Minimizing Inverter Capacity Design and Comparative Performance Evaluation of SVC-Coupling Hybrid Active Power Filters

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DOI: 10.1109/TPEL.2018.2828159
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Abstract
In this study a minimizing inverter capacity design is proposed, and the characteristics and performance of static VAR compensator (SVC) coupling hybrid active power filters (SVC-HAPFs) are analyzed. The cost of the SVC part is much lower than that of the active inverter part, and thus the reduction of inverter part capacity can lead to a decrease in the total cost of SVC-HAPF. To demonstrate the advantages of the proposed minimizing inverter capacity design of SVC-HAPF, comparisons are given between the conventional and the proposed SVC-HAPF designs. Comparisons are also provided among active power filters (APFs), hybrid active power filters (HAPFs), and the proposed SVC-HAPF in terms of DC-link voltage, compensation range, and performance. Finally, simulation and laboratory-scale experimental results are given to validate the minimizing inverter capacity design, and to verify the characteristics and compensation performances of APF, HAPF, and the proposed SVC-HAPF.
0885-8993 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2018.2828159, IEEE
Transactions on Power Electronics
AbstractIn this study a minimizing inverter capacity design is
proposed, and the characteristics and performance of static VAR
compensator (SVC) coupling hybrid active power filters
(SVC-HAPFs) are analyzed. The cost of the SVC part is much
lower than that of the active inverter part, and thus the reduction
of inverter part capacity can lead to a decrease in the total cost of
SVC-HAPF. To demonstrate the advantages of the proposed
minimizing inverter capacity design of SVC-HAPF, comparisons
are given between the conventional and the proposed SVC-HAPF
designs. Comparisons are also provided among active power
filters (APFs), hybrid active power filters (HAPFs), and the
proposed SVC-HAPF in terms of DC-link voltage, compensation
range, and performance. Finally, simulation and laboratory-scale
experimental results are given to validate the minimizing inverter
capacity design, and to verify the characteristics and
compensation performances of APF, HAPF, and the proposed
SVC-HAPF.
Index TermsStatic VAR compensator (SVC), active inverter,
DC-link Voltage, active power filters (APFs), hybrid active power
filters (HAPFs)
I. INTRODUCTION
URRENT quality compensators can be installed to solve
current quality problems such as a low power factor,
harmonic current pollution, and unbalanced problems. A
historical review of the various compensators is given below
and summarized in Table I.
To compensate for reactive power, low-cost passive power
filters (PPFs) [1] and static VAR compensators (SVCs) [2]-[4],
were proposed in the 1940s and 1960s, respectively. In
practical applications, PPFs and SVCs suffer from the potential
resonance problem [5], [6] due to system frequency variation
[7], parameter (inductor and capacitor) values changes [7], [8],
Manuscript received November 01, 2017; revised January 12, 2018 and
March 11, 2018; accepted April 13, 2018
This work was supported in part by the Science and Technology
Development Fund, Macao SAR (FDCT) (025/2017/A1, 109/2013/A3) and in
part by the Research Committee of the University of Macau
(MYGR2017-00038-FST, MYRG2017-00090-AMSV) (Corresponding
author: C.-S. Lam)
L. Wang is with the Department of Electrical and Computer Engineering,
Faculty of Science and Technology, University of Macau, Macao, China.
C.-S. Lam is with the State Key Laboratory of Analog and Mixed Signal
VLSI, University of Macau, Macao, China (e-mail: C.S.Lam@ieee.org;
cslam@umac.mo).
M.-C. Wong is with the Department of Electrical and Computer Engineering,
Faculty of Science and Technology, University of Macau, Macao, China, and
also with the State Key Laboratory of Analog and Mixed Signal VLSI,
University of Macau, Macao, China.
nonlinear dynamics characteristics in thyristor (SVCs only)
[9]-[12], etc. To overcome the resonance problem and to
improve performance, active power filters (APFs) were
proposed in 1976 [13], [14]. For medium voltage applications,
APF requires costly multilevel structures [15]-[19] to reduce
the voltage across the DC-link capacitor and power switches.
