Selective Harmonic Eliminated Pulse-Width Modulation Technique (SHE PWM) applied to Three-level Inverter / Converter

Y. SAHALI', M. K. FELLAH^ Intelligent Control & Electrical Power System Laboratory, University Djillali Liabes , Sidi-Bel-Abbes, Algeria

' email : ya-sahali@yahoo.fr

Abstract-The main interest of this study has been granted to the selective harmonic eliminated pulse width modulation technique, SHE PWM, for the control of single- phase and three-phase full-bridge three-level inverters. This study will be makes for any angle number (even or odd) of switches com- posed the inverter, for purpose of comparison with the results found for only odd number. We explain the resolution method procedure of the nonlinear equation systems in order to achieve the appropriate switching angles. The method of ob- taining the best starting point is also described to overcome the problem of the choice of these points which is one of most difficult tasks associated with the SHE PWM. Finally we will validate the obtained results through some simulations by us- ing MATLAB package.

Index Term- Multilevel inverter (VSIs), three-level in- verter, full-bridge inverter, PWM, selective harmonic elimina- tion technique.

I. INTRODUCTION

Since several years, we attend a renewed interest for the multilevel techniques study and especially (( multilevel in- verters )) [ 1, 2, 31.

These techniques which constitutes a relatively recent research area, still demand and require more developments and optimization in point of view command.

Several controls are used in order to improve the output signals of these converters.

Among these controls, one notes those so-called "Selec- tive harmonic eliminated pulse width modulation SHE PWM or Programmed-PWM method" and whose one makes the object of the present paper. These controls con- sist to calculate the switching instants of the devices in or- der to satisfy certain criteria carrying on the spectrum fre- quency of the wave delivered by the inverter [4]. These se- quences of operation are then memorized and are restored cyclically to assure the control of switches [4].

The criteria ordinarily retained: Eliminating harmonics of specified order ; Eliminating harmonics in a band of specified fre- quencies.

II. PRINCIPLE OF THE MEHOD

This pulse width modulation technique is very important and efficient method, which is widely used in control of the conventional VSIs, in order to improve much more their output voltages quality, will be applied to control a three- level inverter. It consists in forming the output wave

form inverter of a succession of variable width crenels, i.e. chopping the output waveform C times per quarter-cycle [2,5].

Generally, one uses a periodic waveform with quarter- wave symmetry, whose amplitude equals U (unit ampli- tude).

This wave is characterized by the number of pulses (chops or crenels by altemation), C .

Whether C is odd or even, C angles are necessary to determine the width of the whole pulses. C represent also the number of switching angles per quarter-cycle. These switching angles are given in such way to eliminate specific harmonics and to improve the output voltage THD . In the present study we were interested to eliminate the low-order harmonics (3, 5, 7, 9, 11, ... for the single-phase system and 5, 7, 11, 13, 17 .... for the three-phase system) which are most undesirable for the loads such as the electric motors.

Fig. 1 illustrates an example of a generalized three- level SHE PWM waveform generated either by full bridge sin- gle- phase inverter (Fig. 2) or by three-phase inverter (used three full bridge single-phase inverters. Fig. 3).

Because of the odd quarter-wave symmetry property of this waveform, the Fourier series will be simplified and the study will be limited only to the quarter-cycle.

The Fourier series decomposition. that show only the existence of odd harmonic components [6, 7, 81. is given by:

where sin(nwr) dwr (2) 0

After integration and some calculations, an algebraic non linear equations system is obtained. This system sug- gests and admits a possibility of multiple solutions.

The Newton-Raphson iteration method is usually em- ployed to solve such system, which becomes by let ( a = w t ) :

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Fig. 1 Generalized three-level SHE PWM waveform. I

This system, which comprises an additional degree of freedom in order to control the amplitude of the fundamen- tal-frequency component, permits to eliminate C - 1 har- monics from the output voltage. It thus becomes:

Fig 2 Single-Phasefull-bridge inverter

I ! I ,

cos (a1 ) - cos (a2 ) +. . . cos (ac ) = L M 4

(4) Cos(nal)-Cos(i7a2)+ ... f C o s ( n a c ) = 0

h U

where M = 1 is the modulation index.

N The required solution must satisfy the following condi- tion:

where Sp FB is a Single-phase full bridge inverter

Fig. 3 Schematic diagram of three-phase three-level inverter connected in Wye configuration

III. ALGORITHM OF NEWTON'S METHOD [ 5 ]

The resolution of our nonlinear equations system in or- der to find the appropriated switching angles is done by implementation of algorithm of the Newton-Raphson method. This algorithm is characterized by following steps:

cos (al )-cos (a2 ) + ...+ cos (ac ) = z~q 4u

(3) nrt 4u

Cos(nal ) -Cos(na~)+ ...+ Cos(nac) =-I? , ,

1) Propose a set of initial values for a :

n is the odd harmonic order (single-phase system) and odd different from three and from its mul- tiples (three-phase system).

a.i= [ a I ~ , a 2 j , ..., ac jp ( 6 )

where

ai is switching angles. hi is the amplitudes of the odd harmonic compo-

hl is the amplitude of the fundamental component

U is the amplitude of the dc source (dc voltage

with j = O

nents of the output voltage U' ( i = 3,5,. . . ). 2) Calculate the value of:

(7) F ( a j ) = F J of this output voltage U'

where F is the condensed vector format of the non linear

equations system (4). supply).

