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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 56, NO. 2, FEBRUARY 2009
Limitations of Voltage-Oriented PI Current Control
of Grid-Connected PWM Rectifiers
With LCL Filters
Joerg Dannehl, Student Member, IEEE, Christian Wessels, Student Member, IEEE, and
Friedrich Wilhelm Fuchs, Senior Member, IEEE
Abstract—Voltage-oriented PI control of three-phase gridconnected pulsewidth-modulation rectifiers with LCL filters is
addressed. LCL filters require resonance damping. Active resonance damping is state of the art to face the problem, but it is still
under investigation because of the manifold solutions. It is often
realized using many sensors and/or complex control algorithms. In
contrast, pure PI control requires only one set of current sensors,
and its implementation and design are rather simple and well
known from the L filter control. PI control has already been
shown to be a suitable solution also for LCL filters, but there are
limitations. These are investigated in this paper. System stability is
analyzed with respect to different ratios of LCL filter resonance
and control frequencies. The latter are important parameters
for system design and control. Both line and converter current
control are analyzed. For a certain range of frequency ratios, the
voltage-oriented PI control gives stable performance without additional feedback, but for ratios outside this range, stable operation
is impossible. Experimental tests validate the theoretical results.
In addition, an experimentally determined LCL filter transfer
function is shown in this paper, which shows a lower resonance
peak as expected from commonly used filter models.
Index Terms—Active damping (AD), grid-connected
pulsewidth-modulation (PWM) rectifier, LCL filter, PI current
control.
I. I NTRODUCTION
T
HREE-PHASE grid-connected pulsewidth-modulation
(PWM) rectifiers are often used in regenerative energy
systems and in adjustable-speed drives when regenerative braking is required [1]–[5]. Aside from power regeneration, they
offer control of the power factor as well as the dc link voltage
while emitting lower current harmonics to the grid than passive
diode rectifier bridges. A cascaded control structure with an
outer dc link voltage control and inner current control loops
is commonly used. For simple L filter grid connections, the
current control is mostly done with PI controllers in linevoltage-oriented coordinates and is well known [6]. Beyond
L filters, LCL filters are used for grid connections [1]. They
Manuscript received December 10, 2007; revised October 10, 2008. First
published November 7, 2008; current version published January 30, 2009. This
work was supported by the German Research Foundation (DFG).
The authors are with the Institute for Power Electronics and Electrical
Drives, Christian-Albrechts-University of Kiel, 24143 Kiel, Germany (e-mail:
dannehl@ieee.org).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIE.2008.2008774
offer advantages in terms of costs and dynamics, since smaller
inductors can be used than in L filters in order to achieve
the necessary damping of the switching harmonics [7]. One
drawback is that the filters can oscillate with the filter resonance
frequency. Even if passive damping of the resonance using
resistors in series with the filter capacitors is possible [7], [8],
active damping (AD) by control is preferable due to lower
power losses and more flexibility [9]–[24].
Many different AD methods have been discussed in the
literature. In [9]–[12], converter current control with additional
feedback of the voltages across the filter capacitors is shown. In
[13] and [14], line current control with additional feedback of
the current through the filter capacitors is shown. An overview
of different multiloop approaches can be found in [15] and [16].
In [17]–[19], the complete state information is used for control.
All of these approaches require measurements of more than one
state quantity in order to achieve the necessary damping. Due
to the additional hardware required, the costs are increased and
the reliability is decreased. Furthermore, the complexity of the
control algorithms is increased. Even if some values could be
estimated instead of measured [11], [20] the control complexity
would be high.
Because voltage-oriented PI current control is well known
from L-type grid connections and a minimal number of sensors
are desirable for the LCL type as well, some approaches
with only one set of current sensors have been published and
analyzed [21]–[24]. Either the line currents or the converter
currents are controlled. On the one hand, the use of line current
feedback is reasonable since one of the main objectives is the
control of power factor at the point of the grid connection. The
line current phase angle can be directly controlled. On the other
hand, feeding back the converter-side currents is reasonable
if the current sensors are already built into the converter, for
example, in industrial units for protection purpose. In the latter
case, the filter phase shift must be compensated in order to set
the power factor on the grid side. For certain combinations of
filter settings and control frequencies, the applicability of pure
PI control has been confirmed. However, the question of general
applicability and limitations remains open and is studied in this
paper.
