3D SPACE VECTOR MODULATION CONTROL OF FOUR-LEG
SHUNT ACTIVE POWER FILTER USING pq0 THEORY
ALI CHEBABHI1, MOHAMMED-KARIM FELLAH1, MOHAMED-FOUAD BENKHORIS2
Key words: Three-phase four-wire shunt active power filter (SAPF), Hysteresis
current control, 3D space vector modulation (3D SVM), Proportional
integral (PI), pq0 theory.
In this paper, the performances of two techniques for generating switching signals for a
three-phase four-wire four-leg shunt active power filter are compared under balanced
and unbalanced load conditions. The techniques used for this comparative study are the
traditional method of the hysteresis control and the method of 3D SVM control. The
comparisons are made on the basis of simulations in Matlab/Simulink environment.
Through this study, it is shown that the switching losses of the converter and the
harmonics of the source current are reduced for the 3D SVM technique. The regulation
of the current and voltage is accomplished by PI regulators, with use of the pq0 theory
in the αβο-axes for the generation of reference signal harmonic.
1. INTRODUCTION
The widespread use of non-linear loads is a source of great concern of
electric power suppliers. Distortion of electrical quantities derive from the
proliferation of polluting harmonics. Classically, these problems have been treated
with conventional passive LC filters. The main disadvantage of these filters is that
they provide a fixed solution and do not allow fine tuning. They, also, suffer from
congestion problems and aging [1, 2]. An effective solution to these problems is to
use shunt active power filters. This solution consists in using an electronic power
converter that injects currents or harmonic voltages in the electrical network, equal
to that absorbed by the non-linear load, but in phase opposition therewith [3].
The generation of the switching signals and trigger in electronic power
devices, the generation of reference signals, the harmonic current and dc bus
voltage control, are very important for controlling the shunt active power filter.
Two techniques for generating switching signals (Sa, Sb, Sc and Sn) to an four-leg
1
Djillali Liabes University of Sidi Bel-Abbes, ICEPS Laboratory (Intelligent Control &
Electrical Power Systems), Algeria; E-mail: chebabhiali@gmail.com, ali.chebabhi@univ-sba.dz
2
University of Nantes at Saint Nazaire, IREENA Laboratory (Institut de Recherche en
Electronique et Electrotechnique de Nantes Atlantique), France
Rev. Roum. Sci. Techn. – Électrotechn. et Énerg., 60, 2, p. 185–194, Bucarest, 2015
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Ali Chebabhi, Mohammed-Karim Fellah, Mohammed-Fouad Benkhoris
2
SAPF with have been shown in the literature, such as the generation of switching
signals by hysteresis [4, 5], and the generation of switching signals by PWM [6, 7].
In this study, we use a new control technique called 3D space vector modulation
(3D-SVM) technique for controlling the four-leg SAPF using pq0 theory.
This paper presents a comparative study of the switching signals
generation techniques based on the performance of the four-leg SAPF. The
switching techniques that are considered for comparative study are: (i)
Hysteresis Current Control, (ii) 3D-SVM. 3D-SVM ensures precise control
with reduced harmonic distortion and constant the switching frequency
compared to those obtained by the hysteresis. The four-leg SAPF topology
presented in this paper is shown in Fig. 1[6]. The main circuit contains a three
phase non-linear load composed of three single phase rectifier bridge followed
by RL load, connected to a three phase source in a point called the Point of
Common Coupling (PCC) [8–11].
Fig. 1 – Basic configuration of three-phase four-wire four-leg shunt active power filter.
3
3D space modulation control of shunt active power filter
187
2. MATHEMATICAL MODEL OF FOUR-LEG SAPF
The dynamic model of four-leg SAPF is defined in abc-axes [12, 13], as
given in equation (1)
di fa
⎧
+ v cha
⎪v fa = R f i fa + L f
dt
⎪
di fb
⎪
(1)
+ v chb .
⎨v fb = R f i fb + L f
dt
⎪
⎪ v = R i + L di fc + v
f fc
f
chc
⎪ fc
dt
⎩
Form the Eqs. (1), the model of this four-leg SAPF in the αβo-axes consist of
an clarke transformation of the currents and voltages in the abc-axes to the αβoaxes, and is expressed by the following equation:
⎧ di fα
Rf
1
1
=−
i fα +
v fα −
v chβ
⎪
Lf
Lf
Lf
⎪ dt
⎪⎪ di fβ
Rf
1
1
=−
i fβ +
v fβ −
v chβ .
