selective harmonic elimination pulse-width modulation seven-level cascaded h-bridge converter with...

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Published in IET Power ElectronicsReceived on 21st November 2011Revised on 6th March 2012doi: 10.1049/iet-pel.2011.0463

ISSN 1755-4535

Selective harmonic elimination pulse-widthmodulation seven-level cascaded H-bridge converterwith optimised DC voltage levelsM.S.A. Dahidah1 G.S. Konstantinou2 V.G. Agelidis2

1Department of Electrical and Electronic Engineering, The University of Nottingham, Malaysia Campus, Jalan Broga 43500,Selangor, Malaysia2School of Electrical Engineering and Telecommunications, The University of New South Wales, Sydney, NSW, 2052,AustraliaE-mail: [email protected]

Abstract: This study presents a novel formulation of multilevel selective harmonic elimination pulse width modulation (MSHE-PWM) technique with optimised DC voltage levels suitable for high-power voltage source converter-based applications. Thestudy reformulates the problem in a way in which the levels of the DC voltage sources are made variables within certainconstraints. Therefore the degrees of freedom for specifying the cost function are increased when compared to the existingfamily of selective harmonic elimination-based multilevel control for the same physical structure. Moreover, with the proposedapproach, the solution of the switching angles can be sought for the entire range of the modulation index without affecting thenumber of harmonics being eliminated or the number of the output voltage levels. The effectiveness of the proposed method isinvestigated with various waveforms. The theoretical and simulation findings are validated through experimental results.

1 Introduction

Multilevel converters have drawn tremendous interest inrecent years and have been studied for several high-voltageand high-power applications [1–3]. They have also beenused for medium-voltage AC motor drives, as they offerlower semi-conductor stress, reduced harmonic distortionand improved electromagnetic interference. Applications ofmultilevel converters include motor drives such as marinepropulsion, ‘more-electric’ aircraft and traction (locomotive,electric vehicles, hybrid electric vehicles) and gridapplications such as high-voltage direct current transmissionsystem and static synchronous compensator [1].

Several modulation and control techniques have beenreported in the literature including carrier-based pulse widthmodulation (PWM) [3], multilevel space vector modulation[3], stair case modulation and selective harmonicelimination PWM (SHE-PWM) [4]. Since multilevelconverter usually operated with low switching frequency,SHE-PWM offers several advantages over the othermethods such as low switching frequency with a widerconverter’s bandwidth, direct control over low-orderharmonics and better DC source utilisation [4–29]. Themain challenge associated with SHE-PWM techniques is toobtain the analytical solution of the system of non-lineartranscendental equations that contain trigonometric termswhich in turn provide multiple sets of solutions [5, 6].Several algorithms have been reported in the open literaturedescribing methods of solving the resultant non-linear

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transcendental equations such as iterative-based approaches[7], Walsh functions [8] and optimisation-based methods [5,6, 9, 12, 15–20, 27–29].

On the other hand, SHE-PWM methods have also beenextended to multilevel converters in several technicalarticles [4, 10–23, 25–29]. Initially, the switchingfrequency was restricted to line frequency and therefore, thestaircase multilevel waveform was arranged in such a wayas to control the fundamental and eliminate the low-orderharmonic from the waveform [4, 10]. However, suchwaveforms suffer from narrow modulation index range andsmaller number of harmonics that can be eliminated as theyare linked to the number of voltage levels, hence thephysical structure of the converter.

An optimisation method based on general geneticalgorithms has been proposed to solve the same problem ofSHE-PWM [15]. However, the paper shows only thesolutions to the equal DC sources case. Furthermore, hybridgenetic algorithm was applied to multilevel converters withequal and non-equal DC sources in [16–18].