This paper deals with the application of the well-known
voltage-oriented PI control method to LCL filter-based systems
using only one set of current sensors. Either line or converter
currents are used for feedback. Stability is emphasized with
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DANNEHL et al.: LIMITATIONS OF VOLTAGE-ORIENTED PI CURRENT CONTROL OF PWM RECTIFIERS
Fig. 1.
381
PWM rectifier with LCL filter (control based on iL or iC ).
TABLE I
SYSTEM PARAMETERS
Fig. 2.
filters.
respect to different ratios of control frequency and resonance
frequency of the LCL filters. These are important parameters
from a system design and control point of view. The control
frequency corresponds to the maximum achievable control
bandwidth, whereas the resonance frequency influences the
damping of the switching ripple current. With lower resonance
frequencies, higher damping can be achieved. Both are underlying certain constraints such as losses in the power module
or the size of passive components, for example, and cannot be
chosen arbitrarily. However, this subject is beyond the scope of
this paper. In the system design phase, it may be important to
know if control of the designed system can be achieved with one
set of current sensors and without complex control algorithms.
Therefore, the stability of voltage-oriented PI control as applied
to the LCL system with respect to different ratios between control frequency and resonance frequency is investigated in the
following. The first is varied, and the latter is kept constant, but
the results can easily be applied to other filter parameters and
control frequencies. In particular, limitations in the applicability
of PI control are investigated.
System description and modeling are shown in Section II.
The control overview and design are given in Section III. In
Section IV, the stability of voltage-oriented PI current control
is analyzed. Measurement results are presented and analyzed in
Section V. The final section presents the conclusions.
Transfer functions (converter output voltage to line current) of line
measured. The line voltages are measured for the purpose of
synchronizing the control with the line voltage. Line voltage
sensorless operation is possible [11] but beyond the scope of
this paper. The PWM rectifier is loaded by an inverter-fed
induction machine. In this paper, different converter switching
frequencies fc ’s are used.
A. Model for Control Design
For the control design, the grid is modeled as an ideal
sinusoidal three-phase voltage source without line impedances,
although, in reality, there are line impedances and distortions
like harmonics and unbalances in the line voltages [22], [25].
The space vector notation is used [26]. The three-phase values
are transformed into a two-phase stationary reference frame.
These are transformed into the dq-reference frame that rotates
synchronously with the line voltage vector in order to design
the voltage-oriented control. From a control point of view,
it is advantageous to control dc values since a PI controller
can achieve reference tracking without steady-state errors. The
parameters of the LCL filter can be found in Table I. The
series resistances of the inductors, modeling the copper losses,
were measured as Rf g = 50 mΩ and Rf c = 60 mΩ. Modeling
the LCL filters in the dq-reference frame without frequency
dependences of the inductors gives (1). The filter capacitor
voltage is defined as vCf .
Lf g
Cf
dv dq
Cf
II. S YSTEM D ESCRIPTION AND M ODELING
The analyzed system is shown in Fig. 1, and system parameters are listed in Table I. A three-phase IGBT voltage
source converter is connected to the grid through a grid-side
LCL filter. The dc link voltage and, depending on the control
structure, either the line currents or the converter currents are
didq
dq
dq
L
= v dq
L − v Cf − (Rf g + jωLf g )iL
dt
Lf c
dt
dq
dq
= idq
L − iC − jωCf v Cf
didq
dq
dq
C
= v dq
Cf − v C − (Rf c + jωLf c )iC .
dt
(1)
Fig. 2 shows the frequency behavior consistent with (1) from
the converter output voltage to the line current for the LCL
and an L filter. As the inductance of the L filter equals the sum
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of the inductances of the LCL filter, both show the same lowfrequency behavior. Obviously, the LCL filters yield higher
damping of the higher frequency components, whereas a strong
amplification at the resonance frequency fRes appears if the
frequency-dependent losses are neglected
Lf g + Lf c
1
.
