23rd International Conference on Electricity Distribution
Lyon, 15-18 June 2015
Paper 0950
STABILIZATION OF HARMONIC INSTABILITY IN AC DISTRIBUTION POWER
SYSTEM WITH ACTIVE DAMPING
Changwoo YOON
cyo@et.aau.dk
Xiongfei WANG
Claus Leth BAK
Aalborg University – Denmark
xwa@et.aau.dk
clb@et.aau.dk
ABSTRACT
This paper deals stabilizing method of the interaction
problems among the interconnected power electronics
based power devices in a power distribution system. Even
if each of the inverters in the network is stable
individually, the combined network stability cannot be
assured unless holistic stability assessment is performed.
The impedance based stability criterion is used to study
the effect of the active damping on the system stability. A
benchmark of a Cigré power distribution network is
modeled under the impedance based stability criterion
and validated using the PSCAD/EMTDC simulation. The
result shows that not all inverters need to have active
damping for stabilizing the network but some of inverters
can effectively change the impedance of the network, thus
overall network becomes stable.
INTRODUCTION
Today, distributed generation based on renewable energy,
characterized by a large number of inverters, is increasing
rapidly [1]. As a result of this trend, unexpected problems
such as the harmonic interactions among the interconnected inverters in the power system are increasing
[2], [3]. In the inverter, DC electric energies from the
renewable resources is converted to sinusoidal voltages
or currents through the Voltage Source Inverters (VSI)
with Pulse Width Modulation (PWM) techniques. During
the process of converting, power filters are necessary to
remove harmonic voltages or current in the output. These
days, Inductor-Capacitor-Inductor (LCL) filter is a
widespread solution for its high attenuation capability at
high frequencies and with its small filter value [4].
However, its high order filter structure, which fivess the
sufficient attenuation, has a side effect that it is a
destabilizing factor of the VSI and might resonate above
the control bandwidth. In order to deal with this problem,
the resonance peak of the inverter should be lowered in
order not to be affected by the phase drop at the filter
resonance frequency. There are mainly two types of
damping approaches, one is the passive damping method
which inserts real resistors into the filter topology to
dissipate excessive energy from the resonance [5], [6],
and the other one is the active damping, which emulates
the behavior of the passive damping by adding additional
control loops and/or feedback signals [4], [7], [8]. The
passive damping can be implemented simple and
inexpensive ways, but it reduces the overall system
efficiency, so in applications putting emphasis on their
CIRED 2015
Frede BLAABJERG
fbl@et.aau.dk
efficiency like PV systems are not appropriate. Instead of
dissipating it, the energy can be collected by the work of
additional control efforts. However, it requires additional
sensors for feedback signal or state estimators, which
complicates their control loops and sometimes they are
also sensitive to the filter parameter variation. Recently,
research clearly stating the stability of an inverter by the
relation between LCL filter resonance frequency and the
unavoidable time delay from its digital implementation
has been published [9]. This exactly defined the boundary
of the stable region of LCL-filtered inverter without any
damping method; it means the inverter can be stable with
its high resonance peak. However, it is valid only for
single inverter case and if the inverter is somewhere on
the distribution network and connected together with
other devices, this cannot always be stable from the node
impedance [10]–[12]. Therefore, the resonance peak in
the filter may need to be dampened in order not to be
vulnerable from varying impedances in the network. One
important thing to note is that adding a damping function
to the inverter also can be useful to the nearby inverters.
This damping function changes the impedance of the
frequencies of concerns and helps the other converters to
achieve stability. Therefore, not all of the inverters in the
network are needed to have a damping function, but some
of them can successfully stabilize the overall network. In
order to prove this concept, a Cigré benchmark of a
small-scale network [13] is selected for the network
model and five different inverters are designed under the
harmonic emission guideline [14]. The Impedance Based
Stability Criterion (IBSC) is adopted to analyze the effect
of active damping in inverters. Finally, time-domain
simulation is performed in PSCAD/EMTDC environment
to validate the analysis. The result shows two inverters
with active damping among five can stabilize the overall
network.
