abtechnosolutions

A Harmonic Compensation Approach for
Interlinking Voltage Source Converters in Hybrid
AC-DC Microgrids with Low Switching Frequency
Hao Tian, Student Member, IEEE, Xiaohan Wen, and Yun Wei Li, Senior Member, IEEE
Abstract—In recent years, the hybrid AC-DC microgrid has
been well accepted as it combines the advantages of both AC
and DC systems. As the microgrid contains both DC sub-grids
and AC sub-grids, interlinking DC-AC converters are essential.
Meanwhile, considering the nonlinear AC loads may deteriorate
the voltage quality of the AC bus, embedding an ancillary
harmonic compensation function to the interlinking converters is
promising. However, the conventional harmonic control methods
used for active power filters (APFs) may not be suitable for
the interlinking converters due to the main purpose of it is
to exchange real and reactive power between the DC and AC
sub-grids. The switching frequency is preferred to be lower
than the APFs when the capacity of the microgrid is large. At
low switching frequency, harmonic compensation performance
or even the system stability may be affected. In this paper, a
harmonic compensation approach suitable for hybrid AC-DC
interlinking converters at low switching frequency is proposed.
Through feeding the PWM reference signal with the harmonic
compensation component directly to avoid the multi-loop control
path of the fundamental component, the proposed method can
achieve the effective harmonics compensation without being limited
by the closed-loop control bandwidth. The proposed method,
modeling approaches, stability analysis, as well as detailed virtual
impedance design are presented. Experimental verification is also
provided.
Index Terms—Harmonic compensation, hybrid AC-DC
microgrid, virtual impedance.
I. INTRODUCTION
AS renewable DC sources, such as PV arrays and fuel
cells, DC loads and energy storage components are increasingly
being connected to the distribution grid, the concept
of hybrid microgrid has been proposed [1]. Different from the
traditional microgrid, the hybrid AC-DC microgrids have both
AC buses and DC buses, avoiding multiple energy conversions
between AC sources/loads and DC sources/loads [2]. This
structure can obviously increase the reliability and efficiency
Manuscript received October 25, 2016; revised February 19, 2017; accepted
April 1, 2017. Date of publication March 30, 2018; date of current version
December 20, 2017.
H. Tian (corresponding author, e-mail: tianhao1988@gmail.com) and Y.
W. Li are with the Department of Electrical and Computer Engineering,
University of Alberta, Edmonton, AB, T6G 2V4, Canada.
X. Wen was with the Department of Electrical and Computer Engineering,
University of Alberta, Edmonton, AB, T6G 2V4, Canada. He is now with
China Mobile Group Liaoning Co., Ltd., Shenyang, China.
DOI: 10.17775/CSEEJPES.2016.01550
of the system, and thus is considered as a promising future
for the distribution grid. A typical hybrid AC-DC microgrid is
shown in Fig. 1. As shown in Fig. 1, AC components and DC
components are separated and interlinked by an interlinking
converter (ILC). The ILC is the bridge to realize energy
exchange between the AC bus and DC bus. Thus, power
management is one of the key problems to operate the hybrid
system. Several autonomous control methods based on droop
control have been proposed [3], [4].
Grid
G

PV panel

~

=

=
DC-bus AC-bus
Interlinking
converter
(ILC)
DC/DC converter
Storage
components
Zg
DC/DC
converter
DC
loads
AC
generator
AC
loads
Fig. 1. A typical hybrid AC-DC microgrid.
It is worth noting that the rated power of ILCs should
be large enough to realize power balance between the DC
sub-grid and the AC sub-grid due to the complexity of the
operation status. As a result, although the primary purpose
of the ILC is to transfer real power to realize the power
management between the DC bus and AC bus, sufficient
apparent power rating is available for some other functions
in most of the operational status.
Similar ideas have been applied to the interfacing inverter of
these traditional microgrids, including reactive power compensation
[5], voltage unbalance compensation [6]–[8], load current
harmonics compensation [9], [10], flicker mitigation [11],
generation reserve [12], etc. Among these ancillary functions,
harmonics compensation is predicted to play an interesting and
important role as the number of nonlinear loads is growing
rapidly in today’s power distribution systems [9], which also
applies to hybrid AC-DC microgrids. When the AC loads are
nonlinear, the harmonic voltage may appear on the AC bus,
influencing the grid and all the apparatus that are connected
to the AC bus [13]. Meanwhile, it is costly to install an APF
to do the harmonic compensation. Therefore, to embed the
harmonic compensation function to the ILC is also a topic
2096-0042 © 2016 CSEE
40 CSEE JOURNAL OF POWER AND ENERGY SYSTEMS, VOL. 4, NO. 1, MARCH 2018
worth researching. For example, Reference [14] uses the
voltage control method (VCM) by adding proper harmonic
signals into its voltage control reference, improving the AC
bus voltage in a hybrid microgrid. Moreover, the current
control method (CCM) can also be applied to mitigate harmonics
in grid-connected operations, in which the converter
are controlled as a resistive-APF (R-APF) that acts as a
small resistance at the harmonic frequencies [15]–[18]. These
methods have to expand the feedback control bandwidth.
