International Conference on Control, Engineering & Information Technology (CEIT’14)
Proceedings -Copyright IPCO-2014
ISSN 2356-5608
Advanced Nonlinear Control of Three Phase Shunt Active Power
Filter
Y. Abouelmahjoub1, F.Z. Chaoui1, A. Abouloifa2, F. Giri3, M. Kissaoui1
Abstract— The problem of controlling three-phase shunt active
power filter (TPSAPF) is addressed in this paper in presence of
nonlinear loads of rectifier type. The control objective is
twofold: (i) compensation of harmonic currents and the reactive
power absorbed by the nonlinear load; (ii) regulation the DC
bus voltage of the converter. A nonlinear controller which has a
cascade structure comprises two loops, an inner-loop which is
formed by two non-linear regulators is designed, using the
Backstepping technique, to provide the harmonic compensation
and an outer-loop is designed to regulate the DC bus voltage of
the converter. Controller performances are illustrated by
numerical simulation in Matlab/Simulink environment.
currents and reactive power absorbed by the non-linear load.
The outer-loop is designed to regulate the DC bus voltage of
the converter. This theoretical result is confirmed by
numerical simulations.
I. INTRODUCTION
II. ACTIVE POWER FILTER MODELING
Nowadays, with the wide use of nonlinear loads and
electronic equipment in distribution systems, the problem of
power quality (PQ) has become increasingly serious. This
fact has lead to more stringent requirements regarding PQ
which include the search for solutions for such problems [1].
A. Active Power Filter Topology
A reduced-part three-phase shunt active power filter
under study has the structure of Figure 1. It also has two-Leg
less than full-bridge configuration. In the DC side it has two
identical capacitors values of energy storage C f . In the AC
The paper is organized as follows: The system includes
the electric network and the DC/AC converter is modeled in
section 2, the control problem is formulated in section 3
which also includes the design, the stability analysis in
section 4. Performances controller are illustrated by
simulation in section 5. A general conclusion ends the paper
in the last section.
In the case of harmonic pollution, the solution can
consider the use of passive filters however these filters have
the disadvantage of having a fixed compensation and can
generate resonance problems. In this way, the active power
filter (APF) appears as the best dynamic solution for
harmonic compensation [2]. In concrete, active power filters
are devices designed to improve the power quality in
distribution networks. In order to reduce the injection of nonsinusoidal load currents, shunt active power filters can be
connected in parallel with the nonlinear loads.
side the TPSAPF is connected in parallel with a nonlinear
load to the electric network which is modeled by three
sinusoidal AC voltage sources that form a directly balanced
three-phase system. The circuit operates according to the well
known
Pulse
Width
Modulation
principle
(PWM),[7],[8],[9],[10]. Finally the supply network contains
three voltages that form a direct balanced three-phase system
is given by:


 Sin (ωn t )
 vn1 ( t ) 



2π

 vn2 ( t )  = E  Sin  ωn t − 3


 v (t ) 
 n3 

4π
 Sin  ωn t −
3






(1)



 

where E and ωn denote respectively the amplitude and the
frequency of system ( vn1 ( t ) , vn2 ( t ) , vn3 ( t ) ).The resulting
In recent years, three-phase four-wire shunt active power
filters have appeared as an effective method to solve the
problem caused by harmonic and unbalanced currents as well
as to compensate load reactive power. In these filters, three
topologies for current-controlled voltage source inverter are
commonly used, namely the Four-Leg Full-Bridge (FLFB)
topology, the Three-Leg Split-Capacitor (TLSC) topology
and the four-leg split-capacitor (FLSC) topology. These
topologies were presented in the early 90s [3], and the
research work has been done on only those topologies.
Indeed many publications on their control have appeared ever
since [4]-[6].
current in three-phase load is given by its Fourier expansion:


 Sin ( hωn t + ϕ h )
 iL1 ( t )  ∞



2π

I h  Sin  hωn t + ϕ h −
 iL2 ( t )  =
3

 i ( t )  h =1 
 L3 

4π
 Sin  hωn t + ϕ h −
3


In this paper, a control strategy is developed for other
topology named: Three-phase shunt active power filter or
Two-Leg Split-capacitor (Two LSC) in the presence of nonlinear loads. A nonlinear controller that has a cascade
structure has two nested loops, an inner loop which is formed
by two non-linear regulators is designed, using the
Backstepping technique, to ensure compensation of harmonic
∑
95







