A NOVEL ADAPTIVE CURRENT HARMONIC CONTROL STRATEGY
APPLIED IN MULTIFUNCTIONAL SINGLE-PHASE SOLAR INVERTERS
Lucas S. Xavier1 , Allan F. Cupertino2 , Victor F. Mendes3 , Heverton A. Pereira1,3
1 Universidade
Federal de Viçosa, Viçosa – Minas Gerais, Brazil
Federal de Educação Tecnológica de Minas Gerais, Belo Horizonte – Minas Gerais, Brazil
3 Federal University of Minas Gerais, Belo Horizonte – Minas Gerais, Brazil
e-mail: lsantx@gmail.com, allan.cupertino@yahoo.com.br, victormendes@cpdee.ufmg.br, heverton.pereira@ufv.br
2 Centro
Abstract – The inverter multifunctional operation is
based on the harmonic current compensation, generated
by nonlinear loads. The traditional harmonic detection
methods tracks all harmonic contents of the load current
and the control tuning tends to be complex with low
flexibility. In many works, the proportional resonant
(PR) controllers are used to control the inverter current
reference. However, one PR controller needs to be designed
for each harmonic frequency and this fact increases
the control algorithm complexity. Therefore, this work
proposes a novel adaptive current harmonic control
strategy applied in multifunctional single-phase solar
inverters. The strategy is based on a novel detection
method of the harmonic load current. The harmonic
current detection method is frequency adaptive and able to
detect the load harmonic current with higher amplitude.
This method consists in a cascade association of two
phase-locked loop based on second order generalized
integrator (SOGI-PLL). The detected frequency is used
as feedback by the proportional resonant controller.
Therefore, only two PR controller are required: one to
track the fundamental component and another to track the
harmonic component with higher magnitude. Simulation
and experimental results show the performance of the
proposed control strategy, improving significantly the grid
current quality.
Keywords – Adaptive control, harmonic compensation,
harmonic current detection method, multifunctional
operation, proportional resonant controller, SOGI-PLL.
I. INTRODUCTION
For the first time in four decades, the global carbon
emissions related to energy consumption remained stable
in 2014, while the global economy grew [1]. The high
penetration level of renewable energy sources in the global
electrical system is the reason associated with this stabilization
[1]. However, The growth of renewable energy sources
integration (especially wind and photovoltaic sources) with the
electrical system are bringing some concerns for researchers,
such as: quality in the energy injected into the grid due to use
of power electronic converters [2],[3]; the security and support
in an installation under fault in the grid [4]–[6].
Although the use of electronic converters as interface
between renewable source and grid brings challenges in
c
978-1-4799-8779-5/15/$31.00 2015
IEEE
relation to the injected power quality into the electrical
system, these devices have potential to improve the power
quality index of an installation. For example, photovoltaic
inverters have been employed to control the harmonic current
compensation generated by nonlinear loads connected at the
point of common coupling (PCC). In this way, the solar
inverter becomes multifunctional [7]–[10].
In the multifunctional operation of photovoltaic inverters,
the detection method of the nonlinear load harmonic current is
an important issue. Most methods are not designed to detect
just one harmonic component of the load current [7],[8],[11]–
[15]. The proposed strategy can be interesting in order to
compensate only harmonic with higher amplitude, increasing
efficiency and reducing the control complexity.
In single-phase applications, many works use proportionalresonant (PR) controllers [16], [17], because references of the
current loop are sinusoidal and have high index of harmonic
distortion. In these conditions, the conventional proportionalintegral (PI) controller has steady state error due to its
limited current tracking capability [16]. On the other hand,
PR controller can compensate only one frequency. Thereby,
one PR controller needs to be designed for each harmonic
frequency [17]. This fact increases the control algorithm
complexity. Therefore, if the PR controller becomes adaptive
to the system, it is possible to obtain an efficient controller
with low implementation complexity, this fact is approached
in [18], [19] for grid frequency fluctuations.
In this context, this work proposes a novel adaptive current
harmonic control strategy applied in multifunctional singlephase solar inverters. The strategy is based on a novel
detection method of the harmonic load current. The harmonic
current detection method is frequency adaptive and able to
detect the load harmonic current with higher amplitude. The
detection method consists of two-cascaded phase-locked loop
based on second order generalized integrator (SOGI-PLL)
proposed in [2],[20]. This proposed detection method is based
on the frequency adaptive characteristic of the SOGI structure
[2], [13], tuning in the frequency detected by PLL.
