Harmonic rejection strategies for grid converters
Harmonic rejection strategies for grid
converters
Marco Liserre
liserre@poliba.it
Marco Liserre liserre@ieee.org
Harmonic rejection strategies for grid converters
Outline
• Introduction
• Resonant and repetitive controllers
• Models of the non-linear filter: average, picewised, volterra
• Experimental results
• Conclusions
Marco Liserre liserre@ieee.org
Harmonic rejection strategies for grid converters
Introduction
• New power quality standards for distributed power generation (IEEE
1547 and IEC 61727) calls for better current control
• Packaging and cost issues leads to the choice of small grid inductors
• Inductors are often working near to saturation
• In case of saturation the predicted behaviour of current controllers is not
valid anymore
RES DC-DC Grid Converter Non-linear filtering Grid
Boost Module 1-ph VSI inductance 1 x 240 V
DC
RES
single-phase distributed
(PV, FC) generation PV system with
DC
non-linear filtering
inductance
PWM
i
Current
Controller u
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Harmonic rejection strategies for grid converters
Introduction: harmonic limits for PV inverters
In Europe there is the standard IEC 61727
In US there is the recommendation IEEE 929
the recommendation IEEE 1547 is valid for all distributed resources technologies with
aggregate capacity of 10 MVA or less at the point of common coupling interconnected
with electrical power systems at typical primary and/or secondary distribution voltages
All of them impose the following conditions regarding grid current harmonic content
The total THD of the grid current should not be higher than 5%
Marco Liserre liserre@ieee.org
Harmonic rejection strategies for grid converters
Introduction: harmonic limits for WT inverters
In Europe the standard 61400-21 recommends to apply the standard 61000-3-6 valid for
polluting loads requiring the current THD smaller than 6-8 % depending on the type of
network.
harmonic limit
5th 5-6 %
7th 3-4 %
11th 1.5-3 %
13th 1-2.5 %
in case of several WT systems
N
I hi
Ih
i 1 i
in WT systems asynchronous and synchronous generators directly connected to the grid
have no limitations respect to current harmonics
Marco Liserre liserre@ieee.org
Harmonic rejection strategies for grid converters
Harmonic compensation
• The decomposition of signals into harmonics with the aim of monitor and
control them is a matter of interest for various electric and electronic
systems
• There have been many efforts to scientifically approach typical problems
(e.g. faults, unbalance, low frequency EMI) in power systems (power
generation, conversion and transmission) through the harmonic analysis
• The use of Multiple Synchronous Reference Frames (MSRFs), early
proposed for the study of induction machines, allows compensating
selected harmonic components in case of two-phase motors, unbalance
machines or in grid connected systems
Marco Liserre liserre@ieee.org
Harmonic rejection strategies for grid converters
Harmonic compensation
• The harmonic components of power signals can be represented in stationary
or synchronous frames using phasors
• In case of synchronous reference frames each harmonic component is
transformed into a dc component (frequency shifting)
5
i
q7
e j 5
i
q5 d5
7
i
d7
e j 7
i
• If other harmonics are contained in the input signal, the dc output will be
disturbed by a ripple that can be easily filtered out.
REF R. Teodorescu, F. Blaabjerg, M. Liserre and P. Chiang Loh, “A New Breed of
Proportional-Resonant Controllers and Filters for Grid-Connected Voltage-Source
Converters” IEE proceedings on Electric Power Applications, September 2006, Vol.
153, No. 5, pp. 750-762.
Marco Liserre liserre@ieee.org
Harmonic rejection strategies for grid converters
Harmonic compensation by means of
synchronous dq-frames
5
id 5 0
• two controllers should be
i + v
I
+
implemented in two frames rotating at
-5 and 7 e j 5 e j 5
i v
I
+
• or nested frames can be used i.e. +
implementing in the main iq 5 0
synchronous frame two controllers in
two frames rotating at 6 and -6 7
id 7 0
i + v
I
+
• Both solutions are equivalent also in
terms of implementation burden e j 7 e j 7
because in both the cases two i v
I
+
controllers are needed +
iq 7 0
Marco Liserre liserre@ieee.org
Harmonic rejection strategies for grid converters
Harmonic compensation by means of
stationary -frame
Besides single frequency compensation (obtained with the generalized
integrator tuned at the grid frequency), selective harmonic compensation
can also be achieved by cascading several resonant blocks tuned to
resonate at the desired low-order harmonic frequencies to be
compensated.
