peak inrush currents for multiple-step capacitor banks in automatic

Key words: Inrush current, Overvoltage, Switching phenomenon, Low voltage
power factor correction (LV-PFC).
The paper focuses on an accurate predetermination of the peak inrush current that
occurs at switching the multiple step capacitor banks in automatic low voltage power
factor correction systems (LV-PFC). The analytical and numerical approach rely on the
exact solutions of the time second order differential equations which model the entire
dynamic process. The study is also valid in low level harmonic polluted environments
and for both mechanic and static commutation. Some power quality aspects are also
treated along with the estimation of electric stress exerted on electrical equipment. A
case study allows the comparison of the obtained results with those reported in previous
works and International Electrotechnical Commission (IEC) standards.
The constant demand of rational use of electric energy requires its economic
generation, transmission, distribution and usage with minimum losses. That implies
a full control over the factors that may determine these potential losses. The
lagging reactive power flow, which can occur in both industrial and public power
grids, is one of the main causes of energy losses [1–4]. In order to overcome this
problem, power factor correction systems (PFC) are used, destined to compensate
the generated lagging reactive power by supplying leading reactive power at
specific nodes in the network. The required leading reactive power is usually
provided by capacitors connected in parallel and closer to the inductive load. Very
often the grid conditions change due to load variation and thus the required leading
reactive power should be accordingly adapted. That is commonly done by using an
automatic power factor correction system (automatic PFC). This installations,
thanks to a monitoring device and a power factor controller, allow the automatic
switching of multiple capacitor banks and manage to follow the absorbed reactive
power variations and finally to keep a constant power factor at the load [5]. The
paper mainly focuses on a better evaluation of the inrush current absorbed by the
“Politehnica” University of Bucharest, 313 Splaiul Independenţei, 060042 Bucharest, Romania,
[email protected]
Rev. Roum. Sci. Techn. – Électrotechn. et Énerg., 57, 4, p. 341–350, Bucarest, 2012
Emil Cazacu, Iosif Vasile Nemoianu
capacitor banks during various commutation processes. An accurate value of this
parameter is extremely important in designing such automatic PFC systems,
because the entire selection of their main components such contactors, protective
device etc., rely on the capacitors banks inrush current amplitude [6, 7].
Generally, an automatic PFC system contains, as Fig. 1 shows, the following
main parts [1]: current and voltage sensors providing signals proportional to the
absorbed electric energy parameters, a control and command block, an electric
power board comprising switching and protection devices and the capacitor banks.
The control and command block compares the measured power factor with the
prescribed one and operates the connection and disconnection of the capacitor
banks with the necessary reactive power (power factor regulator).
Fig. 1 – Architecture of an automatic power factor correction system (PFC) with three steps.
The modern PFC controllers, which are microprocessor-based, continuously
analyze the signal from the current sensor and commands the commutation of
contactors which add or remove capacitor bank steps in order to compensate the
actual reactive power of the total load. They also ensure the economic usage of
Peak inrush currents for multiple-step capacitor banks
capacitor stages, which means a minimized number of switching operations and
thus an optimized life cycle of the capacitor bank. The number of steps depends on
the number of connected loads. Consequently, the smaller the inductive loads, the
higher the number of steps should be foreseen. It is also useful to mention that
according to IEC 60831-1 standard [8] clause 22 each capacitor bank is provided
with discharge resistors that must be able to “discharge each unit in 3 min. to 75 V
or less, from an initial peak voltage of 2 times rated voltage Un”.
The main issues of the automatic capacitor banks switching process are the
high transient overcurrent required by the banks (also called inrush current) and the
transient overvoltage through the grid [5, 9]. The peak inrush current, overvoltage
amplitude and their frequencies are basically functions of the upstream network
parameters and the characteristic of the capacitor banks. Unlike the case of a single
bank (fixed correction), where the peak inrush current can reach 30 times the
nominal current of the capacitor bank, the inrush current amplitude in the case of
an automatic correction, depends on the power of the steps already connected, and
can reach 100 times the nominal current of the step to be energized [6, 10].
