Elimination of Selective Harmonics in a Multi-Level Inverter

P.K. Dhal Sathyabama University, Chennai, India

E-mail: pradyumna.dhal@rediffmail.com

Abstract-This paper presents a selective harmonics elimination in Multi-Level Inverter. The basic concept of this reduction is to eliminate specific harmonics, which are generally of the lowest orders, with an appropriate choice of switching angles. This paper employs Homotopy algorithm to solve the transcendental equations for finding the switching angles. This method solves the non-linear transcendental equations with a much simpler formulation and without complex analytical calculations for any number of voltage levels. Also several informative simulation results verifY the validity and effectiveness of the proposed algorithm.

Keyword: Multilevel Inverter, Homotopy Algorithm, Optimization Technique.

I. INT RODUCTION

In switch-mode DC-to-AC power inverters are used in AC motor drive and uninterruptible AC power supplies (UPS) etc. [1-3]. In most cases, low distortion sinusoidal output voltage waveforms are required with controllable magnitude and frequency. Moreover, the power semiconductor switching speed has improved dramatically. Modern ultrafast insulated gate bipolar transistors (lGBTs) demand switching frequency as high as 5 kHz. The DSP-based PWM algorithm practically fails on this region where ANNbased PWM can possibly take over [3]. In recent years, multilevel inverters are widely used as static power converter for high-power applications such as FACTS devices, HYDC light transmission, AC drives, and active filters [1-3]. One of the significant advantages of multi-level configuration is the harmonic reduction in the output waveform without increasing switching frequency or decreasing the inverter power output. The output voltage waveform of a multi-level inverter is composed of a number of levels of voltages, typically obtained from capacitor voltage sources. The socalled multi-level starts from three levels and as the number of levels increases, the output total harmonic distortion (THD) decreases. The number of achievable voltage levels, however, is limited by voltage unbalance problems, voltage clamping requirement, circuit layout, and packaging constraints. The computational delay of this mapping becomes negligible if parallel architecture of the network is implemented by an Application-Specific Integrated Circuit (ASIC) chip. The optimal switching pattern Pulse Width modulation (PWM) strategies constitute the best choice for

978-1-4673-4603-0/12/$31.00 2012 IEEE

C. Christober Asir Rajan Pondicherry Engineering College, Puducherry, India

E-mail: asir _70@pec.edu

high power, three-phase, voltage-controlled inverters and for fixed frequency, fixed-voltage UPS systems. For any chosen objective function, the optimal switching pattern depends on the desired modulation index. In the existing practice, the switching patterns are pre-computed for all the required values of this index, and stored in look-up tables of a microprocessor-based modulator [4]. This requires a large memory and computation of the switching angles in real time is, as yet, impossible. To overcome this problem, attempts were made to use approximate formulas, at the expense of reduced quality of the inverter voltage [5-7].

Recently, alternate methods of implementing these switching patterns have been developed. Without using a real time solution of nonlinear harmonic elimination equation, an ANN is trained off-line to output the switching angles for wanted output voltage [8-12]. The greatest disadvantage of this application is the use in training and stage the desired switching angles given by the solving of the harmonic elimination equation by the classical method, i.e., the Newton Raphson method. This algorithm requires starting values for the angles and does not always converge to the required solution. To give a solution to this problem, powers electronics researches always study many novel control techniques to reduce harmonics in such waveforms. For instance, the transforming of the transcendental non-linear equation into polynomial equations [7], produces a simple algebraic equation to define the harmonic-elimination switching angles [6] and by using piecewise constant orthogonal functions [16].

This paper employs Homotopy algorithm to solve the transcendental equations for fmding the switching angles. This method solves the non-linear transcendental equations with a much simpler formulation and without complex analytical calculations for any number of voltage levels. Also, several informative simulation results verify the validity and effectiveness of the proposed algorithm.

