Study of a Precise and Practical Harmonic
Elimination Method for Multilevel Inverters
Full
Damoun Ahmadi and Jin Wang*
Department of Electrical and Computer Engineering
Ohio State University
205 Dreese Labs; 2015 Neil Avenue
Columbus, OH 43210
Phone: 614-688-4041, Fax: 614-292-7596
Email: wang@ece.osu.edu*
Abstract- Multilevel inverters have been widely used in
medium and high voltage applications. Selective harmonics
elimination for the staircase voltage waveform generated by
multilevel inverters has been studied extensively in the last decade.
Most published methods on this topic aimed at solving high-order
multi-variable polynomial equation groups derived from Fourier
series expansion. A totally different approach based on equal
area criteria and harmonics injection in the modulation waveform
is fully studied in this paper. Regardless how many voltage levels
are involved, only four simple equations are involved in the basic
method. The problems of the basic method are identified and
discussed. A full set of solutions is proposed. The results of a case
study with maximum five switching angles show that the proposed
method can be used with excellent harmonics elimination
performance for the modulation index range from 0.2 to 0.9.
The Fourier expansion of the staircase wave can be
expressed as:
Keywords: Equal Area Criteria, Modulation Index, Multilevel
Inverters, Pulse Width Modulation, Total Harmonic Distortion
V(t) =
I.
I\k
Figure 1. The general staircase waveform of multilevel inverters and equal
area criteria.
INTRODUCTION
In high power medium and high voltage applications, the
usage of Pulse Width Modulation (PWM) based two level
inverters is limited by voltage and current ratings of switching
devices, switching losses, and electromagnetic interferences
caused by high dv/dt. Thus, for applications like medium
voltage drives, renewable energy interfaces, and flexible ac
transmission devices (FACTs), multilevel inverters are
gradually becoming the main work force [1-4]. A typical
multilevel inverter utilizes voltage levels from multiple dc
sources. These dc sources can be isolated as in cascade
multilevel structures or interconnected as in diode clamped
structures. In most published multilevel inverter circuit
topologies, the dc sources in the circuits need to be maintained
to supply identical voltage levels. Based on these identical
voltage levels, with proper control of the switching angles of
the switches, a staircase waveform can be synthesized, such as
a 6 level staircase waveform with five switching angles shown
in Fig. 1.
4Vdd,c (cos(m
m=1,3,5.... M)z
0,) + ... cos(mON )) sin(mwt)
(1)
where N is the number of switching angels and m is the
harmonic order. Based on the Fourier expansion, several
methods have been proposed to solve the following equation
groups to realize Selected Harmonics Elimination (SHE) [1119].
(cos(01) + COS(02) + COS(03) + COS(04) + COS(05)) = VF
cos(501) + cos(502) + cos(503) + cos(504) + cos(505) = 0
I z
(2)
cos(I 301) + cos(I 302) + cos(] 303) + cos(] 304) + cos(l 305) = 0
In this equation group, the first equation guarantees the
desired fundamental component. The second equation and
beyond are used to eliminate the selected harmonics. So, it is
clear that with five switching angles, four selected harmonics
One of the greatest benefits of this staircase waveform is can be eliminated. The following is a brief summary of the
that the switches in the inverters only need to switch on and off most quoted and most recently reported methods on solving
once during one fundamental cycle, thus the switching loss of these types of equation groups. A resultants theory based
the device is reduced to minimum. However, with reduced algorithm was reported in 2000 [11]. A unified approach was
switching frequencies, even with additional voltage levels, low presented in 2004 [12]. In the following year, symmetric
frequency harmonics can be found in the staircase voltage [1, polynomials and resultant theory combined solution [13],
2].
power sum based method [14], and generic algorithm based
method [15] for multilevel inverters with uneven dc sources
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871
were proposed. High switching frequency based active Equation 2: the equation to find the switching angle:
harmonics elimination was published in 2006 [16]. Very
recently, a five-level symmetrically defined SHE PWM
Ok = kk - (k - 1)3k- + VF (cos(k ) CoS(k- ))
strategy [18] and a hybrid real coded genetic algorithm [19]
h5 (cos(535k) cos(535k-1))
(4)
were published.
5
One of the main difficulties of applying most of these
hm (cos(mSk) -CoS(mk-l))
...
methods is that when the number of dc level increases, the
m
number of polynomial equations, the number of variables, and Equation 3: the equation to find the harmonics content in the
the order of the equations will all increase accordingly. Thus, staircase waveform:
finding solutions to these equations would become extremely
difficult and often involve advanced mathematical algorithms,
(5)
hm = E V(cos(mok) -cos(m(;T - Ok)))
which make the calculation easy to reach the capability limits
Mk,
of existing computer algebra software tools [12]. Researches
show even for the simplest case like equation groups (2), it still Equation 4: the equation to calculate the new reference
takes special algorithms and long calculation time to solve. waveform:
For larger and higher order equation groups, there would be a
point that to find the solutions becomes not practical [11-18].
