Frequency-Adaptive Repetitive Current Control of Grid-Connected
Inverter for Wind Turbine Applications
Ali Asbafkan*, Behzad Mirzaeian**, Mehdi Niroumand***, Hossein Aboutorabi Zarchi****
*University of Isfahan, a.turbodesign@gmail.com
** University of Isfahan, mirzaeian@eng.ui.ac.ir
*** University of Isfahan, mehdi_niroomand@eng.ui.ac.ir
**** Ferdowsi University of Mashhad, hazarchi@gmail.com
Abstract: this paper is concerned with the current control of
grid-connected inverters based on a simple frequency-adaptive
repetitive controller. A grid-tie inverter is desired to behave as
a robust current source inverter with the high gain in order to
improve both the reference current tracking and disturbance
rejection capabilities. The variations of grid impedance and
frequency in distribution networks are some important
challenges in the design of grid-converters. In this research, the
performance of a frequency adaptive Repetitive Controller
(RC), which is based on the Internal Model Principle (IMP), is
investigated for a single-phase grid-tie inverter.
Keywords: Grid-Connected Inverter (GCI), Repetitive
Control, Resonant Control, Power Quality.
1.
Introduction
Nowadays, Distributed Power Generation Systems
(DPGSs) based on Renewable Energy Sources (RES),
such as wind and solar energy are gaining more and more
attention worldwide. With the exponentially increasing
penetration of wind and solar electrical energy into the
grid systems, utility converters should be compliant to
more stringent standards and regulations than before. Not
only should Grid converters transfer the stochastic
generated DC power to the AC grid, but also be able to
exhibit advanced functions and ancillary services such as
dynamic control of active and reactive power, stationary
operation within a range of voltage and frequency
variations, low voltage ride-through capability, etc. [1].
Being compliant to power quality international
recommendations and standards such as IEEE 519-1992,
IEC 61000-3-12, IEEE-1547 or IEC61727 requires an
advanced current control strategy which is faced to some
unavoidable disturbances and uncertainties. The
variations of grid frequency and impedance seen at the
Point of Common Coupling (PCC) can directly affect the
performance and stability of the converters [2].
Additionally, Due to the proliferation nonlinear loads, the
grid voltage at the PCC is more likely to be distorted
particularly in weak grids with long radial distribution
feeders. The grid voltage distortions or unbalances lead to
the injection of highly distorted low-order harmonic
currents in the output of Voltage-Sourced Inverters
(VSIs). The grid-connected inverter with an LCL filter
has the ability of attenuating the high-frequency current
harmonics. However, the current distortion caused by low
order harmonics in the grid voltage is difficult to be
eliminated. The solution of applying a feed forward grid
voltage can suppress such induced current distortion, but
the result is not satisfactory due to the presence of some
delays and noise amplifications, especially when the
order of the harmonics is high. There have been some
improvements and modifications to grid voltage full feedforward techniques as expressed in [3], [4]. Another
solution is to virtually increase the harmonic impedance
of the converter at some frequencies of interest by
adopting properly designed Harmonic-Compensators
(HC) in the controller.
As shown in Fig. 1, the control strategy applied to the
grid-tie inverter consists mainly of two cascaded loops.
Usually, there is a fast internal current loop, which
regulates the grid current, and an external voltage loop,
which controls the dc-link voltage. The inner currentcontrol loop which is responsible for power quality issues
and current protection is designed with high bandwidth
characteristics aiming to ensure accurate current tracking
with fast dynamic response and force the VSI to
equivalently act as a current source amplifier with high
disturbance rejection against grid distortion. The dc-link
voltage controller is designed for balancing the power
flow in the system.
Fig. 1: Block diagram of typical single-phase RES system [5]
In the following, a brief review on some important
current control strategies is given, and the main properties
of each structure are highlighted.
1.1
PI Controller
PI Controllers are one the most widely used controllers
in the fields of power electronics and industry. Due to the
infinite gain characteristics at zero-frequency, PI
controllers are the best candidates for regulating DC
values with zero steady state error. In the case of
tracking/rejecting sinusoidal signals/disturbances, PI
controllers have steady-state error and limited disturbance
rejection capability. When the current controlled inverter
is connected to the grid, the phase error results in a power
factor decrement and the limited disturbance rejection
capability leads to the need of grid feed-forward
compensation. However the imperfect compensation
action of the feed-forward control results in high
harmonic distortion of the current and consequently noncompliance with international standards. By adopting the
Park/Clark transformation, AC values can be converted to
