![(a) Third harmonic cancellation through a 60-degree phase shift of inverter 2 in a twocoupled inverter system. (b) Harmonic cancellation through a 30-degree phase shift of inverter 2 in a two-coupled inverter system [17].](https://www.researchgate.net/publication/380841202/figure/fig1/AS:11431281246980265@1716559662809/a-Third-harmonic-cancellation-through-a-60-degree-phase-shift-of-inverter-2-in-a_Q320.jpg)
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Citation: Safarishaal, M.; Hemmati,
R.; Saeed Kandezy, R.; Jiang, J.N.; Lin,
C.; Wu, D. Nonlinear Impact of
Topological Configuration of Coupled
Inverter-Based Resources on
Interaction Harmonics Levels of
Power Flow. Energies 2024,17, 2512.
https://doi.org/10.3390/en17112512
Academic Editor: Ahmed Abu-Siada
Received: 29 March 2024
Revised: 10 May 2024
Accepted: 16 May 2024
Published: 23 May 2024
Copyright: © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
energies
Article
Nonlinear Impact of Topological Configuration of Coupled
Inverter-Based Resources on Interaction Harmonics Levels
of Power Flow
Masoud Safarishaal 1,*, Rasul Hemmati 1, Reza Saeed Kandezy 1, John N. Jiang 1, Chenxi Lin 2and Di Wu 3
1School of Electrical and Computer Engineering, University of Oklahoma, Norman, OK 73019, USA;
reza.kandezy@ou.edu (R.S.K.)
2Energy & Power, Jacobs Engineering Group, Orlando, FL 32801, USA; chenxi.lin@jacobs.com
3Electrical and Computer Engineering Department, North Dakota State University, Fargo, ND 58105, USA;
di.mu.3@ndsu.edu
*Correspondence: masoud.safari@ou.edu
Abstract: The increasing level of harmonics in the power grid, driven by a substantial presence
of coupled inverter-based energy resources (IBRs), poses a new challenge to power grid transient
stability. This paper presents the findings from experiments and analytical studies on the impact of
the topological configuration of coupled IBRs on the level of power flow harmonics in a distribution
grid: (i) our findings report that the impact of grid topology on harmonics is nonlinear, which is in
contrast to the common perception that the power grid operates as a large linear low-pass filter for
harmonics; (ii) importantly, this study highlights that the influence of the topological configuration
of inverters on the reduction of system-level harmonics is more substantial than the effect of line
impedance, emphasizing the significance of grid topological configuration; (iii) furthermore, the
observed reduction in harmonics is attributed to a harmonic cancellation effect achieved through
self-compensation by all the coupled inverters without affecting the active power flow in the power
grid. These findings propose a new approach to limit the penetration of complex IBR harmonics
in the power grid from a system-wide perspective. This approach significantly differs from the
component-level or localized solutions used today, such as inverter control, power filtering, and
transformer tap changes.
Keywords: coupled inverter systems; harmonic levels; topological configuration
1. Introduction
1.1. Current State-of-the-Art
High penetration of inverter-based resources will increase the amount of harmonics in
power flows in the power grid, which can cause various complex stability issues. These
are important ways that harmonics from high penetration of inverter-based resources can
negatively impact power system angle stability and voltage stability, as highlighted in
references [
1
–
5
]. The majority of advanced techniques proposed, developed, and imple-
mented to reduce harmonics in response to this challenge are component or local-level
solutions achieved by controlling the IGBT switching strategy and topology of inverters,
blocking high-frequency components with power filters, and shifting the phase of IBRs via
transformer tap changing.
As an example of inverter control, in [
6
], a control strategy was put forth for designing
sequence-asymmetric systems with the aim of locally correcting harmonic voltage issues.
In [
7
], the authors present a technical idea that integrates a nearest-level controller with
selective harmonic elimination control to improve the quality of the inverter’s output
terminal voltage while minimizing harmonic distortion; a new hybrid frame controller is
introduced, where the synchronous frame is utilized for the adaptive harmonic compen-
satory in the grid-connected inverter [
8
]; in [
9
], the article proposes a novel filter, which
Energies 2024,17, 2512. https://doi.org/10.3390/en17112512 https://www.mdpi.com/journal/energies
Energies 2024,17, 2512 2 of 16
effectively attenuates harmonics with small inductance for grid-connected inverters. Gen-
erally, control-based methods focus on active power sources by regulating the switching
patterns of inverters. These methods are designed to provide localized control to reduce
harmonic levels on the source side. The filter-based approach is to design passive filters at
the inverter output, aiming to block high-frequency components from entering the power
lines and the circuit of the power grid. Typically, passive elements such as LCL or LC filters
are employed. These methods normally are not very economical as they will increase the
cost for integration of IBRs and have the risk of inducing undesired or detrimental complex
transients such as harmonic resonance and inrush currents, as reported in [10,11].