Figure 2 qualitatively depicts the inrush current and voltage variation for
the last switched capacitor bank (the third) in a multiple step automatic PFC
system. The numerical values of the used grid parameters are detailed in Chapter 3.
As we already mentioned, the inrush current (pulse) of an automatic PFC
depends on numerous parameters, such as: short-circuit value of power supply
transformer, remaining capacitor voltage due to fast switching banks, ohmic
resistance of capacitor itself and distribution switch gears, connection cables or
conductor resistance and inductance.
Although is rather difficult to take into account all the network and
Fig. 2 – Inrush current and voltage variation for the last switched capacitor bank in a multiple step
automatic PFC whose parameters are presented in Chapter 3.
Emil Cazacu, Iosif Vasile Nemoianu
capacitor bank electric characteristics, the Standards IEC 62271-100 [11] and
IEC 60831-1[8] suggest the relations for determining the inrush current amplitude
in a single and multiple (different or identical) reactive power steps switching
ip_single_step = U n
ip_multiple_steps = U n
2 S sc
2 C
; I nc = C ;
≈ Un
= I nc
3 L0 + L
3 L0
3U n
2 ( C1 + C2 + " Cn −1 ) ⋅ Cn
3 ( C1 + C2 + " Cn ) L +
ip_multiple_equal_steps = U n
1 1
+ +"
L1 L2
Ln −1
, n > 1,
n +1 3 L
where L0 is the network inductance. Also, for the bank step of index k, relations (1)
use the following notations: Lk – the connection inductance; Qk – the reactive
power; Ck – the capacity of the respective bank step; Inc – the rated bank current;
Un – the network rated voltage; Ssc – the short-circuit power at the point where the
capacitor are installed. Switching capacitor banks in automatic PFC is not only
related to high inrush currents, but also to high voltage transients, causing
degradation of power quality by generation of disturbance waves which propagates
throughout the network [5]. In order to evaluate the overvoltage level, one must
also consider the grid parameter alone with the reactive power QC supplied by the
capacitor bank. Often, the automatic PFC system is connected in the proximity of a
distribution transformer of rated apparent power Sn, percentage short-circuits
voltage usc[%] and a per-phase impedance ZT. Thus, the voltage variation is given
by [1, 2]:
∆U RP + XQ XQ usc [%]U n2Q usc [%]QC
≅ 2 =
; Q ≅ QC ;
U n2
S nU n2
3Z T I n Z T S n
usc [%]U n2
ZT ≅ X ; usc [%] = sc =
U n2
Relations (2) use the fact that in a power transformer the winding resistance
is negligible compared with the leakage reactance and the reactive power values
‘seen’ by the transformer, denoted by Q. In practice, the last may be considered as
the reactive power of the capacitor bank QC during the switching phenomenon.
These expressions are only to be used for a single step capacitor bank. Expressions
Peak inrush currents for multiple-step capacitor banks
(1) and even (2) are mostly experimentally obtained and cover almost all the
practical cases, for both single and multiple steps PFC systems. In the case of
automatic multiple steps capacitor banks, where the maximal values of the inrush
current could reach important values, an accurate determination of transient
phenomena becomes very important for the PFC systems optimal design. Beside,
as we already underlined in the previous sections, the selection of all the
commutation and protection devices are based on this values.
For a better estimation of the peak inrush current in multiple step capacitor
banks of an automatic PFC systems, we started from the schematic circuit depicted
in Fig. 1 and modeled each component of the installation with a specific lumped
circuit parameter [13].
The important elements that are involved in the transient process are the
capacities of the capacitor banks, inductance and resistance of the connection
cables between the stages and, for the first bank only, the inductance of the nearby
power transformer. Figure 3 presents the per-phase equivalent circuit diagram of a
three stages capacitor banks automatic PFC system. One can easily indentify the
above-mentioned circuit parameters and then use them to establish the
corresponding time differential transient equations.