DC quantities. In order to compensate for grid induced
low order harmonics Multiple Synchronous Reference
Frames strategy (MSRFs) can be used. However, high
computational efforts, large amount of data memory and
strong dependency to the Phase-Locked-Loop (PLL)
operation, are some limiting factors for using VoltageOriented Control (VOC) method with MSRFs. Also in
case of single-phase systems it is of more complexity to
implement the aforementioned control strategy.
1.2
Proportional Resonant (PR) Controller
Recently, Proportional Resonant (PR) controller
attracted an increased interest due to its superior
behaviour over the traditional PI controllers, when
regulating sinusoidal signals. Basically, the functionality
of the PR controller is to introduce an infinite gain at a
selected resonant frequency for eliminating steady state
error at that frequency, and is therefore conceptually
similar to an integrator whose infinite DC gain forces the
DC steady-state error to zero. The resonant portion of the
PR controller can therefore be viewed as a generalised
AC integrator (GI), as proven in [6]. Generally, a PR
controller is used to control the fundamental current and
several GIs that resonate for each harmonic frequency of
interest (3rd, 5th and 7th) are used in a harmonic
compensator in order to reduce the current THD for
compliance with IEEE 929 standard, as shown in Fig. 2.
In fact, harmonic compensators do not affect the
dynamics of the PR controller unless they are tuned at the
proximity of the cross-over frequency. This structure is a
successful solution in applications with distributed
generation systems, where harmonics compensation,
especially low order harmonics, is required.
Some performance degradations can usually be
observed due to round-off errors, when implementing
resonant controller on a fixed-point DSP. Round-off
errors cause the voltage or current wave shape to change
slightly from cycle to cycle, resulting in significant
fluctuations in its RMS value [7]. Although the abovementioned problem could be resolved by using delta
operator [8, 9], the performance of the PR controller
degrades significantly in the case of grid frequency
variations. With the addition of damping to resonant
controllers, non-ideal quasi-resonant controllers with
wider bandwidth and the sacrificed lower gain are
produced in order to preserve the control loop gain as
high as possible in the face of grid frequency variations.
Another solution is to make the PR filter coefficients
adaptive with respect to the grid frequency variations
[10]. But in the case of implementing multiple harmonic
compensators, each filter must be retuned separately
leading to the increase of computational burden and
errors. In this paper, a kind of adaptive Repetitive
Controller (RC) is introduced, which can easily tune all
resonant frequencies by adapting a simple time delay.
Fig. 2: PR controller with harmonic compensators [1]
1.3
Deadbeat Controller (DB)
It has been shown recently that deadbeat current
control is well suited for grid-connection application if
adequate robust control performance is provided to
mitigate the sensitivity of deadbeat controllers to system
parameter variation and disturbances. Disturbance
observers, such as Luenberger observer, reduced-order
observers, and time-delay estimators are proposed to
enhance the robustness of the conventional deadbeat
controllers [11]. However, the nature of these observers
as low-pass filters reduces the robustness of control
performance, particularly against relatively high
frequency grid harmonics and disturbances.
1.4
Other Controllers
The details for using other controllers based on
Hysteresis Band (HB), robust, sliding mode and ANNbased control can be addressed in [12-15] respectively.
2.
Analysis of Repetitive Controller
As discussed in the last section, repetitive controller
which is based on the internal model principle [16] can be
a good candidate for the control of utility converters with
periodic reference signals or disturbances. The internal
model principle states that a controlled output can track a
class of reference commands without a steady error if the
generator (or the model) of the reference is included in
the stable closed-loop system. Therefore, it can be used to
provide exact asymptotic output tracking of periodic
inputs or to reject periodic disturbances. It is well known
that the generator of a sinusoidal signal that contains only
one harmonic component is a harmonic oscillator, in
other words, a resonant filter. Thus, following this idea, if
a periodic signal has an infinite Fourier series (of
harmonic components), then an infinite number of
harmonic oscillators are required to track or reject such a
periodic signal. Fortunately as shown in Fig. 3, in the
repetitive-control approach, a simple delay line in a
proper feedback array can be used to produce an infinite
number of poles, thereby simulating a bank of an infinite
number of harmonic oscillators, which leads to a system
dynamics of infinite dimension.
transfer function of the system without RC is stable and is
calculated as equation (4):
H ( z) 
Gc ( z )G p ( z )
1  Gc ( z )G p ( z )
And consequently the transfer function of the error
signal can be calculated as equation (5) :

