in Case 3 FinallyVSC3 is a three-level VSC with unbalanced parameters asdelineated in Case 4 Employing (32) one can obtain the accurrents and capacitor voltages for each converter Figures20 21 and 22 show the ac current spectra for VSC1 VSC2and VSC3 respectively To show that the presence of theneighboring converters can have a significant impact on the

harmonic spectra of the converter we resimulate Case 4 withthe grid voltage taking the form of (27) and its currentspectra are shown in Figure 23 Comparing Figure 22 withFigure 23 one can see that VSC3 exhibits a positive sequencenegative sequence and zero sequence of the fundamentalcurrent of 455 A 77 A and 17 A respectively when itis the only converter connected to the grid On the otherhand those components reduce to 37 A 53 A and 122A which correspond to 18 31 and 26 reductionrespectively under the microgrid environment Moreover ascan be observed from Figures 22(b) and 23(b) one can noticethat a negative sequence of second harmonic is injected from


+-

+-

+-

-+

+

+-

-

+-+-

1

2

1

2

1

2

3 33

VSC1

VSC2

VSC3

1

2

3

4

5

6

7

8

Figure 19 An unbalanced microgrid


0

10

20

30

40


）６３

11

(A)

(a)



0

2

4

6

8

）６３

12

(A)

(b)

Figure 20 Current spectra of VSC1 (a) positive sequence of 119868V1199041198881 (b) negative sequence of 119868V1199041198881


0

2

4

6

8


）６３

21

(A)

(a)



0

5

10

15

）６３

22

(A)

(b)


other converters under the microgrid environment whereasno even harmonics exist in Figure 23(b) All the resultspredicted by the proposed method are consistent with thoseof PSCADEMTDC

5 Conclusion

An AC-DC converter has been more conveniently charac-terized by the ABCD matrix for harmonic modeling and

analysis This paper presented a time-domain-based ABCDmatrix for a diode-clamped multilevel voltage source con-verter The derived model produced accurate predictionsof characteristic and all the uncharacteristic harmonics forvarious unbalanced conditions Also the method yieldedaccurate results and does not suffer from harmonic trunca-tion errors Moreover all the predicted results are in greatagreement with those obtained by PSCADEMTDC with thecalculation time being much shorter



0

10

20

30

40


）６３

31

(A)

(a)



0

2

4

6

）６３

32

(A)

(b)



0

05

1

15

2）６

３30

(A)

(c)

Figure 22 Current spectra of VSC3 (a) positive sequence of 119868V1199041198883 (b) negative sequence of 119868V1199041198883 (c) zero sequence of 119868V1199041198883



0

20

40

60

）６３

31

(A)

(a)



0

2

4

6

8

）６３

32

(A)

(b)



0

05

1

15

2

）６３

30

(A)

(c)

Figure 23 Only VSC3 is connected to the unbalanced ac grid (a) positive sequence of 119868V1199041198883 (b) negative sequence of 119868V1199041198883 (c) zero sequenceof 119868V1199041198883

Appendix

Expressions of Submatrices Ω 119865 119866Ω = diag (1198691 1198692 1198693 1198694)119865 = [1198651 11986521198653 0 ]


1198651 =[[[[[[[[

minus119877119886119871119886119886 + 119877119873119871119887119871119888119871 119905119877119887119871119888119871119873 minus 119877119873119871119887119871119888119871 119905

119877119888119871119887119871119873 minus 119877119873119871119887119871119888119871 119905119877119886119871119888119871119873 minus 119877119873119871119886119871119888119871 119905 minus119877119887119871119887119887 + 119877119873119871119886119871119888119871 119905119877119888119871119886119871119873 minus 119877119873119871119886119871119888119871 119905119877119886119871119887119871119873 minus 119877119873119871119886119871119887119871 119905

119877119887119871119886119871119873 minus 119877119873119871119886119871119887119871 119905 minus119877119888119871119888119888 + 119877119873119871119886119871119887119871 119905

]]]]]]]]

1198652 = [11986521 11986522]

11986521 =[[[[[[[[

minus119871119886119886Γ1198861 + 119871119888119871119873Γ1198871 + 119871119887119871119873Γ1198881119871 119905minus119871119886119886Γ1198862 + 119871119888119871119873Γ1198872 + 119871119887119871119873Γ1198882119871 119905 sdot sdot sdot minus119871119886119886Γ119886120593 + 119871119888119871119873Γ119887120593 + 119871119887119871119873Γ119888120593119871 119905119871119888119871119873Γ1198861 minus 119871119887119887Γ1198871 + 119871119886119871119873Γ1198881119871 119905119871119888119871119873Γ1198862 minus 119871119887119887Γ1198872 + 119871119886119871119873Γ1198882119871 119905 sdot sdot sdot 119871119888119871119873Γ119886120593 minus 119871119887119887Γ119887120593 + 119871119886119871119873Γ119888120593119871 119905119871119887119871119873Γ1198861 + 119871119886119871119873Γ1198871 minus 119871119888119888Γ1198881119871 119905119871119887119871119873Γ1198862 + 119871119886119871119873Γ1198872 minus 119871119888119888Γ1198882119871 119905 sdot sdot sdot 119871119886119871119873Γ119886120593 + 119871119886119871119873Γ119887120593 minus 119871119888119888Γ119888120593119871 119905

]]]]]]]]

(A1)

11986522

=[[[[[[[[

119871119886119886Λ 119886(120593+2) minus 119871119888119871119873Λ 119887(120593+2) minus 119871119887119871119873Λ 119888(120593+2)119871 119905119871119886119886Λ 119886(120593+3) minus 119871119888119871119873Λ 119887(120593+3) minus 119871119887119871119873Λ 119888(120593+3)119871 119905 sdot sdot sdot 119871119886119886Λ 119886119898 minus 119871119888119871119873Λ 119887119898 minus 119871119887119871119873Λ 119888119898119871 119905minus119871119888119871119873Λ 119886(120593+2) + 119871119887119887Λ 119887(120593+2) minus 119871119886119871119873Λ 119888(120593+2)119871 119905minus119871119888119871119873Λ 119886(120593+3) + 119871119887119887Λ 119887(120593+3) minus 119871119886119871119873Λ 119888(120593+3)119871 119905 sdot sdot sdot minus119871119888119871119873Λ 119886119898 + 119871119887119887Λ 119887119898 minus 119871119886119871119873Λ 119888119898119871 119905minus119871119887119871119873Λ 119886(120593+2) minus 119871119886119871119873Λ 119887(120593+2) + 119871119888119888Λ 119888(120593+2)119871 119905minus119871119887119871119873Λ 119886(120593+3) minus 119871119886119871119873Λ 119887(120593+3) + 119871119888119888Λ 119888(120593+3)119871 119905 sdot sdot sdot minus119871119886119871119873Λ 119886119898 minus 119871119886119871119873Λ 119887119898 + 119871119888119888Λ 119888119898119871 119905

]]]]]]]]

1198653 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

Γ11988611198621Γ11988711198621

Γ11988811198621Γ11988621198622Γ11988721198622

Γ11988821198622 Γ119886120593119862120593

Γ119887120593119862120593Γ119888120593119862120593

minusΛ 119886(120593+2)119862(120593+1) minusΛ 119887(120593+2)119862(120593+1) minusΛ 119888(120593+2)119862(120593+1)minusΛ 119886(120593+3)119862(120593+2) minusΛ 119887(120593+3)119862(120593+2) minusΛ 119888(120593+3)119862(120593+2) minus Λ 119886119898119862(119898minus1) minus Λ 119887119898119862(119898minus1) minus Λ 119888119898119862(119898minus1)

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

Γ119895119896 =119896sum119894=1

119878119895119894Λ 119895119896 =

119898sum119894=119896

119878119895119894119895 = 119886 119887 119888

119866 =

[[[[[[[[[[[[[[[[

119866119886119886 119866119886119887 119866119886119888 0119866119887119886 119866119887119887 119866119887119888 0119866119888119886 119866119888119887 119866119888119888 00 0 0 11986611986210 0 0 1198661198622 0 0 0 119866119862(119898minus1)

]]]]]]]]]]]]]]]]

119866119886119886 = 119871119886119886119871 119905 times [1 1 sdot sdot sdot 1 sdot sdot sdot 1 1]119866119887119887 = 119871119887119887119871 119905 times [1 1 sdot sdot sdot 1 sdot sdot sdot 1 1]


119866119888119888 = 119871119888119888119871 119905 times [1 1 sdot sdot sdot 1 sdot sdot sdot 1 1]119866119886119887 = minus119871119888119871119873119871 119905 times [1 1 sdot sdot sdot 1 sdot sdot sdot 1 1]119866119886119888 = minus119871119887119871119873119871 119905 times [1 1 sdot sdot sdot 1 sdot sdot sdot 1 1]119866119887119888 = minus119871119886119871119873119871 119905 times [1 1 sdot sdot sdot 1 sdot sdot sdot 1 1]119866119887119886 = 119866119886119887119866119888119886 = 119866119886119888119866119888119887 = 1198661198871198881198661198621 = 1

1198621 times [1 1 sdot sdot sdot 1 sdot sdot sdot 1 1]1198661198622 = 1

1198622 times [1 1 sdot sdot sdot 1 sdot sdot sdot 1 1]119866119862(119898minus1) = 1

119862(119898minus1) times [1 1 sdot sdot sdot 1 sdot sdot sdot 1 1]120593 = 119898 minus 12119871119886119886 = 119871119887119871119888 + 119871119887119871119873 + 119871119888119871119873119871119887119887 = 119871119886119871119888 + 119871119886119871119873 + 119871119888119871119873119871119888119888 = 119871119886119871119887 + 119871119886119871119873 + 119871119887119871119873119871 119905 = 119871119886119871119887119871119888 + 119871119886119871119887119871119873 + 119871119886119871119888119871119873 + 119871119887119871119888119871119873

(A2)

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to sincerely thank the editor andanonymous reviewers for their valuable comments and sug-gestions which improved the quality of the paper This workwas financially supported by the Ministry of Science andTechnology (under contract No 106-2221-E-011-100) and bythe Taiwan Building Technology Center from The FeaturedAreas Research Center Program within the framework ofthe Higher Education Sprout Project by the Ministry ofEducation in Taiwan

References

[1] J Segundo-Ramırez R Pena-Gallardo A Medina C Nunez-Gutierrez andNVisairo-Cruz ldquoAComprehensiveModeling of

a Three-Phase Voltage Source PWM ConverterrdquoMathematicalProblems in Engineering vol 2015 Article ID 426245 11 pages2015