The worst case regarding the capacitor banks peak inrush current value
occurs when the last capacitor bank switches on, while the others are already
Fig. 3 – Main component model of the three-stage automatic PFC system.
energized by lying at the maximum network voltage. That corresponds in our
model to the situation when the contactors K1 and K2 are closed and K3 introduces
the last bank into the circuit. The entire process could be mathematically expressed
by using nodes and loops equations for the circuits. That leads to the following
differential system of equations:
Emil Cazacu, Iosif Vasile Nemoianu
 d i1 (t ) 1
= [u (t ) − R1i1 (t ) − u C1 (t )]
 dt
 d i 2 (t ) 1
[u C1 (t ) − u C 2 (t ) − R2 i 2 (t )]
 dt
 d i3 (t ) 1
[u C 2 (t ) − u C 3 (t ) − R3 i3 (t )]
 d t
 d u C1 (t ) 1
[i1 (t ) − i2 (t )]
 dt
 d u C 2 (t ) 1
[i2 (t ) − i3 (t )]
 dt
 d u C (t ) 1
i3 (t )
 d t
u (t ) = U 2 sin (ωt + ϕ).
Similar equations could be used for a larger number of stages, with no
restriction imposed on the reactive power energy level of the capacitor bank.
The main advantage of this approach is that, by solving the differential
equations system, an accurate time-dependent variation for both the inrush current
and all the capacitor banks overvoltages can be rapidly obtained just after the
commutation process. Numerical solving of the differential equations describing
the entire process can easily be assembled in a very flexible software package
allowing the user to simulate a great number of parameter variations, finally
leading to a minimal peak inrush current value.
In order to quantitatively illustrate the accuracy of the proposed method for
evaluating the inrush peak current and overvoltage variation, we consider a three
stages automatic PFC system connected to the secondary windings of a 400 kVA
power transformer with percentage short-circuit voltage of usc = 4% and 400 V
rated voltage. Assume that the capacitor banks have different reactive power
values, namely Q1 = 5 kvar, Q2 = 10 kvar, Q3 = 15 kvar, and the connection
conductors are made of copper with the cross-sectional areas of 2.5, 4 and 6 mm2,
but all with the same length of 1.5 m. The connection between the power
transformer and the PFC system is realized by using a 5 m length copper busbar
having a 10 mm2 per phase cross-sectional area.
All these numerical data lead to the following value of the lumped circuit
parameters: R1 = 9.4 mΩ, R2 = 7 mΩ, R3 = 9.4 mΩ, L1 = 55 µH, L2 = L3 = 5 µH,
C1 = 100.28 µF, C2 = 200.57 µF and C3 = 300.86 µF. Thus, the numerical solution
of the differential equations system gives the current and voltage variations over a
period of only 2 ms, just after the commutation process, for all the circuit model
capacitors. These variation curves are plotted in Fig. 4 and Fig. 5, respectively.
The variation of the same parameters, but only for the last capacitor bank (the
third), is showed in Fig. 2 during an extended time period of 20 ms.
The initial values of the capacitor voltages for the both on-duty capacitor
banks (already connected) at the very switching moment were considered maximal,
namely the amplitude of network voltage. That forces the computation to analyze
even a worse model case. Studying the inrush current variation absorbed by the last
capacitor bank, one can notice that its peak values reach more than 1700 A, at a
Peak inrush currents for multiple-step capacitor banks
frequency of more than 5.2 kHz, while the bank rated current is of only 37 A at 50
Hz. The highest overvoltage exerted on the first capacitor bank terminals does not
exceed more than two times its rated value.
Fig. 4 – Inrush current variation through capacitors banks immediately after commutation.
Fig. 5 – Overvoltage variation on the capacitors banks immediately after commutation.
Emil Cazacu, Iosif Vasile Nemoianu
The initial values of the capacitor voltages for the both on-duty capacitor
banks (already connected) at the very switching moment were considered maximal,
namely the amplitude of network voltage. That forces the computation to analyze
even a worse model case. Studying the inrush current variation absorbed by the last
capacitor bank, one can notice that its peak values reach more than 1700 A, at a
frequency of more than 5.2 kHz, while the bank rated current is of only 37 A at 50
Hz. The highest overvoltage exerted on the first capacitor bank terminals does not
exceed more than two times its rated value.