However, since the main role of the ILC is to control the active
and reactive power injection, a low switching frequency is
preferred to reduce switching losses (e.g. a few kHz) especially
in relative large capacity hybrid AC-DC microgrids. However,
the low switching frequency will result in larger filter size
and a longer delay in the control loop, which subsequently
limits the voltage or current control loop bandwidth. This low
current or voltage control bandwidth will affect the harmonics
compensation, especially when the compensation scheme is
realized by modifying the current or voltage reference as in the
traditional CCM and VCM based methods. To overcome this
drawback, a harmonic compensation approach under a gridtied
mode is proposed in this paper, which adjusts the PWM
reference signal directly instead of modifying the current
reference and therefore is not limited to the current control
bandwidth. As a result, high bandwidth for the harmonic
compensation can be achieved. This study also reveals that the
performance of the traditional virtual resistance based compensation
(R-APF) is limited due to stability when the converter
switching frequency is low, and therefore the virtual resistiveinductive
impedance based compensation is recommended in
the proposed scheme.
In the following sections, the proposed virtual impedance
based harmonic compensation approach is developed and
presented, after which the stability analysis are also presented.
Detailed design of the virtual impedance for harmonics compensation
is illustrated in Section IV. Finally, the experimental
results are provided in Section V.
II. PROPOSED HARMONIC COMPENSATION METHOD AT
LOW SWITCHING FREQUENCY
The block diagram of the proposed harmonic compensation
method is shown in Fig. 2. The most widely used CCM based
method is adopted to control the fundamental current, whose
references are given by the active and reactive control module.
The power command P∗,Q∗ can be obtained from the outer
control loop, such as the approaches presented in [1], [3],
[4]. In the multi-loop current controller, the PR controller is
used as the outer current loop and the inverter output current
tracking loop with a proportional controller is added as the
inner loop control. The extracted AC bus harmonic voltage
VAC Bus h is used to control the PWM reference signal directly
instead of modifying the harmonic component of the reference
current as the conventional CCM based harmonic control
method (R-APF method) does. To make comparisons, the RAPF
method, shown in the shadowed area in Fig. 2, extracts
the AC bus harmonic voltage and scales it by 1/Rh, where Rh
is a virtual resistor, to generate harmonic current references.
Then the harmonic current reference I∗
ILC h, along with the
fundamental current reference I∗
ILC f, are regulated by the
fundamental current control loop. Therefore, the bandwidth of
the control loop will definitely influence the harmonic control
performance. In comparison to this conventional approach,
the proposed harmonic compensation branch can be realized
with a much higher bandwidth by avoiding the multi-loop
control path, meaning accurate compensation of relatively high
order harmonics. Moreover, the PR controller in the multi-loop
current control scheme only needs to control the fundamental
component which also reduces the current control bandwidth
requirement.
Grid
ILC
AC
loads
++
f
PWM modulator
C
DC bus LCL filter
L1 L2
VACBus
AC bus AC
subgrid
Zg
VACBus_h
Voltage
extraction
Proposed
harmonic
control
VACBus
P*
Q*
Real and
reactive power
Conventional CCM control
based harmonic control
Iinv
IILC
Vinv_h IILC
Multi-loop
current control
DC
subgrid
1/Rh
IILC_h ++ IILC_ f
Fig. 2. Control strategy of the proposed harmonic compensation method.
To extract the AC bus harmonic voltage accurately with
selective harmonic compensation, the notch filter is used,
which is designed as:
SHD(s) =
(h · 120π/Qp)s
s2 + (h · 120π/Qp)s + (h · 120π)2 (1)
where h is the harmonic order and QP is the quality factor.
Then the inverter output harmonic voltage Vinv h is controlled
by the extracted AC bus harmonic voltage VAC Bus h as
Vinv h = −G · VAC Bus h. (2)
As the ILC output voltage harmonics are not directly
regulated, the Thevenin equivalent circuit of a hybrid AC-DC
microgrid system with an LCL output filter can be expressed
as Fig. 3 for harmonic analysis. According to Thevenin’s
+
−
+
−
ILC Grid
+
−
IILC_h
VAC Bus_h
ZILC_h,eq
Equivalent harmonic impedance
at installation point
AC loads
IILC
Zeq
Veq
ZLoad ILoad
VAC Bus
Zg
Vg
Fig. 3. Thevenin equivalent circuit of a hybrid AC-DC microgrid system
with the proposed compensation method.
TIAN et al.: A HARMONIC COMPENSATION APPROACH FOR INTERLINKING VOLTAGE SOURCE CONVERTERS IN HYBRID AC-DC MICROGRIDS WITH LOW SWITCHING FREQUENCY 41
theorem, the ILC is described as a voltage Veq with the output
series impedance Zeq.
Veq = Vinv · ZC
ZL1 + ZC
(3)
Zeq =
ZL1ZL2 + ZL1ZC + ZL2ZC
ZL1 + ZC
(4)
where Vinv is the inverter output voltage, ZL1, ZL2 and ZC are
the impedance of the LCL filter. The subscript L1 indicates the
inverter side inductor while L2 indicates the grid side inductor.
