Grid-Connected Three-Phase Inverter System with

LCL Filter: Model, Control and Experimental Results

Gabriel E. Mejía-Ruiz J.R. Rodríguez Nicolás Muñoz-Galeano A. Zamora
Corporación Universitaria and M.R.A. Paternina Grupo GIMEL, Dpto Eléctrica U. Michoacana de SNH
Minuto de Dios-UNIMINUTO UNAM Universidad de Antioquia Michoacan, Mexico
Bello, Colombia Mexico City, Mexico Medellín, Colombia azamoram@umich.mx
gmejiaruiz1@uniminuto.edu.co jr rodriguez@fi-b.unam.mx nicolas.munoz@udea.edu.co
mra.paternina@fi-b.unam.mx

Abstract—This paper implements a grid-connected two-level Grid-connected inverters are designed according to the
three-phase inverter with both active and reactive power flow norm of power quality and grid synchronization (Std. 1547,
capabilities. This inverter is an effective power electronic 2005 and 2015) [12], [13]. By taking these standards into
interface for renewable energy systems. An average model is
proposed for the inverter system, meanwhile the design of the consideration, Total Harmonic Distortion (THD) for the grid
current controllers is performed taking the dq reference frame current must be less than 5 %. Technical literature reports various
into account. The system has been implemented using a NI SB- control strategies and modulation techniques for three-phase
Rio development board, a high voltage DC-link, a Phase Locked grid-connected systems. One of the well-known techniques is the
Loop (PLL) algorithm, and a LCL filter. The experimental results PWM, which enables a constant switching frequency, being less
validate and corroborate the theoretical analysis carried out.
Index Terms—Three-phase inverter, grid connected inverter, complex the design of the output filters and the cooling system.
SVPWM three phase inverter, LCL filter, micro-grid. One of the basic requirements for grid-connected inverters is
the synchronization of the injected current with the grid voltage,
I. I NTRODUCTION which is performed by the Phase-Locked Loop (PLL) algorithm,
since it is possible to obtain the phase angle information. This
The growing of the electricity demand, the reduction of the information is also used for voltage and current transformation
fossil fuel reserves and the initiatives taken by the governments in the dq reference frame [2], [5], [14].
for reducing the air pollution emissions have led to increase the By following the standards criteria and the above-mentioned
generation from renewable energy sources [1]. These systems requirements, this paper is focused on the implementation of a
can manage the production of energy from photovoltaic (PV) grid-connected three-phase inverter system. The laboratory-scale
and fuel cells. These sources provide direct current (DC) energy. prototype is implemented by using a Single Board Rio platform
The electricity generated by other energy resources, such as: and a two-level VSC. Average models are considered for carrying
wind turbines and tidal stream generators, is rectified to enable out the inverter and LCL filter in Section II. Subsequently, the
the connection to the network. Consequently, the injection of design of the control system in d-q reference frame is detailed in
electricity toward the grid is necessarily transferred through Section III. Finally, the laboratory-scale prototype and experimen-
DC-AC inverters [2], [3]. Therefore, grid-connected three-phase tal results of inverter, LCL filter and PLL algorithm are discussed
inverters are key elements for the conversion of the power at in Section IV. The results demonstrate that the PLL system can
small and medium scale in the distribution system [4]. synchronize the inverter with the grid in less than 8 cycles. More-
The energy obtained from renewable energy sources is over, these results show that the current control system responds
usually transferred toward the grid with limitations due to the to the changes of 75% in the set-point in less than 3 cycles.
cost, lifetime, and efficiency of the storage systems. Besides,
the energy from the DC-link toward the grid is generally II. S YSTEM M ODELING
transferred by means of an electronic power converter. DC-AC The dynamic average model of the two-level three-phase
converters are generally used for this purpose, in which can inverter is obtained based on Kirchhoff’s Laws and the transfor-
be accomplished generating single-phase or multi-phase AC mations of the abc reference frame to the αβ and dq reference
voltage from a DC source. frames, by the Clark and Park’s transformation. The inverter, in
The inverter allows the control of amplitude, frequency Fig. 1, is composed of three identical branches of half-bridge
and phase of the generated voltage [2], [4]. Some inverter inverters (branch a, branch b and branch c), six controlled IGBTs
applications are: control of motor speed [5], wind and solar (Q1 to Q6 ), six anti-parallel diodes (D1 to D6 ), one DC voltage
power generators [6], [7], static compensators [8], active source (vs ), and a capacitor bank (C). This inverter can provide a
filters [9], flexible AC transmission systems (FACTS) [10] and bidirectional power flow between the DC-bus and the grid, which
Dynamic Voltage Restorer (DVR) [11]. In such applications, allows to exchange active or reactive power. The connection of the
the inverter can work as a current source or as a voltage inverter with the grid is made through the LCL filter at the Point
source. Voltage Source Converter (VSC) is one of the most of Common Coupling (PCC). The model in Fig. 1 considers
common technologies used at the industrial level for the DC-AC the following assumptions: (i) IGBTs and diodes work as a
conversion, due to its simplicity and good performance [2]. short-circuit in ON state; (ii) IGBTs and diodes work as an open

