Download Harmonic Analysis of Static var Compensator

Harmonic Analysis of Static var Compensator
Operating Under Distorted Voltages
E. V. Liberado and J. A. Pomilio
F. P. Marafão
School of Electrical and Computer Engineering
University of Campinas
Campinas, Brazil, 13083-852
Emails: [email protected],
[email protected]
Group of Automation and Integrated Systems
Unesp - Univ Estadual Paulista
Sorocaba, Brazil, 18087-180
Email: [email protected]
Abstract—The static var compensators (SVC) are still in
use in both transmission and distribution grids, due to their
reduced costs and high reliability. Especially in distribution
grids, on which the voltage harmonic distortion limits are less
restrictive, it may cause a considerable harmonic distortion
impact when operating under non sinusoidal voltage conditions.
Therefore, this paper describes a harmonic analysis methodology
for the SVC, considering non-ideal source voltages, in order to
determine harmonic power terms that may be used to design and
also control harmonic power filters. Application examples are
presented and the operation of an experimental SVC prototype
is also analysed.
I. I NTRODUCTION
Static var compensator (SVC) is a well-known technology
that has been studied and used for years in voltage control applications [1]–[3]. Many SVC units are indeed being repaired
in order to extent their lifetime [4], which shows that it is still
a cost-effective and reliable solution.
Due to its capacity of absorbing a variable capacitive or
inductive reactive power, voltage and load balancing applications has been also proposed for the SVC, considering its
installation in distribution grids [5]–[10].
However, a well-know drawback of the SVC is the harmonic
distortion caused by its operation. On the SVC traditional
application scenario, i.e., the tranmission and medium-voltage
distribution grids [11], the voltage harmonic distortion is
highly limited, thus the harmonic distortion caused by the SVC
is mainly due to the thyristor valves’s switching, for which
passive filters are applied.
Nevertheless, in low-voltage-level distribution grids the limits for voltage harmonic distortion (HDu ) are higher than the
ones from medium- and high-voltage- levels. In this scenario,
the SVC capacitors may cause more harmonic distortion in the
currents than the thyristors’ switching. Moreover, the passive
filters may be useless in grids with high short-circuit level [12]
and, unlike the active power filter (APF), their current cannot
be adjusted to achieve determined goals.
Therefore, this paper presents a methodology of SVC
harmonic analysis in order to estimate the harmonic power
demand to be addressed for compensation, either by a passive
filter or an APF to be installed closer to the SVC [13]. Recommended harmonic distortion limits are used to determine
the harmonic source voltages [14], [15], in order to evaluate
the harmonic current drained by the SVC capacitors, as well
as the influence of harmonic voltages in the operation of the
thyristor-controlled reactor (TCR).
II. SVC H ARMONIC L INE C URRENTS
Consider the SVC topology represented in Fig. 1, composed
by three delta-connected fixed capacitors (CSV C ) and three
delta-connected TCR branches (LT CR ). This is the simplest
SVC topology and it is suitable for distribution grid applications, due to their reduced number of switching components.
The SVC is fed by a three-phase three-wire voltage source
through three balanced line impedances Zs = Rs + jωLs . As
the voltages in distribution grids may be distorted, unbalanced
and asymmetrical, the source voltages are generically defined
from the fundamental frequency to a determined harmonic
order Kh as:
um =
Kh
√
k
2 · Um
sin kωt + φkm
m = 1, 2, 3.
(1)
k=1
On the SVC side, the line voltages can be written as:
uSV C,1 = u1 − Rs · iSV C,1 − Ls · ı̆SV C,1
uSV C,2 = u2 − Rs · iSV C,2 − Ls · ı̆SV C,2
uSV C,3 = u3 − Rs · iSV C,3 − Ls · ı̆SV C,3
(2)
on which ı̆SV C,m is the time-derivative of iSV C,m , m =
1, 2, 3. Considering that the sum of the line voltages and
currents is zero at any instant (which is also valid for their
time derivatives and integrals) on the proposed circuit, the line
currents can be written in terms of the currents of each SVC
branch as:
c
978-1-5090-3792-6/16/$31.00 2016
IEEE
299
iSV C,1 = iSV C,12 − iSV C,31
iSV C,2 = iSV C,23 − iSV C,12
iSV C,3 = iSV C,31 − iSV C,23 ,
(3)
u3
u2
Rs
Rs
uZs,3
Ls
u1
uZs,2
Ls
uZs,1
iTSVCR
C,mn =
2
on which
iSVC,1
LTCR
uSVC,12
φkT CR,mn
CSVC
CSVC
CSVC
iSVC,31
iSVC,12
LTCR
Fig. 1. SVC circuit for harmonic current analysis.