TEST SYSTEM AND MODELING
This section describes the test system for validating the
effects of active damping in a distribution network. Also
it discusses about the average switching modeling
method for the inverter modeling. Then the time-domain
implementation method is introduced as well. Finally, the
network parameters and the inverter model are unified
from the IBSC method and so as to perform the stability
analysis.
Test system
Fig. 1 shows the Cigré benchmark model of the European
LV distribution network. In order to show clearly show
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23rd International Conference on Electricity Distribution
Lyon, 15-18 June 2015
Paper 0950
the effect of the interacting inverters, only the cables and
the inverters are considered. The voltage of the network
is a three-phase line-to-line 400 V with 50 Hz. This
distribution network is connected from the 20 kV
medium voltage feeder with 400 kVA transformer. Each
node is connected with underground cable and their
positive sequence impedances are given in Table I.
Passive loads, which cause unbalances in the network, are
neglected for the simplicity. Grid inverters are designed
under the harmonic emission guideline IEEE-519 [14]
and their specifications are denoted in Table II.
400kVA
6%
Rx
R15
R6
Inv.1
R8
R17
R9
R18
Inv.2
Fig. 1. Simplified benchmark model of European LV distribution
network. (Inv.1 ~ Inv.5 are Voltage Source Inverters)
TABLE I. POSITIVE SEQUENCE IMPEDANCE OF UNDERGROUND CABLE.
Node
Length Resistance Inductance
(From-To)
[m]
[mΩ]
[uH]
R1-R2
35
10.04
18.62
R2-R3
35
10.04
18.60
R3-R4
35
10.04
18.60
R4-R6
70
20.09
37.21
R6-R9
105
30.13
55.81
R9-R10
35
10.04
18.60
R4-R15
135
155.52
196.8
R6-R16
30
34.56
43.73
R9-R17
30
34.56
43.73
R10-R18
30
34.56
43.73
Transformer
3.2
40.74
TABLE II. GRID INVERTER SPECIFICATIONS AND THEIR PARAMETERS.
Inverter name
Inv. 1 Inv. 2 Inv. 3 Inv. 4 Inv. 5
Power rating [kVA]
35
25
3
4
5.5
Base Frequency, f0 [Hz]
50
Switching Frequency, fs [Hz]
103
1.6×103
103
(Sampling frequency )
DC-link voltage, Vdc [kV]
0.8
Harmonic emission recommendation
Parasitics
values
Controller
gain
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Lf [mH]
Cf [uF]
Lg [mH]
rLf [mΩ]
rCf [mΩ]
rLg [mΩ]
KP
KI
ig
αβ
÷
den
ig
PLL
trigger
θ
S&H
Control
Gc(z)
num
1
2
vPCC
i*αβ
iαβ αβ
Vdc
i*g,dq
αβ
dq
ig
abc
Fig. 3. PSCAD implementation of an inverter with active damping.
Inv.5
R10
abc
R14
R5
R7
1/ZLg
CB
Lg
rLg
iCf
Limiter
Node
R16
Filter
values
rLf
rCf
Cf
PWM
R3
Inv.3
Lf
vM
Vdc
R11
R13
ZCf
Fig. 2. Averaged switching model of an inverter with active damping.
z-1
R12
1/ZLf
KAD
Supply point
Inv.4
Gd
Bus
R2
R4
Gc
vM
KAD
20kV : 0.4kV
vPCC
i*g
Inverter Model
Averaged switching model
On the part of the IBSC for stability analysis, lumped
admittance models of two interconnected systems are
required [12]. One is the object of stability analysis called
source admittance (Ysx), and the other one is the existing
stable system that provides a stable datum called the load
admittance. Normally, the source admittance becomes an
output admittance of the inverter and the load admittance
can be an input admittance of the stable network. Fig. 2.