The novel detection method is applied in a single-phase
photovoltaic system shown in Figure 1. The control system
is based on proportional multi-resonant (PMR) controller
composed by: a proportional controller, a resonant controller
tuned at fundamental frequency and another resonant
controller dynamically tuned at harmonic frequency detected
by the proposed detection method. The PMR controller
transfer function is:
PCC
PV Array
Boost
Cdc
dc
dc
dc
ipv
vpv
Nonlinear
load
iL
Inverter
ZG
PMR
i *S (t)
2
vdc
iS
vPCC
i S (t)
iG
Fig. 1. Grid-connected photovoltaic system based on multifunctional
inverter.
1
ω2f
1
s
K!f
v*
KP
1
s
Ipv
vpv
MPPT
s
vpv
*
PI
*
Iind
vpv
Rh
z }| { z
}|
{
s
s
+ KIh 2
Gc (s) = KP + KI f 2
s + ω 2f
s + hω 2f
s
Fig. 2. Block diagram of the proportional-multiple resonant (PMR)
controller.
Control
Strategy
Rf
hωf
Grid
ac
K!h
1
PWM
PI
Iind
(a)
(1)
where:
KP - proportional gain;
KI f - integral gain tuned to fundamental frequency;
KIh - integral gain tuned to harmonic order h;
- harmonic order, where {h ∈ Z / h > 2};
h
ω f - fundamental frequency [rad/s].
The term R f and Rh are responsible for tracking the
fundamental frequency signal and the harmonic frequency
signal hω 2f , respectively. Therefore, this controller presents an
infinite gain in the resonant frequencies ω 2f and hω 2f [21]. The
proposed harmonic detection method is responsible for tuning,
dynamically, the PMR controller at harmonic frequency hω 2f
to be compensated. Block diagram of the PMR controller
Gc (s) is shown in Figure 2.
II. CONTROL STRATEGY
The complete control strategy is presented in Figure 3. The
dc/dc stage is responsible for maximum power point tracking
(MPPT) of the solar panels. The electrical model of the solar
panel is based on the mathematical model proposed in [22].
The MPPT algorithm is based on perturb and observe (P&O)
method [23]. The boost stage is used to ensure inverter bus
stability [2]. As shown in Figure 3(a), the boost control loop
has an outer loop, responsible for dc-bus voltage control of
the solar plant v pv and an inner loop, responsible for current
control of the dc/dc stage inductor Iind .
The inverter control strategy is shown in Figure 3(b). A
PI compensator is used in the dc-bus voltage control. This
compensator calculates the active current amplitude I ∗ , which
needs to be injected into the power system. This signal
is synchronized with PCC voltage through the SOGI-PLL
structure [2]. The synchronization results in a sinusoidal wave
i∗ (t) for the internal control loop. This current reference is
added to harmonic component of the load current detected by
proposed detection structure. As result, i∗S (t) is compared with
the inverter current iS (t) and the PMR compensator calculates
the modulation index v∗ of the inverter.
Only two resonant controllers are required: The first one
(R f ) is responsible to track the fundamental component
and the second one (Rh ) is adaptive to track the harmonic
component with higher magnitude. The Figure 3(b) shows the
detected frequency hω f being used as feedback for (Rh ).
vPCC
SOGI-PLL
cos(θf )
vdc*
PI
vdc
I*
i*(t)
iS(t)
iG(t)
i *S (t)
ihf (t)
Harmonic
Detection
PMR
v*
PWM
iS(t)
hωf
u
2
(b)
Fig. 3. Complete control strategy. (a) Boost control loop. (b) Inverter
control loop.
III. ADAPTIVE CURRENT HARMONIC DETECTION
METHOD
In order to compensate just one harmonic component of the
load current, this work proposes a detection method able to
detect the harmonic component of the load current with higher
amplitude. This strategy is based on SOGI-PLL structure, as
shown in Figure 4. The complete description about SOGI-PLL
can be found in [2], [20].
The adaptive filter SOGI is tuned at resonant frequency
detected by PLL, the closed-loop transfer functions of the
SOGI are defined in (2) and (3). This feedback frequency is
achieved locking the PLL in order to ensure the rotation of the
space phasor orthogonal to the q-axis.
Hd (s) =
iLα(s)
kωs
= 2
iL (s)
s + kωs + ω 2
(2)
Hq (s) =
iLβ (s)
kω 2
= 2
iL (s)
s + kωs + ω 2
(3)
where:
iLα - generated phase current;
iLβ - generated quadrature current;
ω
- adaptive filter resonance frequency;
- adaptive filter damping factor.
k
SOGI has a bandpass filter characteristic, with adjustable
resonance frequency. These transfers function shows that the
bandwidth of the SOGI only depends on the damping factor
[2]. These SOGI characteristics are explored in the current
proposed detection method.
