Analytical Assessment of DC Components Generated by Renewable Energy Resources with Inverter-Based Interconnection System due to Even Harmonics

Abstract

This paper deals with the assessment of DC components generated by renewable energy resources with inverter-based interconnection system to the electric grid. DC injection is a critical issue related to power quality of distribution network systems with high penetration of inverter-based interconnection systems. This type of interface systems may improve the performance of the electric generation unit and affect positively or negatively the power quality of the distribution network depending on the proper or improper designation. The investigation of the various causes of DC components and the analytical assessment of their maximum levels are crucial for the proper operation of inverter-based interface systems and the limitation of DC injection. A method based on analytical calculations using a computer software has been implemented for the assessment of DC components contained on an inverter's output voltage when even harmonics are present on the network voltage. Moreover, a simulation package was used to demonstrate the existence of DC components under various conditions. It was proved by the current analysis that the amounts of DC components generated when even harmonics are present on the network voltage can be high under abnormalities on the power grid but they are not considerable under normal operating conditions.

Full text

International Scholarly Research Network
ISRN Renewable Energy
Volume 2012, Article ID 261325, 12 pages
doi:10.5402/2012/261325
Research Article
Analytical Assessment of DC Components Generated by
Renewable Energy Resources with
Inverter-Based Interconnection System due to Even Harmonics
Marios N. Moschakis,1Vasilis V. Dafopoulos,1
Emmanuel S. Karapidakis,2and Antonis G. Tsikalakis2
1Department of Electrical Engineering, Technological Educational Institute of Larissa, 41110 Larrisa, Greece
2Department of Natural Resources & Environment, Technological Educational Institute of Crete,
Romanou 3 Street, 73133 Chania, Crete, Greece
Correspondence should be addressed to Marios N. Moschakis, marios.moschakis@gmail.com
Received 29 April 2012; Accepted 7 August 2012
Academic Editors: E. R. Bandala, S. Dai, and G. Namkoong
Copyright © 2012 Marios N. Moschakis et al. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
This paper deals with the assessment of DC components generated by renewable energy resources with inverter-based
interconnection system to the electric grid. DC injection is a critical issue related to power quality of distribution network systems
with high penetration of inverter-based interconnection systems. This type of interface systems may improve the performance
of the electric generation unit and affect positively or negatively the power quality of the distribution network depending on the
proper or improper designation. The investigation of the various causes of DC components and the analytical assessment of their
maximum levels are crucial for the proper operation of inverter-based interface systems and the limitation of DC injection. A
method based on analytical calculations using a computer software has been implemented for the assessment of DC components
contained on an inverter’s output voltage when even harmonics are present on the network voltage. Moreover, a simulation package
was used to demonstrate the existence of DC components under various conditions. It was proved by the current analysis that the
amounts of DC components generated when even harmonics are present on the network voltage can be high under abnormalities
on the power grid but they are not considerable under normal operating conditions.
1. Introduction
The need for analysis and assessment of DC components on
voltage and/or current, which may appear at electric grids,
aroused mainly due to the significant problems they result
in. Power transformers are the power system element which
is mainly affected by the presence of DC components. Satura-
tion, generation of harmonics, power losses, reduction in life
cycle, and increasing noise levels are some of the effects on
transformers introduced by the presence of DC components.
Moreover, the saturation of current transformers used by
measurement devices of control and protection of electric
grid equipment may cause inaccurate measurements with
considerable consequences.
In cases where many DC injection sources are connected
to the grid, the need for limitation and standardization
becomes urgent. Such cases include the interconnection
of several distributed generation units at Medium-Voltage
(MV) or Low-Voltage (LV) level through inverters based
on the technology of power electronics. Those inverters
may generate DC components on the output voltage during
abnormal conditions that will lead to DC current injection
flowing to the electric grid if no power transformer is used
between the inverter and the grid (transformer-less inverter).
Such interface systems are used mainly at LV distribution
networks. In every case, the assessment of DC components
is necessary not only for a proper design of the inverter’s
control system but also for meeting immunity limits of the
interconnection transformer.
The generation of DC components by the inverter
under normal operating conditions is mainly due to the
faulty operation of the control circuits. One of the basic