Applying the second relation from (1) to this particular case, one will get a
value of only 1288 A for the peak inrush current, which does not cover the worst
possible situation and may be misleading especially for the optimal design of the
switching and protective devices. Moreover, the formulas presented in (1) did not
consider all the network parameters and the initial voltage values on the already
connected capacitor banks before commutation takes place. It indicates only an
absolute value that could be reached after a half-period time of network supply
while the transient phenomena occur much faster.
Such extreme high values of the peak inrush currents determine important
electric stresses on the capacitor banks and reduce significantly their lifetime span.
It could also cause welding or fast wearing out of the main contacts of contactors
and also have negative effects on power quality network by generating transit
overvoltage into the grid. Consequently, the necessity of damping these currents
One of the simplest and costless methods of achieving this goal is to connect
inductances in the circuit made by winding the cables designed to be connected to
the capacitor banks, onto a cylinder shape. Only a few turns determines an
additional inductance of 5–10 µH that could reduce with at least 30–50 % the
inrush current amplitude. On the other hand, longer or additional cables could
cause higher Joule losses within the PFC system.
Another solution is offered by the usage of special design contactors
equipped with pre-switching auxiliary contacts that close before the main contacts
do and insert, for a very short time (2–10 ms), damping resistors into circuit [14].
Thus pre-load the capacitors and avoid the current peak values effects. This
procedure decreases with more the 40 % the inrush current amplitude having also
positive influence on the capacitor bank life expectancy and increases power
quality by limiting transients and voltage sags to propagate through the network.
When also a harmonic polluted environment is involved, and voltage and
current total harmonic distortion (THD) are less than 10 % and 5 %, respectively,
the common solution of damping the inrush current and also suppressing the 5th
harmonic and above is to use anti-resonance inductances in the circuit. These so
called detuned-reactors [15, 16] are iron base inductances of several mH connected
in series with the capacitor banks and sized so that a resonance frequency is below
Peak inrush currents for multiple-step capacitor banks
the lowest frequency of the harmonic voltage in the network. Thus, a “detuned
automatic PFC” manage to significantly reduce the inrush current amplitude (with
more than 60 %) and also to filter the network from superior harmonics.
We have to notice that other methods and procedures were also developed in
order to suppress the capacitor bank switch-on transients’ consequences [17, 18].
Even with the above mention damping solution the importance of knowing
an accurate value of the peak inrush current does not diminish. Only a precise
value all the components of the PFC systems and damping devices could be
designed in an optimal way anticipating and avoiding important electrodynamic
stresses or poor power quality effects.
The problem of determining an accurate evaluation for the peak inrush
current and overvoltages (at switching) that occur in multiple stages LV automatic
PFC systems was treated. The circuit model of the main PFC installation
components allowed, by numerically solving the associated differential equations
system, to extract the exact solutions for both the peak inrush currents and the
overvoltages of all the involved capacitor banks during the switching phenomena.
Moreover, the time-variations just after the commutation process of the capacitor
banks’ electric parameters are also investigated. The proposed approach could be
easily adapted to any multiple step PFC system, considering also the network
parameters such as the nearby power transformers characteristics or connecting
cables inductances and resistances. For a particular case, where only three steps
were used, the numerical value for the peak inrush current of the last switched
capacitor was compared with the one obtained according a known formula
provided by IEC standards. The difference between the two values is quite
considerable and appears as a consequence of different parameters considered by
the analytical approach and also by the various initial value assumed by the already
connected (on duty) capacitors. Inserting all the differential equations into a
general purpose software environment, could be a very useful tool in designing
different components of the PFC systems.
The entire study requires further investigations in at least the following
directions: developing and improving the circuit model for the PFC installation,
suggesting an optimal procedure of selection the commutation and protection
devices correlated to the switched capacitor banks and most important, a detailed
study over the of commutation phenomena in automatic PFC systems that work in
installations with intermediate level of harmonic distortion.
Received on July 13, 2012
Emil Cazacu, Iosif Vasile Nemoianu
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