⎨
Lf
Lf
Lf
⎪ dt
⎪ di f 0
Rf
1
1
=−
if0 +
vf0 −
v ch 0
⎪
Lf
Lf
Lf
⎪⎩ dt
(2)
3. SWITCHING SIGNAL GENERATION TECHNIQUES
3.1. HYSTERESIS CURRENT CONTROL TECHNIQUE
The principle of current control by the hysteresis is to maintain each of the
currents generated in a band surrounding the reference current as shown in Fig. 2 [5],
Fig. 2 – Principle of hysteresis band current control.
where S i and S i (i=a, b, c, n) are the upper and lower legs switching signals
respectively; i f αβ 0 and i *fαβ 0 are the references and injects currents, respectively.
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Ali Chebabhi, Mohammed-Karim Fellah, Mohammed-Fouad Benkhoris
4
3.2. THE 3D-SVM WITH NULL VECTOR
In 3D-SVM, there are 16 possible switching vectors: fourteen non-zero and
two null. The 3D-SVM is implemented in the following sequence [14, 15].
3.2.1. PRISM IDENTIFICATION
Six prisms in the 3D space can be identified and numbered as Prisms I
through VI Fig. 3. Within the selected prism, there are six non-zero switching state
vectors and two zero switching state vectors. Fig. 4 shows the physical positions of
the switching state vectors in αβ0-axes [7, 16].
Fig. 3 – Voltage vectors diagram of four-leg inverter in the αβο-axes, and the selection of prism.
3.2.2. TETRAHEDRON IDENTIFICATION
The tetrahedron m is formed by three non-zero voltage vectors and two other
zero vectors, as shown in Fig. 4a. Figure 4b describes the representation of
tetrahedron 1 plans belonging to prism I [16].
Fig. 4 – a) Representation of tetrahedron in the prism I; b) projection of the reference vector on the
adjacent vectors; c) duty cycles for the active vectors.
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3D space modulation control of shunt active power filter
189
3.2.3. DUTY CYCLE CALCULATION
The principle of 3D-SVM is to project the desired reference voltage vector on
three αβο-axes, as shown in Fig. 4c. We use these projections to calculate the
switching time [14, 16].
For the tetrahedron m of the prism j, the states vectors of the inverter (Vx, Vy,
Vz, V1, V16) are the adjacent vectors to the reference voltage vector. These vectors
G
are applied individually for certain periods tx, ty, tz and t0, such that the vector v *f is
equal to the mean value of those vectors for a period of switching.
3.2.4 THE CONTROL PULSES GENERATION
The Fig. 5 shows, on a switching period Th, the distribution of voltage vectors to be
applied in the first prism tetrahedron [15], where: Sa, Sb, Sc, Sn are the upper legs
switching signals; V1,…,V16 are the adjacent vectors to the reference voltage vectors.
Fig. 5 – Principle of generation of the pulses by 3D-SVM.
4. THE pq0 THEORY
This theory exploits the transformation of Concordia voltages and line
currents to calculate the instantaneous real, imaginary and zero-sequence powers
[5]. Figure 6 shows the block diagram of the pq0 theory [5].
The real power pch, the imaginary power qch and the zero-sequence power
pch0 are expressed by the following matrix:
⎡ p ch ⎤ ⎡ v chα
⎢ q ⎥ = ⎢− v
⎢ ch ⎥ ⎢ chβ
⎢⎣ p ch 0 ⎥⎦ ⎢⎣ 0
v chβ
v chα
0
0 ⎤ ⎡i chα ⎤
0 ⎥⎥ ⎢⎢ ichβ ⎥⎥
v cho ⎥⎦ ⎢⎣ ich 0 ⎥⎦
(3)
Components of instantaneous real, imaginary and zero-sequence powers can
be expressed as the sum of a dc and an ac components [19]
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Ali Chebabhi, Mohammed-Karim Fellah, Mohammed-Fouad Benkhoris
p ch ⎤
⎡ p ch ⎤ ⎡ p ch + ~
⎢ q ⎥ = ⎢ q + q~ ⎥ .
ch ⎥
⎢ ch ⎥ ⎢ ch
⎢⎣ p ch 0 ⎥⎦ ⎢⎣ p ch 0 + ~
p ch 0 ⎥⎦
6
(4)
When p ch , q ch and p ch 0 are dc components and ~
p ch , q~ch and ~
p ch 0 are ac
components, this dc and ac components are extracted by the low pass filter (LPF).
Fig. 6 – Block diagram of the pq0 theory.
From equation (3), we can deduce the corresponding current components:
⎡ i *fα ⎤
⎡v chα − v chβ
⎤⎡ ~
0
p ch ⎤
⎢ * ⎥
1
⎢
⎥⎢ ⎥
v
vchα
(5)
0
⎢ i fβ ⎥ = 2
⎥ ⎢ q ch ⎥ ,
2 ⎢ chβ
v
v
(
)
+
*
2
2
ch
ch
⎢i ⎥
⎢ 0
v chα + v chβ ⎥⎦ ⎢⎣ich 0 ⎥⎦
0
⎣
⎣ f0⎦
where p *f and q *f are the real and imaginary powers components, respectively
*
*
*
( p *f = ~
p ch
+ p dc
and q *f = q ch
).