ZC is the impedance of the capacitor in the LCL filter. The
AC nonlinear load is represented as a passive load that is only
related to the fundamental power consumed by the load and
harmonic current source, and the grid can be regarded as a
voltage source Vg and grid impedance Zg. Then the equivalent
impedance of the ILC at the harmonic frequencies can be
obtained as
IILC h =
Veq h − VAC Bus h
Zeq
= −(G · ZC/(ZL1 + ZC) + 1) · VAC Bus h
Zeq
(5)
ZILC h, eq = −VAC Bus h

IILC h

1
G · ZC/(ZL1 + ZC) + 1

· Zeq

1
K
· Zeq (6)
It can be seen that by controlling the inverter output
harmonic voltage with a gain G, the equivalent harmonic
impedance of the ILC will be scaled down by a factor K,
which is defined as G · ZC/(ZL1 + ZC) + 1. When the factor
K is set as 1, the system will be a standard CCM controlled
ILC without any harmonic control. When K is greater than
1, the AC bus harmonic voltage will be reduced because the
ILC absorbs more harmonics due to the reduced equivalent
impedance. This will benefit all the sources and loads that are
connected to the AC bus, including the grid. On the contrary,
when K is greater than 0 and less than 1, the ILC works in
the harmonic rejection mode. The ILC equivalent impedance
increases at harmonic frequencies to reject harmonics. Thus
if the harmonic compensation operation is preferred, the gain
G should be designed to reduce the equivalent impedance,
i.e., the virtual impedance. In addition, it is worth noting that
the gain G will change the output equivalent impedance at
the specific harmonic order, but the complex gain G itself is
not a combination of virtual resistors and inductors. Instead,
the whole output impedance of the interfacing inverter is
considered as the virtual impedance.
In the α-β frame, the harmonic regulation gain G can
be transferred as a complex number (Gα + jGβ), and the
extracted AC bus harmonic voltage also can be regarded as two
orthogonal components in α-β frame. In order to obtain the
harmonic compensation signals, the corresponding G and extracted
AC bus harmonic voltages are multiplied together, then
the harmonic compensation signals for different harmonics are
combined to get the final harmonic compensation signal.
However, the harmonic impedances introduced by the multiloop
controller are not taken into account in (5) and (6). If
the gain of the multi-loop controller is not small enough at
harmonic frequencies, it may have an impact on harmonic
compensation performance. So some analysis about the harmonic
impedance should be done, which will be illustrated in
the following sections.
III. ANALYSIS OF PROPOSED CONTROL STRATEGY
A. System Models
According to the LCL filter shown in Fig. 2, the ILC current
IILC and the inverter output current Iinv can be expressed as
IILC = (Vinv − H1 · VACBus) · H2 (7)
Iinv = H3 · IILC + H4 · VACBus (8)
where
H1 =
s2CL1 + sRdC + 1
sRdC + 1
H2 =
sRdC + 1
s3CL1L2 + s2RdC(L1 + L2) + s(L1 + L2)
H3 =
s2CL2 + sRdC + 1
sRdC + 1
H4 =
sC
sRdC + 1
.
L1, L2, C are the parameters of the LCL filter, Rd is the
damping resistor which is series connected with the capacitor
C in the LCL filter.
The transfer function block of the ILC using the proposed
harmonic compensation method is shown in Fig. 4.
In addition, since the proposed harmonic control method
is applied at low frequencies, the switching delay produced
+_ PR +_ +_ Delay +_ +_ +_
+_
++
G
IILC
I* m ILC Iinv KC Vinv
2/VDC VDC/2
Vinv V1
1/L1s
Iinv
VC IILC
1/Cs+Rd
VC
1/L2s
IILC
Iload
Igrid
Lgs
Vgrid
VACBus
Harmonic
extraction
Iinv
Fig. 4. Control scheme for the ILC system using the proposed harmonic compensation method.
42 CSEE JOURNAL OF POWER AND ENERGY SYSTEMS, VOL. 4, NO. 1, MARCH 2018
during modulation in the digital control should be taken
into account in practice. The delay mechanism is shown
in Fig. 5 in which the delay consists of two components:
computation delay and PWM delay [19]. Considering the fact
that the switching frequency of the proposed method is low,
multisampling is applied to the proposed method to reduce
the computation delay TC, as shown in Fig. 5(b). As a result,
when the sampling frequency fs is much higher than PWM
frequency fPWM, the computation delay TC is negligibly small
(fs = 10fPWM in this study).
Sampling (k-1) Sampling (k) Sampling (k+1)
TC
TC
Computation delay
TPWM
TPWM
PWM period
PWM (k-1) PWM (k) PWM (k+1)
(a)
fs=fpwm
fs>fpwm
PWM (k-1) PWM (k) PWM (k+1)
(b)
…… ……
Fig. 5. Delay mechanism. (a) The sampling frequency is same as PWM
frequency. (b) Sampling frequency is much higher than PWM frequency.