978-1-5386-8218-0/19/$31.00 ©2019 IEEE

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    
ma m̂a cos(ε(t))
 mb  = m̂b cos(ε(t)+120o ) (5)

AC side limit

PCC
mc m̂c cos(ε(t)+240o )

DC side limit
D1 Q3 D3 Q5 where ε(t) has the embedded information about the frequency
Vdc /2 Q1 D5
ia Vta AC Grid and phase angle of the three-phase system. Meanwhile, (4)
a ib Vtb LCL and (5) describe the relationship between the modulation signal
0 b ic Vtc FILTER and the phase voltages on the AC side of the three-phase VSC
c
[15], [16]. Based on these equations, the spatial vector of the
Vdc /2 P
Q2 D2 Q4 D4 Q6 D6 three-phase system can be transformed into the αβ reference
Q frame, such as:
Branch a
Vdc
vtα = mα (6)
2
Vdc
vtβ = mβ (7)
Fig. 1. Schematic diagram sof two-level three-phase VSC. 2
It is notable to indicate that the αβ voltage components on
the AC side are proportional to the αβ components of the
modulation signal. Accordingly, the real and reactive power
circuit in the OFF state; (iii) the transition between the switching of the VSC can be expressed respectively in terms of the
states (ON, OFF) is immediate; (iv) voltage ripple and parasitic quantities mapped in the αβ frame as: Pt = 23 [vtα iα + vtβ iβ ]
resistance are not present in the DC-link; and (v) L2 << L1 , and Qt = 3/2[−vtα iβ +vtβ iα ].
then L can be approximated by L = L2 +L1 in the LCL filter. In order to perform the control, the variables in both αβ
On the other hand, the complementary commutation of the and abc reference frames are assumed sinusoidal functions.
IGBTs Q1 and Q2 , in branch a, allows to control magnitude, fre- Whereas the linear control systems are designed as low-pass
quency and phase of the current injected toward the network. The filter with a bandwidth that rejects high-frequency noise signals.
switching signal (uk ) is generated via the SPWM technique. The This configuration makes difficult the implementation of the
frequency of the carrier signal determines the switching frequency PI control system to adequately track the sinusoidal reference
(fsw ) of the IGBTs. The carrier signal (triangular waveform) is signals. In contrast, the variables to be controlled in the dq
compared with the sinusoidal reference signal to generate fsw . reference frame are constant signals, easily regulated by the
By applying Kirchoff’s laws on the AC side of the circuit PI linear controller. The phase voltages on the AC side of the
in Fig. 1, equation (1) is obtained. vsa is the voltage on the three-phase VSC can be represented in the dq reference frame,
AC side and vta is a signal with period Ts , Rwhich can be such as is displayed in (8) and (9).
T Vdc
described using the Fourier transform. vta = 1/Ts 0 s vta (τ )dτ + vtd = md (8)
Ph=+∞ 2
h=1 [ah cos(hωs t)+bh sin(hωs t)]. Where h is the order of Vdc
the harmonic, ωs = 2π/Ts , ah andR bh are given by: ah = vtq = mq (9)
2
RT T
2/Ts 0 s vta (τ )cos(hωs t), bh = 2/Ts 0 s vta (τ )sin(hωs t) and Active and reactive power of the VSC can be expressed in
τ is auxiliary variable for integration process. Expressions for the terms of the quantities mapped to the dq reference frame as:
3
branch a are presented in the following, using positive sequence. Ps = [vsd id +vsq iq ] (10)
di 2
L +Ri+vs −vt = 0 (1) 3
dt Qs = [−vsd iq +vsq id ] (11)
The average function of the electric power difference in (2) is 2
obtained, ignoring the high frequency terms in Fourier transform A. LCL Filter Model
[15].
di 1 Ts
Z The inverter is the most common power electronic converter
L +Ri = (vt (τ )−vs (τ ))dτ (2) used to connect the renewable energy sources to the utility grid.
dt Ts t−Ts
The pulse-width of the IGBTs switching signals changes from a However, the high frequency switching of semiconductors can
switching period to the next. The duty cycle (d) can assume any cause the harmonic injection to the grid. Since grid filter reduces
value between 0 and 1 and it can be obtained using the average the harmonic injection to the utility grid. LCL filters consists
function of the switching signals. The amplitude modulation of two inductors and one capacitor per branch. The attenuation
index can be defined as m = 2d−1. The average current model of the LCL filter is -60dB/dec for the higher order harmonics
is presented in (3).
dī L1 L2 igrid
L +Rī+v̄s −v̄t = 0 (3)
dt
Voltages in the terminals of each branch of the inverter are
denoted in (4). In order to obtain the balanced three-phase
voltage on the AC side, the control system must be fed by a vVSC C vgrid
balanced three-phase signal for the generation of the switching
signals, as shows in(5).  
vta ma (vdc /2)
 vtb  =  mb (vdc /2)  (4)
vtc mc (vdc /2) Fig. 2. Schematic diagram of LCL filter.