2ISRN Renewable Energy
causes of DC injection by inverters is the asymmetry in
driving signals of the power electronics. A small amount
of DC components is unavoidable as there is always a
small asymmetry on the voltages of the three-phase electric
grids. Such asymmetries are transferred on the waveforms
used for pulse modulation when the filtering process or
the extraction of the fundamental voltage component fails.
Another cause is measurement errors, especially errors that
indicate nonzero values while they are zero. Such errors make
the control system generate a small DC current to balance
the erroneous non-zero value indicated by the measurement
instrument. Obviously, such dc current values are very
low (a few mAs) and difficult to be precisely measured,
so such errors are very likely when improperly designed
measurement instruments are used. Thus, especially under
abnormal operating conditions, errors in control system and
electronic circuits will allow for a small dc current to flow
to the grid before the error is detected and the inverter’s
operationisshutdown[1].
Moreover, DC components may appear in transient
conditions in which the inverter’s response must be rapid.
Such operation is common only in specific inverter types
used for the control and mitigation of transient electro-
magnetic phenomena such as rapid voltage changes (e.g.
small and sharp voltage sags, flicker, etc.), voltage waveform
distortions (harmonics, transient overvoltages), or rapid
power oscillations. Such devices use a special control system
of the DC components that may appear on the inverter’s
output by any cause. Under normal conditions, they use
filtering so the reference waveform be purely sinusoidal but
no filtering is used at abnormal (transient) conditions where
a rapid response is needed [2]. Such a device is the Static
Compensator (Statcom), which is used for mitigation of
flicker generated by some load types (e.g. arc furnaces) and
for mitigation of rapid voltage and power fluctuations on
the output of wind parks [3]. It should be noted here that
this case does not involve the inverters used for small-scale
distributed energy resources as the concept of using such
units for the control of fast voltage fluctuations and/or power
oscillation damping it is not currently adopted.
The aim of the current paper is the analysis of the
way DC components are generated on the output of three-
phase inverters using the Sinusoidal Pulse Width Modulation
(SPWM) technique due to even harmonics. A novel method
for the analytical assessment of DC components is presented,
which is based on the zero-crossing points of the reference
waveform (sinusoidal) and the carrier’s waveform (triangu-
lar) of SPWM technique. Using realistic values for the SPWM
parameters and for the even harmonics, the percentage of DC
components of the three line voltages in the inverter’s output
is calculated. Finally, the values of those parameters, which
result in the maximum percentage of DC components, are
investigated.
2. Even Harmonics and DC-Components on
Three-Phase Inverter’s Output
2.1. SPWM Technique with Pure Sinusoidal Reference Wave-
form. Under normal operating conditions of a three-phase
Ud
Ud
2
+
−
Ud
2
+
−
O
C
B
A
Figure 1: Standard three-phase voltage source inverter topology
[4].
Triangle (t)
0t2π
ArAc
Mf=Ar
Ac
=Fc
Fr
UrUr( ) Ur
Fnc
t
( )t( )t
C
B
A
Figure 2: Basic waveforms and parameters of SPWM technique.
inverter (Figure 1) that uses SPWM technique, the reference
waveform used for the generation of power electronics’ firing
pulses is formulated based on the fundamental component
of the measured network voltage. Thus, any asymmetries
or distortions on the network voltage waveform are not
transferred on the reference waveform required by the
SPWM technique.
In Figure 2, the basic waveforms and parameters of the
SPWM technique under normal operating conditions are
shown. The reference waveforms are purely sinusoidal and
given by the following equations for Ac=1:
UA
r(t)=Mf·sin(t),
UB
r(t)=Mf·sint−2·π
3,
UC
r(t)=Mf·sint−4·π
3,
(1)
where UA
r(t), UB
r(t), and UC
r(t) are reference waveform
functions for phases A,B,andC;Triangle(t) is the carrier
waveform (triangular) function; Ar,Frare the amplitude and
frequency of reference waveforms; Ac,Fcare the amplitude
and frequency of carrier waveform; Mf=Ar/Acis the
modulation Amplitude Index; Fnc =Fc/Fris the modulation
Frequency Index.
When a rapid response is required by the inverter, for
example, in fast electromagnetic phenomena, the reference
waveform is normally formulated in a different way. Specif-
ically, the reference waveform is generated by the measured