uSV C,12 = uSV C,1 − uSV C,2
= u1 − u2 − Rs (iSV C,1 − iSV C,2 )
− Ls (ı̆SV C,1 − ı̆SV C,2 )
= u12 − (3 · Rs · iSV C,12 + 3 · Ls · ı̆SV C,12 )
= u23 − (3 · Rs · iSV C,23 + 3 · Ls · ı̆SV C,23 )
uSV C,23
uSV C,31 = u31 − (3 · Rs · iSV C,31 + 3 · Ls · ı̆SV C,31 )
(4)
From (4), one may observe that in case of unbalanced line
impedances, their delta-equivalent can be obtained by using
a wye-delta transformation and then used in (4) and in the
following formulation.
By using the superposition theorem, the current of the SVC
branches mn = 12, 23, 31 can be split into iCap
SV C,mn and
iTSVCR
C,mn in order to determine each of these components
separately. Therefore, the current of each SVC capacitor
branch can be expressed by:
2
αmn
αmn − π/2
, tb,mn = ti0,mn −
,
ω
ω
on which αmn is the conduction angle of the TCR branch
mn and ti0,mn may be determined iteratively for each semicycle of iTSVCR
C,mn by incrementing t until reach the following
conditions:
T CR
T CR
• iSV C,mn = 0 and ı̆SV C,mn < 0 for the positive semicycle and
T CR
T CR
• iSV C,mn = 0 and ı̆SV C,mn > 0 for the negative semicycle.
Additionally, iTSVCR
C,mn = 0 for t < ta,mn and t > tb,mn .
Term Cmn is the integration constant and may be determined
for each ta as:
Cmn =
Kh
√
2 · ITk CR,mn sin kωta + φkT CR,mn
k=1
Finally, the current of SVC branches are expressed by:
T CR
iSV C,12 = iCap
SV C,12 + iSV C,12
T CR
iSV C,23 = iCap
SV C,23 + iSV C,23
iSV C,31 =
k
2 · ICap,mn
sin kωt + φkCap,mn ,
iCap
SV C,31
+
(6)
iTSVCR
C,31
and the SVC line currents can be determined by using (3).
k=1
on which
III. SVC H ARMONIC C OMPLEX P OWER
k
ICap,mn
=
k
Umn
In the sequel, the fundamental and harmonic positive- and
negative-sequence components of the SVC line currents can
be calculated using a Fast Fourier Transform algorithm and
the Fortescue’s symmetric components matrix.
Assuming that the harmonic voltages will not be compensated by the shunt harmonic filters, only the fundamental
positive-sequence voltages are used in the complex power
calculation and it is expressed as:
2 ,
1
(3 · Rs ) + 3 · kωLs −
kωCSV C
⎞
⎛
1
3 · kωLs −
⎜
kωCSV C ⎟
= φkmn − arctan ⎝
⎠.
3 · Rs
2
φkCap,mn
2
(3 · Rs ) + (kω (LT CR + 3 · Ls ))
kω (LT CR + 3 · Ls )
= φkmn − arctan
.
3 · Rs
ta,mn =
Thus, the SVC line-to-line voltages can be expressed as:
√
k
Umn
Terms ta and tb are the instants that delimit the conduction
time interval of each TCR thyristor. If a phase reference signal synchronized with the fundamental-frequency line-to-line
voltages is used to control the thyristors switching, the TCR
current semi-cycles are symmetrical. Thus, the conduction
time interval of each TCR thyristor is the same and
1
uSVC,31
iCap
SV C,mn =
2ITk CR,mn sin kωt + φkT CR,mn − Cmn ,
ITk CR,mn = LTCR
Kh
Kh
√
k=1
iSVC, 2
iSVC,23
3
Rs
Ls
iSVC,3
uSVC,23
Correspondingly, the current of each TCR branch valid in
the interval ta,mn ≤ t ≤ tb,mn is expressed by:
From the SVC capacitor’s equivalent impedance, the resonant frequency between CSV C and Ls is calculated by:
fo =
1
.
