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MITIGATION OF FERRORESONANCE INDUCED BY
SINGLE-PHASE OPENING OF A THREE-PHASE TRANSFORMER FEEDER
Patrick Picher
Léonard Bolduc
Bruno Girard
Van Nhi Nguyen
Institut de recherche
Institut de recherche
TransÉnergie
TransÉnergie
Hydro-Québec
Hydro-Québec
Hydro-Québec
Hydro-Québec
[email protected] [email protected] [email protected] [email protected]
Abstract
This paper presents a practical mitigation technique for the
ferroresonance induced by the single-phase primary opening
of a three-phase 5-legged (wound core) station service
transformer. This ferroresonance problem was causing erratic
behaviour of the automatic transfer switch between the main
and reserve station service transformers.
A nonlinear three-phase transformer model, based on the
magnetic-circuit theory, was used to reproduce and acquire a
deeper understanding of the ferroresonance phenomenon. The
model, validated by field measurements, allowed comparison
of the effectiveness of resistive and inductive loads for
reducing the voltage on the open phase.
In previous literature, it has been generally recommended
to use a resistive load to damp ferroresonance. This paper
demonstrates that an inductive load is more effective and has
the advantage of being a more energy-efficient solution. The
inductive-load mitigation technique is now in service in HydroQuébec’s Deschambault substation.
Keywords: Ferroresonance, mitigation, transformer.
causing the voltage to drop again. These erratic permutations
will persists until the automatic transfer switch fails.
In this case, replacing the overhead line fuses by a threephase switching device or using single-phase transformers
would be valid mitigation techniques but economics
discourage both these options. A better solution is to connect a
permanent load on the transformers to ensure that the voltage
remains below the threshold level (in this case 75% of the rated
rms voltage) of the automatic transfer switch. In previous
literature [1-2], it has been generally recommended to use a
resistive load to damp the ferroresonance but this paper
demonstrates that an inductive load is more effective and has
the advantage of being more energy-efficient.
The paper describes the use of a nonlinear three-phase
transformer model, based on the magnetic-circuit theory, to
reproduce the magnetic coupling between the phases and in so
doing acquire a deeper understanding of the ferroresonance
phenomenon.
2. Description of the model
1. Introduction
The main and reserve station service transformers of
Deschambault substation are rated 1.5 MVA, three-phase 5legged wound-core, grounded-wye to grounded-wye,
24940/14400-600/347 V. Both transformers are fed by two
separate feeders consisting of 68 m of underground cables in
series with overhead lines. The total capacitance of these
cables is approximately 20 nF (300 pF/m).
Operating problems occur when a fuse opens in a phase of
the overhead line feeding the main transformer. This induces a
voltage in the open phase because of the closed iron path
provided by the three-phase transformer core. The induced
voltage, back-fed on the cable capacitance, creates a potential
risk of ferroresonance with the nonlinear iron core inductance.
If the load on the main transformer is sufficiently high, the
single-phase opening is detected by the voltage detection
device of the automatic transfer switch and the load is
transferred to the reserve transformer connected to the second
feeder. Elimination of the load on the first transformer initiates
a ferroresonance situation which causes the voltage on the
open phase to increase, reaching or exceeding, the rated
voltage. The automatic transfer switch detects this recovered
voltage and transfers the load back on the main transformer,
In order to reproduce the ferroresonance phenomenon, the
transformer model must represent the transformer core
correctly. The model preferred for such analysis is based on
the magnetic-circuit theory and an electrical equivalent circuit
is obtained by the duality principle [3]. This model was
successfully used for previous investigations into direct-current
saturation [4] and load loss unbalance [5].
The magnetic circuit representing the 5-legged core type is
shown in Figure 1. Sources of flux represent primary and
secondary windings. Each section of the core is represented by
a nonlinear reluctance. Linear reluctances are added to
represent the leakage flux between windings (leakage
reactance). A slightly different version of this model was
developed by Mork [6] to better represent the wound core. In
the study presented here, the stacked-core model was used and
validated by field measurements.