ISRN Renewable Energy 3
0π2π
(a)
0π2π
(b)
0π2π
C
B
A
(c)
0π2π
C
B
A
(d)
Figure 3: (a) Waveform with the fundamental and a 2nd harmonic. (b) Waveform with the fundamental and a 3rd harmonic. (c) Waveform
with the fundamental and a 2nd harmonic (negative sequence). (d) Waveform with the fundamental and a 2nd harmonic (positive sequence).
network voltage, which is also subjected to 3-phase-to-2-
phase transformation. The measured network voltage is
subjected to filtering only when a rapid response of the
inverter is not required, that is, only under normal operating
conditions. Therefore, if network voltage contains harmonics
and a rapid response of the inverter is needed, no filtering
will be applied and the reference waveforms will also contain
harmonics [2].
2.2. SPWM Technique with Asymmetrical Even Harmonics
on the Reference Waveform. A case of asymmetry which can
lead to the generation of DC components by an inverter is
the presence of even harmonics, especially when there is an
asymmetry between the positive and negative half period of
the reference waveform [5]. This means that the positive and
negative half periods differ at the peak values and duration.
This will cause the generation of a DC component on the
inverter’s output voltage VAO(t)betweenphaseAand the
hypothetical point O(Figure 1).
In Figures 3(a) and 3(b), the waveforms include a 20%
of a 2nd and 3rd harmonic respectively. The harmonic
angle φh, which is the angle between the harmonic, and the
fundamental component, was set at 60◦and 0◦,respectively.
The asymmetry can be observed only for the 2nd harmonic
not for the 3rd harmonic.
The three reference waveforms of the SPWM technique
with the presence of even harmonics without amplitude and
angle asymmetry are given by the following equations:
UA
r(t)=Mf·sin(t)+uA
h(t)
=Mf·sin(t)+Mf·Ah·sinh·t+φh
h,
UB
r(t)=Mf·sin(t)+uB
h(t)
=Mf·sint−2·π
3
+Mf·Ah·sinh·t+φh
h−2·π
3,
UC
r(t)=Mf·sin(t)+uC
h(t)
=Mf·sint−4·π
3
+Mf·Ah·sinh·t+φh
h−4·π
3,
(2)
where his the harmonic order; uA
h(t), uB
h(t), and uC
h(t)
are harmonic functions for phases A,B,andC;Ahis the
harmonic amplitude as a percentage of the fundamental
component; φhis the angle between the harmonic and the
fundamental component.
Figure 3(c) shows three waveforms with a symmetric 2nd
harmonic. The presence of such a harmonic means that the

4ISRN Renewable Energy
Waveform with harmonics
Waveform with a 2nd harmonic
Carrier waveform
0π2
π
(a) Reference and carrier waveforms
0π2π
(b) Line voltage for a reference waveform with the presence of a
2nd harmonic, DC =−18.88%
0π2
π
(c) Line voltage for a reference waveform without harmonics
0π2π
(d) “Difference” (b)-(c)
Figure 4: SPWM technique for reference waveform with or without the presence of a 2nd harmonic. Ah(%) =20%, Fnc =3, φh=60◦,and
Mf=0.9.
DC component on voltages VAO(t), VBO(t), and VCO(t)will
be equal, thus in line voltages VAB(t), VBC (t), and VCA(t)
(Figure 1), the DC component will be zero. On the other
hand, asymmetrical even harmonics introduce a different
asymmetry on the three phases and, consequently, a DC
component will appear in line voltages.
The current analysis will consider only the asymmetry
in harmonic amplitude. This means that the harmonic
amplitude Ahof the three waveforms will differ. The bigger
the difference is, the higher the asymmetry and the DC
component generated by the inverter are. The difference
in harmonic amplitude means that the three-phase system
contains also a component of inverse (negative) sequence.
When a three-phase harmonic component is completely
of inverse sequence, the asymmetry is maximized and DC
component on the inverter’s output is also maximized.
Obviously, the odd or even harmonics are very likely
to be asymmetrical in amplitude. This may happen due to
DC injection by the inverter for reasons mentioned in the
previous section, which will result in generation of even
harmonics by the inverter’s interconnection transformer [6,
7]. The injected DC current will probably be different in
every phase because DC component is different in every
line voltage. Thus, the even harmonics generated by the
transformerwillhavedifferent amplitude in every phase,
thus they will be asymmetrical.
Figure 3(d) shows a 2nd harmonic of positive (direct)
sequence, thus inverse sequence of the normal sequence as
2nd harmonic is by its nature a negative sequence harmonic.
This means that the three phases rotate in the A-C-Border
and not in the A-B-Corder. The 4th harmonic is normally of
positive sequence, the 6th of zero sequence, and so forth. The
same concepts apply also for the odd harmonics. Tabl e 1 lists
the normal sequence of even harmonics from the 2nd to the
10th order.
Tab l e 1: Normal sequence of the first five even harmonics.
Harmonic order Sequence of phase components
2nd Negative sequence
4th Positive sequence
6th Zero sequence
8th Negative sequence
10th Positive sequence
DC component of output line voltages will be difficult
to be observed in the voltage waveform unless it is very
high. This could be possible only for asymmetrical even
harmonics of inverse sequence and for low Modulation
Frequency Index Fnc.InFigure 4, the SPWM technique and
one of the line voltages are shown for a purely sinusoidal
waveform and for that case it contains a 2nd harmonic.
A high harmonic percentage (20%) is used for a clear
observation of the DC component, and the asymmetries.
The Modulation Amplitude Index Mfwas arbitrarily set to
0.9 and the harmonic angle equals 60◦. By observing the
line voltages formulated for modulation with a reference
waveform containing or not the particular harmonic, we can
see a non-zero (Figure 4(b))andazero(Figure 4(c))amount
of DC component respectively. This is due to the positive
and negative parts which are not identical in pulse series of
Figure 4(b) and identical in pulse series of Figure 4(c). This
means that the mean value (DC component) will have a non-
zero value in the first case and a zero value in the second case.
Another way to verify the existence of a DC component
in the line voltage is to obtain the “difference” between the
generated voltages formulated when an even harmonic exists
or not in the reference waveform by using a simulation
package such as the PSCAD/EMTDC [8]. In other words,