Appropriate shifting of the phases of inverters is an emerging technical approach that
has shown its effectiveness. By strategically adjusting the relative phase angles among
coupled IBRs via the tap setting of the coupling transformer, this technique enables inter-
connected inverters to mitigate each other’s harmonics, resulting in a substantial reduction
in power flow harmonics. The effectiveness of this technique was reported in [
12
]. Many
works following this phase-shifting idea have been reported. For example, in [
13
,
14
], a
different low-frequency hybrid modulation technique is proposed to incorporate phase
shift in pulse width modulation (PS-PWM) and asymmetric selective harmonic current
mitigation PWM (ASHCM-PWM) to reduce harmonics.
The concept of phase-shifting is specifically discussed in [
15
], which explains the
relation between harmonics reduction and phase displacements from a perspective of
coupled IBRs, as well as the nonlinear effect of cancellation interactions of IBRs.
1.2. Contribution of the Paper
Most existing methods to address the challenge of limiting the proliferation of harmon-
ics to power grids to meet predefined design and compatibility standards for integrating
IBRs to power grids are typically
resource-side
technologies using either
component
-level
or
local
solutions. This paper presents the findings of experimental results as well as ana-
lytical studies, suggesting a possible approach
from a system perspective
. This approach
is novel in that it offers a solution to limit the penetration of the harmonics from the system
perspective, different from the component level or local solutions used today. Adjusting the
topological configuration does not need to increase the cost of the power grid and will not
change the active power flows. Thus, it can be seen as a low-cost and
f ree
technical option
for harmonic reduction as it engineers self-compensation interactions of coupled inverters
in the same sense of harmonics cancellation through phase-shifting as described in [15].
1.3. Paper Organization
The remainder of the paper is organized as follows: Section 2provides an analytical
explanation of how the topological configuration affects harmonics. Section 3introduces
the experimental hardware employed in this paper, along with the primary experimental
findings and discussions regarding the influence of electrical distance on total harmonic
distortion. Section 4is dedicated to the presentation of further experimental studies. Finally,
in Section 5, the concluding remarks of this paper are presented.
2. Analytical Explanation of the Relationship between Harmonics and Topological
Configuration in Power Systems
In this section, we start by explaining the relationship between the phase and the
topological configuration of the power system. Subsequently, we explore how phase shifts
in coupled inverter-based resources could influence harmonic content.
2.1. Explanation of the Relationship between Topological Configuration and Phase in Power
Systems Using the Power Flow Equation
To provide a comprehensive explanation of the relationship between topological
configuration and phase in power systems, it is essential to elucidate the phase relationship
and topology through the basic power flow equation between the two points. The power
flow equation for a basic power line can be expressed as Equation (1).
Energies 2024,17, 2512 3 of 16
pωi
ij =ViVj
ωiLωi
ij
(θωi
i−θωi
j), (1)
where
pij
is the active power output from point
i
to point
j
, and
ωi
is the average frequency
of switching of the inverter. To reflect the influence of operation frequency on this equation,
the parameters are defined at
ωi
frequency.
Lωi
ij
is the reactance between point
i
and point
j
,
θωi
i
is the phase angle at point
i
, and
θωi
j
is the phase angle at point
j
at frequency
ωi
. Also,
Viand Vjare the amplitudes of the voltage of the two points.
At a given frequency and load level, assuming nearly constant voltage, the electrical
connection between points is typically characterized by line reactance and phase differences.
In a power network including
N
buses connected by
L
power lines, the relationship
between net active power (
P
), nodal voltage magnitude (
V
), and voltage phase (
θ
) is
governed by Kirchhoff’s circuit laws. It can be written as follows [16]:
Pm=Vm
N
∑
n=1
Vn(Gmn cos(θm−θn)Bmn sin(θm−θn), (2)
for bus
m=
1, 2,
. . .
,
N
.
Gmn
is the magnitude of the real component of the
(m
,
n)
element
of the bus admittance matrix
Y
and,
B
is the the magnitude of imaginary component of it.
We can make three assumptions about the system:
(1) Flat voltage profile, i.e., Vm≈Vn≈1.0 p.u.;
(2) Approximately homogeneous bus phases across the network, i.e.,
cos(θm−θn)≈
1
and sin(θm−θn)≈(θm−θn); (because (θm−θn)is near to 0);
(3) The reactive property of a line is much more significant than its resistive property,
i.e., Bmn ≫Gmn. Under these assumptions, Equation (2) reduces to the following:
Pm=
N
∑
n=1
Bmn(˜
θmn), (3)
where
˜
θmn
denotes the approximate system phase angle at the observation point. Consid-
ering the above-mentioned assumption and given the fact that
(Pm≫Qm)
, it possible to
approximate the above expression according to (Pm∝I2
m):
I2
m=
N
∑
n=1
B′
mn(˜
θmn), (4)
where
Im
is the net electrical current follow of bus
m
. For a balanced power system
with no active power mismatch,
I2
m
is a constant value. In a steady-state system with
a constant electrical current fundamental component, a direct relationship between the
phase difference and the power network’s topological configuration can be established by
examining how the admittance matrix is constructed based on the reactance of power lines.