The current harmonic detection method is composed
by two-cascaded SOGI-PLL, as detailed in Figure 5. The
first stage consists to detect the load fundamental current
component i f (t). In steady state, the measured load current
iL (t) is filtered by the SOGI at fundamental frequency detected
by PLL. Usually, as the fundamental component has a higher
amplitude in relation to the other
√ harmonics, the adaptive filter
damping factor (k) is set to 2 which results in an admissible
settling time [2]. The current component of the load i f (t) is
detected by (4). Therefore, the load current harmonic content
ih f (t) is detected subtracting the measured load current by the
fundamental component.
iLhα (t) = I3 cos(3ω f t) +
s
ωf
s
SOGI
iq
iLα
iLβ
dq
(5)
15I5 k[16sin(5ω f t) + 15kcos(5ω f t)]
225k2 + 256
(6)
9I5 k[16cos(5ω f t) − 15ksin(5ω f t)]
225k2 + 256
(7)
q
ILh,peak = i2Lhα (t) + i2Lhβ (t)
(8)
iLhβ (t) = I3 sin(3ω f t) +
Therefore, the harmonic component with higher amplitude
iLh (t) presents in the load current is detected by (9) and it
is sent to the control strategy for harmonic compensation.
ω
PI
αβ
1
id
θf
1
s
ωf
ωf
LPF
id2 + iq2
If,peak
LPF
Fig. 4. General structure of a single-phase SOGI-PLL [2].
(4)
The second stage consists to detect the load current
harmonic component with higher amplitude. The second
stage PLL extracts the angle and frequency of the harmonic
component with higher amplitude of the signal. This frequency
is used as feedback to the SOGI and it provides two quadrature
signals filtered at this frequency. This process eliminates the
harmonics of low amplitude. The SOGI-PLL structure for
harmonic component detection stage is similar to Figure 4.
For analysis, it is considered the load current content as
iLh (t) in (5). It is composed of 3rd harmonic with amplitude
I3 and a 5th harmonic with amplitude I5 , wherein I3 > I5 . From
(2) and (5), the phase current (in steady state) of the SOGI
output results in (6), considering the detected frequency by
PLL as 3ω f . In the same way, from (3) and (5), the quadrature
axis current (in steady state) results in (7). Note that, the
SOGI generates two quadrature signals filtered at the PLL
feedback frequency, as can be seen in the first terms in (6)
and (7). However, the second terms indicates that unwanted
frequencies are not fully suppressed.
To allow tracking of the harmonic component with higher
magnitude, as will be discussed,
√ the adaptive filter damping
factor at second stage is set to 2. Thus, Figure 6 shows the
time response (in steady state) of (6) and (7) for (ω f = 2π60
rad/s). The amplitude (ILh ) of the current component filtered
by SOGI is given in (8). Note that, once the signal is being
filtered in 3ω f , the expected amplitude value is of I3 . However,
due to the second terms of (6) and (7), oscillations appear
around I3 , as illustrated in Figure 6. These fluctuations can
influence in the harmonic detection and unwanted harmonics
can be injected at the system. For this reason, it is used a lowpass filter (LPF) in the detected amplitude (Figure 6) and at
PLL feedback frequency.
iLh (t) = I3 cos(3ω f t) + I5 cos(5ω f t)
1
k
ωhf
cos(θf )
iL(t)
if (t)
SOGI-PLL
iLh(t)
cos(θhf )
SOGI-PLL
ihf (t)
Ihf, peak
If, peak
Fundamental Component
Detection
Harmonic Component
Detection
Fig. 5. Current harmonic detection method based on SOGI-PLL.
iLhα
iLhβ
Amplitude
Amplitude+LPF
I3
Amplitude [A]
i f (t) = I f ,peak cos(θ f )
Adaptive Filter
iL(t)
-I3
2
2.01
2.02
2.03
Time [s]
2.04
2.05
2.06
Fig. 6. Direct iLhα and quadrature iLhβ current component generated
√
by SOGI with input of iLh for damping factor k = 2.
