Generalized formulation of multilevel selective harmonic elimination PWM: Case I-Non-Equal DC Sources

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The definitive version is available at

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Dahidah, M.S.A. and Agelidis, V.G. (2006) Generalized

formulation of multilevel selective harmonic elimination PWM:

Case I-Non-Equal DC Sources. In: Power Electronics Specialists

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

http://researchrepository.murdoch.edu.au/12038/

Copyright © 2006 IEEE

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Generalized Formulation of Multilevel Selective Harmonic Elimination PWM:

Case I

-Non-Equal DC Sources

Mohamed S. A.Dahidah Vassilios G.

Agelidis

Faculty of Engineering andTechnology School of

Engineering

Science

Multimedia University,75450,JalanAyerKeroh Murdoch

University,

Dixon

Road, Rockingham,

6168

Lama-Melaka,MALAYSIA WESTERN AUSTRALIA

mohamed.dahidahgmmu.edu.my

v.agelidisgmurdoch.edu.au

Abstract- The paper presents optimal solutions for Fourier theory [9]-[10]. Sets of non-linear transcendental eliminating harmonics from the output waveform of a equations are then derived, and the solution is obtained using multilevel staircase pulse-width modulation (PWM) method an iterative procedure, mostly by Newton-Raphson method with non-equal dc sources. Therefore, the degrees of freedom

[2].

This method is

derivative-dependent

and may end in for specifying the cost function increased without physical

lo2]. optisa;

merthod

i drvies

epedeth

and

may

end in changes as compared to the conventional stepped waveform.

loca oima furter

aouiioscce

othe

alpvale

The paper discusses an efficient hybrid real coded genetic alonewillguaranteeconvergence [6]-[10].Another approach algorithm (HRCGA) that reduces significantly the uses Walsh functions [7] where solving linear equations,

computational burden resulting in fast convergence. An instead of solving non-linear transcendental equations, objective function describing a measure of effectiveness of optimizes the switching angles. More

recently,

atechnique

eliminating selected order of harmonics while controlling the based on a combination of an interval-search procedure and

fundamental for any number of levels and forany number of Newton's method is proposed to find all solutions to the switching angels is derived. It is confirmed that multiple nonlinear transcendental equations [12]. However, the independentsets of solutions exist and the ones that offer better method was only applied to the two-level inverters. harmonic performance are identified. Different operating References

[13]

and

[14]

generalized

the

problem

pointsincludingfive- and seven-level invertersareinvestigated

and simulated. Selected experimental results are reported to formulation where

constraint

of quarter-wave symmetry is verify andvalidate the effectiveness

of the proposed

method. relaxed to half-wave symmetry where all the even harmonics are zerobut theharmonic

phasing

isfree tovary.

However,

I. INTRODUCTION only the two- and three-level

waveforms

werereported and thetechnique hasnotbeen extendedtomultilevelcasewhere Multilevel inverters have drawn tremendousinterest in the the problem becomes more

challenging.

field ofhigh-voltage and high-power applications such as On

the

other hand, SHE-PWM methods have been

laminators, mills,

_{laiaos mils} conveyors, pumps,_{cove ors pups}

fans, blowers,

_{fas blwrs}

iO

_{ntroduced to}th ohehadSH

multilevel

_convertersPW

in

mtoshvebn_several

technical

compressors, and so on. The termmultilevel starts with the

artices

tialltheswchngerequen

was

restrict

o introduction of the three-level inverter[1].Byincreasing the le a

frequency

tef,hsta

smtevel

number of levels in a given topology, the output voltages

waveformqwas

and in s he

sawa cast

contrlethe

have more steps

generating

a staircase

waveform,

which

wavefomn

was

arrangedin

such a way as to control the

approches losel the esire sinuoidalwavefrm an als fundamental andeliminatethe loworderharmonic from the

approachesclosely thedesired sinusoidalwaveform and also waveform [3], [4], [6] and [8]. In references [4], [5] and

[16],

has reduced harmonic distortion [1]-[6]. One promising

tanscendental ans arecered

into

pnomial

technology to interface battery packs in electric and

hybrid

equations

wre

tory

ed to

determine

electric vehicles are multilevel inverters because of the

eqswting anglesuto

elmiat spe

ho

nics

posibiityofighVA atig nd ow armnicdisorton the

switching

angles

to eliminate

specific

harmonics.