[2] V Valouch J Skramlık Z Muller et al ldquoPower Control atGrid Connected Converters and Analytical Solution of SteadyStatesrdquoMathematical Problems in Engineering vol 2015 ArticleID 601916 12 pages 2015

[3] R K Subroto and K L Lian ldquoModeling of a multilevel voltagesource converter using the fast time-domain methodrdquo IEEEJournal of Emerging and SelectedTopics in Power Electronics vol2 no 4 pp 1117ndash1126 2014

[4] S A Khajehoddin A Bakhshai and P K Jain ldquoA simplevoltage balancing scheme for m-level diode-clampedmultilevelconverters based on a generalized current flow modelrdquo IEEETransactions on Power Electronics vol 23 no 5 pp 2248ndash22592008

[5] A Yazdani and R Iravani ldquoDynamic model and control of theNPC-based Back-to-Back HVDC systemrdquo IEEE Transactionson Power Delivery vol 21 no 1 pp 414ndash424 2006

[6] A Nabae I Takahashi and H Akagi ldquoA new neutral-point-clamped PWM inverterrdquo IEEE Transactions on Industry Appli-cations vol 17 no 5 pp 518ndash523 1981

[7] P W Lehn and K L Lian ldquoFrequency coupling matrix of avoltage-source converter derived from piecewise linear differ-ential equationsrdquo IEEE Transactions on Power Delivery vol 22no 3 pp 1603ndash1612 2007


[8] E Acha and M Madrigal Power System Harmonics ComputerModelling and Analysis vol 1 Wiley 2001

[9] C Collins N Watson and A Wood ldquoUPFC modeling in theharmonic domainrdquo IEEE Transactions on Power Delivery vol21 no 2 pp 933ndash938 2006

[10] C F Nascimento E H Watanabe O Diene et al ldquoAnalysisof Noncharacteristic Harmonics Generated by Voltage-SourceConverters Operating under Unbalanced Voltagerdquo IEEE Trans-actions on Power Delivery vol 32 no 2 pp 951ndash961 2017

[11] Q Zhong L Lin G Wang Y Zhang and Z Wu ldquoHarmonicanalysis model for voltage source converter under unbalancedconditionsrdquo IET Generation Transmission amp Distribution vol9 no 1 pp 12ndash21 2015

[12] S Busquets-Monge A Filba-Martinez S Alepuz and A Calle-Prado ldquoA Modulation Strategy to Operate Multilevel Multi-phase Diode-Clamped and Active-Clamped DC-AC Convert-ers at LowFrequencyModulation IndiceswithDC-LinkCapac-itor Voltage Balancerdquo IEEE Transactions on Power Electronicsvol 32 no 10 pp 7521ndash7533 2017

[13] Y Wang H Liu W Wang and K Wang ldquoA Novel Neutral-Point Potential Balance Strategy for Three-Level NPC Back-to-Back Converter Based on the Neutral-Point Current InjectionModelrdquoMathematical Problems in Engineering vol 2015ArticleID 736828 12 pages 2015

[14] U-M Choi F Blaabjerg and K-B Lee ldquoControl Strategy ofTwo Capacitor Voltages for Separate MPPTs in PhotovoltaicSystems Using Neutral-Point-Clamped Invertersrdquo IEEE Trans-actions on Industry Applications vol 51 no 4 pp 3295ndash33032015

[15] M M Hashempour M-Y Yang and T-L Lee ldquoA DPWM-controlled three-level T-type inverter for photovoltaic genera-tion considering unbalanced neutral-point voltagerdquo in Proceed-ings of the 9th Annual IEEE Energy Conversion Congress andExposition ECCE rsquo17 pp 3856ndash3862 2017

[16] M Saeedifard R Iravani and J Pou ldquoControl and DC-capacitor voltage balancing of a space vector-modulated five-level STATCOMrdquo IET Power Electronics vol 2 no 3 pp 203ndash215 2009

[17] Y Liu and G T Heydt ldquoPower system even harmonics andpower quality indicesrdquo Electric Power Components and Systemsvol 33 no 8 pp 833ndash844 2005

[18] A Yazdani and R Iravani ldquoA generalized state-space averagedmodel of the three-level NPC converter for systematic DC-voltage-balancer and current-controller designrdquo IEEE Transac-tions on Power Delivery vol 20 no 2 pp 1105ndash1114 2005

[19] N Jayaram P Agarwal and S Das ldquoMathematical model offive-level acdc converter in abc reference framerdquo InternationalJournal of Electrical Power amp Energy Systems vol 62 pp 469ndash475 2014

[20] Z He H Hu Y Zhang and S Gao ldquoHarmonic resonanceassessment to traction power-supply system considering trainmodel in China High-Speed Railwayrdquo IEEE Transactions onPower Delivery vol 29 no 4 pp 1735ndash1743 2014

[21] E V Larsen D H Baker and J C McIver ldquoLow-order har-monic interactions on ACDC systemsrdquo IEEE Transactions onPower Delivery vol 4 no 1 pp 493ndash501 1989

[22] A A Gonzalez S A Verne and M I Valla Multilevel Con-verters for Industrial Applications vol 7 CRC Press 2013

[23] M Chaves E Margato J F Silva and S F Pinto ldquoGeneralizedstate-space modeling for m level diode-clamped multilevelconvertersrdquo Mathematical Methods in Engineering pp 67ndash852014

[24] T-H Huang Harmonic Modeling of a Diode-Clamped Multi-level Voltage Source Converters for Predicting NoncharacteristicHarmonics [Master thesis] National Taiwan University of Sci-ence and Technology 2018

[25] Z Jinghua M Yongqing L Zhengxi and L Kun ldquoResearchon control method of three-level NPC voltage source rectifierrdquoin Proceedings of the 2008 IEEE Vehicle Power and PropulsionConference VPPC 2008 pp 1ndash8 2008

[26] S Kouro M Malinowski K Gopakumar et al ldquoRecent ad-vances and industrial applications of multilevel convertersrdquoIEEE Transactions on Industrial Electronics vol 57 no 8 pp2553ndash2580 2010

[27] B M Wilamowski and J D Irwin Eds Power Electronics andMotor Drives CRC Press 2017

[28] B WuHigh-Power Converters and AC Drive IEEE Press 2006[29] N Hatti Y Kondo and H Akagi ldquoFive-level diode-clamped

PWM converters connected back-to-back for motor drivesrdquoIEEE Transactions on Industry Applications vol 44 no 4 pp1268ndash1276 2008

[30] N W A Lidula and A D Rajapakse ldquoMicrogrids research areviewof experimental microgrids and test systemsrdquo Renewableamp Sustainable Energy Reviews vol 15 no 1 pp 186ndash202 2011

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Submit your manuscripts atwwwhindawicom


+

AC side

-

DC side

Figure 1 Equivalent external network of DCM-VSC

harmonic model of the DCM-VSC has not been developedyet Thus the focus of this paper is on the DCM-VSCs

As VSCs may be subjected to various power qualitydisturbance the derived harmonic models of DCM-VSCsshould be general and able to account for various conditionssuch as unbalanced grid voltage [10 11] and unbalanced dcvoltages [12 13] The imbalance in the dc voltage comes from(i) the DGs with different voltage levels connected to thedc link of the converters [14 15] and (ii) the inherent dcvoltage imbalance of DCM-VSCs [16] It will be demonstratedin the paper that the two cases will lead to drasticallydifferent harmonic profiles The first dc voltage imbalancemay result in low-order even current harmonics at the ac side[14] which in turn may cause undesired dc component andunequal positive and negative peak values [17] However inthe second case no even harmonics are generated but someuncharacteristic odd harmonics are generated Moreover aswill be demonstrated in the paper the harmonic profileof converters will also alter significantly in a microgridenvironment or in the presence of neighboring convertersTherefore it is essential to have an accurate harmonic modelcapable of predicting uncharacteristic harmonics under vari-ous conditions

The current models for DCM-VSCs are inadequate forharmonic modeling Khajehoddin et al [4] proposed acurrent flow model for the DCM-VSCs They formulatedthe capacitor currents as well as the input dc current asa function of the switching functions and output currentsNevertheless they ignored the high-order harmonics of theswitching functions whichmay result in inaccurate side bandharmonic arising with converter modulation Generalizedaveraged models of DCM-VSCs have been proposed in [18]However all the high-order harmonics are effectively elimi-nated in averaging operations thereby producing inaccurateharmonic results Jayaram et al [19] derived a five-levelDCM-VSC in abc frame for which the switching functions areobtained using curve fitting However such amodel becomesvery complex when the number of level is high He et al[20] have modeled the DCM-VSC as a Norton equivalentcircuit However they have assumed that the capacitors arelarge enough so that the dc link voltage can be assumedto be constant Such models are only accurate in predictingcharacteristic harmonics and are not able to account for theuncharacteristic harmonics under unbalanced conditions

Based on the work of Larsen et al [21] the FM of an AC-DC converter is more conveniently modeled by means of anABCD matrix A DCM-VSC can be visualized in Figure 1The harmonic injections caused by external networks canbe modeled as ac voltage harmonics 119881119886119888 (in this papercapital letters represent harmonic domain variables whereas

lower-case letters represent time-domain variables) and dccurrent harmonics 119868119889119888 at the point of common couplingThe converter in turn produces ac current harmonics and dcvoltage harmonics 119881119889119888 via the ABCD matrix as stated in (1)

[119868119886119888119881119889119888] = [119860119861119862119863][119881119886119888119868119889119888] (1)

Therefore the main goal of this paper is to derive anaccurate ABCDmatrix for a DCM-VSC under various condi-tions The proposed method essentially consists of two stepsthe first is to derive the state-space equations of the DCM-VSC and the second is to treat the harmonics of interestas state variables and solve the entire augmented state-spaceequations piecewise with the periodicity constraint imposedThen the ABCDmatrix is obtained by mapping the resultingsubmatrix of the state transition matrix to the harmonicdomain by a transformation matrix The advantage of theproposed method is that

(i) calculation times are much shorter than the brute-force time-domain method as it directly bypasses thetransient state

(ii) different from the harmonic domain methods thereis no need to know how many harmonics need to beincluded to obtain the harmonics of interest

(iii) the predicted harmonics are exact and do not sufferfrom harmonic truncation errors [7]

(iv) it can be easily expanded to study a microgrid systemcontaining multiple numbers of DCM-VSCs

To validate the proposed model all the predicted harmonicsare verified with those obtained from PSCADEMTDC

