Full Study of a Precise and Practical HarmonicElimination Method for Multilevel Inverters

Damoun Ahmadi and Jin Wang*Department of Electrical and Computer Engineering

Ohio State University205 Dreese Labs; 2015 Neil Avenue

Columbus, OH 43210Phone: 614-688-4041, Fax: 614-292-7596

Email: wang@ece.osu.edu*

Abstract- Multilevel inverters have been widely used inmedium and high voltage applications. Selective harmonicselimination for the staircase voltage waveform generated bymultilevel inverters has been studied extensively in the last decade.Most published methods on this topic aimed at solving high-ordermulti-variable polynomial equation groups derived from Fourierseries expansion. A totally different approach based on equalarea criteria and harmonics injection in the modulation waveformis fully studied in this paper. Regardless how many voltage levelsare involved, only four simple equations are involved in the basicmethod. The problems of the basic method are identified anddiscussed. A full set of solutions is proposed. The results of a casestudy with maximum five switching angles show that the proposedmethod can be used with excellent harmonics eliminationperformance for the modulation index range from 0.2 to 0.9.

Keywords: Equal Area Criteria, Modulation Index, MultilevelInverters, Pulse Width Modulation, Total Harmonic Distortion

I. INTRODUCTION

In high power medium and high voltage applications, theusage of Pulse Width Modulation (PWM) based two levelinverters is limited by voltage and current ratings of switchingdevices, switching losses, and electromagnetic interferencescaused by high dv/dt. Thus, for applications like mediumvoltage drives, renewable energy interfaces, and flexible actransmission devices (FACTs), multilevel inverters aregradually becoming the main work force [1-4]. A typicalmultilevel inverter utilizes voltage levels from multiple dcsources. These dc sources can be isolated as in cascademultilevel structures or interconnected as in diode clampedstructures. In most published multilevel inverter circuittopologies, the dc sources in the circuits need to be maintainedto supply identical voltage levels. Based on these identicalvoltage levels, with proper control of the switching angles ofthe switches, a staircase waveform can be synthesized, such asa 6 level staircase waveform with five switching angles shownin Fig. 1.

One of the greatest benefits of this staircase waveform isthat the switches in the inverters only need to switch on and offonce during one fundamental cycle, thus the switching loss ofthe device is reduced to minimum. However, with reducedswitching frequencies, even with additional voltage levels, lowfrequency harmonics can be found in the staircase voltage [1,2].

I\k

Figure 1. The general staircase waveform of multilevel inverters and equalarea criteria.

The Fourier expansion of the staircase wave can beexpressed as:

4VdV(t) = d,c (cos(m 0,) + ... cos(mON )) sin(mwt) (1)m=1,3,5.... M)z

where N is the number of switching angels and m is theharmonic order. Based on the Fourier expansion, severalmethods have been proposed to solve the following equationgroups to realize Selected Harmonics Elimination (SHE) [11-19].

(cos(01) + COS(02) + COS(03) + COS(04) + COS(05)) = VFI z

cos(501) + cos(502) + cos(503) + cos(504) + cos(505) = 0

cos(I 301) + cos(I 302) + cos(] 303) + cos(] 304) + cos(l 305) = 0

(2)

In this equation group, the first equation guarantees thedesired fundamental component. The second equation andbeyond are used to eliminate the selected harmonics. So, it isclear that with five switching angles, four selected harmonicscan be eliminated. The following is a brief summary of themost quoted and most recently reported methods on solvingthese types of equation groups. A resultants theory basedalgorithm was reported in 2000 [11]. A unified approach waspresented in 2004 [12]. In the following year, symmetricpolynomials and resultant theory combined solution [13],power sum based method [14], and generic algorithm basedmethod [15] for multilevel inverters with uneven dc sources

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were proposed. High switching frequency based activeharmonics elimination was published in 2006 [16]. Veryrecently, a five-level symmetrically defined SHE PWMstrategy [18] and a hybrid real coded genetic algorithm [19]were published.

One of the main difficulties of applying most of thesemethods is that when the number of dc level increases, thenumber of polynomial equations, the number of variables, andthe order of the equations will all increase accordingly. Thus,finding solutions to these equations would become extremelydifficult and often involve advanced mathematical algorithms,which make the calculation easy to reach the capability limitsof existing computer algebra software tools [12]. Researchesshow even for the simplest case like equation groups (2), it stilltakes special algorithms and long calculation time to solve.For larger and higher order equation groups, there would be apoint that to find the solutions becomes not practical [11-18].Other limitations of the proposed methods often involve thecapability of dealing with wide range of modulation index [19],and uneven dc voltage levels [15, 17].

