Download No.4 - Transaction on electrical engineering

TRANSACTIONS
ON ELECTRICAL ENGINEERING
CONTENTS
Pástor, M., Dudrik, J.: Stability of Grid-Connected Inverter
with LCL Filter . . . . . . . . . . . . . . . . . . . . . . . . . . .
91 – 96
Bendík, J., Cenký, M., Eleschová, Ž.: 3D Numerical Calculation of
Electric Field Intensity under Overhead Power Line Using
Catenary Shape of Conductors . . . . . . . . . . . . . . . . . .
97 – 103
Cintula, B., Eleschová, Ž., Beláň, A., Janiga, P.: Comparison of
Reconfigurations Using Deterministic Approach for Global
Assessment of Operational State in Power System . . . . . . . .
104 – 111
Eleschová, Ž., Ivanič, M.: Impact of Asymmetry of Over-Head
Power Line Parameters on Short-Circuit Currents . . . . . . . .
112 – 115
Murín, J., Hrabovský, J., Gogola, R., Goga, V., Janíček, F.:
Numerical Analysis and Experimental Verification of
Eigenfrequencies of Overhead ACSR Conductor . . . . . . . . .
116 – 121
Vol. 5 (2016)
No.
4
ERGO NOMEN
pp.
91 – 121
TRANSACTIONS ON ELECTRICAL ENGINEERING
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International scientific journal of electrical engineering
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Prof. DOLEZEL Ivo, The Academy of Sciences of the Czech Republic, Czech Republic
Prof. DUDRIK Jaroslav, Technical University of Kosice, Slovakia
Prof. NAGY Istvan, Budapest University of Technology, Hungary
Prof. NOVAK Jaroslav, University of Pardubice, Czech Republic
Prof. ORLOWSKA-KOWALSKA Teresa, Wroclaw University of Technology, Poland
Prof. PEROUTKA Zdenek, University of West Bohemia, Czech Republic
Prof. PONICK Bernd, Leibniz University of Hannover, Germany
Prof. RICHTER Ales, Technical University of Liberec, Czech Republic
Prof. RYVKIN Sergey, Russian Academy of Sciences, Russia
Prof. SKALICKY Jiri, Brno University of Technology, Czech Republic
Prof. VITTEK Jan, University of Zilina, Slovakia
Prof. WEISS Helmut, University of Leoben, Austria
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Transactions on Electrical Engineering, Vol. 5 (2016), No. 4
91
Stability of Grid-Connected Inverter
with LCL Filter
Marek Pástor 1) and Jaroslav Dudrik 2)
1) 2)
Dept. of Electrical Engineering and Mechatronics, Faculty of Electrical Engineering and Informatics,
Technical University of Košice, Košice, Slovakia,
e-mail: 1) [email protected], 2) [email protected]
Abstract — The paper analyses oscillations in a singlephase grid connected inverter with the LCL output filter.
Passive and active damping techniques are designed and
compared by simulations. The inverter is controlled by a
proportional resonant controller.
Keywords — control, dc/ac converter, filtering
I. INTRODUCTION
Grid connected pulse width modulated (PWM) voltage
source converters require an inductive filter for its
operations. These converters are used in many
applications such as the active power filters, PWM
rectifiers or PWM inverters. To decrease the weight and
volume of the filter it is desirable to use a higher-order
filter. Usually an LCL filter is used [2–14]. The LCL filter
has a drawback of possible resonance if excited at a
resonant frequency. This resonance introduces the system
instability and increases THD of current. To suppress the
oscillations a damping technique is used. There are two
ways of the oscillation damping in the LCL filter: the
passive and active damping. Both of them have its
advantages and disadvantages and are still analysed and
designed. The passive damping [1] uses damping resistors
which introduces additional system losses. An active
damping is used to remove these extra losses but keep
system stability. The active damping of the LCL filter is
studied in the active power filters [2–4], PWM rectifiers
[5–7], PWM inverters [8–12] and in general grid
interacting converters [13–15]. This paper analyses
various damping techniques for the single-phase grid
connected inverter with the LCL filter. Several damping
techniques are analysed and designed for a single phase
inverter with the proportional-resonant (PR) controller.
II. LCL FILTER
The grid-connected voltage source inverter (VSI) needs
an inductive load. This load is provided mainly by the
output filter. The filter topology depends on the harmonics
attenuation requirements. The harmonics are produced by
the pulse-width modulation (PWM). Usually higher order
filters, such as the LCL filter (Fig. 1), are used. The LCL
filter is a third order filter with attenuation of 60 dB/dec.
The advantage of the high attenuation of a third order
system has a drawback of resonance. The LCL filter has
three resonant frequencies defined by the reactive
components. The grid influences the resonant frequency
of the LCL filter as well. Connecting the LCL filter to the
output of a PWM modulated inverter causes driving the
LCL filter input with a spectrum of various highfrequency voltages.
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DOI 10.14311/TEE.2016.4.091
Fig. 1. LCL filter topology and its dynamical model
(RS and RG are parasitic resistances).
The inverter output voltage VS with a frequency equal
to a resonant frequency of the LCL filter defined by (1)
causes a resonance in the output current IG of the LCL
filter. The resonance causes system instability and
increases the THD of the grid current and thus it is
undesirable.
1
2
LS  LG
LS LG C