The rest of the paper is organized as follows Section 2describes the state-space model of a DCM-VSC Section 3shows how the harmonic ABCD coupling matrix is derivedIn Section 4 various scenarios of imbalance for a DCM-VSC and DCM-based microgrid are studied and the pre-dicted results are verified with those of PSCADEMTDCdemonstrating the validity of the proposed method Finallya conclusion is given in Section 5

2 Generalized State-Space Representationof a DCM-VSC

Different from other conventional multilevel converters suchas flying capacitors or cascaded multilevel converters DCM-VSCs have forbidden states [22] in which the output voltageis different if the current direction is different under the sameswitching function state Figure 2 shows the three-level VSCwhere the upper switches are assumed to be complementarywith the lower switches in each leg

To illustrate the forbidden states Figure 3 is referred toand without loss of generality only phase A leg of the VSCis shown When 11990411 turns on and 11990412 turns off and if thecurrent flows as shown in Figure 3(a) V119886119900 = minusV1198882 Howeverwhen the current flows in the reverse direction the circuitbecomes Figure 3(b) and V119886119900 = V1198881 Consequently 11990411 = 1 and



11990411 11990412 V1198861199000 0 V11986210 1 01 1 minusV1198622


119904119886 997888rarr 1198901198862 0 1 0119904119886 997888rarr 1198901198863 0 0 1




119889119889119905 [

119909119906] = [

119865 1198660 Ω][

119909119906] (2)

where


(3)




119889119909ℎ119889119905 = 119895119899120596119909ℎ + 1119879119890minus119895119899120596119879119909 (119905) (4)



119889119889119905 [[[

119909119906119909ℎ]]]= [[

[119909119906119909ℎ]]]

(5)

where

= [[[119865 119866 00 Ω 0119863 0 119869

]]]

119863 =[[[[[[[[

119863111986321198632+119898

]]]]]]]]

1198631 =

[[[[[[[[[[[[[[[[[[[[[[[[[

119890119895119899120596119879119879 0 sdot sdot sdot 0



]]]]]]]]]]]]]]]]]]]]]]]]](2lowast119899+1)times(2+119898)

1198632+119898 =

[[[[[[[[[[[[[[[[[[[[[[[[[

0 sdot sdot sdot 0 119890119895119899120596119879119879



119879

]]]]]]]]]]]]]]]]]]]]]]]]](2lowast119899+1)times(2+119898)


(6)


+

-

+

-

sa

sb

sc

ao

bo

co

a

b

c

C

C

11 21 31

12 22 32

31

32

21

22

11

12


+

-

+

-

+

-

+

-

ao ao

C

C

C

C


+

-

+

-

ao

C

C

(a)

+

-

+

-

ao

C

C

(b)


+

-

+

-

C

C

sa

sb

sc

a

b

c

aa

bb

cc

a

b

c





119909 (119879)119906 (119879)119909ℎ (119879)

]]]= Φ[[

[

119909 (0)119906 (0)119909ℎ (0)

]]]

(7)

where



Φ = [[[

120572 120573 00 120574 0120589 120578 120585

]]]

(9)


119909 (119905 + 119879) = 119909 (119905) (10)


[119868119886119887119888119881119889119888] = [119872][V119904119886119887119888 (0)119894119889119888 (0) ] (11)

where

[119872] = 120589 [(119868 minus 120572)minus1 120573] + 120578 (12)



[119881119904119886119887119888119868119886119887119888 ] = [V119904119886119887119888 (0)119894119889119888 (0) ] (13)

Hence

[119868119886119887119888119881119889119888] = [119860119861119862119863]119886119887119888 [119881119904119886119887119888119868119889119888 ] (14)


119868119886119887119888 = 119875 sdot 119868012 (15)

119881119904119886119887119888 = 119875 sdot 119881119904012 (16)

where

119875 = [[[[

1 1 11 1198862 1198861 119886 1198862

]]]]

119886 = 1ang120∘(17)


[119875 00 119868] [

119868012119881119889119888] = [119860119861119862119863]119886119887119888 [

119875 00 119868] [

119881119886012119868119889119888 ] (18)


[119868012119881119889119888] = [119860119861119862119863]012 [119881012119868119889119888 ] (19)

where [119860119861119862119863]012 = [ 119875 00 119868 ]minus1 [119860119861119862119863]119886119887119888 [ 119875 0


4 Case Studies





01

1

10

40

）＞

(A)


120

100

80

60

40

20

0

minus20

minus40

minus60

Iabc

(Am

pere

)

0 0002 0004 0006 0008 001 0012 0014 0016 0018

time (seconds)





cos (120596119905 + 120579) = 119890119895(120596119905+120579) + 119890minus119895(120596119905+120579)2 (20)







0

20

40

60


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)




follows


]]]

(21)


]]]

(22)


119863 = [00] (24)


[[[[[[[[[[

11986811198861198681119887119868111988811988101198881

11988101198882

]]]]]]]]]]

=[[[[[[[[[

2 0 0 0 00 2 0 0 00 0 2 0 00 0 0 1 00 0 0 0 1

]]]]]]]]]

[[[[[[[[[

2276 minus 034j1109 + 1988j1167 minus 1954j

2920329203

]]]]]]]]]

=[[[[[[[[[[


2920329203

]]]]]]]]]]

(25)


1198881 and11988101198882 are




0

20

40

60


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)



0

20

40

60


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)



[[[[[[[[[[

11986810119868111198681211988101198881

11988101198882

]]]]]]]]]]

=[[[[[[[[[

04553 minus 068j

02920329203

]]]]]]]]]

(26)


119881119904119886 = 119881119904119890119895120596119905 + 005119881119904119890119895120596119905119881119904119887 = 119881119904119890119895120596119905minus120∘ + 005119881119904119890119895120596119905+120∘119881119904119888 = 119881119904119890119895120596119905+120∘ + 005119881119904119890119895120596119905minus120∘

(27)




119898119886 cos (120575) gt 055 (28)


0 0002 0004 0006 0008 001 0012 0014 0016 0018time (seconds)

120

140

160

180

200

220

240Vc

(vol

tage

)



(a)

0 0002 0004 0006 0008 001 0012 0014 0016 0018time (seconds)

120

140

160

180

200

220

240

Vc (v

olta

ge)



(b)


0

0

100

1

Har

mon

ic M

agni

tude

(V)

2

200

3 4

Harmonics Number

5 6 7 8 9


６1

６2

６3

６4




30

25

20

15

10

5

0

minus5

minus10

minus15

Iabc

(Am

pere

)

0 0002 0004 0006 0008 001 0012 0014 0016 0018







time (seconds)






0

2

4

6

8


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)



030

002

20 30

abs(

Out

putI

nput

)

10

004

200 10

006







119881119901119888119888 = 119881119904 minus 119885119904119868119904 (29)



[119868V119904119888119896119881119889119888119896] = [119860119896 119861119896119862119896 119863119896

][119881119901119888119888119868119889119888119896] (31)



[[[[[[[[[[

11986811990411988111988911988811198811198891198882119881119889119888119899

]]]]]]]]]]

= [119872]minus1 [119873][[[[[[[[[[

11988111990411986811988911988811198681198891198882119868119889119888119899

]]]]]]]]]]

(32)

where

119872 =[[[[[[[[[[


119862119899119885119904 0 0 sdot sdot sdot 119868

]]]]]]]]]]

119873 =[[[[[[[[[[


]]]]]]]]]]

(33)



0

20

40

60


） 1(A

)

(a)



0

5

10

） 2(A

)

(b)



0

05

1

15

2） 0

(A)

(c)


1

2

1

2

1

2

VSC1

VSC2

VSCn

1

2

1

2





+-

+-

+-

-+

+

+-

-

+-+-

1

2

1

2

1

2

3 33

VSC1

VSC2

VSC3

1

2

3

4

5

6

7

8



0

10

20

30

40


）６３

11

(A)

(a)



0

2

4

6

8

）６３

12

(A)

(b)



0

2

4

6

8


）６３

21

(A)

(a)



0

5

10

15

）６３

22

(A)

(b)



5 Conclusion





0

10

20

30

40


）６３

31

(A)

(a)



0

2

4

6

）６３

32

(A)

(b)



0

05

1

15

2）６

３30

(A)

(c)




0

20

40

60

）６３

31

(A)

(a)



0

2

4

6

8

）６３

32

(A)

(b)



0

05

1

15

2

）６３

30

(A)

(c)


Appendix



1198651 =[[[[[[[[

minus119877119886119871119886119886 + 119877119873119871119887119871119888119871 119905119877119887119871119888119871119873 minus 119877119873119871119887119871119888119871 119905


119877119887119871119886119871119873 minus 119877119873119871119886119871119887119871 119905 minus119877119888119871119888119888 + 119877119873119871119886119871119887119871 119905

]]]]]]]]

1198652 = [11986521 11986522]

11986521 =[[[[[[[[


]]]]]]]]

(A1)

11986522

=[[[[[[[[


]]]]]]]]

1198653 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

Γ11988611198621Γ11988711198621

Γ11988811198621Γ11988621198622Γ11988721198622

Γ11988821198622 Γ119886120593119862120593

Γ119887120593119862120593Γ119888120593119862120593


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

Γ119895119896 =119896sum119894=1

119878119895119894Λ 119895119896 =

119898sum119894=119896

119878119895119894119895 = 119886 119887 119888

119866 =

[[[[[[[[[[[[[[[[

119866119886119886 119866119886119887 119866119886119888 0119866119887119886 119866119887119887 119866119887119888 0119866119888119886 119866119888119887 119866119888119888 00 0 0 11986611986210 0 0 1198661198622 0 0 0 119866119862(119898minus1)

]]]]]]]]]]]]]]]]







(A2)

Data Availability




Acknowledgments


References








































Journal of












Journal of







Volume 2018






Volume 2018










11990411 11990412 V1198861199000 0 V11986210 1 01 1 minusV1198622


119904119886 997888rarr 1198901198862 0 1 0119904119886 997888rarr 1198901198863 0 0 1




119889119889119905 [

119909119906] = [

119865 1198660 Ω][

119909119906] (2)

where


(3)




119889119909ℎ119889119905 = 119895119899120596119909ℎ + 1119879119890minus119895119899120596119879119909 (119905) (4)



119889119889119905 [[[

119909119906119909ℎ]]]= [[

[119909119906119909ℎ]]]

(5)

where

= [[[119865 119866 00 Ω 0119863 0 119869

]]]

119863 =[[[[[[[[

119863111986321198632+119898

]]]]]]]]

1198631 =

[[[[[[[[[[[[[[[[[[[[[[[[[

119890119895119899120596119879119879 0 sdot sdot sdot 0



]]]]]]]]]]]]]]]]]]]]]]]]](2lowast119899+1)times(2+119898)