In summary, though many methods have been proposed tosolve the SHE problem in multilevel inverters, a simple andpractical method is still needed.

II. THE FoUR-EQUATION BASED HARMONICELIMINATION METHOD

The four-equation method [20] is based on equal areacriteria (shown in Fig. 1) and the harmonics injection in themodulation waveform. The basic approach in the four-equation method is summarized in five steps: 1) use the desiredfundamental voltage as a reference to find the initial switchingangles with equal area criteria; 2) identify the selectedharmonics in the staircase waveform, which comes from theswitching angle; 3) subtract the harmonics from the originalreference waveform to get a new reference waveform; 4) usethe new reference waveform to find a new set of switchingangles; 5) repeat steps 2 to 4 until the selected harmonics areeliminated. The diagram of this method is shown in Fig. 2.The four basic equations used in this method are:

Equation 1: the equation to calculate the junction point of thereference and the voltage level. The Newton-Ralphson methodwill be used to find the numerical solution of the method:

8k = arctan( k.Vdc+ h5 sin(5Sk)..hm sin(mSk) (3)k ~~~~~~VFcos ('5k

Equation 2: the equation to find the switching angle:

Ok = kk - (k - 1)3k- + VF (cos(k ) - CoS(k- ))h5

- (cos(535k) - cos(535k-1))5

(4)

hm... (cos(mSk) -CoS(mk-l))

mEquation 3: the equation to find the harmonics content in thestaircase waveform:

hm = E V(cos(mok) -cos(m(;T - Ok)))Mk,

(5)

Equation 4: the equation to calculate the new referencewaveform:

Vref = VF sin(c) - hms sin(mc) (6)

where, hms is the sum of hm calculated in every iteration:

iter

hms=Zhm(i)i=1,2,....

(7)

Originally, the proposed method was verified with testresults from a 17 level cascade multilevel inverter. Switchingangles of a six level waveform for limited modulation indexrange was also shown [20].

For the work presented in this paper, to identify possibleproblems with the basic four-equation based method, the basicmethod was tested with 6-level waveform with modulationindex sweeping from 0.16 to 0.94. The main problemidentified from this process is the amplitude differencebetween the desired and resulted fundamental voltages.

With the direct implementation of the proposed method, thefundamental voltage of the staircase waveform often divertsfrom the desired value, as shown in Table 1. The reason is thatfor most cases, it is difficult to find a good solution for theswitching angle for the top dc level to satisfy the equal areacriteria. When it is expressed in terms of modulation index,the difference between resulted modulation index and desiredmodulation index can be higher than 0.1 at certain points. Thisalso means sudden changes in the resulted modulation indexes.

The modulation indexes shown in Table 1 are defined bythe following equation:

DesiredFundamental +Component

Figure 2. The four-equation based method.

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Table 1. Sample points from the direct implementation of the four-equation method.

Reference Resulted Switching Angles (rad.) Harmonics (%)

Ml 0I 02 03 04 50 5th 7th 1th I 3th I 7th

0.92 0.8408 0.1147 0.2577 0.4121 0.6465 1.0134 0 0 0 0 0. 584

0.88 0.7923 0.1433 0.3398 0.5275 0.8417 1.1057 0 0 0 0 0.4239

0.84 0.7818 0.1434 0.3406 0.5283 0.8433 1.1062 0 0 0 0 0.5128

0.80 0.7715 0.1435 0.3411 0.5289 0.8443 1.1065 0 0 0 0 0.7062

0.76 0.7251 0.0815 0.4256 0.6818 0.8583 N/A 0 0 0 0.4847 N/A

MI V (8)

4N.Vdc

where VF is the reference ac voltage in the output, N is thenumber of dc levels, and VdC is the dc magnitude for eachvoltage level in multilevel inverter output waveform.

As Table 1 just shows portion of the results, the full table ofmodulation index from 0.16 to 0.94 shows that for the six-level(maximum) waveform, the change of resulted modulationindex is not continuous but more like a staircase.

Earlier, multiple solutions of 8k for one dc level wereexpected to be an issue. But studies show that the equal areacriteria automatically settled on the best 8k . So the study in thepaper mostly is focusing on solving the problems related withthe accuracy of the resulted fundamental components.