Other approaches have also been reported including onewhere multilevel SHE-PWM is defined by the well-knownmulticarrier phase-shifted PWM [19] and another was basedon single-carrier sinusoidal PWM-equivalent SHE for afive-level waveform [20]. In these methods the modulationindex defines the distribution of the switching angles andthen the problem of SHE-PWM is applied to a particularoperating point aiming to obtain the optimum position ofthese switching transitions that offer elimination to a

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Fig. 1 The proposed multilevel voltage source converter

a Three-phase cascaded multilevel converterb Generalised multilevel SHE-PWM waveform (quarter-cycle shown)

selected order of harmonics. A generalised formulation formultilevel SHE-PWM converters for any number of levelsand any number of switching angles with both equal andnon-equal DC voltage levels was also reported in [18].Recently, generalisations of quarter- and half-wave symmetrySHE-PWM method for multilevel waveform were reportedin [21–23], respectively. Particle swarm optimisationtechnique was adopted for SHE multilevel waveformsfor both fundamental switching output waveforms withequal [24] and non-equal DC sources [25] and PWMmultilevel waveform [26]. However, multiple solution setswere not presented and the solution of the switchingangles was not continuous. Furthermore, the solution for theovermodulation region was not sought. A more generalisedformulation with no symmetry requirements in thewaveform was reported in [17, 27] where the switchingangles are relaxed between 0 and 2p. A new variation ofthe problem was recently reported in [28, 29] where theDC voltage levels were made variables increasing thenumber of freedoms; hence the number of harmonics tobe eliminated without changing the physical structure ofthe converter.


The main objective of this paper is to reformulate theproblem of multilevel SHE-PWM assuming variable levelsof the DC voltage sources [28]. Therefore the degrees offreedom are increased when compared with other multilevelSHE_PWM converters with fixed DC sources. Furthermore,the proposed method enables the realisation of themultilevel output waveform with a lower total harmonicdistortion (THD) for low modulation index withoutaffecting the number of eliminated harmonics. Theproposed method possesses interesting features of constantswitching angles over the modulation index range andlinear pattern of DC voltage levels against the modulationindex range which eliminate the need of large lookup tablesand the tedious steps of the off-line calculation of theswitching angles. This makes the proposed method apromising approach for grid-connected and FACTSapplications where usually dynamic operation of theconverter is required.

Moreover, the paper extends the work of Dahidah et al. [28]to various seven-level waveforms. Several solution sets are alsoacquired and different scenarios for the DC voltage levels arealso analysed. Furthermore, two quality indices, that is,

Table 1 Summarised solution sets for seven-level SHE-PWM waveform (all-variable) with N ¼ 11 and different distribution ratios

N ¼ 3/5/3 N ¼ 3/3/5 N ¼ 1/5/5 N ¼ 1/3/7

number of solution sets 6 4 5 3

modulation index range set 1: 1.91 , mi , 2.14 set 1: 2.16 , mi , 2.47 set 1: 2.48 , mi , 2.58 set 1: 2.55 , mi , 2.75

set 2: 1.91 , mi , 2.47 set 2: 2.16 , mi , 2.72 set 2: 2.43 , mi , 2.57 set 2: 2.55 , mi , 3.00

set 3: 1.91 , mi , 2.16 set 3: 2.16 , mi , 2.62 set 3: 2.42 , mi , 2.58 set 3: 2.55 , mi , 2.70

set 4: 1.91 , mi , 2.11 set 4: 2.16 , mi , 2.43 set 4: 2.56 , mi , 2.76

set 5: 2.05 , mi , 2.66 set 5: 2.55 , mi , 2.77

set 6: 2.04 , mi , 2.18

number of eliminated

non-triplen harmonics

13

first non-eliminated harmonic 43rd

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Fig. 2 Switching angle and DC voltage levels solutions for seven-level waveform with N ¼ 1/1/9 (all variables)

a Switching angles solution, set 1 (1.91 , mi , 2.33)b DC voltage levels solution, set 1 (1.91 , mi , 2.33)c Switching angles solution, set 2 (1.90 , mi , 2.55)d DC voltage levels solution, set 2 (1.90 , mi , 2.55)e Switching angles solution, set 3 (1.91 , mi , 2.84)f DC voltage levels solution, set 3 (1.91 , mi , 2.84)g Switching angles solution, set 4 (1.273 , mi , 2.724)h DC voltage levels solution, set 4 (1.273 , mi , 2.724)

854 IET Power Electron., 2012, Vol. 5, Iss. 6, pp. 852–862

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Fig. 3 Switching angle and DC voltage levels solutions for seven-level waveform with N ¼ 1/1/9 (top level variable)

a Switching angles solution (2.74 , mi , 2.89)b DC voltage levels solution (2.74 , mi , 2.89)

harmonic distortion factor (HDF) and THD are also consideredto evaluate the performance of the different formulations.