II. HARMONICS IN ELECTRICAL SYSTEMS

In power quality aspects are the harmonic contents in the electrical system. Generally, harmonics may be divided into two types: 1) voltage harmonics, and 2) current harmonics. Current harmonic are usually generated by harmonics contained in voltage supply and depends on the type of the load such as resistive load, capacitive load, and inductive

Elimination of Selective Harmonics in a Multi-Level Inverter

load. Both harmonics can be generated by either the source or the load side. Harmonics generated by load are caused by nonlinear operation of device, including power converters, arc-furnaces, gas discharge lighting devices, etc. Load harmonics can cause the overheating of the magnetic cores of transformer and motors. On the other hand, source harmonics are mainly generated by power supply with nonsinusoidal voltage and non-sinusoidal current waveforms. Voltage and current source harmonics imply power losses, Electro-Magnetic Interference (EMl) and pulsating torque in AC motor drives [13-15]. Any periodic wave form can be shown to be the superposition of a fundamental and a set of harmonic components. By applying Fourier transformation, these components can be extracted. The frequency of each harmonic component is a multiple of its fundamental [4]. There are several methods to indicate the quantity of harmonics contents. The most widely used measure is the total harmonics distortion (THD), which is defmed in terms of the magnitudes of harmonics, H at pulsation n OJ , where OJ is the pulsation of the fundamental component whose magnitude is H, and n is a an integer [7], [16]. The THD is mathematically given by

a L Hn2 THD = !.!:.n=::;2,,--_

HI

s, '-1 tJ.

,1

t

n :i, -1

. . . (1)

+

'Vd. -

FIG. 1: SINGLE PHASE THREE-LEVEL INVERTER STRUCTURE

III. SELECTIVE HARMONIC ELIMINATION (SHE) STRATEGY

For the described study, the classic harmonic elimination strategy was selected. It consists in determining s optimal switching angles. The primary angles are limited to the fIrst quarter cycle of the inverter output line voltage (phase a) "Figure I". Switching angles in the remaining three quarters are referred to as secondary angles. The full-cycle switching pattern must have the half-wave and quarter-wave symmetry in order to eliminate even hannonics. Hence, the secondary angles are linearly dependent on their primary counterparts "Figure 2". The resultant optimal switching pattern yields a fundamental voltage corresponds to a given value of the modulation index, whereas s - 1 low-order, odd, and triple harmonics are absent in the output voltage [8], [11-12].

FIG. 2: OUTPUT VOLTAGE WAVEFORM OF A THREE-LEVEL INVERTER ANGLE SHE-PWM

21

Note that, each of the waveforms have a Fourier series expansion of the form [8].

4V a 1 s V(wt) =sin(nwt) I -I(-I),+1 cos(n8;) 11: n=I,3,5, .. n 1=1

. . . (2)

Where 0 8, 82 'If 8, n/ 2, the Fourier series is summed over only the odd harmonics. Again, the aim here is to use these switching schemes to achieve the fundamental voltage. And to eliminate the fIfth, seventh and 11 th harmonics, etc, for those values of the modulation index m,. m

= HI (4vd/n). The harmonic elimination technique is very suitable for inverters control. By employing this technique, the low THD output waveform without any fIlter circuit is possible. Switching devices, in addition, turn on and off only one time per cycle. "Figure 2" shows a general quarter symmetric inverter waveform.

In the three-phase inverter, the aim is to use this switching scheme to achieve the fundamental voltage and eliminate the fIfth, seventh and 11th harmonics, etc (n = 1, 5, 7, 11, 13, .. ). For those values of the modulation index m, the switching angles 8" 82, es are chosen to satisfy

These equations are nonlinear, contain trigonometric terms and are transcendental in nature. Consequently, mUltiple solutions are possible. A Newton Raphson method has to be fIrst applied to obtain a linearized set of equations [8]. The solution of these equations is achieved by means of the Gauss-Jordan iterative method. In order to obtain convergence with this method, the starting values of switching angles should be close to the exact solution. A great deal of effort has been done in this technique. However after a great computational time and efforts, no optimal solution is usually reached and convergence problems are highly arising especially when the number of equations is increased [4][16]. The application of the ANN to obtain the switching angles can be introduced to overcome the aforementioned difficulties.

11 1 2

12 1 2

11 2

cos( ) cos( ) ... ( 1) cos( )

cos(5 ) cos(5 ) ... ( 1) cos(5 ) 0

cos( ) cos( ) ... ( 1) cos( ) 0

ss

ss

sn s

h m

h

h n n n

++

+

= + + = = + + = = + + =#

(3)

22 Proceedings of7'h International Conference on Intelligent Systems and Control (ISCO 2013)

IV. CASCADED H-B RIDGE MULTI-LEVEL INVE RTE R

"Figure 3" shows the single-phase structure of a cascade multilevel inverter [8]. It consists of a series of H-bridge (single-phase full-bridge) inverter cells, for the output voltage Vi(i = 1,2, V,S with S number of cells employed) three different values (levels), +U , 0, -U by connecting the DC source Ui to the AC output side by different combinations of the four devices [13-15], Noting in this level that the voltages U of the DC sources supplied inverter cells may be different. So, they can or cannot be equal. The output voltage Vi can be expressed as,

." (4) Where " 2 are, respectively, the connection or switching functions of the upper switches (Ki I, Ki 2) of each cell, which defme its states (switch on or off).

