A Seven-level defined selective harmonic elimination PWM strategy

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Agelidis, V.G. and Balouktsis, A.I. (2006) A Seven-level defined selective

harmonic elimination PWM strategy. In: Power Electronics Specialists

Conference (PESC '06) 2006, 18 - 22 June 2006, Korea.

http://researchrepository.murdoch.edu.au/12042/

Copyright © 2006 IEEE

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A

Seven-Level

Defined Selective Harmonic Elimination PWM Strategy

VassiliosG.Agelidis Anastasios I. Balouktsis

Schoolof Engineering Science DepartmentofInformatics&Communications

Murdoch University, Dixon Road, Rockingham, 6168 TechnologicalInstitution of Serres WESTERNAUSTRALIA TermaMagnesias,62124,Serres,GREECE

v.agelidisgmurdoch.edu.au

tasosboteiser.gr

Abstract-Selective harmonic elimination pulse-width

modulation(SHE-PWM) techniques offer an optimized control

V3

approach for a given converter and are therefore suitable for

the low switching frequency high-power applications. V

Optimization techniques can be successfully used to obtain the solutions of the equations defining the SHE-PWM waveform. In

this paper, a seven-level multilevel strategy (MSHE-PWM) 0.0

defined on the line-to-neutral basis and based on a ratio of a t t t r 0

variable number of angles distributed over three levels to be al a2 a3 2

able to calculate thetransition points is reported.Thetechnique Fig. 1. The generalized staircase waveform suitable for multilevel systems

provides eighteen switching transitions for every quarter period and related angles oftransition between the various voltage levels.

in thestandard modulation index range. In theovermodulation

region, this can be changed in order to increasethe gain of the is combined with a programmed method [18] and another

modulator which in turn results in a compromised bandwidth. where a criterion based on power equalization between

Theswitching angles as a function of themodulation index are various cascaded connected H-bridge converters is used to

reported for thestandard as well as theovermodulation range. obtain the angles of the harmonic elimination method [19].

Selected simulation results are presented to verify the A recent paper [20] reported a MSHE-PWM strategy

effectiveness andfeasibilityof the proposedmethod. defined for five-levels as

shown in Fig.

2.

This method

did

I. INTRODUCTION not only seek single transitions

(Fig.

1)

but rather found

Selective harmonic elimination pulse-width modulation multiple switching angles in order to establish a PWM

(SHE-PWM) techniqueshave been extensively studied for a waveform (Fig. 2). The switching angleswerereported and

two-level, three-level and recently for multilevel converters the comparison of the MSHE-PWM technique against the

[1]-[20]. There have been many approaches to the well-know sinusoidal PWM employing phase-shifted

SHE-PWM problem reported in the technical literature

carriers

confirmed the superiorityof theformer.

including: sequential homotopy-based computation [6] Theobjectiveofthispaper is to report switching angles for

resultants theory [7], optimization search [8], Walsh a seven-level waveform. This extends our knowledge of

functions [9]-[10], optimal methods [11]. The bipolar

MSHE-PWM

techniquesfor the first time into a seven-level

waveform has been treated in detail in [12] where a case (Fig. 3). Solutions for the switching angles for the entire

minimization technique is employed along with a biased range of modulation indices, i.e. standard and

optimization search method to get the multiple sets as overmodulation are reported in the paper and verified to

predictedin[6]. confirmthe effectiveness of the proposed strategy.

These methodswereextended in multilevel systems where The paper is organized as follows. Section II presents in

the staircase waveform wasused to find the angles and per detail the proposed seven-level MSHE-PWM. Switching

unit valuesinordertominimize a number ofharmonics and angles as a function of the modulation indices are reported.

synthesize multilevel waveforms [13]. Such awaveform is The standard modulation index range and the

showninFig. 1.Thiswasfurtherinvestigated and reported in overmodulationare investigated and results are presented in

[14],

[15], [16] introducing an extra switching angle to Section III and conclusions are summarized in

Section

IV.

address the limitations of the previousmethods [14] in areas II. PROPOSED

SEVEN-LEVEL

MSHE-PWM

where

optimized switching angles

cannot be found. The proposed MSHE-PWM strategy is definedaccording

Specifically,

thetheoryofresultants andits performancefor a

multileve stics waefr

wa.eotdi

1] nfe to the waveform shown

in

Fig.

3 andrepresents the line-to-apprachwaspreentd i [1]. Mre ecetly th us of neutral waveform of the converter. The number of levels of

.. . . . '. .

