International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 10, October 2012)237
THD Analysis for Different Levels of Cascade Multilevel
Inverters for Industrial Applications
Dr. Jagdish Kumar
Department of Electrical Engineering, PEC University of Technology, Chandigarh
Abstract —Cascade Multilevel Inverters are very popular and have many applications in electric utility and for industrial drives. When these inverters are used for industrial drive directly, the THD contents in output voltage of inverters is very significant index as the performance of drive depends very much on the quality of voltage applied to drive. In this article, the THD contents of 7, 11 and 15 level cascade multilevel inverters have been analysed. The THD depends on the switching angles for different units of multilevel inverters, therefore, the switching angles are calculated first by using N-R method where certain number of harmonic components has been eliminated. Using calculated switching angles, THD analysis is carried out analytically as well as using MATLAB simulation (both results are in close agreement). As per IEEE-519 standard, the output voltage produced should satisfy the limit on THD. It has been found that the fifteen level CMLI satisfies this limit while seven and eleven level CMLI violates this limit.
Keywords — Cascade multilevel inverter, harmonic
elimination, modulation index, switching angles, total harmonic distortion.
I. INTRODUCTION
Since past decade, multilevel inverters have drawn increasing attention because of their promising applications in power systems and industrial drives. They can be efficiently used in the distributed energy systems in which, output ac voltage is obtained by connecting dc sources such as batteries, fuel cells, solar cells, rectified wind turbines etc at input side of the inverters. The ac output voltage obtained from the inverters can be fed to a load directly or interconnect to the ac grid without voltage balancing problems. In addition, the multilevel inverters are used as voltage source inverters (VSIs) in the static synchronous compensator (STATCOM), a reactive power compensating device used for voltage regulation in power systems [1]-[7].
The multilevel inverters offer several advantages as compared to the hard-switched two-level pulse width modulation inverters, such as their capabilities to operate at high voltage with lower dv/dt per switching, high efficiency, low electromagnetic interference etc [2]-[4].
High magnitude sinusoidal voltage with extremely low distortion at fundamental frequency can be produced at output with the help of multilevel inverters by connecting sufficient number of dc levels at input side. There are mainly three types of multilevel inverters; these are a) diode- clamped, b) flying capacitor and c) cascade multilevel inverter (CMLI). Among these three, CMLI has a modular structure and requires least number of components as compared to other two topologies, and as a result, it is widely used for many applications in electrical engineering [1], [5].
To produce multilevel output ac voltage using different levels of dc inputs, the semiconductor devices must be switched on and off in such a way that the fundamental voltage is obtained as desired along with the elimination of certain number of higher order harmonics in order to have least harmonic distortion in the ac output voltage. For switching the semiconductor devices, proper selection of switching angles is must. The switching angles at fundamental frequency, in general, are obtained from the
solution of non linear transcendental equations
characterizing harmonics contents in the output ac voltage; these equations are known as selective harmonic elimination (SHE) equations. As the SHE equations are non linear transcendental in nature, their solutions may have simple, multiple and even no roots for a particular
value of modulation index (m), moreover, it is difficult to
solve these equations. A big challenge is how to get all possible solution sets where they exist using simple and less computationally complex method. Once these solution sets are obtained, the switching angles producing minimum total harmonic distortion (THD) in the output ac voltage are selected for switching of the power electronics devices.
In [3]-[4], iterative numerical techniques have been implemented to solve the SHE equations producing only one solution set, and even for this, a proper initial guess
and starting value of m for which the solutions exist are
required, in general, it is difficult to guess the initial
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 10, October 2012)238
In [8]-[9], theory of resultants of polynomials and the theory of symmetric polynomials has been suggested to solve the polynomial equations (these polynomial equations are obtained from the transcendental equations).
A difficulty with these approaches is, for several H-bridges connected in series, the order of the polynomials become very high, thereby making the computations of the solutions of these polynomials very complex. Moreover, these techniques have been applied up to 11-level multilevel inverters only due to the computational complexity associated with these techniques. Optimization technique based on Genetic Algorithm (GA) is proposed for computing switching angles only for 7-level inverter in [6], and even for the implementation of this method, proper selection of certain parameters such as initial population
size, mutation rate etc are required, thereby
implementation of this method becomes difficult for higher level multilevel inverters.