1198632+119898 =

[[[[[[[[[[[[[[[[[[[[[[[[[

0 sdot sdot sdot 0 119890119895119899120596119879119879



119879

]]]]]]]]]]]]]]]]]]]]]]]]](2lowast119899+1)times(2+119898)


(6)


+

-

+

-

sa

sb

sc

ao

bo

co

a

b

c

C

C

11 21 31

12 22 32

31

32

21

22

11

12


+

-

+

-

+

-

+

-

ao ao

C

C

C

C


+

-

+

-

ao

C

C

(a)

+

-

+

-

ao

C

C

(b)


+

-

+

-

C

C

sa

sb

sc

a

b

c

aa

bb

cc

a

b

c





119909 (119879)119906 (119879)119909ℎ (119879)

]]]= Φ[[

[

119909 (0)119906 (0)119909ℎ (0)

]]]

(7)

where



Φ = [[[

120572 120573 00 120574 0120589 120578 120585

]]]

(9)


119909 (119905 + 119879) = 119909 (119905) (10)


[119868119886119887119888119881119889119888] = [119872][V119904119886119887119888 (0)119894119889119888 (0) ] (11)

where

[119872] = 120589 [(119868 minus 120572)minus1 120573] + 120578 (12)



[119881119904119886119887119888119868119886119887119888 ] = [V119904119886119887119888 (0)119894119889119888 (0) ] (13)

Hence

[119868119886119887119888119881119889119888] = [119860119861119862119863]119886119887119888 [119881119904119886119887119888119868119889119888 ] (14)


119868119886119887119888 = 119875 sdot 119868012 (15)

119881119904119886119887119888 = 119875 sdot 119881119904012 (16)

where

119875 = [[[[

1 1 11 1198862 1198861 119886 1198862

]]]]

119886 = 1ang120∘(17)


[119875 00 119868] [

119868012119881119889119888] = [119860119861119862119863]119886119887119888 [

119875 00 119868] [

119881119886012119868119889119888 ] (18)


[119868012119881119889119888] = [119860119861119862119863]012 [119881012119868119889119888 ] (19)

where [119860119861119862119863]012 = [ 119875 00 119868 ]minus1 [119860119861119862119863]119886119887119888 [ 119875 0


4 Case Studies





01

1

10

40

）＞

(A)


120

100

80

60

40

20

0

minus20

minus40

minus60

Iabc

(Am

pere

)

0 0002 0004 0006 0008 001 0012 0014 0016 0018

time (seconds)





cos (120596119905 + 120579) = 119890119895(120596119905+120579) + 119890minus119895(120596119905+120579)2 (20)







0

20

40

60


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)




follows


]]]

(21)


]]]

(22)


119863 = [00] (24)


[[[[[[[[[[

11986811198861198681119887119868111988811988101198881

11988101198882

]]]]]]]]]]

=[[[[[[[[[

2 0 0 0 00 2 0 0 00 0 2 0 00 0 0 1 00 0 0 0 1

]]]]]]]]]

[[[[[[[[[

2276 minus 034j1109 + 1988j1167 minus 1954j

2920329203

]]]]]]]]]

=[[[[[[[[[[


2920329203

]]]]]]]]]]

(25)


1198881 and11988101198882 are




0

20

40

60


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)



0

20

40

60


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)



[[[[[[[[[[

11986810119868111198681211988101198881

11988101198882

]]]]]]]]]]

=[[[[[[[[[

04553 minus 068j

02920329203

]]]]]]]]]

(26)


119881119904119886 = 119881119904119890119895120596119905 + 005119881119904119890119895120596119905119881119904119887 = 119881119904119890119895120596119905minus120∘ + 005119881119904119890119895120596119905+120∘119881119904119888 = 119881119904119890119895120596119905+120∘ + 005119881119904119890119895120596119905minus120∘

(27)




119898119886 cos (120575) gt 055 (28)


0 0002 0004 0006 0008 001 0012 0014 0016 0018time (seconds)

120

140

160

180

200

220

240Vc

(vol

tage

)



(a)

0 0002 0004 0006 0008 001 0012 0014 0016 0018time (seconds)

120

140

160

180

200

220

240

Vc (v

olta

ge)



(b)


0

0

100

1

Har

mon

ic M

agni

tude

(V)

2

200

3 4

Harmonics Number

5 6 7 8 9


６1

６2

６3

６4




30

25

20

15

10

5

0

minus5

minus10

minus15

Iabc

(Am

pere

)

0 0002 0004 0006 0008 001 0012 0014 0016 0018







time (seconds)






0

2

4

6

8


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)



030

002

20 30

abs(

Out

putI

nput

)

10

004

200 10

006







119881119901119888119888 = 119881119904 minus 119885119904119868119904 (29)



[119868V119904119888119896119881119889119888119896] = [119860119896 119861119896119862119896 119863119896

][119881119901119888119888119868119889119888119896] (31)



[[[[[[[[[[

11986811990411988111988911988811198811198891198882119881119889119888119899

]]]]]]]]]]

= [119872]minus1 [119873][[[[[[[[[[

11988111990411986811988911988811198681198891198882119868119889119888119899

]]]]]]]]]]

(32)

where

119872 =[[[[[[[[[[


119862119899119885119904 0 0 sdot sdot sdot 119868

]]]]]]]]]]

119873 =[[[[[[[[[[


]]]]]]]]]]

(33)



0

20

40

60


） 1(A

)

(a)



0

5

10

） 2(A

)

(b)



0

05

1

15

2） 0

(A)

(c)


1

2

1

2

1

2

VSC1

VSC2

VSCn

1

2

1

2





+-

+-

+-

-+

+

+-

-

+-+-

1

2

1

2

1

2

3 33

VSC1

VSC2

VSC3

1

2

3

4

5

6

7

8



0

10

20

30

40


）６３

11

(A)

(a)



0

2

4

6

8

）６３

12

(A)

(b)



0

2

4

6

8


）６３

21

(A)

(a)



0

5

10

15

）６３

22

(A)

(b)



5 Conclusion





0

10

20

30

40


）６３

31

(A)

(a)



0

2

4

6

）６３

32

(A)

(b)



0

05

1

15

2）６

３30

(A)

(c)




0

20

40

60

）６３

31

(A)

(a)



0

2

4

6

8

）６３

32

(A)

(b)



0

05

1

15

2

）６３

30

(A)

(c)


Appendix



1198651 =[[[[[[[[

minus119877119886119871119886119886 + 119877119873119871119887119871119888119871 119905119877119887119871119888119871119873 minus 119877119873119871119887119871119888119871 119905


119877119887119871119886119871119873 minus 119877119873119871119886119871119887119871 119905 minus119877119888119871119888119888 + 119877119873119871119886119871119887119871 119905

]]]]]]]]

1198652 = [11986521 11986522]

11986521 =[[[[[[[[


]]]]]]]]

(A1)

11986522

=[[[[[[[[


]]]]]]]]

1198653 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

Γ11988611198621Γ11988711198621

Γ11988811198621Γ11988621198622Γ11988721198622

Γ11988821198622 Γ119886120593119862120593

Γ119887120593119862120593Γ119888120593119862120593


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

Γ119895119896 =119896sum119894=1

119878119895119894Λ 119895119896 =

119898sum119894=119896

119878119895119894119895 = 119886 119887 119888

119866 =

[[[[[[[[[[[[[[[[

119866119886119886 119866119886119887 119866119886119888 0119866119887119886 119866119887119887 119866119887119888 0119866119888119886 119866119888119887 119866119888119888 00 0 0 11986611986210 0 0 1198661198622 0 0 0 119866119862(119898minus1)

]]]]]]]]]]]]]]]]







(A2)

Data Availability




Acknowledgments


References








































Journal of












Journal of







Volume 2018






Volume 2018









+

-

+

-

sa

sb

sc

ao

bo

co

a

b

c

C

C

11 21 31

12 22 32

31

32

21

22

11

12


+

-

+

-

+

-

+

-

ao ao

C

C

C

C


+

-

+

-

ao

C

C

(a)

+

-

+

-

ao

C

C

(b)


+

-

+

-

C

C

sa

sb

sc

a

b

c

aa

bb

cc

a

b

c





119909 (119879)119906 (119879)119909ℎ (119879)

]]]= Φ[[

[

119909 (0)119906 (0)119909ℎ (0)

]]]

(7)

where



Φ = [[[

120572 120573 00 120574 0120589 120578 120585

]]]

(9)


119909 (119905 + 119879) = 119909 (119905) (10)


[119868119886119887119888119881119889119888] = [119872][V119904119886119887119888 (0)119894119889119888 (0) ] (11)

where

[119872] = 120589 [(119868 minus 120572)minus1 120573] + 120578 (12)



[119881119904119886119887119888119868119886119887119888 ] = [V119904119886119887119888 (0)119894119889119888 (0) ] (13)

Hence

[119868119886119887119888119881119889119888] = [119860119861119862119863]119886119887119888 [119881119904119886119887119888119868119889119888 ] (14)


119868119886119887119888 = 119875 sdot 119868012 (15)

119881119904119886119887119888 = 119875 sdot 119881119904012 (16)

where

119875 = [[[[

1 1 11 1198862 1198861 119886 1198862

]]]]

119886 = 1ang120∘(17)


[119875 00 119868] [

119868012119881119889119888] = [119860119861119862119863]119886119887119888 [

119875 00 119868] [

119881119886012119868119889119888 ] (18)


[119868012119881119889119888] = [119860119861119862119863]012 [119881012119868119889119888 ] (19)

where [119860119861119862119863]012 = [ 119875 00 119868 ]minus1 [119860119861119862119863]119886119887119888 [ 119875 0


4 Case Studies





01

1

10

40

）＞

(A)


120

100

80

60

40

20

0

minus20

minus40

minus60

Iabc

(Am

pere

)

0 0002 0004 0006 0008 001 0012 0014 0016 0018

time (seconds)





cos (120596119905 + 120579) = 119890119895(120596119905+120579) + 119890minus119895(120596119905+120579)2 (20)







0

20

40

60


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)




follows


]]]

(21)


]]]

(22)


119863 = [00] (24)


[[[[[[[[[[

11986811198861198681119887119868111988811988101198881

11988101198882

]]]]]]]]]]

=[[[[[[[[[

2 0 0 0 00 2 0 0 00 0 2 0 00 0 0 1 00 0 0 0 1

]]]]]]]]]

[[[[[[[[[

2276 minus 034j1109 + 1988j1167 minus 1954j

2920329203

]]]]]]]]]