shows an averaged switched model of grid inverter with
active damping. The output admittance of the inverter YO
can be obtained by rearranging the block diagram with
the following condition [15]
YSx  YO 

vPCC
ig
Z Lf  Z Cf  K ADGd
Z Lf Z Lg  ( Z Lf  Z Lg ) Z Cf  K ADGd Z Lf  GC Gd Z Cf
where, Gc represents the current controller, Gd is the
equivalent model of digitizing delay, KAD is the active
damping gain, and the filter impedances ZLf, ZLg and ZCf
are as follows,
GC  K P 
3.8
2.8
3
4
1.3
0.9
119.3 88
14.5 11
40.8 28.3
16.6 18
1500 1500
KI s
s  w02
2
Gd  e1.5Ts s
(2)
(3)
1
sC f
(4)
Z Lf  rLf  sL f
(5)
Z Lg  rLg  sLg
(6)
ZCf  rCf 
IEEE519
0.87 1.2
5.1
22
15
2
0.22 0.3
1.7
27.3 37.7 160.2
7.5
11 21.5
6.9
9.4 53.4
5.4 8.05 28.8
1000 1000 1500
(1)
i*g  0
Also the open loop gain TOL can be derived as follows,
TOL 
GC Gd ZCf
Z Lf Z Lg  ( Z Lf  Z Lg )ZCf  K ADGd Z Lf
(7)
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23rd International Conference on Electricity Distribution
Lyon, 15-18 June 2015
Paper 0950
inverse of diagonal impedances.
Time-domain simulation model
Fig. 3. shows the time-domain model for PSCAD/
EMTDC simulation. In order to model the real system of
the digital controller, sample and hold function is made
and a digital resonance controller Gc(z) is used from (2)
with the Tustin discretizing method, giving:
K sin(2 f 0TS )
z 2 1
(8)
GC ( z )  K P  I
2  2 f 0
z 2  2cos(2 f 0TS ) z  1
This controller is driven by an external triggering signal
for sample and hold. The controller operates in the
stationary reference frame.
YLx  1/ Zxx  YSx
where, ‘x’ indicates the inverter numbering.
The Impedance Based Stability Criterion
The two components for the IBSC are obtained, one is
the source admittance of grid inverter (1) and the other
one is the load admittance of stable network (11). If the
ratio of these two quantities called the minor loop gain
Tmx satisfies the Nyquist stability criterion, then the
system with the grid inverter with the network is stable.
Tmx 
Network model for the load admittance
Previously, the source admittance of the inverter is
obtained in (1). However, the load admittance on that
node needs to be calculated. The Kirchhoff’s Current
Law (KCL) admittance matrix is used to solve the
admittance relation [11]. So the reference nodes,
including the nodes of interest in the network, are set and
named as a vector [V] and the current sources (grid
inverters) attached to the nodes are named as a vector [I].
By arranging all the relations of the reference node
voltages and the currents are derived as follows.
Y17
Y27
Y72
Y82
Y77
Y87
 VR 6   I Inv.1 
V   I 
Y18   R10   Inv.2 
V   I 
Y28   R18   Inv.3 
V
I
  R16    Inv.4 



V
I 

Y78   R15   Inv.5 
I
V 
Y88   R 3   R 3 
VR 4
0

 

V
 R 9   0 
(9)
 VR 6 
V 
 R10   Z
VR18   11

  Z 21
VR16   
VR15  

  Z 71
 VR 3  
 V   Z81
 R4 
 VR 9 
[Z ]
Z12
Z 22
Z17
Z 27
Z 72
Z82
Z 77
Z87
40
30
20
10
-10
0
-180
-360
100
Frequency (Hz)
1000
Fig. 4. Characteristics of individually stable inverters given in Table II.
Inv.1
Inv.2
Inv.3
Inv.4
Inv.5
15
(10)
The diagonal elements in [Z] are representing the
equivalent impedances of the node, which produce the
node voltage by the current of the connected grid inverter.