(6)
Vref = VF sin(c) - hms sin(mc)
Other limitations of the proposed methods often involve the
capability of dealing with wide range of modulation index [19], where, hms is the sum of hm calculated in every iteration:
and uneven dc voltage levels [15, 17].
iter
In summary, though many methods have been proposed to
=Zhm(i)
(7)
hms
solve the SHE problem in multilevel inverters, a simple and
i=1,2,....
practical method is still needed.
Originally, the proposed method was verified with test
II. THE FoUR-EQUATION BASED HARMONIC
results from a 17 level cascade multilevel inverter. Switching
ELIMINATION METHOD
angles of a six level waveform for limited modulation index
was also shown [20].
range
The four-equation method [20] is based on equal area
criteria (shown in Fig. 1) and the harmonics injection in the
For the work presented in this paper, to identify possible
modulation waveform. The basic approach in the four- problems with the basic four-equation based method, the basic
equation method is summarized in five steps: 1) use the desired method was tested with 6-level waveform with modulation
fundamental voltage as a reference to find the initial switching index sweeping from 0.16 to 0.94. The main problem
angles with equal area criteria; 2) identify the selected identified from this process is the amplitude difference
harmonics in the staircase waveform, which comes from the between the desired and resulted fundamental voltages.
switching angle; 3) subtract the harmonics from the original
With the direct implementation of the proposed method, the
reference waveform to get a new reference waveform; 4) use fundamental voltage of the staircase waveform often diverts
the new reference waveform to find a new set of switching from the desired value, as shown in Table 1. The reason is that
angles; 5) repeat steps 2 to 4 until the selected harmonics are for most cases, it is difficult to find a good solution for the
eliminated. The diagram of this method is shown in Fig. 2. switching angle for the top dc level to satisfy the equal area
The four basic equations used in this method are:
criteria. When it is expressed in terms of modulation index,
the
difference between resulted modulation index and desired
Equation 1: the equation to calculate the junction point of the
modulation
index can be higher than 0.1 at certain points. This
reference and the voltage level. The Newton-Ralphson method
also
means
sudden
changes in the resulted modulation indexes.
will be used to find the numerical solution of the method:
The modulation indexes shown in Table 1 are defined by
sin(mSk)
(3) the following equation:
8k = arctan(
k k.Vdc + h5
cos ('5k
~~~~~~VF
-
-
sin(5Sk)..hm
Desired
Fundamental
Component
+
Figure 2. The four-equation based method.
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-
Table 1. Sample points from the direct implementation of the four-equation method.
Reference
0.92
0.88
0.84
0.80
0.76
Resulted
Ml
0.8408
0.7923
0.7818
0.7715
0.7251
Harmonics (%)
Switching Angles (rad.)
0I
0.1147
0.1433
0.1434
0.1435
0.0815
MI
02
0.2577
0.3398
0.3406
0.3411
0.4256
03
0.4121
0.5275
0.5283
0.5289
0.6818
50
04
0.6465
0.8417
0.8433
0.8443
0.8583
V
1.0134
1.1057
1.1062
1.1065
N/A
(8)
4N.Vdc
where VF is the reference ac voltage in the output, N is the
number of dc levels, and VdC is the dc magnitude for each
voltage level in multilevel inverter output waveform.
As Table 1 just shows portion of the results, the full table of
modulation index from 0.16 to 0.94 shows that for the six-level
(maximum) waveform, the change of resulted modulation
index is not continuous but more like a staircase.
Earlier, multiple solutions of 8k for one dc level were
expected to be an issue. But studies show that the equal area
criteria automatically settled on the best 8k . So the study in the
paper mostly is focusing on solving the problems related with
the accuracy of the resulted fundamental components.
SOLUTIONS To THE PROBLEMS IN THE BASIC METHOD
A. PI Controller based Fundamental Voltage Correction
III.
5th
7th
0
0
0
0
0
0
0
0
0
0
I 3th
I 7th
0
0
0
0
0.4847
0. 584
0.4239
0.5128
th
1
0
0
0
0
0
0.7062
N/A
to add a simple PI controller in the iteration process to keep
compensating the fundamental component. With this approach,
the modulation waveform of this modified method can be
expressed as
Vref = (VF -his) * (KP + A)-hms sin(mot)
The overall diagram of the modified method is shown in
Fig. 3. The added process is shown in dotted line.