possibility of high VA rating and low harmonic distortion However, as the number ofDC sources increases, the degrees

w

ritoutstemuslevftansormers

[3].cturesarereportedint

of the

polynomials

in

these

equations

are

large

which

Various multilevel inverters

arreprte

i e

increase

the

computation difficulty

and furthermore if one literature and the cascaded

multi-cell

inverterapp

ears

t be

wanted

to

apply

this method to multilevel converter with

superior

to other

multilevel

inverters

in

application

at

high

non-equal DC source which is always the case in most powerrating [3].Itis worth

noting

thatinmostof the works app l th seof t

sc

aleats

to

be

solved

reported

in the

literature,

the level of the dc sources was

applrcatlons,

thesetof transcendental

equations

tobesolved assumedtobeequal andconstant,which will

probably

notbe h re e

equations

ise

the

capblt

of the case in applications even if the sources are nominally contem poray ompte algebra. A new acti ni

equal [3]. elimination technique was recently introduced to the line

Selective harmonic elimination (SHE-PWM) has been frequency method aiming to eliminate higher order of mainly developed for two- and three-level inverters in order hroisb ipygnrtn h poieo h to achieve lower total harmonic distortion (THD) in the hamnctoacetem[5.Hwvrtedidatgen voltage output waveform

[2], [9],

[10] and

[12]-[14].

The ta sta tue ihsicigfeunyt lmnt

combined witha

programmed

method

[5]

and another where multilevel SHE-PWMdefinedby the well-know multicarrier

phase-shifted PWM (MPS-PWM) was proposed in [11]

vdv

where the modulation index defines the distribution of the switching angles and then the problem of SHE-PWM is appliedto aparticular operating point aiming to obtain the optimum position of these switching transitions that offer

eliminationto aselected order of harmonics. Inthispaper a L 2

minimization techniquewhich has been reportedin [9], [10] is once again applied to find the solution for the resultant

non-linear equations. However, the SHE-PWM technique 1 _ V2Vdc wasonly reported for five-level and withequalDC sources.

More recently, a general genetic algorithm using MATLAB GA Optimization Toolbox was applied to solve

the same problem of SHE [6]. However, the paper shows

SXl

only the solutions to the equal DC sources and with the

fundamental frequency switching method which has lower

vIvd

degrees of freedom forspecifying thecostfunctioncompared

s13

with the proposed PWM method for the same physical structure. Moreover, the authors havenotidentified the best

solutionsetsamongthose found. (a)

In this paper, a cascade multilevel PWM inverter with v non-equal dc sourcesisinvestigated. The main objectives of

this paper are: first to reformulate the problem ofSHE for NM

multilevel inverters based on PWM waveform using the VM staircase method. Where individual cellsare operated witha

VM,_1

frequency that is closetothe fundamentalfrequency (two or N2

aN-1

aN three

times)

sothat theswitchinglossesare

relatively

low and v2

higher degrees of freedom for specifying the cost function v

NI

and therefore higher bandwidth whencompared to existing a4(a5 a6 family oftechniques suchasconventional stepped waveform

is

gained. Secondly,

to introduce a minimization

technique

Al

a2 a3

assisted with ahybrid genetic algorithm inorder to greatly

reduce the computational burden associated with the 2

nonlinear transcendentalequations of the selective harmonic _{Fig.1.Phase circuitofamultilevel inverter}(b) _{system.(a) CascadedH-bridge} elimination method [8]. based inverter. (b) Generalized staircase PWM waveform.