 

(2)
International Conference on Control, Engineering & Information Technology (CEIT’14)
Proceedings -Copyright IPCO-2014
ISSN 2356-5608
˜
N
vn1
in1
iL1 Lo
vn2
in2
iL2 Lo
in3
iL3 Lo
vn3
S2
u2
i1
Cf
S1
u
v1
The switching function µk of the inverter is defined by:
Non
Linear
Load
+1
−1
µk = 
is ON
If
Sk
is OFF
and
and
Sk + 2 is OFF
Sk + 2 is ON
Rf Lf
1
Rf Lf
vf2
Rf
vf3
S4
Sk
k ∈ {1,2}
if1 if2 if3
vf1
i2
Cf
If
M
Figure 1.
(5a)
v f 2 = v f 2M + vMN
(5b)
v f 3 = v f 3M + vMN
(5c)
Lf
S3
v2
v f 1 = v f 1M + vMN
(
)
as v f 1 , v f 2 , v f 3 is a symmetric system then
Three-phase shunt APF
B. Active Power Filter Modeling
vf 1 + vf 2 + vf 3 = 0
(5d)
The modeling of the three-phase shunt APF is performed
by applying the usual Kirchhoff’s laws. Doing so, one easily
gets:
1
 µ + µ2  
vMN = −  vo + v2 +  1
vo 
3
2  

(5e)
(5f)
Lf
Lf
Lf
Cf
di f 1
dt
di f 2
dt
di f 3
dt
+ R f i f 1 = v f 1 − vn1
(3a)
 2 µ − µ2
vf 1 =  1
6

+ R f i f 2 = v f 2 − vn2
(3b)
v
 µ + µ2 
vf 2 =  1
vo − d

3
 6

(5g)
+ R f i f 3 = v f 3 − vn3
(3c)
v
 2 µ − µ1 
vf 3 =  2
vo + d

6
6


(5h)
dv1
= i1
dt
(3d)
dv
C f 2 = i2
dt
(3e)
(4a)
v f 2M = v2
(4b)
di f 2
dt
di f 3
dt
 1 + µ2 
=
 ( v1 + v2 )
 2 
 1 + µ1 
 1 + µ2
i1 = − 
if 1 −  2
2



Substituting (4d-e) and (5f-h) in (3) one obtains the
instantaneous model of the filter:
dt
 1 + µ1 
v f 1M = 
 ( v1 + v2 )
 2 
v f 3M
with vo = v1 + v2 and vd = v1 − v2
di f 1
The three-phase shunt active power filter undergoes the
equations:
(4c)