The example of the harmonic spectrum of the LCL
filter output current is shown in Fig. 2. The resonance
peak is clearly visible and its contribution to the THD of
the grid current IG is more significant than the contribution
from the inverter switching frequency.
Suppression of the switching frequency is guaranteed
by a proper design of the LCL filter reactive components.
Suppression of the LCL filter resonance is guaranteed by a
proper damping technique design. However, as it is
shown, some damping techniques influence the harmonics
suppression.
f0 I
GVS

Fig. 2. Example of LCL filter output current (with rms value 1.409 A)
spectrum without proper oscillation damping.
Transactions on Electrical Engineering, Vol. 5 (2016), No. 4
The attenuation of the switching frequency in the
voltage VS with respect to the grid current IG is described
by the transfer function (2).
IG ( s)
1

VS ( s) s3 LS LGC  s 2  LS CRG  RS CLG   s  RS CRG  LS  LG   RS  RG

A. Passive Damping
The passive damping is ensured by adding a dissipative
component to the LCL filter topology. The properly
placed resistor will decrease the resonance peak. The
damping resistor is usually placed in series with the filter
capacitor (Fig. 3).
b) dynamical model
Fig. 4. LCL with various damping resistor values.
B. Active Damping
The frequency characteristics of the system can be
changed also by modifying the controller. The proper
transfer function of the controller can damp the LCL filter
oscillations. The main idea is to remove the undesired
frequencies from the inverter output voltage by modifying
the PWM modulation signal. This approach is called the
active damping.
C. Active Damping – Notch Filter
There are several ways how to achieve the active
damping. Probably the most straightforward one is to
remove the resonant frequency from the inverter output
voltage by a filter (Fig. 5).
Fig. 5. Simplified control structure of PR current control with active
damping by notch filter.
Fig. 3. LCL filter with passive damping resistor.
The damping resistor will change the transfer function
of the LCL filter (3).
IG ( s)
1  sCRC

VS ( s) s3 LS LG C  s 2  LS CRG  RS CLG  RC CLG  LS CRC  
frequency of the LCL filter is not too far from the
switching frequency, this decrease is minimal.


III. RESONANCE DAMPING
There are two ways how to damp the LCL filter. The
so-called passive damping employs an extra resistor added
in series with a filter capacitor. This resistor influences the
transfer function (2) and suppresses the resonance by
modifying the transfer function for frequencies around and
above the resonant frequency defined by (1).
The frequency characteristics of the system can be
changed also by modifying the controller. The proper
transfer function of the controller can damp the LCL filter
oscillations. The main idea is to remove undesired
frequencies from the inverter output voltage by modifying
the modulation signal of the PWM. This approach is
called an active damping.
a) LCL filter with damping resistor
92
The suitable type of the filter is a notch (negative peak)
filter. The simple notch filter consists of the LC resonant
tank (Fig. 6).

