2013_4_4_383_393

Journal of Network & Information Security 4:4 (2013) 383-393

2160-9462 / Copyright © 2013 Binary Information Press

September 15, 2013

SHE of the Optimal PWM Inverter based on CFT

Zeying ZHI, Rucheng HAN

School of Electronic Information Engineering, Taiyuan University of Science and Technology, Taiyuan 030024, China

Abstract

Due to the difficulty of calculating switching angles, the values of the angles were achieved by using the CFT

and programs of MATLAB. Cascaded multilevel inverter was adopted in the structure of the main circuit .This

design can implement the multiple-phase shifting and meet the need of using the high voltage and large power

equipments. In this paper a method to SHE (Selected Harmonic Elimination) choosing initial values was

presented in a full-bridge voltage source inverter .The model of the SHEPWM and the results were given. And

the online real-time control of SHE was realized. The simulation and the experimental results were separately

done by MATLAB and DSP. So the rationality of the design was verified.

Keywords: Cascaded Multilevel Inverter; Curve Fitting Technique (CFT); Selected Harmonic Elimination

(SHE); Optimal PWM

1. Introduction

With widespread application of the inverter, the harmonic generated by the inverter will severely

harm the power grid, electrical equipment and communications network, thus the research of

harmonic elimination inverter is getting more and more important. Cascaded multilevel inverter[1]

was adopted in the structure of the main circuit to meet the need of using the high voltage and large

power equipments and implement the multiple-phase shifting and overlapping. Compared with the

other PWM technique under the same waveform distortion level, SHEPWM can reduce the

switching frequency, selected eliminate the certain lower harmonics, reduce the switching loss and

the current pulsation, save energy and improve the output quality of power system[1-3], so the

research on SHE technique is not only the important meaningful in theory but also valuable in

application[4-5].

- This work is supported by the Coship Electronics Science and Technology Innovation Fund project (NO.TZ201318) Corresponding author.

Email addresses: [email protected] (Zeying ZHI)

mailto:[email protected]

384 Z. Zhi et al. / Journal of Network & Information Security 4:4 (2013) 383-393

The optimal PWM technique is a common used method in inverter controlling. But it is difficult to

online calculate and real time control for the mathematic model of harmonic elimination PWM

inverter is nonlinear transcendental equations. So optimal PWM switching angles is always off-line

calculation, it use switching-node-preset to control which is unfavorable to online voltage regulation.

The online calculation of optimal PWM switching angles based on curve fitting technique make it

possible to regulate the voltage of inverter using optimal PWM technique under the lower switching

frequency. The optimal PWM technique purse the total harmonic content THD of output voltage

minimum. The optimal PWM technique can effective restrain harmonic of output voltage and get

ideal output waveform under lower switching frequency in theory.

2. Configuration of Cascaded Multilevel Inverter

It shows a single-phase topological structure of cascaded multilevel inverter in Figure 1, the

configuration of each inverter is same, which is expressed by a single-phase full bridge inverter. The

configuration of cascaded multilevel inverter allows every module to separation control. The number

of module is equal to the number of needed voltage source. The relationship between the number of

level NL and the number of module M can be show as:

M= (NL-1)/2 (1)

The number of module is the number of H-bridge inverter, if they are series connection a

multilevel output voltage would be produced, the total output voltage is equal to the sum output

voltage of every module, we have:

V0=V_M1+V_M2+V_M3+…+V_Mm (2)

In which V_M is the output voltage of the m module.

Each module has its own DC voltage( DCV ) and 4 switching devices, if switching device of the m

module is S1m, S2m, S3m and S4m, (switch on is “1”, switch off is “0” in Tab.1). As show in Figure

2, each module produce a three level output + DCV , 0, - DCV , DC voltage can be sequential connected

to AC by these 4 switching devices. As show in Fig.1, output voltage is corresponding to power

machine switch states of a multilevel inverter with 5 level module configuration. For each output

voltage contains 5 level: +2 DCV , + DCV , 0, - DCV , -2 DCV and needs 2 modules.