The paper is organised as follows. Section 2 presents theanalysis and formulation of the generalised case. Solutionsets for different seven-level SHE-PWM waveforms arereported in Section 3. Selected simulation results arediscussed in Section 4. Experimental verification ispresented in Section 5. Finally conclusions are summarisedin Section 7.

2 Waveform analysis and problemformulation

Fig. 1 shows three-phase multilevel voltage source converterand its associated generalised SHE-PWM output voltagewaveform. Let K be the number of cascaded H-bridge cells(i.e. number of isolated DC sources with variant voltagelevels) and N be the total number of switching angles per-quarter period of the multilevel output voltage waveform,where N1, N2, . . . , NK are the number of switching anglesfor each voltage level of the waveform within a quarter-period of the waveform. Furthermore, to formulate themulti-level waveform with dv/dt being limited to onevoltage level step, N1, N2, . . . , NK21 must be odd numbers.However, there is no restriction to the number of switchingangles in the top level; hence N can be either odd or even.

The typical half-wave and quarter-wave symmetries arerespected for the waveform, that is, simply when theswitching angles for all modulation indices are obtained forthe angles between zero and p/2, the usual reflection occursto find the rest of the angles.

The generalised set of equations for any number ofswitching angles that describe the fundamental componentand the higher order harmonics for a symmetrically definedK-level SHE-PWM waveform with variable DC voltagelevels is given as follows [12, 18–20]

M = 4

np(VDC1

∑N1

i=1

(−1)i+1 cosai + VDC2

∑N2

i=N1+1

(−1)i cosai

+ · · · + VDCK

∑N

i=NM−1+1

(−1)i cosai) (1)


0 = 4

np(VDC1

∑N1

i=1

(−1)i+1 cosnai +VDC2

∑N2

i=N1+1

(−1)i cosnai

+ ·· ·+VDCK

∑N

i=NM−1+1

(−1)i cosnai) (2)

where (1) describes the fundamental component and (2) thehigher order harmonics.

Where n ¼ 1, 5, . . . , 3(N + K ) 2 1, for (N + K ) is an oddn ¼ 1, 5, . . . , 3(N + K ) 2 2, for (N + K ) is an even. N1, N2,. . ., NK are the number of pulses per-quarter cycle at H-bridgecell 1, 2, . . . , K, respectively (i.e. N ¼ N1 + N2 +L+ NK)and ai is the ith switching angle of the multilevel waveformshown in Fig. 1b.

Equations (2) and (3) possess N + K unknown variables(i.e. a1, a2, . . . , aN and VDC1,VDC2, . . . , VDCM) and a setof solutions is obtainable by equating (N + K ) 2 1harmonics to zero and assigning a specific value for thefundamental component.

b1 and bn [i.e. n ¼ 1, 5, . . . , 3(N + K ) 2 1] represent thenormalised fundamental and low-order harmonicscomponents, respectively. Hence, an objective functiondescribing a measure of effective elimination of selectedorder of harmonics while controlling the fundamental isdefined by

F(a1, a2, . . . , aN , VDC1, VDC2, . . . , VDCM )

= (b1 − M )2 + b25 + b2

7 + · · · + b2n (3)

where

M = pmi

4(4)

and mi is the modulation index (0 . mi . 3, for the seven-level waveform).

The optimal switching angles are obtained by minimising(3) when it is subject to the constraints of (5) and (6)

0 , a1 , a2 , · · · , aN ,p

2

( )(5)

Vmin ≤ VDCj ≤ Vmax (6)

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Fig. 4 Switching angle and DC voltage levels solutions for seven-level waveform with N ¼ 1/1/9 (2 top levels variable)

a Switching angles solution, set 1 (2.22 , mi , 2.57)b DC voltage levels solution, set 1 (2.22 , mi , 2.57)c Switching angles solution, set 2 (1.96 , mi , 2.19)d DC voltage levels solution, set 2 (1.90 , mi , 2.55)e Switching angles solution, set 3 (2.05 , mi , 2.71)f DC voltage levels solution, set 3 (2.05 , mi , 2.71)

where Vmin and Vmax are the minimum and maximum valuesof the DC voltage source and VDCj is the jth DC voltage level( j ¼ 1, 2, . . . , K ).

