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Generalized Formulation of Multilevel Selective Harmonic Elimination PWM:Case I -Non-Equal DC Sources

Mohamed S. A. Dahidah Vassilios G. AgelidisFaculty of Engineering and Technology School of Engineering Science

Multimedia University, 75450, Jalan Ayer Keroh Murdoch University, Dixon Road, Rockingham, 6168Lama- 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 transcendentaleliminating harmonics from the output waveform of a equations are then derived, and the solution is obtained usingmultilevel staircase pulse-width modulation (PWM) method an iterative procedure, mostly by Newton-Raphson methodwith non-equal dc sources. Therefore, the degrees of freedom [2]. This method is derivative-dependent and may end infor specifying the cost function increased without physical lo2]. optisa;merthod i sdrvie epedethand may end inchanges as compared to the conventional stepped waveform. loca oima furter aouiioscce othe alpvaleThe paper discusses an efficient hybrid real coded genetic alone will guarantee convergence [6]-[10]. Another approachalgorithm (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, a techniqueeliminating selected order of harmonics while controlling the based on a combination of an interval-search procedure andfundamental for any number of levels and for any number of Newton's method is proposed to find all solutions to theswitching angels is derived. It is confirmed that multiple nonlinear transcendental equations [12]. However, theindependent sets 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 problempoints including five- and seven-level inverters are investigatedand simulated. Selected experimental results are reported to formulation where constraint of quarter-wave symmetry isverify and validate the effectiveness of the proposed method. relaxed to half-wave symmetry where all the even harmonics

are zero but the harmonic phasing is free to vary. However,I. INTRODUCTION only the two- and three-level waveforms were reported and

the technique has not been extended to multilevel case whereMultilevel inverters have drawn tremendous interest in the the problem becomes more challenging.

field of high-voltage and high-power applications such as On the other hand, SHE-PWM methods have beenlaminators, mills, conveyors, pumps, fans, blowers, iO th ohehadSH PW mtoshvebnlaiaos mils cove ors pups fas blwrs ntroduced to multilevel converters in several technicalcompressors, and so on. The term multilevel starts with the artices tialltheswchngerequen was restrict ointroduction of the three-level inverter [1]. By increasing the le frequencya tef,hsta smtevelnumber of levels in a given topology, the output voltages waveformqwas and in s he sawa cast contrlethehave more steps generating a staircase waveform, which wavefomn was arrangedin such a way as to control the

approches losel the esire sinuoidalwavefrm an alsfundamental and eliminate the low order harmonic from theapproaches closely the desired sinusoidal waveform 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 pnomialtechnology to interface battery packs in electric and hybrid equationswretory ed to determineelectric vehicles are multilevel inverters because of the eqswting anglesuto elmiat spe ho nics

posibiityofighVA atig nd ow armnicdisortonthe switching angles to eliminate specific harmonics.

possibility of high VA rating and low harmonic distortion However, as the number ofDC sources increases, the degreesw ritoutste muslevftansormers [3].cturesarereportedint of the polynomials in these equations are large whichVarious 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 withsuperior to other multilevel inverters in application at high non-equal DC source which is always the case in mostpower rating [3]. It is worth noting that in most of the works app l th seof t sc aleats to be solvedreported in the literature, the level of the dc sources was applrcatlons, the set of transcendental equations to be solved

assumed to be equal and constant, which will probably not be h re e equations ise the capblt ofthe case in applications even if the sources are nominally contem poray ompte algebra. A new acti niequal [3]. elimination technique was recently introduced to the line

Selective harmonic elimination (SHE-PWM) has been frequency method aiming to eliminate higher order ofmainly developed for two- and three-level inverters in order hroisb ipygnrtn h poieo hto achieve lower total harmonic distortion (THD) in the hamnctoacetem[5.Hwvrtedidatgenvoltage output waveform [2], [9], [10] and [12]-[14]. The ta sta tue ihsicigfeunyt lmnt

common~~~~chrcersi ofteslciehroi .lmnto higher order harmonics. Other approaches have also beenmethod iS that the waveform analysis is performed using rpre nldn n hr h amnceiiaini

combined with a programmed method [5] and another wheremultilevel SHE-PWM defined by the well-know multicarrierphase-shifted PWM (MPS-PWM) was proposed in [11] vdvwhere the modulation index defines the distribution of theswitching angles and then the problem of SHE-PWM isapplied to a particular operating point aiming to obtain theoptimum position of these switching transitions that offerelimination to a selected order of harmonics. In this paper a L

