resonance-free shunt capacitor for utility systems
TRANSCRIPT
1
Abstract—Harmonic resonance is one major concern in the
application of shunt capacitors. A consensus on the solution to
this issue is to configure shunt capacitors into passive filters.
However, there is a lack of information on a technically-sound
guide how to do so. In response to this situation, this paper first
introduces the resonance-free concept as well as the
corresponding index. Based on it, the methods to configure the
shunt capacitors into the two most promising passive filter types,
i.e., C-type and 3rd order High-Pass (HP), are developed. These
two filter configurations are compared via a real case study. It is
found that the proposed methods these two filter configurations
can achieve similar performance. In addition, the robustness of
these two filter configuration is investigated by Monte Carlo
simulation method, which indicates the robustness of these two
filter configurations are relatively comparable.
Index Terms— Shunt capacitor, resonance, harmonics, filters.
I. INTRODUCTION
HUNT capacitors are extensively used in electric power
systems due to their well-known benefits, such as power
factor improvement, voltage support, release of system
capacity, and reduced system losses [1-6]. However, as with
any piece of electrical equipment, there are a number of issues
in the application of the shunt capacitors. A critical one is that
adding shunt capacitors to the system can potentially result in
resonance which significantly amplifies the harmonic currents
and voltages. This cannot be tolerated since the amplified
harmonics will not only damage the shunt capacitors
themselves, but also cause insulation breakdown of nearby
electrical equipment, nuisance trip of relay, and excessive
harmonic torque generation, etc [5-7].
One simple solution to avoid resonance is to configure the
capacitor as a single-tuned filter by adding an inductor in
series with it [6, 8, 9]. Its tuning frequency is usually set below
the lowest characteristic harmonic frequency of the system
(3rd or 5th depending on the system grounding condition). On
condition that the system is pure inductive over the whole
frequency range, this will successfully free the capacitor and
the system from the resonance issue. However, the system
This work was supported by the Natural Sciences and Engineering
Research Council of Canada and Alberta Power Industry Consortium.
The authors are with the department of Electrical and Computer
Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada.
impedance is not always pure inductive especially for systems
which have other existing shunt capacitors and long
transmission lines or cables [6, 7, 10]. For these systems, the
resonance risk still exists even if the capacitor is configured to
a single-tuned filter. Damping resistor may be added in series
in the single-tuned filter topology to limit the resonance
severity in such cases. But it will induce considerably large
power losses.
Another widely used solution is to configure the capacitor
into a C-type filter [6, 11-13]. Similarly, its tuning frequency is
usually set below the lowest characteristic harmonic frequency
of the system (3rd or 5th depending on the system grounding
condition). The unique topology makes the C-type filter a
heavily damped high pass filter with zero fundamental
frequency power loss. Several literatures discussed about the
basic characteristics of the C-type filter [6, 11, 12]. However,
how to choose the damping resistor used in the C-type filter is
still not clear especially for the purpose of limiting the severity
of the system resonance.
Due to the same number of components with C-type filter
and the ability to damp parallel resonance, 3rd
order high-pass
(HP) filter [7, 12, 14, 15] is another promising candidate
topology that can be used to adapt shunt capacitors for the
purpose of limiting the resonance severity. However, to
authors' best knowledge, the corresponding design method as
well as the performance analysis has not been reported in the
existing literatures.
This paper is concerned about how to adapt shunt
capacitors to make them resonance-free for utility systems. It
is structured as follows. Section II first discusses the
characteristics of the power system frequency response and
then introduces the concept of resonance free condition and its
corresponding index. Design methods for C-type filter
configuration and 3rd
order HP filter configuration to realize
resonance-free shunt capacitor are given in Section III and
Section IV respectively. Comparison of these two filter
configurations with regard to component parameters and
performance is presented in Section V. Section VI further
compares the robustness of these two filter configurations.
Section VII concludes this paper.
Resonance-Free Shunt Capacitor for Utility
Systems – Configurations, Design Methods and
Comparative Analysis (V1.0) J. W. Hagge, Senior Member, IEEE, and L. L. Grigsby, Fellow, IEEE
S
2
II. RESONANCE FREE CONDITION
A. Power System Frequency Response
Due to the combination of various different frequency
dependent components, such as transmission lines, capacitors,
inductors, etc., power system frequency response (the
corresponding equivalent Thevenin impedance seen form a
certain location of the system versus frequency) is usually not
a simple linear function. Normally, the system impedance is
inductive at fundamental frequency, its value representing the
stiffness of the system [16]. With the increasing of frequency,
the system impedance may change from inductive to capacitive
and back as shown in Fig. 1 (a). This makes the R-X plot a
spiral shape as shown in Fig. 1(b). Such a spiral-shaped power
frequency response is especially common for high voltage
systems [7, 10].
(a) Separate R and X plot (b) Sprial R-X plot
Fig. 1. Power system frequency response [6].
Moreover, to satisfy the various loading conditions, the
power system may operate under different scenarios which
may have different components online such as different
number of capacitors and different transmission tie lines [17-
20]. This results in a varying power system frequency
response.
B. Concept of Resonance Free and Index Established
Fig. 2 shows the equivalent circuit of the system and filter
adapted from the shunt capacitor. The harmonic voltage
amplification ratio ( ) is defined as the ratio of the
harmonic voltages at the filter installation after and before the
filter's installation given by
0( ) ( ) / ( )F FV V (1)
( )FZ
( )FV ( )SZ
( )SV
System Filter
F
( )SV : System background
harmonic voltage
( )SZ : System equivalent
harmonic impedance
( )FZ : Filter equivalent
harmonic impedance
Fig. 2. Equivalent circuit of the system and filter.
According to Fig.2, before the filter's installation the
harmonic voltage at the filter installation location equals to the
system background harmonic voltage, i.e., 0 ( ) ( )F SV V .
Take it into (1), then we can derive
0( ) ( ) / ( ) ( ) / ( ) ( )F F F F SV V Z Z Z (2)
which can be further represented by
2 2
2 2
( ) ( )( )
( ) ( ) ( ) ( )
F F
F S F S
R X
R R X X
(3)
where ( ) Im( ( )), ( ) Re( ( ))S S S SX Z R Z .
As can be seen from (3), for a certain frequency, the most
serious amplification condition is when system impedance is
pure inductive or capacitive and equals to the negative filter
equivalent reactance, that is
( ) ( ), ( ) 0S F SX X R (4)
In such a condition, the denominator in (3) is minimum,
which results in the largest amplification ratio given by
2
( ) 1 ( ) / ( )res F FX R (5)
As could be seen from equation (5), for different harmonic
frequency, the largest amplification ratio is different. And it is
only determined by the filter's reactance to resistance ratio at
this harmonic frequency ( ) / ( )F FX R . Since harmonic
resonance only happens when there is corresponding harmonic
at the resonance frequency, by tuning the filter to a frequency
H below the lowest order of harmonics in the system, the
resonance is only a concern for the frequencies above the
tuning frequency. Further, if for the designed filter,
( ) / ( )F FX R has a maximum over the frequency range
H . Accordingly, the largest amplification ratio ( )res
also has a peak over the same frequency range, that is the
worst voltage amplification induced by the addition of the
filter into the system given by max ( )H
res
.
Intuitively, in the design process, if we set this ratio
max ( )H
res
to a value Safe that is safe for the system based
on the operation experience by the proper selection of the filter
components' parameters, then the addition of the filter would
cause no problem under various system operation conditions.
This has led us to propose the following resonance-free
criterion for the filter design
max ( )H
res Safe
(6)
III. DESIGN METHOD FOR C-TYPE FILTER CONFIGURATION
The C-type filter is a modified version of 2nd
order HP filter
for low tuning and heavy damping with reduced power losses
[6, 7, 12, 13]. In this filter, the resistor is short-circuited for the
fundamental frequency by means of an extra capacitor
3
connected in series with the reactor; see Fig. 3(a). By doing
this, the fundamental losses in the resistor can be nearly
eliminated. For low order harmonics, its 2C L branch
dominates, so C-type filter behaves like a single tuned filter
(see Fig. 3(b)), while for high order harmonics, the R branch
dominates, so C-type filter behaves as a resister R in series
with 1C (see Fig. 3(c)).
1C
2C
LR
1C
2C
L
1C
R
(a) (b) (c)
Fig.3. (a) Topology of C-type filter and its equivalent circuit: (b) low-
frequency equivalent circuit and (c) high-frequency equivalent circuit.