2π 3 · Ls CSV C
√
(5)
f,p
f,p
f,p
− Żsf,p · I˙SV
U̇SV
C = U̇
C
(7)
on which U̇ f,p and Żsf,p are the fundamental-positive sequence
components of the source voltages and line impedances. Then,
300
SV C1
SV C2
QCap
SV C
CR
QT
SV C
CSV C
LT CR
57.6 kvar
67.6 kvar
81.9 kvar
98.8 kvar
1052 μF
1235 μF
4.7 mH
3.9 mH
the complex positive- and negative-sequence power components for each harmonic k are calculated by:
k,p
k,p
k,p
f,p
˙k,p
ṠSV C = PSV
C + jQSV C = U̇SV C · ISV C
(8)
Ṡ k,n = P k,n + jQk,n = U̇ f,p · I˙k,n
SV C
SV C
SV C
SV C
SV C
Therefore, the SVC harmonic complex power to be addressed for compensation can be defined for a determined set
of harmonic orders Ks ≤ k ≤ Kh as [16]:
c
c
c
c
c
ṠSV
C = SSV C φSV C = PSV C + jQSV C ,
(9)
on which
QcSV C
2
2
c
c
c
(PSV
,
C ) + (QSV C ) , φSV C = arctan
c
PSV
C
Kh Kh 2
2
k,p
k,n
c
PSV
PSV
+
,
PSV C = 3.0 · C
C
c
SSV
C =
QcSV C
k=Ks
k=Ks
k=Ks
k=Ks
harmonic current [A], frequency [Hz]
TABLE I
E VALUATED SVC S .
Kh Kh 2
2
Qk,p
Qk,n
= 3.0 · +
.
SV C
SV C
Eqs. (8) allow the selective compensation of unbalanced
harmonic currents [17], which may be useful in applications
on which the SVC operation may be unbalanced (i.e., TCR
branches with different conduction angles). Thus it is interc
esting to calculate ṠSV
C considering a large set of harmonic
orders, in order to evaluate which are the critical ones.
Also, for applications on which the SVC operation may be
c
unbalanced, ṠSV
C should be calculated for a set of combinations of TCR
angles, then only its highest value is
conduction
c,max
considered ṠSV
.
This
value, as well as the conduction
C
angle combination that results in it, may vary according to the
number of harmonic orders considered in the calculation of
c
ṠSV
C.
IV. A PPLICATION EXAMPLES
Table I presents two SVCs designed for load balancing in
a 220 V/60 Hz/75 kVA distribution grid.
The line impedances were defined in terms of the
low-voltage winding impedances of a three-phase-75 kVAdistribution transformer, thus Rs = 6.5mΩ and Ls =
100.0μH. In this case, the resonant frequency of the SVC
capacitors is 283.3 Hz (CSV C1 ) and 261.5 Hz (CSV C2 ). If
the cable impedances are considered, the resonant frequencies
can be reduced. Fig 2 presents the resonant frequency and
harmonic current magnitudes of one capacitor branch of SV C1
when considering the transformer impedances plus a variablelength cable impedance equal to 0.34 + j0.56Ω/km. The
300
5
ICap1
7
ICap1
11
ICap1
fo
250
200
150
100
50
0
0
25
50
75
100
125
150
175
line length [m]
200
225
250
275
300
Fig. 2. Resonant frequency and harmonic currents of one SV C1 capacitor
branch in terms of a variable-length line impedance.
f
k
line voltages were Um
= 127V and Um
= 6.35V, k =
5, 7, 11; m = 1, 2, 3.
