Design and Control for Three-phase Grid-connected
Photovoltaic Inverter with LCL Filter
Yun Chen
Fei Liu
School of Electrical Engineering
Wuhan University
Wuhan, China
hncy1023@tom.com
School of Electrical Engineering
Wuhan University
Wuhan, China
dyj_lf@163.com
Abstract—As the traditional resources have become rare,
photovoltaic generation is developing quickly. The gridconnected issue is one of the most importance problem in this
field. The voltage source inverter usually uses LC or LCL as the
filter. LCL filter, which can reduce the required filtered
inductance and save the cost, is adopted to connect the grid in
this paper. However, reasonable design of LCL filter needs
consideration in order to achieve optimal effect. Meanwhile,
systems with LCL filter need more complex control strategies.
This paper firstly introduces a whole scheme of LCL filter
design, then analyzes a two-loop control method in detail which
uses the capacitor current and grid current as the inner and
outer control variable respectively. Detailed discussion about the
role of the loops and the influence on the system caused by every
parameter of the controller are given out. Finally experimental
results verify the correctness of the analysis and availability of
the control strategy.
Keywords-photovoltaic generation; grid-connected; LCL filter;
design; control strategies
I.
INTRODUCTION
Photovoltaic (PV) generation as one of renewable resources
is developing quickly. PV systems connected to the grid
usually uses LC or LCL as the filter. LCL filter, because of less
inductance and less cost, is chosen in this paper.
However, single grid current feedback control can not
ensure the stability of the system [1]. More complex strategies
are needed. An inner capacitor current feedback loop and an
outer capacitor voltage control loop are adopted in [2] to
decrease the harmonic content of the grid current. But this
strategy cannot directly control the grid current so that it cannot
change the grid power flow. And damping resistor can be
added in the capacitor path to increase the degree of damping
of the system to improve the stability of the system [3]. But the
added resistor will cause loss especially in large capacity
systems. The pole-assignment method is also employed to
control the complex system. Though the method can assign the
poles as wish to get the required system performance, it needs
three sensors to provide the information of the system which
makes the system more costly. A simple and practical method
is to adopt the grid current as the outer loop and choose another
variable as the inner loop in this paper.
II.
MODEL OF THE SYSTEM
Fig.1 gives out the main circuit topology, where R1 and R2
are the resistors associated with inverter side and grid side
inductances respectively. In order to establish the state-space
mathematic model of three-phase grid-connected inverter with
LCL filter, it is assumed that the DC capacitance is large
enough to keep a constant dc voltage. In such a case, it is
further assumed that the system is balanced and the switching
frequency is sufficiently high. The ABC state equations are
shown as (1).
1
1
i1k = Vk − (i1k R1 + Vck )
L1
L1
(1)
1
1
1
i2 k = Vck − U sk − i2 k R2
L2
L2
L2
1
1
VCk = i1k − i2 k
C
C
where k = a, b, c .
Based on 3/2 coordinate transformation principle, the state
equations of PV grid-connected inverter under αβ coordinate
can be obtained as (2).
1
1
i1x = Vx − (i1x R1 + Vcx )
L1
L1
(2)
1
1
1
i2 x = Vcx − U sx − i2 x R2
L2
L2
L2
1
1
VCx = i1x − i2 x
C
C
where x = α , β .
So, the three-phase system can be modeled by two
independent single-phase systems. The control block of single
phase is seen as fig.2.
iPV
PV
array
id c
T1
ic
C1
T4
T5
i1
V
a
V dc
N
T3
b
T6
T2
L1
L2
R1
R 2 i2
U sa
U sb
c
O
U sc
iC
C
VC
Figure 1. Topology of the three-phase grid-connected photovoltaic inverter
978-1-4244-2587-7/09/$25.00 ©2009 IEEE
1
CS
1
L1 S + R1
B. Choice of the Filtering Inductance L1 and L2
Fig.3 shows the schematic diagram of single-phase gridconnected inverter. uh is harmonic voltage source of three-
1
L2 S + R2
phase inverter control bridge leg. We know
Figure 2. Control block of the LCL filter
III.