As there are four components in C-type filter, four design
conditions or equations are needed to determine their
parameters.
A. Accepted Design Equations
For C-type filter, two design conditions are well understood
and accepted by industry and research community:
Condition 1: The reactive power output of the filter shall be
equal to the required amount QF. This condition yields the
following design equation:
2
1 1/FC Q V (7)
where 1 is the power frequency and V is the rated
voltage.
Condition 2: C2 and L are tuned to the fundamental
frequency to eliminate the fundamental frequency power
loss, which leads to
2
1 2 =1/L C . (8)
B. Design Condition Based on Resonance Possibility
Minimization
The objective of adapting shunt capacitors into filters is to
avoid potential harmful resonance. It is, therefore, logical to
use the resonance possibility minimization to establish the 3rd
design equation.
As shown in Fig. 1, for most systems, they are more likely
to be inductive over a wide frequency range. Thus if the filter
is all inductive after the tuning frequency, the likelihood of the
filter to be resonant with system is greatly reduced. This leads
us to choose to set the equivalent reactance of the C-type filter
at tuning frequency equals to zero as the third design equation,
i.e.,
0F HX . (9)
And this leads to
2
1
2
2 2 1 2
1
H C C C
h
h C CR
(10)
where 1/Hh .
It can be further proven that with this tuning frequency setting
and by selecting component parameters constrained by
inequality (11), the condition to minimize the filter-system-
resonance possibility (i.e., 0, F HX ) can be easily
satisfied.
1/R L C (11)
C. Design Equation Based on Resonance-Free Criterion
If the worst case of harmonic amplification condition is
within the limited range, the harmonic amplification caused by
resonance will not be a concern. Hence, the resonance-free
criterion (i.e., (6)) discussed in Subsection II-B is selected as
the fourth design condition.
Through extensive mathematical operations, the frequency
at which C-type filter reaches its maximal max ( )H
res
has
been found. It is shown below.
1/3
1max /1 3
1
1
DA
AB
B
(12)
where
2 2
1 2
22 2
2
2 2 2 3 2
3
2 2
2
/ / , / /
4 9
18 1
2 2 7 4
2 1
2 3 7
3 1
324 4 3 8
D
R L C R L C
A
B
.(13)
Accordingly, substituting (12) into (6), the fourth design
equation for C-type filter can be further represented by
24 2
max max
42 2 2 2
1 m
2
1 ax max
1 2 11
1Safe
h h
C R h h
(14)
4
where max max 1/h .
D. Summary of Design Procedure
To summarize, C-type filter can be designed in the
following way:
1) Determine 1C using (7);
2) Substitute (8) and (10) into (14), and determine 2C using
Bisection Method;
3) Determine L using (8), R using (10);
4) Select the rating of each component for different possible
operating conditions and provide enough margins for
contingency operations.
IV. DESIGN METHOD FOR 3RD ORDER HP FILTER
CONFIGURATION
3rd
order HP filter is another modified version of 2nd
order
HP filter, which is most widely used to filter high order
harmonics such as 11th
, 13th
etc., for both industrial systems
and HVDC links [6, 7, 21, 22]. Compared to the 2nd
order
filter, 3rd
order HP filter also has less fundamental frequency
power loss due to the inserted auxiliary capacitor in series with
the damping resistor. This filter is shown in Fig. 4(a). At low
frequencies below the tuning frequency, the filter’s L branch
dominates so it behaves as a single-tuned filter (see Fig. 4(b)),
while at high frequencies, the 2C R branch dominates hence
it behaves as a first order high pass filter (see Fig. 4(c)).
1C
2C
LR
1C
L
1C
2C
R
(a) (b) (c)
Fig.4. (a) Topology of 3rd order HP filter and its equivalent circuit: (b) low-
frequency equivalent circuit and (c) high-frequency equivalent circuit.
Similar to C-type filter, 3rd
order HP filter also has four
components. Thus, it also needs four design conditions or
equations to determine the corresponding parameters in the
filter design.
A. Accepted Design Equations
Same to C-type filter, there are also two well understood
and accepted design conditions for 3rd order HP filter.
Condition 1: The reactive power output of the filter shall be
equal to the required amount QF. This condition yields the
following design equation shown as (7).
Condition 2: The filter is tuned to have a low non-capacitive
impedance at frequency H . This can be achieved by
selecting L that is resonant with 1C at frequency
H ,
which establishes the 2nd
design equation shown as below.
2
11/ HL C (15)
B. Design Condition Based on Loss Minimization
The main purpose of introduction of the auxiliary capacitor
in the 3rd
order HP filter is to reduce the filter loss at the
fundamental frequency. It is, therefore, logical to use loss
minimization to establish the 3rd
design equation. Through
extensive derivations, on condition that the filter impedance
must be inductive for frequencies higher than the tuning
frequency which minimizes the possibility of the filter-system
resonance, the minimal fundamental power loss is obtained
when
2
2 1 1/ ( )C C L R C L . (16)
Equation (16) establishes the 3rd
design equation. Detailed
derivations of (16) can be referred to [14]. It should be noted
that to have a feasible 2C , inherently the damping resistor in
3rd
order HP filter also needs to satisfy (11).
C. Design Equation Based on Resonance-Free Criterion
As discussed in C-type filter design, to limit the filter-
system harmonic resonance severity with little information
about the system condition, the resonance-free criterion (i.e.,
(6)) discussed in Subsection II-B is selected as the fourth
design condition for 3rd
order HP filter design.
Through extensive mathematical operations, the frequency
at which 3rd
order HP filter reaches its maximal max ( )H
res
has been found. It is shown below.
2 2
ax2 2 4
m4
5
3H
R
R R
(17)
where 1/L C .
Accordingly, substituting (17) into (6), the fourth design
equation for 3rd
order HP filter can be further expressed as
4
6
54 2 2
62 2 2
1081
3125Safe
R R
R R
. (18)
D. Summary of Design Procedure
To summarize, 3rd
order HP filter can be designed in the
following way:
1) Determine 1C using (7) and L using (15).
2) Determine R by (18) using Bisection Method;
3) Determine 2C using (16);
4) Select the rating of each component for different possible
operating conditions and provide enough margins for
contingency operations.
5
V. COMPARISON OF FILTER CHARACTERISTICS
This section compares the performance of adapting shunt
capacitor into C-type filter and 3rd
order HP filter through a
real case.
A. Case Description
Fig. 5 shows the simplified single line diagram of a part of
240/144kV, 60Hz transmission system in northeast Fort
McMurray, Alberta, Canada. As it shows, this area
transmission system is mainly composed of three 240kV
substations (i.e., Kinnosis, Kettle River, Leismer), five 144kV
substations (i.e., Engstrom, Quigley, Egg Lake, Bohn, Waddel
and Chard) and one 180MW cogeneration plant (Long Lake).
Three 240kV transmission lines 9L85, 9L930 and 957L
connect this area transmission system with other parts of the
Alberta transmission system through substations Kinosis and
Leismer. Inside this area transmission system, the three 240kV
substations are connected via two 240kV lines 9L45 and 9L62.
The cogeneration plant Long Lake is connected to 240kV
substation Kinosis via 144kV transmission line 7L15. Quigley
and Engstrom along with Kinosis form a loop via three 144kV
lines 7L104, 7L167 and 7L183. Egg Lake and Bohn are
connected to Kettle River via 144kV lines 7LX1 and 7LX2
respectively, while Waddel and Chard are connected to
Leismer via 144kV line 7L114 and its branch 7LA114
respectively.
Quigley 989S
Bohn 931S
Long Lake 841S
856SKinosis
Engstrom2060S
Egg Lake2021S
Kettle River2049S
Leismer72S
Waddel907S
7L
10
4
9L
990
7L
16
7
(9L45
)
9L62
9L93
0
957L
7L183
7L114
7LA114
Chard 656S
Legend240 kV Line
138 kV Line
Generation
Substation
9L85
Fig. 5. Single line diagram of the studied system.
The main shunt reactive power supports in this system are
as follows:
One 144kV 30MVAr C-type filters at Engstrom;
Two 144kV 30MVAr C-type filters at Kinosis;
Two 144kV 30MVAr C-type filters at Kettle River;
Two 138kV 30MVAr capacitor banks at Leismer.