To reduce the voltage capacity of the inverter, large
LC-impedance coupling hybrid active filters (HAPFs) were
proposed in 2003 [20]. However, HAPFs have a narrow
compensation range, which limits their compensation ability
[20], [21]. In the same year, J. Dixon et al. [22] proposed an
SVC structure in parallel with APF (SVC//APF). The SVC part
of SVC//APF is used to compensate for most of the reactive
power, thus the current rating of the APF can be significantly
reduced. However, the voltage rating of the APF part remains
high, so the SVC//APF still requires a costly multilevel
structure for medium voltage level applications. Another
hybrid structure of PPF in parallel with APF (PPF//APF) was
proposed after 2010 in [23] and [24]. The PPF part of PPF//APF
can share the large compensating current and thus the current
rating of APF can be reduced. However, when the loading
reactive power is capacitive or has a wide reactive power
variation, the APF loses its small current rating characteristic.
In 2012, Luo et al. [25] proposed a novel hybrid system
consisting of a thyristor-controlled reactor (TCR) and a
resonant impedance-type hybrid active power filter
(RITHAPF) (TCR//RITHAPF). The RITHAPF part is an APF
crossing over a matching transformer connected in parallel with
a fundamental series resonant circuit. As most of the loads are
inductive and the TCR part is also inductive, the RITHAPF part
is required to provide a large capacitive compensating current
for inductive loads compensation, so that the current/power
rating of RITHAPF can be high. However, the matching
transformer in TCR//RITHAPF can also drive up the total
system cost. After 2014, SVC-coupling hybrid active power
filters (SVC-HAPFs) were widely studied [26]-[31], which
have the characteristics of a wider compensation range than
HAPF and a lower DC-link voltage than the APF. Of these
compensators, the PPF and HAPF have poor dynamic reactive
power tracking performance compared to the others, and the
tracking performances of APF, SVC//APF, PPF//APF,
TCR//RITHAPF, and SVC-HAPFs are better than those of
SVC.
The structure of SVC-HAPF was first proposed in [26],
and the different aspects of SVC-HAPFs discussed in [27]-[31].
The control strategies of SVC-HAPFs for balanced loads
compensation have been discussed in [26] and for unbalanced
Minimizing Inverter Capacity Design and
Comparative Performance Evaluation of
SVC-Coupling Hybrid Active Power Filters
Lei Wang, Member, IEEE, Chi-Seng Lam, Senior Member, IEEE, and Man-Chung Wong, Senior Member, IEEE
C
0885-8993 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2018.2828159, IEEE
Transactions on Power Electronics
loads in [27]. To overcome the potential over-capacity problem
of the SVC-HAPFs, the selective compensation control method
was proposed in [28] for harmonic distortion, and unbalanced
and reactive power. The focus of [26]-[28] is on the control
methods, while the focus of [29] and [30] is on the system
design of the SVC-HAPFs. Although the final selected
parameters are the same in [29] and [30], the SVC-HAPF
design approaches differ. In [29], the SVC-HAPF design was
for unbalanced loads compensation. In [30], the SVC-HAPF
was designed to compensate for fundamental reactive power,
while the current harmonic components were not taken into
consideration. The hardware implementation of the
SVC-HAPF was discussed in [31]. The minimizing inverter
capacity design of the SVC-HAPF was not considered in
[26]-[31], although a reduction in SVC-HAPF inverter capacity
can bring about decreases in the compensation device capacity,
power loss, and cost. Therefore, the parameter design
procedures for reducing the DC-link operation voltage of the
SVC-HAPFs are explored in this study. The proposed
SVC-HAPF design is also compared with the conventional
designs in [29] and [30].