2.2. Exploring the Relationship between Harmonics and Phase
In the preceding subsection, we explicated that by adjusting the reactance, phase
difference is induced in power lines. In this section, we aim to elucidate how this phase
difference between two lines impacts the harmonics level of the coupled IBR system.
The relationship between harmonics and phase at the interaction point of the coupled
system could be explained with the harmonic cancellation effect. The abstract concept of
harmonic cancellation involves separating the power supply into two, shifting one source
of harmonics 180 degrees to the other, and then combining outputs.
For instance, the output voltage of a full-bridge inverter with a sinusoidal pulse width
modulation (SPWM) switching method is mathematically described by equation [15]:
Energies 2024,17, 2512 4 of 16
V(t) = VDC Mcos(ω0t+θ0)
| {z }
Fundamental component
,
+
∞
∑
m=1
Wmcos(m(kω0t+ˆ
θc))
| {z }
H= {High-order intra-harmonic components}
,
+
∞
∑
m=1
∞
∑
n=−∞
Wmn cos((m(ωct+θc) + n(ω0t+θ0))
| {z }
{Harmonic components of carrier and sideband harmonics /∈H(inter-harmonics)},
(5)
where
θc
is the phase of carrier waveform,
θ0
is the phase angle of fundamental waveform,
and
Wm
and
Wmn
are magnitudes of intra-harmonics and inter-harmonics, respectively.
Without losing generality,
ˆ
θc
in Equation (5) is the characteristic phase of the inverter carrier
of the frequencies that belong to the synchrophasor family of the fundamental frequency
where all harmonic components fully commute.
VDC
is the DC link voltage,
M∈[
0 1
]
is the
modulation index,
ωc
is the carrier angular frequency, and
ωs
is the fundamental angular
frequency. The angular carrier to angular frequency ratio is an integer (
ωc
ω0=k
), the carrier
and baseband harmonics would be inter-harmonics to the fundamental frequency [15].
When the fundamental component of output electrical voltage shifted by
∆θ
, the
term
(n(ω0t+θ0+∆θ))
, the appropriate phase shifts in inter-harmonics components are
multiplied by
n
. If two identical inverters with similar lines are connected in parallel,
there may be a situation where a 180-degree phase shift in one line results in a shift in the
harmonic. In this case, there would be two current harmonics with a 180-degree phase
difference at the connection point from the two coupled inverters. The corresponding
currents would have the same magnitude in the opposite direction, resulting in their
cancellation.
Figure 1a shows how a
π/
6 degree phase shift in output two results in the third
harmonic of two outgoing waves that cancel each other. However, this is not limited to
only one harmonic. Figure 1b shows how some of the harmonics are removed with this
30-degree phase shift. The set of these removed harmonics finally determines the impact of
phase change on the total harmonic distortion.
Figure 1. (a) Third harmonic cancellation through a 60-degree phase shift of inverter 2 in a two-
coupled inverter system. (b) Harmonic cancellation through a 30-degree phase shift of inverter 2 in a
two-coupled inverter system [17].
This explanation can be extended to the number of inverters. For example, Figure 2
shows Ncoupled inverters.
Energies 2024,17, 2512 5 of 16
Figure 2. Topological configuration of Ncoupled inverters.
A simplified model of
N
-coupled inverters is used as an example to illustrate the
relationship between reactance and harmonics level. This model is based on the unfiltered
voltage of an H-bridge and was originally presented in [
15
]. In this study, the equation is
rewritten under the assumption that the inductance of each line may not be equal and that
the inverters are identical as follows:
IL(t) =
N
∑
i=1
Ycos(ωs(t)−Leq ωs
R),
N
∑
i=1
∞
∑
n=2
Xin cos(nωs(t)−φn),
+
N
∑
i=1
∞
∑
m=1
∞
∑
n=−∞
Wmni cos [2mωc+ [2n−1]ωs](t)
−φmn !,
Y=4VDC
π
R
qR2+ (Leq ωs)2
1
[ωs
ωc]J1(ωs
ωc
π
2M),
Xin =4VDC
π
R
qR2+ (Leq nωs)2
1
[nωs
ωc]Jn(nωs
ωc
π
2M)sin(nπ
2),
Wmni =4VDC
π
R
qR2+ (Leq (2mP + [2n−1])ωs)2,
×1
h2m+ [2n−1]ωs
ωciJ2n−1(2m+ [2n−1]ωs
ωcπ
2M),
×cos([m+n−1]π),
tan(φn) = Leqnωs
R,
tan(φmn ) = Leq(2mP + [2n−1])ωs
R,
Leq =1
N
∑
i=1
1
Li
,
(6)
where
VDC
is the DC link voltage,
N
is the number of coupled inverters,
M∈[
0 1
]
is
the modulation index,
ωc
is the carrier angular frequency,
ωs
is the fundamental angular
Energies 2024,17, 2512 6 of 16
frequency,
Jn(x)
is a Bessel function of order
n
and argument
x
,
m
is the carrier index,
n
is
the beseband index,
R
is the load resistance,
Leq
is the equivalent inductance, and
P=fc
fs
is
the carrier to reference frequency ratio.