On the other hand, since the PWM reference is needed to
hold on and compared to the triangular carrier to generate the
modulation signal after the PWM reference is updated. This
behavior introduces PWM delay and it can be expressed as a
zero-order hold (ZOH) [19], which is defined as
HZOH(s) =
1 − e−sTPWM
s
(9)
The equation HZOH(s) is approximately equal to
TPWMe−0.5sTPWM when s = jω applied, which introduces
the PWM delay that is equal to 0.5TPWM. Therefore, when
multisampling is applied, the delay of proposed method is
approximately equal to one half the PWM period. In this
study, the second-order Pad´e approximation, which is a
rational function, is used to approximate the delay in the
calculation. Therefore, according to (7), (8) and Fig. 4, the
equivalent impedance of the ILC at the harmonic frequencies
can be obtained as
ZILC h, eq = −VAC Bus h

IILC h

1 + KCH2D · PR + KCH2H3D
KCH2H4D + H1H2 + H2G · D
(10)
where PR is the transfer function of the PR controller,
KC is the proportional gain of the inner loop control, G is
the harmonic compensation gain, H1 to H4 are the transfer
functions as described previously. D is the transfer function
of the switching delay. In addition, as mentioned previously,
the equivalent harmonic impedance ZILC h, eq can be controlled
to be a virtual resistance Rh by setting G as a proper complex
number to provide better damping to the system. For the virtual
resistance Rh, the corresponding complex G can be obtained
as
G =
1 − H1H2Rh
H2RhD
(11)
To further evaluate the harmonic compensation performance
under different parameter conditions, the attenuation between
the grid harmonic current Igrid h and nonlinear load harmonic
current Iload h (|Igrid h/Iload h|) is used to show how much
harmonic current flows from the nonlinear load into the grid.
In the simplified system model, the relationship between Igrid h
and Iload h can be obtained as
Igrid h
Iload h
= (12)
−H2KCD(PR + H3) − 1
1 + H2sLg(G · D + H1) + H2KCD(PR + H3 + sLgH4)
B. System Stability Analysis
As discussed in the above, according to (6), it is shown that
the equivalent harmonic impedance of ILC is scaled down by
the factor K which contains the harmonic compensation gain
G. The larger the modulus of G with a proper phase angle, the
smaller the corresponding equivalent harmonic impedance. As
a result, more harmonics introduced by the nonlinear load are
absorbed by the ILC, leading to better AC bus voltage quality.
However, the larger modulus of G will reduce the stability
margin, and therefore it is necessary to investigate how the
phase angle of G affects the stability.
With the control scheme for the hybrid system using the proposed
harmonic compensation method, the close-loop transfer
function of the system can be obtained as
IILC
I∗
ILC
= (13)
H2PR · KCD
1 + H2Zg(G · SHD · D + H1) + H2KCD(PR + H3 + H4Zg)
where Zg is the transfer function of the grid impedance which
is defined as Lgs, and SHD is the harmonic extraction block.
Based on (13), Fig. 6 shows the pole positions of the control
scheme when the modulus of the 5th harmonic compensation
gain G5 is set as a constant (|G5| = 5) with different phase
angles. The pole positions of the control scheme are calculated
every degree in the theoretical analysis, but they are marked
every 45 degrees in Fig. 6 in order to facilitate the observation.
As shown in Fig. 6, with the phase angle of G5 increasing, the
pair of poles closest to the right half plane move anticlockwise
and become an oval finally. According to the obtained pole
positions, when the phase angle of G5 is 0 degrees, this pair
of poles is farthest away from the imaginary axis in the left
half plane, which means the system is the most stable at this
time. Table I shows the modulus range of G5 which makes the
system stable under different phase angles. It is can be seen
that with the phase angle of G5 increasing from 0, the system
stability deteriorates quickly. The modulus range when the
phase angle is 0 degrees is much greater than the other phase
TIAN et al.: A HARMONIC COMPENSATION APPROACH FOR INTERLINKING VOLTAGE SOURCE CONVERTERS IN HYBRID AC-DC MICROGRIDS WITH LOW SWITCHING FREQUENCY 43
2,000
6,000
4,000
2,000
−2,000
−4,000
−6,000
−8,000
0
−1,500 −1,000 −500 0 500
1,960
1,920
1,880
1,840
1,800
−100 −80 −60 −40 −20 0 20 40 60
−2,000
−1,960
−1,920
−1,880
−1,840
−1,800
−100 −80 −60 −40 −20 0 20 40 60
8,000
Pole-Zero Map
Imaginary Axis (rad/s)
Real Axis (rad/s)
Imaginary Axis (rad/s) Imaginary Axis (rad/s)
Real Axis (rad/s)
Real Axis (rad/s)
Pole-Zero Map
315 deg 270 deg
225 deg
180 deg
135 deg
90 deg
45 deg
0 deg
Pole-Zero Map
270 deg 225 deg
180 deg
135 deg
90 deg
45 deg
0 deg
315 deg
Fig. 6. Pole positions when compensating 5th harmonic under different phase angles (|G5| = 5).
angles. In conclusion, the system can be well stabilized when
G5 is a real number without changing harmonic compensation
signal phase angle.
