7/29/2019 IJEST12-04-06-202

1/5

Selective Harmonics Elimination of PWM

Cascaded Multilevel Inverter

ANIKET ANAND1, K.P.SINGH

2

Department of Electrical Engineering

Madan Mohan Malaviya Engineering College Gorakhpur-273010, Indiaaniketanand.mmmec@yahoo.in1, kp1960@rediffmail.com2

Abstract For cascaded multilevel inverter switching angles are calculated by solving nonlinear harmonicsequation to eliminate the selective lower order harmonics. This paper presents a generalized method to calculate

the switching angles based on Newton Raphson method for solving the nonlinear equation which istranscendental in nature. Calculation are performed by taking any random initial guesses and numbers ofiterations are performed for each value to solve the equation in to eliminate a particular harmonics. The anglesobtained are then used to decide the switching pulses for 9-level cascaded multilevel inverter. The simulationresults reveal that method can effectively eliminate the selective lower order harmonics in the output waveformof the inverter, and also low THD (Total Harmonics Distortion). The computational results have been shown

graphically to prove the effectiveness of the method.

Keywords-Multilevel Inverter, Selective Harmonics Elimination, Total Harmonics Distortion (THD), Pulse

Width Modulation(PWM).

I. Introduction

Multilevel inverter is considered as one the most recent and popular type of advances in power electronics. Itsynthesizes desired output voltage waveform from several dc sources used as input for the multilevel inverter.As the number of dc sources is increased, the output voltage waveforms obtained is closer to the sinusoidal

voltage waveform. These multilevel inverters found there application in induction motor drives, static varcompensation, UPS system, laminators, mills, conveyors and compressors. To obtain the sinusoidal voltagewaveform from multiple dc sources the semiconductors switches such as MOSFET/IGBT are switched on and

off in such a way to keep the THD % to its minimum value. These semiconductor switches are of low powerratings but of high switching speed. The multilevel inverter configurations include; flying capacitor topology

[10], diode clamped topology [11] and H-bridge topology. The commonly used switching technique is selectiveharmonics elimination method at fundamental frequency, the harmonics equation obtained is nonlinear in natureand thus different methods are available to solve the harmonics equation to eliminate the particular harmonics.The available methods to solve these transcendental equations include; resultant theory [1]-[3], Nelder Meadmethod based on Mathematica optimization toolbox [9] and Newton Raphson method [4]. The maindisadvantage of using flying capacitor based multilevel inverter is that more number of capacitors are required,

moreover diode clamped multilevel inverter topology [10], [11], which restricts the use of it to the high powerrange of operation. The topology cascaded H-bridge multilevel inverter is advantageous with respect to other

topology as voltage level can be easily increased in steps by in increasing the number of dc sources. An apparentdisadvantage of this topology is that large numbers of switches are required. The topology proposed formultilevel inverter has nine levels associated with a power switches.

II. Cascaded Multilevel Inverter

The general structure of cascaded multilevel inverter is shown in Figure 1. Nine level inverter is discussed herewith four input dc sources equal in magnitude V1=V2=V3=V4. Circuit consists of four H-bridge each having fourswitches to control the output voltage waveform. If number of sources are S in number the output level

associated with it is given as,

N level = 2S+1 (1)

If S = 4, the output waveform has nine levels (4V4, 3V3, 2V2, V1 and 0).

The number of switches used in this topology is expressed as

Aniket Anand et al. / International Journal of Engineering Science and Technology (IJEST)

ISSN : 0975-5462 Vol. 4 No.06 June 2012 2743

7/29/2019 IJEST12-04-06-202

2/5

N switch = 4S (2)

The output voltage of the nine level inverter is given as

V0 = V1+ V2 +V3 +V4 (3)

III. Selective Harmonics Elimination

Pulse width modulation technique is extensively used to eliminate harmful low-order harmonics in inverters. In