s  RS CRG  LS  LG  RS RC C  RC CRG   RS  RG



The damping resistor adds losses to the LCL filter. The
value of the resistor is usually chosen as one third of the
capacitor reactance at the resonant frequency:
RC 
1
1
3 2 f0 I

C
GVS



The advantage of the passive damping is its robustness.
However there are also disadvantages. The damping
resistor has losses which decrease the overall system
efficiency. The losses on the damping resistor RC
connected in series with the capacitor can be calculated
[16] where h represents harmonic order:
PRC  RC  iS  h   iG  h 
b) notch filter frequency characteristics
Fig. 6. Notch filter.
2

h



The second main disadvantage is decrease of the
higher frequency damping (Fig. 4). If the resonant
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DOI 10.14311/TEE.2016.4.091
a) notch filter
topology
The transfer function of a notch filter is defined by (6).

Fnotch ( s) 
LnCn s 2  1
LnCn s 2  RnCn s  1



Transactions on Electrical Engineering, Vol. 5 (2016), No. 4
93
where:


IV. CONTROL SYSTEM DESIGN
Ln  1mH 
1
Cn 
2
4 Ln f 02I V
G S
Rn 




 2 f n 2 LnCn  1
 2 f nCn 2 



The frequency fn defines the notch filter bandwidth
around the LCL filter resonant frequency f0IGVS.
The advantage of the active damping by the notch filter
is its sensorless concept. When compared to the passive
damping, the system transfer function for higher
frequencies is not influenced by inserting the notch filter.
However, the design of the notch filter depends on the
LCL filter parameters.
D. Active Damping – Virtual Resistance
Another active damping method uses the concept of a
virtual resistor. The virtual resistor method is based on the
LCL filter capacitor current sensing and multiplying of
this current by the virtual damping resistor resistance. The
resulting virtual voltage is then subtracted from the
inverter PWM modulating voltage (Fig. 7).
The virtual resistor value can be calculated using (2).
As can be seen from Fig. 8, the virtual resistor does not
change the system frequency characteristic for higher
frequencies than the f0IGVS resonant frequency.
The transfer function of the LCL filter damped by
virtual damping resistor RC is changed to:

TABLE I. LCL FILTER PARAMETERS
LCL filter parameter
Apparent power S
Switching frequency fSW
Current ripple of IS
Resonant frequency f0IGVS
Inductance LS
Inductance LG
Capacitance C
Value
Unit
1.2
5
20
2
3.6
1.8
5.3
kW
kHz
%
kHz
mH
mH
µF
B. Proportional Resonant Controller
The inverter is a single-phase system. It is thus
beneficial to use the proportional resonant (PR) controller
(Fig. 9). This approach will remove the need to transform
the single-phase system into the rotating reference frame
in dq coordinates for the PI controller.
Fig. 9. PR controller.
IG ( s)
1

VS ( s) s3 LS LG C  s 2  LS CRG  RS CLG  RC CLG  
s  RC RG C  RS RG C  LS  LG   RS  RG
A. LCL Filter Parameters
The performance of three damping methods is
compared by simulation. The system consists of a singlephase PWM voltage source inverter with switching
frequency of 5 kHz and dc link voltage of 420 V. The
inverter is connected to the grid through the LCL filter
with parameters shown in Table I.


Fig. 7. Simplified control structure of PR current control with active
damping by virtual resistor.

The PR controller has a transfer function (11).
FPR ( s)  K P 
2 K I PR s
s 2  2PR s  g2




The frequency ωPR defines the PR controller bandwidth
around the grid frequency ωg. To design the PR controller
it is necessary to know the transfer function from the
inverter voltage VS (manipulated variable) to the grid
current IG (controlled variable). The grid voltage VG is a
measured disturbance and is compensated in the PR
controller. The grid current is a measured controlled
variable. The high frequency transfer function from VS to
IG of the LCL filter is:
IG (s)

VS ( s )
1

s3 LS LGC  s 2  LS CRG  RS CLG   s  RS CRG  LS  LG   RS  RG


The PR controller controls the grid current IG with the
grid frequency and thus generates the manipulated
variable VS with the grid frequency. The transfer function
(3) is simplified by omitting the high-frequency terms:
Fig. 8. Frequency characteristics of LCL filter with active damping by
virtual resistor.
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IG ( s)
VS ( s)
1
s  LS  LG   RS  RG

