Relaying 230 kV, 100 Mvar C-Type Filter Capacitor Banks
Randy Horton, Ted Warren
Alabama Power Company
Timothy Day, Jack McCall, Arvind Chaudhary
Cooper Power Systems
These levels
guidelines.
Abstract:
Shunt capacitor banks are considered an economical
source of reactive power and are installed in many
locations throughout the power system. The addition of
a capacitor bank creates a new, and lower resonant
frequency as the capacitance interacts with the inherent
system inductance. Recent years have seen greater
numbers of large industrial customers, with their nonlinear loads, connected to high-voltage portions of the
power system. If the harmonic current injected by these
nonlinear loads has appreciable components at the
resonant frequency of the power system, a severe overvoltage situation can occur. As a result, the ability to
filter harmonic current with adequate damping over a
selected frequency range has become a necessity in
some shunt capacitor bank designs. The C-type
harmonic filter can be designed to meet these
requirements. This paper describes the design,
application, and novel protection of a C-type harmonic
filter bank.
were
obviously
outside
IEEE-519
Following a preliminary analysis, it was concluded that
the harmonic currents would need to be filtered by
converting the capacitors into tuned harmonic filters. A
single-tuned filter design was considered along with its
disadvantages when connected to a transmission
system. As the system changes; i.e. new lines are built,
other capacitor banks are installed, etc., the resonant
frequency will shift. Changes to the filter would need to
be made in order for it to work properly. Since the main
goal of this project was to provide var support to the
system, it was decided to find a method that would
allow the bank to be connected to the system without
causing a resonance condition. The C-type filter, which
has a flat frequency response over the frequency range
of interest, is capable of performing this function and
was selected as the design of choice. Frequency scans
of the considered options are shown below.
5
10
Introduction:
Southern Company Services Transmission Planning
Department determined that 200 Mvars of capacitors
needed to be installed at a 230 kV substation on the
Southern System to provide voltage support. The 200
Mvar was split into two equal 100 Mvar banks. Both
banks were originally designed to be split-wye
grounded banks.
Harmonics were monitored at the station bus for one
week. With the collected data, it was determined that
there was an appreciable magnitude of current at the 5th,
7th and 11th harmonics.
Preliminary calculations
indicated that the application of 100 Mvars and/or 200
Mvars of capacitors at this particular location would
create a harmonic resonance near the 7th harmonic.
Both present and future short-circuit strengths were
considered. Calculated voltage THD levels at this
particular location without filtered capacitor banks
ranged from 7.1 – 23.2%, depending on system strength
and how many area capacitor banks were on line.
Bus Voltage (per Amp Injected Harm. Current)
4
10
Single-tuned Filter
3
Capacitor Only
10
2
C-Type Filter
10
1
10
0
10
2
3
4
5
6 7 8 9 10 11 12 13 14 15
Harmonic (60 Hz base)
Figure 1. Frequency Scans of Various Filters
The plot shows the amount of voltage appearing at the
230 kV bus for each Ampere of harmonic current
injected. Peaks indicate undesirable parallel-resonance
conditions, and the 7th harmonic resonance for the
capacitor only design is clearly seen. The single-tuned
filter permits a shift of the resonance to a non-offensive
Page 1
resonant combination results in a net impedance of zero
Ohms. Therefore, under normal 60 Hz operation, the
damping resistor is effectively shunted thus preventing
costly losses. Under these conditions, the only circuit
element effectively in service is the main capacitor