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3) Linearize equation (7) about ctJ :

F j +[%]"&.j = H (8)

where H is the amplitude of the harmonic components.

f is the functions connecting harmonics with switching angles.

and

where E , is the angular gap, and

C is the number of switching angles.

For the three-phase system (Fig. 5):

Fig. 5 Switching angle trajectories as a function of modulation index

Modulation index

4) Solve dcl' from equation (8) by :

where

(9)

l"[ E].' is the inverse matrix of [e- 5) Change the initial values of each step by :

6) Repeat the process, equation (7) to equation (10)

until da.' satisfied the desired degree of accuracy.

Since the major problem of Newton's method is the knowledge of starting points of switching angles, we will give here and as an example a guide of choice of these val- ues (Fig. 4 and Fig. 5).

For the single-phase system (Fig. 4):

d 2 +

Fig. 4 Switching angle trajectories as a function of modulation index

h e

.- b

CA

5 0

Modulation index

L7L where Eo,lg = - for (C = 3, 5. 7;..) 3 (C+l)

IV. SIMULATION

To verify our study, MATLAB is used as a program- ming and simulation tool. Firstly and after execution of the programs some results giving the variations of the switch- ing angles as function of modulation index are shown in Fig. 6 for the single-phase system and in Fig. 7, Fig. 8 for the three-phase system.

For simulation three-level inverters fed by a 156V source are employed to generate 110Vrms-5OHz output phase voltage. The simulated waveforms are the SHE Pwhl three-level waveforms with different switching an- gles.

The example of a 9-switching angle SHE PWM wave- form generated by the three-phase three-level inverter is shown in Fig. 9

After several simulations and an analysis of the found results we make the following observations:

0 Generally and moving off the 2-switching angles waveform, the output voltage THD is inversely proportional to the modulation index (Fig. 10) ;

0 Increasing the number of switching angles does not necessarily decrease the voltage THD of the three-level inverter, since the total harmonic dis- tortion THD of the three-level inverter using SHE PWh4 varies in an arbitrary manner as func- tion of the number of switching angles per quar- ter-cycle (Fig. 10).

These results confirms very well those found in [5] for only odd number of switching angles.

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3ok==--y ,-j 2o0 0.2 0 4 0.6 0.8 1

'Ob 012 0:4 0.6 0 8 1!2

-- 90

1 '0 0'2 0 4 0.6 0'8 i

20b 0:2 0'4 0:6 0:8 i 112

Fig. 6 Range of switching angles as a function of modulation index : single-phase system ( c is even or odd number).

-200 1- 0.2 0.4 0.6 0.8 1

loo/

'0 - 0.2 0.4 0.6 0.8 1

70w 60

0.2 0.4 0.6

40 50E 30

20

10 i L

OO 0.2 0.4 0.6 0.8

Fig. 7 Range of switching angles as a function of modulation index: three-phase system ( c is even number) g.=:

B z c a ' ' Modulation

index

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90

80

70

60

50

40

30

20

"0 0.2 0.4 0.6 0.8 1 1.2

1.2

I Oo 0.2 04 0.6 0.8 i i 2

Odd I Jnzbe r

90

80

70

60

50

40

30

20

I

90

80

70

60

50

40

30

20

10

'0 0 2 04 06 0 8 i 1 2

90

80

70

60

50

40

30

20

"0 0 2 04 06 0 8 1 1 2

90

80

70

60

50

40

30

20

i o I

Oo 0.2 0.4 0.6 0.a 1 1.2 I

Fig. 8 Range of switching angles as a function of modulation index : three-phase system ( c is odd number).

400-1

200

0

-200

0 10 20 30 40 -400

Fig. 9 The 9-switching angles SHE PWM wavefonn and his fre- quency spectrum, THD = 36.69 %

Other founded remarks are counted by the following points:

A . Maximum Value of the niodzilation index

%Case of three-phase system : Concerning the maxi- mum modulation index value as function of the number of switching angles, we notes, from the founded solutions (in addition to that of 16- switching angle) and from Fig. 11, that this parameter changes in an arbitrary manner if the number angles is even. On the other hand, it tends to be constant and stable in the contrary case (odd case, Fig. 11). Thus it is better to choose the solutions with an odd num- ber.

%Case of single-phase system: for this case the problem does not arise any more because the maximum modulation index remains generally constant whatever the parity of the number angles (odd or even).

