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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 18, NO. 3, MAY 2003
Grid Current Regulation of a Three-Phase Voltage
Source Inverter With an LCL Input Filter
Erika Twining, Student Member, IEEE, and Donald Grahame Holmes, Member, IEEE
Abstract—Many grid connected power electronic systems, such
as STATCOMs, UPFCs, and distributed generation system interfaces, use a voltage source inverter (VSI) connected to the supply
network through a filter. This filter, typically a series inductance,
acts to reduce the switching harmonics entering the distribution
network. An alternative filter is a LCL network, which can achieve
reduced levels of harmonic distortion at lower switching frequencies and with less inductance, and therefore has potential benefits for higher power applications. However, systems incorporating
LCL filters require more complex control strategies and are not
commonly presented in literature.
This paper proposes a robust strategy for regulating the grid
current entering a distribution network from a three-phase VSI
system connected via a LCL filter. The strategy integrates an outer
loop grid current regulator with inner capacitor current regulation
to stabilize the system. A synchronous frame PI current regulation strategy is used for the outer grid current control loop. Linear
analysis, simulation, and experimental results are used to verify
the stability of the control algorithm across a range of operating
conditions. Finally, expressions for “harmonic impedance” of the
system are derived to study the effects of supply voltage distortion
on the harmonic performance of the system.
Index Terms—Current regulation, grid connection, harmonic
distortion, LCL filter, linear analysis, voltage source inverter .
I. INTRODUCTION
P
OWER electronic converters are now used in many
grid-connected applications including STATCOMs,
UPFCs, and active interfaces for distributed generation systems
(e.g., PV, wind etc.). These converters are commonly based on a
voltage source inverter (VSI) connected to the supply network,
operated to achieve objectives such as power flow regulation
or power factor optimization by regulating the current into the
grid using schemes such as synchronous frame controllers, Predictive Current deadbeat control, or hysteresis-based strategies.
Typically, simple series inductors are used as the filter interface
between the VSI and the grid network. However, these filters
require high switching frequencies to acceptably attenuate
switching harmonics, particularly in weak-grid applications
where the supply is sensitive to these harmonics.
In contrast, the alternative LCL form of low-pass filter
offers the potential for improved harmonic performance at
lower switching frequencies, which is a significant advantage
Manuscript received June 14, 2002; revised November 1, 2002. This work
was supported by the Australian Research Council. This paper was presented
at PESC’02, Cairns, Australia, June 23–27, 2002. Recommended by Associate
Editor J. H. R. Enslin.
The authors are with the Department of Electrical and Computer Systems
Engineering, Monash University, Clayton Campus, Victoria 3800, Australia
(e-mail: erika.twining@eng.monash.edu.au).
Digital Object Identifier 10.1109/TPEL.2003.810838
in higher-power applications [1], [2]. However, systems incorporating LCL filters require more complex current control
strategies to maintain system stability, and are more susceptible
to interference caused by grid voltage harmonics because
of resonance hazards and the lower harmonic impedance
presented to the grid.
Reference [3] has shown how an inner “lag-lead” compensation loop on the capacitor voltage of the LCL filter of a threephase grid-connected VSI actively damps the filter resonance
and improves the stability of the control system. Similar results have been achieved for single and three-phase grid-connected VSI systems using an inner capacitor current feedback
loop [4], and in a number of single-phase uninterruptible power
supply (UPS) applications [5]. However, these systems control
the filter capacitor output voltage rather than the grid current
and are therefore not directly applicable for grid power flow
control. Multi-variable control strategies have been proposed to
regulate the grid current for VSIs connected through LCL filters
[1], but these strategies are complex and sensitive to variations
in system parameters. More recently, an analytical study of grid
connected active rectifiers with LCL input filters incorporating
PI-based controls has been presented [2]. However, this work
still only considers the regulation of the ac current out of the inverter, rather than the current into the grid after the filter.
This paper proposes a robust strategy for regulating the grid
current of a converter connected to an electrical network through
a LCL filter. The essence of the scheme is to use a synchronous
frame PI (SRFPI) controller to regulate the grid current, together
with a simple inner capacitor current regulating loop to stabilise
the system.
To determine the transient performance of the system, a
P Resonant controller is considered first. Unlike the SRFPI
controller, the P Resonant controller can be easily reduced to
a single-phase equivalent system so that conventional stability
analysis techniques may be applied. Stable operation of this
controller at fundamental frequency is confirmed using a
linearised model of the inverter/grid system. Then, using the
knowledge that the P Resonant controller has similar performance characteristics to the SRFPI controller [6], stability
analysis for this controller is shown to be sufficient to predict
the stability of a SRFPI controller.