However, even with the PI controller in place, the numbers
of utilized dc levels are still not optimized for most modulation
indexes. Thus, there is still a slight difference between resulted
and desired modulation index for most cases. And since the PI
controller is only used for fundamental compensation, the
performance of harmonics elimination went bad especially at
high modulation index points. This can be seen in the results
listed in Table 2.
Since the performance of modified method with PI
controller does not have good harmonics cancellation,
alternative solutions are proposed and validated as following.
To solve the problem of the difference between the desired
and resulted fundamental component, one possible solution is
|Switching angle calculations
_ _ _
Desired fundamental
L+' D d fl a F7
*\
component
Xk --
|Harmonics calculation|
with Equation 3
with Equation 1 and 2
New modulation
-- +
LL---> waveform synthesizing
4
with Equld
so
Calculated fundamental from the resulted switching angles
Reference
Figure 3. Modified method with PI controller to adjust the fundamental component.
Table 2: Sample points with PI controller based modified method.
Resulted
Harmonics
Switching Angles (rad.)
Ml
0
0
o
0
o
1,th
5th
7th
m
1
02
3
04
05
13th
17th
0.92
0.9112
0.1794
0.2567
0.315
0.4993
0.7304
0.3976
0.285
0.0085
0.613
0.4304
0.88
0.8467
0.1043
0.2647
0.394
0.6277
0.9994
0.0179
0.0554
0.25
0.4643
0.1339
0.84
0.8308
0.1146
0.2577
0.4119
0.6464
1.0133
0.0008
0.0019
0.0019
0.0038
0.0017
0.80
0.7931
0.1399
0.3182
0.5149
0.802
1.0932
0.3026
0.1336
0.1316
0.7332
0.6585
0.76
0.7351
0.0005
0.2402
0.4088
0.6905
N/A
0.0062
0.0723
0.5558
0.7025
N/A
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873
2)
Then, the switching angle of the additional voltage
level is calculated based on the difference between the desired
In the final solutions, PI controller is no longer used in the iterations.
Instead, either an additional voltage level or additional adjustment of fundamental voltage, VF, and the resulted fundamental voltage,
the switching angle at the highest voltage level is used depending on Vim, with the following equation
whether an extra voltage level is available at the defined modulation
(1 1)
Om+, a cos( 4V (VF -VIm))
index.
B. Final Solutionsfor the Problems in the Four-equation Method
1.
Harmonic Elimination with No Extra Voltage Levels
In multilevel inverters, when the desired modulation index
becomes smaller, fewer dc levels will be used to synthesize the
staircase waveform. In this case, with the four-equation
method, at the fundamental frequency, the difference between
the desired voltage and generated voltage will become larger.
But since an extra voltage level is available, it can be utilized
to realize the fundamental voltage compensation. Based on
this idea, the same five steps in the four-equation method
would be used. The difference is that an extra switching angle
would be calculated for an extra voltage level to achieve
desired fundamental voltage. However, additional harmonics
will be present in the extra voltage level. So in this method,
the additional harmonics content generated by the extra voltage
level would be added to the reference waveform which is used
to calculate the other switching angles. This means that the
extra harmonics generated in the additional voltage level would
be compensated by the switching angles for all the other
voltage levels.
This modified method is illustrated in the diagram shown in
Fig. 4. The additional process is shown in dotted line.
The following is the procedure on the calculation of the
additional "m+l" switching angle:
1)
First, the total fundamental voltage based on
switching angles from 01 to O,m is calculated with the
following equation
VIm =
4Vdc cos(Oi),
m<N
2.
Harmonics Elimination with No Extra Voltage Levels
For larger modulation indexes, where all dc levels are
already used for staircase generation, there is no additional
voltage level for fundamental voltage compensation.
Therefore, in this proposed method, the switching angle of the
last dc level will be adjusted to achieve desired fundamental
voltage. The "adjustment" switching angle is calculated with:
(12)
ON =acos(4 (VF -17VN))
I
where VIN is the total fundamental voltage generated by
switching angles from 01 to O,m
This "adjustment" angle is used to modify the switching
angle for the last voltage level:
(13)
0N(mod ified) a cos(cos(ON) + cos(0 ))
Therefore, based on the switching angle adjustment for the
last dc level, the desired voltage magnitude in the fundamental
frequency can be achieved. However, if not compensated, the
"adjustment" switching angle would bring in the additional
harmonics in the resulted staircase waveform. Thus, the
selected harmonics caused by the "adjustment" angle would
also be calculated and be added to the final modulation
waveform. The total process of this modified method is
illustrated in Fig. 5.