The constraints are imposed on the formulation to ensurethat the resultant waveform is realisable and physicallycorrect. Specifically (6) assures the optimal DC level iswithin the range. In this work, the DC sources are assumedto have nominal values of 0.1 and 1.0 p.u. as minimum andmaximum values, respectively.

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3 Multilevel SHE-PWM waveform: analysisand solutions

To demonstrate the feasibility of the proposed formulationvarious waveforms are considered and studied with adifferent combination of switching instants distribution andDC voltage levels. The mathematical model (i.e. costfunction) of the proposed multilevel SHE-PWM techniquewith variant DC sources and for N switching angles per


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quarter period is first developed and then the optimisationtechnique presented in [6, 12] is employed to find thesolutions to the non-linear equation system defined by (3).

A seven-level waveform with N ¼ 11 angles per quarterperiod is considered in this paper for various distributionratios of the switching instants as well as different DCvoltage level combinations. Table 1 summarises thesolution sets for different switching angle distributions. Anexample of seven-level waveform with N1/N2/N3 ¼ 1/1/9switching angles is chosen and analysed. Initially, all thethree DC sources are considered variables within theconstraint imposed by (6). A large pool of solution sets isobtained with different ranges of the modulation index andthe selected sets are presented in Fig. 2. It is found that thesolutions for both switching angles and DC voltage levelsmaintain the merits of the linear relationship of DC voltagelevels as well as fixed switching angles across the entirerange of the modulation index.

For justifiable comparison, various arrangements for thesame waveform are also investigated, where it was firstreconsidered with fixing two DC voltage sources at 1 p.u.value and leaving the third one (i.e. top level) variant.Similarly, various solution sets are calculated for this case;however, it is found that neither the switching angles arefixed nor the DC voltage levels are perfectly linearcompared with the previous case. This can be observedfrom the sample solution set illustrated in Fig. 3. Secondly,another case when only one of the DC sources is kept


constant and the other two left variant was alsoinvestigated. Selected sets of solutions are shown in Fig. 4,where it is once again noted that the linearity of thesolution has deteriorated. This has also been confirmed withanother seven-level waveform where N1/N2/N3 ¼ 3/3/5 andthe sample solution sets are illustrated in Fig. 5.

4 Simulation results

The feasibility of the proposed approach is first investigatedwith extensive simulation studies using PSIM softwarepackage. An operating point for each case is randomlyselected and reported. Fig. 6 shows the implementation ofthe seven-level waveform with variable DC levels. Thephase voltage of the converter, normalised to the maximumvalue of each voltage level for a distribution of 1/1/9 withall levels variable and its associated harmonic spectrum ispresented in Fig. 6a. With this arrangement, 13 low-ordernon-triplen harmonics are eliminated while maintaining thefundamental at a pre-defined value. The spectrum of thewaveform illustrates that the next significant non-triplenharmonic to appear in the line-to-line output voltage is the43rd. The other two cases of the same seven-levelwaveform with only one and two DC voltage levels aremade variables and are also simulated and presented inFigs. 6b and c, respectively. However, the number of low-order harmonics to be eliminated is reduced as comparedto the former case. For instance, the next significant

Fig. 5 Switching angle and DC voltage levels solutions for seven-level waveform with N ¼ 3/3/5 (two top levels variable)

a Switching angles solution, set 1 (1.91 , mi , 2.51)b DC voltage levels solution, set 1 (1.91 , mi , 2.51)c Switching angles solution, set 2 (2.13 , mi , 2.92)d DC voltage levels solution, set 2 (2.13 , mi , 2.92)

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non-triplen harmonic appearing in the spectrum of the line-to-line output voltage when only the top DC voltage level wasmade variable is the 37th.

Furthermore, Fig. 7 illustrates another example of theseven-level waveform with N1/N2/N3 ¼ 3/3/5 and

mi ¼ 2.92. In this case, the two top DC voltage levels arevariant, which allow only 11 harmonics to be eliminatedwhile controlling the fundamental at a pre-definedvalue. For this waveform, the next non-triplen harmonic isthe 41st.