Next, reduced quality current regulation caused by grid
supply voltage harmonics is investigated by determining
the harmonic impedance of the proposed control strategy,
and methods of tuning the current regulator to mitigate this
distortion are considered.
Finally, the experimental results obtained using a DSP control
platform are presented to verify the robustness of the proposed
0885-8993/03$17.00 © 2003 IEEE
TWINING AND HOLMES: GRID CURRENT REGULATION
Fig. 1.
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Three-phase VSI system.
control algorithm and to study the system’s harmonic performance at a practical level.
II. SYSTEM MODELLING
Fig. 1 shows the converter system considered in this investigation, comprising a standard three-phase VSI driven from a
constant voltage DC bus and connected to the grid through a
LCL filter. Note that the assumption of a constant dc voltage
is reasonable if the dc capacitance is large or if dc bus voltage
ripple compensation is included within the PWM control algorithm. For this investigation, it is further assumed that the system
is balanced, and that the VSI switching frequency is sufficiently
high that it will have negligible effect on the inverter control
loop dynamics.
Under these assumptions, the converter system can be represented using a linearized “average switching model (ASM),”
where the inverter switches are replaced by a function representing their averaged value over each carrier interval. Provided
the controller does not saturate the VSI output, this linearised
inverter model has been shown to achieve very accurate results
in this type of application [7]. Furthermore, the ASM approach
allows classical stability analysis techniques to be used to investigate the system fundamental, transient and harmonic responses, and it is easily implemented in a simulation package
such as Matlab Simulink with greatly reduced simulation times
compared to full switched models.
III. CONTROL STRATEGY
The primary aim of the control scheme is to modulate the
inverter to regulate the magnitude and phase angle of the grid
supply current, so that the real and reactive power entering the
network can be controlled. It is known already that direct feed-
back control of the grid current is unstable [4]. However, control
of this current should be possible in the same way as voltage
regulation for a UPS is achieved [5], by making an outer grid
current feedback controller drive an inner capacitor current regulating loop.
Fig. 2 shows an ASM single-phase equivalent representation
of the controller/inverter system, where
represents the grid
current reference signal. From this figure, it can be seen how the
“outer” grid current feedback loop provides a reference value,
, to the “inner” capacitor current feedback loop and the output
of the inner loop then determines the VSI output voltage, .
Representing the system with this single-phase model allows it
to be analyzed using conventional stability analysis techniques.
The inner loop has a simple proportional gain transfer function, since it only stabilises the control system and its steadystate errors do not affect the accuracy of the outer control loop.
The outer loop is shown in Fig. 2 as a generic transfer func. This transfer function could be the single phase
tion
equivalent of a SRFPI controller, but this has the complication
of being difficult to model because of dependencies between
the three phases. For simplicity, it would be preferable to use a
current regulation scheme that is independent between phases,
such as a P Resonant controller, which is equally applicable
to single or three phase systems. This also has the benefit that
since these two controllers are known to have almost identical
stationary frame performance characteristics [6], single-phase
stability analysis of a P Resonant scheme can then directly be
applied to determine the transient performance of a SRFPI controller. This is the approach used here.
A. SRFPI Controller Model
SRFPI controllers are commonly used in three-phase systems
and operate by transforming the three-phase ac currents into
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Fig. 2.
IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 18, NO. 3, MAY 2003
Single-phase representation of proposed control strategy (notation defined in Fig. 1).
DC components in the synchronously rotating frame so that the
steady-state error that is normally associated with the application of PI control to ac quantities can be eliminated [6]. The
strategy also has the particular advantage of independent control of the real and reactive current components, which translates
directly to real and reactive power flow control. This is advantageous for this application, where the grid supply currents can be
directly regulated in the synchronous frame and transferred back
into the stationary frame to provide references for the simple
inner loop (single-phase) capacitor current controllers as shown
in Fig. 3.
The transfer function of a SRFPI controller in the rotating
frame is given by
(1)
Fig. 3. Synchronous reference frame PI control strategy (notation defined in
Fig. 1).
transient performance can be readily analyzed using classical
control theory
(3)
Using the transformation techniques described in [6], the
equivalent stationary - - frame representation of the SRFPI
controller can be developed as (2). It is this form that is used
to compare the transient performances of the SRFPI and the
P Resonant controllers in Section IV. The main concern
with (2), shown at the bottom of the page, is the significant
off-diagonal terms, which represent cross coupling between
phases and make an ASM single-phase representation difficult.