III. SOLUTIONS To THE PROBLEMS IN THE BASIC METHOD

A. PI Controller based Fundamental Voltage Correction

to add a simple PI controller in the iteration process to keepcompensating the fundamental component. With this approach,the modulation waveform of this modified method can beexpressed as

Vref = (VF -his)* (KP + A)-hms sin(mot) (9)

The overall diagram of the modified method is shown inFig. 3. The added process is shown in dotted line.

However, even with the PI controller in place, the numbersof utilized dc levels are still not optimized for most modulationindexes. Thus, there is still a slight difference between resultedand desired modulation index for most cases. And since the PIcontroller is only used for fundamental compensation, theperformance of harmonics elimination went bad especially athigh modulation index points. This can be seen in the resultslisted in Table 2.

Since the performance of modified method with PIcontroller does not have good harmonics cancellation,alternative solutions are proposed and validated as following.

To solve the problem of the difference between the desiredand resulted fundamental component, one possible solution is

|Switching angle calculations |Harmonics calculation|D d f a _ _ _ with Equation 1 and 2 with Equation 3Desired fundamental L+' l

component *\ Xk -- F7

- - + New modulationLL---> waveform synthesizing

withEquldso4

Calculated fundamental from the resulted switching angles

Figure 3. Modified method with PI controller to adjust the fundamental component.Table 2: Sample points with PI controller based modified method.

Reference Resulted Switching Angles (rad.) Harmonics

Ml 0 0 o 0 o 5th 7th 1,th 13th 17thm

1 02 3 04 050.92 0.9112 0.1794 0.2567 0.315 0.4993 0.7304 0.3976 0.285 0.0085 0.613 0.4304

0.88 0.8467 0.1043 0.2647 0.394 0.6277 0.9994 0.0179 0.0554 0.25 0.4643 0.1339

0.84 0.8308 0.1146 0.2577 0.4119 0.6464 1.0133 0.0008 0.0019 0.0019 0.0038 0.0017

0.80 0.7931 0.1399 0.3182 0.5149 0.802 1.0932 0.3026 0.1336 0.1316 0.7332 0.6585

0.76 0.7351 0.0005 0.2402 0.4088 0.6905 N/A 0.0062 0.0723 0.5558 0.7025 N/A

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B. Final Solutionsfor the Problems in the Four-equation Method

In the final solutions, PI controller is no longer used in the iterations.Instead, either an additional voltage level or additional adjustment ofthe switching angle at the highest voltage level is used depending onwhether an extra voltage level is available at the defined modulationindex.

1. Harmonic Elimination with No Extra Voltage LevelsIn multilevel inverters, when the desired modulation index

becomes smaller, fewer dc levels will be used to synthesize thestaircase waveform. In this case, with the four-equationmethod, at the fundamental frequency, the difference betweenthe desired voltage and generated voltage will become larger.But since an extra voltage level is available, it can be utilizedto realize the fundamental voltage compensation. Based onthis idea, the same five steps in the four-equation methodwould be used. The difference is that an extra switching anglewould be calculated for an extra voltage level to achievedesired fundamental voltage. However, additional harmonicswill be present in the extra voltage level. So in this method,the additional harmonics content generated by the extra voltagelevel would be added to the reference waveform which is usedto calculate the other switching angles. This means that theextra harmonics generated in the additional voltage level wouldbe compensated by the switching angles for all the othervoltage levels.

This modified method is illustrated in the diagram shown inFig. 4. The additional process is shown in dotted line.

The following is the procedure on the calculation of theadditional "m+l" switching angle:

1) First, the total fundamental voltage based onswitching angles from 01 to O,m is calculated with thefollowing equation

VIm= 4Vdc cos(Oi), m<N (10)

2) Then, the switching angle of the additional voltagelevel is calculated based on the difference between the desiredfundamental voltage, VF, and the resulted fundamental voltage,Vim, with the following equation

Om+, a cos( 4V (VF -VIm)) (1 1)

2. Harmonics Elimination with No Extra Voltage Levels

For larger modulation indexes, where all dc levels arealready used for staircase generation, there is no additionalvoltage level for fundamental voltage compensation.Therefore, in this proposed method, the switching angle of thelast dc level will be adjusted to achieve desired fundamentalvoltage. The "adjustment" switching angle is calculated with:

(12)ON =acos(4 (VF -17VN)) I

where VIN is the total fundamental voltage generated byswitching angles from 01 to O,m

This "adjustment" angle is used to modify the switchingangle for the last voltage level:

(13)0N(mod ified) acos(cos(ON) + cos(0 ))