...

the waveform iS assumed to be seven, i.e., lp.u., 2p.u., 3p.u.,

symmetric polynomials iS combined

with theresultanttheory Opu,-pu.-2u.ad3pu

fraMultilevel conerter

M)sstm

hv

be

The problem is formulated in a generalized form as

MultievelSHE-WM (MHE-PM) sstemshavebeen follows. Let P be the number of levels of thewaveform and

contolld usnghe nipoar thre-leel)waveormand the case where

this

number

iS

odd is considered although an

phase-shifted techniques

[17]. Other approaches have also

for the angles betweenzero and /-T/ 2 , the usual reflection

occurs to find the rest of the

angles.

Since there are N

switching

angles

(i.e.,aj,a2,

a32

....aN),

N-I

harmonics can

1.0 be eliminated if solutions can be found. For a three-phase

a6 a9 a10 a13a a

inverter,

the harmonicsto be eliminated from the

waveform

0.0 are assumed to be non-triplen odd harmonics (i.e.,

5th,

7th,

a~2aa a 2 11th

13th nthwhere

n=3N-1 whenN=evenandn=3N-2when

N=odd).Thestrategythen reliesonthestructureofthepower

Fig. 2. Afive-level defined(line-to-neutral) SHE-PWM waveform shown for circuit in order to remove the triplen ones from the

adistribution ratioof 5/12

(z1=5,

z2=12,N=z1+z2=17). line-to-line voltagewaveforms. For the seven-level case we

3.0 have:

_{_O<M<3}

(6)

2.0

UIf

vL

is the

amplitude

of the fundamental

component

to

be

2.0v _ Z l | , generated, then

1.0 a13 a16 a18

VIl=V4M

(7)

m|a6 a9 aio whereas the square-waveform of 3p.u.

amplitude

can

0.0 generate (12/T) p.u. maximum value at fundamental

t t gT2 0

al a3 a5 2

frequency.

Fig. 3.Aseven-level defined(line-to-neutral)SHE-PWM waveform shown

111.

RESULTS ANDDISCUSSION

for a distributionratio of 5/7/6

(zj=5,

z2=7, z3=6,N=z1+z2+z3=18). The minimization technique

proposed

in [8] has been

applied and software is usedto investigate the method [21].

considerP=7. Let Nbe the number of

switching

transitions For this paper N=1 8. However, this number and the various

(angles) of the waveform

sought

within the quarter of the ratios can be changed as desired and theproposed method

period of the waveform. Let zi be the

switching

angles

would provide therespective solutions provided they exist.

(transitions)in everylevel and

P-I (1 A. StandardModulationRange

i=1,2,3...

(2)

) The standard modulation rage includes areas of M where

The equation that describes the Fourier

analysis

of the solutions can be found. The maximum value found is M=2.58

multilevelwaveform is then: for the case of 1/1/16 (Fig.

4(p)).

It should be noted that the

Z- result obtained from the

proposed

method whenthe ratio is

fh

=

(-I)k-I

cos(

h

.a

k) + chosen to be 18/0/0 (Fig. 4(a)) provides solutions for all

k=1 (2 modulation indices up to 0.73

(continuous solutions)

and

(P- Zq ) other sets can increase the maximum obtainable M to 0.83

/Iq=1 k-1-2,zq

(Fig. 4(b)).

Further results for all combinations of ratios are

+nt

,& (-1) h

ak)

plotted in Fig. 4. The results are summarized in Table 1. n=2

k=l+2_

Zq Scanning the resultspresentedin

Fig.

4,one can seethatthere

whr

is

an

overlap

between the

regions, ensuring

the method can

where be implemented for the entire range of modulation indices.

(P-1

Y

N The last harmonic that can be eliminated

according

to

E

zq

= N (3)

equation (4)

is the

53p.u.

and this is

confirmed

in

Fig.

6. The

problem

is solved

through

constraint minimization

[8]

Specifically,

Fig. 6(a)

shows this

waveform

and its

spectrum

and the

following

function whenN=evenis considered:

confirms

that it does have only multiple of triplen harmonics

l(f

_M)2

+f2 2 2 (Fig. 6(a)). These triplen harmonics are cancelled out in a

MINt(fi-+

5 J7 + 3N-1

(4)

three-phase

system

from the line-to-line waveforms

(Fig.

with the constraints:

6(c)).

This is

confirmed through

the associated

spectrum

of

2T

(5)

theline-to-line

voltage

waveform shown in

Fig.

6(d).

0<al

<a2 <.