(2)
fRes =
2π Cf Lf g Lf c
As shown in [24], if the losses in the filter and converter are
neglected, the dc link voltage dynamics can be expressed by
dvDC
3 iLd vLd
CDC
= iDC − iLoad =
− iLoad .
dt
2 vDC
(3)
B. Expanded Model for Stability Analysis
The system model in (1) is commonly used for stability
analysis as well, but experimental results will show a more
stable behavior of the LCL filter than expected from the simple
model shown in (1). For this reason, the frequency behavior of
the LCL filter is experimentally determined. For this purpose,
the LCL filter is connected to separate measurement equipment. The grid-side inductors are shortened at the grid side,
and at the converter side, a voltage is supplied by a converter
that can deliver voltages with fundamental frequencies of up to
several kilohertz. The current through Lf g at the corresponding
frequency is adjusted to reach a peak current of 5 A. The
resulting voltage on the converter side is measured, and the
impedance at the measured frequency is calculated.
The measured data are shown in Fig. 2, and it is clear that
there is a lower resonance peak. Therefore, the model from
(1) is expanded by resistances (RFE,g = 140 Ω and RFE,c =
140 Ω) in parallel to the ideal inductances in order to model the
additional frequency-dependent losses. The values are determined by parameter fitting to the measured frequency behavior.
The frequency behavior of the expanded model of the LCL
filter is shown in Fig. 2 as well (referred to as LCL extended).
Due to the discrete-time nature of the control algorithm
implementation, the stability analyses in this paper are also performed discretely in the z-domain [27]. The expanded model of
the LCL filter, including losses as well as the line impedance
and PI controller, is discretized by a zero-order hold [27]
with the sampling frequency fc . As will be shown later, the
control contains simple decoupling terms in order to decouple
the d- and q-current dynamics. Even if no perfect dynamic
decoupling can be achieved due to delays in the loop and the
filter resonance, the couplings between the d- and q-axes are
neglected for the stability analyses. These are performed in
voltage-oriented coordinates for the d-axis.
Fig. 3.
Overview of complete control structure (γL : line voltage phase angle).
synchronous reference frame that is aligned to the line voltage
vector. Either the line currents or the converter currents will
be controlled. A detailed design and analysis is shown in the
following sections. The determination of the line voltage phase
angle is done by a phase-locked loop (PLL) algorithm. A survey
of different PLL solutions can be found in [30]. Note that
the orientation to the so-called virtual flux is also possible,
particularly if voltage-sensorless operation is desired [11].
The control algorithm is executed once per switching period,
and the sampling is performed at the same rate at the middle
of each switching period in order to eliminate switching ripple
in the measured data. Therefore, there is no difference between
the switching and the control frequency in this analysis.
As the dc link voltage and dq currents are constant in steady
state, the PI controller can achieve reference tracking with zero
steady-state errors. Here, the PI controllers with proportional
gains k’s and integrator time constants Ti ’s are used as defined
in (4). In addition, an antiwindup mechanism is used for each
PI controller to prevent windup problems in case of limitation
of the current or voltage references [28]
GPI (s) = k
sTi + 1
.
sTi
(4)
B. DC Link Voltage Control
The design of the dc-link voltage PI controller parameters
(kDC , TDC ) takes into account the inner current loop in terms
of a four-sample delay (Tinner = 4 Tc ). Assuming that the
dc link voltage is close to its constant reference, the PI controller can be tuned with symmetrical optimum [29]
kDC =
∗
CDC
2 · VDC
3 · aDC Tinner vLd
TDC = a2DC Tinner
aDC = 3.
(5)
C. Voltage-Oriented Current Control
III. C ONTROL O VERVIEW AND D ESIGN
A. Overview
The cascaded control structure is shown in Fig. 3. The outer
∗
loop regulates the dc link voltage to a constant reference VDC
.
The inner loops control the active and reactive currents in
The current control structure under investigation is shown in
Fig. 4 (left). In order to decouple the d- and q-current dynamics,
decoupling terms are added to the PI controller.
The PI controller design is done with an L approximation of
the LCL filter (Lf = Lf g + Lf c ) as shown in [7]. The lowfrequency behavior of the LCL filter is similar to an L filter as
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DANNEHL et al.: LIMITATIONS OF VOLTAGE-ORIENTED PI CURRENT CONTROL OF PWM RECTIFIERS
383
Fig. 4. Current control structures. (Left) PI current control structure with decoupling. (Right) Converter PI control structure including additional damping and
decoupling (Lf = Lf g + Lf c ).
can be seen in Fig. 2. Assuming that the d- and q-current dynamics are decoupled, the filter behaves like a first-order delay
element for both components. In addition, the line voltage is
treated as disturbance and is not taken into consideration during
the control design process. Therefore, the same parameters can
be used for the d- and q-current controllers. For the control
design, the delays in the loop caused by analog-to-digital conversion, computation, and PWM are taken into account in terms
of a one-sample delay. The PI controller parameters (kI , TI )
are first tuned using the symmetrical optimum [29], and after
that, kI is slightly increased in order to achieve the maximum
bandwidth. This gives
kI = kI,opt = −Lf /(2Tc ) TI = a2I Tc
aI = 3.