DESIGN THE PARAMETER OF LCL FILTER
A. Choice of the Total Inductance
At first work mode of the three-phase grid-connected
inverter needs consideration in the choice of the total
inductance. If photovoltaic system has energy-storage
components, grid-connected inverter needs work not only in
inverter mode, but also in PWM rectifier mode. So, gridconnected inverter needs realization of four-quadrant running.
It should meet the following expression [4].
L=
2
2
2
2
Emp sin ϕ + U mp
+ Emp
sin 2 ϕ + U mp
− Emp
ω I mp
(3)
In (3), L = L1 + L2 is the total inductance of filter. Emp is
peak of power grid phase voltage. ϕ is angle of power factor.
Ump is peak of fundamental wave of phase voltage outputted by
three-phase grid-connected inverter. In SVPWM modulation,
the largest peak of fundamental wave of phase voltage
achieved is U mp = U dc / 3 , and Udc is voltage on DC side. So
U
2
dc
3
L ≤
ω I
In (5),
2
m p
(4)
m p
1 U dc
4 3 Lf sw
(5)
f sw is switch frequency. So we can get
U dc
(6)
4 3ΔI ripple − max f sw
According to (4) and (6), the range of total inductance is
L=
X L1 and X C 2 // L 2 .The parallel branch circuit of C2 and L2
brings in increscent series impedance, which decreases i1 .
However, X C 2 // L 2 is finite, so the decrease of i1 is limited. So,
the amount of output ripple of inverter is mainly determined by
L1.
When the total inductance L=L1+L2 is invariable, the ratio
a = L1 / L2 should be considered comprehensively. Based on
harmonic model in Fig.3, the transfer function between gridconnected current i2 and inverter bridge leg output voltage uh
can be expressed as formula (8), in which h is the order of
harmonics.
I 2 ( jhω )
1
(8)
=
U h ( jhω ) hω ( − h 2ω 2 L1 L2C2 + L1 + L2 )
According to formula (8), Fig.4 shows the relationship
between the harmonic current amplitude igrid ( jhω ) and L1 / L2
in fixed L, C2 and h. It can be seen that igrid ( jhω ) reaches the
least when
L1 / L2 ＝ 1, and that igrid ( jhω ) is increasing
L1 / L2 is increasing. So when C2 and
are L1 + L2 fixed with L1 = L2 , the harmonic suppression
correspondingly when
− E
When the above transistor in A phase and the underside
transistor in B and C phase are turn-on, the A phase current of
three-phase grid-connected inverter has the largest current
ripple[4],[5].
ΔI ripple−max =
i1 is determined by
U dc
4 3ΔI ripple− max f sw
U dc2
2
− Emp
3
≤L≤
ω I mp
effect to grid-connected current of LCL filter reaches best.
When C2 , L1 + L2 , h and ω are fixed with L1 = L2 ,
H LCL ( jhω ) reaches least, namely
effect to switch frequency reaches best. Meanwhile ， the
resonant frequency of LCL filter ωres =
( L1 + L2 )
L1L2C2
also reaches least, so the cut-off frequency of LCL filter can
overlap the bandwidth of grid-connected current controller
,which will influence the dynamic tracking performance of
controller. Besides，if L1 is chosen too little，the suppression
effect of ripple current Δiripple can be impacted. However , too
large L1 as L1 > 5 L2 will lead to cost increase for more
magnetic core material. Above all, choosing the ratio of L1 and
L2 about 5 will be reasonable.
(7)
In order to increase current tracking ability and system
response speed, the less L, the better. But the larger L, the
better the filter effect is. Considering adopting LCL filter
,whose filtering effect of high frequency is better than L-type
filter, the total inductance is chosen as least as possible.
harmonic suppression
jhω L1
i1 ( jhω )
uh ( jhω )
jhω L2
1
jhωC2
ic ( jhω) i2 ( jhω )
Figure 3. Harmonic model of LCL Figure 4. The relationship between
abs ( igrid ) and L1 / L2 with fixed C2
filter in current tracking control
C. Choice of the Filtering Capacitance
As above mentioned, in order to shunt ripple weight of
switch frequency and make high-frequency weight flow as
most as possible in capacitance branch circuit, X C 2 << X L 2
must be insured in the design. Here, it can choose
1 1
X C 2 = ( ~ ) X L2
10 5
(9)
If X C 2 is chosen too large, the high-frequency weight of
switch frequency ripple won’t be shunted enough in
capacitance branch circuit and more high-frequency harmonic
current will flow into grid. If X C 2 is chosen too little, the filter
capacitance will increase, which leads to more reactive power
current flowing into filtering capacitance. Besides, in order to
avoid excessive low power factor of grid-connected inverter,
fundamental wave reactive power absorbed by filtering
capacitance shouldn’t be larger than 5 percent of system rated
active power[4]. So
C2 ≤
λP
3 × 2π f1 Em2
(10)
In (10), P is the rated active power outputted by gridconnected inverter. Em is effective value of grid phase voltage.
