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Simple Model for Understanding Harmonics
Propagation in Single-Phase Microgrids
Geir Kulia
Signal Analysis Lab,
Trondheim
geir.kulia@signalanalysislab.com
Lars M. Lundheim
Dept. of Electronic Systems
NTNU, Trondheim
lars.lundheim@ntnu.no
Marta Molinas
Dept. of Engineering Cybernetics
NTNU, Trondheim
marta.molinas@ntnu.no
Olav B. Fosso
Dept. of Electric Power Engineering
NTNU, Trondheim
olav.fosso@ntnu.no
Abstract—This paper presents an analytical method developed
to explain the mechanism of harmonic transfer between the ac
and dc sides of a single-phase inverter in a PV microgrid. The
model explains how the feed-forward of the current from the
ac side of the PV-inverter into the control system, causes even
harmonics on the dc voltage. It further shows how the controller’s
feedback of the dc bus voltage carrying even harmonics results in
odd harmonics on the ac side of the PV-inverter. This harmonic
propagation model is verified with a simulation of the PV
microgrid system. The results of this simulation study provides
consistency by verifying that odd harmonics on the ac voltage
causes even harmonics on the dc bus, and that even harmonics
on the dc bus again causes more odd harmonics on ac voltage.
Keywords—Power Spectrum, Harmonic propagation, Harmonic
Distortion, Single-phase microgrid, Time-frequency analysis.
I. MOTI VATI ON
Today, one in five people still lacks access to modern
electricity while 3 billion people rely on wood, coal, charcoal
or animal waste for cooking and heating. The United Nations
7th Sustainable Development Goal (SDG) is to ensure access
to affordable, reliable, sustainable and modern energy for all.
Electricity lays the foundation for most aspects of modern
life, being modern business, medicine, education, agriculture,
infrastructure, and communications. Stand-alone microgrids
have been outlined as an alternative for affordable electricity
to the 1.2 billion people lacking it today, as extending the main
grid will be too expensive in many rural areas.
However, these microgrids operate in a different way com-
pared to conventional power generation systems. Conventional
power generation systems rely on mechanical or thermal con-
version to generate electricity directly. These systems consist
of large rotating masses with considerable inertia that helps
them maintain a stable frequency. Modern power generation
systems (e.g. photovoltaic power, wind power) frequently rely
on solid-state conversion devices, which do not have the inher-
ent inertia of the conventional generation systems. Maintaining
a stable frequency in these modern power electronics systems
represents a new challenge for the provision of stable supply
of electricity. Therefore, understanding the mechanism be-
hind frequency related phenomena (e.g. harmonic propagation,
frequency variations) in these systems lacking inertia is a
prerequisite to a stable, reliable and affordable electricity using
microgrids.
The paper is organized around a case study of a photo-
voltaic plant where considerable distortion is observed, al-
though the load is linear. Arguments are given for how this
distortion can be explained by a feed-back mechanism through
the controller of the system.
II. SY ST EM DESCRIPTION
In this paper, a stand-alone PV microgrid is used as case-
study to investigate the mechanism of harmonic transfer in
systems lacking inertia. The PV microgrid at the campus of
the Royal University of Bhutan’s was the basis for this inves-
tigation. Measurements of voltage and currents were taken on
this microgrid to further analyze their harmonic content and
any frequency related phenomena.
A. Single phase microgrid description
A single-phase microgrid with a photovoltaic source repre-
sents a power generation system in an isolated community as
indicated in Figure 1. A boost controller is used to maintain a
stable dc voltage vd(t). The dc voltage is converted to ac by
a solid-state dc/ac inverter based on pulse-width modulation
(pwm) that is supervised by a control unit. The pwm volt-
age vpwm(t)is filtered using a low-pass filter to achieve a
smooth ac voltage waveform vac(t)to the load. The inverter’s
controller keeps the grid frequency stable on the ac voltage
vac(t)by using the ac current iac (t), the dc voltage vd(t), and
the reference angle frequency ωoas a reference. The load is
assumed linear. The output voltage should ideally be given by
vac(t) = Vac ·cos(ωot).(1)
Vac is the desired amplitude of vac(t)and is 230√2Volt
in Bhutan. The angle frequency is given by
ωo= 2πfo(2)
PV power source with
boost controller
Load
+ ×
Switch-mode
inverter and filter
+
vd(t)
Vr
vpi(t)
iac(t)
vac(t)
vcontrol(t)
PI Controller RC Controller
vin,rc(t)
+
+
-
-
cos(
ω
ot)
Linear subsystem H2
Linear subsystem H1
ve(t)
Fig. 1: Schematics of the microgrid at RUB College of Science and Technology.
fois the grid-frequency. The grid-frequency in Bhutan is
fo= 50 Hz.