section. It is this element which provides the needed
var support. When the bank is subjected to harmonics,
the tuning section no longer presents a zero impedance
branch within the network, and harmonic currents are
“spilled” into the resistor where they are dissipated.
frequency, but this resonance point may vary somewhat
depending on system conditions.
The C-Type filter relies upon resistive damping to yield
the favorable scan characteristics of essentially no
problematic resonance across the frequencies of
interest. The figure below shows the main components
of this type of filter.
Main Capacitor
Tuning Capacitor
Filter Bank Design and Protection
Damping
Resistor
Tuning Reactor
As is the norm with high-voltage capacitor banks,
system voltage and var requirements typically exceed
the ratings of individual capacitor units. The design
therefore involves connection of individual units in a
matrix array in which the connectivity compliments
both capacitor unit and system requirements. Figure 3
shows the single-phase connectivity for the 100 Mvar
filter bank. Complete bank protection is also shown
comprising the numerous voltage and current
transducers placed within the bank as well as the
protective elements indicated by their device numbers.
Figure 2. Simplified 1-Line Diagram of the C-Type
Filter
The operation of the C-Type harmonic filter is quite
different than a typical tuned filter bank. At 60 Hz, the
reactance of the tuning capacitor will cancel the
inductive reactance of the tuning reactor. This series-
Each Unit: 577 kvar, 17465 V
8 Series Sections
123 kV CT
Main
Capacitor
51 (CM)
Each Unit: 401 kvar, 6072 V
3 Series Sections
360 Ω (x 2)
36 kV CT
Tuning
Section
51 (CT)
37 (L)
49 (L)
51 (L)
59 (L)
36 kV CT
49 (R1)
51 (R1)
49 (R2)
51 (R2)
21 (L)
60.98 mH
34.5 kV PT
Sta. Class
Arrester
15 kV CT
Figure 3. 230 kV 100 Mvar C-Type Harmonic Filter Bank: Design and Protection Detail
Page 2
87 (R)
Main Capacitor Section (51-CM)
Ze =
Protection for the main capacitor section is provided by
a definite-time over current relay (51-CM) that monitors
the unbalance in the H-connected main capacitor
section.
The intent is to utilize an indirect
measurement, in this case the fundamental frequency
bridge current, to yield an indication of bank imbalance,
i.e., failure of individual capacitor sections. Ideally, no
bridge current flows when the healthy capacitors
produce a perfect balance. Bridge current flows when a
section fails and the balance is lost. The main concern
in the unbalanced case is the voltage impressed upon
the remaining capacitor sections. The altered capacitive
voltage divider of the bank yields elevated voltages
across some remaining sections. Calculations can relate
this internal voltage stress to the amount of bridge
current.
Two levels of definite time over current protection were
utilized. One element was designated as the alarm set
point, while the other was designated as the trip set
point. Both over current elements were biased to accept
a steady state vectorial correction current to compensate
for inherent unbalance within the capacitor section.
This nulling of the unbalance allows for a more
sensitive setting.
The capacitor units that made up the main capacitor
section were fuseless. The internal construction of the
capacitor unit (the can) with its 8 series sections is
shown below.
1 of 8 Series
Sections
Total Unit (can) Rating:
577 kvar
17,465 V
Figure 4. Internal Series Section Construction of the
Main Capacitor Unit
The anticipated failure mode of a capacitor section is a
short-circuit. The unbalance current flowing in the
current transformer shown in Figure 3 for “N” failed
capacitor sections can be calculated using (1– 10).
Zc =
(Vc )2
Qc
(1)
Page 3
Zc
Nc
(2)
Z f = Z e (N c ⋅ N u − N s )
(3)
Z s = Nu Zc
(4)
Z top =
Zs ⋅ Z f
(5)
Z f (N str − 1) + Z s
Z bottom =
Zs
N str

