IEEE TRANSACTIONS ON MAGNETICS, VOL. 48, NO. 4, APRIL 2012
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Design Optimization of Gapped-Core Shunt Reactors
Abbas Lotfi and Mohsen Faridi
Electrical Engineering Group, Higher Education Institute of Roozbeh, Zanjan, Iran
Department of Electrical Engineering, Khodabandeh Branch, Islamic Azad University, Khodabandeh, Iran
Shunt reactors are important components for the development of UHV power systems since they compensate large capacitive currents generated by HV transmission lines over great distances. Shunt reactors can also limit over voltages resulting from load shedding
operations or from a line-to-ground fault. With regard to construction, shunt reactors are designed in distributed gapped-core which
has one winding and a core divided into several laminated magnetic steel disks separated by air-gap wedges. The Volume of the gap is
in relation with reactive power
of the reactor and maximum magnetic flux density
as well. After calculating the gap volume,
the way for separating of the gap area
and length
is an important matter. It plays main role in reactor dimensions, used
materials and power losses. In this paper a method based on minimizing the cost is presented for selecting the gap area and length by
ratio with constant volume constraint.
finding an optimum value for
Index Terms—Gapped core, optimization, shunt reactor.
I. INTRODUCTION
HUNT reactors are important components for the development of UHV power systems since they compensate large
capacitive currents generated by HV transmission lines over
great distances. Shunt reactors can also limit over voltages resulting from load shedding operations or from a line-to-ground
fault [1]. Concerning the construction, shunt reactors are designed in distributed gapped-core which has one winding and
a core divided into several laminated magnetic steel disks separated by air-gap wedges [2]. This construction is shown in Fig. 1.
This type of reactor is usually oil-immersed construction.
The volume of the gap is in relation with reactive power
and maximum value of magnetic flux density
. After calculating the gap volume by a simply obtained equation which is
presented in Section II, the way for separating of gap area
and length
is an important matter. It plays main role in
reactor dimensions, used materials and power losses. There are
no remarkable works about the mentioned problem and in point
of fact, most of the researches are related to interaction between
shunt reactors and power system in different transient conditions [3]–[5].
In this paper, a method based on minimizing the cost is
presented for selecting the gap area and length by finding
an optimum value for
ratio with constant volume
constraint. As a matter of fact, increasing the gap area in
constant volume, the gap length will be decreased. This is
corresponding to increase of
ratio that leads the
used iron to be increased. In other view, increasing that ratio
causes the reluctance of magnetic path to be decreased. So, it
is necessary to decrease the number of turns for tuning of the
inductance. In the same way, decreasing of the
ratio
S
Manuscript received August 15, 2011; accepted October 11, 2011. Date
of current version March 23, 2012. Corresponding author: A. Lotfi (e-mail:
a_lotfi_ee@live.com).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TMAG.2011.2173180
Fig. 1. Gapped core shunt reactor.
can result increase of winding turns and decrease in iron. As
ratio has
a result, choosing different value for the
considerable effect on used main materials (iron and copper).
Therefore it seems that, we can find the optimum value for the
mentioned ratio to minimize the iron and copper consuming
and their losses as well. At the following sections, formulating
the gap volume as a function of reactive power as well as
copper and iron mass and losses as function of the
ratio, we try to find optimum value of that ratio by minimizing
a suitable cost function.
The method used in this paper is based on the magnetic circuit’s theory. It should be noted that, although we can use other
methods such as finite element method (FEM) to obtain very
accurate calculation of magnetic flux density and inductance
[6], [7], but such methods are very time consuming that makes
them unsuitable for an optimization problem. Whiles relations
resulted from magnetic circuit’s theory provide simple equations for obtaining the best beginning point for optimum design
process. In addition, it is important to mention that the fringing
effect has been ignored in this work. In practice the large gap
length of the reactor is divided to several smaller gaps which
lead to lower fringing effects. So, it is very simple to control
its effect on the total inductance by number of core blocks (see
to
Fig. 1). Therefore, we can use a compensation factor
correct the fringing effect on inductance. Anyway, if it is necessary, there are some useful equations for calculating the fringing
factors [8], [9].
0018-9464/$31.00 © 2012 IEEE
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 48, NO. 4, APRIL 2012
II. FORMULATION
All of the structural parameters used in following subsections
are depicted in Fig. 1.
A. Required Gap Volume
In order to obtain an equation for gap volume, we start from
the following well-known relations:
(1)
(2)
Fig. 2. Winding cross sectional area (disc type).
(3)
C. Copper Mass and Losses
By manipulating the above equations:
The copper loss is calculated by following laws:
(4)
is the phase effective voltage,
is the current peak,
where
is the reactive power,
is the number of turns,
is the
maximum flux density, is the frequency,
is the fringing
effect factor, and
is free space permeability. The left side of
(4) is the gap volume and will be constant if the reactive power,
flux density and frequency are kept constant.