TABLE I CHARACTERISTICS OF DIFFERENT CURRENT QUALITY COMPENSATORS
Year
Resonance
prevention
capability
Comp.
range
Tracking
performance
Power
loss
PPF [1]
1940s
Week
Narrow
Poor
Low
SVC [2]-[12]
1960s
Week
Wide
Medium
Low
APF [13]- [19]
1976
Strong
Wide
Good
High
HAPF [20],
[21]
2003
Strong
Narrow
Poor
Medium
SVC//APF
[22]
2003
Strong
Wide
Good
High
PPF//APF
[23], [24].
2010
Strong
Wide
Good
High
TCR//RITHA
PF [25]
2012
Medium
Medium
Good
Medium
Conventional
SVC-HAPFs
[26]-[31]
2014
Strong
Wide
Good
Medium
Proposed
SVC-HAPF
2017
Strong
Wide
Good
Low-
Medium
Based on the limitations of the conventional SVC-HAPFs in
[27]-[31], this study aims to:
enable a reduction in the power rating ratio between the
active inverter part and the SVC part through
mathematical analysis;
propose a minimizing DC-link voltage design for
SVC-HAPFs reactive power and harmonic currents
compensation (Section III);
compare the conventional design of the SVC-HAPFs [29],
[30] with the proposed one through analysis and
simulations (Section III);
perform a comprehensive study of DC-link voltage,
compensation, and tracking performances of the APF,
conventional HAPF, and SVC-HAPF (Section IV); and
provide power loss and efficiency analyses of the APF, the
conventional HAPF, and the proposed SVC-HAPF
(Section V).
The layout of this paper is as follows. In Section II, the
structures, modeling, and control are introduced for the APF,
HAPF, and SVC-HAPF. In Section III, the power rating ratio
analysis is proposed between the active inverter part and the
coupling SVC part of the SVC-HAPF, based on the structures
and modeling. The required DC-link voltage is also deduced
and analyzed for both the conventional and the proposed
SVC-HAPFs. The conventional HAPF and the proposed
SVC-HAPF are compared in Section IV. The experimental
verifications are provided in Section V for the conventional
HAPF and the proposed SVC-HAPF. Finally, a summary is
given in Section VI.
II. STRUCTURES, MODELING, AND CONTROL OF HAPF AND
SVC-HAPF
Figs. 1 and 2 show the structures of the APF, HAPF, and the
SVC-HAPF. The subscript “xstands for phases a, b, and c in
the following analysis. vsx and vx are the source and load
voltages; isx, iLx, and icx are the source, load, and compensating
currents, respectively. Ls is the transmission line impedance. In
Fig. 1, L is the coupling inductor of APF, and Lc and Cc are the
coupling inductor and capacitor of HAPF, respectively. Lc, LPF,
and CPF in Fig. 2 are the coupling inductor, SVC part inductor,
and capacitor, respectively.
Fig. 1 shows that the inverter voltage of APF is high due to
the small coupling inductive impedance L. The coupling LC
part (Lc and Cc) of HAPF is used to provide the fixed reactive
power compensation, and the active inverter part is used to
enlarge the reactive power compensation range and compensate
the harmonic current. Compared with HAPF, the SVC part of
SVC-HAPF provides a wide and continuous inductive and
capacitive reactive power compensation range, which is
controlled by the firing angles (
xof the thyristors. As the
SVC part can provide the fundamental voltage drop, the rating
of the active inverter part can be small and the use of a costly
multi-level structure is thus avoided [29]. In addition, the
isolation transformer is used to protect the active inverter part
from the inrush current.
The control model of the SVC-HAPF for simulation and
experiments is provided in Fig. 2. The inputs are load voltage
vx, load current iLx, and compensating current icx, and the
outputs are the trigger signals for the thyristors in the SVC part
and IGBTs in the active inverter part. For the SVC part control,
the first step is to calculate the load reactive power qLx via the
single phase instantaneous p-q theory [19], [20]. The required
fundamental impedance XSVCx of the SVC part can then be
obtained based on the load reactive power qLx and load voltage
vx. The corresponding firing angle
x of the SVC part can then
be obtained. Finally, by comparing the firing angle αx with the
phase angle of the load voltage
x, the trigger signals for the
thyristor of the SVC can be obtained.