PWM control the inverter switches are turned ON and OFF during each half cycle and inverter output voltage iscontrolled by varying the pulse width. Duty cycle for each pulse is decided to control the width of each pulse. Inthis method of selective harmonics elimination the switching angles are calculated and used to control the outputvoltage waveform. The harmonics equations are obtained by performing the Fourier analysis [7]. The derivednonlinear equations are then solved by different available solving tools and methods. Solving tools such asMathcad, Mathematica [9], Maple etc are used to solve these nonlinear equations. The harmonics equations in

this paper are solved by using Newton Raphson method. In order to control the fundamental output voltage andeliminate n harmonics, n+1 equations are needed. The method of elimination will be presented for 9-level

inverter such that the solution of four angles is achieved. The switching angles can be found by solving thefollowing equations.

cos cos cos cos =4 mIcos3 cos3 cos3 cos3 = 0cos5 cos5 cos5 cos5 = 0cos7 cos7 cos7 cos7 = 0 (4)Where modulation index, mI=

THD%=.

100 (5)

The switching angles must lie under given range. 0 .Generalized Method to Solve the Harmonics Equation

The set of equation in (4) are nonlinear in nature Newton Raphson method is used to solve these equation [7],initially guesses are taken and then following steps are perform to calculate the switching angles and iterationsare perform till the harmonics content is reduce up to four decimal places, here three harmonics are taken that is3rd, 5 th and 7th and angles are calculated to remove these three lower order harmonics. The system of nonlinearequation in M variables can be represented as

, , .M I=1, 2, 3M (6)These M equations are obtained for the problem by equating (4) to zero for any M harmonics desired to beeliminated. Equation (4) is written in vector notation as

F () =0 where (7)F= [f1 f2 f3fM]

T an M1 matrix= [ M] T an M1 matrixEquation (23) can be solved by using a linearization technique, where the nonlinear equations are linearalizedabout an approximate solution .the steps involved in computing a solution are as follows

Guess a set of values for call them

= [

] T

Determine the values ofF () =f0 (8)

Aniket Anand et al. / International Journal of Engineering Science and Technology (IJEST)

ISSN : 0975-5462 Vol. 4 No.06 June 2012 2744

7/29/2019 IJEST12-04-06-202

3/5

Neglecting the higher order terms from Taylors series expansion+ [/=0 (9)

/=

(10)

Evaluated at and d= [ d d . d M] T solve (9) for d Repeat above as improved guesses

= +dAlgorithm to Solve the Harmonics Equation

Fig: 1 Steps performed to calculate the switching angles to eliminate selective harmonics

The algorithm shown in figure: 1 is used to calculate the switching angle in MATLAB 7.10.0(R2010a). Steps

given in algorithm are used to perform each iteration and then numbers of iterations are performed to reduce theharmonics value up to four decimal places. The calculated value of switching angles:-

1=0.0409 2=0.5055 3=0.7830 4=1.4534

Simulation Results

Switching angles are calculated by following the above discussed algorithm, and then the calculated values areused to give the firing pulses to the nine level inverter, the staircase voltage waveform is obtained which isshown in figure: 5 when resistive load is applied. The output waveform further improved when RL load is

Start

0.0577, 0.5033, 0.7867, 1.455Taking the initial guesses

= [;;;

a=[-sin(1),-sin(2),-sin(3),-sin(4);-3*sin(3* 1),-3*sin(3* 2),-3*sin(3* 3),-3*sin(3* 4);-5*sin(5* 1),-

5*sin(5* 2),-5*sin(5* 3),-5*sin(5* 4);-7*sin(7* 1),-7*sin(7* 2),-7*sin(7* 3),-7*sin(7* 4)]

b=inv a

c=[cos(1)+cos(2)+cos(3)+cos(4);cos(3* 1)+cos(3* 2)+cos(3* 3)+cos(3* 4);cos(5*1)+cos(5*

2)+cos(5* 3)+cos(5* 4); cos(7* 1)+cos(7* 2)+cos(7* 3)+cos(7* 4)]

k=s+d

Stop

t=[2.69;0;0;0], j=t-c, d=b*j

Aniket Anand et al. / International Journal of Engineering Science and Technology (IJEST)

ISSN : 0975-5462 Vol. 4 No.06 June 2012 2745

7/29/2019 IJEST12-04-06-202

4/5

considered. Current waveform across the RL load is easily depicted in figure: 4. The FFT analysis is done inboth the cases. The THD calculated in case of resistive load is 14.49% while in case of RL load is furtherreduced to 8.08%. The calculation revealed that 3rd,5th and 7th harmonics is significantly reduced up to fourdecimal places and thus overall THD% is reduced.