However, these diagonal elements include the output
impedance of the grid inverter, which is not identified.
Therefore, a stable load admittance can be obtained by
subtracting the unidentified source admittance from the
CIRED 2015
Inv.1
Inv.2
Inv.3
Inv.4
Inv.5
50
0
[I ]
 I Inv.1 
I 
Z18   Inv.2 
I 
Z 28   Inv.3 
I
  Inv.4 
  I Inv.5 
Z 78  

I
Z88   R 3 
0


 0 
ACTIVE
As explained in [9], the stable region of the inverter with
LCL filter is determined by the sample frequency of the
inverter and the resonance frequency of the LCL filter. If
the LCL filter resonance frequency is higher than 1/6 of
the switching frequency fs, the inverter becomes stable
without any damping method. Therefore, in TABLE II,
the grid inverters are designed according to this stability
region, (7) is used for each inverter the stability analysis
as shown in Fig. 4. All inverters are designed stable
individually, and Fig. 5 shows time-domain simulation of
the respective inverter currents with ideal grid voltage.
60
So all the admittances connected to the node are included
in the matrix [Y]. In order to get the relation of the node
voltage created by the node currents, [Y]-1 is multiplied
both sides of (9) and the impedance matrix [Z] is
obtained.
[V ]
WITH
Stable inverter in ideal grid w/o damping
Magnitude (dB)
Y12
Y22
[I ]
(12)
Phase (deg)
Y11
Y
 21


Y71
Y81
[V ]
YSx
YLx
SYSTEM STABILIZING
DAMPING
10
Current (kA)
[Y ]
(11)
5
0
-5
-10
-15
0.1
0.12
0.14 Time (sec) 0.16
0.18
0.2
Fig. 5. Operation of stable inverters connected to the ideal grid voltage.
Necessity in damping with grid impedance
However, if these inverters are located in the network in
Fig. 1, their stable operation cannot be assured from the
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23rd International Conference on Electricity Distribution
Lyon, 15-18 June 2015
Paper 0950
6
4
Inv.1
Inv.2
Inv.3
Inv.4
Inv.5
Inv.1
Inv.2
8
4
Imaginary Axis
additional impedance at their connected node. These
cases can be estimated by the IBSC. Each node in the
system can have its own minor loop gain (12), and its
stability is assessed by the Nyquist stability criterion. Fig.
6 shows the Nyquist diagram of the inverters. One thing
to note is that Fig. 6 is drawn at one inverter in the
network at a time in order not to include the effect of the
other inverters. The interactions between the inverters are
presented in the next section. Inv. 5 is the only stable
inverter when the grid impedance changes, and the rest of
them become unstable. Thus, all the inverters except the
Inv. 5 may need active damping in order to be stable. Fig.
7 shows a stabilized effect from active damper.
0
-4
-8
-8
-6
-4
-2
Real Axis
0
2
4
Fig. 9. Unstable inverters in the network without active damping.
20
Inv.1
Inv.2
Inv.3
Inv.4
Inv.5
Current (A)
Imaginary Axis
2
0
0
-2
-20
-4
-6
-20
-10
Real Axis
0
0
Fig. 6. Unstable inverters in the network without active damping.
4
3
Imaginary Axis
2
0
-2
-3
-3
-2
-1
Real Axis
0
1
Fig. 7. Individually stabilized inverters with active damping.
Inv.1
0
-20
20
Inv.2
0
Current (A)
-20
20
20
-20
Inv.1
Inv.2
Inv.3
Inv.4
Inv.5
Inv.4
0
Current (A)
0
-20
20
0.1
Even if the inverter is stable when it operates alone, it
may easily become unstable, when the other inverters are
participating in the network. As shown in Fig. 9, inverters,
especially Inv. 1 and Inv. 2 are becoming unstable, when
they operate together at the same time. The time domain
simulations in Fig. 10 show the unstable inverters. Inv. 1
and Inv. 2 are no longer stable sinusoidal waveforms as
compared to the stable cases in Fig. 8. Therefore, the
active damping gains in the inverters are increased in
order to stabilize the overall network. There are many
possible gains to obtain stability. In this case Inv. 3 and
Inv. 4 are changed.