In the present work, 7-level, 11-level and 15-level CMLIs are employed to generate ac output voltage producing different magnitudes of THD at different values of modulation indices for comparison purpose. The THD produced by 7-level inverter is more than IEEE-519
standard [10] for all values of m, THD produced by
11-level inverter is satisfying this standard for some values
of m (not for all values) while THD produces by 15-level
inverter is well within the limits imposed by IEEE-519
standards for complete working range of m. The switching
angles have been computed by the implementation of Newton Raphson (N-R) numerical technique in a particular way producing complete solutions for working range of modulation index without much computational complexity.
II. CASCADE MULTILEVEL INVERTER
The cascade multilevel inverter consists of a number of H-bridge inverter units with separate dc source for each unit and is connected in cascade or series as shown in Fig. 1. Each H-bridge can produce three different voltage
levels: +Vdc,0 and –Vdc by connecting the dc source to ac
output side by different combinations of the four switches
S1, S2, S3, and S4. The ac output of each H-bridge is
[image:2.612.326.555.142.530.2]connected in series such that the synthesized output voltage waveform is the sum of all of the individual H-bridge outputs. By connecting sufficient number of H-bridges in cascade and using proper modulation scheme, a nearly sinusoidal output voltage waveform can be synthesized.
Fig. 1. Single-phase cascade multilevel inverter topology.
The number of levels in the output phase voltage and
line voltage are 2s+1 and 4s+1 respectively, where s is the
number of bridges used per phase. For example, three H-bridges, five H-bridges and seven H-bridges per phase are required for 7-level, 11-level and 15-level multilevel inverter respectively. Fig. 2 shows a typical waveform produced by 7-level CMLI.
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 10, October 2012)239
In the Fig. 2, α1, α2 and α3 are the switching angles for
three H-bridges in each phase, and β1, β2 and β3 are
corresponding supplementary angles for α1, α2 and α3. The
magnitude and THD content of output voltage depends very much on these switching angles, therefore, these angles need to be selected properly.
2 t
3Vdc
-3Vdc
1
[image:3.612.50.291.213.335.2] 2 3 3 2 1
Fig. 2. Output phase voltage waveform for 7-level CMLI.
III. SELECTIVE HARMONIC ELIMINATION EQUATIONS
In general, the Fourier series expansion of the staircase output voltage waveform as shown in Fig. 2 is given by [4].
)
1
(
)
sin(
))
cos(
...
)
cos(
)
(cos(
4
)
(
2 ,... 5 , 3 , 1 1t
k
k
k
k
k
V
wt
v
s k dc an
Where s is the number of H-bridges connected in
cascade per phase and k is order of harmonic components.
For a given desired fundamental peak voltage V1, it is
required to determine the switching angles such that 0 ≤ α1
< α2 … < αs ≤ π/2 and some predominant lower order
harmonics of phase voltage are zero. Among s number of
switching angles, generally one switching angle is used for
fundamental voltage selection and the remaining (s-1)
switching angles are used to eliminate certain
predominating lower order harmonics. In three-phase power system, triplen harmonic components are absent in line-to-line voltage, as a result, only non-triplen odd harmonic components are present in line-to-line voltages [4-5].
From equation (1), the expression for the fundamental voltage in terms of switching angles is given by
)
2
(
))
cos(
...
)
cos(
)
(cos(
4
1 2 1V
V
sdc
Moreover, the relation between the fundamental voltage and the maximum obtainable voltage is given by modulation index.
The modulation index, m, is defined as the ratio of the
fundamental output voltage (V1) to the maximum
obtainable fundamental voltage. The maximum
fundamental voltage is obtained when all the switching
angles are zero i.e. V1max = 4sVdc/π, therefore,
m = πV1/4sVdc [1].