TABLE I
MODULUS RANGE OF G5 UNDER DIFFERENT PHASE ANGLES
Degree 0 45 90 135 180 225 270 315
Modulus Range 68 35 3 1 1 1 3 35
Similarly, the pole positions of the control scheme when
compensating 7th and 11th harmonics under the different G
phase angles are shown in Fig. 7 and Fig. 8 respectively.
The modulus ranges of G7 and G11 under different phase
angles are shown in Table II and Table III respectively.
Likewise, the most stable phase angle for G7 is also 0 degree
based on Fig. 7 and the corresponding calculation results.
However, the most stable phase angle for G11 is 180 degrees.
6,000
4,000
2,000
0
−4,000
−6,000
−1,500 −1,000 −500 0 500
−150 −100 −50 0 50 100
−2,800
−2,700
−2,600
−2,500
−150 −100 −50 0 50 100
2,800
2,700
2,600
2,500
−2,000
Pole-Zero Map
Imaginary Axis (rad/s)
Imaginary Axis (rad/s)
Imaginary Axis (rad/s)
Real Axis (rad/s)
Real Axis (rad/s)
Real Axis (rad/s)
Pole-Zero Map
Pole-Zero Map
0 deg 315 deg
270 deg
225 deg
180 deg
135 deg
90 deg
45 deg
225 deg
180 deg
135 deg
45 deg 90 deg
0 deg
315 deg
270 deg
Fig. 7. Pole positions when compensating 7th harmonic under different phase angles (|G7| = 5).
44 CSEE JOURNAL OF POWER AND ENERGY SYSTEMS, VOL. 4, NO. 1, MARCH 2018
6,000
4,000
2,000
0
−2,000
−4,000
−6,000
−1,500 −1,000 −500 0 500
4,500
4,300
4,100
3,900
−500 −400 −300 −200 −100 0 100 200
−500 −400 −300 −200 −100 0 100 200
−4,500
−4,300
−4,100
−3,900
Pole-Zero Map
Imaginary Axis (rad/s)
Imaginary Axis (rad/s) Imaginary Axis (rad/s)
Real Axis (rad/s)
Real Axis (rad/s)
Real Axis (rad/s)
Pole-Zero Map
45 deg 0 deg
315 deg
270 deg
225 deg
180 deg
135 deg
90 deg
Pole-Zero Map
135 deg
180 deg
225 deg
270 deg
315 deg 0 deg
45 deg
90 deg
Fig. 8. Pole positions when compensating 11th harmonic under different phase angles (|G11| = 5).
TABLE II
MODULUS RANGE OF G7 UNDER DIFFERENT PHASE ANGLES
Degree 0 45 90 135 180 225 270 315
Modulus Range 31 5 1 1 2 1 1 5
TABLE III
MODULUS RANGE OF G11 UNDER DIFFERENT PHASE ANGLES
Degree 0 45 90 135 180 225 270 315
Modulus Range 1 1 1 2 17 2 1 1
It is because that when the phase angle of G11 is set as
0 degree, the 11th harmonic equivalent impedance of ILC
is calculated to be an impedance which contains negative
resistance according to (12). The negative resistance tends
to make the system unstable. 0 and 180-degree phase angle
both make the imaginary part of G be 0. Therefore, a real
number G can always be applied to achieve a stable system
even though it may not be the best for stability. Between the
0 degree and 180 degrees, the most stable phase angle can
be determined by (12) by checking if the harmonic equivalent
impedance contains negative resistance. In addition, the sign
of G can also be determined easily by (6). Since the harmonic
equivalent impedance is scaled by a factor K, which equals
[G · ZC/(ZL1 + ZC) + 1], G and [ZC/(ZL1 + ZC)] need to be
both positive/negative.
IV. VIRTUAL IMPEDANCE DESIGN
A. Virtual Impedance Using Real Number G
Fig. 9 shows the pole positions of the control scheme when
real number G5 is increased from 0 to 70. The G5 maximum
modulus that makes the system stable is 68. According to (13),
the harmonic compensation performances when the modulus
of real number G is set as its maximum value are obtained and
shown in Table IV. It is can be observed that the compensation
performance is very effective because the good system stability
allows the modulus of G to be set as a relatively large value.
lmaginary Axis (rad/s) lmaginary Axis (rad/s)
Pole-Zero Map
Pole-Zero Map
6,000
4,000
2,000
−2,000
−4,000
−4,000 −3,000 −2,000 −1,000
−1,000
−4,800
−4,600
−4,400
−4,200
−4,000
−3,800
−3,400
−3,600
−800 −600 −400 −200 0 200
0 1,000
−6,000
0
Real Axis (rad/s)
G5=68
Fig. 9. Pole positions when real number G5 is increased from 0 to 70.