Z top
Vtop = Vmax 
 Z top + Z bottom

(6)





Z bottom
Vbottom = Vmax 
 Z top + Z bottom

(7)




Vtop  N str
V
⋅
I unbalance =  bottom −
Z s  2
 Zs
Vtop ⋅ N c
%OV =
⋅ 100
Vc [( N c ⋅ N u ) − N s ]
(8)
(9)
(10)
where,
Zc = Impedance of capacitor can, Ω
Ze = Impedance of capacitor element, Ω
Zf = Impedance of string with failed capacitor
element, Ω
Vc = Rated voltage of each capacitor unit, kV
Qc = Reactive power rating of capacitor unit, Mvars
Nc = Number of series sections per capacitor unit
Ns = Number of shorted series sections in one string
Nu = Number of series-connected capacitor units per
string
Zs = Impedance of healthy string, Ω
Ztop = Impedance of top half of main capacitor
section, Ω
Zbottom = Impedance of bottom half of main capacitor
section, Ω
Vtop = Voltage across top half of main capacitor
section, kV
Vbottom = Voltage across bottom half of main capacitor
section, kV
Vmax = Maximum line-to-neutral system voltage, kV
Iunbalance = Unbalance current flowing through current
transformer, A
%OV = Percent overvoltage on remaining series
sections, %
Typically, a table is developed to show the unbalance
current and %OV for multiple series section failures.
Table 1 includes the calculation results for the main
section of a 242 kV – 100 Mvar C-Type harmonic filter
bank. A detailed example calculation for the main
section unbalance current is included in the Appendix.
# Failed
Series
Sections
Zf
Zs
Ztop
Zbottom
Vtop
Vbottom
Unbalance
Current
Elements
0
1
2
3
4
5
6
7
(Ω)
2114.6
2048.5
1982.4
1916.3
1850.2
1784.2
1718.1
1652.0
(Ω)
2114.6
2114.6
2114.6
2114.6
2114.6
2114.6
2114.6
2114.6
(Ω)
264.4
263.3
262.1
260.9
259.7
258.3
256.9
255.4
(Ω)
264.4
264.3
264.3
264.3
264.3
264.3
264.3
264.3
(V)
69859.4
69718.8
69571.6
69412.6
69243.2
69062.1
68868.1
68659.9
(V)
69859.4
69999.9
70151.3
70310.2
70479.7
70660.8
70854.7
71063.0
(A)
0.00
0.53
1.10
1.70
2.34
3.02
3.76
4.55
%OV
(% of
rated)
100.0%
103.0%
106.2%
109.6%
113.3%
117.2%
121.3%
125.8%
Table 1. Main Capacitor Section Unbalance Calculation Results
The alarm setting was chosen to detect a single series
section failure. A setting of 80% of the calculated value
will provide adequate margin. Thus a setting of 0.4 A
primary was used.
The trip setting was chosen to detect multiple series
section failures. The bank should be taken off line
before the remaining capacitor sections are subjected to
a voltage greater than 110% of their rating. Typically, a
string is rated at nominal system voltage; however for a
filter bank the string is rated at a voltage higher than
nominal. To account for possible overvoltage due to
harmonics, a setting that correlated to two series section
failures was chosen. From Table 1 it can be seen that
the bank would still be operating below 110% of its
rating for two series section failures. However, it was
decided to go with the more conservative approach due
to the unknown voltage developed by harmonics. Thus
a setting of 0.8 A primary was selected.
Zc =
Ze =
(Vc )2
Zc
1000 ⋅ N c
Page 4
(12)
Z f = Z e (N c ⋅ N u − N s )
(13)
Z s = Nu Zc
(14)
Z top _ tune =
I main =
I tune =
Zs ⋅ Z f
(15)
Z f ⋅ (N str − 1) + Z s
Z bottom _ tune =
Tuning Capacitor Section (51-CT)
A capacitor unit failure is detected in the tuning section
of the filter bank in the same manner as that of the main
capacitor section. A separate 51-CT (capacitor, tuning)
element is used, one per phase. However, since the
tuning section consists of capacitors, reactors and
resistors, the equations involved in calculating the
unbalance current due to a failed series section is
somewhat different than for the main capacitor section.
Equations (11-22) may be used to calculate this
unbalance current.
(11)
Qc
Zs
N str
(16)
Vmax
− jZ main
(17)
R
 
2
R
  − jZ top _ tune − jZ bottom _ tune + jωL
2
(
Vtop _ tune = I tune ⋅ − jZ top _ tune
(
(18)
)
Vbottom _ tune = I tune ⋅ − jZ bottom _ tune
⋅ I main
(19)
)
(20)
 Vbottom _ tune Vtop _ tune
−
I unbalance = 
Zs
Zs