λ is the ratio of fundamental wave reactive power absorbed by
filtering capacitance to P.
IV.
TWO-LOOP CONTROL STRATEGY
A. Selection of the Inner Loop Variable
Single grid current loop controller is not sufficient for the
stability of the system. Higher level control loops are required.
Another inner loop is employed to provide fast dynamic
compensation for system disturbances and improve stability.
For the LCL filter, apart from the grid current, i C and i1 can
be chosen as the inner loop variable. Moreover, since the inner
loop does not affect the accuracy of the outer loop, a simple
proportional controller is adopted as the inner loop. And the
outer loop employs a PI controller. Fig.5 and fig.6 give out the
control block of the system with i1 inner loop and iC inner
loop respectively.
In order to judge which one is more suitable for the inner
loop, root locus is used as the analysis tool. In fig.7 and fig.8, it
is evident from the plot that choosing iC as the feedback
variable can shift the root locus left. On the contrary,
i1 feedback makes the system stability worse. As i1 = iC + i2 ,
i1 can be modeled by vC through a sC block and v 2 through
a (1 /( L2 s + R2 )) block. The additional (1 /( L2 s + R2 )) block
along the feedback path is dangerous for the system stability.
Furthermore, iC is proportional to the change rate of vC so it
can reflect the capacitor voltage change caused by the load
changes immediately. Above all, iC is chosen as the inner loop
variable.
B. One Criterion for the Parameters of the Controller
Here the Routh’s Stability Criterion is used for giving out
the scope of the three parameters of the controller. From (12),
we can easily get the characteristic equation as (13).
a1 S 4 + a 2 S 3 + a3 S 2 + a 42 S + a5 = 0
(13)
According to the Routh’s Method, in order to ensure the
stability of the system, (14) and (15) must be fulfilled. And
then all known parameters in the equations are substituted with
its values. L1 = 5.5mH , L2 = 1mH , C = 20 μF . Define
K C = K C K PWM in the following (14) and (15) to simplify the
analysis.
Δ 2 = a 2 a 3 − a1 a 4 > 0
(14)
−12
2
C
or (250.1664+ 130.48KC − 0.11K P KC + 0.16K ) ×10
Δ 2 > a 2 a5 / a 4
or Δ 2 >
(15)
−9
−9
2
(52 × 10 + 20 × 10 K C ) K I K C
0.8 + K P K C
So (14) and (15) are the basic requirements for the stability
of the system, and the final chosen parameters will be tested by
this criterion later.
KP +
KI
S
K PWM
1
L1S + R1
1
L2 S + R2
1
CS
And their closed-loop transfer functions are shown as (11)
and (12).
I2
K K K S + KI KCKPWM
(11)
= P4 C PWM
*
I2 a1S + a2S3 + a3S2 + a41S + a5
I2
K K K S + KI KC KPWM
(12)
= P4 C PWM
*
I 2 a1S + a2 S 3 + a3S 2 + a42S + a5
where
a1 = L1L2C , a 2 = L1 R2 C + L2 R1C + K C K PWM L2 C
a 3 = R1 R2 C + L1 + L2 + K C K PWM R2 C ,
a 41 = R1 + R2 + K P K C K PWM + K C K PWM ,
a 42 = R1 + R2 + K P K C K PWM , a5 = K I K C K PWM .