=[[[[[[[[[[


2920329203

]]]]]]]]]]

(25)


1198881 and11988101198882 are




0

20

40

60


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)



0

20

40

60


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)



[[[[[[[[[[

11986810119868111198681211988101198881

11988101198882

]]]]]]]]]]

=[[[[[[[[[

04553 minus 068j

02920329203

]]]]]]]]]

(26)


119881119904119886 = 119881119904119890119895120596119905 + 005119881119904119890119895120596119905119881119904119887 = 119881119904119890119895120596119905minus120∘ + 005119881119904119890119895120596119905+120∘119881119904119888 = 119881119904119890119895120596119905+120∘ + 005119881119904119890119895120596119905minus120∘

(27)




119898119886 cos (120575) gt 055 (28)


0 0002 0004 0006 0008 001 0012 0014 0016 0018time (seconds)

120

140

160

180

200

220

240Vc

(vol

tage

)



(a)

0 0002 0004 0006 0008 001 0012 0014 0016 0018time (seconds)

120

140

160

180

200

220

240

Vc (v

olta

ge)



(b)


0

0

100

1

Har

mon

ic M

agni

tude

(V)

2

200

3 4

Harmonics Number

5 6 7 8 9


６1

６2

６3

６4




30

25

20

15

10

5

0

minus5

minus10

minus15

Iabc

(Am

pere

)

0 0002 0004 0006 0008 001 0012 0014 0016 0018







time (seconds)






0

2

4

6

8


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)



030

002

20 30

abs(

Out

putI

nput

)

10

004

200 10

006







119881119901119888119888 = 119881119904 minus 119885119904119868119904 (29)



[119868V119904119888119896119881119889119888119896] = [119860119896 119861119896119862119896 119863119896

][119881119901119888119888119868119889119888119896] (31)



[[[[[[[[[[

11986811990411988111988911988811198811198891198882119881119889119888119899

]]]]]]]]]]

= [119872]minus1 [119873][[[[[[[[[[

11988111990411986811988911988811198681198891198882119868119889119888119899

]]]]]]]]]]

(32)

where

119872 =[[[[[[[[[[


119862119899119885119904 0 0 sdot sdot sdot 119868

]]]]]]]]]]

119873 =[[[[[[[[[[


]]]]]]]]]]

(33)



0

20

40

60


） 1(A

)

(a)



0

5

10

） 2(A

)

(b)



0

05

1

15

2） 0

(A)

(c)


1

2

1

2

1

2

VSC1

VSC2

VSCn

1

2

1

2





+-

+-

+-

-+

+

+-

-

+-+-

1

2

1

2

1

2

3 33

VSC1

VSC2

VSC3

1

2

3

4

5

6

7

8



0

10

20

30

40


）６３

11

(A)

(a)



0

2

4

6

8

）６３

12

(A)

(b)



0

2

4

6

8


）６３

21

(A)

(a)



0

5

10

15

）６３

22

(A)

(b)



5 Conclusion





0

10

20

30

40


）６３

31

(A)

(a)



0

2

4

6

）６３

32

(A)

(b)



0

05

1

15

2）６

３30

(A)

(c)




0

20

40

60

）６３

31

(A)

(a)



0

2

4

6

8

）６３

32

(A)

(b)



0

05

1

15

2

）６３

30

(A)

(c)


Appendix



1198651 =[[[[[[[[

minus119877119886119871119886119886 + 119877119873119871119887119871119888119871 119905119877119887119871119888119871119873 minus 119877119873119871119887119871119888119871 119905


119877119887119871119886119871119873 minus 119877119873119871119886119871119887119871 119905 minus119877119888119871119888119888 + 119877119873119871119886119871119887119871 119905

]]]]]]]]

1198652 = [11986521 11986522]

11986521 =[[[[[[[[


]]]]]]]]

(A1)

11986522

=[[[[[[[[


]]]]]]]]

1198653 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

Γ11988611198621Γ11988711198621

Γ11988811198621Γ11988621198622Γ11988721198622

Γ11988821198622 Γ119886120593119862120593

Γ119887120593119862120593Γ119888120593119862120593


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

Γ119895119896 =119896sum119894=1

119878119895119894Λ 119895119896 =

119898sum119894=119896

119878119895119894119895 = 119886 119887 119888

119866 =

[[[[[[[[[[[[[[[[

119866119886119886 119866119886119887 119866119886119888 0119866119887119886 119866119887119887 119866119887119888 0119866119888119886 119866119888119887 119866119888119888 00 0 0 11986611986210 0 0 1198661198622 0 0 0 119866119862(119898minus1)

]]]]]]]]]]]]]]]]







(A2)

Data Availability




Acknowledgments


References








































Journal of












Journal of







Volume 2018






Volume 2018











119909 (119879)119906 (119879)119909ℎ (119879)

]]]= Φ[[

[

119909 (0)119906 (0)119909ℎ (0)

]]]

(7)

where



Φ = [[[

120572 120573 00 120574 0120589 120578 120585

]]]

(9)


119909 (119905 + 119879) = 119909 (119905) (10)


[119868119886119887119888119881119889119888] = [119872][V119904119886119887119888 (0)119894119889119888 (0) ] (11)

where

[119872] = 120589 [(119868 minus 120572)minus1 120573] + 120578 (12)



[119881119904119886119887119888119868119886119887119888 ] = [V119904119886119887119888 (0)119894119889119888 (0) ] (13)

Hence

[119868119886119887119888119881119889119888] = [119860119861119862119863]119886119887119888 [119881119904119886119887119888119868119889119888 ] (14)


119868119886119887119888 = 119875 sdot 119868012 (15)

119881119904119886119887119888 = 119875 sdot 119881119904012 (16)

where

119875 = [[[[

1 1 11 1198862 1198861 119886 1198862

]]]]

119886 = 1ang120∘(17)


[119875 00 119868] [

119868012119881119889119888] = [119860119861119862119863]119886119887119888 [

119875 00 119868] [

119881119886012119868119889119888 ] (18)


[119868012119881119889119888] = [119860119861119862119863]012 [119881012119868119889119888 ] (19)

where [119860119861119862119863]012 = [ 119875 00 119868 ]minus1 [119860119861119862119863]119886119887119888 [ 119875 0


4 Case Studies





01

1

10

40

）＞

(A)


120

100

80

60

40

20

0

minus20

minus40

minus60

Iabc

(Am

pere

)

0 0002 0004 0006 0008 001 0012 0014 0016 0018

time (seconds)





cos (120596119905 + 120579) = 119890119895(120596119905+120579) + 119890minus119895(120596119905+120579)2 (20)







0

20

40

60


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)




follows


]]]

(21)


]]]

(22)


119863 = [00] (24)


[[[[[[[[[[

11986811198861198681119887119868111988811988101198881

11988101198882

]]]]]]]]]]

=[[[[[[[[[

2 0 0 0 00 2 0 0 00 0 2 0 00 0 0 1 00 0 0 0 1

]]]]]]]]]

[[[[[[[[[

2276 minus 034j1109 + 1988j1167 minus 1954j

2920329203

]]]]]]]]]

=[[[[[[[[[[


2920329203

]]]]]]]]]]

(25)


1198881 and11988101198882 are




0

20

40

60


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)



0

20

40

60


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)



[[[[[[[[[[

11986810119868111198681211988101198881

11988101198882

]]]]]]]]]]

=[[[[[[[[[

04553 minus 068j

02920329203

]]]]]]]]]

(26)


119881119904119886 = 119881119904119890119895120596119905 + 005119881119904119890119895120596119905119881119904119887 = 119881119904119890119895120596119905minus120∘ + 005119881119904119890119895120596119905+120∘119881119904119888 = 119881119904119890119895120596119905+120∘ + 005119881119904119890119895120596119905minus120∘

(27)




119898119886 cos (120575) gt 055 (28)


0 0002 0004 0006 0008 001 0012 0014 0016 0018time (seconds)

120

140

160

180

200

220

240Vc

(vol

tage

)



(a)

0 0002 0004 0006 0008 001 0012 0014 0016 0018time (seconds)

120

140

160

180

200

220

240

Vc (v

olta

ge)



(b)


0

0

100

1

Har

mon

ic M

agni

tude

(V)

2

200

3 4

Harmonics Number

5 6 7 8 9


６1

６2

６3

６4




30

25

20

15

10

5

0

minus5

minus10

minus15

Iabc

(Am

pere

)

0 0002 0004 0006 0008 001 0012 0014 0016 0018







time (seconds)






0

2

4

6

8


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)



030

002

20 30

abs(

Out

putI

nput

)

10

004

200 10

006







119881119901119888119888 = 119881119904 minus 119885119904119868119904 (29)



[119868V119904119888119896119881119889119888119896] = [119860119896 119861119896119862119896 119863119896

][119881119901119888119888119868119889119888119896] (31)



[[[[[[[[[[

11986811990411988111988911988811198811198891198882119881119889119888119899

]]]]]]]]]]

= [119872]minus1 [119873][[[[[[[[[[

11988111990411986811988911988811198681198891198882119868119889119888119899

]]]]]]]]]]

(32)

where

119872 =[[[[[[[[[[


119862119899119885119904 0 0 sdot sdot sdot 119868

]]]]]]]]]]

119873 =[[[[[[[[[[


]]]]]]]]]]

(33)



0

20

40

60


） 1(A

)

(a)



0

5

10

） 2(A

)

(b)



0

05

1

15

2） 0

(A)

(c)


1

2

1

2

1

2

VSC1

VSC2

VSCn

1

2

1

2





+-

+-

+-

-+

+

+-

-

+-+-

1

2

1

2

1

2

3 33

VSC1

VSC2

VSC3

1

2

3

4

5

6

7

8



0

10

20

30

40


）６３

11

(A)

(a)



0

2

4

6

8

）６３

12

(A)

(b)



0

2

4

6

8


）６３

21

(A)

(a)



0

5

10

15

）６３

22

(A)

(b)



5 Conclusion





0

10

20

30

40


）６３

31

(A)

(a)



0

2

4

6

）６３

32

(A)

(b)



0

05

1

15

2）６

３30

(A)

(c)




0

20

40

60

）６３

31

(A)

(a)



0

2

4

6

8

）６３

32

(A)

(b)



0

05

1

15

2

）６３

30

(A)

(c)


Appendix



1198651 =[[[[[[[[

minus119877119886119871119886119886 + 119877119873119871119887119871119888119871 119905119877119887119871119888119871119873 minus 119877119873119871119887119871119888119871 119905


119877119887119871119886119871119873 minus 119877119873119871119886119871119887119871 119905 minus119877119888119871119888119888 + 119877119873119871119886119871119887119871 119905