TIAN et al.: A HARMONIC COMPENSATION APPROACH FOR INTERLINKING VOLTAGE SOURCE CONVERTERS IN HYBRID AC-DC MICROGRIDS WITH LOW SWITCHING FREQUENCY 45
TABLE IV
HARMONIC COMPENSATION PERFORMANCE WHEN THE MODULUS OF
REAL NUMBER G IS SET AS THE MAXIMUM VALUE
Item 5th 7th 11th
G maximum
modulus
68
(G5 = 68∠0)
31
(G7 = 31∠0)
17
(G11 = 17∠180)
Attenuation
(|Igrid h/Iload h|)
0.010 0.018 0.019
B. Virtual Resistance
The feasibility of virtual resistance is discussed in this
section. To begin with, the ranges of virtual resistance Rh
that ensures the system stability are provided. As we know,
with the decrease of virtual resistance, the amplitude of the
corresponding G increases and the system stability gets worse.
Fig. 10 shows the pole positions of the control scheme when
the virtual resistance at the 5th harmonic frequency R5 is
decreased from 4 Ω to 1 Ω. The minimum R5 that makes
the system stable is 2.3 Ω. Then the corresponding harmonic
compensation performances when the virtual resistance R5,
R7 and R11 are controlled to be their minimum values are
shown in Table V. Compared with Table IV, the harmonic
compensation performances using virtual resistance are much
worse than the ones using real number G at the 5th and 7th
harmonics. However, the phase angle corresponding to virtual
resistance is determined by the grid, the ILC, switching delay
and controller together, which is hard to control through the
virtual impedance design. In conclusion, virtual resistance is
not flexible for the proposed harmonic compensation method
at a relatively low switching frequency.
lmaginary Axis(rad/s) lmaginary Axis(rad/s)
Pole-Zero Map
4,000
3,000
2,000
1,000
−1,000
−2,000
−3,000
−4,000
−1,000
−1,200
−1,400
−1,600
−1,800
−2,000
−2,200
−2,400
−2,600
−2,800
−3,000
−20 −15 −10 −5 0 5 10 15 20 25 30
−100 −50 0 50
0
Pole-Zero Map
Real Axis (rad/s)
Real Axis (rad/s)
R5=2.2 Ω
Fig. 10. Pole positions when virtual resistance R5 is decreased from 4 Ω
to 1 Ω.
TABLE V
HARMONIC COMPENSATION PERFORMANCE WHEN THE VIRTUAL
RESISTANCE AT HARMONIC FREQUENCY IS SET AS ITS MINIMUM VALUE
Item 5th 7th 11th
Minimum Rh
2.3 Ω
(G5 =
2.27∠101.2)
4.4 Ω
(G7 =
1.27∠143.6)
0.5 Ω
(G11 =
13.56∠177.2)
Attenuation
(|Igrid h/Iload h|)
0.237 0.317 0.024
C. Virtual Impedance Using Complex Number G
When the modulus of G is fixed, the G phase angle that
provides optimal harmonic compensation performance can be
obtained. Therefore, the feasibility of virtual impedance using
the complex number G which provides optimal harmonic
compensation performance when its modulus is constant will
be discussed in this section.
Fig. 11 shows the steady-state compensation performance
for the 5th harmonic when |G5| = 5, 7, and 15. The X-axis is
the phase angle of G5 and the Y -axis is the attenuation gain
|Igrid h/Iload h| which reflects the 5th harmonic compensation
performance. The three lines represent |G5| = 5, 7, and
11 respectively. Based on Fig. 11 and the corresponding
calculation results, when the modulus of G5 is set as 5, the
phase angle that provides the minimum attenuation gain is
27 degrees. As the modulus of G5 increased, the phase angle
that provides the minimum attenuation gain changes little as
shown in Fig. 11. As a result, for the different fixed modulus
of G5, the phase angle which achieves the optimal harmonic
compensation performance can be approximated as a constant
value. Similar conclusions can also be made for 7th and 11th
harmonics compensation gains.
Steady-state Compensation Performance
0.24
0.16
0.2
0.12
0.08
0.04
60 120 180 240
Phase of G5
300 360
Attenuation Gain
G5=5<27
G5=7<27
G5=15<27
Fig. 11. Steady-state compensation performance for 5th harmonic when
|G5| = 5, 7 and 15 under different phase angles (attenuation = |Igrid h/
Iload h|)
When the modulus of G is fixed, the optimal phase angle
can provide a better harmonic compensation performance than
real number G. However, the optimum angle for the harmonic
compensation is unnecessarily the best angle for stability. As a
result, a tradeoff should be made, making the design procedure
more complicated. Since the real number G with a positive or
negative sign can fulfill requirements on both the stability and
46 CSEE JOURNAL OF POWER AND ENERGY SYSTEMS, VOL. 4, NO. 1, MARCH 2018
the harmonic compensation performance, it is preferred to use
the real number to make the design more straightforward.
D. Comparison of the Three Scenarios
Through the discussion above, it can be concluded that the
proposed harmonic compensation method using a real number
harmonic control gain is not only easy to implement but also
has good harmonic compensation performance and stability.