E ( z )  1  Q( z ) z  N  
1



R( z ) 1  Gc ( z )G p ( z )  1  Q( z ) z  N (1  k RG f ( z ) H ( z )) 
Fig. 3: repetitive controller structure
According to Fig. 3, the transfer function of RC can be
written as equation (1):
Grc ( s) 
 1 1  1  e 
e
1
 sT
 -  

1- e- sT
e -1  2 2  1- e- sT  
- sT
(5)
According to small gain theorem, the stability
condition that makes the system asymptotically stable is
given as:
(6)
Q( z )(1  k G ( z ) H ( z ))  1
R
f
- sT
)1(
Using equation (2), the RC transfer function can be
expressed as equation (3):

(4)

e x  e x
1
1 
2x
x 2
  2
x
 x
2
e e
x k 1 x  k 2
k  x  k
 1 1 1

2s
2s
Grc ( s)  kR      2
 2
 ... 
2
2

 2 T  s s   s  (2 )
)2(
)3(
Comparing equation (3) and Fig. 2, it is easily
concluded that RC is equivalent to a parallel combination
of an integral controller, proportional controller and many
resonant controllers which can be tuned simultaneously
by a simple time delay parameter. It should be noted that
due to the infinite dimension characteristics of the RC, it
is very probable to have an unstable system. This is why
RC controllers must be implemented with a proper low
pass filter in cascade in order to preserve the system
phase and gain margins within the acceptable range [17].
As indicated in Fig. 4, a digital RC is added to a
conventional stable tracking controller Gc(z) in a plug-in
fashion so as to improve the system steady state
performance. In this figure KR, Q(z) and Gf (z) are the RC
gain, low pass filter and the compensating filter
respectively [18].
Fig. 4: digital implementation of RC in a plug-in mode [18]
In Fig. 4, N is the number of samples per period and
Gp(z) is the plant model. It is assumed that closed loop
According to equation (6), the best stability
performance of RC occurs when KR.Gf(z)=H-1(z).
However, since there is error in modeling, it is impossible
to compensate the plant model completely. The phase of
the vector 1− KR.Gf (e jωt ).H(e jωt ) could exceed (−90 ,90)
in high frequency. Therefore, it is possible that the trace
of 1− KR.Gf (e jωt ).H(e jωt ) gets out of the unit circle.
Therefore, a modified repetitive control is adopted in the
system which makes Q(z) < 1, as shown in Fig. 5. Also it
is noteworthy to mention that there is always a trade-off
between the stability margin (the circle radius) and the
error convergence rate (KR). In this paper a zero-phase
moving average low pass filter of Q(z)=0.25z-1 +0.5+0.25z is
chosen as proposed in [19].
Fig. 5: Stability condition for RC
In the case of grid frequency variation, the RC
performance would be degraded leading to the injection
of highly distorted currents exceeding the THD threshold
of 5%. This condition is more likely to happen in weak
grid systems connected to wind farms. Normally the
demands for WT systems are more severe compared to
those applying to PV systems. For example, both voltage
and frequency ranges of operation are much larger in the
case of wind turbines, thus such systems should have a
control strategy that allows them to ride-through when
larger grid variations occur.
3.
Frequency-Adaptive RC
In the repetitive control technique, the ratio of the
sampling frequency to the grid frequency is an integer.
However, this ratio cannot maintain an integer in the case
of frequency variations. In 2002, a kind of adaptive RC
with the fictitious sampling points was proposed [20].
However, the fictitious sampling points are calculated
from practical sampling points by linear-calculation, and
there are errors between them. The repetitive control
method proposed in [13] uses a variable one-order lowpass filter to adapt to the varied grid frequency. Though
its resonant point for the fundamental frequency can be
perfectly matched, the rest resonant points still deviate
from the harmonics. This will decrease the harmonics
compensation performance. Another idea is to design a
proper low pass FIR filter of Q(z) with adjustable linear
characteristics, which means that its delay time is linear
to the frequency [21]. In this paper an efficient, still
simple method of frequency adaptation which was used
in Dynamic Voltage Restore (DVR) [22] is presented. As
shown in Fig. 6, the delay compensator of L(z) is added
to the low pass filter in order to force the time delay to be
equal to the signal period.
KHz which can easily satisfy the IEEE 1547 standard.
detailed parameters of the system are tabulated as below:
Table 1: Parameter values for single phase grid-tie inverter
Utility Voltage (vu)
110V, 50 Hz
Grid Inductance(uh)
50<Lg<200
L1f
1mh
L2f
0.3mh
Cf
6uF
CDC
560uF
Switching Frequency
20 KHz
Sampling Frequency
40 KHz
Fig. 7 shows the system model and the main current
control loop .The main controller of Gc(z) is chosen to be
a simple proportional constant (Kp). Also the filter
capacitor current is fed-back to the controller as a kind of
active damping signal. The main loop control parameters
are optimally designed to yield the damping ratio of
ζ=0.7, with the gain and phase margin of 10.6dB and 57.7
degrees respectively.
Fig. 6: Adaptive frequency RC
L(z) can be simply calculated as equation (9) using
Pade's and Tustin's approximations.
T  NTs  leTs , 0  le  1
(7)
lTs
1 e s
e  NTs S e  leTs s
 leTs s
2
Grc ( s ) 
,e