(10)
r-- -- -- -- -- --
I
Calculate the
I
additional"m+1"."
switching angle I
_ _ _ __
Switching angle calculations
Calculate the
/''
r->\'
harmonics in the
T
additional
voltage level
L--____-__-____-__-__
New modulation
- ----- - --> waveform synthes'izing
:~~
~
~~~~
4
> ~~~~~~~~~with
| |
Equation
4 -Ao
nO iuncamenta
_
Tr
oespre
component
Calculated fundamental from the resulted "m" switching angles
selectedwihEutoIan2
,
Figure 4. Modified method with "additional" switching angle.
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874
Harmonics calculation
with Equation 3
Calculatethe
>-.-.--------------------------.--"adjustment"
S
a
cui
switchimg angle
___
_ _
Switching angle calculations
with Equation 1 and 2
Calculate the selected
_/+ ")
\ -, |
harmonics caused by
4
| X
thffe "Cadjustment"
angle
New modulation
:~~~~ L_-- -- -- -- ---:
~
~~~~
-._____ -- - --> waveform synthesizing
with Equation 4
Desired fur1kdamentalI
Calculated fundamental
compc)rnent
+
Harmonics c,alculation
with Equiation 3
Figure 5. Modified method with "adjustment" switching angle for the highest voltage level.
IV.
VERIFICATION OF PROPOSED SOLUTIONS WITH A CASE
STUDY
With the final solutions 1 and 2, switching angles, resulted
modulation index, and selected harmonics content were
calculated with the modulation index sweeping from 0.2 to 0.9
for staircase waveforms with maximum six levels. For all the
calculated modulation points, the resulted modulation index
follows the desired value well. The selected harmonics
elimination can be 100% except at modulation indexes close to
0.9. Table 3 shows some sample points of the calculation
results. At the first sample point MI=0.9, since no extra
voltage is available, the "adjustment" angle in method 2 is used.
For the other four points in the table, an additional voltage
level is used to achieve desired modulation index. The results
show that both solution 1 and 2 work well in terms of
achieving desired fundamental voltage and harmonics
elimination.
The overall optimized switching angles based on the
proposed method for 6-level (maximum) waveform is shown in
Fig. 6. It is noticeable that with the modification of the
switching angle of the highest voltage level, the 05 sometimes
becomes smaller than other switching angles. It may look
strange, but this will not cause any problem. In the real
inverter, the final voltage is the summation of the voltage from
all the dc sources, instead of getting the waveform shown in
Fig. 7 (a), the output voltage of the inverter would always look
like the waveform in Fig. 7 (b).
0.2
0.3
0.4
0.5 0.6
0.7 0.8
Modulation Indexes (MI)
V.
CONCLUSIONS
In this paper, a simple method using four equations for
harmonic elimination in multilevel inverters is reviewed and
further studied. The problems with direct implementation of
the method is identified and discussed. Then, a full set of
solutions is provided and verified with a case study.
Switching Angles (rad.)
Resulted
Mml
Ml Ml
0a1,5th02
0.90
0.0641 0.1984
63
04
Harmonics (%)
05
7th
11Ith
13th
17th
0.3395
0.7714
0.5319
0.069
0.0291
0.0375
0.0474
0.0215
0.76
0.76
0.1878
0.3618
0.5922
0.9231
1.091
0
0
0
0
N/A
0.60
0.60
0.1971
0.4689
0.8051
1.1216
N/A
0
0
0
N/A
N/A
0.46
0.46
0.2175
0.5954
1.0522
N/A
N/A
0
0
N/A
N/A
N/A
0.20
0.20
0.3889
1.4961
N/A
N/A
N/A
0
N/A
N/A
N/A
N/A
0.90
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Figure 6. Optimized switching angles of the proposed method for 6-Level
waveform.
Table 3. Sample points for six (maximum) level waveform with the modified four-equation method.
Reference
0.9
875
(a)
(b)
Figure 7. Explanation of the top switching angle in Fig.6.
Comparing with other SHE methods proposed for
multilevel inverters, the harmonics elimination method studied
in this paper has the following advantages: 1) only four simple
equations are involved in the basic method; 2) for different
numbers of switching angles, the equations remain the same,
no huge increasing of calculation time is expected when the
number of switching angles increases; and 3) in some cases,
this method can eliminate more than N-1 harmonics with only
a small difference between the desired and resulted modulation
index [20]. With the simple modifications proposed in this
paper, this method has become not only precise in harmonics
elimination but also practical in terms of simplicity and
realization by field engineers.
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