Considering
(Leq =
1
/N
∑
i=1
1
Li)
, the inductance of all lines determines the equivalent in-
ductance. Therefore, adjusting each power line produces a change in equivalent inductance.
As a result, the amplitudes of
Xin
and
Wmni
would be changed. Simultaneously, due to the
impact of equivalent inductance in
φn
and
φmn
, the phases of all harmonic components
would be shifted. The impact of changing the equivalent inductance is primarily observed
on the higher-order harmonics, as they experience a higher phase shift based on
φn
and
φmn equations. Figure 3represents a summary of the topics discussed in this section.
Figure 3. Summary of the discussed topics regarding the relationship between topological configura-
tion and harmonics level).
Two key characteristics of phase shift and the harmonic cancellation effect on coupled
systems, which we aim to illustrate in an experimental setup, could be as follows:
•
It does not exert a substantial influence on the fundamental component; however, at
higher frequencies, it notably affects the harmonic component.
•
The process of harmonic alteration does not follow a linear relationship between har-
monic levels and frequency because it may lead to a decrease or increase in harmonic
level between two consecutive harmonics orders.
2.3. Terminology and Definitions Used in this Work
2.3.1. Definition of Electrical Distance
The term “electrical distance” carries various definitions depending on the context [
18
,
19
].
Generally, it represents the proximity or separation of components within an electrical or
power system, forming its topological configuration. In power system analysis, electrical
distance describes the effective impedance between network points, considering electrical
characteristics such as resistance, capacitance, and inductance that influence energy flow.
The definition of electrical distance remains non-unified. It is often defined as the mutual
impedance between two points based on their electrical characteristics [
20
]. Researchers
have used this concept in various power system studies to describe the distance between
Energies 2024,17, 2512 7 of 16
points based on measurements [
21
,
22
]. Despite these differing definitions, electrical distance
plays a significant role in shaping power system topology. In this study, we adjusted
parameters between two points, effectively altering the electrical distance between them
and resulting in changes to the topological configuration of the power system.
2.3.2. Relative Distance Ratio
Relative electrical distance refers to the comparison of electrical distances between
different points in a power system. Unlike local parameters that describe characteristics at
specific locations, relative electrical distance pertains to the system as a whole. It involves
assessing the ratio of electrical distances between two points. This comparison helps in
understanding the relative impact of electrical distances on various aspects of power system
behavior, taking into account the interconnected nature of the entire system.
While local or component parameters focus on the characteristics of individual el-
ements, relative electrical distance looks at the system-level relationship between these
elements. It provides valuable insights into how the arrangement and configuration of
components affect the overall system performance.
2.3.3. Total Harmonic Distortion (THD)
The level of total harmonic distortion of the electrical voltage is used to comprehen-
sively determine how electrical distance in coupled inverter systems impacts harmonics
and is defined as follows:
THD =v
u
u
t
N
∑
n=2
(Vn
V1
)2∗100, (7)
where
Vn
is the magnitude of
n
th harmonic component, and
V1
is the magnitude of fun-
damental component. In this work, the total harmonic distortion was calculated for the
electrical voltage signal.
3. Experiment Design and Primary Findings
3.1. Test Scenarios and Experiment Setup
This subsection introduces the experimental test setup and designed scenario utilized
to illustrate the relationship between the topological configuration of the power system
and harmonics levels in a power system integrated with IBR.
3.1.1. Design of Test Scenarios
To demonstrate the impact of the topological configuration on the interaction harmon-
ics of the coupled IBR system, a power system was designed with two coupled power lines
of IBR clusters, facilitating adjustments to the relative impedance.
A slight mismatch was created between the output of two inverter clusters to instigate
the interaction of harmonics. It should be noted that the inverter output is filtered, and
the total harmonic distortion (THD) of an individual system is less than 1%. However, the
THD increased by over 5% after coupling the two clusters. This change in THD, despite the
presence of filters, is attributed to the mismatch between the two clusters. The analyses
conducted in this work consider these harmonics, which are termed interaction harmonics.