%OV =
Vtop _ tune ⋅ N c
Vc [(N c ⋅ N u ) − N s ]
 N str
⋅
 2

(21)
⋅ 100
(22)
where,
Zs = Impedance of healthy string, Ω
Ztop = Impedance of top half of main capacitor
section, Ω
Zbottom = Impedance of bottom half of main capacitor
section, Ω
Vtop = Voltage across top half of main capacitor section,
kV
Zc = Impedance of capacitor can, Ω
Ze = Impedance of capacitor element, Ω
Zf = Impedance of string with failed capacitor element,
Ω
Vbottom = Voltage across bottom half of main capacitor
section, kV
Vmax = Maximum line-to-neutral system voltage, kV
Ztop_tune = Impedance of the top half of the tuning
section, Ω
Iunbalance = Unbalance current flowing through current
transformer, A
Zbottom_tune = Impedance of the bottom half of the tuning
section, Ω
%OV = Percent overvoltage on remaining series
sections, %
Vc = Rated voltage of each capacitor unit, kV
Qc = Reactive power rating of capacitor unit, Mvars
Nc = Number of series sections per capacitor unit
Ns = Number of shorted series sections in one string
Nu = Number of series-connected capacitor units per
string
As with the main capacitor section, a table is developed
to analyze the behavior of the string for multiple series
section failures.
Table 2 shows the %OV and
unbalance current produced by multiple series section
failures. A detailed calculation for the tuning section is
included in the Appendix.
Nstr = Number of parallel strings per phase
# Failed
Series
Sections
Zf
Zs
Ztop
(tune)
Zbottom
(tune)
Vtop
(tune)
Vbottom
(tune)
Unbalance
Current
Elements
0
1
2
3
(Ω)
91.94
61.30
30.65
0.00
(Ω)
91.94
91.94
91.94
91.94
(Ω)
11.49
10.82
9.19
0.00
(Ω)
11.49
11.49
11.49
11.49
(V)
3037.6
2858.9
2440.5
0.0
(V)
3037.6
3037.6
3050.7
3031.4
(A)
0.00
7.77
26.5
131.88
%OV
(% of
rated)
50.0
70.6
120.6
0.0
Table 2. Tuning Capacitor Section Unbalance Calculation Results
The alarm setting is chosen to detect a single series
section failure. A setting of 80% of the calculated value
will provide adequate margin. Thus a setting of 6.2 A
primary will be used.
impact on the tuning of the bank. The tuning section
capacitors are un-fused and have a voltage rating of
twice nominal voltage. This is evident in the %OV data
for zero failed sections. This rating minimizes the
probability of a failure and allows operation with one
shorted series section.
The trip setting is chosen to detect multiple series
section failures. Typically, a bank is tripped off line
when the voltage across the remaining capacitor cans
exceeds 110% of their nominal rating. However, for the
case of the tuning section, the bank should be tripped
after the second series section failure regardless of the
percent overvoltage. This is done because a failure of a
series section within the tuning section can have an
Page 5
The tuning section bridge current used to indicate
capacitor unit failures is ideally zero under healthy bank
(balanced) conditions. As discussed above with the
Main Capacitor, in practice there will be some small
current flow in the bridge circuit due to tolerances in the
capacitors that yield bank structures not perfectly
balanced. Accommodating this error would require de-
sensitizing the protective settings.
To increase
sensitivity, nulling logic was incorporated into the
relay: during bank commissioning, this error signal is
measured, committed to relay memory and used to
compensate the real-time protective algorithms.
Because this compensation process involves both the
magnitude and phase angle of the inherent error signal,
a reference phasor is required for coherent phase
accounting. The relay uses the reactor current, of the
associated phase, as that reference.
+XL
Impedance
Plane
Alarm Radius
Nominal Impedance
Trip Radius
Alarm Impedance
Trip Impedance
R
Figure 5. Tuning Reactor Impedance Protection (21-L)
Tuning Reactor (L)
The tuning reactor protection consists of an
undercurrent relay (37-L); a fundamental frequency
definite time over current relay (51-L); thermal image
protection (49-L), employing harmonics and thermal
time constants; fundamental frequency impedance
protection (21-L); and a summed harmonic over voltage
element (59-L).
When the reactor begins to fail (short turns, etc.), the
total impedance of the filter bank begins to increase.
Assuming a nearly constant bus voltage, this increase in
bank impedance will cause a decrease in the amount of
current flowing through the reactor. As a result, an
undercurrent relay (37-L) monitoring reactor current is
required to detect this particular failure mode of the
tuning reactor.
Additional protection based upon the reactor impedance
is used to enhance the detection of reactor failures. The
impedance-based
reactor
relay
monitors
the
fundamental frequency voltage across the reactor and
fundamental frequency reactor currents to determine the
actual reactor impedance on a per-phase basis.
Although the bus work and CTs contribute some
resistance, this value is neglected by the real-time
measurement via a compensation set when the unit is
commissioned. Figure 5 shows the manner in which the
impedance-based method works. The offset mho
elements are inward looking. That is, the elements only
operate when the reactor impedance falls outside of a
given mho circle. Essentially, the sensitivity of the
protection is such that a single shorted series section can
be detected and an alarm issued with identification of
phase-involvement. If the number of shorted series
sections is large enough to cause an impedance range