As an example, the transfer
functions of a non-ideal idd Kp
harmonic compensator (HC) u
i
designed to compensate for the
3rd, 5th and 7th harmonics is
reported
Ki s
i
s2 2
2 K ih c s Kih s
Gh ( s)
h 3,5,7 s 2 2 c s h 2
h 3,5,7... s ( h)
2 2
Marco Liserre liserre@ieee.org
Harmonic rejection strategies for grid converters
Resonant Controllers
Bode Diagram GI res pons e
200 2
in
s out
input and output of the resonant controller
100 1.5
Magnitude (dB)
0
s2 2 1
-100 0.5
only changing the
-200
180
0
parameters of the
90
-0.5
controllers
Phase (deg)
0 -1
-90 -1.5
-180
1 2 3 -2
10 10 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Frequency (Hz) time ec]
time [s
[s]
Bode Diagram Bode Diagram
400 400
300
1000s 1000s 1
10
300
10 2
Magnitude (dB)
Magnitude (dB)
200
100 s2 2 200
s 2 1 0.1s
100
0
-100 0
-200 -100
180 180
90 90
Phase (deg)
Phase (deg)
0 0
-90 -90
-180
1 2 3 -180
PM
10 10 10 1 2 3
10 10 10
Frequency (Hz) Frequency (Hz)
Marco Liserre liserre@ieee.org
Harmonic rejection strategies for grid converters
Resonant Controllers
open loop
1 2
60 closed loop
without harm. comp. fund 5th
50 with harm. comp. 7th
4 5th
3rd
3rd
40 fund
Magnitude [db]
7th
2
Magnitude [db]
30
20
0
10 -2
650 Hz
BW=650Hz
-3dB
0
cross-over freq=460 Hz -4
-10 1 2 3
with harm. comp.
without harm. comp. bandwidth
10
stability margin 10 10 -6 1
10 10
2
10
3
0
72°
-20 without harm. comp.
with harm. comp.
-40 0
Phase [Grad]
-60
PM=72 grd -20
Phase [Grad]
-80
-100 -40
-120
-60
-140
-160 -80
-180 1 with harm. comp.
10 10
2
10
3 without harm. comp.
-100
Frequency [Hz] 2 3
10 10
Frequency [Hz]
P
3
0
i dd Kp
PI
-50
PR+HC
PI
i u P
-100
PR+HC
Ki s -150
i 1. Open loop Bode -270
s 2
2
diagram -360
Kih s 2. Closed loop Bode
-450
h 3,5,7... s 2 ( h)2 diagram
-540
3. Disturbance rejection
1 2 3
10 10 10
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Harmonic rejection strategies for grid converters
Harmonic compensation by means of
stationary -frame
Open-loop PR current control system with and Closed loop PR current control system with and
without harmonic compensator without harmonic compensator
• Having added the harmonic compensator, the open-loop and closed-loop
bode graphs changes as it can be observed with dashed line. The change
consists in the appearance of gain peaks at the harmonic frequencies, but
what is interesting to notice is that the dynamics of the controller, in terms
of bandwidth and stability margin remains unaltered.
Marco Liserre liserre@ieee.org
Harmonic rejection strategies for grid converters
Hybrid solution: generalized integrator in
dq frame
Instead of using two
nested frames rotating
at 6 and -6 in
the main synchronous
frame one resonant
controller can be used
REF M. Liserre, F. Blaabjerg, R. Teodorescu, “Multiple harmonics control for three-
phase systems with the use of PI-RES current controller in a rotating frame” IEEE
Transactions on Power Electronics, May 2006, vol. 21, no. 3, pp. 836-841.