The schematic diagram illustrating the experimental setup for the coupled configuration
of IBR clusters is depicted in Figure 4. Here,
˜
θmn
denotes the approximate system phase
angle at the observation point,
θ1
represents the phase of the output voltage of IBR cluster
1, and
θ2
signifies the phase angle of the output of IBR cluster 2. Furthermore,
Xline1
,
r1
,
and
c1
denote the impedance values of the
Π
model corresponding to line 1, while
Xline2
,
r2
, and
c2
signify the impedance values of the
Π
model associated with line 2. Additionally,
the capacitance
C
, resistor
R
, and inductance L are considered in the coupling interface
circuit of two power lines. The study involved ten different test configurations, each at
a different characteristic impedance, where the line inductance was adjusted to reflect
changes in the electrical distance as well. The inductance of
Xline1
remained constant in all
Energies 2024,17, 2512 8 of 16
test configurations. In contrast, the inductance of line 2 increased by almost equal amounts
with each subsequent test configuration. In the first configuration, the inductance of
ˆ
Xline2
was equal to
X2
. For the second configuration, an
∆X
and related
Π
model elements were
added to
ˆ
Xline2
. This approach was repeated for the remaining test configurations, where
the inductance value of line 2 increased using the same methodology. Table 2 lists the
applied inductance values for each test configuration.
Figure 4. Schematic of the initial experimental setup: exploring 10 test configurations, where
ˆ
Xline2
is
altered but Xline1remains the same (Table 1).
Table 1. Inductance values in 10 configurations of primary test setup (Figure 4).
Test Number XLine1(mH) ˆ
XLine2(mH) Ratio ( ˆ
XLine2/XLi ne1)
10.280 0.279 0.99
20.280 0.570 2.03
30.280 0.864 3.08
40.280 1.136 4.05
50.280 1.429 5.10
60.280 1.708 6.10
70.280 1.989 7.10
80.280 2.245 8.01
90.280 2.528 9.02
10 0.280 2.797 9.98
3.1.2. Hardware Configuration of the Experimental Test Setup
Based on the designed test, the hardware configuration included two coupled IBR
clusters coordinated within a specific grid integration coordination unit, as illustrated in
Figure 5. To set up the power line, a fixed resistance and a series of reactances connected in
series were used, allowing for adjustment of the power system’s topological configuration
by altering the reactance or electrical distance of power systems.
By setting the reactance of the lines during the tests, the system’s impedance charac-
teristics were changed for various test scenarios. The power line included a sequence of
inductances and one resistor. These components were positioned after the IBR clusters.
A one-to-one transformer was implemented between lines and equivalent power (P) and
voltage (V) load. Table 2presents the parameters used in the test setup. In our experimental
setup, the test was designed to investigate interaction harmonics, where the mismatch
between two coupled systems creates harmonics in the network after the filters on the
inverter side. Our objective was not to intentionally increase or decrease reactance in a
manner that alters active and reactive power. Extrapolating scaled-down experimental
results to full-scale systems with confidence requires careful consideration, especially for
Energies 2024,17, 2512 9 of 16
nonlinear phenomena like harmonics. Some of the key conditions we tried to satisfy include
the following:
•
Using representative distribution line parameters and rodding geometries scaled
down appropriately;
•
Ensuring harmonic sources/loads had comparable behavior to full-scale inverter/rectifier
harmonics operating at frequencies high enough that skin-effect scaling is maintained;
• Avoiding excessive thermal transients by using a short test duration.
However, there are inevitably limitations in capturing all performances at full scale.
Figure 5. The hardware configuration of the experimental test setup.
Table 2. Parameters used in hardware test setup
Parameter Value Explanation
DC source 12 V Dc voltage resource
IBR clusters 120 V, 60 Hz, 0.5 kW Pure sine wave inverters
R1 = R2 = R3 100 ΩLines resistance
Req 100 ΩEquivalent load resistance
Leq 0.540 Hz Equivalent load inductance
Transformer 120 V, 60 Hz 1:1
3.2. Demonstration of the Relationship between the Level of Total Harmonic Distortion and the
Electrical Distance
The results of this study demonstrate that the change in electrical distance can signif-
icantly impact the output total harmonic distortion level in the coupled inverter system.