B. P Resonant Controller Model
In contrast, the P Resonant controller transfer function,
given by (3), is already in the stationary frame and is independent between phases. This transfer function has infinite gain
at fundamental frequency and therefore also eliminates steady
state error [6], while the phase independence means that its
IV. STABILITY ANALYSIS OF SINGLE PHASE MODEL
The open-loop and closed-loop transfer functions of a single
phase ASM of the system with a P resonant controller are given
by (4) and (5), respectively, as shown at the bottom of the next
,
,
page, where:
,
.
Analysis of this system has determined that its stability is pri, as
marily determined by the outer loop proportional gain
illustrated in Fig. 4 which shows how the position of the closed
for three difloop poles of the P Resonant system vary with
ferent values of (see Section VI for component values). In all
between zero and approxthree cases the system is stable for
imately one (per unit). Based on these results, an initial value of
(2)
TWINING AND HOLMES: GRID CURRENT REGULATION
891
(a)
(b)
(c)
Fig. 4. Variation of P+resonant VSI system closed loop poles with control parameter K
0 < K < 10. (c) K = 100, K = 50, 0 < K < 10.
was chosen for the simulation and experimental work
presented here. This value was later increased to achieve an improved harmonic response as described in Section V.
The primary function of the inner loop gain, , is to damp
the resonant peak introduced by the LCL input filter as shown
also contributes to the overall loop
in Fig. 5. However, as
gain, increasing its value can compromise the system stability.
was chosen to achieve a suffiTherefore a value of
. (a)
K = 20, K = 50, 0 < K < 10. (b) K = 50, K = 50,
ciently damped response at the resonant frequency of the LCL
filter whilst maintaining an acceptable phase margin.
The integral gain,
, of the controller acts to eliminate
steady state error at the fundamental frequency. Due to the
complexity of the plant transfer function, simulation and experimental techniques were used to tune this value. The integral
gain required to achieve a good steady-state and transient
.
response was found in this way to be
(4)
(5)
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 18, NO. 3, MAY 2003
Fig. 5. Variation of open-loop frequency response with control parameter
(
0:5, K = 50).
K =
K
From (2), it can be seen that the diagonal terms of the SRFPI
controller are the same as the (single-phase) transfer function
of the P Resonant controller (3) apart from a scaling factor.
It is these diagonal terms that lead to the same steady-state response of zero steady-state error at the fundamental frequency.
However, it would be expected that the cross coupling terms
of the SRFPI controller might cause some differences between
the transient performances of the two controllers, and hence this
issue needs to be considered further, as follows:
Fig. 6 shows the open-loop frequency response for both the
P resonant and SRFPI controllers, where it can be seen that
the cross-coupling terms of the SRFPI system have only a small
influence on the performance of the controller around the fundamental frequency. Outside of this region, the frequency response
of the two controllers is practically identical. This suggests that
the transient analysis of the single-phase P Resonant controller
can be directly applied to the SRFPI controller, and hence that
the stability of the SRFPI control algorithm is also primarily de.
termined by outer loop proportional gain
V. GRID CURRENT CONTROL UNDER DISTORTED SUPPLY
CONDITIONS
Previous experimental investigations have shown that even
small levels of supply voltage distortion can result in significant current distortion using current regulators that are tuned
for a fundamental response [7]. The work reported here has
identified that this distortion comes about because the control
schemes described in the previous section have a limited bandwidth and are therefore unable to adequately compensate for
grid supply voltage harmonics. It is noted in passing that most
previous studies in this area have assumed a sinusoidal fundamental supply, and appear not to have considered the effect of
grid network harmonic voltages.
A. Calculating Harmonic Impedance
The sensitivity of each controller to grid harmonic distortion
can be investigated by calculating its “harmonic impedance,”
Fig. 6.
Open-loop frequency response of proposed control strategies.
i.e., the relationship between a harmonic voltage disturbance
into the system, and the resultant injected grid current harmonic
component. This impedance provides a simple measure of the
harmonic sensitivity of a current regulation scheme, and is a
useful tool to assist with the design of AC filters to achieve acceptable levels of harmonic distortion.
With some manipulation, the harmonic impedance for each
phase of the P Resonant system described in Section III-B can
be shown to be (6). Similarly, from (2), the harmonic impedance
matrix for the SRFPI controller can be derived as (7). With further algebra, the diagonal terms of (7) can be shown to be equivalent to that of the P Resonant converter (6). Finally, the results
from the previous section suggest that the cross-coupling terms
will have negligible influence on the performance of the SRFPI
controller at harmonic frequencies. Hence it is reasonable to
conclude that analysis of the P Resonant system is sufficient
to predict the harmonic performance of the SRFPI system also.
See (6) and (7) shown at the bottom of the next page, where ,
, and are defined in Section IV.
refers to the
th element of the transfer function matrix given in (5).