*--(5

<l

vlN

<w

2 B. Overmodulation

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1.1 1.2

90 __________ __________90

80 80 80 80

70 70 70 ____-_ ,70

60 60 _u6

40 _ . 40 40 40

30300_

20 20 20 20

10_ 10 10 _ _ 10

0 __0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1.1 1.2

(a) (e)

0.7 0.8 1.2 1.3 1.4

90 _ _ _ -- -90

-80 _ 80 80 = 80

70 70 700 70

60 60 60- - _ _ _ 60

50

_ ~~~~~~~~~~~~~~~~~~~~5050 -50

40 40 40 4

30 30 30 30

20 20 20 20

10 10 10 10

_~ 0 0

0.7 0.8 1.2 1.3 1.4

(b) (t)

0.8 0.9 1.4 1.5

90 90 __ __

80 80 80 80

70 770 70 = 70

60 60 60 260

50 - 50 50 - - 50

40 40 40 40

30 30 30 30

20 -'I_5 _ _ _-__ _ _ _ _ 0 20 __ -l_ l50_ __

20 20 20 = 20

10 10 10 10

0. 9 1.4 1.5

0.9 1 1.5 1.6 1.7

90 90

80 80 80 80

70 __ __ __ __ __70 70 _ _ _ _ _ _ __ 70

60 60 60 60

50 50 50 - 50

40 40 40 40

30 -30 30 30

20 20 20 -- - 2

10 -1010-0

90

__ _ _ __ __ __ _ _ __90_______

80 - 80 80 -_ 80

70 - - 70 70 70

40 40 40 40

330 30 _ _- _- 30

20 20 20 20

10 10 10 10

-~~~~~~~~~~~~~~~~-~~~~~~~~~~ 5~~~~~0

40 - 40 40 40

30 30 30 30

0 620 0 6_

10 _ _

1. 1.9 2. 2.32.42.

9030 30 30__

80 80 80 80

50 50 50 50

0 0

1.8 ~~~~~~~~~~~~1.9 222.3 2.4 2.52.

90 90

80 80 80 _ 80

70 ~~~~~~~~~~~~7070 - - -70

60 60 60 60

50 50 50 50

40 40 40 40

30 30 30 30

20 20 20 20

1010 10 __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 10

1. 9 2 2. 1 2.65

90 90~~~~() p

Fi.4eete0wthn tastosfr h ee-evlSEPMtehiu o aroscmiatoso wtcigtaston essmduainidx

3 0 30 3 0

2 0 20 2 0 -_ 7

60 60 60 60~~~~~~~~~~~2625

90 900 __ __ __ _

80 4__ __ ___ _ _ 0 8 0 80

7 0 3--- 0 7 0 70

0 2__ 0 0 60

50 --- 0 50 - 0

40 ---- 40 40 ______ ______ _______ ______ ______40~~~~~~~~0______ ______

30 ....803 30 8__ 0

20 7- 0 20 20

10 6__ 0 10 10

4 0 4 0~~~~~~~~~~~~~~~~~~~~~~24 2.5

90309303

8020820 20

50 500 10

40

40 4~~~~~00-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

-2.5 2.6 ~~~~~~.72.42.7 2.5 2.6

4.00 4.00

.0

2.00--1.00 a) 1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.00

---.00 -- --0---

--.00--.0 0 ---1 .0

0-

---0---0.00 ---0---0.00 1---0---0.00 10.00 20.00 Time(me)~~~~~~~~~~~~~~~~~~-200

0.20 - - -- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

--2 .0 ---

-4.00---.,-04.00

240 .05.01 01.00 2.000

0 00 00Sd 1.001502.0 im (s

3.20~~ ~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~r

0.00~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~40

0.00a0.00tal

4.00

.0 -2

--2.0 --0. 0

1.00---I---L--- 5 ---

---0.8 -I- --.0 )- -I- -I- -I --J.-

---1.00 _-1.0

-0.01 A A0.00

-0.00 000 100 1000 1000 200.00 0.00 0.5 0.5 .500 1000 1.5 100 20 0 25 20.500

Timeqec (KHz) Freqec(m Hz)

7.00~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~00

d) .00 d)---0- .00

.00 1 .00

---.00 000---000--- --000----

-000---120---100---2----40--0.00 00-100-200-Freque-m--(0Hz)

Freqoenco (0Hz) 00 ---

---Fig.6:-Imlementatto-of-the-prposed-teehtque-for-te-followtn-ease Fig-7:-Implemetatton-of-he-propose teehntquefor-the-folowtng-eas within the standard modulatton tndex range 1/3/14 M 2 2 (a) Ltne to within the overmodulatton--- tndex--- range 1/1/9---M--2-7---ontrolltng---10 neutral -oltage-wveform- ()-Spectrm-of-theltne-to-eutral-vltage-no-trip-enharmont--(a)-Ltn-to-neutal-voltae-wavefom-(h)-Spctrum-o waveform. (e) Line-to-line voltage waveform. (d) Spectrum of the the line-to-neutral voltage waveform (e)---Line-to-line--voltage-waveform-- (d)

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