Furthermore, modifying the distance can lead to achieving a minimum output total har-
monic distortion level. Figure 6a shows the output voltage waveforms of the ten tested
configurations with different inductance values. Each line represents a different configura-
tion, and the x-axis represents time in seconds, while the y-axis represents the configuration
number from 1 to 10. The time window of interest is between 0 and 0.04 s. Figure 6b shows
the FFT estimations of the same ten configurations in dB, with the x-axis representing
frequency in Hz and the z-axis representing the magnitude in dB. The frequency range is
limited to 10 kHz. These figures provide a visual representation of the output characteristics
of each configuration, which are analyzed in this section.
Energies 2024,17, 2512 10 of 16
Figure 6. Experimental results of the impact of electrical distance on total harmonic distortion in
coupled inverter systems. (a) Voltage waveform of ten tested configurations with varying inductance.
(b) Fast Fourier transform (FFT) analysis of ten tested configurations showing the levels of harmonic
distortion (dB).
Figure 7illustrates the output THD levels of the experimental results at each induc-
tance value (electrical distances), along with a fit curve of output THD level versus electrical
distance. Three points for each inductance value represent the results of three repeated tests.
The graph clearly shows a decreasing trend of output THD level with increasing electrical
distance. However, this trend is reversed after the seventh test configuration position,
where
ˆ
Xline2
is 1.989 mH (
Xline1=
0.280 mH). Contrary to the conventional prescription
that THD decreases with increasing electrical distance, this result shows a different trend in
that area.
Figure 7. Experimental result of output THD level vs. electrical distance for initial experimental
study (test repeated 3 times at any distance).
These observations highlight the impact of increasing electrical distance on output
THD levels and demonstrate the nonlinear relationship between them. Therefore, appropri-
ate selection of electrical distance can lead to improved current quality in terms of reduced
total harmonic distortion and mitigation of relevant harmonics.
4. Further Experimental Investigation Inspired by Analytical Studies
4.1. Design of Test Scenarios
The purpose of further experimental investigation is to examine the topological config-
uration impact on harmonics by altering the entire structure of the coupled inverter system
when the electrical distances of both power lines are changed concurrently as well as the
ratio of relative distance (
ˆ
Xline2/ˆ
Xline1
). Figure 8displays the designed circuit schematic
for further investigation. In contrast to the previous experiment, the distances of both lines
have been modified in this experimental study. The ratio of the distance of line 2 to line 1
Energies 2024,17, 2512 11 of 16
increased in each configuration from 1 to 10. The reactance values used in the lines for each
of the 10 configurations are listed in Table 3.
Figure 8. Schematic of the extended experimental setup: incorporating 10 test configurations
(Figure 4); both ˆ
Xline1and ˆ
Xline2have been changed (Table 3).
Table 3. Inductance values in 10 configurations for further investigation (Figure 8).
Test Number ˆ
XLine1(mH) ˆ
XLine2(mH) Ratio ( ˆ
XLine2/ˆ
XLine1)
12.61 0.279 0.10
22.253 0.570 0.25
31.997 0.864 0.43
41.717 1.136 0.66
51.436 1.429 0.99
61.142 1.708 1.49
70.878 1.989 2.26
80.574 2.245 3.91
90.270 2.580 9.55
10 0.091 2.797 30
4.2. Further Experimental Results and New Findings
4.2.1. Nonlinear Relationship between Topological Configuration and Harmonics
This section demonstrates that the total level of harmonic distortion is impacted by the
entire topological configuration of the power system and is not dependent on just a single
line characteristic impedance. Figure 9illustrates the output THD level versus electrical
distance diagram in the experimental test configuration for further investigation, where
changes in the distance were applied to both lines. Table 3lists the inductance values
applied to each line for the ten configurations. In comparison to the diagram in Figure 7,
the diagram in Figure 9shows a shift in the point of minimum output THD level. In this
experimental setup, where the inductance of line 2 is 1.136 mH, the output THD level is at
a minimum value, whereas in the initial experimental setup, it was at 1.989 mH. Moreover,
the fit curve of the results shows a sharper decrease in the output THD level versus the
electrical distance relationship at the beginning of the graph in the second setup. These
differences indicate that changes in the distances of both lines have distinct effects on
harmonics and that the total harmonic distortion is affected by the entire structure of the
coupled inverter system rather than just a single line.
Energies 2024,17, 2512 12 of 16
Figure 9. Experimental result of output THD level vs. electrical distance for further investigation
(ratio = ˆ
Xline2/ˆ
Xline1).
Also, some other test scenarios have been done, which are presented in Table 4. This
finding indicates that the relationship between electrical distance and total harmonics
distortion is nonlinear. This nonlinearity highlights the distinct nature of the harmonic
relationship with electrical distance compared to the variations arising from passive filters.
In this work, our focus is on adjusting reactance rather than just increasing reactance.
By modifying the distances between components or adding/removing IBR lines, we aimed
to determine optimal configurations. It is worth noting that optimal configurations were
occasionally achieved by reducing or increasing reactance or making adjustments to IBR
lines.