2

minimization technique which has been reported in [9], [10]is once again applied to find the solution for the resultantnon-linear equations. However, the SHE-PWM technique 1 _ V2Vdcwas only reported for five-level and with equal DC sources.More recently, a general genetic algorithm using

MATLAB GA Optimization Toolbox was applied to solvethe same problem of SHE [6]. However, the paper shows SXlonly the solutions to the equal DC sources and with thefundamental frequency switching method which has lower vIvddegrees of freedom for specifying the cost function compared s13with the proposed PWM method for the same physicalstructure. Moreover, the authors have not identified the bestsolution sets among those found. (a)

In this paper, a cascade multilevel PWM inverter with vnon-equal dc sources is investigated. The main objectives ofthis paper are: first to reformulate the problem of SHE for NMmultilevel inverters based on PWM waveform using the VMstaircase method. Where individual cells are operated with a VM,_1frequency that is close to the fundamental frequency ( two or N2 aN-1 aNthree times) so that the switching losses are relatively low and v 2higher degrees of freedom for specifying the cost function vNIand therefore higher bandwidth when compared to existing a4(a5 a6family oftechniques such as conventional stepped waveformis gained. Secondly, to introduce a minimization technique Al a2 a3assisted with a hybrid genetic algorithm in order to greatlyreduce the computational burden associated with the 2

nonlinear transcendental equations of the selective harmonic (b)Fig. 1. Phase circuit of a multilevel inverter system. (a) Cascaded H-bridgeelimination method [8]. based inverter. (b) Generalized staircase PWM waveform.

The paper is organized as follows. Section II presents theformulation along with analysis for the generalized staircase With the assumption of non-equal dc level of each inverterPWM voltage waveform. Section III discusses the cell, a generalized expression of Bn (i.e. for the single-phaseimplementation ofthe hybrid genetic algorithm. Results for a

numbr ofseleted asesareprovded n Setion~ d case) and for any number of switching angles (even or odd,n ot and provided that the waveform is physically correct and can befinally conclusions are summarized in Section V. implemented) is given by2N-1 Pi1

II. PROBLEM FORMULATION AND ANALYSIS Bn I[V E(VIN1 (-P ) cos nak +

Fourier series expansion ofthe generalized staircase PWM w1 n=1,3,5, k=1output waveform of the single-phase multilevel inverter P2 k PMshown in Fig. I(b) can be easily expressed as follows: V2 (1)cos nk 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 PWM waveform characteristics of odd function n =1,3,5, . .., 2N -1, for single-phase systemsymmetry and half-wave symmetry, the output voltage can be n =1, 5,7, . .., 3N -2, for three-phase system and N is oddreduced to: n =1, 5,7, . .., 3N -1, for three-phase system and N is even

=,H E Bn sin nO (2) M =the numnber of dc sources (i.e. inverter cells) and then=4,3,5,.. product VM VdC is the value of the M dc source

where, Bn is the Fourier coefficient. P1 =N1, P2 =N1+N2,., PM =N1 +N2 +***--+NM,

N1, N2,...,NM are the number of pulses per-quarter cycle at III. HYBRID GENETIC ALGORITHM IMPLEMENTATIONinverter 1,2,...,M , respectively. Genetic algorithms (GAs) are highly suited to searchN = N1 +N2 +... +NM (Number ofpulses per-quarter cycle spaces which are not well defined or have a high number of

local minima, which plague more traditional calculus-basedof them tilevelcinv output ag) search methods. By removing the need for auxiliaryack isthe k switching angle, and information regarding the optimization surface theIn eqn. (3), the polarity ± is positive if PM-1 is an odd computational requirements are greatly reduced.number otherwise it is negative. The GA necessitates the need for the optimizationEqn. (3) has N unknown variables (a, a2 ,..., aN) and a variables ( a,, a 2,..., aN ) to be coded as population ofset of solutions is obtainable by equating N -1 harmonics to strings transformed by three Genetic operators: selection,zero and assigning a specific value to the fundamental. crossover, and mutation.Solutions to eqn. (3) can be obtained through any of those The presented hybrid genetic algorithm combines amethods highlighted in section 1. standard real coded GA and the phase-2 of conventional