if 3

 1 − µ1 
 1 − µ2 
i2 = 
if 1 + 

if 3
 2 
 2 
vd

 vo + 6

=−
=−
=−
 2 µ − µ2
if 1 +  1
 6 Lf
Lf

Rf
 µ + µ2
if 2 −  1
 6 Lf
Lf

Rf

v
v
 vo − d − n2

3L f L f

 2 µ − µ1 
v
v
if 3 +  2
 vo + d − n3
 6 Lf 
Lf
6 Lf Lf


Rf
(4d)
 µ1i f 1 + µ2 i f 3
dvo
= −

dt
Cf

(4e)
 if 1 + if 3
dvd
= −
 Cf
dt

96

v
v
 vo + d − n1

6
L
Lf
f









(6a)
(6b)
(6c)
(6d)
(6e)
International Conference on Control, Engineering & Information Technology (CEIT’14)
Proceedings -Copyright IPCO-2014
ISSN 2356-5608
x4
vn3 x
The binary control signals µ 1 , µ 2 are produced by a
PWM generator and take its values in the finite set {-1, 1},
the switched converter model (6) is a variable structure
system and therefore inadequate for the design of a
continuous control laws. To overcome this difficulty we
resorted to using the averaged model where the different state
variables and control laws are replaced by their average
values over cutting intervals [11]. Since equation (6b) is a
linear combination of the equations (6a) and (6c) then the
obtained averaged model of the three-phase shunt APF is the
following:
x1 = −
Rf
Lf
x1 +
v
U1
1
x3 +
x4 − n1
Lf
6 Lf
Lf
x
x3
x1
x2
x3
vn1
Regulator 1
Regulator 2
x*2
U2 (u1, u2)
x*1
U1 (u1, u2)
+ iL1
iL3 +
vn3
x4
x
u2
-
u1
-
Three-phase Shunt
Active Power
Filter
x2
(7a)
vn1
x1
x
+
β-
Regulator 3
Rf
v
U
1
x2 = −
x2 + 2 x3 +
x4 − n3
Lf
Lf
6 Lf
Lf
u x +u x
x3 = −  1 1 2 2

Cf

x +x
x4 = −  1 2
 Cf

(7b)
+




*
y
( )2
(7c)
y
( )
2
x32
(
x3*2




(7d)
Figure 2.
The structure of the controller
A. Current inner loop design
with
 2u − u 
U 1 =  1 2  and
6


 2u − u 
U2 =  2 1 
6


The currents in1 , in2 and in3 delivered by the supply
network must be a sinusoidal signals in phase respectively
with the network voltage vn1 , vn2 and vn3 . Indeed the
currents x1 , x2 injected by three-phase shunt active power
filter should follow the best as possible respectively the
reference x*1 , x*2 defined by:
where x1 , x2 , x3 , x4 , u1 and u2 respectively denote the
average values over
cutting periods of the signals
i f 1 , i f 3 , vo , vd , µ1 and µ2 . The system (7) is clearly
nonlinear because of the product between the state
variables x1 , x2 , x3 , x4 and the control inputs u1 and u2 . It
is assumed that all signals are measurable.
 x1* = iL1 − β vn1 


 x* = i − β v 
n3 
 2 L3
III. CONTROLLER DESIGN
(8)
Let’s introduce the tracking errors on the currents x1 , x2 :
The control objective of the three-phase shunt APF is to
compensate the harmonics and reactive power required by the
nonlinear load. The controller synthesis is carried out by
cascading two loops. First an inner current loop which is
formed by two non-linear regulators is designed to
compensate the reactive and harmonic power. Second an
external loop is constructed to ensure control of the voltage
in the DC side of TPSAPF. The controller has the structure
of Figure 2. The currents i f 1 , i f 3 are controlled in the system
and due to symmetry of the system i f 2 is being controlled.
*
e   x − x 
e= 1= 1 1 
*
 e2   x2 − x2 
(9)
Given (7a) and (7b), time derivative of errors e1 , e2 are
written as follows:
 Rf

U
v
1
x1 + 1 x3 +
x4 − n1 − x*1 
−
Lf
6 Lf
Lf
 e1   L f

(10)
 =

e
R
U2
vn3
1
f
 2 
*
−
x +
x +
x −
− x2
 Lf 2 Lf 3 6 Lf 4 Lf



Since the controls U1 and U 2 appear in (10) after a
single derivation of the tracking errors, then the control laws
97
International Conference on Control, Engineering & Information Technology (CEIT’14)
Proceedings -Copyright IPCO-2014
ISSN 2356-5608
of the currents controller will be elaborated in a single step
using Lyapunov function [12].
V=
1 T
1
e e = e12 + e22
2
2
(
)

2
4π 
3E 2
ϕ
 ϕ 
k =  ko + k1 L f ωn I1Cos  1 +
Sin  1   , ko =

3
3 
Cf
 2
 2 

(11)
k1 =
Its dynamic is given by:
V = e1e1 + e2 e2
(12)
3E
Cf
q ( t ) = q1 ( t ) + q2 ( t ) + q3 ( t )
The choice V = −c1e12 − c2 e22 ensuring the asymptotic
stability of (10) with respect to the Lyapunov function (11)
where c1 0 and c2 0 are a design parameters. Indeed
this choice would imply:
where q1 ( t ) , q2 ( t ) and q3 ( t ) are given in the appendix 1.
 e1   −c1e1 
 =

 e2   −c2 e2 
law so that the squared voltage y = x32
The ratio β stands as a control input in the first-order
system (16). The problem at hand is to design for β a tuning
(13)
*
Bearing in mind the fact that β
reference signal y =
and its derivative should be available (Proposition 1), a
filtered PI control law is considered:
Comparing (13) and (10) gives the following:

R x +v −
 U 1  1  f 1 n1

=
 
 U 2  x3  R x + v −
 f 2 n3

x4

+ L f x1* − L f c1e1 
6

x4

*
+ L f x2 − L f c2 e2 
6

tracks a given
x3*2 .
(14)
β = e5 =
b
( c3 e3 + c4 e4 )
b+s
(17)
One deduce the expressions of effective control laws u1 and
t
u2
with
 u1   4U 1 + 2U 2 
 =

 u2   2U 1 + 4U 2 
0
(15)
At this point, the regulator parameters ( c3 , c4 , b ) are any
positive real constants. The next analysis will make it clear
how these should be chosen for the control objectives to be
achieved.
Proposition 1: Consider the three-phase shunt APF of
Fig.1 described by its averaged model (7) with the control
laws (15). If
∫
e3 = y* − y , e4 = e3 dτ
diL
is measurable, the signal β and its first
dt
time-derivative are available then the inner loop system is
globally asymptotically stable and its dynamic is described
by (13).
IV. ANALYSIS OF THE CLOSED-LOOP CONTROL SYSTEM
In the following Theorem, it is shown that the control
objectives are achieved (in the mean).
B. Voltage outer loop design
Theorem 1 (main result). Consider the three-phase shunt
active power filter shown by Figure 1 represented by its
average model (7), together with the control system
consisting of the inner-loop regulator (15) and the outer-loop
regulator (17). Then the closed-loop system has the following
properties:
The purpose of the outer loop is to generate a tuning law
for the ratio β in a way that makes the output voltage x3
regulated to a given reference value x3* . To this end, the
relation between β and x3 must be made clear.
1) Let the reference signal y* be any positive constant signal.
Replacing in (7c) the effective laws u1 and u2 defined by
their expressions (15) while replacing U1 and U2 by (14) an
supposing negligible parasitic resistance R f , one obtains:
There exists a positive real ε * such that if ε ≤ ε * then, the
tracking errors e1 , e2 , e3 , e4 and the ratio β are harmonics
signals that continuously depends on
y = k β − k1 I1Cos (ϕ1 ) + q ( t )
(16)
e1 = e1 ( t,ε ) ,
β = β ( t ,ε )
with
98
e2 = e2 ( t,ε ) ,
e3 = e3 ( t ,ε ) ,
ε = 1 / ωn , i.e.
e4 = e4 ( t,ε ) ,
International Conference on Control, Engineering & Information Technology (CEIT’14)
Proceedings -Copyright IPCO-2014
ISSN 2356-5608
2) Furthermore, when ε → 0 , errors ( e1 ,e2 ,e3 ) vanishes and
e4 , β converges:
T
P = ( 0 0 k1 I1Cos (ϕ1 ) 0 0 )
lim e1 ( t ,ε ) = 0 , lim e2 ( t,ε ) = 0 , lim e3 ( t,ε ) = 0 ,
ε →0
ε →0
lim e4 ( t,ε ) =
c4 k
ε →0
In order to get stability results regarding the system of
interest (19), it is sufficient (thanks to averaging theory) to
analyze the following averaged system:
ε →0
k1 I1Cos (ϕ1 )
, lim β ( t ,ε ) =
k1 I1Cos (ϕ1 )
ε →0
(20)
Z = ε AZ + ε P
k
(21)
To this end, notice that (21) has a unique equilibrium at:
Proof. First, notice that (17) guarantees that β and its
first time-derivative are available. Then, by Proposition 1, the
errors e1 , e2 undergoes (13). This together with (16) and
(17) show that the augmented state vector:
Er = ( e1
e2
e3
e4

Z* =  0 0 0


k1 I1Cos (ϕ1 )
c4 k
T
k1 I1Cos (ϕ1 ) 

k

(22)
On the other hand, as (21) is linear, the stability
properties of its equilibrium are fully determined by the statematrix A More specifically, the equilibrium Z * will be
globally exponentially stable if the matrix A is Hurwitz. To
this end, we note that the eigen values are zeros the following
(18) characteristic polynomial:
T
e5 )
undergoes the following state equation:
E r = AEr + P ( t )
where
det ( λ I − A ) = a5 λ 5 + a4 λ 4 + a3 λ 3 + a2 λ 2 + a1λ + a0
 −c1