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 to the resonant frequency. Fig. 2 depicts the schematic diagram
VSC POWER CONVERTER

PPC
of the LCL filter without damping resistance [15]. Mathematical P
deduction of the LCL filter model of Fig. 1 is presented in (12). Q Relay
igrid 1 LCL FILTER Sensors connectors
= (12) L1 L4 ia Vsa
vvsc L1 L2 CS 3 +S ∗(L1 +L2 ) Vta

Resonance frequency of the LCL r filter (f0 ) is presented in:

AVERAGE
V L2 C1 C3 L5 ib Vsb
Vs C MODEL tb
1 L1 +L2 THREE-PHASE
L3 C2 L6 ic Vsc
f0 = (13) VSC V tc
2π L1 L2 C
Capacitive reactance of the LCL filter (XC ) can be calculated Vab
ia ib ic

{
using the expression (14) Z-120° a bc
SPWM Generator
1
XC = (14) αβ
ωC abc
PLL
The design criteria of LCL filter of L1 , L2 and C are: Ѳ αβ
Ѳ
L1 ≤ 0,05V line δ 1 1 1 1
∆iout fsw , 10 L1 ≤ L2 ≤ 5 L1 and 10 XL2 ≤ XC ≤ 5 XL2 .
αβ dq
Where Vline is the grid voltage amplitude, δ is the maximum αβ Ѳ
d Control -
width of the PWM modulation signal, fsw is the switching d

+
-
+
q
frequency and ∆iout is the maximum current ripple expected. q Control

id _ref iq _ref
III. C ONTROL S TRATEGY Fig. 3. Current control diagram in the dq reference frame.