For 7, 11 and 15-level cascade multilevel inverters,
s = 3, s = 5 and s = 7 respectively. Number of degrees of
freedom available is equal to s; one degree of freedom is
used to choose the value of V1 and the remaining degrees of
freedom are used to eliminate the lower order harmonic components. For example, in case of 7-level CMLI, only
two harmonic components (in general, 5th and 7th) can be
eliminated, similarly for 11-level CMLI, four harmonic
components (i.e. 5th, 7th, 11th and 13th) and in 15-level
CMLI, six harmonic components (i.e. 5th, 7th, 11th, 13th, 17th
and 19th) can be eliminated.
From equation (1), the expressions for fundamental
voltage in terms of m, and lower order harmonic
components, when they are eliminated, can be written as for 7-level CMLI:
m
3
)
cos(
)
cos(
)
cos(
1
2
3
0
)
5
cos(
)
5
cos(
)
5
cos(
1
2
3
0
)
7
cos(
)
7
cos(
)
7
cos(
1
2
3
(3)For 11-level CMLI, corresponding equations are as follows:
m
5
)
cos(
...
)
cos(
)
cos(
1
2
5
0
)
5
cos(
...
)
5
cos(
)
5
cos(
1
2
5
0
)
7
cos(
...
)
7
cos(
)
7
cos(
1
2
5
0
)
11
cos(
...
)
11
cos(
)
11
cos(
1
2
5
0
)
13
cos(
...
)
13
cos(
)
13
cos(
1
2
5
(4)Similarly for 15-level CMLI equations are given below:
m
7
)
cos(
...
)
cos(
)
cos(
1
2
7
0
)
5
cos(
...
)
5
cos(
)
5
cos(
1
2
7
0
)
7
cos(
...
)
7
cos(
)
7
cos(
1
2
7
0
)
11
cos(
...
)
11
cos(
)
11
cos(
1
2
7
0
)
13
cos(
...
)
13
cos(
)
13
cos(
1
2
7
0
)
17
cos(
...
)
17
cos(
)
17
cos(
1
2
7
0
)
19
cos(
...
)
19
cos(
)
19
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 10, October 2012)240
The equations (3-5) are transcendental equations, known as selective harmonic elimination (SHE) equations, where unknown parameters are switching angles. The first equation of the set of equations given by (3-5) determines the magnitude of fundamental voltage for a given value of
m, and the remaining equations eliminate selective
harmonic components [11]. The equations (3-5) are to be solved by employing N-R method in such a way that all
possible solutions for a given value of m are obtained
without much computational effort. The algorithm for the solution of equations (3-5) is given in [11].
IV. COMPUTATIONAL RESULTS
By using N-R method, all possible solution sets for a 7, 11 and 15-level CMLIs are computed and a complete analysis is presented.
A. 7-Level CMLI
For 7-level CMLI, two switching angles have been calculated as shown in Fig. 3.It can be seen from Fig. 3 that
for some values of m solutions do not exist, for some other
values of m, multiple solutions exist and simple solution
exists for most of the values of m. Some typical values of
switching angles (in radians) are given in Table I.
0.4 0.5 0.6 0.7 0.8 0.9 1
0 10 20 30 40 50 60 70 80 90 100
Modulation Index(m)
S
w
it
c
h
in
g
A
n
g
le
s
(
D
e
g
re
e
s
)
[image:4.612.318.569.276.545.2]1 2 3
Fig. 3. Switching angles verses modulation index for 7-level CMLI.
For each of the multiple solution sets as computed above, total harmonic distortion (THD) in percentage is computed according to equation (6), for 7-level CMLI x=11. The set of switching angles among multiple solutions which produce least THD is selected for switching of semiconductor devices, and these are termed as combined solutions.
The THD plots for solution are plotted as a function of m
in Fig. 4. It can be seen from the Fig. 4, that THD is more than 5% for all values of modulation indices, hence not satisfying IEEE-519 standard. This type of output voltage cannot be used directly for power system and industrial drive applications due to high THD.