 0
A= 0

 0
 0

0
−c2
0
0
0
0
0
0
1
bc3
0
0
0
0
bc4
0 

0 
−k 

0 
−b 
Where
a5 = 1, a4 = b + c1 + c2 , a3 = c1c2 + b ( c1 + c2 ) + kbc3
a2 = bc1c2 + kbc3 ( c1 + c2 ) + kbc4 ,
a1 = kbc1c2 c3 + kb ( c1 + c2 ) c4 , a0 = kbc1c2 c4
T
P ( t ) = ( 0 0 k1 I1Cos (ϕ1 ) − q ( t ) 0 0 )
Putting
The stability of the above system will now be analyzed
using the averaging theory. Now introduce the time-scale
change τ = ωn t . It is readily seen from (18) that
Z( τ ) ≡ Er ( τ / ωn ) undergoes the differential equation:
dZ (τ )
dτ
= ε AZ (τ ) + ε P (τ )
(19)
where
The application of algebraic Routh criterion implies that the
system is stable at the condition that:
T


 τ 
P (τ ) =  0 0 k1 I1Cos (ϕ1 ) − q 
 0 0 

 ωn 


a4 > 0 , b1 > 0 ,
Now, let us introduce the average functions:
1
ε →0 2π
Z = lim
1
P = lim
ε →0 2π
a4 a3 − a5 a2
a a −a a
, b2 = 4 1 5 0 , b3 = 0 ,
a4
a4
b1 a2 − a4 b2
b1a0 − a4 b3
b4 =
, b5 =
, b6 = 0 ,
b1
b1
b b −b b
b b −b b
b7 = 4 2 1 5 , b8 = 0 , b9 = 0 , b10 = 7 5 8 4 ,
b4
b7
b11 = 0 , b12 = 0
b1 =
b4 > 0 , b7 > 0 , b10 > 0
The equilibrium Z * of the linear system (21) is actually
globally exponentially stable. Applying Theorem 4.10 in
[13], one concludes that there exists a ε * > 0 such that for
ε < ε * , the differential equation (19) has a harmonic solution
Z = Z (t , ε ) that continuously depends on ε . Moreover, one
2π
∫ Z (τ )dτ
0
2π
∫ P (τ )dτ
has
0
lim Z (t , ε ) = Z * . This, together with (22), yields in
ε →0
particular that:
It follows from (19) that :
99
International Conference on Control, Engineering & Information Technology (CEIT’14)
Proceedings -Copyright IPCO-2014
ISSN 2356-5608
lim e1 ( t ,ε ) = 0 , lim e2 ( t,ε ) = 0 , lim e3 ( t,ε ) = 0 ,
ε →0
ε →0
lim e4 ( t,ε ) =
sinusoidal and in phase with the network voltages vn1 which
shows that the power factor correction is achieved. Fig. 7
shows the control laws u1 and u2 in inner-loop. Fig. 8 shows
the waveform of the load current iL1 we observe that the wave
current are almost square in shape so it is rich in harmonics.
The compensation current i f 1 delivered by the three-phase
ε →0
k1 I1Cos (ϕ1 )
c4 k
ε →0
Then, using (17) one gets
lim β ( t ,ε ) =
shunt active power filter is shown in Figure 9. Fig. 10 shows
that the network currents in1 is sinusoidal and therefore
testifies the good performances of the inner loop.
k1 I1Cos (ϕ1 )
k
ε →0
1520
The Theorem is thus established.
1500
V. NUMERICAL SIMULATIONS
1480
In order to simulate the behavior of the three-phase shunt
active power filter shown in Figure 1, the chosen nonlinear
load is a three-phase full bridge rectifier which supplies an
inductive load comprising a resistor RL and an inductor LL ,
the coil Lo is inserted to the input of three-phase Rectifier
diL
. The performances of the proposed
dt
controller are now numerically evaluated using a TPSAPF
based system with the following characteristics:
1460
1440
1420
1400
1380
0
0.1
0.2
Figure 3.
Bridge to limit the
0.3
Time(s)
0.4
0.5
0.6
Output voltage vo
1400.4
1400.2
TABLE I.
PARAMETERS SIMULATION VALUES
1400
Parameters
Network
Symbol
E
f
Rf
Three-phase active shunt
power filter
Lf
Cf
Currents regulators
Voltage regulator
1399.8
220 2 V
1399.6
50 Hz
1399.4
0.2
80 mΩ
3 mH
9000 µF
Lo
c1 = c2
10000
0.4
c3
2 × 10
c3
b
6 × 10
1000
0.24
0.26
0.28
0.3
Figure 4. Ripple in the Output voltage vo
0.8
LL
0.22
Time(s)
20 Ω
500 mH
5 mH
RL
Rectifier
Values
0.6
−6
0.2
−5
0
0
0.1
0.2
Figure 5.
The objective of the simulation is to illustrate the
behavior of the controller in response to progressive changes
of the voltage reference x3* . More specifically, the voltage
reference goes from 1400V to 1500V. The performances of
the controller are illustrated by Figs 3 to 10. As predicted by
Theorem 1, the output voltage vo converges, in the mean, to
its reference value with a good accuracy Figure 3, a zoom is
made in Figure 4 it is observed that the voltage ripples
oscillates at the frequency 2ωn , but their amplitude are too
small compared to the average value of the signals,
confirming thus Theorem 1. Fig. 5 shows that the ratio β
always takes (in the mean) a constant value after transitional
periods. Fig. 6 clearly shows that the network current in1 is
100
0.3
Time(s)
0.4
0.5
0.6
Control signal β
80
(vn1) / 5
in1
60
40
20
0
-20
-40
-60
-80
0
0.05
0.1
Time(s)
0.15
Figure 6. PFC checking: in1 in phase with vn1
0.2
International Conference on Control, Engineering & Information Technology (CEIT’14)
Proceedings -Copyright IPCO-2014
ISSN 2356-5608
1
APPENDIX
u1
u2
0.5
2L f β E
q1 ( t ) =
0
-1
0
2L f β 
4π