However, these methods suffer from various drawbacks, search technique.such as prolonged and tedious computational steps [2], [12] A. Phase-i (Real Coded) Algorithmconvergence to local optima, increase the degrees of the Real coded genetic algorithm is implemented as follows,polynomials if the technique in [3], [4] is considered, etc. 1- A population of Np trail solution is initialized. EachFurther, the number of harmonics eliminated by thesetechniques is linked to the number of switching angles, soluions take n to

a number wiableshence, the more the number of harmonics to be eliminated, dimensions corresponding to the number of variablesthe larger the computational complexity and time. ( a1, a2,... ,aN ). The initial components of a i are

To alleviate such problems, a hybrid genetic algorithm selected in accordance with a uniform distributionapproach is proposed, which solves the same problem with a ranging between 0 and 1.simpler formulation, with any number of levels or switching 2- The fitness score for each solution vector ai isangles and without extensive derivation of analytical evaluated, after converting each solution intoexpressions. corresponding switching instants aa using upper andAn objective function describing a measure of lower bounds.

effectiveness of eliminating selected order of harmonics3- Roulette wheel based selection method is used to producewhile controlling the fundamental must be defined. Two

predominate methods in choosing the switching angles, Np offspring from parents.ac,,a ....aN can be considered: 1) eliminate the lower 4- Arithmetic crossover and non-uniform mutation2~ ~~~~~~~~~~~N

operators are applied to offspring to generate nextfrequency dominant harmonics or 2) minimize the total operator area to tharmonic distortion (THD). The first method is the most generation parents.popular and straightforward of the two techniques, that is to The algorithm proceeds to step 2, unless the best solutionpopla an stagtowr of th tw tehius tha is to does not change for a pre-specified interval of generations.eliminate the lower dominant harmonics and the filter or thenature of the load will take care of the higher residualfrequencies. In this paper the first method is also chosen. PhaHence, the objective function (for the single-phase case) is Algorithmdefined as After the phase- I is halted, satisfying the halting condition

2 B2 +B2 + B2 4 described in the previous section, optimization by directF(a1, a2,---., a N) (B1I 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 solution4 accuracy, the success rate, and the computation time the best

_ VF vector obtained form the phase- I is used as an initial point forma=F,is the modulation index tepae2a Vdc the phase-2.

dc The optimization procedure based on direct search andNtntaVFstefdm m ne systematic reduction in search region is found effective in

and 0<ma <M. solving various problems in the field of nonlinearThe optimal switching angles are obtained by minimizing programming [16]. This direct search optimization procedure

eqn. (4) when it is subject to the constraint of eqn. (6). is implemented as followsConsequently selected harmonics are eliminated. These 1- The best solution vector obtained from the first phase ofswitching angles are generated for different operating points the hybrid algorithm is used as an initial point ax(0) forand then stored in look-up tables to be used to control the phase-2 and an initial range vector is defined asinverter for certain operating point. R (0) =RM~F x Range (7)

(0. a1 <1X2 .< .a-<(N .-2) (6) Where, Range is defined as the difference between theupper and lower bound (the upper and lower bound for each

assumed here that each individual inverter cell is operating atswitching angle here are /2 and 0, respectively) and RMF is frequency of three times of the fundamental one. As a result,a range multiplication factor. The value ofRMF varies there are three switching angles per-quarter cycle at eachbetween 0.0 and 1.0. level of the output waveform. Hence, there are six switching2- Ns trail solution vectors around a(0) are generated angles per-quarter cycle of the output waveform of the

using following relationship, multilevel inverter, which offer elimination of five low ordera a (0) + a (0). * rand (1, n) (8) of harmonics and controlling the fundamental at a certain

th vector, (*) represents value. The switching angles variation against ma for the saidwhere, ai is the i trail solution vector, )represents case is plotted in Fig. 2. It is confirmed that more than one setelement-by-element multiplication operation, and of solutions are exist and for a certain range of ma (i.e. set 1,rand (1, n) is a random vector, whose element value 1.1<ma<1.94, set 2, l.l<ma<2.0 and set 3,varies from - 0.5 to 0.5. 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 thefunction value and find the trail solution set, which proposed method and the results (i.e. simulation andminimizes F(a) and equate it to a(0) as follows, experimental) are shown in Fig. 3 (a)-(d). It is evident that thea(O) =abest (9) targeted harmonics (3rd, 5th, 7th, 9th, and 1 Ith) are eliminated

and the next significant harmonic appearing in the outputwhere, abest is the trail solution set with voltage is the 13th. Good agreement between the simulationminimum F(a). and the experimental one is obtained.