The AC output voltage Van(U 2) is, therefore, the sum of all the individual inverter outputs,

s

Van=V,+V2+V+VS= I i=1

Using the connection functions, equation (5) becomes,

." (5)

Van=UI(fll-f'2)+V +US(fS l-fS2) ." (6) For the three-phase system, the output of three identical structure of single- phase cascaded inverter can be connected in either wye or delta configuration. In this case, line voltage can be expressed in term of two phase voltages. For example the line voltage Vah is the potential between phase a phase b which can be expressed as,

Vab = Van -Vbn . . . (7)

The maximum number of the phase voltage levels that can be achieved is 3s, where S is the number of cells or H-bridges used. "Figure 2" illustrates one of the more possible low frequency switching scheme of the output voltage wavefonns

H- irMrter cell.

Q

U, Vi

u v,

u?=v."

U".\.'I P:-;n

U.I v,

FIG. 3: THE SINGLE-PHASE STRUCTURE OF THE CASCADE MULTILEVEL INVERTER

that can be synthesized by the cascaded multilevel inverter of "Figure 1", This switching scheme is designed as a fundamental switching scheme producing a staircase wavefonn U 2 to approximate the desired sinusoid. It represents the typical or generalized multilevel output voltage wavefonn involving pre-calculated or predetennined switching angles modulation methods.

This work is centered on this waveform chosen here for the study. It is a periodic waveform which presents the odd half and quarter-wave symmetric characteristic. It contains 4S switching angles namely aI, az,V, as per cycle (period) and structured by several voltage levels which are equal or not.

t 1'=1,71

FIG. 4: CHOSEN GENERALIZED MULTILEVEL OUTPUT VOLTAGE WAVEFORM

V. REVIEW OF OPTIMIZATION GENERALIZED MULTILEVEL WAVEFORM TECHNIQUES

Elimination of Selective Harmonics in a Multi-Level Inverter

1

FIG. 5: A GENERALIZED WAVEFORM WITH EQUALL Y WIDTH STEPS

t ":

FIG. 6: A REGULAR STAIRCASE GENERALIZED WA VEFORM

FIG. 7: AN ARBITRARY GENERALIZED WAVEFORM

VI. SIMULATION RESULTS

To verify the proposed Homotopy algorithm, a simulation model for a three-phase 7-level cascaded H-bridge inverter is implemented. 5th and 7th harmonics are selected to be eliminated from the output voltage and the fundamental component is specified by the modulation index m. DC source voltages are selected to be VI = VI,vdc = 63.00V, V2 = V2,vdc = 51.00 V, and V3 = V3 Vdc = 60.60 V. The results for phase a are plotted in "Figure 8" which shows the switching angles {8" 82, 83} versus m. Comparing "Figure 8" with the simulation and experimental results of [2] confirms validity of the proposed algorithm. A three-phase induction motor model with the following parameters is attached to the multi-level inverter:

Rated Power = 1/3 hp Rated Current = 1.5 A Rated Speed = 1425 rpm Rated Voltage = 208 V line to line rms at 50 Hz

With the switching angles corresponding to m = 0.52, i.e., 81= 40.0978, 82= 54.3146, 8J= 75.6119 (taken from Figure 3), the simulation results of the 50 Hz three-phase output voltages, both phase and line to line voltages, are presented in "Figure 9". Normalized FFT of the phase a voltage and line to line voltage between phases a and bare shown in "Figure 10" and "Figure I I". Note that 5th and 7th harmonics are zero in phase and line-to-line voltages. Also, it is noted that although phase voltage contains triplen harmonics such as 3rd and 9th, these harmonics do not appear in line to line voltage. THD for the phase voltage and line to line voltage can be computed from the FFT given in "Figure

23

lO" and "Figure 11" which are found to be 46.36%, and 11.50%, respectively.

"Figure 12" shows the three-phase motor currents resulting from applying the voltages of "Figure 9" to the motor. The normalized FFT of phase a current waveform is shown in "Figure 13". The harmonic content of the current is significantly reduced compared to that of the voltage because of filtering by the motor's inductance. THD of the current waveform of phase a, computed using the FFT data of "Figure 13", is found to be 0.76%.