The control system in the dq reference frame reduces

the number of plants to be controlled from three to two, in diq
comparison with the abc frame, allowing to control the active L = −Riq −v̂s sin(ω0 t+θ0 )+vtq (17)
dt
(P ) and reactive (Q) power flow between the grid and the where, the proper selection of ω0 and θ0 allows to represent the
inverter. The control system in the dq reference frame tracks system in the dq reference frame, as presents in (18) and (19).
constant reference signals, enabling the accurate regulation of did
L = Lω0 iq −Rid −v̂sd +vtd (18)
electrical variables with linear controllers. The PI control system dt
can reduce the error in steady state to zero, when it follows diq
L = −Lω0 id −Riq −v̂sq +vtq (19)
constant references. Some of the advantages of the control in dq dt
Active and reactive powers can be expressed by (20) and (21),
reference frame in comparison with the control in the αβ and abc using the dq components of the system and these equations
reference frames are: the control and filtering of DC magnitudes can be simplified recognizing that vsq = 0 in steady state.
is most simple than the control of AC magnitudes, using linear 3
controllers in this system allows to uncouple the control by Ps = [vsd id ] (20)
2
active and reactive power. The current control system in the 3
dq reference frame enables to regulate P and Q, modifying Qs = − [vsd iq ] (21)
2
the phase angle and amplitude of line current with respect to The desired reference currents can be obtained by (22) and (23).
2
the grid voltage measured at the PCC. Moreover, this current id−ref = Ps−ref (22)
control system protects the system against overcurrent [15]. 3vsd
2
Fig. 3 exhibits the schematic diagram of the current control iq−ref = − Qs−ref (23)
system proposed in this paper. This control strategy regulates 3vsd
Voltage on the AC side of the VSC can be expressed as:
P and Q powers, modifying the direct (id ) and quadrature vtd = md (Vdc /2) and vtq = mq (Vdc /2). In (24) and (25), the
(iq ) current components in rotating frame. The measurements presence of the terms Lω0 iq and Lω0 id represents a coupling
of the grid currents are transformed to the dq reference frame. of the dynamics between id and iq . Which can be decoupled
Subsequently, these transformed signals are compared to the by using the pre-feed loops in the implementation of the control
desired reference current. The result of these comparisons is system. However, this way of decoupling control involves the
processed by PI controllers. Control signals are transformed to measuring of two line to line voltages to complete the pre-feed
the abc reference frame, in turn these are used to generate the loops. This work implements the control system avoiding the
PWM signal. The synchronization of the inverter and grid needs pre-feeding loops with satisfactory results. To decouple the
the PLL system. The information of the phase angle of the grid terms Lω0 iq and Lω0 id for the design of the control system,
is provided by the PLL. The spatial phasor of the three-phase md and mq can be determined, as give in (24) and (25).
system in the dq rotary reference frame can be expressed as
Xd +JXq = Xe ~ −jθ and the dynamic behavior of the inverter 2
(µd −Lω0 iq +vsd )
md = (24)
AC side can be represented by (15), replacing the phasor Vdc
voltage measurement defined by ~vs = v̂s ej(ω0 t+θ0 ) in (3). 2
d~i mq = (µq −Lω0 id +vsq ) (25)
L = −R~i−v̂s ej(ω0 t+θ0 ) +~vt (15) Vdc
dt where µd and µq are two new input control signals. (25) and
The spatial phasor of the three-phase system can be expressed (26) can be obtained by replacing md and mq .
in the dq reference frame, as is illustrates in (16) and (17).
did did
L = −Rid −v̂s cos(ω0 t+θ0 )+vtd (16) L = −Rid +µd (26)
dt dt

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 diq
L = −Riq +µq (27)

Magnitude (dB)
dt
The simplified diagram of the control system is exhibited
in Fig. 3, which indicates that the average model of the
grid-connected inverter is equal for components d and q. By this
fact, the control system for the components of the current d and
q can also be identical. In this work, the control loops for the
component d and q are implemented using proportional-integral
(PI) controllers. The PI controller parameters are shown in Table

Phas (deg)
I. These parameters are obtained using root locus method.
Fig. 4 depicts the location of the system roots in open and
closed loop. The location of the pole that belongs the open loop
system is on the left side of the complex half-plane, in turn
indicating that the system is inherently stable. For the closed
loop system, poles are located making the sub-damped system Frequency (Hz)
and reaching the unitary gain in stable state.
Bode diagram in Fig. 5 illustrates the magnitude and phase of Fig. 5. Bode diagram of the open-loop and closed-loop systems.
the open-loop and closed-loop systems. The closed loop system
has a unit gain in steady state. The integral action reduces the TABLE II
error in stable state. The cutoff frequency of the closed loop PARAMETERS OF EXPERIMENTAL PROTOTYPE
system is of 400 Hz, filtering the high frequency components
caused by the IGBTs switching. Parameter Value
IGBTs IRG4PC50UD
IV. E XPERIMENTAL R ESULTS Snubber diodes FR154
Experimental results are presented in this section. The Snubber capacitor 100nF
dynamic performance of the two-level three-phase VSC is Snubber resistor 56Ω
tested in the presence of changes in operating conditions. The LCL inductors 675µH, 100µH
prototype implementation is shown in Fig. 6. The GDS-2000 LCL capacitors x3 10µF
oscilloscope, AP031 isolated voltage probes (100V / div) from DC link voltage 400Vdc
the Teledyne and A622 oscilloscope current probes (100mV/A) Grid voltage 120Vrms
from Tektronics have been used to acquire the measurements. Grid frequency 60Hz
Table II shows the parameters used in the prototype. Switching frequency 40kHz
Efficiency 95%
TABLE I
PARAMETERS OF THE C ONTROL S YSTEM .