100 (6)
1 2 49
V V THD
n n
x n
TABLE I
Some Values of Switching Angles (in radians) for 7-level CMLI
m α1 α2 α3
0.2710 0.8120 1.4755 1.5410 0.5000 0.6881 0.9818 1.3980 0.6000 0.2064 0.7280 1.4960 0.8400 0.2729 0.3273 0.9146 0.9200 0.1394 0.2672 0.6348
0.4 0.5 0.6 0.7 0.8 0.9 1
5 10 15 20
Modulation Index (m)
T
H
D
(
%
)
Fig. 4. THD verses m for combined solution of 7-level CMLI.
B. 11-level CMLI
In similar way as discussed in case of seven-level CMLI, switching angles are calculated by using equation (4). In
this case five switching angles are obtained eliminating 5th,
7th, 11th and 13th (x= 17 in equation 6) order harmonic
[image:4.612.50.288.419.599.2]International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 10, October 2012)241
0.4 0.5 0.6 0.7 0.8 0.9
0 10 20 30 40 50 60 70 80 90
Modulation Index (m)
S
w
it
c
h
in
g
A
n
g
le
s
(
D
e
g
re
e
s
)
[image:5.612.54.285.108.315.2]1 2 3 4 5
Fig. 5. Switching angles verses modulation index for 11-level CMLI.
0.4 0.5 0.6 0.7 0.8 0.9
3 4 5 6 7 8 9 10 11
Modulation Index (m)
T
H
D
(%
)
Fig. 6. THD verses m for combined solution of 11-level CMLI. C. 15-Level CMLI
Lastly, seven switching angles were calculated for 15-level CMLI for its 7 H-bridges using N-R method. The calculated switching angles (including multiple solutions) are shown in Fig. 7. In this case, more number of harmonic components (six) as compared to 7 and 11 level CMLIs THD contents in output voltage reduces appreciably. The
harmonic components eliminated are 5th, 7th, 11th, 13th, 17th
and 19th. The output voltage THD content is calculated
using equation (6) with x=23 and the corresponding plot is shown in Fig. 8. It is to be noted here that among different solution sets of switching angles computed as above, the THD is computed by using those switching angles which are producing minimum THD (this applies for the values of
m where multiple solution exist).
It can be observed from Fig. 8 that THD level in output voltage is strictly satisfying IEEE-519 standard for normal working range of modulation indices. Some of calculated switching angles have been shown in Table III for reference. By bringing THD level below threshold value is very significant as THD produces various losses and undesirable effects in electric power systems and equipments [12-14].
[image:5.612.53.285.337.520.2] [image:5.612.326.564.381.684.2]It can be observed from Figs. 3, 5 and 7 that the range of modulation index for which solution for switching angles exist decreases with the increase in number of levels but the occurrence of multiple solutions increase with the increase in number of level i.e. for 7-level inverter only two sets of solution exist, for 11-level four sets of solution exist and for 15-level nine sets of solution are obtained thereby making computation more complex with number of levels. Fig. 9 shows the comparison of THD produced by 7, 11 and 15 level CMLIs. It can be observed from this figure that the THD reduces with the increase in number of levels.