(6)
IV. S TABILITY A NALYSIS
In this section, the PI controller that has been designed is
first applied to an L filter-based system and, afterward, to the
LCL filter-based system either with line or converter current
feedback. Stability analyses are carried out in the discrete
z-domain using the expanded system model that incorporates
the frequency dependences of the filter in accordance with
Section II-B.
A. L Filter-Based System
The application of the PI control to an L filter-based PWM
rectifier yields the root locus of the discretized d-current loop
shown in Fig. 5. It shows the location of the closed-loop poles
depending on the proportional gain kI of the controller. The
pole locations with kI = kI,opt are highlighted in Fig. 5. One
pole is compensated by the PI controller zero, and the two
others together build a complex conjugate pole pair. The grid
of the root locus shows the frequency to which the poles are
shifted by the control as well as the damping factor of the
poles. Fig. 5 is valid for all of the analyzed control frequencies
since the frequency scale is related to the control frequency.
The control bandwidth is determined to approximately fc /5.
Increasing the proportional gain would cause more oscillatory
behavior without increasing the control bandwidth significantly.
Therefore, the bandwidth of fc /5 is considered the maximum
achievable for each control frequency for the L filter system.
Fig. 5. Root locus of PI-controlled d-current dynamics of the PWM rectifier
with L filter valid for different control frequencies (sampling period T = Tc ;
highlighted pole locations: kI = kI,opt ).
B. LCL Filter-Based System With Line Current Feedback
Applying the voltage-oriented PI control to the line current
control of the LCL filter-based PWM rectifier gives the root
loci shown in Fig. 6. In Fig. 6(a), the closed-loop poles for
low control frequency are shown. Comparison with the root
locus obtained with the L filter (see Fig. 5) shows that the lowfrequency pole branches are only slightly different. In addition,
the resonance poles are visible at the left of the figure, and
they are attracted inside the unity circle. Thus, stable operation
can be achieved. The control bandwidth achieved with an
L filter can be calculated as 700 Hz. In this frequency range,
the transfer functions of the different filters in Fig. 2 show
almost the same behavior. The approximation of the LCL filter
as an L filter is justified in this case. Note that the losses are
advantageous with respect to resonance damping. With lower
losses, the resonance poles would get closer to the unity circle
for zero gain. For line current feedback, this would not cause
any problems as the poles are attracted inside the unity circle
for nonzero PI gains.
It is expected that while increasing the control bandwidth by
increasing the control frequency, the stability behavior might
change due to the lower accuracy of the L filter approximation.
A control frequency of 5 kHz yields a bandwidth of 1 kHz for
an L filter. In this frequency range, the LCL filter behavior
deviates considerably from the L filter, as can be seen in Fig. 2.
Fig. 6(b) shows that a higher control frequency moves the
resonance poles toward the low-frequency poles in the root
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Fig. 6. Root loci of the PI-controlled d-axis dynamics of the PWM rectifier with LCL filter using the line currents as feedback signals for different control
frequencies. (a) fc = 3.5 kHz, (b) fc = 5 kHz, and (c) fc = 7 kHz (sampling period T = Tc ; highlighted pole locations: kI = kI,opt ).
Fig. 7. Root loci of the PI-controlled d-axis dynamics of the PWM rectifier (without additional AD) with LCL filter using the converter currents as feedback
signals for different control frequencies. (a) fc = 3.5 kHz, (b) fc = 5 kHz, and (c) fc = 7 kHz (sampling period T = Tc ; highlighted pole locations:
kI = kI,opt ).
Fig. 8. Root loci of the PI-controlled d-axis dynamics of the PWM rectifier (with additional AD) with LCL filter using the converter currents as feedback signals
for different control frequencies. (a) fc = 3.5 kHz, (b) fc = 5 kHz, and (c) fc = 7 kHz (sampling period T = Tc ; highlighted pole locations: kI = kI,opt ; for
clarification, poles in range of resonance are not highlighted for case c).
locus (note that the frequency grid is scaled by the control
frequency). As a result, the low-frequency branches get more
affected by the resonance, and vice versa. However, still a stable
system is obtained. The gain margin is slightly reduced.