 iL 3 ( t ) − I1Sin  ωn t + ϕ1 −
Cf 
3

+
-0.5
0.02
0.04
0.06
0.08
2L f β
0.1
Time(s)
+
vn 3 ( t )
Cf
Figure 7. Inner-loop Control
2
30
−
−
0
-10
−
-20
-30
0.1
0.11
0.12
0.13
Time(s)
0.14
0.15
0.16
Cf
Cf
Cf
2 L f c1e1
Cf
−
q2 ( t ) =
0
-5
-10
0.11
0.12
0.13
Time(s)
0.14
0.15
0.16
+
0.15
iL 3 ( t )
(
Cf
Cf
4L f β
−
VI. CONCLUSION
−
Cf
vn1 ( t )
4 L f e1β
Cf
Cf
−
+
2L f
Cf
4L f
Cf
4L f β
Cf
iL1 ( t )
(
dt
+
dt
−
(
iL1 ( t ) iL1 ( t ) − I1Sin (ωnt + ϕ1 )
Cf
) − 4L f β 2 v
2 L f e2
Cf
4π 

Sin  2ωnt + 2ϕ1 −
3 

2 L f β EI1ωn
dt
Cf
2 L f β e1 dvn3 ( t )
Cf
diL1 ( t )
( iL1 ( t ) − I1Sin (ωnt + ϕ1 ) )
d iL1 ( t ) − I1Sin ( ωnt + ϕ1 )
dt
L f I12ωn
vn1 ( t ) −
Sin ( 2ωnt + ϕ1 ) +
2 L f e1 diL 3 ( t )
Cf
)
)
iL1 ( t ) +
4 L f β EI1ωn
Figure 10. Network current in1
101
Cf
vn3 ( t )
Cf
4 L f e1
0.16
In this paper, a nonlinear controller is proposed for the
three-phase shunt active power filters. It aims to compensate
the harmonic current and reactive power caused by non-linear
loads. The problem is processed using a non-linear controller
with two cascaded loops. The inner loop which is formed by
two non-linear regulators is designed, using the Backstepping
technique, to compensate the current harmonics. The outer
loop is designed to regulate the DC bus voltage of the
inverter. The simulation results show that it performs
perfectly the objectives of pursuit. In addition, it is robust
against changes in setpoint DC bus voltage.
4 L f c2 e2
4 L f e1 diL1 ( t ) 4 L f e1β dvn1 ( t )
x4
βx
+
iL1 ( t ) − 4 vn1 ( t ) −
Cf
Cf
Cf
dt
Cf
dt
-20
0.14
Cf
vn 3 ( t ) +
2 EI1
4
Cos ( 2ωnt + ϕ1 ) −
vn1 ( t ) iL1 ( t ) − I1Sin (ωnt + ϕ1 )
Cf
Cf
-10
0.13
Time(s)
2 L f c1e1β
+
+
0.12
vn3 ( t )
8L f β d ( vn3 ( t ) iL 3 ( t ) ) 16 L f dvn23 ( t )
+
−
Cf
dt
Cf
dt
(
−
0.11
Cf
2
4π 
β E2

vn3 ( t ) iL1 ( t ) − I1Sin (ωnt + ϕ1 ) −