One may observe that the longer are the cables, the lower is
the resonant frequency and the capacitor harmonic currents are
considerably reduced. Alternatively, a small series inductance
may be inserted between the delta-connected SVC capacitors
and the line impedances in order to atenuate the harmonic
currents that are close to the resonant frequency [18]. However,
it is important to observe that the power losses increase
proportionally to the line impedance. Therefore, by adopting
the transformer impedance plus a 25 m cable impedance,
Rs = 15.0mΩ and Ls = 137.1μH and the resonant frequencies of SV C1 and SV C2 capacitors are 241.9 Hz and 223.3
Hz in the next two application examples.
A. Example 1: IEEE recommended limits
In the first application example the recommended limits
from IEEE Std-519-2014 [14] were adopted to define the
harmonic source voltages, expressed in terms of the individual harmonic distortion factor HDu and the fundamental
k
f
= HDu · Um
, k = 5, 7, 11. As the
frequency voltage as Um
nominal voltage was 220 V, the limit of individual harmonic
voltages is 5% [14], thus HDu = 0 · · · 5%.
The harmonic currents and complex power to be compensated were calculated for SV C1 and SV C2 considering all the
odd harmonic orders between the 3rd and 51st . Also, these
calculations were repeated for a set of 51 values of TCR
conduction angles into the [π/2, π] interval, resulting in the
evaluation of 513 = 132, 651 combinations of SVC operation.
Fig. 3 presents the odd harmonic (3rd -13th ) current magnitudes of one capacitor branch of SV C1 in function of the
source voltages T HDu , which varied from 0 to 8.66%. Note
that the capacitor harmonic currents that correspond to the
harmonic source voltages dramatically increases in function
of T HDu . The fundamental-frequency capacitor current remained fixed and equal to 92.94 A.
Fig. 4 presents the harmonic currents of one TCR branch
of SV C1 . Every curve of each harmonic order is plotted in
function of the conduction angles for one value of (T HDu ). In
301
40
35
100
3
5
7
9
11
13
h=
h=
h=
h=
h=
h=
80
25
h
ISV
C,1[A]
ICh ap,12[A]
30
h=
h=
h=
h=
h=
h=
20
15
10
3
5
7
9
11
13
60
40
20
5
0
0
1
2
3
4
T HDu %
5
6
7
0
1.5708
8
Fig. 3. Harmonic currents of one capacitor of SV C1 in function of the source
voltage’s T HDu .
16
h=
h=
h=
h=
h=
h=
14
ITh C R,12[A]
12
3
5
7
9
11
13
2.1991
2.5133
αSV C,12[rad]
2.8274
3.1416
Fig. 5. SV C1 harmonic line currents in function of the TCR conduction
angles and the source voltage’s T HDu .
c,max
ṠSV
C
TABLE II
OF SV C1 AND SV C2 AND C ORRESPONDING TCR C ONDUCTION
A NGLES FOR E ACH C ASE OF S OURCE VOLTAGE T HDu .
T HDu %
10
c,max
ṠSV
C [kV A]
α12 [rad]
α23 [rad]
α31 [rad]
1.9478
1.9164
1.8535
1.8535
1.8535
1.8535
1.8535
1.8535
1.8535
1.8535
1.8535
1.9792
1.9164
1.8535
1.8535
1.8535
1.8535
1.8535
1.8535
1.8535
1.8535
1.8535
1.9478
1.9164
1.9164
1.8221
1.8535
1.8535
1.8535
1.8535
1.8535
1.8535
1.8535
1.9478
1.9478
1.9164
1.8535
1.8535
1.8535
1.8535
1.8535
1.8535
1.8535
1.8535
SV C1
8
6
4
2
0
1.5708
1.885
1.885
2.1991
2.5133
αSV C,12[rad]
2.8274
3.1416
Fig. 4. Harmonic currents of one TCR branch of SV C1 in function of the
conduction angles and the source voltage’s T HDu .
relation to the capacitors behavior, it is possible to observe that
the TCR currents do not vary significantly with the increasing
of T HDu .
Fig. 5 presents the harmonic components in the line current
of SV C1 , calculated only for the SVC balanced operation
cases, i.e., considering the same conduction angle for the
three TCR branches. Also in this Figure every curve of each
harmonic component is function of one value of T HDu and
the TCR conduction angles. The triplen harmonics are absent
in the line currents, thus the most relevant harmonics in the
line currents are the ones that are also present in the source
voltages (5th , 7th and 11th ), due to the currents drained by
the SVC capacitors.