3. Optimal PWM Control Strategy

It is a controlling design of 5 level cascaded multilevel inverter in Figure 2, if DC voltage of each

cascaded multilevel inverter module VDC=20V, as show in Tab.2, the operation of cascaded

multilevel inverter is most depend on module 1, unless q>0.5, module 2 could participate in

operation. Although 5 level cascaded multilevel inverter need 8 power machines, but these 8 power

machines could not be fully utilized, thus reduce the voltage stress of power machine. As

single-phase type inverter shown in Figure 2, maximum of fundamental amplitude q is 1.0, too large

will lead to some iterative methods (such as the Newton iteration method to convergence).

In which , VDC is DC-side line voltage

Z. Zhi et al. / Journal of Network & Information Security 4:4 (2013) 383-393 385

V01 is output fundamental voltage of invert

q is fundamental amplitude

V1-M1 (V2-M2) is output fundamental voltage of module 1(2)

q-M1 (q-M2) is output fundamental voltage of module 2(1)

V-M2

V-Mm

V-M1

Module m

Module 2

Module 1

Vphase (Vo)

S4m

S32

S22

S1m

S42

S2m

S3m

S12

S41S31

S21S11

VDC

VDC

VDC

Fig. 1 Single-phase Cascaded Multilevel Inverter

Fig. 2 Single-phase Voltage Type Inverter Circuit

Table 1 Switch state of 5 level cascaded multilevel inverter power machine

Output Voltage S11（ 31S ） S21（ 41S ） S12（ 32S ） S22（ 42S ）

+2 VDC, 1 0 1 0

+ VDC, 1 0 1 1

0 1 1 1 1

0 0 1 1 0

- VDC, 0 1 1 1

-2 VDC, 0 1 0 1


Table 2 Multilevel Controlling Design of 5 Level Cascaded Multilevel Inverter

V01（peak）（V） q V1-M1（peak）（V） q-M1 V2-M2（peak）（V） q-M2

0 0 0 0 0 0

0.2VDC 0.1 0.2 VDC 0.2 0 0

0.4 VDC 0.2 0.4 VDC 0.4 0 0

0.6 VDC 0.3 0.6 VDC 0.6 0 0

0.8 VDC 0.4 0.8 VDC 0.8 0 0

VDC 0.5 VDC 1 0 0

1.2 VDC 0.6 0.2 VDC 0.2 VDC 1

1.4 VDC 0.7 0.4 VDC 0.4 VDC 1

1.6 VDC 0.8 0.6 VDC 0.6 VDC 1

1.8 VDC 0.9 0.8 VDC 0.8 VDC 1

2 VDC 1 VDC 1 VDC 1

4. The Foundation of the Harmonic Elimination Model

The fundamental method of harmonic elimination PWM controlling is: we can get PWM control

Fourier series expansion through Fourier series analysis, draw Fourier series expansion, we take

pulse phase angle as the unknown, make certain harmonic zero, then we could get a set of nonlinear

equations, which is the harmonic elimination model, According to the results from solving the model

to control, these certain lower harmonic would be produced. The foundation of harmonic elimination

model is related to the control method of PWM, there we take voltage type inverter for an example,

model can be summarized into two based different PWM: unipolar impulse control model and

bipolar impulse control model. In this paper we only discuss unipolar impulse control model. As

show in Figure 2, in order to reduce switching loss, we can take a switch of the same bridge arm

(such as S2 or S4) maintaining conducting state during the half cycle under this control mode. We

can get PWM waveform by switching of another switch of bridge arm. Figure 3 is output voltage

wave of unipolar impulse control model, Fourier series expansion of unipolar wave is:

1

1

[ ( 1) ( )],

0, 1,2,3,...

0, 2,4,6,...

41,3,5,...