Cos  2ωnt −

Cf
Cf
3 

10
-30
0.1
Cf
4 L f e2 β
iL 3 ( t ) +
−
30
0
4 L f e2
2β E 2
2 EI1
4π 

Cos ( 2ωnt ) +
Cos  2ωnt + ϕ1 −
Cf
Cf
3 

Figure 9. Compensation current i f 1
20
−
 4 L f e2 diL 3 ( t )
− C
dt

f
−
-15
-20
0.1
4π

Sin  2ωn t −
3

iL 3 ( t ) −
4 L f c2 e2 β
10
5
dt
dt
dt
  dvn1 ( t )
  dt

d ( iL1 ( t ) − I1Sin (ωn t + ϕ1 ) )
4 L f e2 β dvn3 ( t )
4 L f diL23 ( t )
Figure 8. Load current iL1
15
2
L f β E ωn
20
10
4π 

I1ωn Sin  2ωn t + 2ϕ1 −
3 

Cf
4π 

Sin  2ωnt + ϕ1 −

3 

)
n1
iL1 ( t ) −
(t )
dvn1 ( t )
dt
dvn1 ( t )
dt
2 L f e2 β
Cf
vn1 ( t )
International Conference on Control, Engineering & Information Technology (CEIT’14)
Proceedings -Copyright IPCO-2014
ISSN 2356-5608
REFERENCES
q3 ( t ) = +

4π  

d  iL 3 ( t ) − I1Sin  ωn t + ϕ1 −

3  


vn1 ( t )
dt
2L f β
Cf
L f β 2 E 2ωn
−
4π

Sin  2ωn t −
3

Cf
−
4 L f c1e1β
Cf
vn1 ( t ) +
2 L f c2 e2
Cf
[2]
 2 L f c1e1
iL1 ( t )
+ C

f
iL1 ( t ) −
2 L f c2 e2 β
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[3]
vn1 ( t )
[4]
EI
4π 

+ 1 Cos  2ωn t + ϕ1 −
3 
Cf

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[5]

2
4π  

vn1 ( t )  iL 3 ( t ) − I1Sin  ωn t + ϕ1 −

3  
Cf


β E2
Cf
4π

Cos  2ωn t −
3

−
+
−
2β E 2
8π

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2L f
Cf
−
vn3 ( t ) −
iL 3 ( t )
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f
f
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


f

4π
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dt
−
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[9]
[10]
iL 3 ( t )
4e1
vn1 ( t )
Cf
2
x e 4 L f e1e1 2 L f e1e2 4 L f c1e1
4e
− 1 vn3 ( t ) + 4 1 −
−
+
Cf
Cf
Cf
Cf
Cf
+
[1]
[11]
[12]
[13]
2 L f e2 e1
xe
2e2
2e
vn1 ( t ) − 2 vn3 ( t ) + 4 2 −
Cf
Cf
Cf
Cf
2 L f c1e1e2
Cf
+
4 L f c2 e22
Cf
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