Fig. 6 Implementation of line-to-neutral seven-level waveform and its associated spectra with N ¼ 1/1/9

a All-variables and mi ¼ 1.273b One-variable and mi ¼ 2.89c Two-variable and mi ¼ 2.5

Fig. 7 Implementation of line-to-neutral seven-level waveform and its associated spectra with N ¼ 3/3/5 (two-variable) and mi ¼ 2.92

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5 Performance indices

The seven-level waveform example with N ¼ 1/1/9 is chosento evaluate the performance of different formulations andcompare the different configurations of the waveform for


variable and constant voltage levels. Additional aspects inthe comparison of the solutions include the range of themodulation index, number of solution sets, the number ofeliminated harmonics as summarised in Table 2 and the twoquality indices, previously defined by (7) and (8).

Table 2 Seven-level SHE-PWM waveform with different formulation and N ¼ 1/1/9: a comparison

All-constant DC

voltage levels

One-variable DC

voltage level

Two-variable DC

voltage levels

All-variable DC

voltage levels

number of solution sets 4 1 5 6

modulation index range set 1: 3.16 , mi , 3.39 set 1: 2.74 , mi , 2.89 set 1: 2.22 , mi , 2.57 set 1: 1.90 , mi , 2.33




set 5: 1.96 , mi , 2.19 set 5: 1.27 , mi , 2.25

set 6: 1.27 , mi , 2.12

number of eliminated

non-triplen harmonics

10 11 12 13

first non-eliminated harmonic 35th 37th 41st 43rd

Fig. 8 Performance indices evaluation

a THD versus Modulation indexb HDF versus Modulation index

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As can be seen from Table 2, with the proposed method,more harmonics can be eliminated and the approachpossesses more continuous solutions with a wider region ofmodulation index. Assuming variable DC-voltage levelsmake the solutions obtainable in a wider range ofmodulation index. On the other hand, when the DC voltagelevels are kept equally constant at 1 p.u. the solution will belimited to a narrower range within the high modulationindex region. This explains the gap seen in the solutionspace of Fig. 8; however, this can be improved oreliminated by redistributing the switching angles over thevoltage levels of the waveform. The THD of the line-to-linevoltage is calculated and depicted in Fig. 8a using (7) andup to 99th order of harmonics is taken into account for allcases. The proposed method where all the DC voltagelevels were variables shows a lower value of THDcompared to the conventional one and that is because itoffers more harmonics to be eliminated. Owing to the linearsolution patterns the THD remains constant over the entiremodulation index range. On the other side, the case whenonly one DC voltage level is variable shows a better THD

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for the selected solution set, however it has relativelynarrower modulation index range.

THD =

��∑99n=2 V 2

n

√

V 1

(p.u.) (7)

A simple HDF is another performance factor considered inthis investigation and it is illustrated in Fig. 8b using (8).The same solution sets are chosen and the graphs are drawnbased on the consideration of the two most significantharmonics appearing in the output voltage waveform (i.e.43rd and 47th for all-variable levels, 41st and 43rd for two-variable levels, 37th and 41st for one-variable level and35th and 37th for all-constant levels). It can be noted thatthe proposed method still exhibits a better harmonic profilecompared to constant and equal level formulation

HDF =��V 2

h1 + V 2h2

√V 1

(p.u.) (8)

Fig. 9 Experimental implementation of line-to-neutral seven-level waveform and its associated spectra with N ¼ 1/1/9

a All-variables and mi ¼ 1.273b One-variable and mi ¼ 2.89c Two-variable and mi ¼ 2.5


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Fig. 10 Experimental implementation of line-to-neutral seven-level waveform and its associated spectra with N ¼ 3/3/5 (two-variable) andmi ¼ 2.92

where Vh1 and Vh2 are the first and second non-triplen, non-eliminated harmonics in the output voltage harmonicspectrum.

6 Experimental validation

A seven-level single-phase inverter laboratory prototypebased on three cascaded H-bridge topology has beendeveloped in order to validate the theoretical considerationsand simulation results. The inverter is operated with afundamental frequency of 50 Hz and is connected to aresistive load. 12 insulated gate bipolar transistor (IGBT)switches with internal anti-parallel diodes (IRG4BC20FD)are used to construct the one-leg inverter circuit. Six high-voltage high-speed IGBT drivers (IR2112) were used toprovide proper and conditioned gate signals to the powerelectronic switches. The pre-calculated PWM signals areimplemented using low-cost high-speed Texas InstrumentsTMS320F2812 DSP board.