Thepaper isorganizedasfollows. SectionII presentsthe

formulationalong withanalysis for the generalized staircase With the assumption of non-equal dc level of each inverter PWM voltage waveform. Section III discusses the cell, a generalized expression of Bn (i.e. for the single-phase implementation ofthe hybrid_{numbr ofseleted asesareprovded n Setion~}genetic algorithm. Results for_da _{case) and for any number ofswitching angles (even or odd,}

n otand

provided

that the

waveform

is

physically

correct andcanbe

finally conclusions

are

summarized

in

Section

V.

implemented)

is

given by

2N-1 Pi1

II. PROBLEM FORMULATION AND ANALYSIS

Bn

[V

I

(VIN1

E

(-P

) cos

nak

+

Fourier seriesexpansion ofthegeneralized staircasePWM

w1

n=1,3,5,

k=1

output waveform of the single-phase multilevel inverter P2 k PM

shown in

Fig. I(b)

canbe

easily expressed

asfollows:

V2

(1)cosnk VM

(_)

COS

ak)]

oo k=P1+1 k=PM-l+1

V0,t =ZAn

cos

nO

+

Bn

sinnO

(1)

(3)

n=1

where,

Owing to the PWMwaveform characteristics of odd function n=1,3,5,..., 2N -1, for single-phase system

symmetry and half-wave symmetry, the output voltage can be n

=1,

5,7,..., 3N -2, for three-phase system and N is odd

reduced to: n

=1,

5,7,..., 3N -1, for three-phase systemand N is even

=,H

E

Bn

sin

nO

(2) M

=the

numnber

of dc sources (i.e. invertercells) and the

n=4,3,5,.. product VM VdC is the value of the M dc source

N1, N2,..., NM arethe number of pulses per-quarter cycle at III. HYBRIDGENETIC ALGORITHM IMPLEMENTATION inverter

1,2,

...,M ,

respectively.

Genetic

algorithms (GAs)

are

highly

suited to search

N=N1+N2 +...+NM (Number ofpulsesper-quartercycle spaces which are not well

defined

or have a high number of

local

minima,

which plague more traditional calculus-based of m

the

tilevelcinv

output

ag)

search methods. By removing the need for auxiliary ack isthe k switchingangle, and information regarding the optimization surface the In eqn. (3), the polarity ± is positive if

PM-1

is an odd computational requirementsaregreatly reduced.

number otherwise it is negative. The GA necessitates the need for the optimization Eqn. (3) has N unknown variables

(a,

a2,...,aN) and a variables (a,,a2,...,aN ) to be coded as population of

setof solutions is obtainableby equating N -1 harmonics to strings transformed by three Genetic operators: selection, zeroand assigning aspecificvalue to the fundamental. crossover, and mutation.

Solutions to eqn. (3) can be obtained through any of those The presented hybrid genetic algorithm combines a methods highlightedinsection1. standard real coded GA and the phase-2 of conventional

However, these methods suffer from various drawbacks, search

technique.

suchasprolonged and tedious computational steps [2], [12] A. Phase-i (Real Coded) Algorithm

convergence to local

optima,

increase the degrees of the Real coded

genetic algorithm

is

implemented

as

follows,

polynomials

ifthe

technique

in

[3], [4]

is

considered,

etc. 1- A

population

of Np trail solution is initialized. Each Further, the number of harmonics eliminated by these

techniques is linked to the number of switching

angles,

soluions take

n a to number

wiables

hence, the more the number of harmonics to be eliminated, dimensions corresponding to the number of variables the

larger

the

computational complexity

and time. ( a1,a2,... ,aN

).

The initial

components

of ai are

To alleviate such problems, a hybrid genetic algorithm selected in accordance with a uniform distribution approach is proposed, which solves thesameproblem with a ranging between 0 and 1.

simpler formulation, withanynumberof levels or switching 2- The fitness score for each solution vector ai is angles and without extensive derivation of analytical evaluated, after converting each solution into expressions. corresponding switching instants

aa

using upper and

An

objective

function describing a measure of lower bounds.

effectiveness of

eliminating

selected order of harmonics 3- Roulette wheel based selection method is used to produce

while controlling the fundamental must be defined. Two

predominate methods in choosing the switching angles, Np offspring from parents.

ac,,a

....aN can be considered:

2~

1)

~~~~~~~~~~~

eliminate the lower 4- Arithmetic crossover and

non-uniform

mutation

N

operators

are

applied

to

offspring

to

generate

next

frequency dominant harmonics or 2) minimize the total

operator area

to t harmonic distortion (THD). The first method is the most generation parents.

popular and

_popla

_an

_stagtowr

straightforward of the two techniques, that is to_{ofth tw tehius tha is to} The_doesalgorithm_not

_change

proceeds to step 2, unless the best solution_fora

_{pre-specified}

_{interval of}

_generations.

eliminate the lower dominant harmonics and the filterorthe

nature of the load will take care of the higher residual

frequencies.

In this paper the first method is also chosen.

Pha

Hence, the objective function (for the single-phase case) is Algorithm

definedas After the

phase-

I ishalted,satisfyingthehaltingcondition

2

B2 +B2

+

B2

4 described in the

previous section, optimization

by direct

F(a1,

a2,---.,

a N)

(B1

I

AO)

+

B3

5 n

(4)

search and systematic reduction of the size of search region

M=

a , method is employed in the phase-2.

In

the light ofthe solution

4 accuracy,thesuccessrate,and thecomputation time the best

_

VF

vectorobtained form thephase-I is usedas aninitialpointfor

ma=F,is

the modulation index tepae2

a

Vdc

the

phase-2.

dc

The optimization procedure based on direct search and

NtntaVFstefdm m

ne

systematic

reduction in

search

region

is found

effective in

and 0<ma <M. solving various problems in the field of nonlinear The optimal switching angles are obtained by minimizing programming [16]. Thisdirectsearch optimization procedure eqn. (4) when it is subject to the constraint of eqn. (6). is implemented as follows

Consequently selected harmonics are eliminated. These 1- The best solution vectorobtained from the firstphase of switching angles aregenerated for different operating points the hybrid algorithm is used as an initialpoint ax(0) for and then stored in look-up tables to be used to control the phase-2 and an initial range vector isdefinedas

inverter for certain operating point. R(0)

=RM~F

xRange (7)

(0.

a1

<1X2

.< .a-<(N .-2)

(6)

Where,

Range is defined as the difference between the

assumed here that each individual inverter cell isoperatingat

switching angle hereare

/2

and 0,respectively) andRMF is frequency of threetimes of thefundamental one. As a

result,

arangemultiplication factor. The value ofRMFvaries there are three switching angles per-quarter cycle at each between0.0and 1.0. level ofthe output waveform. Hence, there are six switching 2-

Ns

trail solution vectors around a(0) are generated angles per-quarter cycle of the output

waveform

of the usingfollowingrelationship, multilevel inverter,which offer elimination offivelow order

a a(0)+a(0).*rand(1, n) (8) ofharmonics and controlling the fundamental at a certain

th vector, (*) represents value. Theswitchingangles variationagainst mafor the said where, ai is the i trail solutionvector,

)represents

case isplotted in Fig. 2.

It

isconfirmedthatmore than one set element-by-elementmultiplicationoperation, and ofsolutions are exist and for a certain range of ma (i.e. set 1, rand(1,n) isarandomvector,whose element value

1.1<

ma<1.94,

set 2,

l.l<ma<2.0

and set 3, varies from-0.5to0.5. 1.94<ma <2.0). As an example, an operating point when

3- For each feasible trail solution vector find objective ma =1.48 was chosen to illustrate the effectiveness of the

function value and find the trail solution set, which proposed method and the results (i.e. simulation and minimizes F(a) andequateitto a(0) asfollows, experimental) are shown in Fig. 3 (a)-(d). It is evident that the a(O)=abest

(9)

targetedharmonics

(3rd,

5th, 7th,

9th,

and 1

Ith)

are eliminated and the next significant harmonic appearing in the output

where,

abest is the trail solution set with voltage is the

13th.

Good agreement between the simulation minimum F(a). and theexperimentaloneis obtained.

Onthe otherhand, thenext

significant

harmonic could be 4- Reduce the range by an amount given considerably shifted further (i.e.

19th)

ifthree-phase system is byR(0)=R(O)*(1-

0,).

considered where all triplen harmonics are eliminated by the

where, ,6 is therangereductionfactor, whose

typical

phase shift between thetwo

phases.

value is0.05. B. Case

II.

Seven-level inverter with

V1

=1,

V2=

0.95 and 5- Thealgorithmproceedstostep2, unless the best solution V3=

0.85

p.u.

does not change for a

pre-specified

interval of Three inverter cells with

unequal

dc sources arecascaded generations. inthiscase,onceagain each inverter cell is

operated

at athree times of the fundamental

frequency.

Therefore,

there are

IV. RESULTSAND DISCUSSION three

switching angles

per-quarter

cycle

atoutputwaveform In ordertoverify thevalidity of the

proposed

algorithm,

a of each inverter cell. As a result there are nine degrees of program was developed using

M\ATLAB

6.0 software freedom

offering

the elimination of eight low order of package [17]. The program is run for a number of harmonics and controlling the fundamental component. The independent trials. Theproposedtechnique has been

applied

solution

sets for the

switching

angles for a certain range of successfully to different cases in order to indicate its ma (set

1,

2.1<ma<2.9, set 2, 2.1<ma<2.9 and set ruggedness. The simulation results are obtained

accordingly

3,2.92<

ma

<3.0

)

is illustrated in

Fig.

4.

Fig.

5

(a)

depicts

usingPSIMsoftwarepackage [18]. the output voltage

waveform

and its spectrum in Fig. 5 (b) Alow-powerlaboratory

five-level

inverterprototypebased where the absence of selected harmonics (3rd, 5th,. . ,t 7h) iS onthetwoIGBTH-bridges

configuration

was

developed

and clearlyevident for the given fundamental value which is 2.90 tested to verify the

feasibility

and the

validity

of the p.u. It is worth noting that the most significant order of theoretical and the simulation

findings.

The

pre-calculated

harmonicappears in the output

waveform

is the19thand once PWM signals are implemented using low-cost

high-speed

again itcould be further shifted to the

31St

if the three-phase Texas Instruments TMS320F2812 digital

signal

processor systemwas considered.

(DSP) board with an accuracy of 20ptseconds. A

digital

90 Set real-time

oscilloscope (Tektronix

TDS210)

was used to 8 Set2

display and capture the output waveforms and with the feature of the fast Fouriertransform

(FFT),

the

spectrum

of - 70

each of theoutput

voltage

wasobtained. t

60p

A discussion on the results is presented in the following g50 < sections. It is worth noting that in the following discussion,

40-the levels of 40-the dc sources are non-equal and can be t 0-

-measured;

fiurthermore,

each dc source has anominal value M 20 t X e

oftCap.u. Fielvlivrewt0V 1V=08pu

A. CaseI: ive-level iverter withV1 ~1, V2 080p.

U.1.1

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

In this case, two-cell inverter with different DC source Modulation index

(ma)

90 - -Set -Set 2

80 ~ ~ ~~~~~~~~~~~-Set2

50

740-.020

40-2.3. 23 24 .0.6 27 2.-.

2~~~~~~~a _{20 -~~~~~oulto ndx(n}

TekS

o~~~~M ~~o 2. 00m5 0H2 Fig. 4: Switching angles vs. maIi)~~~~~~~~~~~~~~~~_4(case

DO ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

00 2 00 400 a 00IN) I a GO 14 00 IE; 00 I Et 00 00I0~~~I

Tim ilml~~~~~Fc~

Thlk SL. Sto Pv 2.00n MAH(b