Due to the load growth, 30MVAr reactive power support is
needed at Quigley. The system frequency response at this
substation is shown as Fig. 6. As we can see, this system has a
complex frequency response with multiple series and parallel
resonance due to that it includes an assembly of transmission
lines, transformers, and several shunt reactive power support
devices such as C-type filters and shunt capacitors. Moreover,
there is a significant amount harmonic voltage over a wide
frequency range in this system (see Table I). Therefore, if not
carefully designed, the adding of reactive power support
usually realized by shunt capacitors has a high potential to
induce serious harmonic amplification. As a result, one critical
requirement for the added reactive power support is that it
should not cause more than 110% amplification of the existing
background voltage distortion.
500 1,000 1,500 2,000 2,500 3,000-100
-50
0
50
100
150
200
Frequency (Hz)
R o
r X
R
X
(a) Separate R and X plot
20 40 60 80-100
-80
-60
-40
-20
0
20
40
60
80
100
300Hz660Hz
1020Hz
1380Hz
1886Hz
1880Hz
1892Hz
R ohms
X o
hm
s (
-)
X o
hm
s (
+)
(b) Sprial R-X plot
Fig. 6. System frequency response at Quigley.
TABLE I
BACKGROUND HARMONIC VOLTAGE AT QUIGLEY
Harmonic
Order %
Harmonic
Order %
Harmonic
Order %
1 100 18 0.2 35 0.6
2 1.5 19 1.2 36 0.2
3 2.5 20 0.2 37 0.5
4 1.0 21 0.2 38 0.2
5 3.0 22 0.2 39 0.2
6 0.5 23 0.8 40 0.2
7 2.5 24 0.2 41 0.5
8 0.4 25 0.8 42 0.2
9 1.0 26 0.2 43 0.5
10 0.4 27 0.2 44 0.2
11 1.7 28 0.2 45 0.2
12 0.2 29 0.6 46 0.2
13 1.7 30 0.2 47 0.5
14 0.2 31 0.6 48 0.2
15 0.3 32 0.2 49 0.5
16 0.2 33 0.2 50 0.2
17 1.2 34 0.2 THD 4.0
B. Design Results for Different Filter Configurations
The proposed methods are applied to design C-type filter
6
and 3rd order HP filter, which can provide the required
30MVAr reactive power support without causing any
harmonic amplification over 110% in the worst case. As can
be seen from Table I, the most significant harmonic is the 5th
order. Tuning the filter to this order can help reduce the system
harmonic distortion level. Hence, 300Hz is selected as the
tuning frequency of the filter.
The design results are shown as Table II and Fig.7 to Fig. 9.
It should be noted that here, the worst harmonic amplification
ratio is determined without considering the damping resistor of
the system and it is used as a design criterion for the filter
parameter determination. As for the actual maximum harmonic
amplification, it takes into consideration the impedance of the
system characteristic shown as Fig. 6.
As can be seen from Table II and Fig. 7 to Fig. 9, with the
proposed design methods, both filter configurations can
achieve similar performance in terms of the loss, harmonic
reduction at tuning frequency and the actual maximum
harmonic amplification ratio, while the C-type filter
configuration requires larger auxiliary components (C2, L and R)
with respect to the capacitance, inductance and resistance.
0 500 1000 1500 2000 2500 30000
200
400
600
800
Frequency (Hz)
Impedance (
ohm
)
C-type
3rd order HP
0 500 1000 1500 2000 2500 30000
100
200
300
Frequency (Hz)
Resis
tance (
ohm
)
C-type
3rd order HP
0 500 1000 1500 2000 2500 3000-800
-600
-400
-200
0
200
Frequency (Hz)
Reacta
nce (
ohm
)
C-type
3rd order HP
Fig. 7. Frequency response of designed filters.
500 1000 1500 2000 2500 3000100
105
110
115
Frequency (Hz)
Wors
t harm
onic
am
plif
ication r
atio (
%)
C-type
3rd order HP
Fig. 8. Worst harmonic amplification ratio of designed filters.
500 1000 1500 2000 2500 300050
60
70
80
90
100
110
Frequency (Hz)
Actu
al harm
onic
am
plif
ication r
atio (
%)
C-type
3rd order HP
Fig. 9. Actual harmonic amplification ratio of designed filters.
TABLE II
DESIGN RESULTS FOR DIFFERENT FILTER CONFIGURATIONS
C-type 3rd Order
HP
Component
C1 Value (uF) 4.06 4.06
Rating (kV) 140.37 146.40
C2 Value (uF) 71.41 2.65
Rating (kV) 7.98 9.84
L Value (mH) 98.53 69.32
Rating (A) 130.27 142.24
R R (Ω) 295.31 208.00
Rating (A) 22.29 26.46
Performance
Reactive power support (MVAr) 30.00 31.28
Worst harmonic
amplification ratio (%) 110 110
Actual maximum harmonic
amplification ratio (%) 100 100
5th harmonic voltage reduction (%) 2.30 3.94
Fundamental frequency loss (kW) 0.00 7.12
Harmonic frequency loss (kW) 440.21 429.59
Total loss (kW) 440.21 436.72
VI. ROBUSTNESS INVESTIGATION
In practice, the parameters of all the components in the
filter have manufacturing errors [8, 11]. This section
investigates the impact of the manufacturing errors on the
performance of different filter configurations through Monte
Carlo simulation (MCS) method. The manufacturing error
ranges for different components are given as:
Capacitor: 0~+10%
Reactor: -5%~+5.0%
Resistance: -5%~+5.0%
A. MCS Method Description
The investigation method is described as follows. First, take
the design results in Section V as the base case. Then
randomly generate cases with filter components whose
parameters are deviated from that of this base and evenly
distributed within the typical manufacturing error ranges as
aforementioned. Thereafter evaluate the filter performance of
7
these randomly generated cases. Indices to evaluate include the
loading index of each component, power loss, reactive power
support, worst harmonic amplification ratio. Once this is done
for each random case, aggregate the evaluation results for all
random cases using statistical indices.
B. Robustness Investigation Results
In our investigation, 100,000 MCS trials are conducted.
The investigation results are shown as Table III. As can be
seen from Table III, both the component loading and filter's
performance index will not derivate from the base case
significantly. The maximum mean derivation is the maximum
harmonic amplification ratio for both filters. However, it is
only 10.89% for C-type filter, and 10.43% for 3rd
order HP
filter, which means the manufacturing error of the component
parameter may cause the maximum harmonic amplification
ratio rise to around 121% for both filter under the worst
scenario on average. Fig. 10 and Fig.11 also gives to the
histogram of the histogram of the derivation of the filter
performance caused by the components' parameter variation.
As we can see, the component' parameter variation induced
performance derivation have the similar pattern for both
filters: the reactive power support derivation are evenly
distributed, the maximum harmonic amplification and power
loss are more likely to be norm distributed. This can be
attributed to that, the reactive power support are mainly
determined by the main capacitor's capacitance C1, while the
other two index are complex functions of all components'
parameters which make them are more likely to be norm
distributed.
TABLE III
FILTER PARAMETER AND PERFORMANCE VARIATION
C-type filter 3rd order HP filter
Mean Std Mean Std
Parameter
Main
Capacitor
Loading
Index (%)
△Qrms 5.46% 2.89% 5.33% 3.11%
△Vrms 0.27% 0.23% 0.21% 0.17%
△Vpeak -0.04% 0.22% -0.11% 0.17%
△Irms 4.91% 2.71% 4.83% 2.84%
Auxiliary
Capacitor
Loading
Index (%)
△Qrms 5.63% 6.33% -1.92% 3.23%
△Vrms 0.36% 3.81% -0.37% 2.11%
△Vpeak 0.04% 3.70% -0.94% 2.48%
△Irms 4.91% 2.71% 0.26% 1.97%
Inductor
Rating
(%)
△Srms 6.97% 4.12% 5.55% 4.05%
△Vrms 0.70% 1.98% -0.04% 1.19%
△Irms 4.91% 2.71% 4.65% 2.77%
Resistor
Rating
(%)
△Prms 0.48% 2.03% 0.54% 3.40%
△Vrms 0.23% 1.51% 0.26% 2.46%
△Irms 0.26% 1.99% 0.26% 1.97%
Performance
Reactive Power Support 5.29% 2.91% 5.23% 3.03%
Maximum harmonic
amplification ratio 10.89% 1.50% 10.43% 1.59%
Power Loss 0.48% 2.03% 0.54% 3.40%
-2 0 2 4 6 8 10 12
1
2
3
4
5
6
7
Reactive Power Support Deviation Ratio(%)
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8
10
12
Maximum Harmonic Amplificatin Deviation Ratio (%)
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Filter Power Loss Deviation Ratio (%)
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)
(a) Reactive Power Support (b) Maximum Harmonic Amplification (c) Power Loss
Fig. 10. C-type filter performance index derivation histogram.