On the other hand, the next significant harmonic could be4- Reduce the range by an amount given considerably shifted further (i.e. 19th) ifthree-phase system is

by R (0) = R(O) * (1- 0,). considered where all triplen harmonics are eliminated by thewhere, ,6 is the range reduction factor, whose typical phase shift between the two phases.value is 0.05. B. Case II. Seven-level inverter with V1 =1, V2= 0.95 and

5- The algorithm proceeds to step 2, 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 are cascadedgenerations. in this case, once again each inverter cell is operated at a three

times of the fundamental frequency. Therefore, there areIV. RESULTS AND DISCUSSION three switching angles per-quarter cycle at output waveform

In order to verify the validity of the proposed algorithm, a of each inverter cell. As a result there are nine degrees ofprogram was developed using M\ATLAB 6.0 software freedom offering the elimination of eight low order ofpackage [17]. The program is run for a number of harmonics and controlling the fundamental component. Theindependent trials. The proposed technique has been applied solution sets for the switching angles for a certain range ofsuccessfully to different cases in order to indicate its ma (set 1, 2.1 <ma<2.9, set 2, 2.1 <ma<2.9 and setruggedness. The simulation results are obtained accordingly 3, 2.92 < ma < 3.0 ) is illustrated in Fig. 4. Fig. 5 (a) depictsusing PSIM software package [18]. the output voltage waveform and its spectrum in Fig. 5 (b)A low-power laboratory five-level inverter prototype based where the absence of selected harmonics (3rd, 5th,. . ,t 7h) iS

on the two IGBT H-bridges configuration was developed and clearly evident for the given fundamental value which is 2.90tested to verify the feasibility and the validity of the p.u. It is worth noting that the most significant order oftheoretical and the simulation findings. The pre-calculated harmonic appears in the output waveform is the 19th and oncePWM signals are implemented using low-cost high-speed again it could be further shifted to the 31St if the three-phaseTexas Instruments TMS320F2812 digital signal processor system was considered.(DSP) board with an accuracy of 20ptseconds. A digital 90 Setreal-time oscilloscope (Tektronix TDS210) was used to 8 Set2display and capture the output waveforms and with thefeature of the fast Fourier transform (FFT), the spectrum of - 70

each of the output voltage was obtained. t 60pA discussion on the results is presented in the following g 50 <

sections. It is worth noting that in the following discussion, 40-the levels of the dc sources are non-equal and can be t 0- -measured; fiurthermore, each dc source has a nominal value M 20 t X e

oftCap.u. Fielvlivrewt0V 1V=08puA. CaseI: ive-level iverter withV1 ~1, V2 080 p.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)levels (i.e. V1 1,V2 0.80 p.u. ) was considered. It is Fig. 2: Switching angles vs. ma (case 1).

90 - -Set- Set 2

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

50

740-

.020

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

2~~~~~~~a 20 -~~~~~oulto ndx(nTekSo~~~~M ~~o 2. 00m5 0H2 Fig. 4: Switching angles vs. ma(caseIi)~~~~~~~~~~~~~~~~_4

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

00 2 00 4 00 a 00IN) I a GO 14 00 IE; 00 I Et 00 00I 0~ ~~I

Tim ilml~~~~~Fc~

Thlk SL. Sto Pv 2.00n MAH(b~~~ ~~Fig. 5: CSewitcingSevn-lesvel minvererwihmR .0)a)Otuotg

6CMH

#0 i 4a 9 I e imprtntthaIte ptiumse beidntfieawthresec tWindow harmonic~~~~~~~~~~performance...... For.hispurpse,the.ota.haroni

____ dstrton(TD)ispltedinFi...fr.hetw.wdCH1 5XV 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~~~~~~~Tureoftehihyinutvela

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

(experimental)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~s

39

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