shift beyond the alarm region, then a trip command is
issued.
Figure 5 illustrates the concept of offset impedance
circles of normal range and of the alarm zone and the
trip zone. Note that during the commissioning process,
any errors due to the CTs and VTs are included in the
initial measured impedance and are compensated. This
error-nulling process allows the setting of a smaller,
more sensitive, radius impedance circle around the
initial measured impedance.
To protect the tuning reactor from a possible overload
condition caused by excessive exposure to harmonic
current, thermal image relays are provided. A 100
Mvar filter bank at the 230 kV level is an effective sink
for system harmonics. Harmonics can cause thermal
damage to the reactors. By nature, harmonic overloads
are not constant over time: rising and falling over a
fixed detection window.
Therefore, a thermal
protection algorithm is employed that estimated the
temperature of both the reactor’s insulator cap and
winding hot-spot.
The ambient temperature, an
important input to this algorithm, is measured via a
conventional thermo-couple and a 4-20 mA transducer
circuit.
The relay’s 49-L (inductor) thermal element is based
upon the following equation:
TL= 0.00069 (IL
2+
-t
47
2
2
(ILo - IL )e )
+ Tambient (°C)
which estimates the inductor winding temperature
where,
IL =
∑αi Ii2 .
ILo = IL calculated at the previous iteration.
I = rms inductor current of the ith harmonic.
t = calculation iteration time (minutes).
i = harmonic index (1, 3, 5, 7).
α1,3,5,7 = 1.0, 1.12, 1.35, 1.69.
The reactor manufacturer provided the above analytic
expression along with the conversion factors, time
constant and harmonic gains. Response from the relay
Page 6
is recommended when a calculated inductor winding
temperature of 150 °C is reached. This calculated
temperature is considered an absolute value, not a
temperature rise.
In addition to the above, the manufacturer recommends
estimating the temperature of the critical insulating cap
of the inductor assembly. The relay included this via its
49-C (cap) element by using the following expression.
TC= 0.00037
(IC2 +
-t
24
2
2
(ICo - IC )e )
+ Tambient (°C)
which estimates the inductor insulation cap temperature
where,
IC =
2∑i2 Ii2 .
ICo = IC calculated at the previous iteration.
I = rms inductor current of the ith harmonic.
t = calculation iteration time (minutes).
i = harmonic index (1, 3, 5, 7).
The inductor manufacturer recommends relay response
when calculated insulation cap temperature reaches
120 °C
Harmonic reactor currents cause instantaneous
harmonic reactor voltages which can stress the turn-toturn insulation of the reactor. Note that higher
harmonics of same magnitude produce a higher voltage
due to the higher angular frequency (ω). The harmonic
voltage is summed arithmetically and compared to a set
value which provides sufficient turn-to-turn overvoltage
protection. The analytic expression used in this over
voltage element is as follows:
VL= 0.377 L ∑i Ii
which estimates the instantaneous inductor harmonic
voltage stress where,
L = inductance (mH)
I = rms inductor current.
i = harmonic index (1, 3, 5, 7).
The inductor manufacturer recommends relay response
when calculated voltages reach 20 kV.
Although a single device would suffice, two parallelconnected resistors for damping out harmonics are used
in order to provide redundancy and create a quasidifferential zone of protection. Note that in a situation
of perfect resonance between the tuning L and the
tuning C, there will be only harmonic current in the two
resistors. The currents that flow through each resistor
are measured and the rms current computed. The
protection provided is a simple rms (harmonics
included) definite time over current element on a perresistor and per-phase basis: the 51-R. However, the
possibility of short-term (below the time-delay setting
of the 51 elements) repeated rms current overloads
necessitates the use of thermal tracking elements
(49-R), employing thermal time constants of the
resistors and permissible temperature rise allowed. The
ambient thermal temperature is also used in the resistor
thermal model. The analytic expression for the thermal
model of the resistors follows a form similar to that
discussed above for the filter bank inductor.
One component of the protection problem involves
detecting either an opened or shorted resistor when
typically only very small harmonic current flows
through the component. Normally, differential (87)
protection is applied for unit protection, where the
current in is equal to the current out. In this particular
case it is applied to protect two separate resistors, since
currents in the two resistors are expected to be identical
as identically rated resistors and CTs are used. The
objective is to detect a difference of resistor currents
which may indicate failure of either unit. The relay
calculates a restraint current, which is an average of the
two resistor currents and the differential current, which
is the difference between the two resistor currents.
RMS sensing is utilized to incorporate the harmonicrich signals. Figure 6 depicts the slope characteristic of
this differential element.
Differential
Current
IR1− IR2
Slope Setting
Operate
Region
Non-Operate
Region
Damping Resistors
The two damping resistors are protected using a rms
definite time overcurrent relay (51), thermal image
protection (49) employing rms currents and resistor