The detected harmonic frequency is used as feedback to the
resonant controller, as shown in Figure 3(b).
iLh (t) = Ih f ,peak cos(θh f )
(9)
For analysis of SOGI bandwidth influence at detection
method, considers that in a time t, the harmonic content
amplitudes changes from I3 > I5 to I5 > I3 in (5). However,
in the next sample time the SOGI frequency feedback is still
tuned in 3ω f by PLL. For this reason, if the SOGI bandwidth
is too small, the fifth harmonic is suppressed. Thus, the PLL
continues to detect the third harmonic component in steady
state. For second stage damping factor equal to 0.1, the Figure
7(a) shows the SOGI output and the unwanted suppression of
fifth harmonic in the next sample time after of the load change.
The ideal damping factor does not allows that the SOGI
bandwidth suppresses the harmonic with higher amplitude
during the load harmonic component variations, as depicted
in
√
Figure 7(b). In this case, the damping factor is set to 2. The
fifth harmonic has higher amplitude than third harmonic even
in the next sample time after of the load change. Therefore,
in the next samplings, the frequency detected by PLL tends to
the fifth harmonic. These interactions, between of SOGI-PLL
bandwidths suppress the third harmonic at steady state.
IV. RESULTS
A. Simulation Results
The case study presents a solar array with 3 parallel strings
of 13 panels of 48 W in series. The solar irradiance is
maintained in 500 W /m2 during the study. The boost and
i Lhα
I5
i Lhβ
Amplitude [A]
3
5
Harmonic Order
(a)
7
0 1 2
0
5
7
9
11
13
3
1
3
5
Harmonic Order
(b)
7
15
17
30
19
0<t<2 seconds
2<t<4 seconds
4<t<6 seconds
20
4
10
2
0
0 1 2
0
3
5
7
9
11
13
15
17
19
Harmonic Order
(b)
Amplitude [A]
Fig. 7. SOGI output in the next sample time after of the load
√ change
from I3 > I5 to I5 > I3 in (5). (a) For k = 0.1. (b) For k = 2.
6
30
20
10
0
4
2
0<t<2 seconds
2<t<4 seconds
4<t<6 seconds
0 1 2
0
3
5
7
9
11
13
15
17
19
Harmonic Order
(c)
Frequency [ Hz ]
Fig. 8. Current spectra during harmonic compensation of load
harmonic current component with higher amplitude. (a) Grid current
spectrum. (b) Inverter current spectrum. (c) Load current spectrum
100
50
30
20
10
0
0
0
2
4
6
0
2
Time [ s ]
(a)
Frequency [ Hz ]
300
180
0
2
4
6
Time [ s ]
(b)
420
4
5
4
3
2
0
6
0
2
Time [ s ]
(c)
4
6
Time [ s ]
(d)
Fig. 9. Fundamental and harmonic component of the load current
detected. (a) Fundamental current frequency. (b) Fundamental current
amplitude. (c) Harmonic current component frequency. (d) Harmonic
current component amplitude
1
0.8
0.6
0.4
PMR error [A]
inverter switching frequency is 10 kHz. Voltage at point of
common couple is 220 V. The inverter dc-bus voltage (Vdc) is
controlled in 420 V. The discrete simulation was implemented
in Matlab/Simulink environment with sample time of twice the
switching frequency.
The adaptive PMR controller is discrete by Tustin with
prewarping method to avoid deviations at resonant frequency
caused by the discretization, as depicted in [24]. The PMR
parameters design is done using stability analysis by means of
Nyquist diagrams, as proposed in [21], [24]. Therefore, The
PMR parameters is set to KP = 0.9, KI f = 1000 and KIh =
5000. The resonance frequency of R f is fixed at fundamental
frequency (ω f = 2π60) and the resonance frequency of Rh
(hω f ) is provided by the proposed detection method of the
load harmonic current with higher amplitude.
In this simulation, the load current harmonic contents
are changed to validate the proposed harmonic detection
method and the proposed adaptive control strategy. In the
first 4 seconds, the nonlinear loads connected to the PCC
are represented by current sources, injecting harmonics in the
system. Between 0 < t < 2 seconds, the load current has a 3rd
and 5th harmonic of 2 A and 0.5 A, respectively. Between 2 <
t < 4 seconds, the load current has a 3rd and 5th harmonic of 1
A and 3 A, respectively. After 4 seconds, a load with rectifier
diodes is connected at the PCC, generating odd harmonics on
the grid.
Current spectra during harmonic compensation of load
harmonic current component with higher amplitude are shown
in Figure 8. Between 0 < t < 2 seconds, the 3rd harmonic of
the load current is detected and strongly reduced in the grid
current. Between 2 < t < 4 seconds, the 5th harmonic of the
load current is detected and compensated by inverter. After 4
seconds, the 3rd harmonic of the rectifier diodes is detected
and compensated.