As can be seen in Fig. 6(c), the system finally gets unstable
for almost all proportional gains by further increasing the
control frequency. The bandwidth achieved with the L filter at
a control frequency of 7 kHz equals 1.4 kHz, which is in the
same range as the resonance frequency of the filter. The L filter
approximation is no longer valid, and the PI control becomes
unsuitable. Without the filter losses, no stable pole locations
can be identified.
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385
The analysis of line current feedback showed that, as long
as the maximum achievable control bandwidth is lower than
the resonance frequency, the PWM converter with LCL filters
can be controlled with the voltage-oriented PI control with
line current feedback. It can be concluded that the following
condition must hold for stability (including a suitable safety
margin):
fc
fc
> fRes > .
2
4
(7)
Stability problems arise if higher control frequencies are used
in order to achieve a higher control bandwidth. Similar analyses
(not shown in this paper) show that the same problem occurs if
the control frequency is kept constant and an LCL filter with
a lower resonance frequency is used in order to achieve better
damping of the switching ripple.
C. LCL Filter-Based System With
Converter Current Feedback
In the same manner as previously discussed for line current
control, the voltage-oriented PI control in Fig. 4 (left) can be
applied to the converter current control. As already explained,
in this case, the filter phase shift has to be compensated for
the unity power factor on the grid side, and therefore, the
reference value of the q-current becomes nonzero [7]. Using
the control from Fig. 4 (left) with converter current feedback
yields the root loci shown in Fig. 7. It becomes clear that at all
possible ratios of control and resonance frequencies, the system
stability is worse. The resonance poles are pushed toward the
outside of the unity circle. Without losses, the system would
be unstable for all proportional gains in any case. Therefore,
additional resonance damping is necessary. In [22] and [24],
an additional damping approach is used, which is applied here
as well.
The modified control structure is shown in Fig. 4 (right).
For a unity steady-state gain of the additional damping block,
the gain kAD is set to (1 − 2Re{p0 } + |p0 |2 )/(1 − 2Re{z0 } +
|z0 |2 ) in order to enable the use of the proportional gain shown
in (6). The main idea of the control modification is to add in
the open-loop zeros and poles around the resonance poles and
zeros of the system. By doing this, the resonance poles are
attracted inside the unity circle (see Fig. 8). It is clear that the
low-frequency pole branches are less affected by the resonance
poles and zeros. However, there is a limited area around the
system resonance poles and zeros in which the AD poles and
zeros can be placed for the purpose of improving the system
stability. It is this area that is affected by the low-frequency
poles. With fc = 3.5 kHz, the area is large. By increasing the
control frequency, the interactions of the AD poles and zeros
with the system poles and zeros increase. This, in turn, reduces
the area in which the AD poles and zeros can be placed. Finally,
the AD poles and zeros have to be placed more or less directly
next to the system poles and zeros.
In order to place the damping poles and zeros appropriately,
the locations of the system resonance poles and zeros have
to be known accurately. Hence, accurate information about
the effective LCL filter resonance frequency taking the line
Fig. 9. Line current feedback: Measured steady state with fc = 3.5 kHz and
optimally tuned PI gain (kI = kI,opt ; N = 1450 r/min). (Upper) Waveforms
of converter current (Ch 2, 10 A/Div) and line current (Ch 4, 10 A/Div). (Lower)
Line current frequency spectrum.
inductance into account, as well as their damping factors, which
corresponds to the copper- and frequency-dependent losses of
the filter elements, is needed. However, such data are commonly
not available. The closer the AD poles and zeros have to be
placed to the system poles and zeros, the higher the required accuracy of system data. Parameter uncertainties can lead to instability very easily, for instance, if the line impedance varies [24].
The analysis of converter current feedback showed that the
voltage-oriented PI control as derived from the L filter control
cannot be applied to the converter current control without
modification. However, using an additional active resonance
damping function yields a stable system for a certain range
of ratios of control and resonance frequencies. For the range
already given in (7), the system can be stabilized using the
approach demonstrated. However, it becomes very difficult to
design the AD if higher control frequencies are used because
the locations of the system poles and zeros have to be known
very accurately. In addition, the sensitivity to parameter variations increases considerably.