8
-2 0 2 4 6 8 10 12
1
2
3
4
5
6
Reactive Power Support Deviation Ratio(%)
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un
cti
on
(%
)
6 8 10 12 14 16
2
4
6
8
10
12
Maximum Harmonic Amplificatin Deviation Ratio(%)
Pro
ba
bilit
y D
en
sit
y F
un
cti
on
(%
)
-8 -6 -4 -2 0 2 4 6 8 10
1
2
3
4
5
6
7
8
9
10
Filter Power Loss Deviation Ratio(%)
Pro
ba
bilit
y D
en
sit
y F
un
cti
on
(%
)
(a) Reactive Power Support (b) Maximum Harmonic Amplification (c) Power Loss
Fig. 11. 3rd order HP filter performance index derivation histogram.
VII. CONCLUSION
In this paper, the concept of resonance-free is introduced
through the analysis of the filter-system resonance interaction.
The corresponding index to quantify the anti-resonance ability
of the filter or the resonance-free criterion for the filter is
established. Based on it, the methods to configure shunt
capacitors to the two most promising passive filter types to
avoid harmful harmonic resonance, i.e., C-type and 3rd
order
High-Pass (HP), are developed. By the developed methods, the
component parameters of both filters can be determined by
four formulas using Bisection Method. Moreover, comparative
studies of these two filter configurations are conducted via a
real shunt capacitor application case. . It is found that the
proposed methods these two filter configurations can achieve
similar performance. In addition, robustness studies dealing
with the component parameter manufacturing errors are also
carried out for these two filter configurations. And they show
that the robustness of these two filter configurations is
relatively comparable.
VIII. REFERENCES
[1] T. H. Fawzi, S. M. El-Sobki and M. A. Abdel-Halim, “New
approach for the application of shunt capacitors to the primary
distribution feeders,” Power Apparatus and Systems, IEEE
Transactions on, vol. PAS-102, no. 1, pp. 10-13, Jan. 1983.
[2] Y. Baghzouz and S. Ertem, “Shunt capacitor sizing for radial
distribution feeders with distorted substation voltages,” IEEE
Trans. Power Del., vol. 5, no. 2, pp. 650-657, 1990.
[3] B. C. Furumasu and R. M. Hasibar, “Design and installation of
500 V back-to-back shunt capacitor banks,” IEEE Trans. Power
Del., vol. 7, no. 2, pp. 539-545, Apr. 1992.
[4] A. A. Sallam, M. Desouky and H. Desouky, “Shunt capacitor
effect on electrical distribution system reliability,” IEEE Trans.
Reliab., vol. 43, no. 1, pp. 170-176, Mar. 1994.
[5] T. M. Blooming and D. J. Carnovale, “Capacitor application
issues,” IEEE Trans. Ind. Appl. , vol. 44, no. 4, pp. 1013-1026,
Jul./Aug. 2008.
[6] J. C. Das, Power System Harmonics and Passive Filter Designs:
John Wiley & Sons, Inc, 2015.
[7] J. Arrillaga, D. A. Bradley and P. S. Bodger, Power System
Harmonics, p.^pp. 1-336: John Wiley & Sons, 1985.
[8] “IEEE Guide for the Application of Shunt Power Capacitors,”
IEEE Std 1036-2010 (Revision of IEEE Std 1036-1992), pp. 1-88,
2011.
[9] D. J. Carnovale, “Power factor correction and harmonic resonance:
A volatile mix,” EC&M Magazine, pp. 16-19, Jun. 2003.
[10] N. R. Watson and J. Arrillaga, “Frequency-dependent AC system
equivalents for harmonic studies and transient convertor
simulation,” IEEE Trans. Power Del., vol. 3, no. 3, pp. 1196-
1203, Jul. 1988.
[11] “IEEE Guide for Application and Specification of Harmonic
Filters,” IEEE Std 1531-2003, pp. 1-60, 2003.
[12] A. B. Nassif, W. Xu and W. Freitas, “An investigation on the
selection of filter topologies for passive filter applications,” IEEE
Trans. Power Del., vol. 24, no. 3, pp. 1710-1718, Jul. 2009.
[13] R. Horton, R. Dugan and D. Hallmark, "Novel design
methodology for C-type harmonic filter banks applied in HV and
EHV networks." pp. 1-6.
[14] T. Ding, W. Xu and H. Liang, “Design method for 3rd order High-
Pass filter,” IEEE Trans. Power Del., vol. PP, no. 99, pp. 1-1,
2015.
[15] Y. Xiao, "The method for designing the third order filter." pp. 139-
142 vol.1.
[16] S. Jian, “Impedance-based stability criterion for grid-connected
inverters,” IEEE Trans. Power Electron. , vol. 26, no. 11, pp.
3075-3078, Nov. 2011.
[17] K. Y. Lee, J. L. Ortiz, M. A. Mohtadi et al., “Optimal operation of
large-scale power systems,” IEEE Trans. Power Syst., vol. 3, no.
2, pp. 413-420, May 1988.
[18] E. Vaahedi and H. M. Z. El-Din, “Considerations in applying
optimal power flow to power system operation,” IEEE Trans.
Power Syst., vol. 4, no. 2, pp. 694-703, May 1989.
[19] N. Amjady, D. Farrokhzad and M. Modarres, “Optimal reliable
operation of hydrothermal power systems with random unit
outages,” IEEE Trans. Power Syst., vol. 18, no. 1, pp. 279-287,
Feb. 2003.
[20] J. J. Thomas and S. Grijalva, “Flexible Security-Constrained
Optimal Power Flow,” IEEE Trans. Power Syst., vol. 30, no. 3, pp.
1195-1202, May 2015.
[21] CIGRE WG 14.03, “AC harmonic filter and reactive compensation
for HVDC: a general survey,” Electra, no. 63, 1979.
[22] C. Chih-Ju, L. Chih-Wen, L. June-Yown et al., “Optimal planning
of large passive-harmonic-filters set at high voltage level,” IEEE
Trans. Power Syst., vol. 15, no. 1, pp. 433-441, Feb. 2000.
1
Resonance-Free Shunt Capacitors -
Configurations, Design Methods and
Comparative Analysis (Final Version)
Wilsun Xu, Fellow, IEEE, Tianyu Ding, Student Member, IEEE, Xin Li, and Hao Liang, Member, IEEE
Abstract-- Harmonic resonance has become an important con-
cern for the application of shunt capacitors in recent years. A
potential solution to address this challenge is to convert a shunt
capacitor into a passive filter. This paper presents design methods
to configure a shunt capacitor as a C-type filter or a 3rd order
high pass filter with guaranteed resonance-free performance. The
concept of resonance-free condition is first introduced in this pa-
per. It is then used to develop filter design methods that always
meet the resonance-free condition. The two filter configurations
are also compared. It was found that the 3rd order high pass filter
has more advantages than the C-type filter. Another useful find-
ing of this work is that the filter parameters as determined using
the proposed design are independent of the system conditions. As
a result, a lookup table for the filter parameters has been created
to facilitate immediate use by industry.
Index Terms-- Shunt capacitor, resonance, harmonics, filters.
I. INTRODUCTION
HUNT capacitors, as the simplest form of reactive power
compensation, have been widely used in utility networks,
industrial systems and wind farms. In recent years, the prolif-
eration of harmonic-producing loads in power systems have
led to increased incidences of shunt-capacitor related harmonic
resonance [1-5]. In some cases, for example, a shunt capacitor
cannot be energized due to excessive harmonic currents, lead-
ing to reactive power shortage in the system [6].
In respond to this situation, the idea of configuring a shunt
capacitor as a single-tuned filter has been proposed. This is
done by inserting a series inductor to the shunt branch [7]. The
tuning frequency is normally the harmonic frequency closest to
the capacitor-system resonance frequency, such as the 5th
har-
monic frequency. This approach does not always work since
the newly configured capacitor may resonate at other frequen-
cies due to the complex frequency response of the power sys-
tem. Adding a damping resistor is not an option since it can
significantly increase 60Hz losses of the shunt device.