The nonlinear characteristic can be obtained by
measurements or, as in our case, by using a typical transformer
steel B-H curve.
The electrical circuit of the transformer is obtained using the
principle of duality. Figure 2 shows the complete simulation
circuit with the transformer model, the cables (represented by
shunt capacitors) and the load.
S im u la tio n
c a s
D e s c h a m b a u lt - P h a s e
B
o u v e rte
- X L
b ra n c h é e
(9
A )
2 .5 0 E + 0 4
2
2
2 .0 0 E + 0 4
1 .5 0 E + 0 4
f
3
1
A1
f
B1
1
1
(V )
1 .0 0 E + 0 4
T e n s io n
f
C1
5 .0 0 E + 0 3
0 .0 0 E + 0 0
3
-5 .0 0 E + 0 3
-1 .0 0 E + 0 4
-1 .5 0 E + 0 4
A2
B2
S im u la tio n c a s D e s c h a m b a u lt - P h a s e B o u v e r te - X L b r a n c h é e (9 A )
C2
-2 .0 0 E + 0 4
2 .5 0 E + 0 4
-2 .5 0 E + 0 4
2 .0 0 E + 0 4
Lf
4 .0 0 E -0 1
4 .0 5 E -0 1
4 .1 0 E -0 1
4 .1 5 E -0 1
4 .2 0 E -0 1
4 .2 5 E -0 1
T e m p s
4 .3 0 E -0 1
4 .3 5 E -0 1
4 .4 0 E -0 1
4 .4 5 E -0 1
(s )
1 .5 0 E + 0 4
V a
V b
V c
1 .0 0 E + 0 4
T e n s io n (V )
Figure 1: Magnetic circuit representing 5-legged core.
L3
L1
5 .0 0 E + 0 3
0 .0 0 E + 0 0
-5 .0 0 E + 0 3
-1 .0 0 E + 0 4
L2
-1 .5 0 E + 0 4
L1
-2 .0 0 E + 0 4
-2 .5 0 E + 0 4
Lf
Figure 3: Comparison of simulations and measurements. Top:
open-phase A (no load). Bottom: open-phase B (2.6 kvar load).
4 .0 0 E -0 1
4 .0 5 E -0 1
4 .1 0 E -0 1
4 .1 5 E -0 1
4 .2 0 E -0 1
4 .2 5 E -0 1
L2
L1
Lf
L3
Figure 2: Simulation circuit (transformer, cables and
mitigation load).
The nonlinear inductances of the transformer are adjusted to
match the corresponding core dimensions (wound legs f,
yokes 2 and return legs 3). The leakage reactance is
modeled by reluctance 1. The cross section of the yokes is
the same as the return legs and half of the wound legs. The
yoke and the return leg lengths were assumed to be 0.375 and
1.75 times the length of the wound legs respectively. Lastly,
the magnetizing current was set at 0.5% and the no-load loss at
0.1%.
3. Validation of the model
The model was validated using field measurements.
Comparisons were done with measurements taken with openphase A, B and C, with and without load. The simulated
waveforms were similar to the experimental measurements as
shown in Figure 3. Table 1 compares simulated and measured
peak and rms voltage values on open-phase A and B. The
correlation of the results is better for phase B, the measured
values being somewhat higher than the simulated ones for
phase A. These small differences can be explained by the
various assumptions made for the transformer model (core
dimensions, saturation characteristics, core losses, rated
magnetizing current, stacked-core model). However, the main
objective of the study, to compare active and reactive loads
mitigation techniques, did not justify adding more complexity
to the transformer model.