B. Mitigation of Harmonic Distortion
In order to minimize the harmonic current distortion
produced by the converter system under distorted supply
conditions, the harmonic impedance should ideally be infinite
for all low order harmonics. In other words, the inverter should
produce zero harmonic current for all values of harmonic
supply distortion. In practice, Fig. 7 shows the harmonic
impedance magnitude and phase verses frequency for two
. It can be seen from this figure that while the
values of
harmonic impedance goes through a minimum close to the
, its absolute magnitude generally
LCL resonant frequency,
increases. Hence the grid current distortion
increases as
caused by grid supply harmonics will decrease for a higher
. But the transient system response will
proportional gain
also become more underdamped as this gain increases.
Fig. 8 shows the SRFPI controller transient response for
and
with a 2% 5th harmonic
values of
TWINING AND HOLMES: GRID CURRENT REGULATION
Fig. 7. Harmonic
(ohms/degrees).
system
impedance
for
+
P resonant
893
(a)
controller
and a 1% seventh harmonic in the grid supply voltage. While
the current distortion has been reduced from 7.8% to 5.2% as
increases, the transient response is correspondingly more
oscillatory. These results show the tradeoff that can be made
between a regulator’s harmonic performance and its transient
response when operating into a distorted supply.
In [8] it was shown that converter devices can be used as “harmonic current sinks” within a distribution system and thereby
reduce the levels of harmonic voltage distortion at the point of
common coupling. In contrast to the harmonic current mitigation
described above, this filtering application requires that the
harmonic impedance of the converter system be minimized.
However, careful design is required to avoid overloading the
inverter and causing harmonic voltage magnification at other
points within the distribution system. The harmonic impedance
derived above also provides a useful tool for further investigating
(b)
Fig. 8. VSI transient response for 2% fifth and 1% seventh supply distortion.
( = 20
= 50). (a)
= 0:5. (b) K = 1:0.
K
K
K
(6)
(7)
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 18, NO. 3, MAY 2003
Fig. 10. Grid current harmonic spectra for
Fig. 9. Grid current waveform for
K.
K.
these effects of converter systems on the harmonic voltage
distortion levels within distribution systems. However, such
investigation is beyond the scope of this paper.
VI. EXPERIMENTAL VERIFICATION
An experimental platform based on a DSP controller was used
to confirm the accuracy of the ASM analysis described above,
and to test the practical robustness and harmonic performance
of the control algorithms developed. The particular circuit pamH,
mH,
F,
rameters are
F (the supply variac inductance was used as inof the LCL filter). The system is rated at 10 kVA and
ductor
has a 415 V supply voltage and a 700 V dc bus.
A SRFPI controller was implemented for the outer grid current regulation, with an additional PI controller to maintain the
dc link voltage at the specified value. This controller acts as
an outer control loop, providing the real current demand to the
SRFPI controller. (Note that in a complete system, the bulk of
the real and reactive current references would be generated by
higher-level control loops. However, the operation of these control loops is beyond the scope of this paper, so simple default
values were used.) The operation of the dc voltage control loop
was also decoupled from the current regulator by giving it a significantly longer time constant.
To load the system, the dc link of the converter system was
simply connected to a resistor, and the inverter then acted as
an active rectifier. However, since a VSI is of course implicitly
bidirectional, the results are readily applicable to any type of
grid-connected application.
Fig. 9 shows the measured supply phase current for a step
change in the reactive current reference, . The real current
reference, , is supplied by the dc voltage controller. These
results confirm that the proposed control algorithm is stable at
full supply voltage, achieves zero steady state error at fundamental frequency and has a good transient response as anticipated. However, it can be seen that the measured phase currents
have significant levels of harmonic distortion. For this test, the
supply voltage distortion was measured at approx. 2.3% with
the fifth and seventh harmonics dominating. Fig. 10 shows the
K.
Fig. 11.
Grid current waveform for 2
Fig. 12.
Grid current harmonic spectra for 2
K.
harmonic spectra of the phase current distortion, with fifth and
seventh harmonic distortion caused by the supply voltage harmonics and a total harmonic distortion (THD) of 10.5%. Figs. 11
and 12 show the improvement that was achieved by doubling
, to achieve a reduced harmonic curthe proportional gain
is still well
rent distortion of THD 6.7%. This increased
within the stability margin, since there is no significant transient
TWINING AND HOLMES: GRID CURRENT REGULATION
oscillation visible in Fig. 9. It is further noted that the reference
signal in both cases contains a small level of harmonic distortion (THD 2%). This is, in part, due to the ripple in the dc bus
voltage. It is expected that this distortion would be minimized by
signal through a low pass filter. This would furpassing the
ther reduce the levels of grid current distortion without affecting
the performance of the dc voltage controller significantly.