Therefore, based on the switching angle adjustment for thelast dc level, the desired voltage magnitude in the fundamentalfrequency can be achieved. However, if not compensated, the"adjustment" switching angle would bring in the additionalharmonics in the resulted staircase waveform. Thus, theselected harmonics caused by the "adjustment" angle wouldalso be calculated and be added to the final modulationwaveform. The total process of this modified method isillustrated in Fig. 5.

r------------Calculate the I

additional"m+1" I."switching angle I

_ _ _ __ Switching angle calculations

/'' Calculate the selectedwihEutoIan2r->\' harmonics in the

T additional voltage levelL--____-__-____-__-__ New modulation

:~~ ~ ~~~~------ ---> waveform synthes'izing| | > ~~~~~~~~~withEquation 4

Harmonics calculationwith Equation 3

oespreoiuncamentacomponent

Tr _

Calculated fundamental from the resulted "m" switching angles

Figure 4. Modified method with "additional" switching angle.

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4nO -A,

874

Calculatethe >-.-.--------------------------.--- I"adjustment" +

switchimg angle S a cui_ _ _ _ _ Switching angle calculations Harmonics c

_/+ ") Calculate the selected with Equation 1 and 2 with Equi\ -, |harmonics caused by 4

| X thffe "Cadjustment" angle:~~~~ L_---- -------- Newmodulation: ~ ~~~~-._____ -- ---> waveform synthesizing

with Equation 41kdamentalI

,alculationation 3

nent Calculated fundamental

Figure 5. Modified method with "adjustment" switching angle for the highest voltage level.

IV. VERIFICATION OF PROPOSED SOLUTIONS WITH A CASESTUDY

With the final solutions 1 and 2, switching angles, resultedmodulation index, and selected harmonics content were

calculated with the modulation index sweeping from 0.2 to 0.9for staircase waveforms with maximum six levels. For all thecalculated modulation points, the resulted modulation indexfollows the desired value well. The selected harmonicselimination can be 100% except at modulation indexes close to0.9. Table 3 shows some sample points of the calculationresults. At the first sample point MI=0.9, since no extravoltage is available, the "adjustment" angle in method 2 is used.For the other four points in the table, an additional voltagelevel is used to achieve desired modulation index. The resultsshow that both solution 1 and 2 work well in terms ofachieving desired fundamental voltage and harmonicselimination.

The overall optimized switching angles based on theproposed method for 6-level (maximum) waveform is shown inFig. 6. It is noticeable that with the modification of theswitching angle of the highest voltage level, the 05 sometimesbecomes smaller than other switching angles. It may lookstrange, but this will not cause any problem. In the realinverter, the final voltage is the summation of the voltage fromall the dc sources, instead of getting the waveform shown inFig. 7 (a), the output voltage of the inverter would always looklike the waveform in Fig. 7 (b).

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Modulation Indexes (MI)

Figure 6. Optimized switching angles of the proposed method for 6-Levelwaveform.

V. CONCLUSIONS

In this paper, a simple method using four equations forharmonic elimination in multilevel inverters is reviewed andfurther studied. The problems with direct implementation ofthe method is identified and discussed. Then, a full set ofsolutions is provided and verified with a case study.

Table 3. Sample points for six (maximum) level waveform with the modified four-equation method.

Reference Resulted Switching Angles (rad.) Harmonics (%)

Mml ,5th 7th 11Ith 13th 17thMl Ml 0a1 02 63 04 050.90 0.90 0.0641 0.1984 0.3395 0.7714 0.5319 0.069 0.0291 0.0375 0.0474 0.0215

0.76 0.76 0.1878 0.3618 0.5922 0.9231 1.091 0 0 0 0 N/A

0.60 0.60 0.1971 0.4689 0.8051 1.1216 N/A 0 0 0 N/A N/A

0.46 0.46 0.2175 0.5954 1.0522 N/A N/A 0 0 N/A N/A N/A

0.20 0.20 0.3889 1.4961 N/A N/A N/A 0 N/A N/A N/A N/A

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Desired furcompc)r

875

(a)

(b)

Figure 7. Explanation of the top switching angle in Fig.6.

Comparing with other SHE methods proposed formultilevel inverters, the harmonics elimination method studiedin this paper has the following advantages: 1) only four simpleequations are involved in the basic method; 2) for differentnumbers of switching angles, the equations remain the same,no huge increasing of calculation time is expected when thenumber of switching angles increases; and 3) in some cases,this method can eliminate more than N-1 harmonics with onlya small difference between the desired and resulted modulationindex [20]. With the simple modifications proposed in thispaper, this method has become not only precise in harmonicselimination but also practical in terms of simplicity andrealization by field engineers.

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