4 .3 0 E -0 1
4 .3 5 E -0 1
4 .4 0 E -0 1
4 .4 5 E -0 1
4 .5 0 E -0 1
T e m p s (s )
V a
V b
V c
Table 1: Comparison of simulations with measurements:
peak and rms voltages in p.u. (measurements in parentheses)
kvar
Open Phase: A
Open Phase: B
Peak
RMS
Peak
RMS
0
1.04 (1.02)
0.91 (0.86)
1.07 (1.02)
0.83 (0.79)
2.6
0.53 (0.67)
0.52 (0.58)
0.57 (0.58)
0.54 (0.50)
4.6
0.39 (0.52)
0.39 (0.47)
0.42 (0.47)
0.39 (0.36)
9
0.23 (0.44)
0.27 (0.37)
0.28 (0.34)
0.26 (0.27)
4. Simulation results
All simulations were performed using MATLAB. The
simulation length was kept to 0.5 s with variable time steps.
The peak voltage and flux in the wound legs were calculated
for the period between 0.25 to 0.5 s. A series of simulations
was performed to investigate the influence of cable capacitance
on the peak voltage obtained on the open phase. Figure 4
shows the results obtained. The simulations demonstrate that
the peak flux does not follow the peak voltage. At 40 nF, for
instance, the peak voltage reaches 1.8 p.u., and the peak flux is
limited to 1.2 p.u., which is a very limited over-excitation.
This observation is in accordance with the conclusions
presented in [1]. From this graph, it is impossible to identify
the mode of ferroresonance (‘period 1’, ‘period 2’, ‘chaotic’,
etc.) [7]. For instance, period 2 refers to a waveform with a
period twice that of the forcing function (applied voltage).
Another representation of waveforms is the phase plane plot
(flux versus voltage). The trajectory on the phase plane
provided a unique signature of the waveform. Periodic
waveforms have trajectories that close back on themselves and
repeat. Period 2 can easily be inferred by carefully following
the waveform’s trajectory on the phase plane. Figure 5 depicts
the waveforms obtained for cable capacitances of 20 and
25 nF.
These graphs show clearly that, if the cable
4 .5 0 E -0 1
2
Max voltage, flux (pu)
1,8
1,6
1,4
1,2
1
0,8
1
Vmax on open phase (pu)
capacitance is raised from 20 nF to 25 nF, the ferroresonance
mode shifts from period 1 to period 2. In the case studied, all
field measurements showed ferroresonance in period 1.
R-load (peak)
0.75
R-load (RMS)
0.5
0.25
L-load (peak)
L-load (RMS)
0,6
0
0,4
0
0,2
5
10
Load (kVA)
0
0
10
20
Cable capacitance (pF)
RMS voltage
Peak flux
30
40
Peak voltage
Figure 4: Effect of cable capacitance on maximum voltage
and flux obtained on open phase.
1.5
The effect of reactive load was simulated for different
capacitance values. The results are shown in Figure 7. This
type of representation could be used to select an appropriate
reactive load level for various lengths of feeder cable.
5. Discussion
0.5
0
-0.5
-1
-1.5
0.3
0.32
0.34
0.36
0.38
0.4
Time (s)
1.5
25 nF
20 nF
1
The following intuitive explanation for the case studied is
offered. When one phase of the three-phase transformer is
opened, the total amount of flux generated by the connected
phases returns via the open wound leg and both return legs.
Normally, when there is no ferroresonance, the voltage on the
open phase should be around half the rated voltage. When a
feeder cable is connected to the transformer, a shunt
capacitance is then connected in parallel with the transformer’s
magnetizing impedance, forming a parallel resonant circuit.
This shunt capacitive impedance is approximately 130 kΩ as
compared to 70 kΩ for the magnetizing impedance. Without
additional load, the parallel resonant impedance is the
inductive type and can be calculated using (1).
0.5
1.8
0
1.6
-0.5
-1
-1.5
-1.5
-1
-0.5
0
0.5
1
1.5
Voltage (pu)
20 nF
25 nF
Figure 5: Waveforms for cable capacitances of 25 and 20 nF.
Top: voltage. Bottom: phase plane diagram of the voltage.
A series of simulations was performed to compare the
attenuation of the voltage by active and reactive loads. Figure
6 illustrates the results for peak and rms voltage in p.u.
Voltage on open phase (pu)
Voltage (V)
20
Figure 6: Comparison of inductive and resistive loads.