The microgrid’s controller in this article is based on a
typical PV inverter controller [7], [8]. In the controller, the
dc voltage, vd(t)is compared with a predefined reference dc
voltage Vrand results in an error signal
ve(t) = Vr−vd(t).(3)
The controller has two main components. A proportional-
integral controller (PI) serves to maintain a stable output
amplitude of vac(t)while a resonant controller (RC controller)
ensures a stable grid frequency.
The output vpi(t)of the PI controller is multiplied with a
pure sinusoid and compared with the ac current iac(t)so that
vin,rc(t) = Rkiac (t)−vpi(t)·cos(ωot)(4)
Rkis a constant to scale iac(t).vin,rc (t)is used as the input
to the Resonant Controller, that is basically a narrow bandpass
filter centered around the grid frequency f0.
The output signal vcontrol(t)is used as a reference for the
dc/ac inverter so that
vac(t)∝vcontrol (t)(5)
III. MEASUREMENTS
Measurements of vac,m(t)and iac,m(t)were taken from
a real PV microgrid at the Royal University of Bhutan’s
College of Science and Technology. The letter mis added
in subscript to indicate that the waveform is measured. The
voltage waveform vac,m(t)of the stand-alone microgrid is
shown in Figure 2.
The power spectrum, shown in Figure 3 is calculated by
Pac(ω) = 1
N
N−1
X
n=0
vac(nTs)e−j ω·nTs
2
(6)
where ωis the angular frequency, Tsis the sampling period,
nis the discrete time variable, and Nis the total number of
samples [5, pp 960-1046].
20 40 60 80
Time [ms]
-1
-0.5
0
0.5
1
Voltage [pu]
vac(t)
Fig. 2: The measured ac voltage vac,m(t).
The power spectrum has a fundamental frequency ωo, with
odd harmonics of considerable amplitude. From the power
spectrum we find that the waveform has a total harmonic
distortion of THD = −13 dBc,which is substantially more
than what is to be found on the main Grid of Bhutan.
The distortions are also visible in time domain as seen
in Figure 2. The section IV proposes an explanation of these
distortions.
IV. SYS TE M ANALYSI S
Harmonic distortions on power grids are often attributed to
non-linear loads. This section deals with how such distortions
can occur even with a linear load, as is the case in the reported
measurements.
In previous works on the same system, it was shown how
oscillations on the dc side can explain oscillations on the ac
side [2], [3].
By grouping the subsystems in H1and H2as shown in
Figure 1, the system can be abstracted to two linear operations,
0 100 200 300 400 500
Frequency [Hz]
-80
-60
-40
-20
0
Power [dB]
Pac(ω)
Fig. 3: Power spectrum of vac,m(t). The spectrum is calculated
using Equation 6 where N= 178 and Fs= 8.9KHz.
H1H2
×
cos(
ω
ot)
[⋅]²
Inverter
ac side
dc side
vpi(t)vin,rc(t)
vac(t)vd(t)
Fig. 4: Abstraction of the microgrid model.
one multiplication, and the inverter. An abstraction of Figure 1
where the assumption that vac(t)∝vcontrol (t)has been made,
is shown in Figure 4. The subsystems H1and H2are linear and
time invariant. This means that no new frequency components
will be generated by these. From subsection A it is seen that
influence from the ac voltage vac(t)through the inverter on
vd(t)is given by
vd(t) = a·v2
ac(t) + Vd(7)
where ais a constant. Note that this influence from the ac
side on the dc side is in opposite direction to the energy flow.
Appendix A states that the oscillatory part of the dc voltage
vd(t)is given by
˜vd(t)∝v2
ac(t)(8)
The feedback effect from the dc bus vd(t)to the ac voltage
vac(t)through the controller is given by
vac(t) = (vd(t)∗h1(t)) ·cos(ωot)∗h2(t)(9)
where h1(t)and h2(t)are the impulse reponses of H1and H2
respectivly.