5. PROPOSED PI CONTROL
The correctors used are proportional integral (PI) these are the most widely
used because they are easy to implement. However, the parameters of this corrector
depends on the settings of the system to regulate therefore a precise estimation of
these is necessary for good performance [17].
6. SIMULATION RESULTS AND DISCUSSION
Simulation results for the proposed control strategies are shown in Figs. 7–10.
The unbalanced in the loads is performed by the insertion at t = 0.4 s of a new
single-phase load (R = 5Ω, L = 10 mH) in parallel with the single-phase non-linear
load connected to the first rectifier bridge.
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3D space modulation control of shunt active power filter
191
Figures 7a and 8a show the currents waveforms of the source after filtering.
They are sinusoidal and still in phase with the corresponding voltage and the power
factor is unitary as shown in Figs. 7 and 8c. Figures 7b and 8b represent the load
current form, it is a non-sinusoidal and highly deformed current. Figures 7d and 8d
shows that after compensation the neutral current is in the range of ± 6 A with the
traditional hysteresis control and the range of ± 5 A with the 3D-SVM The dc bus
voltage is stabilized at its reference value with a small static error as shown in Figs.
7e and 8e. Figures 7f and 8f shows the real and imaginary power of load.
Fig. 7 – The hysteresis control simulation results: a) source currents; b) load current; c) voltage and
current source of the first phase; d) neutral current; e) dc link voltage; f) real and imaginary powers.
Fig. 8 – The 3D-SVM control simulation results: a) source currents; b) load current; c) voltage and
current source of the first phase; d) neutral current; e) dc link voltage; f) real and imaginary powers.
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Ali Chebabhi, Mohammed-Karim Fellah, Mohammed-Fouad Benkhoris
8
Figures 9a, b, c and d show the first phase source current's THD before and
after unbalanced load in (t < 0.4s) with the two techniques.
Figures 9a and b show the THD with hysteresis control technique before and
after unbalanced load respectively, the THD before unbalanced load is 1.4 % (Fig.
9a) and 1.97 % after unbalanced (Fig. 9b).
Figures 9 c and d show the THD with 3D-SVM control technique before and
after unbalanced load respectively, the THD before unbalanced load is 1.22 %
(Fig. 10a) and 1.94 % after unbalanced (Fig. 10b).
Figures 10 a and b show the switching signals in four-leg inverter. It is
constant in the 3D-SVM control. Further, it should be noted that control by 3DSVM is the best because it reduces the rate of THD and operates at constant
switching frequency.
Fig. 9 – THD of the source currents before and after unbalanced load with the two techniques.
Fig.10 – Switching pulses: a) the hysteresis current control; b) 3D-SVM control.
9
3D space modulation control of shunt active power filter
193
7. CONCLUSION
In this paper, the proposed switching technique for three-phase four-wire
four-leg SAPF in load conditions balance and unbalance is compared with
traditional hysteresis current control. From the results obtained, it is seen that the
compensation performance of 3-phases four-wire four-leg SAPF is almost similar
for the two methods. The switching frequency is constant in the 3D-SVM control,
unlike the hysteresis control wherein the frequency varies over a wide range.
The results from the simulation of the two switching techniques of control
show that these two switching techniques can compensate harmonic currents in the
network, can extract the zero-sequence current and reduce the magnitude of the
neutral current. The two switching techniques, performed into αβ0-axes, are
implemented using pq0 theory. However, we demonstrate that the 3D-SVM is
better because it reduces the rate THD of network side, reduces the magnitude of
the neutral current and compensate the reactive power to the system.
The Table 1 gives the THD and amplitude of the neutral current comparison
with 3D-SVM and hysteresis control using Matlab/Simulink.
Table 1
Comparison of the two techniques
Source current THD
The amplitude of the neutral
current
Hysteresis Control
3D-SVM Control
Balanced load Unbalanced load Balanced load Unbalanced load
1.4 %
1.97 %
1.22 %
1.94 %
6A
5A
APPENDIX 1. SIMULTION PARAMETERS
The system parameters considered for simulation are given in Table 2.
Table 2
System parameters for simulation and load specifications.
Capacitance of the capacitor
dc bus voltage Vdc
Coupling impedance Rf ,Lf
The source voltage and frequency
Source parameters Rs ,Ls
Line parameters Rch ,Lch
Load characterisitics Rl ,Ll
Parameters of unbalanced load R , L
Received on April 28, 2014
5 mF
800 V
0.1 mΩ, 0.1 mH
220 V, 50 Hz
1 mΩ, 1 mH
1 mΩ, 1 mH
5 Ω, 10 mH
5 Ω, 10 mH
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Ali Chebabhi, Mohammed-Karim Fellah, Mohammed-Fouad Benkhoris
10
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