N

k

k

n

n

DCn COS k

b n

a n

Ua n n

n

(3)

Fig. 3Unipolar Impulse Control Output Voltage Wave (Forward S1, S4 on-off transform, inverse S2, S3 on-off

transform)


5. The Curve-fitting from MATLAB and Results

5.1. The Initial Value and Calculation Result of Switching Angle for the

Single Phase Inverter

How to give initial value of switching angle in single-phase inverter is founded by tentative

successive approximating to low order nonlinear equations. Some literatures presented initial value

selection method of switching angle such as [6]. In this paper the curve fitting method is using to

solve[7]. As show in Fig.4 it is the initial value of first switching angle variation curve (when

fundamental amplitude q=0) 1 with the number of switch angle.

In which ordinate 1 is in degrees “ ”, abscissa N is a dimensionless quantity. In Figure 4 four

points is obtained by using tentative successive approximating, then we can get fitting curve by

MATLAB curve-fitting.

Fig. 4 The Distribution Diagram of Value of the First Switching Angle in Inverter

The fitting results:

General model Power1:

f(x) = a*x^b in which a, b is undetermined coefficient

Coefficients (with 95% confidence bounds):

a = 64.0504 (64.0504, 64.0504)

b = -0.8321 (-0.8321, -0.8321)

Goodness of fit:

SSE: 3.504e-013

R-square: 1

Adjusted R-square: 1

RMSE: 4.186e-007

Regression curve is f(x) = 64.0504*x^(-0.8321)

1 = 64.0504*N^(-0.8321)


We have: 1 = 2/2 = 3/3 …= /N N , (4)

Then we know 1 the fitting results is exponential function distribution, the variation curve of

first switching angle initial value with the number of switch angle is exponential function. Each

switch angle initial value is interval distribution by tentative successive approximating.

And 0〈 1 〈 2 〈 3 〈 4 〈 5 〈 6 〈 7 〈 N 〈 / 2 , we can take all switching angle initial value by

these 2 conditions. The more of the switching angle number N, the small of first switching angle

value. In practical engineering application, switching angles number should not get too much, or it

will bring hardware unnecessary trouble and even impossible to achieve[8]. The interval between

switching angle become more and more small with the increasing of fundamental amplitude q.

Table 3 The Undetermined Coefficient in Exponential Function when the Switching Angle Number N is Different

Undetermined Coefficient N=2-8 N=9-22 N=23-79 a 64.0504 74.3803 84.7724

b -0.8321 -0.9375 -0.9871

The initial value of the first switching angle between angle 2 and 79can be obtained according to

the given undetermined coefficients a and b in the table 3 and equation (4) ,then the initial value of

all switching angle can be got by equation (4).The value of twenty one switching angle is showed in

Table 4(q=0.9) and in Table 5(q=1.0). The value of 35 switching angle is showed in Table 6(q=0.9) .

The value of 57 switching angle is showed in Table 7(q=0.85).

Table 4 The Value of Twenty One Switching Angles ( 0.9q )

3.935 8.392 11.814 16.786 19.711 25.187

27.637 33.601 35.605 42.035 43.635 50.505

51.753 59.036 59.978 67.652 68.362 76.403

76.947 85.299 85.757

Table 5 The Value of Twenty One Switching Angles ( 1.0q )

3.845 8.245 11.542 16.486 19.245 24.722

26.971 32.949 34.719 41.167 42.496 49.366

50.308 57.547 58.164 65.705 66.058 73.837

73.995 81.936 81.975

5.2. Initial Value Selection and Calculation Results of Three-phase

Inverter Switching Angle


Two groups solution of low order 5,7 switching angles of three-phase inverter can be obtained by

test successive approximation when fundamental amplitude q=0.For each in equations (5)is

differentiable at any time, then the solutions of the equations should be continuously differentiable.