TABLE II
Some Values of Switching Angles (in radians) for 11-level CMLI
m α1 α2 α3 α4 α5
0.4410 0.6247 0.8391 1.0615 1.3305 1.5706 0.5490 0.0718 0.6486 0.7353 1.3836 1.5480 0.6000 0.4650 0.7667 0.8994 1.0891 1.2654 0.6580 0.1563 0.3292 0.6048 1.0123 1.5675 0.7320 0.0782 0.2101 0.4618 0.7124 1.5378 0.8460 0.1604 0.2027 0.4227 0.6225 1.0017
0.450 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9
10 20 30 40 50 60 70 80 90
Modulation Index (m)
S
w
it
c
h
in
g
A
n
g
le
s
(
D
e
g
re
e
s
)
1
2
3
4
5
6
7
[image:5.612.324.563.381.683.2]International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 10, October 2012)242
0.450 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85
1 2 3 4 5 6
Modulation Index (m)
T
H
D
(
%
[image:6.612.55.288.135.328.2])
[image:6.612.326.562.287.415.2]Fig. 8. THD verses m for combined solution of 15-level CMLI. TABLE III
Some Values of Switching Angles (in radians) for 15-level CMLI
m α1 α2 α3 α4 α5 α6 α7
0.42 0.601 0.757 0.920 1.093 1.292 1.540 1.554 0.47 0.590 0.733 0.873 1.028 1.190 1.386 1.565 0.60 0.249 0.585 0.681 0.914 1.027 1.164 1.474 0.70 0.108 0.374 0.558 0.747 0.866 1.073 1.300 0.75 0.119 0.319 0.427 0.621 0.867 0.994 1.185 0.80 0.126 0.228 0.364 0.484 0.683 0.952 1.095
0.4 0.5 0.6 0.7 0.8 0.9 1
2 4 6 8 10 12 14 16 18 20
Modulation Index (m)
T
H
D
(
%
)
[image:6.612.44.295.366.667.2]7-level 11-level 15-level
Fig. 9. Comparison of THD for different levels of CMLI.
V. SIMULATION RESULTS
In order to validate the analytical results, three-phase, 7, 11 and 15-level CMLIs have been simulated on
MATLAB/SIMULINK platform using SimPower Systems
BlockSets [15]. For each of the H-bridges in the CMLI, 12V dc source is used. The switching device used is 400V, 10A MOSFET.
Simulated line voltage and THD analysis is presented in Fig. 10. The analytical and simulated values of THD are
11.17% and 11.51% respectively at m = 0.8400. These two
values of THDs are in close agreement hence validating analytical result.
Fig. 10. (a) Simulated line voltage and (b) THD analysis for 7-level CMLI at m = 0.8400.
Similarly, the simulated phase and line voltage waveforms for 11-level CMLI for the modulation index equals to 0.6200 are shown in Figs. 11 and 12 respectively. Again there is a close agreement between analytical and simulated values of THDs and these are 5.93% and 5.97% respectively. It is to be noted that in Fig. 11 (b), THD shown is 32.90%, the reason for this is that the THD shown is for phase voltage which includes triplen harmonic components while analytical value is for line voltage which excludes triplen harmonic components.
[image:6.612.325.571.570.661.2]International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 10, October 2012)243
Fig. 12. (a) Simulated line voltage and (b) THD analysis for 11-level CMLI at m = 0.6200.
The simulated result for line voltage and its THD spectrum is shown in Fig. 13 for a 15-level CMLI at modulation index equals to 0.7400. The calculated and simulated values of THD are 3.97% and 4.02% respectively. These values of THD are in close agreement hence validating results. It is to be noted that the modulation index for different level of CMLIs have been chosen randomly.
Fig. 13. (a) Simulated line voltage and (b) THD analysis for 15-level CMLI at m = 0.7400.
VI. EXPERIMENTAL RESULTS
A prototype single-phase 11-level CMLI has been built using 400V, 10A MOSFET as the switching device. Five separate dc supply of 12V each is obtained using step down transformers with rectifier unit. Pentium 80486 processor based PC with clock frequency 2MHz with timer I/O card is used for firing pulse generation. Firing pulses to the switching devices are given through a delay circuit which provides 5µsec delay.
In order to validate the analytical and simulated
results, an 11-level single-phase output voltage at m =
0.6200 was synthesized at fundamental frequency (f = 50Hz) producing output voltage.
The experimentally obtained wave form for phase
voltage and its THD spectrum is shown in Fig. 14 for m =
0.6200. It can be seen from Figs. 11 and 14 that waveforms as well as harmonic spectrum of simulated and experimental results are approximately identical hence validating analytical and simulated results. Further it is to be noticed that in Figs. 11 and 14 triplen harmonics are present as these waveforms are for phase voltage. The analytical, simulated and experimentally obtained values of
THD upto 13th order and 49th order have been shown in
Table IV for comparison purpose. It can be seen from the Table IV that all results are in close agreement.