thermal time constants, and a unique rms percentage
restraint biased differential protection (87).
Page 7
Restraint
Current
IR1+ IR2
2
Figure 6. Differential Slope Characteristic
During the commissioning process the inherent
differential current is determined and input into the
relay. This inherent differential current is subtracted
from the calculated differential current which nulls out
all the measurement errors and permits a tighter setting
for the differential current element operation. Classical
differential protection is applied, where the differential
current pickup is a percentage of the average current.
This unique method provides for greater security in the
event of a large temporary over current and consequent
CT saturation. Also both resistors are protected by this
one differential (87) element employing only two CTs.
Traditional differential protection would have required
two CTs to protect one resistor.
Conclusions
The C-Type filter bank provides an economical means
of applying shunt capacitor banks in harmonic rich
environments with confidence that resonance conditions
will be avoided. The protection of such a filter bank
can be complex due to the fact that the filter is
comprised of many components. The schemes shown
in this paper provide a reliable and cost effective means
of thoroughly protecting a C-Type harmonic filter bank.
The availability of a flexible protection hardware
platform, in terms of graphical programmability and
expandable voltage/current inputs, permitted the
incorporation of all protective elements within a single
relay device.
Bibliography
1.
IEEE
Recommended
Practices
and
Requirements for Harmonic Control in Electrical Power
Systems, IEEE Std. 519-1992
2.
IEEE Guide For The Protection Of Shunt
Capacitor Banks, IEEE Std. C37.99-2000
3.
Horton, et al “Unbalance Protection of
Fuseless, Split-Wye, Grounded, Shunt Capacitor
Banks” IEEE Transaction on Power Delivery July 2002
pp. 698-701
L. Fendrick, et. Al., “Complete Relay
4.
Protection for Multi-String Fuseless Capacitor Banks”
Georgia Tech Protective Relaying Conference, May
2002.
5.
ANSI/IEEE C37.015-1993, IEEE Application
Guide for Shunt Reactor Switching
Randy Horton received the B.S.E.E. degree from the
University of Alabama at Birmingham and the M.E.E. degree
from Auburn University all with specialization in electric
power systems. He is currently employed at Alabama Power
Company as a Senior Engineer in the Protective Equipment
Page 8
Application group. Randy is a member of several IEEE
PSRC working groups and is a registered professional
engineer in the state of Alabama.
Ted Warren is currently employed as a Senior Engineer in the
Protective Equipment Application group at Alabama Power
Company in Birmingham, Alabama. He has also worked as a
protection engineer with Alabama Electric Cooperative and
has worked in the industrial automation field. He received a
B.E.E. degree from Auburn University in 1993. Ted is a
registered professional engineer in the State of Alabama.
Timothy R. Day is a Senior Relay Application Engineer in
the Protective Relay Group of Cooper Power Systems,
Franksville, Wisconsin. His present professional endeavors
include modeling and analysis of electrical power systems to
assess and optimize protection schemes. Timothy enhances
existing protective algorithms and develops customized
schemes for the Edison Idea line of relays and incorporates
Cooper’s power system simulator to verify scheme
modifications. He received a M.S.E.E. from Washington
State University in 1991.
Jack McCall is currently the director for Cooper Power
Systems’ Protective Relay Group, located in South
Milwaukee, WI.
He has an MS in Electric Power
Engineering from Rensselaer Polytechnic Institute and a
BSEE from Gannon University. Previous positions within
Cooper include Marketing Manager for Cooper’s Power
Capacitor Group and Power Systems Engineer for Cooper’s
Systems Engineering Group. He is a Member of the IEEE
Power Engineering Society and has authored numerous
papers on transients, harmonics, and capacitor applications in
power systems.
Arvind Chaudhary received the B.S.E.E. degree from the
Indian Institute of Science, Bangalore, India; the M.S.E.E.
degree from North Carolina State University, Raleigh; and
the Ph.D. degree with a concentration in electric power
engineering from Virginia Polytechnic Institute and State
University, Blacksburg. He is a Staff Engineer with the
Protective Relays, Cooper Power Systems, South Milwaukee,
WI. He is responsible for relay applications for the Cooper
line of relays and relay settings for power system equipment.
He is the recipient of the 2000 IEEE PES Chicago Chapter
Outstanding Engineer Award. Also, he is a member of the
Technical Committee of the International Power Systems
Transients Conferences of 1999, 2001, and 2003. His
previous experience has included Sargent & Lundy
consulting engineers (1991–1998) and Bharat Heavy
Electricals Limited, India (1979–1983).
APPENDIX
Example Main Section Unbalance Current Calculation
Given:
Vmax =
242
kV
3
Vc = 17.465 kV
Qc = 0.577 Mvar
N u = 4 Number of capacitors per string
N c = 8 Number of series sections per capacitor can
N s = 1 Number of failed series sections
N str = 8 Number of strings per phase
Zc =
Ze =
(Vc )2
Qc
= 528.64 Ω
Zc
= 66.08 Ω
Nc
Z f = Z e ( N c ⋅ N u − N s ) = 2048.49 Ω
Z s = N u Z c = 2114 .57 Ω
Z top =
Zs ⋅ Z f
Z f (N str − 1) + Z s
Z bottom =
= 263.26 Ω
Zs
= 264.32 Ω
N str