To validate the harmonic current detection method, the
Figure 9 shows the detection of load fundamental component
and harmonic component with higher amplitude. Note that, the
control does not destabilize even at frequency feedback during
load harmonic content variation, as illustrated in Figure 10.
The Table I details the total harmonic distortion (THD) of
the grid current (iG ) with and without harmonic compensation.
Note that, a strong improvement in the grid power quality
index. Between 0 < t < 2, e.g., the grid current THD reduces
from 15.38% to 4.19%.
6
Amplitude [ A ]
1
2
0<t<2 seconds
2<t<4 seconds
4<t<6 seconds
Harmonic Order
(a)
I
3
30
20
10
0
4
Amplitude [ A ]
I
6
3
Amplitude [A]
Amplitude [A]
i Lhβ
Amplitude [A]
i Lhα
I5
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
Time [ s ]
Fig. 10. Adaptive PMR controller error.
TABLE I
Total Harmonic Distortion (THD) of the Grid Current (iG )
Time
Intervals
0<t <2
2<t <4
4<t <6
iG THD without Harmonic
Compensation (%)
15.38
23.00
19.52
iG THD with Harmonic
Compensation (%)
4.09
9.06
7.93
This work presented a novel adaptive current harmonic
control strategy applied in multifunctional single-phase
photovoltaic inverters with proportional multi-resonant (PMR)
controller. This strategy is based on the proposed harmonic
current detection method of the nonlinear loads which consists
in a cascade association of two phase-locked loop based on
second order generalized integrator (SOGI-PLL).
The harmonic current detection method is frequency
adaptive and able to detect the load current harmonic with
higher amplitude and its frequency. Thereby, the PMR current
controller can be tuned for specific harmonics, increasing
efficiency and reducing the control algorithm complexity.
Simulation and experimental results show performance of
the proposed control strategy and the improvements in the
grid power quality index through harmonic compensation
generated for nonlinear loads connected to the point of
Is [A]
5
0
Grid Current
Inverter Current
Load Current
3.5
Amplitude [ A ]
3
2.5
2
1.5
1
0.5
0
1
3
5
7
9
Time [ s ]
11
13
15
Fig. 12. Spectrum of the grid, inverter and load current at experiment
initial state.
4
Grid Current
Inverter Current
Load Current
3.5
3
Amplitude [ A ]
V. CONCLUSIONS
4
2.5
2
1.5
1
0.5
0
1
3
5
7
9
Time [ s ]
11
13
15
Fig. 13. Spectrum of the grid, inverter and load current during
compensation of the 3rd harmonic.
4
Grid Current
Inverter Current
Load Current
3.5
3
Amplitude [ A ]
B. Experimental Results
To validate the adaptive current harmonic control strategy,
experiments are carried for single-phase inverter. The initial
load is an rectifier diodes. Figure 11 shows the initial state
of the grid, inverter and load current waveforms. Figure
12 illustrates the spectrum of these currents. Note that, the
inverter supplies a peak fundamental current of 1 A for load
and 2 A for grid, approximately. The nonlinear load harmonic
content has a 3rd harmonic component with higher amplitude
and there is no harmonic compensation, as can been seen in
the grid current spectrum.
The harmonic compensation is enabled and the harmonic
detector tracks the 3rd harmonic component. Figure 13 shows
the harmonic mitigation of the 3rd harmonic on the grid
current.
In a second scenario, the load is changed and the 5th
harmonic becomes the component with higher amplitude. The
harmonic detector tracks the 5th harmonic and tunes the PMR
at the related frequency. Figure 14 illustrated the harmonic
mitigation of the 5th harmonic on the grid current.
2.5
2
1.5
1
-5
0
20
40
60
80
100
0.5
(a)
0
IL [A]
5
1
-5
0
20
40
60
80
100
(b)
IG [A]
3
0
5
7
9
Time [ s ]
11
13
15
Fig. 14. Spectrum of the grid, inverter and load current during
compensation of the 5rd harmonic.
common coupling.
5
0
-5
ACKNOWLEDGEMENTS
0
20
40
60
80
100
(c)
Time [ms]
Fig. 11. Experiment initial state of the system currents waveforms.
(a) Inverter current waveform. (b) Load current waveform. (c) Grid
current waveform.
The authors would like to thank the Brazilian agencies
CAPES, FAPEMIG and CNPQ, which supported this work.
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