Modulation mdev

Fig 10 Mammum value of modulation index as function of the num- ber of switching angles

0 . 5 1 5 i o 15

Number of switching angles

Fig. 11 Maximum value of modulation index as function of the num- ber of switching angles

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B. Choice between solutions

In the majority of the cases, several adequate solutions have to be obtained in order to eliminate the same harmon- ics from the output voltage inverter (Fig. 12). It is the rea- son for which and until these last years, we needed to cal- culate the total harmonic distortion value, which is chosen as criteria of the performance harmonic, for the different solutions for purpose of comparison. The final solution and which must be adopted is that presenting the low value, i.e. presenting the minimum harmonic content. Currently, it is necessary to have a compromise between this value and the value of the first remaining harmonic that is not cancelled.

BO, , , I 1001

60

50

.20;------- 02 04 06 n e i 40b 0'2 04 06 0 8 1

Fig 12 Two solutions for eliminating 5"' harmonic of the output voltage

Remark- It exist another method called (( Resulting the- ory )) for the determination of switching angles, whose ad- vantage dwell in the fact that the lowest remaining harmon- ics have the reduced amplitudes compared to that cited in this paper. This method, that made the object of another study in our research group. will be published subsequently and compared with the precedent method.

V. CONCLUSION

This study permitted us to well understand the principle of an alternative control among several existing alternative techniques to ensure the control of the three-level inverter devices.

Since each pulse-width modulation technique presents advantages and inconveniences, and the selection depends on given performance specifications, one notes that the Se- lective Harmonic Eliminated technique SHE PWM is char- acterized by the three following advantages:

The command instants of the devices that are pre- viously known ; It permits to selecting the undesirable and un- wanted harmonics which can be eliminated. It also permits to control the amplitude of the fun- damental-frequency component.

This improves the efficiency of the inverter-load associa- tion by reducing the torque ripple, for instance, in the ad- justable speed drives applications.

The main inconvenience of the applied Newton's tech- nique for the resolution of non linear equation systems as- sociated with this technique resides in the choice difficulty

-

-

-

of the initial values of switching angles. It is the reason for which this paper comes to contribute to give a choice guide of these values of some solutions.

At the beginning of control development using a pro- grammed-PWM method, one worried to determine se- quence orders (sequences of control) that eliminate the lowest harmonic components from the output voltage be- cause that facilitates filtering (harmonics to filter will be shifted to higher frequency). Now the application is taken into consideration, on seek, for example, to minimize the total harmonic distortion, which is an extension to the pre- sent work. It is deserved to be also another axis of research in our laboratory to try to complete works on three-level inverters and whose results indicates that the harmonic dis- tortion is significantly reduced. These results will be pub- lished subsequently.

VI. RERERENCES

[ l ] L. M. Tolbert, F. Z. Peng et T. G. Habetler. (( Multilevel Converters for Large Electric Drives )) IEEE Transactions on Indirstr?; Applicotions. T701.3j. No. 1. Jrniiioi?/Febriiai>, IYYY.

[2] G.Seiguier, F.Labrique et R.Baussiere, (( Les convertisseurs de l'elec- tronique de puissance )) T. 4 second edition Tec ond Doc 1995.

[3] A. NABAE, I. TAKAHASHI et H. AKAGI, (( A New Neutral-Point- Clamped PWM Inverter )) IEEE Transactions on fiidiish? Applico- tions. T701.1A-l 7. No.5, Septeniber/October 1981.

[4] G. Grellet, G. Clearc, G Actionneurs electriques )) Edirion 2000.

[5] SSirisukprasert, (( Optimized Harmonic Stepped-Waveform for Multi- level Inverter )) Moster thesis 1999 at Virginia Polj,technic Institiire

[6] H.S.Pate1, R.G.Hofi, (( Generalized Techniques of Harmonic Elimina- tion and Voltage Control in Thyristor Inverters: Part I1 - Voltage Control Techniques )) IEEE Transactions on Indiistq, Applications, Vol.IA-IO, No.5, Septeniber/October 1971

[7] H.L.Liu, G.H.Cho et S.S.Park. Optimal PWh4 Design for High Power Three-Level Inverter Through Comparative Stumes B fEEE Trans- actions On Power Elechonics, Vol.10, No. 1. Janiioiyl995.

[SI P.N. Enjeti et R. JaMi, (( Optimal Power Control Strategies for Neutral Point Clamped W C j Inverter Topology )) IEEE Transactions on Indinti?. Applications, VoI.28, No.3, Mqihine 1992.

Y. SAHALI was bom in Sidi-bel-Abbes, Algeria. She received the Eng. degree in Electro technical engineering from the Djillali LIABES Univer- sity of Sidi-Bel-Abbes, Algeria. She is currently worlung the Master de- gree. She is a member of the Intelligent Control and Electrical Power Sys- tems Laboratory at this University.

M. K. FELLAH was bom in Oran, Algeria, in 1963. He received the Eng. degree in Electrical Engineering from University of Sciences and Technol- ogy, Oran, Algeria, in 1986, and The Ph.D. degree from National Poly- technic Institute of Lorraine (Nancy, France) in 1991. Since 1992, he is Professor at the University of Sidi-bel-Abbes (Algeria) and Director of the Intelligent Control and Electrical Power Systems Labo- ratory at this University. His current research interest includes Power Elec- tronics, HVDC links, and Drives.

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