]]]]]]]]

1198652 = [11986521 11986522]

11986521 =[[[[[[[[


]]]]]]]]

(A1)

11986522

=[[[[[[[[


]]]]]]]]

1198653 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

Γ11988611198621Γ11988711198621

Γ11988811198621Γ11988621198622Γ11988721198622

Γ11988821198622 Γ119886120593119862120593

Γ119887120593119862120593Γ119888120593119862120593


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

Γ119895119896 =119896sum119894=1

119878119895119894Λ 119895119896 =

119898sum119894=119896

119878119895119894119895 = 119886 119887 119888

119866 =

[[[[[[[[[[[[[[[[

119866119886119886 119866119886119887 119866119886119888 0119866119887119886 119866119887119887 119866119887119888 0119866119888119886 119866119888119887 119866119888119888 00 0 0 11986611986210 0 0 1198661198622 0 0 0 119866119862(119898minus1)

]]]]]]]]]]]]]]]]







(A2)

Data Availability




Acknowledgments


References








































Journal of












Journal of







Volume 2018






Volume 2018









[119868012119881119889119888] = [119860119861119862119863]012 [119881012119868119889119888 ] (19)

where [119860119861119862119863]012 = [ 119875 00 119868 ]minus1 [119860119861119862119863]119886119887119888 [ 119875 0


4 Case Studies





01

1

10

40

）＞

(A)


120

100

80

60

40

20

0

minus20

minus40

minus60

Iabc

(Am

pere

)

0 0002 0004 0006 0008 001 0012 0014 0016 0018

time (seconds)





cos (120596119905 + 120579) = 119890119895(120596119905+120579) + 119890minus119895(120596119905+120579)2 (20)







0

20

40

60


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)




follows


]]]

(21)


]]]

(22)


119863 = [00] (24)


[[[[[[[[[[

11986811198861198681119887119868111988811988101198881

11988101198882

]]]]]]]]]]

=[[[[[[[[[

2 0 0 0 00 2 0 0 00 0 2 0 00 0 0 1 00 0 0 0 1

]]]]]]]]]

[[[[[[[[[

2276 minus 034j1109 + 1988j1167 minus 1954j

2920329203

]]]]]]]]]

=[[[[[[[[[[


2920329203

]]]]]]]]]]

(25)


1198881 and11988101198882 are




0

20

40

60


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)



0

20

40

60


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)



[[[[[[[[[[

11986810119868111198681211988101198881

11988101198882

]]]]]]]]]]

=[[[[[[[[[

04553 minus 068j

02920329203

]]]]]]]]]

(26)


119881119904119886 = 119881119904119890119895120596119905 + 005119881119904119890119895120596119905119881119904119887 = 119881119904119890119895120596119905minus120∘ + 005119881119904119890119895120596119905+120∘119881119904119888 = 119881119904119890119895120596119905+120∘ + 005119881119904119890119895120596119905minus120∘

(27)




119898119886 cos (120575) gt 055 (28)


0 0002 0004 0006 0008 001 0012 0014 0016 0018time (seconds)

120

140

160

180

200

220

240Vc

(vol

tage

)



(a)

0 0002 0004 0006 0008 001 0012 0014 0016 0018time (seconds)

120

140

160

180

200

220

240

Vc (v

olta

ge)



(b)


0

0

100

1

Har

mon

ic M

agni

tude

(V)

2

200

3 4

Harmonics Number

5 6 7 8 9


６1

６2

６3

６4




30

25

20

15

10

5

0

minus5

minus10

minus15

Iabc

(Am

pere

)

0 0002 0004 0006 0008 001 0012 0014 0016 0018







time (seconds)






0

2

4

6

8


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)



030

002

20 30

abs(

Out

putI

nput

)

10

004

200 10

006







119881119901119888119888 = 119881119904 minus 119885119904119868119904 (29)



[119868V119904119888119896119881119889119888119896] = [119860119896 119861119896119862119896 119863119896

][119881119901119888119888119868119889119888119896] (31)



[[[[[[[[[[

11986811990411988111988911988811198811198891198882119881119889119888119899

]]]]]]]]]]

= [119872]minus1 [119873][[[[[[[[[[

11988111990411986811988911988811198681198891198882119868119889119888119899

]]]]]]]]]]

(32)

where

119872 =[[[[[[[[[[


119862119899119885119904 0 0 sdot sdot sdot 119868

]]]]]]]]]]

119873 =[[[[[[[[[[


]]]]]]]]]]

(33)



0

20

40

60


） 1(A

)

(a)



0

5

10

） 2(A

)

(b)



0

05

1

15

2） 0

(A)

(c)


1

2

1

2

1

2

VSC1

VSC2

VSCn

1

2

1

2





+-

+-

+-

-+

+

+-

-

+-+-

1

2

1

2

1

2

3 33

VSC1

VSC2

VSC3

1

2

3

4

5

6

7

8



0

10

20

30

40


）６３

11

(A)

(a)



0

2

4

6

8

）６３

12

(A)

(b)



0

2

4

6

8


）６３

21

(A)

(a)



0

5

10

15

）６３

22

(A)

(b)



5 Conclusion





0

10

20

30

40


）６３

31

(A)

(a)



0

2

4

6

）６３

32

(A)

(b)



0

05

1

15

2）６

３30

(A)

(c)




0

20

40

60

）６３

31

(A)

(a)



0

2

4

6

8

）６３

32

(A)

(b)



0

05

1

15

2

）６３

30

(A)

(c)


Appendix



1198651 =[[[[[[[[

minus119877119886119871119886119886 + 119877119873119871119887119871119888119871 119905119877119887119871119888119871119873 minus 119877119873119871119887119871119888119871 119905


119877119887119871119886119871119873 minus 119877119873119871119886119871119887119871 119905 minus119877119888119871119888119888 + 119877119873119871119886119871119887119871 119905

]]]]]]]]

1198652 = [11986521 11986522]

11986521 =[[[[[[[[


]]]]]]]]

(A1)

11986522

=[[[[[[[[


]]]]]]]]

1198653 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

Γ11988611198621Γ11988711198621

Γ11988811198621Γ11988621198622Γ11988721198622

Γ11988821198622 Γ119886120593119862120593

Γ119887120593119862120593Γ119888120593119862120593


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

Γ119895119896 =119896sum119894=1

119878119895119894Λ 119895119896 =

119898sum119894=119896

119878119895119894119895 = 119886 119887 119888

119866 =

[[[[[[[[[[[[[[[[

119866119886119886 119866119886119887 119866119886119888 0119866119887119886 119866119887119887 119866119887119888 0119866119888119886 119866119888119887 119866119888119888 00 0 0 11986611986210 0 0 1198661198622 0 0 0 119866119862(119898minus1)

]]]]]]]]]]]]]]]]







(A2)

Data Availability




Acknowledgments


References








































Journal of












Journal of







Volume 2018






Volume 2018












0

20

40

60


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)




follows


]]]

(21)


]]]

(22)


119863 = [00] (24)


[[[[[[[[[[

11986811198861198681119887119868111988811988101198881

11988101198882

]]]]]]]]]]

=[[[[[[[[[

2 0 0 0 00 2 0 0 00 0 2 0 00 0 0 1 00 0 0 0 1

]]]]]]]]]

[[[[[[[[[

2276 minus 034j1109 + 1988j1167 minus 1954j

2920329203

]]]]]]]]]

=[[[[[[[[[[


2920329203

]]]]]]]]]]

(25)


1198881 and11988101198882 are




0

20

40

60


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)



0

20

40

60


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)



[[[[[[[[[[

11986810119868111198681211988101198881

11988101198882

]]]]]]]]]]

=[[[[[[[[[

04553 minus 068j

02920329203

]]]]]]]]]

(26)


119881119904119886 = 119881119904119890119895120596119905 + 005119881119904119890119895120596119905119881119904119887 = 119881119904119890119895120596119905minus120∘ + 005119881119904119890119895120596119905+120∘119881119904119888 = 119881119904119890119895120596119905+120∘ + 005119881119904119890119895120596119905minus120∘

(27)




119898119886 cos (120575) gt 055 (28)


0 0002 0004 0006 0008 001 0012 0014 0016 0018time (seconds)

120

140

160

180

200

220

240Vc

(vol

tage

)



(a)

0 0002 0004 0006 0008 001 0012 0014 0016 0018time (seconds)

120

140

160

180

200

220

240

Vc (v

olta

ge)



(b)


0

0

100

1

Har

mon

ic M

agni

tude

(V)

2

200

3 4

Harmonics Number

5 6 7 8 9


６1

６2

６3

６4




30

25

20

15

10

5

0

minus5

minus10

minus15

Iabc

(Am

pere

)

0 0002 0004 0006 0008 001 0012 0014 0016 0018







time (seconds)






0

2

4

6

8


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)



030

002

20 30

abs(

Out

putI

nput

)

10

004

200 10

006







119881119901119888119888 = 119881119904 minus 119885119904119868119904 (29)



[119868V119904119888119896119881119889119888119896] = [119860119896 119861119896119862119896 119863119896

][119881119901119888119888119868119889119888119896] (31)



[[[[[[[[[[

11986811990411988111988911988811198811198891198882119881119889119888119899

]]]]]]]]]]

= [119872]minus1 [119873][[[[[[[[[[

11988111990411986811988911988811198681198891198882119868119889119888119899

]]]]]]]]]]

(32)

where

119872 =[[[[[[[[[[


119862119899119885119904 0 0 sdot sdot sdot 119868

]]]]]]]]]]

119873 =[[[[[[[[[[


]]]]]]]]]]

(33)



0

20

40

60


） 1(A

)

(a)



0

5

10

） 2(A

)

(b)



0

05

1

15

2） 0

(A)

(c)


1

2

1

2

1

2

VSC1

VSC2

VSCn

1

2

1

2





+-

+-

+-

-+

+

+-

-

+-+-

1

2

1

2

1

2

3 33

VSC1

VSC2

VSC3

1

2

3

4

5

6

7

8



0

10

20

30

40


）６３

11

(A)

(a)



0

2

4

6

8

）６３

12

(A)

(b)



0

2

4

6

8


）６３

21

(A)

(a)



0

5

10

15

）６３

22

(A)

(b)



5 Conclusion





0

10

20

30

40


）６３

31

(A)

(a)



0

2

4

6

）６３

32

(A)

(b)



0

05

1

15

2）６

３30

(A)

(c)




0

20

40

60

）６３

31

(A)

(a)



0

2

4

6

8

）６３

32

(A)

(b)



0

05

1

15

2

）６３

30

(A)