(9)
(10)
where
is the copper loss, is the winding resistance,
is the mean length of turn and is the specific conductance of
copper. In this paper the variation of specific conductance as a
function of temperature is ignored.
For
calculating we can use the following equation:
B. Winding Area and Number of Turns
(11)
In according to inductance definition, it can be written:
(5)
Finally, using the (9) with (8), (10) and current density definition, the copper losses can be expressed as
(12)
where is inductance (per phase) of the reactor that can be
calculated by
(6)
where
is the eddy loss factor of winding.
And eventually, the copper mass is
For winding area calculation, according to Fig. 2, the winding
area can be simply expressed:
(13)
where
is the copper density.
(7)
D. Iron Mass and Losses
where
is the winding area, is the rated current,
is the
winding filling factor and
is the wire cross sectional area.
By using the current density definition and (5) we have
By using Fig. 1, it is simple to obtain volume of the legs and
yokes as be done in the following equations. Then by multiplying to iron density, the mass of iron and consequently its
power loss can be obtained as follow:
(8)
(14)
where is the current density.
It is necessary to mention that the wire insulations as well as
number of cooling channels between discs for thermal considerations have direct impact on value of filling factor. In the optimization process dimensions of the winding will be changed.
So, the power loss of winding and finally number of cooling
channels will be changed. As a result, the filling factor is not a
constant.
For purposes of this paper and avoiding of complication we
consider the
to be constant.
(15)
Thus:
(16)
(17)
are the cross section area, the
where
volume, the mass and the losses of legs respectively,
is the
LOTFI AND FARIDI: DESIGN OPTIMIZATION OF GAPPED-CORE SHUNT REACTORS
1675
is the iron density,
is increment
lamination factor,
loss factor of core block,
is the specific power
losses of iron.
By the same way, for the yoke (see Fig. 1):
(18)
(19)
(20)
where
is the core structure factor,
are
the cross section area, length, mass, losses of the yoke and
is the specific loss of iron in the yoke.
Note that, according to the above-mentioned relations between yoke and leg structure and the FEM analysis [7], the following approximated equation was presented for magnetic flux
density:
Fig. 3. The cost function versus
.
(21)
where
are magnetic flux density in yoke, leg and
gap, respectively. By calculating the flux density, it is simple to
extract specific power losses of iron using the relevant specific
loss curve.
E. Cost Function
In order to find the optimum value for the
following cost function
is defined:
ratio, the
(22)
where
is the mass, is the cost,
is the fine for excessive losses.
is defined as follows:
(23)
Fig. 4. The cost function versus
In point of fact, if the calculated losses
becomes
smaller than desired value
, guaranteed loss,
is
gotten to be zero. Thus the fine cost is not considered.
According to relations presented in subsections to , all
components of the (22) are function of
ratio. In addition, for a constant value of
ratio,
is also constant. So, we can consider the
(or
) being free
parameter in optimization problem. As a final point, the total
is a function of
and
cost
It is necessary to mention that parameters of
are selected by insulation design
and other parameters comprise
,
,
, J,
and
are selected by manufacturer experience as well as FEM
analysis.
.
III. CASE STUDY
In order to validate the presented method, a 50 MVAR,
400 kV shunt reactor which has been optimized and built
by Iran Transfo Inc. (the largest manufacturer of power
transformers in Zanjan, Iran) has been selected. Using its
construction data the optimum value for
and
is
estimated.
In the first step the Bw is considered to be equal to winding
width of the test object. And by using its construction informaratio. As
tion, we try to find an optimum value for
can be seen in Fig. 3, there is an optimum value of 0.733 for
ratio corresponding to minimum point of cost function versus
.
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 48, NO. 4, APRIL 2012
TABLE I
COMPARISON BETWEEN OPTIMIZE AND BUILT VALUES
ACKNOWLEDGMENT
The authors would like to thank Iran Transfo Inc. for providing the test results.
REFERENCES
In the second step, considering the Bw to be free parameter we try to find minimum points of Cost Function versus
ratio. And then by plotting that Cost Function versus
Bw the best value of Bw is obtained. As can be seen in Fig. 4,
this optimum value for Bw is 0.26 m that is corresponding to
AgLg ratio of 0.8. The Cost values in comparison with test object is given in Table I that shows decrement in Cost Function
at estimated points.
IV. CONCLUSION
In this paper, sets of equations are presented for calculating
the iron and copper mass and loss which are function of dimensional values of the shunt reactor. Using these relations and a
suitable cost function, it is very simple to find an optimum value
for area and length of the gap. The presented method can be
tuned by FEM and used for optimization of shunt reactors. As
be done in this paper, developed method is great congruity with
a built and optimized reactor by empirical and validated equations.
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