Marco Liserre liserre@ieee.org
Harmonic rejection strategies for grid converters
Hybrid solution: generalized integrator in dq frame
cause: 5th inverse 7th direct
Three different harmonic controllers are applied at t=0.5 in three different simulations:
1 use of a standard integrator in a frame rotating at 6;
2 use of two standard integrators implemented in two frames rotating at 6 and -6;
3 use of a 6th harmonic resonant controller
Further compensation due to unfiltered synchronization signal
Marco Liserre liserre@ieee.org
Harmonic rejection strategies for grid converters
Disturbance rejection comparison
Disturbance rejection (current error ratio disturbance) of the PR+HC, PR and P
( s) G f (s)
ug (s) i* 0 1 Gc (s) Gd (s) G f (s)
i
• Around the 5th and 7th harmonics the PR attenuation being around 125 dB and the PI
attenuation only 8 dB. The PI rejection capability at 5th and 7th harmonic is comparable
with that one of a simple proportional controller, the integral action being irrelevant
• PR +HC exhibits high performance harmonic rejections leading to very low current THD!
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Harmonic rejection strategies for grid converters
Repetitive current control
The repetitive controller is able to track any periodic signal of period T1 and it corresponds to
A delay of duration T1 in feedback control loop results in the placements of an infinite number
of poles at j and at all their multiples so that any periodic disturbance of period T1 can be
rejected.
The repetitive controller transfer function is implemented as an
N-samples delay closed in feedback
T
N 1 is the number of samples in a fundamental period T1 and
T repetitive controller
T is the sample time.
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Harmonic rejection strategies for grid converters
Repetitive current control
FRep z
FDFT k FIR ih e
i* i i ' Gc Gi Gp i
i
Gc(s) is a PI controller designed to ensure that the dynamic of the inner loop has a damping
factor of 0.707;
Despite the careful design of Gc(s) the stability is the main issue of this control method;
The repetitive controller amplifies infinite high-order harmonics while the system to be
controlled as a limited bandwidth.
Marco Liserre liserre@ieee.org
Harmonic rejection strategies for grid converters
Repetitive current control
A different solution based on a FIR filter can be chosen:
2 N 1 2
FDFT z
N i 0 hNh
cos h i N a z i
N
FRep z
FDFT k FIR ih e
i* i i ' Gc Gi Gp i
i
The FIR generates the grid harmonic disturbance and it does not lead to instability
since it amplifies only Nh harmonics
Marco Liserre liserre@ieee.org
Harmonic rejection strategies for grid converters
Repetitive current control
The FIR filter employed in positive feedback positive loop summarizes a set of resonant filters
+
+ FDFT
2 N 1 2
FDFT z i 0 hNh cos N h i Na z i
N
Marco Liserre liserre@ieee.org
Harmonic rejection strategies for grid converters
Results: grid voltage background
distortion
Effect of the grid voltage Use of harmonic compensators
background distortion on the
currents
Marco Liserre liserre@ieee.org
Harmonic rejection strategies for grid converters
Results: grid voltage background
distortion
Effect of the grid voltage Use of harmonic compensators
background distortion on the
currents
Marco Liserre liserre@ieee.org
Harmonic rejection strategies for grid converters
Resonant and Repetitive Controllers
k
s
• Resonant control CP Re s ( s) k p ki s ( s)
s
s 2 2 ih
s (h )2
2
h 3,5,7
e
i1 i
i* CPRes Gi Gp
i
resonant controller
2 N 1 2
• Repetitive control based on DFT FDFT z i 0 kNh cos N h i Na z i
N
FRep z
FDFT k FIR ih e
i* i i ' Gc Gi Gp i
i
Marco Liserre liserre@ieee.org
Harmonic rejection strategies for grid converters
Resonant and Repetitive Controllers
Open-loop Bode plot of the
system with the proposed
current controllers: (a) resonant
controller; (b) FIR repetitive-
based controller.
• The difference between resonant and repetitive controllers in normal conditions
(linear behaviour of the inductor) is very small (0.9 % in terms of THD).
• The use of DFT with the running window gives a small advantage to the repetitive
controller.