Thus an easy way to find the gain can be obtained—figure out
the stable range of G for the desired harmonic order first and
then choose a large one with enough stability margin (small
virtual impedance will be obtained when G is large).
For the virtual resistance scenario, to improve the harmonic
compensation capability leads to poor stability margin in low
switching frequency systems. The range of applicable virtual
resistance value is thus limited, resulting in that the harmonic
compensation performance being not as good as the scenario
with real harmonic feedback gains. However, this scenario
enables the harmonic compensation with parallel ILCs as
the paralleled ILCs will absorb the harmonic currents like a
resistor at the desired frequency, the harmonic sharing will be
easily achieved. Considering the stability constraints, G has to
be carefully designed and checked to make sure the effective
harmonic compensation.
Moreover, when the modulus of G is fixed with different
phase angles, the phase angle that provides optimal harmonic
compensation performance can be obtained. And this optimal
phase angle changes little with the modulus of G. Once the
optimal phase angle is obtained, the virtual impedance can
be conveniently determined. However, due to the limit of
the stability, best harmonic compensation angle may not be
applicable, leading to a complicated design and verification
procedure. Since the real number gain has good performance,
this kind of optimization seems to be unnecessary.
V. EXPERIMENTAL RESULTS
To experimentally validate the proposed harmonic compensation
method, experiments have been carried out on an ILC
in the lab, shown in Fig. 12. The experiment parameters are
the same with the simulation parameters as listed in Table VI.
A dSPACE (DS1103) control system is used to control the
ILC. And the grid voltage is provided by a three-phase
programmable power supply. The nonlinear load, which is a
three-phase diode rectifier with parallel-connected capacitor
and resistor at the DC side, is connected to the AC bus.
Fig. 12. Experimental system.
TABLE VI
PARAMETERS OF THE PROTOTYPE
Parameters Values
Grid Lg = 5 mH, Rg = 0.2 Ω
Load (6-pulse diode rectifier) Lac = 2.5 mH, Cdc = 1,000 uF,
Rdc = 8 Ω
LCL filter L1 = 2.5 mH, L2 = 2.5 mH,
C = 40 uF, Rd = 1 Ω
Switching frequency 2 kHz
A. Harmonic Compensation Performance Using a Real Number
G
The performance of the ILC without harmonic compensation
is shown in Fig. 13, and the harmonic analysis of
the AC bus voltage is shown in Fig. 14. As shown, without
harmonic compensation, the AC bus voltage is distorted due
to the harmonic current injected by the nonlinear load. And
the grid side current is also distorted due to the distorted AC
bus voltage. According to Fig. 14, the THD of the AC bus
voltage is 11.03%, and the 5th, 7th, and 11th harmonics are
7.92%, 5.69%, and 4.30% respectively.
Time (10 ms/div)
(c)
(b)
(a)
Fig. 13. Performance of ILC without harmonic compensation. (a) AC bus
voltage (50 V/div). (b) ILC current (5 A/div). (c) Grid current (2.5 A/div).
Fundamental (60 Hz)=37.07, THD=11.03%
10
8
6
4
2
0
0 2 4 6 8 10 12 14 16 18 20
Harmonic Order
Mag (% of Fundamental)
Fig. 14. Harmonic analysis of the AC bus voltage without harmonic
compensation (THD = 11.03%, h5 = 7.92%, h7 = 5.69% and h11 =
4.30%).
Then the performance of the ILC when the proposed harmonic
compensation method is implemented to compensate
5th, 7th, and 11th harmonics is shown in Fig. 15, in which
the harmonic control gains are set as G5 = 20, G7 = 10,
and G11 = −8. The compensation is very effective so that
the AC bus voltage becomes sinusoidal. This is because most
harmonic currents of the nonlinear load are absorbed by the
ILC, which can be observed from waveform (b) in Fig. 15,
leaving an improved AC bus voltage and grid current with
lower THD. As shown in Fig. 16, the THD of the AC bus
TIAN et al.: A HARMONIC COMPENSATION APPROACH FOR INTERLINKING VOLTAGE SOURCE CONVERTERS IN HYBRID AC-DC MICROGRIDS WITH LOW SWITCHING FREQUENCY 47
voltage is reduced from 11.03% without compensation to
4.12% when the proposed method is applied. As a result, when
the control gain G is set as a real number, the effectiveness
of the proposed harmonic control method at a relatively low
switching frequency is verified in the experiments.
Time (10 ms/div)
(a)
(c)
(b)
(G5=20, G7=10, and G11=−8)
Fig. 15. Performance of ILC with harmonic compensation. (a) AC bus
voltage (50 V/div). (b) ILC current (5 A/div). (c) Grid current (2.5 A/div).
Fundamental (60 Hz)=36.51, THD=4.12%
10
8
6
4
2
0
0 2 4 6 8 10 12 14 16 18 20
Harmonic Order
Mag (% of Fundamental)
Fig. 16. Harmonic analysis of the AC BUS voltage with harmonic
compensation (THD = 4.12%, h5 = 1.01%, h7 = 2.02% and h11 = 0.34%).
Fig. 17 shows the transient when the nonlinear load changes.