Parameter Value Fig. 7 presents the synchronization of three-phase inverter,

Proportional gain Kp 3620 using the PLL system. vref is the grid voltage signal reference
and vV SC is the voltage generated by the inverter. The waveform
Integral gain Ki 0,003
depicts that the stabilization time of the PLL control system
depends on the initial phase difference between vref and vV SC .
In this case, the initial phase difference between vref and vV SC
is greater than 180o and the PLL system only lasts 8 cycles of the
fundamental frequency to synchronize the inverter with the grid.
Fig. 8 exhibits that output voltage of the LCL filter vLCL
Imaginary axis (seconds-1)

has a sinusoidal waveform with a fundamental frequency of

60 Hz, in comparison with the voltage at the output of the VSC
before the filter vV SC . The waveform allows to qualitatively
show the reduction of the harmonics of higher order to the
fundamental harmonic, evidencing the effectiveness of the LCL
filter. The measurement of THD (4.5%) in the output currents
allows to verify the accomplishment of the power quality and
grid synchronization norms (Std. 1547, 2005 and 2015).
Fig. 9 displays the three-phase grid-injected currents and the
reference line voltage (vsa ). The digital controller regulates
the dynamic behavior of id and iq . Whereas Fig. 10 illustrates
Real axis (seconds-1) that the current control system responds to the change in the
Fig. 4. Location of the roots of system in open loop and in closed loop. set-point in less than 3 cycles of the grid, allowing to change

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 Fig. 8. Measured output voltage of the LCL filter (Measured grid voltage
Fig. 6. Photograph of the experimental prototype of two-level three-phase VSC. (vref ) and voltage generated by the inverter (vV SC ).

Fig. 9. Measured line voltage of phase a (vsa ) and three-phase grid-injected

Fig. 7. Measured grid voltage (vref ) and voltage generated by the inverter
currents(ia ,ib ,ic ).
(vV SC ) while the PLL system is work.

The two-level three-phase VSC is a simple and good-

the phase of the currents with respect to the reference line
performing converter that can be used in applications with
voltage; which effectively change the active and reactive power
renewable energy sources. The current control system in the dq
injected toward the grid.
reference frame allows the independent control of the direct and
Ps and Qs can be modified controlling the magnitude of id
quadrature current. PI current controllers in the dq reference
and iq . Since the output real signals of the inverter in Fig. 9 are
frame enabled the reduction of the error in steady state to zero.
in the abc reference frame, whereas the digital control system in
The implemented PLL system established the phase and
the dq reference frame is able to modify the amplitude and phase
frequency synchronization of the voltage generated by the
of ia , ib and ic in comparison with va , vb and vc ; consequently,
three-phase inverter with the grid voltage. The phase signal
this control system is also able to regulate Ps and Qs .
supplied by the PLL can be modified, which lets the correction
of the phase shift introduced into the system by the LCL
V. C ONCLUSIONS network filter. The PLL can guarantee the synchronization of
A three-phase inverter prototype has been built with a the three-phase inverter with the grid, even in the presence of
real-time control system to validate the effectiveness of the the phase jumps in the source.
theoretical approaches, verifying that the linear control system
ACKNOWLEDGMENT
is a simple way to achieve local stability of the system and
control of the injected power in presence of small disturbances Authors greatly thank to Corporación Universitaria Minuto
and changes in the point of operation. de Dios (UNIMINUTO), UNAM, Universidad de Antioquia

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