The law two groups solutions to the initial value is derived by lots of trying.

cos( 1)-cos( 2)+cos( 3)-cos( 4)+cos( 5)-cos( 6)+cos( 7)-cos( 8))= 4

q

;

cos(3* 1)-cos(3* 2)+cos(3* 3)-cos(3* 4)+cos(3* 5)-cos(3* 6)+cos(3* 7)-cos(3* 8)=0;


cos(7* 1)-cos(7* 2)+cos(7* 3)-cos(7* 4)+cos(7* 5)-cos(7* 6)+cos(7* 7)-cos(7* 8)=0; (5)


cos(11* 1)-cos(11* 2)+cos(11* 3))-cos(11* 4)+cos(11* 5)-cos(11* 6)+cos(11* 7)-cos(11* 8)=0;



Table 6 The Value of Thirty Five Switching Angles ( 0.9q )

2.441 5.078 7.325 10.154 12.211 15.235

17.102 20.314 21.997 25.397 26.903 30.482

31.817 35.571 36.745 40.665 41.688 45.766

46.648 50.877 51.628 55.997 56.638 61.139

61.674 66.297 66.746 71.488 71.865 76.706

77.028 81.956 82.244 87.228 87.497

Table 7 The Value of Fifty Seven Switching Angles ( 0.85q )

1.534 3.135 4.597 6.272 7.662 9.411

10.731 12.548 13.798 15.684 16.866 18.822

19.936 21.958 23.007 25.097 26.082 28.236

29.156 31.376 32.236 34.519 35.319 37.661

38.403 40.805 41.493 43.948 44.585 47.095

47.683 50.246 50.785 53.397 53.892 56.552

57.005 59.709 60.126 62.868 63.252 66.035

66.385 69.204 69.526 72.377 72.675 75.553

75.833 78.734 78.998 81.916 82.168 85.102

85.345 88.289 88.528

5.2.1. The Law First Groups Solutions to the Initial Value when q=0

1 0


2

120

1N

...

1

60( 1)

1K K

K

N

(6)

…

2

60( 1)

1N

N

N

60N

In which, 13 NK and k is odd. The tracking of every switching angle k changing with q is

close to the segment line.

5.2.2. The Law Second Groups Solutions to the Initial Value when q=0

1

120

1K K

K

N

60N (7)

In which, 11 NK and K is odd. The solution tracking is the same as the first group

solution, and just a little change on distribution.

Practice proves that method these two groups solution to initial value is effective, it is enough to

solve N=100 in engineering application.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

Fig. 5 The First Switching Angles Solutions Trajectories for N=19


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

100

data1

data2

data3

data4

data5

data6

data7

data8

data9

data10

data11

data12

data13

data14

data15

data16

data17

data18

data19

Fig. 6 The Second Switching Angles Solutions Trajectories for N=19

6. Conclusion

Due to the difficulty of calculating switching angles initial value of SHE, we proposed using the CFT

and program of MATLAB to optimal switching angles initial values, then other switching angles are

calculated under different fundamental amplitude. The harmonic elimination PWM model is given

according to needed, low order odd harmonic is eliminated. Some certain harmonics can be removed

by optimal PWM technique. The inverter could get good output voltage waveform under low

switching frequency, and power output quality is improved. The test results of PWM output and

harmonics spectrum are got by MATLAB simulation and experiment using TMS320F2812

development board of DSP system. The output voltage PWM wave and harmonics spectrum figure

of 5 level cascaded multilevel inverter are show in Fig.6 by MATLAB simulation and DSP

experiment when q=0.4, it means some certain low order odd harmonic is eliminated by SHE, output

voltage waveform is improved. Thus verification the initial value by curve fitting technique and the

other switch angle calculating depend on this initial value are effective.


(a) Simulation

(b) Experimental

Fig. 7 PWM Wave Form and Harmonic Spectrum of the 5-level Cascaded Multilevel Inverter Output Voltage(V0) or

q=0.4 (a) Simulation (b) Experimental

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2013_4_4_383_393

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