When the nonlinear current is zero, the AC bus voltage and
the ILC output current are both sinusoidal. Then a nonlinear
load is connected to the AC bus, causing an impulse and
then injecting harmonics to the system. With the help of the
harmonic compensation method, the AC bus voltage maintains
a high quality while the harmonics are absorbed to the ILC,
whose sinusoidal current then becomes distorted. The transient
of the AC bus voltage is smooth although the impulse load
currents are injected, showing good stability.
(a)
(b)
(c)
Time (10 ms/div)
Fig. 17. Transient Performance when nonlinear load changes. (a) AC bus
voltage (50 V/div). (b) ILC current (5 A/div). (c) Load current (2.5 A/div).
B. Harmonic Compensation Performance Using a Virtual Resistance
In order to demonstrate the effectiveness of the proposed
method when the ILC equivalent impedance is controlled
as a resistance, the corresponding experimental results are
also obtained. The performance of the ILC when the virtual
resistances at the harmonic frequencies are controlled properly
to be their minimum values (R5 = 2.3 Ω, R7 = 4.4 Ω,
and R11 = 0.5 Ω) and the harmonic analysis of the AC bus
voltage are shown in Fig. 18 and Fig. 19 respectively. The
THD of the AC bus voltage is reduced from 11.03% to 6.90%.
Compared with the real number G, the harmonic compensation
performance using virtual resistance is limited by insufficient
system stability. Therefore, virtual resistance is not flexible for
the proposed harmonic compensation method at a relatively
low switching frequency.
Time (10 ms/div)
(a)
(b)
(c)
Fig. 18. Performance of ILC with harmonic compensation using virtual
resistance. (a) AC bus voltage (50 V/div). (b) ILC current (5 A/div). (c) Grid
current (2.5 A/div). (G5 = 2.27∠101.17, G7 = 1.27∠143.56 and G11 =
13.56∠177.18).
Fundamental (60 Hz)=36.94, THD=6.90%
10
8
6
4
2
0
0 2 4 6 8 10 12 14 16 18 20
Harmonic Order
Mag (% of Fundamental)
Fig. 19. Harmonic analysis of the AC bus voltage with harmonic compensation
using virtual resistance (G5 = 2.27∠101.17, G7 = 1.27∠143.56 and
G11 = 13.56∠177.18).
VI. CONCLUSION
This paper proposes a harmonic compensation approach
for interlinking converters in the hybrid AC-DC microgrid,
especially for the low switching frequency systems which
have difficulty to expand the control bandwidth. Through
controlling the PWM reference signal directly for the harmonic
component to avoid the multi-loop control path of the
fundamental component, the proposed method can achieve
the effective high order harmonics compensation. In this
48 CSEE JOURNAL OF POWER AND ENERGY SYSTEMS, VOL. 4, NO. 1, MARCH 2018
study, three design schemes are analyzed respectively: virtual
impedance using real number G, virtual resistance using a
proper complex number G, and complex number G with
optimal phase angle for harmonic compensation performance.
By comparing the harmonic compensation performance of
these schemes, the harmonic compensation performance using
the real number G is found to be much better because the
good system stability allows the modulus of G to be set to
a relatively large value. Meanwhile, the real number makes
the design procedure more simple and straightforward. The
proposed harmonics compensation method has been verified
through experiments in a lab prototype.
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Hao Tian (S’12) received his B.S. and M.Eng.
degrees in electrical engineering from Shandong
University, Jinan, China, in 2011 and 2014, respectively.
He is currently pursuing the Ph.D. degree with
University of Alberta, Edmonton, Canada.
His research interests include distributed generation
and power quality.
Xiaohan Wen received the B.Sc. degree in automation
from Dalian University of Technology, Dalian,
China, in 2010, and the M.Sc. degree in energy
systems from University of Alberta, Edmonton, AB,
Canada in 2015.
His research interests include microgrids, distributed
generation and harmonic compensation.
Yun Wei Li (M’05–SM’11) received the B.Sc.
degree in electrical engineering from Tianjin University,
Tianjin, China, in 2002, and the Ph.D. degree
from Nanyang Technological University, Singapore,
in 2006.
In 2005, Dr. Li was a Visiting Scholar at Aalborg
University, Denmark. From 2006 to 2007, he was
a Postdoctoral Research Fellow at Ryerson University,
Canada. In 2007, he also worked at Rockwell
Automation Canada before he joined University of
Alberta, Canada in the same year. Since then, Dr.
Li has been with University of Alberta, where he is a Professor now. His
research interests include distributed generation, microgrid, renewable energy,
high power converters and electric motor drives.
Dr. Li serves as an Associate Editor for IEEE Transactions on Power
Electronics, IEEE Transactions on Industrial Electronics, IEEE Transactions
on Smart Grid, and IEEE Journal of Emerging and Selected Topics in Power
Electronics. Dr. Li received the Richard M. Bass Outstanding Young Power
Electronics Engineer Award from IEEE Power Electronics Society in 2013
and the second prize paper award of IEEE Transactions on Power Electronics
in 2014.