V. M EASUREMENT R ESULTS
Measurements are carried out in order to validate the theoretical analysis of the voltage-oriented control of PWM rectifiers
with LCL filters. For this purpose, the PWM rectifier system
shown in Fig. 1 was constructed in a laboratory environment.
The control algorithm is implemented on a dSPACE DS 1006
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Fig. 10. Line current feedback: Measured steady-state currents with fc =
3.5 kHz and increased PI gain (kI = 1.7 kI,opt ; N = 1450 r/min). (Upper)
Waveforms of converter current (Ch 2, 10 A/Div) and line current (Ch 4,
10 A/Div). (Lower) Line current frequency spectrum.
Fig. 11. Line current feedback: Measured steady-state currents with fc =
7 kHz and small PI gain (kI = 0.45 kI,opt ; N = 1000 r/min). (Upper)
Waveforms of converter current (Ch 2, 10 A/Div) and line current (Ch 4,
10 A/Div). (Lower) Line current frequency spectrum.
board. The PWM rectifier is loaded by an inverter-fed fourpole induction motor. For this purpose, the inverter part of
an industrial drive is connected to the dc link. The effective
dc link capacitance is 4450 μF. Motor switching is executed
not synchronously with the PWM rectifier but with the same
frequency. Tests are performed at line-to-line voltages of 400 V
(rms). The dc link voltage is controlled to 700 V. The LCL filter
parameters from the theoretical analysis are used (see Table I).
Unless otherwise stated, all tests are carried out using a nominal
motor speed of 1450 r/min. The load torque generated by a
dc machine is adjusted in order to maintain the nominal line
current. Harmonic compensators on the grid side, as shown in
[31], are not used.
resonance poles. This would also be the case for an L filter
system (see Fig. 5).
Fig. 11 shows the steady-state current waveforms and the line
current spectrum obtained with a 7-kHz control frequency. As
expected from the root locus in Fig. 6, stable operation cannot
be achieved with reasonable proportional gains. Even with half
the optimal proportional gain, heavy oscillations at 1.1 kHz are
clearly visible in Fig. 11. Note that the stability margin would
further decrease with loss-reduced inductors.
A. Line Current Control
Figs. 9 and 10 show the results obtained using a 3.5-kHz
control frequency and various PI gains. In Fig. 9, the steadystate current waveforms and the line current spectrum show
that the resonance is well damped for an optimally tuned PI
gain. Certain low-frequency components (5th and 7th) caused
by distorted line voltages are visible, and this could be reduced
by using an additional harmonic controller [31]. By increasing
the proportional PI gain, the system can approach the stability
limit as shown in Fig. 10. Oscillations at 800–900 Hz appear
due to the low-frequency poles leaving the unity circle, as can
be seen in the root locus in Fig. 6(a). These are not due to the
B. Converter Current Control
Converter current control results obtained at a 3.5-kHz control frequency are shown in Figs. 12 and 13. Due to the high
sensitivity of the control to model uncertainties, such as line
impedances and inductor losses, the AD poles and zeros in
Fig. 8 have to be modified for the experimental tests in order to
stabilize the system. As already mentioned in [22], placing the
AD zeros near the resonance pole and optimizing the location
of the AD poles yield a stable system.
First, the AD is disabled, and with kI = 0.8 kI,opt , the
resonance appears as shown in the left of Fig. 12(a). It is clear
that the resonance gets damped quickly and effectively after
the activation of AD. In Fig. 13, line current spectra with and
without AD also confirm the resonance damping effect. As
expected from Fig. 8(a), the system can approach its stability
limit by increasing the proportional gain of the PI controller.
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Fig. 13. Converter current feedback: Measured line current frequency spectra
with fc = 3.5 kHz. (a) AD off and kI = 0.8 kI,opt . (b) AD on and kI =
0.8 kI,opt . (c) AD on and kI = 1.9 kI,opt .
Fig. 12. Converter current feedback: Measured converter and line current
waveforms with fc = 3.5 kHz (Ch 2: Converter current, 20 A/Div, Ch 4: Line
current, 20 A/Div, Ch 3: Status of AD: high: AD on/ low: AD off). (a) Effect of
AD: Left without AD and right with AD (kI = 0.8 kI,opt ). (b) Steady-state
currents with AD and increased PI gain (kI = 1.9 kI,opt ).