• The repetitive controller exhibits better performances than the resonant one in the
rejection of the fifth and the seventh harmonics
Marco Liserre liserre@ieee.org
Harmonic rejection strategies for grid converters
Average inductor model
• The describing function method has been widely used to determine the
dynamic behaviour of nonlinear systems. The describing functions
method can be used to linearise the nonlinear characteristic of the
inductor and estimate the average inductance value
i
T
1 1
L L
dt
Lsat
L
T
i 0
where the interval of integration T can be
chosen to be one period of the ac input 1 1
current and δ is the portion of fundamental
Leq Lsat L
period (expressed in p.u.) for which the
inductance has value Lsat
REF S.C. Chung, S.R. Huang and E.C. Lee, “Applications of describing functions to
estimate the performance of nonlinear inductance”, IEE Proceedings-Science,
Measurement and Technology, vol.48, no. 3, May 2001, pp.108-114.
Marco Liserre liserre@ieee.org
Harmonic rejection strategies for grid converters
Average inductor model
Grid current (reference, actual and error) with
resonant controller in case of increment of
saturation from δ=0.25 to δ=0.33.
Real and imaginary part of the closed loop of the
PWM inverter system (with PI current controller)
for variations of the degree of filtering inductance
saturation -> instability
saturation from δ=0 to δ=0.4.
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Harmonic rejection strategies for grid converters
Piecewise linearizated inductor model
• A time-variant current dependent model can be developed on the basis of
the piecewise linearization.
• Two different cases of nonlinearities are considered: the saturation of the
inductor, which occurs for high values of current, and a light nonlinearity
of the first portion of the magnetization curve which occurs for very low
value of current.
Lisat i isat
i sat i Li isat i isat
Li i i
sat sat d Li
e t Ri t
dt
L1i i is*
* i sat i
L2 i i is
*
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Harmonic rejection strategies for grid converters
Piecewise linearizated inductor model
Lisat i isat L1i i is*
i sat i Li isat i isat i sat i
*
L2 i i is
*
Li i i
sat sat
resonant
repetitive
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Harmonic rejection strategies for grid converters
Volterra-series expansion inductor model
• The frequency behaviour of the non-linear inductance can be studied
splitting the model in a linear part and a non-linear part in accordance
with the Volterra theory.
5
The Volterra-series expansion of the flux is t i t
i 1
1 t L1i1 t
2 t L2i1 t
2
3 t 2 L2i1 t i2 t L3i1 t
3
4 t 2L2i1 t i3 t L2i2 t 3L3i1 t i2 t L4i1 t
2 2 4
5 t 2 L2 i1 t i4 t 3L3i1 t i3 t 3L3i1 t i2 t 4L4i1 t i2 t L5i1 t
2 2 3 5
1 t is the first order response of the inductor which describes the
behaviour in the linear case
i t is the non-linear response of the inductor obtained using an appropriate
excitation which is function of the lower order excitation
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Harmonic rejection strategies for grid converters
Volterra-series expansion inductor model
Vd 1 i1
1 L1
2
2
L2
i2
1 L1
i
second order model
i3
third order model
i4
fourth order model
i5
fifth order model
implementation of the non-linear inductance model
• The Volterra model allows calculating harmonics which are introduced in
the systems as effect of the filter inductance saturation
• These harmonics can be modelled as external disturbances, hence they
can be compensated by the resonant and repetitive controllers similarly
to grid voltage harmonics
• This explains theoretically the effectiveness of the resonant and repetitive
controllers in case of non-linear inductance
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Harmonic rejection strategies for grid converters
Volterra-series expansion inductor model
L1 i1 i
n i1 ,..., in 1 3 i1 , i2 2 i1
in i3 i2
L1 L1 L1
v e
non-linear inductance
• ii(t) through the non-linear inductor acts as an external source exciting
the linear circuit
• it can be represented as an external source of current which is connected
to the system between the converter and the grid
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Harmonic rejection strategies for grid converters
Volterra-series expansion inductor model
input current at ω1= 50 Hz
input current at ω2= 150 Hz
input current at (ω1 + ω2 )
I15
flux spectrum of the non-linear inductance i t I sen 1t 10sen1t 5sen31t sen51t
5 5 5
1 1
16
• When two sinusoids of different frequencies are applied simultaneously
intermodulation components are generated
• They increase the frequency components in the response of the system
and the complexity of the analysis
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Harmonic rejection strategies for grid converters
Volterra-series expansion inductor model
e ii
i1 i
i* CPRes Gi Gp
i non-linear inductance
resonant controller
REF R. A. Mastromauro, M. Liserre, A. Dell'Aquila, Study of the Effects of Inductor Non-
Linear Behavior on the Performance of Current Controllers for Single-Phase PV
Grid Converter, IEEE Transactions on Industrial Electronics, VOL. 55, NO. 5, MAY
2008.