ISRN Renewable Energy 5
0π2π
(a) “Difference” for a 3rd harmonic, DC =0%
0π2π
(b) “Difference” for a 6th harmonic, DC =−10.24%
0π2
π
(c) “Difference” for an 8th harmonic, DC =−25.03%
0π2π
(d) “Difference” for a 10th harmonic, DC =17.75%
Figure 5: “Differences” obtained by the generated line voltages from the modulation with or without the presence of a 3rd, 6th, 8th, and
10th harmonic. Ah(%) =20%, Fnc =3, φh=60◦(0◦for the 3rd), and Mf=0.9.
taking the result by subtracting waveforms of Figures 4(b)
and 4(c), the pulse series of Figure 4(d) is formulated. In
Figure 5, the “differences” obtained between the generated
line voltages from the modulation with or without the pres-
ence of a 3rd, 2nd, 6th, 8th, and 10th harmonic are shown,
and specifically, the line voltage which gives the maximum
DC component. The corresponding DC component value is
given as a percentage of the rms value of the fundamental
component of the line voltage in the inverter’s output, the
function of which is mentioned in [4] and given in the
following section.
3. Methodology for the Analytical
Calculation of DC Components on
Inverter’s Output Voltages
The calculation method of DC components on a three-phase
inverter’s output voltage is based on the calculation of zero-
crossing points of the carrier waveform Triangle(t)forevery
reference waveform UA
r(t), UB
r(t), and UC
r(t)andcanbe
implemented using a mathematical software package such
as MathCad [9]. The zero-crossing points for one period
(cycle) are calculated by solving arithmetically the following
equation:
Ur(t)−Triangle(t)=0.(3)
In order to solve this equation, an analytical expression
is required for the triangular waveform Triangle(t). It can
be easily observed that the triangular waveform consists of n
direct lines, where n=2·Fnc. The equation that describes the
triangular waveform Triangle(t) for one cycle (t∈[0, 2π])
can be written for Ac=1 as follows:
Triangle(t)=(−1)n+1
+(−1)n·t−n−1
2·π
Fnc ·2·Fnc
π,
n=0, 1, ...,2·Fnc ,t∈(2n−1)·π
2·Fnc
,(2n+1)·π
2·Fnc .
(4)
The zero-crossing points of the triangular waveform with
the reference waveforms define voltages VAO(t), VBO(t), and
VCO(t)(Figure 1) according to the following relations:
V(t)=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
Ud
2:Ur≥Triangle(t)
=⇒t∈[t2n,t2n−1],n=1, 2, ...,Fnc,
−Ud
2:Ur<Triangle(t)
=⇒t∈[t2n+1,t2n],n=0, 1, ...,Fnc −1,
(5)
where Udis the voltage on the dc side of the inverter
(Figure 1)t0,t1,...,t2·Fnc are the time points corresponding
to zero-crossing points of the triangular waveform with the
reference waveforms.
Using the time points given by (5), the inverter’s voltage
pulse series VAO(t), VBO (t), and VCO(t) can be formulated.
From them, the DC component can be calculated by taking
the mean time for a period. Specifically, the pulse duration
for the positive and negative part of the pulse series is added
and multiplied with their amplitude Ud/2. The resulting
summations are subtracted and the result is divided with the
period 2π. The voltage value Uddoes not affect the value of
the DC component, so it can be set equal to 1.
The analytical equations for the DC component ratio of
inverter’s voltages VAO(t), VBO(t), and VCO(t) with the rms
fundamental line voltage at the inverter’s output Vo,1(rms) can
be written as follows:
VAO
dc
=(1/2π)−Fnc−1
n=0tA
2n+1 −tA
2n/2+Fnc
n=1tA
2n−tA
2n−1/2
Vo,1(rms)
,
VBO
dc
=(1/2π)−Fnc−1
n=0tB
2n+1 −tB
2n/2+Fnc
n=1tB
2n−tB
2n−1/2
Vo,1(rms)
,