This is shown in Fig. 12(b). Again, the oscillations are due to
low-frequency poles leaving the unity circle at higher PI gains
[see Fig. 8(a)].
As already explained, designing the AD in a laboratory setup
is difficult. It becomes even more challenging for low ratios
of the resonance and control frequencies. In theory, the AD
poles and zeros have to be placed close to the system poles and
zeros, but model uncertainties lead to instability very readily.
Therefore, no results for 5 and 7 kHz are obtained in practical
application.
VI. C ONCLUSION
Voltage-oriented PI current control has been extensively
studied in the context of grid-connected PWM rectifiers with
L filters. In this paper, its general applicability for current
control of LCL filter-based systems was analyzed. Either
line or converter currents can be used for feedback. Both
possibilities are considered and analyzed. Control design and
analysis are performed separately with different system models.
The stability is analyzed for different ratios between control
and resonance frequencies of the LCL filters. The control
frequency is varied, whereas the resonance frequency remains
constant. Theoretical analyses are performed in the frequency
domain and are validated via experimental tests in a laboratory
environment.
It is concluded that voltage-oriented PI control can be used
even for LCL filters without passive damping, as long as
the resonance frequency is less than half and above a quarter
of the control frequency. For line current control, it can be
used without modification. Converter current control requires
an additional active resonance damping function which can
be implemented without additional sensors. Tuning of the AD
function for converter current control turned out to be difficult
in the practical application. In addition, the control system is
very sensitive to model uncertainties. Moreover, the measured
frequency behavior of the LCL filter indicates a lower resonance peak, as expected from commonly used filter models.
Hence, the model for stability analysis is expanded by including
frequency-dependent losses.
For specific applications, control frequencies higher than
four times the resonance frequency may be necessary in order to
achieve a higher control bandwidth. In this case, advanced control algorithms are necessary. They are commonly implemented
using additional sensors or observer solutions.
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Joerg Dannehl (S’06) was born in Flensburg,
Germany, in 1980. He received the Dipl.-Ing. degree
from the Christian-Albrechts-University of Kiel,
Kiel, Germany, in 2005.
Since 2005, he has been a Research Assistant with
the Institute for Power Electronics and Electrical
Drives, Christian-Albrechts-University of Kiel. His
main research interests include control of power
converters and drives.
Mr. Dannehl is a Student Member of the IEEE
Industrial Electronics Society.
Christian Wessels (S’08) was born in Hamburg,
Germany, in 1981. He received the Dipl.-Ing. degree
from the Christian-Albrechts-University of Kiel,
Kiel, Germany, in 2007.
Since 2007, he has been a Research Assistant with
the Institute for Power Electronics and Electrical
Drives, Christian-Albrechts-University of Kiel. His
main research interests include control of power
converters and renewable energies.
Mr. Wessels is a Student Member of the IEEE
Power and Energy Society.
Friedrich Wilhelm Fuchs (M’96–SM’01) was born
in Minden, Germany, in 1948. He received the Dipl.Ing. and Ph.D. degrees from the RWTH University
of Technology Aachen, Aachen, Germany, in 1975
and 1982, respectively.
In 1975, he carried out research work at the University in Aachen, mainly on ac drives for batterypowered electric vehicles. Between 1982 and 1991,
he was a Group Manager in the field of power electronics and electrical drives in a medium-sized company. In 1991, he joined the Converter Division of
AEG, Berlin, Germany (currently known as Converteam), where he was a Managing Director for research, design, and development of the complete range of
drive products, drive systems, and high-power supplies from 5 kVA to 50 MVA.
Since 1996, he has been with the Faculty of Engineering at the ChristianAlbrechts-University of Kiel, Kiel, Germany, as a full Professor, where he is
currently the Head of the Institute for Power Electronics and Electrical Drives,
which he and his team have expanded. His research interests include power
semiconductor applications, converters and their control, and variable-speed
drives. There is special focus on application to renewable energy, particularly
wind energy, on state-space and nonlinear control as well as on diagnosis and
fault-tolerant drives. He has authored or coauthored more than 80 papers.
Dr. Fuchs is a member of the Association of German Electrical and Electronics Engineers (VDE) and the European Power Electronics Association (EPE).
He is a convener and international speaker of the German standardization
committee K33 1 (TC22) for power electronics. His institute is a member of
the Cewind competence center of wind energy, Schleswig-Holstein, Germany.
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