As a result, some utilities have started to configure shunt
capacitors as the C-type filters [8, 9]. The C-type filter config-
uration possesses good damping characteristics at frequencies
higher than its tuning frequency and has almost zero losses at
the fundamental frequency. However, to the authors’ best
This work was supported by the Natural Sciences and Engineering Re-
search Council of Canada.
The authors are with the department of Electrical and Computer Engineer-
ing, University of Alberta, Edmonton, AB T6G 2V4, Canada (email:
knowledge, design methods to configure a shunt capacitor as a
C-type filter that guarantees a resonance-free performance
have not been developed.
The C-type filter is just one of the modified versions of the
2nd
order high-pass (2nd
HP) filter. Another version is 3rd
order
high-pass (3rd
HP) filter [10]. Both the C-type and 3rd
HP filters
have the same number of RLC components. The main differ-
ence is on the mechanisms of reducing the fundamental fre-
quency loss. In view of this situation, one would naturally
wonder if the 3rd
HP filter can also be used to construct a reso-
nance-free shunt capacitor and which filter configuration has
more advantages.
This paper presents our research findings on the above is-
sues. The resonance-free concept is introduced first. It is then
used to develop design methods for both C-type and 3rd
HP
configured shunt capacitors with guaranteed resonance-free
outcome. Performances of the two configurations are then
compared. A major finding of this work is that the filter pa-
rameters (derived from the proposed design methods) are in-
dependent of the system frequency responses. As a result, a
standard set of filter component parameters are calculated for
direct industry use.
II. RESONANCE FREE CONDITION
Resonance involves the interaction of components with dif-
ferent impedance characteristics. A power system to which a
shunt capacitor is connected can have different impedance
characteristics at various frequencies such as those illustrated
in Fig. 1. The impedance characteristics may also change as a
function of the number and type of network components in
service. As a result, it is very difficult for a shunt capacitor not
to resonate with the system impedance at some frequencies.
0 1000 2000 3000-1000
0
1000
2000
Frequency (Hz)
R o
r X
(
)
RS
XS
0 500 1000 1500 2000-1000
-500
0
500
1000
300Hz1020Hz
1380Hz
1886Hz
1880Hz
1892Hz
R ()
X (
)
(a) Separate R and X plot (b) Sprial R-X plot Fig. 1. A sample power system frequency response.
Fig. 2 shows the equivalent circuit of the system and a pas-
sive shunt capacitor device. For simplicity, this device is called
filter as the capacitor will take the form of either C-type or
3rd
HP filter. Before the filter is connected, the harmonic volt-
S
2
age at the interconnection point is 0 ( )FV , where is the
angular frequency. After connecting the filter, the voltage be-
comes ( )FV . The ratio of these two voltages is defined as the
harmonic amplification ratio (HAR):
0( ) ( ) / ( )F FHAR V V . (1)
( )FZ
( )FV ( )SZ
( )SV
System Filter (Capacitor)
F
( )SV : System background
harmonic voltage
( )SZ : System equivalent
harmonic impedance
( )FZ : Filter equivalent
harmonic impedance
Fig. 2. Equivalent circuit of the system and filter (capacitor).
According to Fig. 2, 0 ( )FV is equal to the system background
harmonic voltage ( )SV . Substituting it into (1) yields
0( ) ( ) / ( ) ( ) / ( ( ) ( ))F F F F SHAR V V Z Z Z (2)
which can be further represented as
2 2
2 2
( ) ( )( )
( ) ( ) ( ) ( )
F F
F S F S
R XHAR
R R X X
(3)
where ( ) Im( ( )), ( ) Re( ( ))S S S SX Z R Z .
As can be seen from (3), for a given frequency, the most se-
rious voltage amplification occurs when the system impedance
is purely reactive and is equal to the negative equivalent filter
reactance
( ) 0, ( ) ( )S S FR X X (4)
Under such a condition, the denominator in (3) is minimum,
which results in the largest (i.e., worst case) amplification ratio
of
2
worst ( ) 1 ( ( ) / ( ))F FHAR X R (5)
It can be seen from (5) the worst or the largest amplification
ratio is different for different harmonic frequencies and it is
only affected by the filter's reactance to resistance ratio at the
harmonic frequencies.
A resonance-free condition is defined as
worst limit( ) , for HHAR HAR (6)
where HARlimit is a user specified threshold. Definition (6)
quantifies a shunt capacitor as a resonance-free capacitor if the
worst case amplification of harmonic voltage caused by the
capacitor is less than a user specified limit HARlimit, for any
harmonic frequencies higher thanH .
For example, a limit value of HARlimit=1.2 means the har-
monic voltages after a capacitor connection are guaranteed to
be less than 1.2 times of the harmonic voltages prior to the
capacitor connection. Since the 5th
harmonic is the lowest or-
der characteristic harmonic in power systems, H can be se-
lected as the frequency of the 5th
harmonic. As a result, a ca-
pacitor satisfying these conditions is guaranteed not to reso-
nate with the system at any frequencies above and equal to the
5th
harmonic.
Based on the above considerations, design methods for the
C-type filter and 3rd
HP filter configured resonance-free shunt
capacitor are developed next.
III. C-TYPE FILTER CONFIGURED CAPACITOR
The C-type filter (Fig. 3b) was first introduced in [11] to
replace multiple single-tuned filters for a HVDC application. It
is a modified version of the 2nd
HP filter (Fig. 3a) [9]. The ob-
jective of modification is to eliminate the fundamental fre-
quency loss. This is achieved through the C2+L branch which
is tuned to series resonance at the fundamental frequency. This
condition leads to the bypass of the R branch and, thereby, the
elimination of power loss at the fundamental frequency. At low
order harmonics, the 2C L branch dominates so the filter
behaves like a single tuned filter (Fig. 3c). At high order har-
monics, the R branch dominates and the filter behaves as a
resister R in series with 1C (Fig. 3d).
1C
L R
(a) 2ndHP filter
1C
2C
L
(c) Low frequency
equivalent circuit
1C
R
(d) High frequency
equivalent circuit
(b) C-type filter
1C
2C
LR
Fig.3. C-type filter and its equivalent circuits at different frequency ranges.
A C-type filter has four components. Four design conditions
or equations are therefore needed to determine their parame-
ters.
A. Basic design equations
For the C-type filter, two design conditions are well under-
stood and accepted by industry and research community:
1) Condition 1: The reactive power output of the filter shall be
equal to the required amount QF. This condition yields the
following design equation:
2
1 1/ ( )FC Q V (7)
where 1 is the power frequency and V is the rated volt-
age.
2) Condition 2: C2 and L are tuned to the fundamental frequen-
cy to eliminate the fundamental frequency power loss,
which leads to
2
1 2 =1/( )L C . (8)
B. Condition of inductive impedance
Depending on the combination of component parameters, a
C-type filter can exhibit a capacitive or inductive impedance
characteristic at any harmonic frequencies. Here we propose
3
the following design condition: the impedance of C-type is
always inductive above its tuning frequency, i.e.,
( ) 0, F HX (9)
This condition is based on the following consideration: If a
system has an inductive impedance characteristic at frequency
and the C-type also has an inductive impedance, the actual
amplification ratio will be much less than HARworst because the
term ( ) ( )F SX X in (3) is additive. Since the system im-
pedance is more likely to be inductive than capacitive at vari-
ous frequencies, it is advantageous to have a C-type filter that
exhibits inductive impedance above its tuning frequencies.
It has been proven mathematically in Appendix A.2 that
condition (9) will always be satisfied if
2 2
2 1( 1) /H HC n n C (10)
and ( ) 0F HX (11)
where 1/H Hn . Equation (11) leads to the third design
equation:
2 2
1 2 2 1 2( ) / (1 ( ))H H HC Cn CR n C C (12)
The derivation is given in Appendix A.1.
C. Resonance-free condition
The C-type filter configured shunt capacitor must be reso-
nance free. Thus, the 4th
design equation is the resonance-free
condition established by (6).