VII. CONCLUSION
This paper has presented a robust control algorithm to regulate the grid current entering a distribution network from a
three-phase VSI system via an LCL input filter. Linear analysis,
simulation and experimental results are used to verify the stability of the algorithm across a range of operating conditions.
Expressions for “harmonic impedance” of the system are derived to study the effects of supply distortion on the harmonic
performance of the system. It is shown that the controller can
be tuned to achieve an improved overall response when operating into a distorted supply, at the expense of some reduction
in transient stability margins. Hence, an acceptable harmonic
performance can still be achieved with a lower value of input
inductance than would be required for a simple inductive filter,
which offers potential for significant reductions in filter cost.
REFERENCES
[1] M. Lindgren and J. Svensson, “Control of a voltage-source converter
connected to the grid through an LCL-filter-application to active
filtering,” in Proc. Power Electron. Spec. Conf. (PESC’98), Fukuoka,
Japan, 1998.
[2] M. Liserre, F. Blaabjerg, and S. Hansen, “Design and control of an LCLfilter based three-phase active rectifier,” in Proc. 2001 IEEE Ind. Applicat. Conf., 2001, pp. 297–307.
[3] V. Blasko and V. Kaura, “A novel control to actively damp resonance
in input lc filter of a three-phase voltage source converter,” IEEE Trans.
Ind. Applicat., vol. 33, pp. 542–550, 1997.
[4] N. Abdel-Rahim and J. E. Quaicoe, “Modeling and analysis of a feedback control strategy for three-phase voltage-source utility interface systems,” in Proc. 29th IAS Annu. Meeting, 1994, pp. 895–902.
[5] P. C. Loh, M. J. Newman, D. N. Zmood, and D. G. Holmes, “Improved
transient and steady state voltage regulation for single and three phase
uninterruptible power supplies,” in Proc. 32nd Ann. IEEE Power Electron. Spec. Conf. (PESC’01), 2001.
[6] D. Zmood, D. Holmes, and G. Bode, “Frequency-domain analysis of
three-phase linear current regulators,” IEEE Trans. Ind. Applicat., vol.
37, pp. 601–610, 2001.
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[7] E. Twining, “Modeling grid-connected voltage source inverter operation,” in Proc. AUPEC’01, 2001, pp. 501–506.
[8] H. Akagi, H. Fujita, and K. Wada, “A shunt active filter based on voltage
detection for harmonic termination of a radial power distribution line,”
IEEE Trans. Ind. Applicat., vol. 35, pp. 638–645, 1999.
Erika Twining (S’02) received the B.Eng. and
M.Eng.Sc. degrees from the University of Melbourne, Parkville, Australia, in 1995 and 2000,
respectively, and is currently pursuing the Ph.D.
degree in the Department of Electrical and Computer
Systems Engineering, Monash University, Clayton,
Australia.
She was a Graduate Electrical/Instrument Engineer at Orica Pty., Ltd., from 1996 to 1998. During
this time, she worked at a number of manufacturing
sites in Victoria and NSW, Australia, where she was
involved in the design and maintenance of power
distribution and process control systems. Her major research interests include
grid connected PWM converters, distributed generation, power quality, and
voltage compensation in weak distribution networks.
Donald Grahame Holmes (M’87) received the
B.S. degree and M.S. degree in power systems
engineering from the University of Melbourne, Melbourne, Australia, in 1974 and 1979, respectively,
and the Ph.D. degree in PWM theory for power
electronic converters from Monash University,
Victoria, Australia, in 1998.
He worked for six years with the local power company developing SCADA systems for power transmission networks, before returning to the University
of Melbourne as a faculty member. In 1984, he moved
to Monash University to work in the area of power electronics, and he is now an
Associate Professor. He currently heads the Power Electronics Research Group,
Monash University, where he manages graduate students and research engineers
working together on a mixture of theoretical and practical R&D projects. The
present interests of the group include fundamental modulation theory, current
regulators for drive systems and PWM rectifiers, active filter systems for quality
of supply improvement, resonant converters, current source inverters for drive
systems, and multilevel converters. He has a strong commitment and interest
in the control and operation of electrical power converters. He has made a significant contribution to the understanding of PWM theory through his publications and has developed close ties with the international research community in the area. He has published over 100 papers at international conferences
and in professional journals, and regularly reviews papers for all major IEEE
TRANSACTIONS in his area.
Dr. Holmes is an active member of the IPC and IDC Committees of the IEEE
Industrial Applications Society.