The simulated results of Section 4 are experimentallyverified for the same operating points and the results arepresented in Figs. 9–10. The exact values of the switchingangles and DC voltage levels used in the experiments aresummarised in Table 3 in the Appendix of this paper.Specifically, Fig. 9a shows the line-to-neutral outputvoltage of seven-level waveform with N ¼ 1/1/9 andall-variable case. As the spectrum shows, only triplenharmonics appeared within the selected bandwidth of theconverter. Other cases for the same waveform when onlythe top voltage level and the two top levels made variableare also experimentally validated as shown in Figs. 9band c, respectively. Once again, the spectra clearly showthat only triplen (uncontrolled) harmonics are present withinthe band width. Finally, the second seven-level waveformwith N ¼ 3/3/5 and the top two voltage levels are variables(i.e. V1 ¼ 1 p.u., V2 ¼ 0.896277 p.u. and V3 ¼ 0.62317 p.u.)is illustrated in Fig. 10.

7 Conclusion

A new variation of SHE-PWM for cascaded multilevelconverters based on the optimisation on the individual


voltage levels is proposed in this paper. The methodprovides an increase in the number of harmonics eliminatedfrom the output spectrum while maintaining a linear relationfor the voltage levels and constant switching pattern overthe range of modulation indices. Multiple solutions fordifferent seven-level waveforms are calculated andpresented in this paper. The effectiveness of the proposedapproach is verified through simulation and experimentallyvalidated results using a laboratory prototype. Additionalformulation considers a combination of variable andconstant voltage levels in order to reduce the complexity ofthe system. It is found that optimising the DC voltage levelsimproves the harmonic performance of the outputwaveforms and the complexity of the modulation scheme atthe cost of reduced dynamics because of the slower DCvoltage regulation in the system. Although, only seven-levelwaveform was presented in this paper, however the methodcan be equally applied for any number of output voltagelevels provided that the solution can be obtained.

† The method is extended to seven-level waveform withdifferent formulation and operating points. Several solutionsets were acquired for a number of waveforms and differentscenarios for the dc voltage levels were also analysed. Acomparison based on different aspects was reported for theseven-level case.† Two quality indices that is, HDF and THD are consideredto evaluate the performance of the different formulations.

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27 Dahidah, M.S.A., Konstantinou, G., Flourentzou, N., Agelidis, V.G.:‘On comparing the symmetrical and non-symmetrical SHE-PWMtechnique for two-level three-phase voltage source converters’, IETPower Electron., 2010, 3, (6), pp. 829–842

28 Dahidah, M.S.A., Konstantinou, G., Agelidis, V.G.: ‘SHE-PWM andoptimized DC voltage levels for cascaded multilevel inverters control’.ISIEA, 2010, pp. 143–148

29 Dahidah, M.S.A., Agelidis, V.G.: ‘Selective harmonic eliminationmultilevel converter control with variant DC sources’. ICIEA, 2009,pp. 3351–3356

9 Appendix

The exact values of the switching angles and DCvoltage levels used in the experiments are summarised inTable 3.

Table 3 Actual values of the switching angles (in degrees) and the DC source voltage levels (p.u.) used in the experiments

Case 1: all-variable DC voltage

levels (N ¼ 1/1/9)

V1 ¼ 0.466, V2 ¼ 0.419, V3 ¼ 0.420

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11

2.705 18.053 34.564 38.371 40.240 43.871 47.556 60.042 72.795 75.562 88.833

Case 2: one-variable DC voltage

level (N ¼ 1/1/9)

V1 ¼ 1, V2 ¼ 1, V3 ¼ 0.715

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11

5.567 17.578 34.090 44.019 46.418 56.541 59.386 66.077 71.441 75.647 88.313

Case 3: two-variable DC voltage

levels (N ¼ 1/1/9)

V1 ¼ 1, V2 ¼ 0.614, V3 ¼ 0.990

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11

12.615 25.926 42.805 45.225 56.647 59.098 61.685 75.362 78.564 85.245 88.002

Case 4: two-variable DC voltage

levels (N ¼ 3/3/5)

V1 ¼ 1, V2 ¼ 0.896, V3 ¼ 0.623

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11

4.242 6.516 9.040 17.628 22.856 25.234 36.597 48.036 49.099 77.397 80.394


selective harmonic elimination pulse-width modulation seven-level cascaded h-bridge converter with...

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