Table II presents the maximum values of harmonic complex
power for each source voltage T HDu among all combinations
of SV C1 and SV C2 operation. The corresponding TCR
conduction angles indicate that in most cases the maximum
harmonic complex power occurs for unbalanced operation of
the TCR, on which the triplen harmonics caused by the TCR’s
switching are present in the line currents.
B. Example 2: PRODIST limits
In the second application example, the HDu limits proposed
in [15] were adopted to the source voltages. The HDu for the
0.00
0.87
1.73
2.60
3.46
4.33
5.20
6.06
6.93
7.79
8.66
5.34 50.26◦
6.92 35.31◦
9.27 9.09◦
12.07 9.16◦
14.93 9.26◦
17.81 9.28◦
20.71 9.29◦
23.61 9.31◦
26.52 9.31◦
29.44 9.32◦
32.36 9.32◦
0.00
0.87
1.73
2.60
3.46
4.33
5.20
6.06
6.93
7.79
8.66
6.30 50.52◦
7.42 41.44◦
9.09 31.17◦
11.10 8.58◦
13.39 8.40◦
15.72 8.25◦
18.07 8.13◦
20.44 8.04◦
22.81 8.03◦
25.20 7.98◦
27.59 7.93◦
3.1102
1.5708
1.8535
1.8535
1.8850
1.8850
1.8850
1.8850
1.8850
1.8850
1.8850
SV C2
3.1102
3.1102
1.5708
1.8535
1.8535
1.8535
1.8535
1.8535
1.8850
1.8850
1.8850
5th harmonic varied from 5% to 7.5%, while the HDu for
the 7th varied from 5% to 6.5% and the 11th HDu remained
constant and equal to 4.5%.
Table III presents the maximum harmonic power of SV C1
and SV C2 and the corresponding TCR conduction angles for
this application example. As expected, the harmonic complex
power continued to increase in relation to the previous example, as the HDu limits of 5th and 7th harmonics are higher
in these cases than the ones adopted before.
302
TABLE III
c,max
ṠSV
C OF SV C1 AND SV C2 AND C ORRESPONDING TCR C ONDUCTION
A NGLES C ONSIDERING THE T HDu L IMITS P ROPOSED IN [15].
T HDu %
c,max
ṠSV
C [kV A]
α12 [rad]
α23 [rad]
α31 [rad]
1.8535
1.8535
1.8535
1.8535
1.8535
1.8535
1.8535
1.8535
1.8850
1.8850
1.8535
1.8535
1.8535
1.8535
1.8535
1.8535
1.8535
1.8535
1.8535
1.8535
1.8535
1.8850
1.8535
1.8535
SV C1
32.28
35.14
38.00
40.86
43.42
45.98
8.38
8.99
9.60
10.23
10.56
10.90
27.51
29.83
32.15
34.48
36.49
38.52
9.34◦
9.36◦
9.42◦
9.43◦
9.45◦
9.51◦
8.38
8.99
9.60
10.23
10.56
10.90
1.8850
1.8850
1.8850
1.8850
1.8850
1.8850
SV C2
7.95◦
7.93◦
7.91◦
7.94◦
7.93◦
7.96◦
1.8850
1.8850
1.8850
1.8850
1.8850
1.8850
Fig. 6. Experimental setup: left: SVC front panel and capacitor banks (on
top), right: TCR and linear loads.
TABLE IV
E XPERIMENTAL R ESULTS FOR THE SVC O PERATION .
V. E XPERIMENTAL R ESULTS
An experimental SVC prototype with QCap
SV C = 7.5 kvar,
QTSVCR
C = 10 kvar, CSV C = 137μF and LT CR = 40mH was
set to compensate reactive power and load imbalance of a 6.2
kVA three-wire linear unbalanced RL load. Fig. 6 presents the
experimental SVC and load, which were fed by a three-phase
programmable voltage source.