Through extensive mathematical operations given in Ap-
pendix A.3, the frequency at which a C-type filter reaches its
maximal HARworst has been found as follows:
when 2 2
2 1( 1) /H HC n n C ,
4 2
ax
2
m 1 25 22 1 5 1)( / 6H H Hn n n ;(13)
when 2 2
2 1( 1) /H HC n n C ,
1/3 1/3
max 1 ( ) ( ) ( 1) / ( 1)A D AB B (14)
where
2 2
1 2
2 2 2
2
2 2 2 3 2
3
2 2 2
/ ( / ), / ( / )
(4( ( ) 9 )( )
18( 1)
2 2 7 4
2( 1)
( 2 3 7 ) /
324 (4 ) 3
(3(
8)
1) )
R L C R L C
B
D
A
(15)
Resonance-free condition means this maximal HARworst must
be less than or equal to HARlimit. Substituting (13) or (14) ac-
cording to the value of C2 into (6) yields the fourth design
equation:
24 2
max max
2 limit22 2 2 4
1 max max1
(1 ) ( 2) 11
( 1)
h hHAR
C R h h
(16)
where max max 1/h .
D. Summary
To summarize, parameters of a C-type filter configured ca-
pacitor can be determined using the following procedure:
1) Determine C1 using (7).
2) Set 2 2
2 1( 1) /H HC n n C .
3) Calculate L and R using (8) and (12), respectively.
4) Substitute the values of C1, C2, L and R into (16) to check if
the fourth design equation is satisfied. If it is satisfied, then
the C2, L and R values are the solutions. Otherwise go to
Step 5).
5) Try another value of C2 using the Bisection Method [12]
and go to Step 3).
IV. 3RD
HP FILTER CONFIGURED CAPACITOR
The 3rd
HP filter (Fig. 4b) is the most widely used filter to
shunt high order harmonics such as 11th
, 13th
etc., for both
industrial systems and HVDC links [10, 13]. It is also a modi-
fied version of the 2nd
HP filter. The objective of modification
is again to reduce the fundamental frequency loss. This is
achieved by inserting 2C into R branch, which increases the
impedance of that branch at fundamental frequency and thus
reduces fundamental frequency loss of component R. At low
frequencies below the tuning frequency, the filter’s L branch
dominates so it behaves as a single-tuned filter (Fig. 4c), while
at high frequencies, the 2C R branch dominates hence it
behaves as a first order high pass filter (Fig. 4d).
(b) 3rdHP filter
1C
L
(c) Low frequency
equivalent circuit
1C
2C
R
(d) High frequency
equivalent circuit
(a) 2ndHP filter
1C
L R
1C
L2C
R
Fig.4. 3rd HP filter and its equivalent circuits at different frequency ranges.
Similar to the C-type filter, a 3rd
HP filter also has four
components. Thus, it also needs four design equations.
A. Basic design equations
The following two design requirements have been well ac-
cepted by industry and research community [14, 15]
1) Condition 1: The reactive power output of the filter shall be
equal to the required amount QF. This condition yields a de-
sign equation that is shown as (7).
2) Condition 2: The filter is tuned to have a low non-capacitive
impedance at frequency H . This can be achieved by se-
lecting L that is resonate with 1C at frequency
H , which
establishes the 2nd
design equation shown as below.
4
2
11/ ( )HL C (17)
B. Condition of loss minimization
The main purpose of the auxiliary capacitor in the 3rd
HP fil-
ter is to reduce the filter loss at the fundamental frequency. It
is, therefore, logical to use loss minimization to establish the
3rd
design equation. Reference [15] has shown that the corre-
sponding design equation is:
2
2 1 1/ ( )C C L R C L . (18)
Note that this design equation is obtained based on the re-
quirement that the filter impedance must be inductive for fre-
quencies higher than the tuning frequency, which requires the
damping resistor must satisfy the condition 12 /R L C [15].
C. Resonance-free condition
It is also evident that this filter configuration must be reso-
nance free, i.e., its maximal HARworst must be less than or equal
to HARlimit. Through extensive mathematical operations similar
as those shown in Appendix A.3, the frequency at which 3rd
HP
filter reaches its maximal HARworst has been found as follows:
2 2 4 2 2 4
max 5( ) / 3( )H R R R (19)
where 1/L C . Accordingly substitute (19) into (6).Then
the fourth design equation for the 3rd
HP filter can be derived
as follows:
4
l
4 2 2 5
2 2 2 imit66
108(1
3125
)
( )
R
RHA
R
RR
. (20)
D. Summary
To summarize, parameters of a 3rd
HP filter configured ca-
pacitor can be determined using the following procedure:
1) Determine 1C using (7) and L using (17).
2) Set 12 /R L C and 1/L C .
3) Substitute the values R and into (20) to check if the
fourth design equation is satisfied. If it is satisfied, then the
R value is the solution. Calculate C2 using (18). Otherwise,
go to Step 4).
4) Try another value of R using the Bisection Method [12] and
go to Step 3).
V. COMPARISON OF FILTER CHARACTERISTICS
This section compares the performances of the two filter
configurations for a shunt capacitor through a case study.
A. Case Description
Fig. 5a shows the simplified single line diagram of a part of
240/144kV transmission system Alberta, Canada. The system
has over 1000 transmission voltage buses and a number of
shunt capacitors. This study involves a 30MVAr reactive pow-
er to be added to substation SX. The system frequency re-
sponse as seen from this substation is shown as Fig. 1. The
response shows multiple series and parallel resonance points.
The background voltage distortion spectra at 144kV bus in
substation SX is shown as Fig. 5b.
SX
S5
S2S1
S3
S6S4
S7
S9
L4
L3
L2
L6
L10
L7
L5
L8
L9
S8
Legend240 kV Line
138 kV Line
Generation
Substation
L1
(a) Single line diagram (b) Background voltage spectra
5 10 15 20 25 30 35 40 45 THD0
1
2
3
4
Harmonic Order
IHD
(%
)
Fig. 5. Single line diagram and background voltage spectra.
B. Design results
The proposed methods are applied to determine the parame-
ters for the C-type and 3rd
HP filters. The resonance free condi-
tion is selected as HARlimit=1.2 and the tuning frequency is
300HzHf (i.e., 5th
harmonic).
The design results are shown in Fig.6 and Fig. 7. Fig. 6
shows the frequencies responses of the filters. It can be seen
that both filter has a large equivalent resistance above the tun-
ing frequency. This resistance is the source of damping that
brings down the amplification ratio. Both filters have similar
frequency response characteristics.
0 500 1000 1500 2000 2500 30000
200
400
600
800
Frequency (Hz)
|ZF| (
)
C-type
3rdHP
0 500 1000 1500 2000 2500 30000
100
200
300
400
RF (
)
Frequency (Hz)
0 500 1000 1500 2000 2500 3000-1000
-500
0
500
XF (
)
Frequency (Hz)
300 Hz
Fig. 6. Frequency response of designed filters.
The theoretical worst-case and the actual system-dependent
harmonic amplification ratios are shown in Fig. 7. The worst-
case ratio is determined according to (5) assuming the system
impedance has no resistance and is in perfect resonance with
the filter. The actual amplification ratio is calculated consider-
ing the system impedance (i.e., using (3)). The amplification
ratio associated with pure capacitor installation is also calcu-
lated. The ratio is close to 2p.u. for the 5th
and 7th
harmonic
and is as high as 14 p.u. around the 11th
harmonic. The results
reveal the following: (1) if not configured as a filter, the shunt
5
capacitor will result in excessive harmonic amplification at
several harmonic frequencies. On the other hand, either C-type
or 3rd
HP filter configured capacitor will not increase harmonic
voltages since the actual amplification ratio is always less than
1 for all harmonics; (2) The C-type and 3rd
HP filters have
similar resonance mitigation performance.
300 500 1000 1500 2000 2500 30000
0.5
1.0
1.2
1.5
Frequency (Hz)
Am
plifica
tio
n r
atio
(p
.u.)
HARworst
--3rd
HP
HARworst
--C-type
HARactual
--C-type
HARactual
--3rd
HP
Fig. 7. Harmonic amplification ratio of filter configured capacitors.
Table I shows filter parameters and loading conditions. In
the table, index “voltage” means the RMS voltage experienced
by the corresponding component. It can be seen that the two
filter configurations have similar parameters and loading con-
ditions. The losses are also comparable. The main difference
is the value of C2. Taking into consideration that C2 is much
smaller, R and L are slightly smaller for the 3rd
HP configura-
tion, it seems that the 3rd
HP configuration is more advanta-
geous.