(c)


Appendix



1198651 =[[[[[[[[

minus119877119886119871119886119886 + 119877119873119871119887119871119888119871 119905119877119887119871119888119871119873 minus 119877119873119871119887119871119888119871 119905


119877119887119871119886119871119873 minus 119877119873119871119886119871119887119871 119905 minus119877119888119871119888119888 + 119877119873119871119886119871119887119871 119905

]]]]]]]]

1198652 = [11986521 11986522]

11986521 =[[[[[[[[


]]]]]]]]

(A1)

11986522

=[[[[[[[[


]]]]]]]]

1198653 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

Γ11988611198621Γ11988711198621

Γ11988811198621Γ11988621198622Γ11988721198622

Γ11988821198622 Γ119886120593119862120593

Γ119887120593119862120593Γ119888120593119862120593


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

Γ119895119896 =119896sum119894=1

119878119895119894Λ 119895119896 =

119898sum119894=119896

119878119895119894119895 = 119886 119887 119888

119866 =

[[[[[[[[[[[[[[[[

119866119886119886 119866119886119887 119866119886119888 0119866119887119886 119866119887119887 119866119887119888 0119866119888119886 119866119888119887 119866119888119888 00 0 0 11986611986210 0 0 1198661198622 0 0 0 119866119862(119898minus1)

]]]]]]]]]]]]]]]]







(A2)

Data Availability




Acknowledgments


References








































Journal of












Journal of







Volume 2018






Volume 2018










0

20

40

60


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)



0

20

40

60


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)



[[[[[[[[[[

11986810119868111198681211988101198881

11988101198882

]]]]]]]]]]

=[[[[[[[[[

04553 minus 068j

02920329203

]]]]]]]]]

(26)


119881119904119886 = 119881119904119890119895120596119905 + 005119881119904119890119895120596119905119881119904119887 = 119881119904119890119895120596119905minus120∘ + 005119881119904119890119895120596119905+120∘119881119904119888 = 119881119904119890119895120596119905+120∘ + 005119881119904119890119895120596119905minus120∘

(27)




119898119886 cos (120575) gt 055 (28)


0 0002 0004 0006 0008 001 0012 0014 0016 0018time (seconds)

120

140

160

180

200

220

240Vc

(vol

tage

)



(a)

0 0002 0004 0006 0008 001 0012 0014 0016 0018time (seconds)

120

140

160

180

200

220

240

Vc (v

olta

ge)



(b)


0

0

100

1

Har

mon

ic M

agni

tude

(V)

2

200

3 4

Harmonics Number

5 6 7 8 9


６1

６2

６3

６4




30

25

20

15

10

5

0

minus5

minus10

minus15

Iabc

(Am

pere

)

0 0002 0004 0006 0008 001 0012 0014 0016 0018







time (seconds)






0

2

4

6

8


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)



030

002

20 30

abs(

Out

putI

nput

)

10

004

200 10

006







119881119901119888119888 = 119881119904 minus 119885119904119868119904 (29)



[119868V119904119888119896119881119889119888119896] = [119860119896 119861119896119862119896 119863119896

][119881119901119888119888119868119889119888119896] (31)



[[[[[[[[[[

11986811990411988111988911988811198811198891198882119881119889119888119899

]]]]]]]]]]

= [119872]minus1 [119873][[[[[[[[[[

11988111990411986811988911988811198681198891198882119868119889119888119899

]]]]]]]]]]

(32)

where

119872 =[[[[[[[[[[


119862119899119885119904 0 0 sdot sdot sdot 119868

]]]]]]]]]]

119873 =[[[[[[[[[[


]]]]]]]]]]

(33)



0

20

40

60


） 1(A

)

(a)



0

5

10

） 2(A

)

(b)



0

05

1

15

2） 0

(A)

(c)


1

2

1

2

1

2

VSC1

VSC2

VSCn

1

2

1

2





+-

+-

+-

-+

+

+-

-

+-+-

1

2

1

2

1

2

3 33

VSC1

VSC2

VSC3

1

2

3

4

5

6

7

8



0

10

20

30

40


）６３

11

(A)

(a)



0

2

4

6

8

）６３

12

(A)

(b)



0

2

4

6

8


）６３

21

(A)

(a)



0

5

10

15

）６３

22

(A)

(b)



5 Conclusion





0

10

20

30

40


）６３

31

(A)

(a)



0

2

4

6

）６３

32

(A)

(b)



0

05

1

15

2）６

３30

(A)

(c)




0

20

40

60

）６３

31

(A)

(a)



0

2

4

6

8

）６３

32

(A)

(b)



0

05

1

15

2

）６３

30

(A)

(c)


Appendix



1198651 =[[[[[[[[

minus119877119886119871119886119886 + 119877119873119871119887119871119888119871 119905119877119887119871119888119871119873 minus 119877119873119871119887119871119888119871 119905


119877119887119871119886119871119873 minus 119877119873119871119886119871119887119871 119905 minus119877119888119871119888119888 + 119877119873119871119886119871119887119871 119905

]]]]]]]]

1198652 = [11986521 11986522]

11986521 =[[[[[[[[


]]]]]]]]

(A1)

11986522

=[[[[[[[[


]]]]]]]]

1198653 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

Γ11988611198621Γ11988711198621

Γ11988811198621Γ11988621198622Γ11988721198622

Γ11988821198622 Γ119886120593119862120593

Γ119887120593119862120593Γ119888120593119862120593


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

Γ119895119896 =119896sum119894=1

119878119895119894Λ 119895119896 =

119898sum119894=119896

119878119895119894119895 = 119886 119887 119888

119866 =

[[[[[[[[[[[[[[[[

119866119886119886 119866119886119887 119866119886119888 0119866119887119886 119866119887119887 119866119887119888 0119866119888119886 119866119888119887 119866119888119888 00 0 0 11986611986210 0 0 1198661198622 0 0 0 119866119862(119898minus1)

]]]]]]]]]]]]]]]]







(A2)

Data Availability




Acknowledgments


References








































Journal of












Journal of







Volume 2018






Volume 2018









0 0002 0004 0006 0008 001 0012 0014 0016 0018time (seconds)

120

140

160

180

200

220

240Vc

(vol

tage

)



(a)

0 0002 0004 0006 0008 001 0012 0014 0016 0018time (seconds)

120

140

160

180

200

220

240

Vc (v

olta

ge)



(b)


0

0

100

1

Har

mon

ic M

agni

tude

(V)

2

200

3 4

Harmonics Number

5 6 7 8 9


６1

６2

６3

６4




30

25

20

15

10

5

0

minus5

minus10

minus15

Iabc

(Am

pere

)

0 0002 0004 0006 0008 001 0012 0014 0016 0018







time (seconds)






0

2

4

6

8


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)



030

002

20 30

abs(

Out

putI

nput

)

10

004

200 10

006







119881119901119888119888 = 119881119904 minus 119885119904119868119904 (29)



[119868V119904119888119896119881119889119888119896] = [119860119896 119861119896119862119896 119863119896

][119881119901119888119888119868119889119888119896] (31)



[[[[[[[[[[

11986811990411988111988911988811198811198891198882119881119889119888119899

]]]]]]]]]]

= [119872]minus1 [119873][[[[[[[[[[

11988111990411986811988911988811198681198891198882119868119889119888119899

]]]]]]]]]]

(32)

where

119872 =[[[[[[[[[[


119862119899119885119904 0 0 sdot sdot sdot 119868

]]]]]]]]]]

119873 =[[[[[[[[[[


]]]]]]]]]]

(33)



0

20

40

60


） 1(A

)

(a)



0

5

10

） 2(A

)

(b)



0

05

1

15

2） 0

(A)

(c)


1

2

1

2

1

2

VSC1

VSC2

VSCn

1

2

1

2





+-

+-

+-

-+

+

+-

-

+-+-

1

2

1

2

1

2

3 33

VSC1

VSC2

VSC3

1

2

3

4

5

6

7

8



0

10

20

30

40


）６３

11

(A)

(a)



0

2

4

6

8

）６３

12

(A)

(b)



0

2

4

6

8


）６３

21

(A)

(a)



0

5

10

15

）６３

22

(A)

(b)



5 Conclusion





0

10

20

30

40


）６３

31

(A)

(a)



0

2

4

6

）６３

32

(A)

(b)



0

05

1

15

2）６

３30

(A)

(c)




0

20

40

60

）６３

31

(A)

(a)



0

2

4

6

8

）６３

32

(A)

(b)



0

05

1

15

2

）６３

30

(A)

(c)


Appendix



1198651 =[[[[[[[[

minus119877119886119871119886119886 + 119877119873119871119887119871119888119871 119905119877119887119871119888119871119873 minus 119877119873119871119887119871119888119871 119905


119877119887119871119886119871119873 minus 119877119873119871119886119871119887119871 119905 minus119877119888119871119888119888 + 119877119873119871119886119871119887119871 119905

]]]]]]]]

1198652 = [11986521 11986522]

11986521 =[[[[[[[[


]]]]]]]]

(A1)

11986522

=[[[[[[[[


]]]]]]]]

1198653 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

Γ11988611198621Γ11988711198621

Γ11988811198621Γ11988621198622Γ11988721198622

Γ11988821198622 Γ119886120593119862120593

Γ119887120593119862120593Γ119888120593119862120593


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

Γ119895119896 =119896sum119894=1

119878119895119894Λ 119895119896 =

119898sum119894=119896

119878119895119894119895 = 119886 119887 119888

119866 =

[[[[[[[[[[[[[[[[

119866119886119886 119866119886119887 119866119886119888 0119866119887119886 119866119887119887 119866119887119888 0119866119888119886 119866119888119887 119866119888119888 00 0 0 11986611986210 0 0 1198661198622 0 0 0 119866119862(119898minus1)

]]]]]]]]]]]]]]]]







(A2)

Data Availability




Acknowledgments


References








































Journal of












Journal of







Volume 2018






Volume 2018










0

2

4

6

8


） 1(A

)

(a)


0

5

10


） 2(A

)

(b)



030

002

20 30

abs(

Out

putI

nput

)

10

004

200 10

006







119881119901119888119888 = 119881119904 minus 119885119904119868119904 (29)



[119868V119904119888119896119881119889119888119896] = [119860119896 119861119896119862119896 119863119896

][119881119901119888119888119868119889119888119896] (31)



[[[[[[[[[[

11986811990411988111988911988811198811198891198882119881119889119888119899

]]]]]]]]]]

= [119872]minus1 [119873][[[[[[[[[[

11988111990411986811988911988811198681198891198882119868119889119888119899

]]]]]]]]]]