In another set of simulations, the modulation index m is considered to be equal to 0.70 and the frequency is set to 50 Hz. The switching angles are taken from Figure 8 with m = 0.70 (81 = 17.4122, 82 = 41.9400, 83 = 2.5332). The resulting three-phase voltages are simulated and both the phase and line to line voltages are shown in "Figure 14". Normalized FFT of the phase a voltage and line to line voltage between phases a and b are shown in "Figure 15" and "Figure 16" respectively. Similar results as those of the previous case can be deduced again, with a considerable reduction in the phase voltage harmonics. The THD of phase a voltage and line to line voltage between phases a and b are computed using the information given in Figures 15 and 16 and found to be 18.36%, and 10.53%, respectively.

To obtain an accurate result, the harmonic components up to the 200th have been considered in calculating the voltage THD. The phase and line to line voltage THD of the 7-level inverter, as a function of the modulation index m, are shown in "Figure 17". It is seen that the phase voltage THD increases dramatically, when m decreases. The line to line voltage THD, however, increases slightly with decreasing m. Also, for a given m, the line to line voltage THD is much less than the phase voltage THD, due to cancellation of the triplen harmonics in the line to line voltage. For example, at m = 0.63, THD of the output phase voltage is 32.97%, whereas, that of the line to line voltage is 9.42%.

80 ------ - ---- ----- ---- .', ----- : --

-----.,: ----- --or

i -1 . .

----i- - - - - - -2 ____ _____ ____ _____ - 3

: : 50 ____ L _____ i ___ __ ' ____ _ --

o:> 40 ---- - -; _o r --- - , - - - - ; 30 ------ - ---- - - - - - -1- - ----i - -- - - - --- - -; ---- - - - - - ---- - - ; -: : :

------- - - - - -------

1 8.3 0.35 OA 0.45 0.5 0.55 0.6 0.65 0.7 0.75 Mod"I, ... IQIl 1".1., .. ( ... )

FIG. 8: SWITCHING ANGLES VERSUS M

i=.'." " " " ""''''' ' '' +

.'''

.'''

. ,- . 'i 0 . . . . . . . . . . . . . . "' :::;1 -.- : . ; . . ,

! .. ... . . .. . ... ... . . ... ... ... . . . . . . ... . . - - - - -

FIG. 9: PHASE AND LINE TO LlNE VOLTAGE WAVEFORMS FORM= 0.52

24 Proceedings of7'h International Conference on Intelligent Systems and Control (ISCO 2013)

100nr---------------------.

80 E 60

lL

'0 40

'" 20 i

Harmonic Order

FIG. 10: NORMALIZED FFT OF THE PHASE A VOLTAGE WA VEFORM

100nr---------------------.

80 E 60

lL

'0 40

'" 20 i OUL __ __ L_I I_. __ _u.L_

o 5 10 15 20 25 30 Harmonic Order

FIG. 11: NORMALIZED FFToF THE LINE TO LINE VOLTAGE BETWEEN PHASES A AND B

Time [s]

35

FIG. 12: OUTPUT CURRENT WA VEFORMS FORM= 0.52

100 nr--------------__ --__ --__ ----_,

80 E 60

.... '0 40 i '" 20 :!l

OUL---------------- o 5 10 15 20 25 so 35 40 Harmonic Order

FIG. 13: NORMALIZED FFT OF THE PHASE A CURRENT WA VEFORM

n-0.4 0.42 0._ 0.""" OA8 0.&

Time [&1

. . . . -- t ::J -3.4 0.4.2 0.._ 0.46 0.48 0.5 TIJne [s)

FIG. 14: PHASE AND LINE TO LINE VOLTAGE IW A VEFORMS FOR M = 0.70

100nr----------------------__.