[image:7.612.50.287.141.250.2]Fig. 14. (a) Experimentally obtained phase voltage and (b) harmonic spectrum for 11-level CMLI at m = 0.6200.
TABLE IV
Analytical, Simulated and Experimental values of THDs for m = 0.6200
THDs Analytical Simulated Experimental
Up to 13th order 0.00 0.15 0.20
Up to 49th order 5.93 5.97 5.80
VII. CONCLUSION
[image:7.612.332.568.289.464.2] [image:7.612.50.286.389.508.2]International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 10, October 2012)244
REFERENCES
[1] Fang Zheng Peng, Jih-Sheng Lai, et al, ―A Multilevel Voltage-Source Inverter with Separate DC Voltage-Sources for Static Var Generation‖, IEEE Trans. on Industry Applications, vol. 32, no. 5, pp. 1130-1138, September/October 1996.
[2] Jih-Sheng Lai, Fang Zheng Peng, ―Multilevel Converters-A New Breed of Power Converters‖, IEEE Trans. on Industry Applications, vol. 32, no. 3, pp. 509-517, May/June 1996.
[3] F. Z. Peng, J. W. McKeever, and D. J. Adams, ―Cascade Multilevel Inverters for Utility Applications‖, IECON Proceedings (Industrial Electronics Conference), vol. 2, pp. 437-442, 1997.
[4] L. M. Tolbert, F. Z. Peng, and T.G. Habetler, ―Multilevel converters for large electric drives‖, IEEE Transactions on Industry Applications, vol. 35, no. 1, pp. 36-44, Jan. /Feb. 1999.
[5] Jose Rodriguez, J S Lai, and F. Z. Peng, ―Multilevel Inverters: A Survey of Topologies, Controls, and Applications‖, IEEE Trans. on Industrial Electronics, vol. 49, no. 4, pp. 724-738, August 2002. [6] Burak Ozpineci, Leon M. Tolbert, John N. Chiasson, “Harmonic
Optimization of Multilevel Converters Using Genetic Algorithms‖, IEEE Power Electronics Letters, vol. 3, no. 3, pp.92-95, September 2005.
[7] Zhong Du, Leon M. Tolbert and John N. Chiasson, ―Active Harmonic Elimination for Multilevel Converters‖, IEEE Trans. on Power Electronics, vol. 21, no. 2, pp. 459-469, March 2006.
[8] John N. Chiasson, Leon M. Tolbert, Keith J. McKenzie, Zhong Du, ―Control of a Multilevel Converter Using Resultant Theory‖, IEEE Transaction on Control Systems Technology, vol. 11, no. 3, pp. 345-353, May 2003.
[9] John N. Chiasson, Leon M. Tolbert, Keith J. McKenzie, Zhong Du, ―A new approach to solving the harmonic elimination equations for a multilevel converter‖, in Proc. IEEE Industry Applications Soc. Annu. Meeting, Salt Lake City, UT, pp. 640-645, Oct.12-16, 2003. [10] C.K. Duffey and R. P. Stratford, ―Update of harmonic standard
IEEE-519: IEEE recommended practices and requirements for harmonic control in electric power systems‖, IEEE Trans. on Industry Applications, vol. 25, no. 6, pp. 1025-1034, Nov./Dec. 1989.
[11]Jagdish Kumar, Biswarup Das, and Pramod Agarwal, ―Optimized Switching Scheme of a Cascade Multilevel Inverter‖, Electric Power Component and Systems, vol. 38, issue 4, pp. 445-464, January 2010.
[12] George J. Wakileh, ―Power Systems Harmonics, Fundamentals, Analysis and Filter Design‖, Springer, 2001
[13] Robert D. Henderson, Patrick J. Rose, ―Harmonics: The Effects on Power Quality and transformer‖, IEEE Transactions on Industry Applications, vol. 30, no. 3, May/June 1994.
[14] Mulukutla et al, ―Power Quality, VAR Compensation in Power Systems‖, CRC Press, 2008.