First it will be illustrated how one harmonic affects the dc
bus. Assuming the ac voltage vac(t)is a pure sine wave on
the form given in Equation 1, the result on the dc bus will be
v0
d(t) = aV 2
ac
2cos(2ωot)+(aV 2
ac
2+Vd)(10)
The harmonic on the dc bus influences the ac bus through the
controller. The effect is found by inserting Equation 10 into
Equation 9, so that
v0
ac(t) = V0
ac,3cos(3ωot) + V0
ac cos(ωt)(11)
This shows that an ideal ac voltage is impossible as it will
cause distortions itself. The distortions will cascade back to the
dc side, and feed forward to the ac side again, causing more
harmonics. All odd harmonics on the ac side will generate
even harmonics on the dc side. All even harmonics on the dc
side will produce odd harmonics on the ac side. To generalize,
it can be said that odd harmonics of ωowill dominate the ac
side, and even harmonics of ωowill dominate the dc side.
Appendix B shows that if the ac voltage vac(t)only contain
odd harmonics, then the dc voltage will be
vd(t) = Vd+
∞
X
i=1
˜
Vd,i cos(2i·ωot−φd,i).(12)
This means that if odd harmonics are present in the ac voltage
vac(t), even harmonics will appear in the dc voltage vd(t).
In general, if a sum of even harmonics is multiplied with its
fundamental harmonics then the following is obtained:
cos(ωot)·
∞
X
i=1
V0
i·cos(2i·ωot) =
∞
X
i=1
V0
i
2·cos (2i+ 1)ωot
+
∞
X
i=1
V0
i
2·cos (2i−1)ωot
=
∞
X
i=1
Vi·cos (2i+ 1)ωot.
(13)
This means that the effect of the feedback from the dc bus
vd(t)through the controller and to the ac side results in an ac
voltage
vac(t) = (vd(t)∗h1(t)) ·cos(ωot)∗h2(t)
=
∞
X
i=1
Vac,i·cos (2i+ 1)ωot−φac,i.(14)
Equation 12 and 14 shows that odd harmonics on the ac voltage
vac(t)causes even harmonics on the dc bus vd(t), and that even
harmonics on vd(t)cause more odd harmonics on vac(t).
A Simulink model of the microgrid at the Royal University
of Bhutan’s College of Science and Technology was used
to examine the statement above further. A screenshot of the
simulation is shown in Figure 7. The power spectra of the
simulated ac voltage vac,s(t)and the dc voltage vd,s(t)using
the Simulink simulation is shown in Figure 5. sis added
in subscript to indicate that the variables are simulated. The
power spectrum of the dc bus voltage vd,s(t)only contains
0 100 200 300 400 500
Frequency [Hz]
-80
-60
-40
-20
0
Power [dB]
Pd,s(t)
Pac,s(t)
Fig. 5: Power spectrum of simulated dc bus vd,s(t)and ac
voltage vac,s(t). The spectrum is calculated using Equation 6
where N= 20 K and Fs= 1 MHz.
even harmonics. The ac side mainly contains odd harmonics.
A numerical model of the system described in this paper and
adjusted for the pulse width modulation effects has previously
been used to explain this [1].
V. CONCLUDING REMARKS
Since the analytical and simulated investigation shows
a dominance of odd harmonics on the ac side and even
harmonics on the dc side, it can be concluded, without loss of
generality, that the feedback loop is the primary cause of the
propagation of the distortions observed in the ac voltage vac(t).
The feedback loop originates on the ac side and cascades
through the inverter to the dc side, and back to the ac side
via the control unit. The degree of the distortion will vary
depending on the tuning and configuration of the controller
and on the size of the capacitor in the inverter. The effect of
these factors was a THD of -13 dBc observed in the setup
at the Royal University of Bhutan’s College of Science and
Technology lab and in the simulations.
ACKNOWLEDGMENT
The system analysis would not have been preformed with-
out the measurements at RUB. It is therefore in place to thank
Tshewang Lhendup, and Cheku Dorji from CST, my travel
partner and coworker H˚
akon Duus and all the staff and students
at CST for their support and assistance in collecting data from
the microgrids. The field trip to Bhutan was partially supported
by Ren-Peace, IUG NTNU, and Department of Electronics and
Telecommunications at NTNU.
REFERENCES
[1] Geir Kulia. Investigation of distortions in microgrids. Norwegian
University of Science and Technology, Trondheim, 2016.