6ISRN Renewable Energy
VCO
dc
=(1/2π)−Fnc −1
n=0tC
2n+1 −tC
2n/2+Fnc
n=1tC
2n−tC
2n−1/2
Vo,1(rms)
.
(6)
The rms value of the fundamental component of the
inverter’s output line voltage is given as follows [4,10]:
Vo,1(rms)=√3
2√2·Mf·Ud.(7)
For the line voltages VAB(t), VBC(t)andVCA(t), the DC
component is calculated as follows:
VAB
dc =VAO
dc −VBO
dc ,
VBC
dc =VBO
dc −VCO
dc ,
VCA
dc =VCO
dc −VAO
dc .
(8)
The SPWM technique with an inverse (positive)
sequence 8th harmonic on the reference waveforms when
Ah(%) =20%, Fnc =9, φh=60◦,andMf=0.8is
shown in Figure 6. The zero-crossing (time) points of the
reference waveforms and the triangular waveform have been
calculated using MathCad software [9] and are given in
Tab l e 2. The values of the DC component for the voltages
VAO(t), VBO (t), and VCO(t) as a percentage of the rms value
of the fundamental component of the inverter’s output line
voltage Vo,1(rms) are approximately as follows:
VAO
dc ∼
=7.177%, VBO
dc ∼
=0%, VCO
dc ∼
=−7.177%.(9)
The corresponding values for the line voltages VAB(t),
VBC(t), and VCA(t)areasfollows:
VAB
dc =VAO
dc −VBO
dc ∼
=7.177%,
VBC
dc =VBO
dc −VCO
dc ∼
=7.177%,
VCA
dc =VCO
dc −VAO
dc ∼
=−14.354%.
(10)
4. Assessment of DC Components
The DC component for every line voltage at the inverter’s
output VAB
dc ,VBC
dc ,andVCA
dc will probably have different
value. It may take positive or negative values as we saw
in the previous section. As the sign is not of significance,
the calculation of DC component will be based on the
maximum absolute value among the three DC values for
every line voltage or the voltage between each phase and the
hypothetical point O(Figure 1), that is,
max

VAO
dc 

,

VBO
dc 

,

VCO
dc 

or
max

VAB
dc 

,

VBC
dc 

,

VCA
dc 

.
(11)
The value of the DC component depends on parameters
of the SPWM technique and the even harmonics. The
assessment of the maximum DC percentage will be done for
Tab l e 2: Time instants based on zero-crossings points of reference
waveforms and triangular waveform.
UA
r(t)∧Triangle(t)UB
r(t)∧Triangle(t)UC
r(t)∧Triangle(t)
t0∼
=0.031 −0.134 0.095
t1∼
=0.318 0.479 0.274
t2∼
=0.829 0.552 0.748
t3∼
=0.912 1.134 1.072
t4∼
=1.559 1.299 1.379
t5∼
=1.581 1.774 1.867
t6∼
=2.202 2.094 1.987
t7∼
=2.322 2.415 2.608
t8∼
=2.810 2.890 2.630
t9∼
=3.117 3.055 3.277
t10 ∼
=3.441 3.637 3.360
t11 ∼
=3.915 3.710 3.871
t12 ∼
=4.094 4.323 4.158
t13 ∼
=4.661 4.434 4.473
t14 ∼
=4.775 4.947 4.958
t15 ∼
=5.351 5.236 5.120
t16 ∼
=5.514 5.525 5.696
t17 ∼
=5.999 6.038 5.811
t18 ∼
=6.314 6.149 6.378
a wide range of possible values of those parameters and the
effect of every parameter will be investigated.
4.1. Parameters of the SPWM Technique. The basic param-
eters and the corresponding range that will be used in the
current analysis are as follows.
Modulation Frequency Index Fnc.This parameter is an odd
multiple of 3 for three-phase inverters [4,10]inorderto
eliminate the undesirable odd harmonics. Its value depends
on the nominal power of the inverter. Inverters of high
power use a low value of Fnc, mainly for power losses
elimination reasons and to increase the efficiency. The
current analysis will include every possible inverter size that
may be connected to MV or LV level. Thus, the Fnc will
take values in the range 9–159 or in the range 450–7950 Hz
(corresponding frequency values).
Modulation Amplitude Index Mf.The range is theoretically
between 0 and 1 but common values are in the range 0.6–1.
4.2. Parameters of the Even Harmonics. The basic parameters
and the corresponding range that will be used in the current
analysis are as follows.
Harmonic Order h . The even harmonics that are more likely
to appear are the low order harmonics between the 2nd
and the 10th harmonic. The 2nd harmonic gathers the most