Inv.3
0
20
0.08
Interaction among inverters
-1
20
0.04 Time (sec) 0.06
Fig. 8 compares the respective unstable and stabilized
inverter output currents, which are estimated by the
Nyquist plots shown in Fig. 6 and Fig. 7. The inverters in
the network are stable when the inverter operates one at a
time for all inverters.
Inv.1
Inv.2
Inv.3
Inv.4
Inv.5
1
-4
0.02
Fig. 10. Instability from the interaction of the inverters.
Inv.5
-20
0
W/O damping
-20
0.1
0.12
0.14
W/ active damping
0.16
Time (sec)
0.18
0.2
Fig. 8. Unstable inverter w/o damping and stabilized w/ damping
(KAD,Inv.1 = 1, KAD,Inv.2 = 1, KAD,Inv.3 = 4, KAD,Inv.4 = 5).
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0
0.02
0.04
Time (sec)
0.06
0.08
0.1
Fig. 11. Stabilized inverters with increased gains
(KAD,Inv.1 = 1, KAD,Inv.2 = 1, KAD,Inv.3 = 10, KAD,Inv.4 = 12).
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23rd International Conference on Electricity Distribution
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Paper 0950
Fig. 11 shows the stabilized waveform of the inverters by
the action of additional gain changes in active damping in
Inv. 3 and Inv. 4. It is related to the impedance changes in
the network caused by the impedance change in
Stable network with reduced active damping
Furthermore, the system stability can be obtained by
reduced efforts, if the gains are properly chosen. In this
case, only the Inv. 1 and Inv. 2 are having active damping
functions with increased gains. Fig. 12 shows the
stabilized waveforms of inverter currents. As compared
to Fig. 11, the transient high frequency oscillations are
much reduced by the proper selection of gains. Also, the
Nyquist plot shows the stable operations of the inverters
in Fig. 13.
20
Inv.1
Inv.2
Inv.3
Inv.4
Inv.5
0
-20
0
0.02
0.04
Time (sec)
0.06
0.08
0.1
Fig. 12. Stabilized inverters with two active damping functions
(KAD,Inv.1 = 2, KAD,Inv.2 = 3).
40
Inv.1
Inv.2
Inv.3
Inv.4
Inv.5
Imaginary Axis
20
0
-20
-40
-30
-20
-10
0
Real Axis
10
20
30
Fig. 13. Stabilized inverters with two active damping functions.
CONCLUSION
This paper demonstrates the role of an active damping
function for stabilizing an unstable inverter. The damping
function can stabilize the inverter itself as well as the
unstable network suffering from the interactions among
other inverters. The result shows that not all the inverters
in the network are needed active damping functions if
some of the inverters are designed properly. Therefore,
the overall network is successfully stabilized.
In order to prove this concept, Cigré small-scale
benchmark case is selected for the network model and
five different inverters are designed under the harmonic
emission guideline. The Impedance Based Stability
Criterion (IBSC) is adopted to analyze the effect of the
active damping in inverters. Finally, time-domain
simulations are performed in the PSCAD/EMTDC tool.
CIRED 2015
Acknowledgments
This work was supported by The European Research
Council (ERC) under the European Union’s Seventh
Framework Program (FP/2007- 2013)/ERC Grant
Agreement no. [321149-Harmony].
REFERENCES
[1] F. Blaabjerg, R. Teodorescu, M. Liserre, and A. V. Timbus,
“Overview of Control and Grid Synchronization for Distributed Power
Generation Systems,” IEEE Trans. Ind. Electron., vol. 53, no. 5, pp.
1398–1409, Oct. 2006.