Table 4. Tested scenarios and conclusions
Sc. No. Definition of Scenarios State of Change Result
1
Increasing the reactance
of line 1 while keeping
line 2 fixed.
Xline1: Fix, Xline2: Increase
The nonlinear relation-
ship between output THD
and topological configura-
tion changes.
2
Increasing the reactance
of line 2 while keeping
line 1 fixed.
Xline1: Increase, Xline2: Fix
Same as in scenario 1: The
nonlinear relationship be-
tween output THD and
topological configuration
changes.
3
Simultaneous increase of
the reactances of both
lines with a constant rate.
Xline1: Increase, Xline2: Increase
The THD vs. distance
curve varies more
smoothly with electrical
distance compared to the
previous two scenarios.
4
Increase in the reactance
of line 2, decrease in the
reactance of line 1, and a
sharp increase in the ratio.
Xline1: Increase, Xline2: Decrease
The THD vs. distance
curve varies more sharply
with electrical distance
compared to the previous
scenarios.
4.2.2. Harmonic Cancellation Effect through Self-Compensation
This subsection is focused on a deeper analysis of the analytical explanation in
Section 3
and the results discussed in Section 3, which confirms that the variation of
total harmonic distortion results from the high-frequency interactions among all coupled
Energies 2024,17, 2512 13 of 16
IBRs, not the change of IBR outputs by some harmonic control or mitigation technologies.
More specifically, changing the electrical distance via phase displacement will cause the
cancellation of some harmonics of certain frequencies, which may lead to a reduction in the
total harmonics distortion level.
In the analysis of Section 2.1, two key characteristics of harmonic cancellation have
been presented. First, the process of harmonic alteration deviates from a nonlinear rela-
tionship between harmonic levels and frequency. This nonlinearity may result in either a
decrease or increase in harmonic levels between consecutive orders within the combined
harmonics produced by coupled inverters. Second, harmonic cancellation minimally im-
pacts both voltage and current fundamental components but significantly influences the
harmonic component, particularly at higher frequencies. In the analysis of this section, two
key characteristics of harmonic cancellation have been illustrated through the experimental
test results.
To illustrate the first characteristic of the harmonic cancellation process, which involves
the deviation from a nonlinear relationship between harmonic levels and frequency in the
process of harmonic alteration, a radar plot analysis was conducted.
The experimental results obtained in each experiment with different test configurations,
from number 1 to 10, were used to plot the radar plot of the harmonics spectrum shown in
Figure 10. The radar diagram in Figure 10 shows the magnitude of all individual harmonics
in the frequency domain up to 10 kHz for each of the 10 different test configurations. The
closer the points are to the center of the circle, the lower the frequency of the harmonic,
while the further away the points are from the center, the higher the frequency of the
harmonic. The color of each point indicates the magnitude of the harmonic at a certain
frequency in dB. Each configuration number corresponds to a specific zone in the radar
diagram, which is shown in a clockwise rotation from test configuration 1 to 10. For
instance, zone 1 in the radar diagram corresponds to test configuration 1.
Figure 10. (a) Radar diagram of frequencies of output experimental results in Section 2, showing the
harmonic content of the inverter-based power system with two coupled inverter systems. (b) The
magnitude of harmonics at 4 kHz, 5 kHz, and 6 kHz frequencies is represented in dB (the results
depicted in this figure represent a single repetition of the test results presented in Section 3.2, Table 2,
and Figure 7).
By looking at the coordinates of a specific point on the radar diagram, one can deter-
mine the test configuration number, frequency, and harmonic level of that particular point.
This helps in understanding how changing the distance between coupled IBRs affects the
harmonic levels in the frequency domain.
The radar diagram analysis helps to explain the phenomenon of harmonic cancellation
and how it affects the output THD level. The electrical distance between IBRs causes
a phase shift in the harmonics, which can lead to the cancellation of certain harmonics.
The variation in the output THD level is, therefore, the result of the cancellation of these
harmonics, and the radar diagram shows that this cancellation effect is not a general trend
for all harmonics like the local filtering method, but rather each harmonic has a unique
level variation.
Energies 2024,17, 2512 14 of 16
For a more precise interpretation of the radar plot, several target points were examined.
Tracking some target points at a specific frequency, from the first to the tenth configuration,
the color variations of the points (representing the magnitude) are not uniform; they do not
decrease or increase at the same time. This implies that each individual harmonic has a
distinct relationship with the change in electrical distance. To provide a clearer analysis,
Figure 10b illustrates the corresponding harmonic level measured at three target points
with 4 kHz, 5 kHz, and 6 kHz frequencies over 10 tested electrical distances. These points
are displayed in yellow in zone 1, corresponding to configuration 1 in Figure 10a. By
examining the trend of harmonic level changes in these three frequencies, it is apparent that
all three harmonics follow a unique trend (Figure 10b). Their increase and decrease do not
occur simultaneously, which is significant because it demonstrates that any harmonics with
a specific frequency are influenced by a change in electrical distance in a distinct way that
corresponds to the harmonic phase change. It indicates that the total harmonic distortion
and electrical distance relationship are produced by the combination of these individual
changes.