ISRN Renewable Energy 7
t0
t1
t2
t3
t4
t5
t6
t7
t8
t9
(t0)
t10
t11
t12
t13
t14
t15
t16
t17
t18
0π2π
Ud
−Udt
V(t)=V(t)−V(t)
V(t)=V(t)−V(t)
Ud
−Ud
V(t)=V(t)−V(t)
Ud
−Ud
0 0.31 0.63 0.94 1.26 1.57 1.88 2.2 2.51 2.83 3.14 3.46 3.77 4.08 4.4 4.71 5.03 5.34 5.65 5.97 6.28
Ud
2
−Ud
2
0 0.31 0.63 0.94 1.26 1.57 1.88 2.2 2.51 2.83 3.14 3.46 3.77 4.08 4.4 4.71 5.03 5.34 5.65 5.97 6.28
0 0.31 0.63 0.94 1.26 1.57 1.88 2.2 2.51 2.83 3.14 3.46 3.77 4.08 4.4 4.71 5.03 5.34 5.65 5.97 6.28
V(t)
V(t)
Ud
2
−Ud
2
Ud
2
−Ud
2
V(t)
Triangle (t)
Ur
Ur( )
Ur
t
( )t( )t
AO
BO
CO
AB AO BO
BO COBC
CA AOCO
A
B
C
Figure 6: Reference waveforms with an 8th harmonic and zero crossings with the triangular waveform obtained using PSCAD/EMTDC
simulation package [8]. Ah(%) =20%, Fnc =9, φh=60◦,andMf=0.8.

8ISRN Renewable Energy
Tab l e 3: Approximate relation for the maximum DC component versus the harmonic percentage.
Harmonic order Fnc Maximum DC component on voltages
VAO(t), VBO (t), and VCO(t)
Maximum DC component on voltages
VAB(t), VBC (t), and VCA(t)
2nd 9–159 ∼
=0.01 ·Ah∼
=0.035 ·Ah
4th 9–159 ∼
=0.008 ·Ah∼
=0.03 ·Ah
6th 9–159 ∼
=0.007 ·Ah∼
=0.03 ·Ah
8th 15–159 ∼
=0.006 ·Ah∼
=0.02 ·Ah
10th 15–159 ∼
=0.005 ·Ah∼
=0.02 ·Ah
8th or 10th 9 ∼
=0.09 ·Ah∼
=0.42 ·Ah
0π2π
C
B
A
Figure 7: Reference waveforms with the fundamental and an
asymmetric harmonic (φh=90◦,Mf=0.8).
references in the literature [5–7,11–13] as it is the most
frequent and usually takes higher values than the other even
harmonics.
Harmonic Amplitude Ah.The amplitude of even harmonics
on network voltage is usually low under normal operating
conditions. Thus, possible values according to some refer-
ences [2,5] are between 1%and3% of the network voltage
fundamental component amplitude, and the 2nd harmonic
is more likely to keep those levels. The other examined
even harmonics (4th to 10th) rarely present those levels
except for abnormal operation of power system elements
or transient conditions. Specifically, in cases of energization
of high power transformers [2,12] or injection of large
amounts of DC current (e.g., by geomagnetically induced
currents) [6,11,13], large amounts of even harmonics are
generated but for a short time period.
Harmonic Angle φh.The angle between the harmonic and
the fundamental component will take values in the range 0◦–
180◦.
Asymmetry of the Harmonics. The asymmetry assumed in
the current analysis considers the amplitude difference of
harmonics in every network phase voltage. Possible values
for this difference can be assumed to be about 20% at
maximum. Such an asymmetry may appear on the even
harmonics generated by a power transformer when affected
by a dc current injected by the inverter. This current will
be different in every phase A,B,andCas it happens with
the DC components of the line voltages. Consequently, the
even harmonics generated by the power transformer will
have different amplitude in every phase, thus they will be
asymmetrical.
All the above parameters and the value range used in the
current analysis are summarized in Tab l e 3. The analytical
equations for the reference waveforms UA
r(t), UB
r(t), and
UC
r(t) when they contain an even harmonic with 20% lower
amplitude for phase Bin relation with phases Aand C,areas
follows:
UA
r(t)=Mf·sin(t)+Mf·Ah·sinh·t+φh
h,
UB
r(t)=Mf·sint−2·π
3
+0.8·Mf·Ah·sinh·t+φh
h−2·π
3,
UC
r(t)=Mf·sint−4·π
3
+Mf·Ah·sinh·t+φh
h−4·π
3.
(12)
The reference waveforms with a 2nd harmonic and such
an asymmetry are shown in Figure 7.Thedifference on the
maximum (peak) value between phases A(or C)andBin
the positive and negative part of the waveform can be easily
observed.
Applying the methodology of the previous section, the
DC component of the inverter’s line voltages is calculated.
By carefully examining (12) and taking into account those
mentioned in the previous Sections, it can be easily proved
that the DC component will be equal in absolute terms for
line voltages VAB and VBC and zero for the line voltage VCA.
Thus, only one of the components |VAB
dc |and |VBC
dc |has to be
calculated.
5. Results and Discussion
The results for the maximum DC component are given as
a percentage of the rms value of the inverter’s line voltage
fundamental component Vo,1(rms).TheDCcomponentwas
calculated using MathCad [9] not only for line voltages
but also for the voltages between every phase and point O
(Figure 1). Among the three calculated values, the maximum
was taken. It should be noted here that the DC component