Fig:2 Cascade multilevel inverter Fig:3 THD in case of RL load

Fig:4 Inverter output current in case of RL load Fig:5 Inverter 9 level output voltage in case of R load

Conclusions

The selective harmonics elimination method is used to eliminate the 3rd, 5th and 7th harmonics and switchingangle calculation is made in that respect. Sugsested algorithm is used to calculate the value of switching anglesand thus switching angles are used to decide the switching pulses for nine level cascaded multilevel inverter .The Total Harmonics distortion is reduced to 8.08% and staircase voltage waveform is obtained which is much

closer to sinsouidal waveform. Therefore, an effective reduction of total harmonics distortation is achieved.

Refrences

[1] Chiasson. J; Tolbert. L; Mckenzie. K; Du. Z(2002). AComplete Solution to the Harmonic Elimination Problem. IEEE.pp. 503-508.[2] Chiasson. J; Tolbert. L; Mckenzie. K; Du. Z(2003). A New Approach to Solving the Harmonic Elimination Equations for a multilevel

Converter. IEEE.pp.640-647.[3] Ghasemi. N; Zare. F; Langton. C; Ghosh. A.(2011). In Proceedings of the 14 th European Power Electronics Confrence, The

International Convention Centre, Birmingham. Pp. 1-9.

[4] Kumar. J; Das. B; Agarwal. P(2008). Selective Harmonic Elimination for a Multilevel Inverter. Fifteenth National Power SystemConfrence(NPSC), IIT Bombay.pp.608-613.

[5] Liang. T.J; Conell; Richard. G(1997). Eliminating Harmonics in a Multilevel Converter using Resultant Theory . IEEE Transactionson Power Electronics , Vol.12.No.6.pp 971-982.

Current

Time (ms)

Vo

lta

e

Time (ms)

Aniket Anand et al. / International Journal of Engineering Science and Technology (IJEST)

ISSN : 0975-5462 Vol. 4 No.06 June 2012 2746

7/29/2019 IJEST12-04-06-202

5/5

[6] Murugesan. M;Sivaranjani. S; Ashokkumar. G; Sivakkumar.R(2011). Seven Level Modified Cascaded Inverter for Induction MotorDrive Application. Journal of Inteernational Engineering and Applications. Pp.36-45.

[7] Patel. H. S; Hoft. R. G(1973). Generalized Techniques of Harmonics Elimination and Voltage Control in Thyristor Inverters: Part I-Harmonic Elimination. IEEE Transactions on Industry Applications, Vol.IA-9, No.3.pp.310-317

[8] Ramani. K; Krishnan. A(2009). An Estimation of Multilevel Inverter Fed Induction Motor Drive. International Journal of Reviews inCompputing. pp.19-24.

[9] Samadi. A; Farhangi. S(2007).A Novel Optimization Method for Solving Harmonic Elimination Equations. The 7th InternationalConfrence on Power Electronics.pp.180-185.

[10] Tolbert. L. M; Khomfoi. S. Chapter31. Multilevel Power Converters. The University of Tennessee.pp.31-1 to 31-50.[11] Yuan. X; Barbi. I(2000). Fundamentals of a New Diode Clamping multilevel Inverter. IEEE Transactions on Power Electronics,

Vol.15,No.4.pp.711- 718.

Aniket Anand et al. / International Journal of Engineering Science and Technology (IJEST)

ISSN : 0975-5462 Vol. 4 No.06 June 2012 2747