[2] J. H. R. Enslin and P. J. M. Heskes, “Harmonic Interaction Between
a Large Number of Distributed Power Inverters and the Distribution
Network,” IEEE Trans. Power Electron., vol. 19, no. 6, pp. 1586–1593,
Nov. 2004.
[3] X. Wang, F. Blaabjerg, and W. Wu, “Modeling and Analysis of
Harmonic Stability in an AC Power-Electronics-Based Power System,”
IEEE Trans. Power Electron., vol. 29, no. 12, pp. 6421–6432, Dec.
2014.
[4] M. Liserre, R. Teodorescu, and F. Blaabjerg, “Stability of
photovoltaic and wind turbine grid-connected inverters for a large set of
grid impedance values,” IEEE Trans. Power Electron., vol. 21, no. 1,
pp. 263–272, Jan. 2006.
[5] W. Wu, Y. He, T. Tang, and F. Blaabjerg, “A New Design Method
for the Passive Damped LCL and LLCL Filter-Based Single-Phase
Grid-Tied Inverter,” IEEE Trans. Ind. Electron., vol. 60, no. 10, pp.
4339–4350, Oct. 2013.
[6] R. Peña-Alzola, M. Liserre, F. Blaabjerg, R. Sebastián, J. Dannehl,
and F. W. Fuchs, “Analysis of the Passive Damping Losses in LCLFilter-Based Grid Converters,” IEEE Trans. Power Electron., vol. 28,
no. 6, pp. 2642–2646, Jun. 2013.
[7] D. Pan, X. Ruan, C. Bao, W. Li, and X. Wang, “Capacitor-CurrentFeedback Active Damping With Reduced Computation Delay for
Improving Robustness of LCL-Type Grid-Connected Inverter,” IEEE
Trans. Power Electron., vol. 29, no. 7, pp. 3414–3427, Jul. 2014.
[8] Chenlei Bao, Xinbo Ruan, Xuehua Wang, Weiwei Li, Donghua Pan,
and Kailei Weng, “Step-by-Step Controller Design for LCL-Type GridConnected Inverter with Capacitor–Current-Feedback ActiveDamping,” IEEE Trans. Power Electron., vol. 29, no. 3, pp. 1239–1253,
Mar. 2014.
[9] S. G. Parker, B. P. McGrath, and D. G. Holmes, “Regions of Active
Damping Control for LCL Filters,” IEEE Trans. Ind. Appl., vol. 50, no.
1, pp. 424–432, Jan. 2014.
[10] X. Wang, F. Blaabjerg, and P. C. Loh, “An Impedance-Based
Stability Analysis Method for Paralleled Voltage Source Converters,” in
Proc. The 2014 International Power Electronics Conference, 2014, pp.
1529–1535.
[11] X. Wang, F. Blaabjerg, M. Liserre, Z. Chen, J. He, and Y. Li, “An
Active Damper for Stabilizing Power-Electronics-Based AC Systems,”
IEEE Trans. Power Electron., vol. 29, no. 7, pp. 3318–3329, Jul. 2014.
[12] J. Sun, “Impedance-Based Stability Criterion for Grid-Connected
Inverters,” IEEE Trans. Power Electron., vol. 26, no. 11, pp. 3075–
3078, Nov. 2011.
[13] “Benchmark Systems for Network Integration of Renewable and
Distributed Energy Resources C06.04.02,” CIGRE, 2014.
[14] D. Committee, I. Power, and E. Society, “IEEE Recommended
Practice and Requirements for Harmonic Control in Electric Power
Systems,” vol. 2014, pp. 1–29, 2014.
[15] X. Wang, X. Ruan, S. Liu, and C. K. Tse, “Full Feedforward of
Grid Voltage for Grid-Connected Inverter With LCL Filter to Suppress
Current Distortion Due to Grid Voltage Harmonics,” IEEE Trans.
Power Electron., vol. 25, no. 12, pp. 3119–3127, Dec. 2010.
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