Two source voltage cases were tested: A) 220 V/60 Hz and
B) 220 V/60 Hz + HDu = 2% for 5th harmonic. The line
impedances were neglected. Table IV presents the measured
RMS line voltages and currents, line T HDu and T HDi
(current total harmonic distortion) factors and the magnitude
c
of ṠSV
C , which were calculated in these cases considering
all harmonic components present in the line currents, by extracting the fundamental-frequency current from the measured
ones.
In the first case (A), both load and SVC were connected to
the voltage source. As the load is purely resistive/inductive, all
the harmonic distortion observed in the line currents may be
accounted to the SVC, particularly the TCR. Fig. 7 presents
the measured line currents in this case.
When the source voltages were distorted (cases B), the line
currents were not much distorted when only the load was
connected to the source (case B.1). However, the currents
became highly distorted when the SVC was turned on (case
B.2), mainly due to the SVC capacitors, as shown in the results
of case B.3, on which both load and TCR were turned off. Fig.
8 presents the waveforms of the line currents in these cases.
Regarding the harmonic power, Table V presents a comc,max
parison between the magnitude of ṠSV
C , calculated using
(9) and considering the odd harmonic orders from the 3rd to
c
the 51st for T HDu = 0 and 2%, and SSV
C of cases A and
B.2. As the conduction angles that compensate the load in
c,max
that cases were different than the ones that result in ṠSV
C ,
c
the experimental SSV
C resulted in lower values in relation to
the maximum expected ones even in case B.2, on which the
A
B.1
B.2
B.3
U1 [V ]
U2 [V ]
U3 [V ]
126.62
126.75
126.53
126.80
126.52
126.41
126.49
126.69
126.54
127.28
127.28
127.10
I1 [A]
I2 [A]
I3 [A]
11.77
11.72
11.28
13.84
15.17
19.19
12.11
11.55
11.50
19.56
19.63
19.80
T HDu,1 %
T HDu,2 %
T HDu,3 %
0.52
0.60
0.48
2.09
2.14
2.18
2.24
2.11
2.07
2.17
2.23
2.22
T HDi,1 %
T HDi,2 %
T HDi,3 %
11.13
10.35
12.33
1.55
2.04
1.51
16.44
10.76
23.18
10.83
10.47
10.98
c
SSV
C [VA]
493.20
104.13
763.36
802.92
TABLE V
c,max
C OMPARISON B ETWEEN T HEORETICAL SSV
C
c
SSV
.
C
T HDu = 0
case A
T HDu = 2%
case B.2
AND THE
E XPERIMENTAL
c
SSV
C [V A]
α12 [rad]
α23 [rad]
α31 [rad]
685.64
493.20
1.5708
1.9208
1.9792
2.1288
2.0106
2.5408
1036.76
763.36
2.5133
1.9288
2.4819
2.1168
2.5133
2.5368
measured line currents included also the harmonic currents
drained by the load.
VI. C ONCLUSION
A methodology to estimate the harmonic power absorbed by
the SVC was presented. It is a relevant parameter to design
SVC harmonic filters, especially in distribution applications,
on which even when the voltage harmonic distortion is under
recommended limits, the SVC capacitors may cause more
current harmonic distortion than the TCR.
Even though the source voltages in the presented examples
and results were balanced and symmetrical, the methodology
allows the using of any sort of voltage conditions, including
303
definition of a selective harmonic compensation for active
filters and can be also addressed to distributed compensators.
ACKNOWLEDGMENT
The authors would like to thank the Research National
Council for Scientific and Technological Development - CNPq,
(grants 302257/2015-2 and 487471/2012-1) and Varixx Co. for
provide the thyristor’s control modules.
R EFERENCES
Fig. 7. Line currents with SVC and load fed by sinusoidal voltages.
Fig. 8. Line currents for cases B.1 (top), B.2 (middle) and B.3 (bottom).
unbalanced and asymmetrical cases.
Some alternatives to reduce the SVC harmonic power
include: to adopt an optimum placement strategy for the
SVC, on which the minimization of the power losses should
be considered; the insertion of a small-rated power series
inductance between the grid and the SVC capacitors; or the
adoption of a complete electronic compensator composed by
TCR and active filter.
Moreover, the harmonic complex power terms allows the
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