TABLE I: FILTER PARAMETERS AND LOADING CONDITIONS
C-type 3rdHP
Sp
ecific
atio
n
C1 Value (uF) 3.84 3.84
Voltage (kV) 144.44 150.58
C2 Value (uF) 75.66 2.06
Voltage (kV) 7.32 9.50
L Value (mH) 92.99 73.34
Voltage (kV) 12.72 12.07
R Value (Ω) 361.02 233.84
Voltage (kV) 10.00 7.45
Perfo
rm
an
ce
Reactive power (MVAr) 30.00 31.28
HARactual
(p.u.)
Maximum 1.00 0.98
Average (among harmonics) 0.80 0.80
Power loss
(kW)
Fundamental frequency 0.00 5.14
Harmonic frequencies 276.78 232.25
Total 276.78 237.40
C. Filter Robustness
In practice, parameters of the filter components are not ex-
act due to manufacturing variations. This section investigates
the impact of this issue on filter performance and loading con-
dition. Based on [16] and the data specification collected from
manufacture, the range of parameter variations for RLC com-
ponents are 1) Capacitance: 0~+10%; 2) Inductance: -
3%~+3%; 3) Resistance: -10%~+10%.
The method of investigation is as follows. The base case is
selected as the design parameters and associated loading con-
ditions of Subsection V-B. The component parameters are then
assumed vary randomly around the design parameters based on
the range of manufacturing variations. A normal distribution is
assumed for the variations. This will result in tens of thousands
of possible combinations or scenarios of component parame-
ters. Each filter performance index is then calculated for all
studied scenarios and a pair of statistical values representing
the 95% variation interval are then determined from the tens of
thousands scenarios.
In this study, 100,000 combinations are calculated using the
Monte Carlo simulation method. The results (the variation
from the base case in percentage) are shown in Table II. As
can be seen from Table II, both the component loading and
filter's performance indices all vary around the base case value
within a small range (below 7%) except for the reactive power
support. It should be noted that the reactive power support is
approximately in a linear relation with C1. So its variation
range is similar with the capacitance manufacturing error tol-
erance. The results show that the performances of both filters
as designed by the proposed method are quite robust. Both
filter types have comparable robustness. The 3rd
HP filter con-
figuration is slightly more attractive from the harmonic ampli-
fication perspective since the variation ranges of its HARworst
and HARactual are all smaller.
TABLE II: FILTER LOADING AND PERFORMANCE VARIATION
C-type 3rdHP
Lo
ad
ing
Vrms (%)
C1 -0.04~0.54 -0.04~0.49
C2 -6.31~6.91 -3.54~4.54
L -1.16~3.64 -1.33~2.06
R -3.45~2.00 -5.49~5.66
Perfo
rm
an
ce
Reactive power support (%) 0.23~10.30 -0.01~10.47
HARworst (%) Maximum -4.20~6.92 -3.31~3.91
HARactual
(%)
Maximum -0.55~0.65 -0.09~0.06
Average (among harmonics) -1.02~0.97 -0.90~0.20
Power
loss (%)
Harmonic frequency -6.45~3.42 -3.85~3.28
Total -6.28~3.61 -3.29~3.69
VI. LOOKUP TABLE FOR FILTER PARAMETERS
A very interesting outcome of the proposed filter design
method is that the filter parameters are independent of the sys-
tem impedances. This is because the parameters are deter-
mined for the worst case system condition. As a result, a
standard set of filter parameters can be calculated and
achieved for direct use by industry. For this purpose, a lookup
table of per-unit filter parameters has been created and is
shown in Table III. The table lists the parameters for two tun-
ing frequencies (nH=3 and nH=5) under three HARlimit values.
To further facilitate the selection of proper HARlimit value in
terms of the size of components, the variations of filter param-
eters as functions of the HARlimit is shown in Fig. 8 for the case
of nH=5 where all component parameters are normalized val-
ues with respect to the case of HARlimit=1.1 for easier compari-
son.
The base values of the Table III are: rated voltage of the
system where the capacitor is to be installed (Vr), rated reac-
tive power of the main capacitor C1 under rated voltage condi-
6
tion (Qr), and system fundamental frequency f1. Thus base val-
ue can be determined as 2
1/ (2 )b r rC Q f V for capacitance, 2
1/ (2 )b r rL V f Q for inductance, and 2 /b r rR V Q for re-
sistance.
TABLE III: LOOKUP TABLE FOR PER-UNIT FILTER PARAMETERS
3Hn 5Hn
C-type 3rdHP C-type 3rdHP
limitHAR
1.1
C1 1.0000 1.0000 1.0000 1.0000
C2 5.9845 0.6519 17.6440 0.6519
L 0.1671 0.1111 0.0567 0.0400
R 0.7678 0.5306 0.4533 0.3184
1.2
C1 1.0000 1.0000 1.0000 1.0000
C2 6.6414 0.5372 19.7164 0.5372
L 0.1506 0.1111 0.0507 0.0400
R 0.8878 0.5639 0.5223 0.3383
1.3
C1 1.0000 1.0000 1.0000 1.0000
C2 6.9799 0.4697 20.7838 0.4697
L 0.1433 0.1111 0.0481 0.0400
R 0.9993 0.5896 0.5871 0.3538
(b) 3rdHP filter(a) C-type filter
1.1 1.2 1.3 1.4 1.5
0.6
0.8
1
1.2
1.4
1.61.6
HARlimit
p.u
.
C1
C2 L R
1.1 1.2 1.3 1.4 1.5
0.6
0.8
1
1.2
1.4
1.61.6
HARlimit
p.u
.
C1
C2 L R
Fig. 8 Filter parameter variations as functions of user selected
limitHAR (nH=5).
Whenever a shunt capacitor needs to be configured into a
resonance-free filter, parameters of the filter components can
be determined using Table III as follows: (1) determine the
based values (Vr, Qr, f1) to be used; (2) calculate the base val-
ues of the capacitance, inductance and resistance; (3) decide
on the harmonic voltage amplification limit HARlimit, tuning
harmonic order Hn , and the filter topology to be used; (4)
locate the p.u. values of the component parameters in Table
III; (5) multiply the p.u. values by the corresponding base val-
ues to get the physical values.
It is worthwhile to point out that although the filter parame-
ters are independent of the system conditions, the component
loading levels are affected by the system impedance and back-
ground harmonic voltages. As a result, the actual physical
components to be manufactured can be slightly different even
for the same sized C1 capacitor. The component loading levels
can be easily determined through harmonic power flow studies
since the filter parameters are known already.
VII. CONCLUSIONS
This paper has presented research findings on how to make
a shunt capacitor resonance free. The concept of resonance-
free is introduced first and is quantified mathematically. Based
on the concept, design methods to configure a shunt capacitor
as a resonance-free C-type filer or 3rd
HP filer are developed.
Through rigorous mathematical analysis, this work has proven
that the proposed design methods can guarantee resonance-free
performance of the reconfigured shunt capacitor. Furthermore,
it has shown that the filter parameters are independent of sys-
tem conditions. As a result, a lookup table for the parameters
has been created for direct use by industry.
Comparative analysis has been conducted on the two filter
configurations. The results show that they have comparable
performance characteristics. The 3rd
HP filter has more ad-
vantages in term of smaller component sizes and higher ro-
bustness against component parameter variations.
APPENDIX: DERIVATION OF DESIGN EQUATIONS
The appendix presents the derivations and proofs of design
equations for the C-type filter. The derivations for the 3rd
HP
filter are similar. They are omitted here due to page limit.