[2] Geir Kulia, Marta Molinas, and Lars Lundheim. Tool for detecting
waveform distortions in inverter-based microgrids: a validation study. In
2016 IEEE Global Humanitarian Technology Conference (GHTC 2016),
Seattle, USA, October 2016.
Rs
vd(t)Vs
id
Fig. 6: Th´
evenin equivalent of dc side of the microgrid. This
circuit is equivalent to the “PV power source with boost
controller” in Figure 1.
[3] Geir Kulia, Marta Molinas, Lars Lundheim, and Bjørn B. Larsen.
Towards a real-time measurement platform for microgrids in isolated
communities. Procedia Engineering, 159C:94–103, 2016.
[4] Ned Mohan, Tore M. Undeland, and William P. Robbins. Power
electronics - Converters, Applications, and Design. Wiley, New York, 3
edition, 2003.
[5] John G. Proakis and Dimitris G. Manolakis. Digital Signal Processing:
Principles, Algorithms, and Applications. Pearson, New Jersey, 4 edition,
2013.
[6] Karl Rottmann. Matematisk Formelsamling (Matematische Formelsamm-
lung). Spektrum forlag, Oslo, Norway, 2006.
[7] Remus Teodorescu, Marco Liserre, and Pedro Rodriguez. Grid Con-
verters for Photovoltaic and Wind Power Systems. Wiley, Sussex, UK,
2011.
[8] Y. Yang, F. Blaabjerg, and H. Wang. Constant power generation of
photovoltaic systems considering the distributed grid capacity. In 2014
IEEE Applied Power Electronics Conference and Exposition - APEC
2014, pages 379–385, March 2014.
APPENDIX
A. The ac voltage’s effect on the dc bus through the inverter
This appendix shows how to obtain a general expression of
the ac voltage vac(t)’s impact on the dc bus voltage vd(t). The
calculations are based on the system presented by Mohan et. at.
[4, p. 200]. The system is assumed to be lossless. Considering
a resistive load, the ac power is given by
Pac(t) = v2
ac(t)
Rload
(15)
and on the dc side we have that
Pd(t) = vd(t)·id(t).(16)
The dc current can be written as
id(t) = vd(t)−Vs
Rs
(17)
where Vsis the supply voltage and Rsis the line impedance
as shown in Figure 6. By assuming Vd˜vd(t)and Vs≈Vd
we obtain
Pd(t) = Vd·vd(t) + V2
d
Rs
(18)
As the system is lossless, the ac power is equal to the dc power:
Pd(t) = Pac(t)(19)
Therefore Vd·vd(t)−V2
d
Rs
=v2
ac(t)
Rload
(20)
The dc voltage can then be written as
vd(t) = Rs
Rload
v2
ac(t)
Vd
+Vd(21)
This can be rewritten as
vd(t) = Va·v2
ac(t) + Vd(22)
where
Va=Rs
Rload ·Vd
(23)
B. Square of sum containing odd harmonics
This appendix shows how the square of a sum of odd
harmonics result in even harmonics. The square of a sum [6]
is given by
∞
X
i=1
vi(t)2
=
∞
X
i=1
v2
i(t)+2
∞
X
i=1,k6=i
vi(t)·vk(t)(24)
It follows that the product of two odd harmonics is given by
cos((2i+ 1)ωot)·cos((2k+ 1)ωot) = 1
2cos 2(i−k)ωot
+1
2cos 2(i+k+ 2)ωot
(25)
By assuming vi(t)to be an odd harmonic of the form
vi(t) = Vicos (2i+ 1)ωot(26)
and inserting (25) into (24) we get that
∞
X
i=1
cos (2n+ 1)ωot2
=1
2
∞
X
i=1
Vi+1
2
∞
X
i=1
Vicos 4(n+ 1)ωot
+
∞
X
i=1,k6=i
ViVkcos 2(i−k)ωot
+
∞
X
i=1,k6=i
ViVkcos 2(i+k+ 2)ωot.
(27)
Equation (28) covers all even harmonics, and then it can be
rewritten as
∞
X
i=1
cos (2n+ 1)ωot2
=1
2
∞
X
i=1
Vi+
∞
X
i=1
Ve,i cos 2i·ωot
(28)
Fig. 7: Simulink schematics of the main blocks of the microgrid simulation.