(32)

where

119872 =[[[[[[[[[[


119862119899119885119904 0 0 sdot sdot sdot 119868

]]]]]]]]]]

119873 =[[[[[[[[[[


]]]]]]]]]]

(33)



0

20

40

60


） 1(A

)

(a)



0

5

10

） 2(A

)

(b)



0

05

1

15

2） 0

(A)

(c)


1

2

1

2

1

2

VSC1

VSC2

VSCn

1

2

1

2





+-

+-

+-

-+

+

+-

-

+-+-

1

2

1

2

1

2

3 33

VSC1

VSC2

VSC3

1

2

3

4

5

6

7

8



0

10

20

30

40


）６３

11

(A)

(a)



0

2

4

6

8

）６３

12

(A)

(b)



0

2

4

6

8


）６３

21

(A)

(a)



0

5

10

15

）６３

22

(A)

(b)



5 Conclusion





0

10

20

30

40


）６３

31

(A)

(a)



0

2

4

6

）６３

32

(A)

(b)



0

05

1

15

2）６

３30

(A)

(c)




0

20

40

60

）６３

31

(A)

(a)



0

2

4

6

8

）６３

32

(A)

(b)



0

05

1

15

2

）６３

30

(A)

(c)


Appendix



1198651 =[[[[[[[[

minus119877119886119871119886119886 + 119877119873119871119887119871119888119871 119905119877119887119871119888119871119873 minus 119877119873119871119887119871119888119871 119905


119877119887119871119886119871119873 minus 119877119873119871119886119871119887119871 119905 minus119877119888119871119888119888 + 119877119873119871119886119871119887119871 119905

]]]]]]]]

1198652 = [11986521 11986522]

11986521 =[[[[[[[[


]]]]]]]]

(A1)

11986522

=[[[[[[[[


]]]]]]]]

1198653 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

Γ11988611198621Γ11988711198621

Γ11988811198621Γ11988621198622Γ11988721198622

Γ11988821198622 Γ119886120593119862120593

Γ119887120593119862120593Γ119888120593119862120593


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

Γ119895119896 =119896sum119894=1

119878119895119894Λ 119895119896 =

119898sum119894=119896

119878119895119894119895 = 119886 119887 119888

119866 =

[[[[[[[[[[[[[[[[

119866119886119886 119866119886119887 119866119886119888 0119866119887119886 119866119887119887 119866119887119888 0119866119888119886 119866119888119887 119866119888119888 00 0 0 11986611986210 0 0 1198661198622 0 0 0 119866119862(119898minus1)

]]]]]]]]]]]]]]]]







(A2)

Data Availability




Acknowledgments


References








































Journal of












Journal of







Volume 2018






Volume 2018










0

20

40

60


） 1(A

)

(a)



0

5

10

） 2(A

)

(b)



0

05

1

15

2） 0

(A)

(c)


1

2

1

2

1

2

VSC1

VSC2

VSCn

1

2

1

2





+-

+-

+-

-+

+

+-

-

+-+-

1

2

1

2

1

2

3 33

VSC1

VSC2

VSC3

1

2

3

4

5

6

7

8



0

10

20

30

40


）６３

11

(A)

(a)



0

2

4

6

8

）６３

12

(A)

(b)



0

2

4

6

8


）６３

21

(A)

(a)



0

5

10

15

）６３

22

(A)

(b)



5 Conclusion





0

10

20

30

40


）６３

31

(A)

(a)



0

2

4

6

）６３

32

(A)

(b)



0

05

1

15

2）６

３30

(A)

(c)




0

20

40

60

）６３

31

(A)

(a)



0

2

4

6

8

）６３

32

(A)

(b)



0

05

1

15

2

）６３

30

(A)

(c)


Appendix



1198651 =[[[[[[[[

minus119877119886119871119886119886 + 119877119873119871119887119871119888119871 119905119877119887119871119888119871119873 minus 119877119873119871119887119871119888119871 119905


119877119887119871119886119871119873 minus 119877119873119871119886119871119887119871 119905 minus119877119888119871119888119888 + 119877119873119871119886119871119887119871 119905

]]]]]]]]

1198652 = [11986521 11986522]

11986521 =[[[[[[[[


]]]]]]]]

(A1)

11986522

=[[[[[[[[


]]]]]]]]

1198653 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

Γ11988611198621Γ11988711198621

Γ11988811198621Γ11988621198622Γ11988721198622

Γ11988821198622 Γ119886120593119862120593

Γ119887120593119862120593Γ119888120593119862120593


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

Γ119895119896 =119896sum119894=1

119878119895119894Λ 119895119896 =

119898sum119894=119896

119878119895119894119895 = 119886 119887 119888

119866 =

[[[[[[[[[[[[[[[[

119866119886119886 119866119886119887 119866119886119888 0119866119887119886 119866119887119887 119866119887119888 0119866119888119886 119866119888119887 119866119888119888 00 0 0 11986611986210 0 0 1198661198622 0 0 0 119866119862(119898minus1)

]]]]]]]]]]]]]]]]







(A2)

Data Availability




Acknowledgments


References








































Journal of












Journal of







Volume 2018






Volume 2018









+-

+-

+-

-+

+

+-

-

+-+-

1

2

1

2

1

2

3 33

VSC1

VSC2

VSC3

1

2

3

4

5

6

7

8



0

10

20

30

40


）６３

11

(A)

(a)



0

2

4

6

8

）６３

12

(A)

(b)



0

2

4

6

8


）６３

21

(A)

(a)



0

5

10

15

）６３

22

(A)

(b)



5 Conclusion





0

10

20

30

40


）６３

31

(A)

(a)



0

2

4

6

）６３

32

(A)

(b)



0

05

1

15

2）６

３30

(A)

(c)




0

20

40

60

）６３

31

(A)

(a)



0

2

4

6

8

）６３

32

(A)

(b)



0

05

1

15

2

）６３

30

(A)

(c)


Appendix



1198651 =[[[[[[[[

minus119877119886119871119886119886 + 119877119873119871119887119871119888119871 119905119877119887119871119888119871119873 minus 119877119873119871119887119871119888119871 119905


119877119887119871119886119871119873 minus 119877119873119871119886119871119887119871 119905 minus119877119888119871119888119888 + 119877119873119871119886119871119887119871 119905

]]]]]]]]

1198652 = [11986521 11986522]

11986521 =[[[[[[[[


]]]]]]]]

(A1)

11986522

=[[[[[[[[


]]]]]]]]

1198653 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

Γ11988611198621Γ11988711198621

Γ11988811198621Γ11988621198622Γ11988721198622

Γ11988821198622 Γ119886120593119862120593

Γ119887120593119862120593Γ119888120593119862120593


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

Γ119895119896 =119896sum119894=1

119878119895119894Λ 119895119896 =

119898sum119894=119896

119878119895119894119895 = 119886 119887 119888

119866 =

[[[[[[[[[[[[[[[[

119866119886119886 119866119886119887 119866119886119888 0119866119887119886 119866119887119887 119866119887119888 0119866119888119886 119866119888119887 119866119888119888 00 0 0 11986611986210 0 0 1198661198622 0 0 0 119866119862(119898minus1)

]]]]]]]]]]]]]]]]







(A2)

Data Availability




Acknowledgments


References








































Journal of












Journal of







Volume 2018






Volume 2018










0

10

20

30

40


）６３

31

(A)

(a)



0

2

4

6

）６３

32

(A)

(b)



0

05

1

15

2）６

３30

(A)

(c)




0

20

40

60

）６３

31

(A)

(a)



0

2

4

6

8

）６３

32

(A)

(b)



0

05

1

15

2

）６３

30

(A)

(c)


Appendix



1198651 =[[[[[[[[

minus119877119886119871119886119886 + 119877119873119871119887119871119888119871 119905119877119887119871119888119871119873 minus 119877119873119871119887119871119888119871 119905


119877119887119871119886119871119873 minus 119877119873119871119886119871119887119871 119905 minus119877119888119871119888119888 + 119877119873119871119886119871119887119871 119905

]]]]]]]]

1198652 = [11986521 11986522]

11986521 =[[[[[[[[


]]]]]]]]

(A1)

11986522

=[[[[[[[[


]]]]]]]]

1198653 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

Γ11988611198621Γ11988711198621

Γ11988811198621Γ11988621198622Γ11988721198622

Γ11988821198622 Γ119886120593119862120593

Γ119887120593119862120593Γ119888120593119862120593


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

Γ119895119896 =119896sum119894=1

119878119895119894Λ 119895119896 =

119898sum119894=119896

119878119895119894119895 = 119886 119887 119888

119866 =

[[[[[[[[[[[[[[[[

119866119886119886 119866119886119887 119866119886119888 0119866119887119886 119866119887119887 119866119887119888 0119866119888119886 119866119888119887 119866119888119888 00 0 0 11986611986210 0 0 1198661198622 0 0 0 119866119862(119898minus1)

]]]]]]]]]]]]]]]]







(A2)

Data Availability




Acknowledgments


References








































Journal of












Journal of







Volume 2018






Volume 2018









1198651 =[[[[[[[[

minus119877119886119871119886119886 + 119877119873119871119887119871119888119871 119905119877119887119871119888119871119873 minus 119877119873119871119887119871119888119871 119905


119877119887119871119886119871119873 minus 119877119873119871119886119871119887119871 119905 minus119877119888119871119888119888 + 119877119873119871119886119871119887119871 119905

]]]]]]]]

1198652 = [11986521 11986522]

11986521 =[[[[[[[[


]]]]]]]]

(A1)

11986522

=[[[[[[[[


]]]]]]]]

1198653 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

Γ11988611198621Γ11988711198621

Γ11988811198621Γ11988621198622Γ11988721198622

Γ11988821198622 Γ119886120593119862120593

Γ119887120593119862120593Γ119888120593119862120593


]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

Γ119895119896 =119896sum119894=1

119878119895119894Λ 119895119896 =

119898sum119894=119896

119878119895119894119895 = 119886 119887 119888

119866 =

[[[[[[[[[[[[[[[[

119866119886119886 119866119886119887 119866119886119888 0119866119887119886 119866119887119887 119866119887119888 0119866119888119886 119866119888119887 119866119888119888 00 0 0 11986611986210 0 0 1198661198622 0 0 0 119866119862(119898minus1)

]]]]]]]]]]]]]]]]







(A2)

Data Availability




Acknowledgments


References








































Journal of












Journal of







Volume 2018






Volume 2018













(A2)

Data Availability




Acknowledgments


References








































Journal of












Journal of







Volume 2018






Volume 2018







































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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