1
Flux (pu)
15
1.4
40
35
30
1.2
1
40 nF
25
0.8 20
0.6 15
5
0.4
5 nF
0.2
0
0
5
10
15
Load (kvar)
Figure 7: Effect of reactive load for different capacitance
values (peak voltage shown).
20
X eq 
6. Conclusion
X C X mag
(1)
X C  X mag
The calculated equivalent reactive load is 150 kΩ. This
increase in the inductance is translated by a proportional
decrease in the leg’s reluctance in the magnetic circuit, thereby
increasing the flux and the voltage in the open phase. To
mitigate this resonance, adding inductance in parallel with the
magnetizing impedance will reduce the resonant equivalent
reactance and increase the leg’s reluctance, forcing the flux out
of the open phase and, hence, reducing the voltage. For
instance, with a load of 5 kvar (shunt reactance of 125 kΩ on
the HV side base), the inductive reactance is reduced to 45 kΩ,
and the parallel equivalent impedance with the cable
capacitance becomes 69 kΩ. If the voltage is assumed equal to
1 p.u. when there is no load, a parameter similar to the voltage
reduction can be obtained by dividing the newly obtained
impedance (69) by the reference impedance without load
(150), which gives 0.46 p.u.
This methodology was used to calculate the reduced voltage
for loads between 1 to 20 kvar: the resulting curve (Figure 8)
is similar to the simulation results, demonstrating the validity
of the proposed explanation.
The main contribution of this paper was to compare active
and reactive load effectiveness in reducing the voltage on the
open phase. In the practical case presented, the automatic
transfer switch operates when the voltage drops below 75% of
the rated rms voltage. The simulations (Figure 6) show that the
minimal inductive and resistive loads required to reduce the
rms voltage to 0.75 p.u. are respectively 1 kvar and 2 kW
(three-phase, wye-connected).
A 4.5-kvar load was
permanently connected to the secondary of the three-phase
transformers (1.5 kvar per phase, 347 V). The load was
connected on the secondary side to reduce the cost, the
effectiveness being the same. A permanent load must be
connected on the three phases to prevent ferroresonance in all
cases of single-phase opening.
1
Calculated voltage (p.u.)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
Load (kvar)
15
20
Figure 8: Calculated voltage on open phase – discussion
methodology.
This contribution has presented a novel use of a transformer
model to compare the active and reactive load mitigation
techniques to deal with the ferroresonance induced by singlephase opening of a 5-legged three-phase transformer. The
model showed how the modification of the cable capacitance
changes the ferroresonance mode and overvoltages. It was also
observed that even in the case of high overvoltages, the crest
flux remains at a much lower level.
A permanent 4.5-kvar reactive load was connected on the
secondary side of the main and reserve 1.5-MVA station
service transformers of Hydro-Québec’s Deschambault
substation. This load reduces the voltage on the open phase to
below the 75% threshold level used by the automatic transfer
switch when a fuse opens on the feeder side. The simulations
showed that the voltage decrease is even higher than what
would have been obtained by an equivalent resistive load, and
the active energy consumption is negligible. This mitigation
approach corrects the erratic commutations caused by
misinterpreting the ferroresonant voltage on the open phase as
re-connection of the transformer feeder.
References
[1] R.A. Walling, K.D. Barker, T.M. Compton, and L.E.
Zimmerman, “Ferroresonant overvoltages in grounded
wye-wye padmount transformers with low-loss siliconsteel cores,” IEEE Trans. Power Delivery, Vol. 8, No. 3,
pp. 1647-1660, July 1993.
[2] M.R. Iravani et al., “Modeling and analysis guidelines for
slow transients-Part III: The study of ferroresonance,”
IEEE Trans. Power Delivery, Vol. 15, No. 1, pp. 255-265,
January 2000.
[3] J.A. Martinez, R. Walling, B.A. Mork, J. Martin-Arnedo,
and D. Durbak, “Parameter determination for modeling
system transients-Part III: Transformers,” IEEE Trans.
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