The active inverter part is used to limit the compensating
current icx to its reference icx*, which includes the harmonic
component and reactive power component. By using the
instantaneous p-q theory [13], the instantaneous active power
and reactive power in

frame can be calculated from the
load voltage and current (vx and iLx). The obtained reactive
power and harmonic active power can then be used to calculate
the reference icx *. Through the current hysteresis pulse width
modulation (PWM) control method, the trigger signals for the
active inverter part can be generated by comparing icx with icx*.
Discussions of the trade-offs of different PWM control
methods are included in the Appendix.
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Transactions on Power Electronics
isa
isb
isc
iLa
iLb
iLc
ica
icb
Loads
A
B
C
va
vb
vc
vsa
vsb
vsc Ls
Ls
Ls
icc
Multi-level
Active
Inverter
part
VDC
+
-
.
.
.
.
.
.
.
.
.
CDC1
CDCn
vinva vinvb vinvc
+
-
+
-
+
-
Conventional
HAPF
Cc
Lc
L
APF
Fig. 1. Structure of the conventional multi-level APF and
HAPF
isa
isb
isc
iLa
iLb
iLc
ica
icb
Loads
A
B
C
va
vb
vc
vsa
vsb
vsc Ls
Ls
Ls
icc
CPF
Lc
LPF
T1a T2a
CPF
Lc
T2b
CPF
Lc
T2c
T1b T1c
SVC
part
CDC
Ta1
Ta2
Tb1
Tb2
Tc1
Tc2
vinva vinvb vinvc VDC
+
-
Active
Inverter
part
Coupling
Transformer
1:1 or
N1:1
iLa, iLb, iLc
va, vb, vc
Calculate
three
phase load
power
Calculate
Reference
Current
Current
PWM
Control
To
IGBTs
ica
*, icb
*,
icc
*
ica, icb, icc
qL
HPF
L
p
~
Comparator
a,
b,
c
qLa, qLb,
qLc
Calculate
SVC
Impedance
Calculate
firing angle
Phase Lock Loop
a,
b,
c
XSVCa,
XSVCb,
XSVCc,
Calculate
individual
load power
pL
To
Thyristors
Fig. 2. Structure and control method of the SVC-HAPF
Fig. 3 shows the single-phase equivalent fundamental and
harmonic frequency circuit models of the APF, HAPF, and
SVC-HAPF. The coupling impedances of the models are found
to be different. For the harmonic models in Figs. 3 (b) and (d),
the L part of APF and the LC part of the HAPF cannot generate
the harmonic current, while the SVC part of SVC-HAPF can
produce the harmonic current Iixn. Therefore, the compensating
current Icxn of SVC-HAPF includes both the load harmonic
current ILxn and the SVC part harmonic current Iixn.
Vsx1
Isx1
x=a,b,c
ILx1
Icx1
(=-ILxq1)
Vinvx1
At fundamental
frequency
Vx1
Ls
Cc
Lc
L
(a)
Isxn
x=a,b,c
ILxn
Icxn
(=-ILxn)
Vinvxn
At harmonic
frequency
Vxn
Ls
Cc
L
Lc
(b)
Vsx1
Isx1
x=a,b,c
ILx1
Icx1
(=-ILxq1)
Vinvx1
At fundamental
frequency
Vx1
Ls
CPF
Lc
LPF
(c)
Isxn
x=a,b,c
ILxn
Icxn
(=Iixn-ILxn)
Vinvxn
At harmonic
frequency
Vxn
Ls
CPF
Lc
LPF
(d)
Fig. 3. Single-phase equivalent circuit models of (a) the APF/HAPF
fundamental frequency model, (b) the HAPF harmonic frequency model, (c)
the SVC-HAPF fundamental frequency model, and (d) the SVC-HAPF
harmonic frequency model
III. RATING RATIO OF THE ACTIVE INVERTER PART AND THE
SVC PART, AND THE MINIMUM REQUIRED DC-LINK VOLTAGE
OF SVC-HAPF
Based on the circuit analysis in Fig. 2, the power rating in the
active inverter part can be calculated as VinvxIcx, while the power
rating of the SVC part can be calculated as VSVCIcx.