J. J. Bussagang, L. Ehrman, J. W. Graham, “Analysis of Nonlinear Systems with
Multiple Inputs”, Proceedings of the IEEE, vol. 62, no. 8, Aug. 1974, pp.1088-1119.
F. Yuan, A. Opal, “Distortion Analysis of Periodically switched Nonlinear circuits
Using time-Varying Volterra Series” IEEE Transactions on Circuits and Systems-I:
Fundamental Theory and Applications, vol.48, no. 6, June 2001, pp.726-738.
E. Van Den Eijnde, J. Schoukens, “Steady-State Analysis of a Periodically Excited
Nonlinear System”, IEEE Transactions on Circuits and Systems, vol.37, no. 2, Feb.
1990, pp.232-241.
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Harmonic rejection strategies for grid converters
Inductors classification
POWDERED METAL CORE FERRITE CORE
• energy is stored in a distributed non- • energy is stored in a discrete gap in
magnetic gap series
• are feasible because of higher saturation • are preferred when core losses dominate
in case of low switching frequency and in case of higher switching frequency
low current ripple and/or current ripple
toroidal inductor with powdered metal core air-gap based inductor with ferrite core
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Harmonic rejection strategies for grid converters
Simulation results: high values of currents
RESONANT CONTROL
4
3.5
3
Magnitude [%]
2.5
2
1.5
1
0.5
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Harmonic order
grid current response grid current harmonic spectrum
Marco Liserre liserre@ieee.org
Harmonic rejection strategies for grid converters
Simulation results: high values of currents
REPETITIVE CONTROL BASED ON DFT
4
3.5
3
Magnitude [%]
2.5
2
1.5
1
0.5
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Harmonic order
grid current response grid current harmonic spectrum
Marco Liserre liserre@ieee.org
Harmonic rejection strategies for grid converters
Simulation results: light non-linearities for low
values of the current
RESONANT CONTROL
1.5
Magnitude [%]
1
0.5
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Harmonic order
grid current response grid current harmonic spectrum
Marco Liserre liserre@ieee.org
Harmonic rejection strategies for grid converters
Simulation results: light non-linearities for low
values of the current
REPETITIVE CONTROL BASED ON DFT
1.5
1
Magnitude [%]
0.5
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Harmonic order
grid current response grid current harmonic spectrum
Marco Liserre liserre@ieee.org
Harmonic rejection strategies for grid converters
Simulation results: remarks
current harmonics
case ampl. 1 % THD (%)
%
a 2;3;4,5;6;7;8 13;15; 17 5,69
;9;10;11;12;1
4;16
b 2;3;4;5;6;7;8 11;13;15;17 1,58
;9;10;12,14;1
6;
c 3,4,5;6;7;8; 2,9;12;13,14;16 11 2,67
10;
15;17
d 3,4,5;6;7;8;9; 2; 16 1,76
1011;12;13;1
4;15,17
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Harmonic rejection strategies for grid converters
Experimental Setup: Polytechnic of Bari
inverter DSpace
1104
Dc filtering
power inductance
supplies
power
analyzer
RLC
load
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Harmonic rejection strategies for grid converters
Experimental results
Three different kind of of single-phase filtering inductance have been tested:
3 mH and 1.5 mH toroidal inductor with a powdered metal core
a 2.6 mH air-gap based inductor with a ferrite core
For low currents the
air-gap based inductor
characteristic is more
non-linear
• a – 3 mH toroidal inductor characteristic
• b – 2.6 mH air-gap inductor characteristic
• c – 1.5 mH toroidal inductor characteristic
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Harmonic rejection strategies for grid converters
Experimental results: inductors characterization
voltage drop of toroidal inductor voltage drop of air-gap based inductor
THE VOLTAGE THD CAUSED BY THE TOROIDAL INDUCTOR IS
LOWER
Marco Liserre liserre@ieee.org