~~~ ~~Fig. 5: CSewitcingSevn-lesvel invererwihmRm .0)a)Otuotg

6CMH

#0 i 4a 9 I e imprtntthaIte ptiumse beidntfieawthresec t

Window

harmonic~~~~~~~~~~

performance... For.hispurpse,the.ota.haroni

____ dstrton(TD)ispltedinFi...fr.hetw.wd

CH15XV CH2 SDOMVFE Zom cntiuou sesC(et an se22)

(10)and up to 39 or Erof harmoicsU14 istake int account.

(d) The low pass filter and the

na~~~~~~~Ture

oftehihyinutvela

Fi.3:CsI iv-eelivetrwihm 14. a utu vlae il aecaeo tehihrore obamois Output voltge waveform(experimenal). (d) Spctrum of ouput voltag

39

Z

v2

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optimality,THDis chosentoevaluate theperformance ofthe programs", Springer, 1996.

proposed method andtoidentify the optimal solutionset.The [17] MATLAB6.0 softwarepackage, http://www.Mathworks.com.

simulation resultspresented fortwodifferentcasesincluding [18] PSIM software package, version

6.0,

Powersim Inc.

five-

and seven-level

inverters

and selected

experimental

http://www.powersJmtech.com.

results are presented to verify the theoretical and simulation