leTs s
1  e NTs S e  leTs s
1
2
(8)
s
(1  le )  (1  le ) z 1
2 z 1
 L( z ) 
Ts z  1
(1  le )  (1  le ) z 1
Fig. 7: Model of grid-tie inverter with active damping
(9)
As shown in equation (8), the compensating delay
function of L(z) has a unit magnitude which does not
affect the stability of the system. In this approach the
coefficients of L(z) can be updated continuously with
regard to the grid frequency variations.
4. System Description
To evaluate the performance of Adaptive RC a single
phase grid connected inverter with an LCL filter is
designed. The resonance frequency of the filter is
designed to be at least 3.69 kHz (designed based on the
maximum grid inductance) which can provide the
possibility of harmonic compensation up to more than
19th order. Adopting the unipolar switching pattern, the
filter attenuation factor is designed to be -90dB at 40
The reference current is produced by the dc-link
voltage regulator which is designed based on the closed
loop model of the internal current loop. A Second Order
Generalized Integrator Frequency Locked Loop (SOGIFLL) with a settling time of 100ms is adopted not only to
detect the grid frequency but also to produce both Inphase (active current reference) and In-Quadrature
(reactive current reference) signals with respect to the
first harmonic of the grid voltage [1]. The designed
parameters values for RC are tabulated in table 2. The
open loop frequency response of the current loop is
shown in Fig. 8.
Table 2: RC parameters value
KR
0.1
Low pass filter
Q(z)=0.25z-1+0.5+0.25z-1
Phase Compensating Filter
Gf(z)=z3
Fig. 8: Open loop frequency response of the designed current loop (P+RC)
5.
Simulation Results
The simulations have been carried out using
Matlab/Simulink. The utility is concerned to have a THD
of 6.57% with the the harmonic components as below:
Table 3: the utility voltage harmonics at the PCC
h
3
5
7
9
11
13
15
17
19
X%
4.4
3.2
2.8
1.5
1.2
1
0.5
0.6
0.4
In order to evaluate the adaptive RC performance, the
grid frequency is abruptly changed in two steps of 49 Hz
to 51 Hz and 51 Hz to 50 Hz. Fig. 11 shows the grid
frequency and the response of SOGI-FLL. Simulation
results are shown for system with RC+P and Adaptive
RC+P in Fig. 12 and Fig. 13 respectively.
Fig. 9 shows the injected current in the case of using a
proportional
controller
without
any
harmonic
compensation. It is evident that the system has a high
steady state error and the THD of the injected current
does not satisfy the IEEE 1547standard.
Fig. 11: SOGI-FLL response to the grid frequency variations
Fig. 9: simulation results in the case of using a proportional controller
Adding the repetitive controller to the system, results
in a zero steady state error with the current THD of
1.94%, as shown in Fig. 10.
Fig. 12: P+RC response to frequency variations
Fig. 10: The inverter output current and spectrum in the case of using
P+RC
Fig. 13: P+Adaptive RC response to frequency variations
The simulation results of output current THD for the
frequency variations are tabulated in table 4.
Table 4: Output Current THDs for P+RC and P+Adaptive RC in the
case of frequency variations.
Frequency
P+RC
P+Adaptive RC
49
7.2%
2.15%
50
2.36%
2.16%
51
7.9%
2.17%
6. Conclusion
In this paper a brief survey on different current control
strategies was given. Having analysed the general
repetitive controllers, a kind simple adaptive repetitive
controller was proposed for grid-tie inverters. The
proposed Repetitive controller can tune all the resonant
frequencies simultaneously via the adaptation of the time
delay. As it was shown in the simulations the
performance of repetitive controllers can degrade
significantly in the case of grid frequency variations
while in the adaptive repetitive scheme the high
performance of the current controller is still preserved.
Acknowledgements
The authors would like to express their sincere thanks
to Isfahan Regional Electric Company for their valuable
financial supports.
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