Harmonic rejection strategies for grid converters
Experimental results: low current non-linearity
resonant controller repetitive controller
a a
THD= 4.8% THD= 3.9%
b b
Grid current with air-gap based inductor and Grid current with air-gap based inductor and
resonant controller: a) (1) grid current [10A/div]; (2) repetitive controller: a) (1) grid current [10A/div];
grid voltage [400V/div]; (A) grid voltage spectrum (2) grid voltage [400V/div]; (A) grid voltage
[10V/div]; (B) grid current spectrum [0.5A/div]; (C) spectrum [10V/div]; (B) grid current spectrum
a period of the grid voltage; (D) a period of the grid [0.5A/div]; (C) a period of the grid voltage; (D) a
current; b) a period of the grid current (simulation period of the grid current; b) a period of the grid
results) [10A/div]. current (simulation results) [10A/div].
Marco Liserre liserre@ieee.org
Harmonic rejection strategies for grid converters
Experimental results: high current non-linearity
resonant controller repetitive controller
a a
THD= 8.1% THD= 4.9%
b b
Grid current with non-linear inductor and resonant Grid current with non-linear inductor and repetitive
controller: a) (1) grid current [10A/div]; (2) grid controller: a) (1) grid current [10A/div]; (2) grid
voltage [400V/div]; (A) grid voltage spectrum voltage [400V/div]; (A) grid voltage spectrum
[10V/div]; (B) grid current spectrum [0.5A/div]; (C) [10V/div]; (B) grid current spectrum [0.5A/div]; (C)
a period of the grid voltage; (D) a period of the grid a period of the grid voltage; (D) a period of the grid
current; b) a period of the grid current (simulation current; b) a period of the grid current (simulation
results) [10A/div]. results) [10A/div].
Marco Liserre liserre@ieee.org
Harmonic rejection strategies for grid converters
Experimental results
• The repetitive controller exhibits better performances than the resonant
controller in the rejection of the 5th and the 7th harmonic
•When the system is supplied with a distorted grid voltage, intermodulation
harmonics are caused by the inductor saturation, hence the repetitive controller
can mitigate also the 9th, 11th, 13th harmonics (caused by intermodulation
between the 1st and the 5th and between the 1st and the 7th)
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Harmonic rejection strategies for grid converters
Conclusions
The effects of non-linear inductance on the performance of current controllers have
been investigated with a frequency-domain model
Resonant and repetitive controllers have been tested in case of a non-linear plant
A current-dependent model of the non-linear inductance has been developed using
the Volterra series expansions
The model allows proving how harmonic compensation provided by resonant and
repetitive controllers can also mitigate the effects of the inductance saturation.
The proposed controllers are able: to compensate grid voltage harmonics
to compensate odd harmonics caused by
plant non-linearity
The repetitive controller is able to comply with the harmonic limits reported in IEEE
1547 and IEC 61727 even in very hard saturation conditions
Marco Liserre liserre@ieee.org
Harmonic rejection strategies for grid converters
Conclusions
In case of high-current saturation, the repetitive controller exhibits better performances
in fact it reduces the fifth and the seventh harmonics more than the resonant one.
For this reason the repetitive controller provides better performances also in
correspondence to the ninth, the eleventh and the thirteenth harmonics since these
harmonics are created as a consequence of the intermodulation effect between the
first and the fifth harmonics and between the first and the seventh harmonics.
The repetitive controller is able to comply with the harmonic limits reported in IEEE
1547 and IEC 61727 even in very hard saturation conditions.
Marco Liserre liserre@ieee.org