>0
Figure 5. Control block of the system with i1 inner loop
KP +
KI
S
K PWM
1
L1S + R1
1
CS
1
L2 S + R2
Figure 6. Control block of the system with ic inner loop
i2*α +
i2,abc
i2α
αβ
i2 β
ic,abc
Figure 7. Root locus with variation
of Kp with i1 as the inner loop
Figure 8. Root locus with variation
of Kp with ic as the inner loop
C. Two-loop Control Scheme of PR Resonant Controller
Nowadays, the normal current control used in fixed switch
frequency instantaneous value control is PI control. However,
the routine PI control in static coordinate has difficulty in
eliminating steady state error of sine reference current. ABC-dq
coordinate transformation is used to solve such problem. In
synchronous rotating coordinate, sine reference instruction
becomes DC instruction, and zero steady error can be realized
using PI control. But the novel controller, namely PR
(proportion + resonant) controller can realize zero steady error
in steady coordinate tracking sine reference current[6]-[8]. For
the gain of PR resonant controller is infinite in fundamental
wave frequency but very little in other frequencies, system can
realize zero steady error in fundamental wave frequency. It
avoids tedious ABC-dq coordinate transformation and
simplifies controller design. The transfer function of PR
controller is
K s
(16)
GP R ( s ) = K p + 2 r 2
s + ω0
where,
ω0 is fundamental wave frequency.
D. Experimental Results
Based on a three-phase photovoltaic platform where
L1 = 5.5mH , L2 = 1mH , C = 20 μF , the controller is realized.
The values of K P , K C and K I are respectively 0.9, 2, 2600.
This group of values is satisfied with the basic criterion (14)
and (15). This controller is successfully used to connect the
grid. The waveforms of the grid current of three phases are
shown in fig.11. The waveforms of grid voltage and grid
current of phase A are given out as fig.12. The total THD of
grid voltage is 5.067% while the THD of grid current is
3.686%. The 3rd, 5th and 7th component of grid current is
respectively 0.787%, 2.051%, 0.937%.
Kp +
KRs
s + ω2
2
ic*α
+
KC
−
Vk*,abc
αβ
*
2β
i +
−
Kp +
αβ
KRs
s2 + ω2
ic*β +
KC
−
Figure 10. Control diagram of three-phase grid-connected inverter using
current two-loop control
Figure 11.The waveforms of grid
current
V.
Figure 12.The waveforms of grid
voltage and current
CONCLUSION
The proposed design scheme of LCL filter is simple and
feasible, and the experiment results also show the filtering
effect is obvious. Meanwhile, this paper proposed a simple and
practical control strategy for the three-phase grid-connected
system with LCL filter. The proposed control strategy is easy
to be implemented using DSP. Further the two loops are
analyzed in detail including the role of every loop and the
selection of the three parameters of the controller. Experiments
are carried out to verify the analysis.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
Figure 9. Amplitude-frequency characteristics comparison between PI
controller and PR controller
−
[8]
P. C. Loh, “Analysis of multiloop control strategies for LC/CL/LCLfiltered voltage-source and current-source inverters,” IEEE Trans. on
Industry Applications, vol.41, pp.644-654, February 2005.
A. R. Naser and E. Q. John, “Modeling and analysis of a feedback
control strategy for three-phase voltage-source utility interface systems,”
in Proc. 29th IAS Annu. Meeting, pp. 895-902,1994.
M. Liserre, A. D. Aquila, and F. Blaabjerg, “Stability improvements of
an LCL-filter based three-phase active rectifier,” PESC, vol.3, pp.11951201,2002.
M. Liserre, F. Blaabjerg, S. Hansen, “Design and control of an LCLfilter based three-phase active rectifier,” thirty-Sixth IAS Annual
Meeting Conference Record of the 2001 IEEE, vol.1, pp. 299-307, 2001.
R. Teodorescu, F. Blaabjerg, M. Liserre, and A. D. Aquila, “A stable
three-phase LCL-filter based active rectifier without damping” IAS,
vol.3, pp.1552-1557, 2003.
R. Teodorescu, F. Blaabjerg, U. Borup, and M. Liserre, “A new control
structure for grid-connected LCL PV inverters with zero steady-state
error and selective harmonic compensation,” APEC, vol.1, pp.580-586,
2004.
D. N. Zmood and D. G. Holmes, “Frequency-domain analysis of threephase linear current regulators,” IEEE Trans on Industry Application,
vol.37, pp.601-610,2001.
T. C. Wang, Z. H. Ye, and G. Sinha, “Output filter design for a gridconnected three-phase inverter,” Power Electronics Specialist
Conference, PESC’ 03 IEEE 34th Annual, 2003.