A.1 Equation to determine R for C-type filter
According to Fig. 3b, the impedance of C-type filter is
1 2
1 1( ) / /FZ j L R
j C j C
. (21)
Accordingly, the reactive component ( )FX and resistive
component ( )FR of C-type filter impedance can be derived
as
4 2
1 1
2 2 2 2
2 2 1
2 2
2
2 2 2 2
2 2
1( ) Im( ( ))
( ( ) ( 1) )
( 1)( ) Re( ( ))
( ) ( 1)
F F
F F
a bX Z
RC LC C
R LCR Z
RC LC
(22)
where
2 2 2 2 2 2 2
1 2 1 2 1 2 1 2 2, 2a R LC C L C b LC R C C R C (23)
As can be seen from (22), the denominator of ( )FX is al-
ways positive, 0F HX is equivalent to the numerator
polynomial of FX equals to 0 when H , i.e.,
4 2
1 1 1 0H Ha b (24)
Take (8) and (23) into (24), then we can get
2
4 2 2 2 21 2
1 2 22 4 2
1 1 1
1 21 0H H
R C CR C C R C
. (25)
Let 1/H Hn . Substitute it into (25) and rearrange the
equation by taking R as the unknown variable. Then we have
2 2 2 2
1 2 2 1
2 2
1 2( ) 1( )H H Hn n C C C C C nR (26)
Since 1Hn , both sides of (26) are positive, which means
2 2
1 2 2 1 2 2 1( ) (0 1)H Hn C C C C C C n C . (27)
Then we can take square root of both sides of (26) and substi-
tute 1Hn by H . By moving all other terms to the left side of
7
the equation except for R, (12) is obtained.
A.2 Proof of inductive impedance condition for C-type filter
Substitute (8) and (12) into (23), then
4
1 2 1 1
4 2 4 2(1( 2 1) / / ) 1H H H Hn n na C C n . (28)
According to (10),
2 1
4 2 4 2(1 / ) 2 1H H H Hn n n nC C (29)
Hence
4 2 4
2
2
1(( 2 1) 1/ 1 / )H H H Hn n n C nC (30)
Therefore,
2 2 2 2
1 2 1 2 10 0 /a R LC C L C R L C (31)
Let 2x . Then the numerator polynomial of ( )FX can be
represented by
2
1 1( ) 1Hg x a x b x (32)
When 1 0a , according to Vita's fomulas, ( ) 0Hg x has
one negative root 1x and one positive root 2
2 Hx . Accord-
ingly ( )Hg x can be further represented as
1 1 2( ) ( )( )Hg x a x x x x (33)
Then it can be seen when 2x x , ( ) 0Hg x .Since as afore-
mentioned that the denominator of ( )FX is always positive,
correspondingly when H , ( ) 0FX .
When 1 0a , i.e., 2 2
2 1( 1) /H HC n n C , 1/R L C , then 2
Hb . Hence
2
( ) 1H
H
xg x
. (34)
As can be seen from (34), when 2
Hx , ( ) 0Hg x . Accord-
ingly when H , ( ) 0FX .
In summary, when (10) and (11) (or (12)) are satisfied,
1 0a and ( ) 0FX for H .
A.3 Frequency to reach maximal HARworst for C-type filter
This section first presents the equivalent condition to reach
maximal HARworst, which simplifies the solve of maximal
HARworst to an easier problem and then derives the frequency
to reach maximal HARworst for C-type filter.
Let
( ) ( ) / ( )F FX R (35)
Then according to (5), 2
worst ( ) 1 ( )HAR . The deriva-
tive of worst ( )HAR with respect to is given by
worst
worst
( )( ) ( )
( )
dHARHAR
d
. (36)
Since ( ) 0FR and ( ) 0FX for H according to the
established design condition, and the derivative of
worst ( )HAR with respect to ( ) is
worst
2
( ) 2 ( )0
( ) 1 ( )
dHAR
d
. (37)
Therefore,
worst
worst
worst
( ) 0, when ( ) 0
( ) 0, when ( ) 0
( ) 0, when ( ) 0
HAR
HAR
HAR
(38)
Accordingly, worst ( )HAR achieve its extrema when ( )
achieves its extrema, that is,
worstarg max ( ) arg max ( )
HH
HAR
. (39)
Therefore, the solve of the frequency to reach maximal
HARworst is transformed into the solve of the frequency to reach
maximal ( ) .
Substitute (22) into (35), then we can obtain
4 2
1 1
2 2
1 2
1( )
( 1)
a b
RC LC
. (40)
The derivative of ( ) with respect to is given by
6 4 2
1 1 1
2 2 3
1 2
( )( )
1
1
3d e
RC LC
g
(41)
where 3 3 2 2 3
1 2 2 1d L C R L C C , 3 2
1 2 2
2 2Re CL L C and 2 22
1 2 1 2 23g C C LCR R C .
As can be seen from (41), the denominator of ( ) is always
positive. Hence whether ( ) is positive, zero or negative is
exclusively determined by its numerator. Let
2
1/ /R L C , 2
2/ /R L C and 2
1/x . Then
the numerator of ( ) can be further represented by
3 2( ) (1 ) 3( 1) ( 3) 1Cg x x x x (42)
When 1 0a , 1/R L C , then, 1 0 . Accordingly,
( ) 0, ( ) 0, (0) 0C C Cg g g . (43)
Further, as ( ) 0F HX , and ( ) 0F HX , when H ;
( ) 0FX , when H per the proof in Appendix A.2, as
can be seen from (40), ( ) 0H , ( ) 0 , when H ;
( ) 0 , when H . Correspondingly, ( ) 0H ,
which also means
2( ) 0C Hg n (44)
8
By (43) and (44), we can see ( ) 0Cg x has three roots 1x ,
2x
and 3x , which are governed by
2
1 2 30 Hx x n x . (45)
Accordingly ( )Cg x can also be represented by its root form as
1 2 3( ) (1 )( )( )( )Cg x x x x x x x (46)
It can be seen from (46), when 3x x , ( ) 0Cg x ; when
3x x , ( ) 0Cg x . Accordingly, when 3 1x , ( ) 0 ,
when 3 1x , ( ) 0 . Therefore, ( ) achieves its
maximum extrema at 3 1x for
H . So is
worst ( )HAR per (39). By the general formula for roots of cu-
bic equation, 3x can be obtained as
1/3 1/3
3 ( ) ( ) ( 1) / ( 1)x A D AB B (47)
Then as max 3 1x , (14) is obtained.
When 1 0a , 1/R L C and 2 2
2 1( 1) /H HC n n C , take
them with (8) into (40), then
2 3 2 2
1
3 2 2 2
1
1 ( )( )
( )
H H
H
n
n
. (48)
The derivative of ( ) with respect to is given by
2 3
1 0
3 2 2 2 3
1
1 ( )( )
( )
H C
H
n g
n
(49)
where 4 2 2 2 2 2
0 1 1( ) 3 ( 5 )C H Hg .
As can be seen from (49), the other terms except 0 ( )Cg in
( ) is always positive. Hence, whether ( ) is positive,
zero or negative is exclusively determined by 0 ( )Cg . Let
2x , then
2 2 2 2 2
0 1 1 1 2( ) 3 ( 5 ) 3( )( )C H Hg x x x x x x x (50)
where 2 2
1 1 2Hx x , since 2
0 1( ) 0Cg , 2
0 ( ) 0C Hg .
It can be seen from (50), when 2x x ,
0 ( ) 0Cg x ; when
2x x , 0 ( ) 0Cg x . Accordingly, when 2x , ( ) 0 ,
when 2x , ( ) 0 . Therefore, ( ) achieves its max-
imum at 2x for H . So is worst ( )HAR per (39).
Further, by the general formula for roots of quadric equation,
we can obtain
2
2
4 22
1 25 22 1 5 1)( / 6H H Hn n nx (51)
Then as max 2x , (13) is obtained.
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Wilsun Xu (M’90-SM’95-F’05) obtained the Ph.D. degree in Electrical and
Computer Engineering from the University of British Columbia, Vancouver,
in 1989. Currently, he is a NSERC/iCORE Industrial Research Chair Profes-
sor at the University of Alberta. His current main research interests are power
quality and power disturbance analytics.
Tianyu Ding (S'12) obtained the B.Sc. degree in Electrical Engineering from
Shandong University, Jinan, China, in 2010. Currently, he is pursuing his
Ph.D. degree in Electrical and Computer Engineering at the University of
Alberta, Edmonton, AB, Canada. His main research interest is power quality.
Xin Li received the B.Sc. degree and M.Sc. degree in Electrical Engineering
from Southwest Jiaotong University, Chengdu, China, in 2010 and 2013
respectively. Currently, he is pursuing his Ph.D. degree in Electrical and
Computer Engineering at the University of Alberta, Edmonton, AB, Canada.
His main research interest is power quality.
Hao Liang (S’09–M’14) received the Ph.D. degree in Electrical and Com-
puter engineering from the University of Waterloo, Waterloo, ON, Canada, in
2013. Since 2014, he has been an Assistant Professor at University of Alberta.
His research interests are in the areas of smart grid, wireless communications,
and wireless networking.