ISRN Renewable Energy 9
0
0.1
0.2
0.3
0.4
0.5
9 21334557698193105
117 129 141 153
Maximum DC component (%)
2nd harmonic
0
0.1
0.2
0.3
0.4
0.5
9 21 33 45 57 69 81 93 105 117 129 141 153
Maximum DC component (%)
4th harmonic
6th harmonic
Modulation Frequency Index Fnc
0
0.1
0.2
0.3
0.4
0.5
9 21 33 45 57 69 81 93 105 117 129 141 153
Maximum DC component (%)
Ah=3%
Ah=2%
Ah=1%
Modulation Frequency Index Fnc
Modulation Frequency Index Fnc
(a)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Maximum DC component (%)
2nd harmonic
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Maximum DC component (%)
4th harmonic
9 21 33 45 57 69 81 93 105 117 129 141 153
921 33 45 57 69 81 93 105 117 129 141 153
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Maximum DC component (%)
921 33 45 57 69 81 93 105 117 129 141 153
6th harmonic
Ah=3%
Ah=2%
Ah=1%
Modulation Frequency Index Fnc
Modulation Frequency Index Fnc
Modulation Frequency Index Fnc
(b)
Figure 8: (a) Maximum DC component on voltages VAO (t), VBO(t), and VCO (t) for a 2nd, 4th, or 6th harmonic and Fnc =9–159. (b)
Maximum DC component on voltages VAB(t), VBC (t), and VCA(t) for a 2nd, 4th, or 6th harmonic and Fnc =9–159.
of the voltages between each phase and point Oincreases as
the harmonic percentage increases while the DC component
on the line voltages increases as the asymmetry of harmonics
increases. If no asymmetry exists, the DC component on the
line voltages becomes zero independently of the harmonic
percentage.
In Figure 8, the maximum DC component is shown for
every possible combination of the parameters discussed in
the previous Section when a 2nd, 4th, or 6th harmonic
is present on the SPWM reference waveforms. It can be
seen that the DC component is higher for the voltages
between each phase and point O(Figure 8) in relation with
the line voltages (Figure 9), which was expected to occur.
It can also be observed that the 2nd harmonic gives the
higher DC component among the others followed by the 4th
harmonic. As it was expected, the DC component increases
as the harmonic percentage increases. For Fnc =15 and
low harmonic percentage (1% or 2%), the DC component is
almost zero and takes non-zero value only for a 3% harmonic
percentage. Moreover, the DC component takes almost equal
values for large values of Fnc.
Figure 9 shows the maximum DC component when an
8th or 10th harmonic is present on the reference waveforms.
Only the values for Fnc between 15 and 159 are given in this