Accordingly, the power rating ratio of the active inverter and
the SVC parts (Sinvx/SSVCx) can be expressed as:
svcx
invx
cxsvcx
cxinvx
svcx
invx
V
V
IV
IV
S
S
R
tot
(1)
where Sinvx and SSVCx are the ratings of the phase active inverter
part and the SVC part; Vinvx and VSVCx are the phase root mean
square (RMS) voltages of the active inverter part and the SVC
part, and Icx is the phase RMS compensating current. The
required DC-link voltage VDC_tot can be expressed as below
[28]-[30]:
xtot_ 6invDC VV
(2)
As shown in Figs. 3(c) and (d), the current passing through
the SVC-HAPFs includes both fundamental and harmonic
components. Therefore, the VSVCx and Vinvx in (1) and (2) can be
expressed by using Ohm’s law as:
 
2
22
11
2
22 1SVC
n
cxnSVCncxSVC
n
SVCxnSVCxx IXIXVVV
(3)
 
2
2
21
2
22 1
n
cxnSVCninvx
n
invxninvxinvx IXVVVV
(4)
Combining (1)-(4) above, the Rtot and VDC_tot can also be
expressed in terms of fundamental and harmonic components
as:
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Transactions on Power Electronics
2
22 1
2
22 1
2
2
2
22 1
21
tot
R
n
totntot
n
SVCxnSVCx
n
invxn
n
SVCxnSVCx
invx RR
VV
V
VV
V
(5)
2
2_
21_
2
22 1tot_ 66
n
totnDCtotDC
n
invxninvxDC VVVVV
(6)
In (1)-(6), the subscript n’ stands for the fundamental (n=1)
and harmonic (n
2) frequency components. XSVC1 and XSVCn
are the impedance of the SVC part at the fundamental and at
each harmonic frequency order. Vinvx1 and Vinvxn are the
fundamental and harmonic inverter output voltage. Icx1 and Icxn
are the fundamental and harmonic compensating currents. The
Icxn includes two parts (Iixn and ILxn). Iixn represents the
self-harmonic injection current components by SVC and ILxn
represents the loading harmonic current components. Rtot1 and
VDC_tot1 along with Rtotn and VDC_totn are the fundamental and
harmonic components of the power rating ratio Rtot and the
DC-link voltage of VDC_tot, respectively.
In the following analysis, the Rtot in (5) and VDC_tot in (6) is
deduced, compared, and discussed through six parts, as
described below:
In Part A, the parameter design and characteristics of SVC
(XSVC1 and XSVCn) are proposed and discussed;
In Part B, the fundamental components (Rtot1 and VDC_tot1) of
Rtot and VDC_tot are deduced;
In Part C, the harmonic components (Rtotn and VDC_totn) of
Rtot and VDC_tot are obtained;
In Part D, the required Rtot and VDC_tot are deduced and
discussed;
In Part E, simulation case studies are provided;
In Part F, a section summary is drawn.
The calculation of the parameters in Rtot (5) and VDC_tot (6)
are explained in Parts A-C. In Part D, the deduced Rtot in (5)
and VDC_tot in (6) are plotted in Figs. 8 and 9 discussions of Rtot
and VDC_tot are also included. The simulation case studies are
provided in Part E to verify the analysis in Parts A-D. Finally, a
section summary is drawn in Part F.
A. The parameter design and characteristics of SVC
The expressions of impedance of SVC (XSVCn) at
fundamental (n = 1) and harmonic frequency order (n = 2, 3,
4…) can be given as:
c
PFPF
PF
SVCn Ln
CLn
Ln
X
2
)()2sin22(
)(
)(
(7)
where
is the firing angle, ω is the angular frequency (ω =
2πf) and f is the fundamental frequency.