80 E 1 60

... (; 40 i1! '" 20 :i1

O-----I I . 5 10 15 20 25 30 35 40 Harmonic Order

FIG. 15: NORMALIZED FFT OF THE PHASE A VOLTAGE WA VEFORM

100nr--------------------------__,

80 E 60

... o 40

OUL-----I --L--- 5 10 15 20 25 30 35 40 H3rmonic Order

FIG. 16: NORMALIZED FFTOF THE LINE TO LINE VOLTAGE BETWEEN PHASES A AND B

30 25

- Ph""se Volt"'g& - Line_to_Line Voltgae

Mod"I .... 'io" I"de .. (I .. )

FIG. 17: THE PHASE AND LINE TO LINE OUTPUT VOLTAGE THD OF A 7-LEVEL INVERTER AS A FUNCTION OF M

VII. CONCLUSION

This paper employs Homotopy algorithm to solve the nonlinear transcendent equations which are formed to fmd switching angles of the devices in a cascaded H-bridge multi-level inverter with unequal DC sources, in order to eliminate some selected harmonics from the output voltage. The proposed algorithm is very effective, efficient and reliable in finding solutions to high-order nonlinear equations. This algorithm solves the nonlinear transcendent equations with a much simpler formulation. Also it can be used for any number of voltage levels without complex analytical calculations. Computer simulations based on a seven-level cascaded Hbridge inverter have been provided for the verification of validity of the proposed algorithm.

REFERENCES [ 1] M. Kojima, K. Hirabayashi, Y. Kawabata, E. C. Ejiogu and

T. Kawabata, "Novel Vector Control System Using Deadbeat Controlled PWM Inverter With Output Le Filter", iEEE Transactions on Industrial Applications, Vol. 40, No. 1, January/February 2004, pp. 132- 169.

[2] P. Z. Grabowski, M. P. Kazmierkowski, B. K. Bose and F. Blaabjerg, "A Simple Direct-Torque Neuro- Fuzzy Control of PWM-Inverter-Fed Induction Motor Drive", IEEE

Elimination of Selective Harmonics in a Multi-Level inverter

Transactions on industrial Electronics, Vol. 47, No. 4, August 2000, pp. 863-870.

[3] J. O. P. Pinto, B. K. Bose, L. E. B. da Silva and M. P. Kazmierkowski, "A Neural-Network-Based Space-Vector PWM Controller for Voltage-Fed Inverter Induction Motor Drive", iEEE Transactions on industrial Applications, Vol. 36, No. 6, November/December 2000, pp. 1628- 1636.

[4] S. sirisukprasert, J. S. Lai and T. H. Liu, "Optimwn Harmonic Reduction With a Wide Range of Modulation Indexes for Multilevel Converters", iEEE Transactions on industrial Electronics, Vol. 49, No. 4, August 2002, pp. 875-88 1.

[5] S. Jian, S. Beineke and H. Grotstollen, "Optimal PWM based on real-time solution of harmonic elimination equations", iEEE Transactions on Power Electronics, Volume: 1 1, Issue: 4, July 1996, Pages:6 12 - 62 1.

[6] S. R Bowes and P. R. Clark, "Simple Microprocessor Implementation of New Regular-Sampled Harmonic Elimination PWM Techniques", iEEE Transactions on industrial Applications, Vol. 28, No. I, January/February 1992, pp. 89-92.

[7] J. N. Chaisson, L. M. Tolbert, K. J. Mckenzie and Z. Du, "A Unified Approach to Solving the Harmonic Elimination Equations in Multilevel Converters", iEEE Transactions on Power Electronics, Vol. 19, No. 2, March 2004, pp. 478-490.

[8] T. H. Abdelhamid, "Application of Artificial Neural Networks to the Voltage Inverters Controlled by Programmed PWM Control Techniques", Proceedings of the iEEA '97, international Conference 7-9 Dec. 1997, Batna University, Algeria. pp. 165-168.

[9] Y. J. Wang and R. M. O'Connell, "Experimental Evaluation of a Novel Switch Control Scheme for an Active Power Line Conditioner", iEEE Transactions on industrial Electronics, Vol. 50, No. 1, February 2003, pp. 243-246.

[ 10] D. Daniolos, M. K. Darwish and P. Mehta, "Optimised PWM inverter Control using Artificial Neural Networks", iEE 1995 Electronics Letters Online No: 19951 186, 14 August 1995, pp. 1739-1740.

25

[ 1 1] A. M. Trzynadlowski, and S. Legowski, "Application of Neural Networks to the Optimal Control of Three-Phase VoltageControlled Inverters", iEEE Transactions on Power Electronics, Vol. 9, No. 4, July 1994, pp. 397-402.

[ 12] M. Mohaddes, A. M. Gole and P. G. McLaren, "A neural network controlled optimal pulse-width modulated STATCOM", iEEE Transactions on Power Delivery, Volwne: 14 Issue: 2, April 1999, pp. 48 1- 488.

[ 13] O. Bouhali, E.M. Berkouk, C. Saudemont and B. Fran