Figure 11a illustrates the fundamental component values obtained for each of the ten
test configurations used to investigate the effect of electrical distance changes (the results
depicted in this subsection represent a single repetition of the test results presented in
Section 3.2, Table 2, and Figure 7). Our goal is to demonstrate that the correlation between
THD levels and electrical distance is not solely due to changes in the fundamental values of
line voltage. The results indicate that the percentage of changes in fundamental components
is relatively small, which could be attributed to measurement accuracy. The maximum
difference observed from the 120 V voltage level is 1.8 V. The difference between the highest
and lowest values is less than 2% of the minimum value, whereas the THD level difference
is 10%. Moreover, Figure 11b illustrates that there is no linear correlation between the
patterns of change of the fundamental components in the ten test configurations and the
THD level variation. This means that there are other factors, in addition to changes in the
fundamental values of line voltage, that contribute to the correlation between THD levels
and electrical distance. Also, this relatively small change in fundamental component values
may not have a significant impact on the correlation between THD levels and electrical
distance.
Figure 11. Experimental results show the change of fundamental component values through test
1–10 (the results depicted in this figure represent a single repetition of the test results presented in
Section 3.2, Table 2, and Figure 7) (a) Fundamental component values across ten test configurations;
(b) relationship between THD and fundamental component values. (The x-axis shows increasing
fundamental component values).
5. Concluding Remarks
The power grid has traditionally been regarded as a large low-pass power filter, with
its system-level impact on harmonic propagation over the power lines viewed as a linear
or first-order effect. The novel aspect of how the topological configuration influences the
level of harmonics accompanying the energy flow in the power grid, primarily driven by
coupled inverter-based energy resources, advances our understanding of the relationship
between the power grid structure and harmonics.
Energies 2024,17, 2512 15 of 16
If we represent the structural impact of the grid configuration with the underlying
characteristic impedance of a cluster of IBRs and express the relative distance between
different IBR clusters through the ratio of characteristic impedance ratios, we experimentally
discovered that the impact of the coupling configuration of IBR clusters on the level of
power flow harmonics, as indicated by the total harmonic distortion index, is nonlinear. As
illustrated in Figures 7and 9for the two IBR cluster setup, the total harmonic level initially
decreases with an increase in the relative distance ratio. However, as we further increase
the ratio, the level of harmonics surprisingly begins to increase significantly.
Hundreds of intensive experiments with different configurations, line impedances,
IBRs, and load levels have been carried out in this way, and every one of them showed the
effect to a greater or lesser degree. These experiments also confirm that this nonlinearity is
associated with the grid structure, as a proportional increase in characteristic impedances
among the IBR clusters has a minor impact on the harmonics. In other words, it is the
structural configuration, represented by the characteristic impedance ratio, that causes this
nonlinear effect.
This discovery suggests a potential new approach to limiting the penetration of
complex IBR harmonics in the power grid from a system-wide perspective. This approach
is distinct from conventional component-level or localized solutions employed today,
such as inverter control, power filtering, and transformer tap changes. It is particularly
attractive because increasing the capacity of power filters significantly raises costs and may
introduce other complex effects like electromagnetic resonance. It is worth mentioning
that reducing the level of power flow harmonics through an appropriate reconfiguration
of the coupling structure is a cost-effective, independent option. This approach leverages
the self-compensation interactions of IBR clusters without altering the active power flow,
making it an attractive alternative.
Author Contributions: Conceptualization, M.S. and J.N.J.; Methodology, M.S.; Software, M.S. and
R.H.; Validation, M.S.; Investigation, M.S. and R.S.K; Writing—original draft, M.S.; Writing—review
& editing, R.H., R.S.K., J.N.J., C.L., and D.W.; Visualization, M.S.; Supervision, J.N.J.; Funding
acquisition, D.W. All authors have read and agreed to the published version of the manuscript.
Funding: Foundation (NSF) EPSCoR RII Track-4 Program under the grant number OIA-2033355. The
findings and opinions expressed in this article are those of the authors only and do not necessarily
reflect the views of the sponsors.
Data Availability Statement: Data are contained within the article.
Conflicts of Interest: Author Chenxi Lin was employed by the company Energy & Power, Jacobs
Engineering Group. The remaining authors declare that the research was conducted in the absence of
any commercial or financial relationships that could be construed as a potential conflict of interest.
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