Z top
Vtop = Vmax 
 Z top + Z bottom


 = 69718.82 V



Z bottom
Vbottom = Vmax 
 Z top + Z bottom


 = 69999.95 V


Vtop  N str
V
⋅
I unbalance =  bottom −
 2 = 0.532 A (primary)
Z
Z
s
s


%OV =
Vtop ⋅ N c
Vc [(N c ⋅ N u ) − N s ]
⋅ 100 = 103.02 %
Page 9
Example Tuning Section Unbalance Current Calculation
Given:
Vmax =
242
kV
3
Vc = 6.072 kV Tuning capacitor voltage rating
Qc = 0.401 Mvar Tuning capacitor Mvar rating
L = 60.98 mH Tuning reactor inductance
ω = 2π f = 377 rad/s
R = 360 Ω Damping resistor
N u = 1 Number of capacitors per string
N c = 3 Number of series sections per capacitor can
N s = 1 Number of failed series sections
N str = 8 Number of strings per phase
Z main = 528.64 Impedance of entire main capacitor section (from above calculations)
Zc =
Z s = N u Z c = 91.94 Ω
Zs ⋅ Z f
Z top _ tune =
= 10.82 Ω
Z f ⋅ (N str − 1) + Z s
−
V max
;  I main = 264.29 A primary.
− jZ main
I main =
(Vc )2
= 91.94 Ω
Z bottom _ tune =
Qc
Z
Z e = c = 30.64 Ω
Nc
Z f = Z e (N c ⋅ N u − N s ) = 61.29 Ω
I tune =
R
 
2
R
  − jZ top _ tune − jZ bottom _ tune + jωL
2
(
V top _ tune = I tune ⋅ − jZ top _ tune
(
)
V bottom _ tune = I tune ⋅ − jZ bottom _ tune
Vtop _ tune ⋅ N c
Vc [( N c ⋅ N u ) − N s ]
I tune = 264.29 A (primary)
V top _ tune = 2858.85 V
)
 V bottom _ tune V top _ tune
−
I unbalance = 
Z
Zs
s

%OV =
⋅ I main
Zs
= 11.49 Ω
N str
V bottom _ tune = 3037.53 V
 N str
⋅
 2

I unbalance = 7.77 A (primary)
⋅ 100 = 70.62 %
Page 10
Page 11
Relaying 230 kV, 100 Mvar C-Type Filter Capacitor Banks
©2003 Cooper Power Systems, Inc.
Bulletin 03019 • June 2003 • New Issue
P.O. Box 1640
Waukesha, WI 53187
www.cooperpower.com
KWP
6/03