At the fundamental frequency, the compensating reactive
power by the SVC part (Qcx_SVC) can be expressed as:
)(
1
2
1
_
SVC
x
SVCcx X
V
Q
(8)
where Vx1 is the RMS value of the fundamental load voltage,
and XSVC1 can be obtained from (7) with n = 1.
From Figs. 2 and 3, the impedance SVC part is controlled
by back-to-back connected thyristors T1x and T2x through the
firing angle
. When

= 180°, the SVC part provides the
maximum capacitive compensating reactive power
Qcx_SVC(MaxCap) for inductive load reactive power compensation.
However, when

= 90°, the SVC part provides the maximum
inductive reactive power Qcx_SVC(MaxInd) for capacitive load
reactive power compensation. Based on (7)-(8) the LPF and CPF
of the SVC part can be designed as:
2
1)(
2
)(
xcMaxIndLx
MaxIndLx
PF VLQ
Q
C
(9)
PFxMaxCapLxPFcMaxCapLx
MaxCapLxcx
PF CVQCLQ
QLV
L
2
1)(
3
)(
)(
2
1
(10)
where QLx(MaxInd) and QLx(MaxCap) are the load maximum
inductive and capacitive reactive power, respectively.
Based on (7), the design criteria of Lc can be expressed as:
PF
cCn
L2
1)(
1
(XSVC(n) =0 with
=180°)
 
PFPF
cLCn
L
1
1
2
2
(XSVC(n) =0 with
=90°)
(11)
According to the research in [29], the Lc in the SVC part is to
filter out the current ripple caused by the power switches of the
active inverter part. In [30], the Lc is designed to tune the n1 and
n2 away from the dominated harmonic orders. In [27]-[31], the
zero harmonic impedance point is in fact tuned at around the
3.7th harmonic order (XSVC(n) = 0 at n1≈n2n≈3.7) [27]-[31].
In this study, the proposed 5th harmonic order (XSVC(n) = 0 at
n1≈n2n≈5) is based on the general comparative performance
evaluation (in Fig. 4) and the minimized Rtot inverter design (in
Fig. 5).
Fig. 4 shows that the SVC-HAPF with XSVC(n) = 0 at n = 5
can provide the best compensation results. The performance of
SVC-HAPF becomes worse with XSVC(n) = 0 at n<5 or n>5.
Fig. 5 shows that the SVC-HAPF with Rtot = 17% can provide
the relatively best THDisx performance. In addition, the
corresponding zero impedance point for Rtot = 17% is n≈5.
In the following, two different designs for the SVC part
with XSVC(n) = 0 at n≈3.7 [27]-[31] and the proposed XSVC(n) = 0
at n≈5 are discussed.
0 5 10 15 20
0
5
10
15
20
25
30
PFL 0.85
PFL 0.95
PFL 0.75
PFL 0.65
THDisx (%)
n
Fig. 4. Simulated source current THDisx with different PFL and tuned at
different n after SVC-HAPF compensation (VDC_tot = 2500V)
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Transactions on Power Electronics
10 20 30 40 50 60 70 80 90 100
0
5
10
15
20
25
30
PFL 0.85
PFL 0.95
PFL 0.75
PFL 0.65
Rtot (%)
THDisx (%)
Rtot =17%(with n=5)
Fig. 5. Simulated source current THDisx with different load power factor (PFL)
after SVC-HAPFs compensation with different Rtot
B. Fundamental frequency analysis of Rtot1 and VDC_tot1
From Fig. 3(c), the fundamental inverter voltage Vinvx1 can
be expressed as:
11SVC11 )( cxxinvx IXVV
(12)
where Vx1, Icx1, and XSVC1 are the fundamental load voltage,
fundamental compensating current, and fundamental SVC part
impedance, respectively. The load reactive power can be
expressed as:
)( 111q1L cxxLxxx IVIVQ
(13)
where QLx is the loading reactive power and ILxq1 is the load
fundamental reactive current, which is equal to -Icx1 during
compensation. Combining (2), (8), (12), and (13), the Rtot1 and
VDC_tot1 can be expressed as:
Lx
SVCcxL
SVC
invx
tot Q
QQ
V
V_x
1
1
1
R
(14)
SVCcx
SVCcxL
xinvxtotDC Q
QQ
VVV
_
_x
111_ 66
(15)
In (14) and (15), Qcx_SVC and QLx are the compensating
reactive power by the SVC part and the load reactive power,
respectively.