10 ISRN Renewable Energy
0
0.1
0.2
0.3
0.4
0.5
15 27 39 51 63 75 87 99 111 123 135 147 159
Maximum DC component (%)
8th harmonic
0
0.1
0.2
0.3
0.4
0.5
15 27 39 51 63 75 87 99 111 123 135 147 159
Maximum DC component (%)
10th harmonic
Ah=3%
Ah=2%
Ah=1%
Modulation Frequency Index Fnc
Modulation Frequency Index Fnc
(a)
8th harmonic
15 27 39 51 63 75 87 99 111 123 135 147 159
Maximum DC component (%)
10th harmonic
15 27 39 51 63 75 87 99 111 123 135 147 159
Maximum DC component (%)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Ah=3%
Ah=2%
Ah=1%
Modulation Frequency Index Fnc
Modulation Frequency Index Fnc
(b)
Figure 9: (a) Maximum DC component on voltages VAO (t), VBO(t), and VCO (t) for an 8th or 10th harmonic and Fnc =15–159. (b)
Maximum DC component on voltages VAB(t), VBC (t), and VCA(t) for an 8th or 10th harmonic and Fnc =15–159.
figure as the DC component follows a different pattern for
Fnc =9. The DC component increases similarly with the low
order harmonics (2nd–6th) decreases as the harmonic order
increases. Moreover, the maximum DC component increases
as the harmonic percentage increases. Similarly with the
lower order harmonics, for a low harmonic percentage (1%
or 2%), the DC component is almost zero. Furthermore, as
for low order harmonics, the DC component takes almost
equal values for large values of Fnc.
Figure 10 depicts the maximum DC component when a
8th or 10th harmonic is present on the reference waveforms
for Fnc =9. In this case, the DC component takes much
higher values than for the other harmonics and the other
values of Fnc and is the same for these two harmonics (8th
and 10th). It can be concluded that these two harmonics
dominate as regards the DC component and they can be
expressed by the relationship Fnc ±1.
By close examination of the results given in Figures 8–10,
an approximate relation can be written for the maximum
DC component versus the harmonic percentage on the
reference waveforms of SPWM technique, which is presented
in Tab l e 3.
Moreover, the value of Modulation Amplitude Index
Mf, at which the maximum DC component occurs, follows
the same pattern for all the examined harmonics except
for the case of 8th and 10th harmonic and for Fnc =9.
Specifically, the maximum (for all the examined parameters)
DC component increases as the Mfincreases and becomes
maximum for Mf=1. In particular, in most cases the DC
component is zero for Mf<0.9. A typical variation pattern
for the DC component versus Mfis shown in Figure 11.
The respective variation for the 8th and 10th harmonic
and for Fnc =9 is shown in Figure 12. Unlike the pattern
shown in Figure 11, in this case the value of DC component
ascends until a maximum value at about Mf=0.98 and
descends until Mf=1.
Finally, Figure 13 depicts the typical pattern followed by
the DC component in relation with the harmonic angle φh
in the majority of cases. For φh=90◦, the DC component
becomes maximum while the values 0◦and 180◦give the
same percentage of DC component in absolute terms.
6. Conclusions
In this paper, the amount of DC components on the output
voltages of inverter-based renewable energy resources was
investigated when one of the first five even harmonics

ISRN Renewable Energy 11
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
8th 10th
Maximum DC component (%)
Harmonic order
Ah=3%
Ah=2%
Ah=1%
(a)
0
0.05
0.1
0.15
0.2
0.25
0.3
8th 10th
Maximum DC component (%)
Harmonic order
Ah=3%
Ah=2%
Ah=1%
(b)
Figure 10: (a) Maximum DC component on voltages VAO (t), VBO(t), and VCO (t) for an 8th or 10th harmonic and Fnc =9. (b) Maximum
DC component on voltages VAB(t), VBC (t), and VCA(t) for an 8th or 10th harmonic and Fnc =9.
0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
DC component
Modulation Amplitude index Mf
Figure 11: Typical variation of the DC component in relation
with the Modulation Amplitude Index Mffor the 2nd, 4th, and
8th harmonics when Fnc =9–159 but also for the 8th and 10th
harmonics when Fnc =15–159.
appears on the network voltage. A calculation method of
the DC components was developed based on the zero
crossings of the SPWM control waveforms. The results for
the DC component were given as a percentage of the rms
fundamental component of the inverter’s output voltage. The
absolute maximum value among the three per phase or line
voltages was taken in order to investigate the maximum levels
of DC component. All the parameters of SPWM technique
and even harmonics were examined and the maximum range
of values was used.
It can be concluded by the current analysis that the
DC components on the inverter’s output voltage are not
considerable under normal operating conditions and do
not constitute a problem even for high penetration of
distributed energy resources to the electric grid. This is
supported by the fact that the values of the DC components
assessed here are the maximum expected and only when a
0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
DC component
Modulation Amplitude index Mf
Figure 12: Typical variation of the DC component in relation
with the Modulation amplitude index Mffor the 8th and 10th
harmonics when Fnc =9.
0 30 60 90 120 150 180
DC component
Harmonic angle ϕh(◦)
Figure 13: Typical variation of the DC component in relation with
the harmonic angle φhfor all even harmonics.

12 ISRN Renewable Energy
series of coincidences occurs, a considerable amount of DC
components may appear. Moreover, as more sophisticated
topologies and control systems are used for the inverter-
based interface system of the renewable energy resources, the
DC injection will be minor. The only exception found by the
current analysis which gives a considerable amount of DC
components is the case for Fnc =9 together with the presence
of an 8th or 10th harmonic.
Acknowledgments
This paper was prepared during a project funded by the
European Union and Greek Government. The authors wish
to acknowledge the significant contribution of the sponsors.
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