C. Harmonic frequency analysis of Rtotn and VDC_totn
As shown in Fig. 3(d), the harmonic compensating current
can be expressed as:
)( Lxnixncxn III
(16)
where Iixn is the self-harmonic injection current by SVC and ILxn
is the load harmonic current.
In this part, the required Rtotn and VDC_totn are discussed
separately in two sub-parts. In Part C.1, the required Rtotn and
VDC_totn to compensate iixn injected by the SVC part are deduced
under assumptions of Icxn= Iixn and ILxn = 0 (linear loading
condition). In Part C.2, the total required Rtotn and VDC_totn are
obtained so that both current ILxn and Iixn (Icxn = Iixn+(-ILxn)) can
be compensated for the non-linear loading condition.
C.1 Rtotn and VDC_totn for compensating self-harmonic injection
current by SVC part (iixn) only
In this part, the required Rtotn and VDC_totn can be obtained to
compensate for iixn injected by the SVC part. To clearly
illustrate the iixn problem, the loading is assumed to be linear
(iLxn = 0). The equivalent single-phase self-harmonic current
rejection analysis model is provided in Fig. 6.
Ls
CPF
isx
Lc
LPF
x=a,b,c
S
vx
on
off
iLPF iCPF
+
-
vsx
iLx1
icx
=(icx1 + iixn)
Fig. 6. SVC equivalent single phase model for harmonic currents rejection
analysis
In Fig. 6, the thyristors (T1x and T2x) in each phase of the
SVCs can be considered as a pair of bidirectional switches,
which can generate the low order harmonic current iixn. As the
switch S turns on and off, the two differential equations of icx(t)
(icx_off and icx_on) can be obtained as:
HarmoniclFundamenta
tKtAiii nixncxoffcx )sin()sin( 1n1111_
(17)
HarmoniclFundamenta
KtKtAiii nixncxoncx 322n221_ )cos()sin(
(18)
where A1 and A2 are the peak values of the fundamental
compensating current during each turn-off and turn-on; K1, K2,
K3,
n1, and
n2 are constants during each switching cycle and
depend on the initial conditions of the compensating current
and capacitor voltages; and ωn1 and ωn2 are the harmonic
angular frequencies, which can be expressed as:
 
PFcs CLL
1
1n
(
 
PFcs CLLf
f
n
2
1
2
1
1
)
(19)
 
PFPFcs
PFcs
CLLL
LLL
2n
(
 
PFPFcs
PFcs
CLLL
LLL
ff
n
2
1
2
2
2
)
(20)
where Ls is the system inductor, which is typically a small
value, and the Lc, LPF and CPF are parameters in the SVC part.
90 100 110 120 130 140 150 160 170 180
0
50
100
150
200
250
300
350
400
Fundermental Icxf
5th Icx5
7th Icx7
11th Icx11
13th Icx13
Firing Angle (Degree)
Compensating Current (A)
(a)
90 100 110 120 130 140 150 160 170 180
0
50
100
150
200
250
300
350
400
Funderment Icxf
5th Icx5
7th Icx7
11th Icx11
13th Icx13
Firing Angle (Degree)
Compensating Current (A)
(b)
Fig. 7. Simulated fundamental and harmonic compensating currents at different
firing angles under linear loading with: (a) SVC with (XSVC(n) = 0 at n≈3.7)
[27]-[31], and (b) SVC with (XSVC(n) = 0 at n≈5)
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