LL



The LCL filter is therefore simplified to a first order
system with the time constant of:

Transactions on Electrical Engineering, Vol. 5 (2016), No. 4

And its gain:
LS  LG
RS  RG
TLCL 



1
RS  RG




The PR controller is designed to compensate the time
constant TLCL. The proportional gain of the PR controller
is set to (τ is the time constant of the required control
dynamics of the whole controlled system):
K LCL 
KP 
TLCL
 K LCL


KP
TLCL




The integral gain of the PR controller is set to:

KI 

94
A. Passive Damping Method
The damping resistor RC for the LCL filter specified in
Table I calculated using (2) is 5 Ω. Figure 11 shows
frequency characteristics of the undamped LCL filter and
LCL filter damped by passive damping resistor.
TABLE II. PR CONTROLLER PARAMETERS
PR Controller
Parameters
Time constant TLCL
Gain KLCL
Proportional gain KP
Integral gain KI
Frequency ωPR
Grid frequency ωG
Value
Unit
13.5
2.5
5.4
400
1
314
Ms
rad/s
rad/s
Fig. 11. LCL filter frequency characteristics with passive damping.
The designed passive damping causes a positive gain
margin of 13.4 dB and a positive phase margin of
90.2 deg. (Fig. 12). The stability of the system is thus
ensured.
V. COMPARISON OF DAMPING METHODS
The stability of the system created by PR controller and
LCL filter is checked by gain and phase margins obtained
from Bode characteristics. The system is considered stable
if the gain margin is at least 10 dB and the phase margin is
at least 60 deg. Besides the control loop, the damping of
the LCL filter frequency characteristic is compared as
well.
The frequency characteristics of the open control loop
with designed PR controller and undamped LCL filter is
shown in Fig. 10. It has a gain margin of -46.7 dB and
phase margin of -89.7 deg. and is clearly unstable. The
negative peak in phase at 315 rad/s is caused by resonant
frequency of PR controller.
Fig. 12. Frequency characteristics of open control loop with PR
controller and LCL filter with passive damping.
Figure 13 shows the dynamic response of the grid
current. The stability is clearly visible in the time domain
as well without influencing the system dynamic
behaviour.
Fig. 10. Frequency characteristics of open control loop with PR
controller and undamped LCL filter.
Fig. 13. LCL filter output current and its reference signal with passive
damping.
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Transactions on Electrical Engineering, Vol. 5 (2016), No. 4
B. Notch Filter
The simulated notch filter transfer function is:
6.36 109 s 2  1
6.36 109 s 2  4.664 105 s  1 



To suppers the resonant frequency the notch filter has a
narrow bandwidth of 1 kHz. Figure 14 shows the
frequency characteristics of the undamped filter and the
filter damped with the notch filter. The resonant peak is
completely removed.
Fnotch ( s) 
95
C. Virtual Resistance
To create a damping in the controlled system a virtual
damping resistor with resistance of 10 Ω (two times the
passive damping resistor) was chosen. Figure 17 shows
the frequency characteristics of the undamped LCL filter
and the LCL filter damped with the virtual resistor.
Fig. 17. LCL filter frequency characteristics with virtual resistance
damping.
Fig. 14. LCL filter frequency characteristics with notch filter damping.
The notch filter damping (Fig. 5) causes a positive gain
margin of 17.2 dB and a positive phase margin of 87.5 deg
(Fig. 15).
After checking the stability of the virtual damping
method only a 8.89 dB gain margin was observed (Fig.
18). The system is stable but it is advisable to use a three
times the value of calculated passive damping resistor (4)
to ensure a gain margin at least 10 dB.
Fig. 18. Frequency characteristics of open control loop with PR
controller and LCL filter damped with virtual resistor.
Fig. 15. Frequency characteristics of open control loop with PR
controller and LCL filter damped by notch filter.
Fig. 16. LCL filter output current and its reference signal with notch
filter damping.
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DOI 10.14311/TEE.2016.4.091
The smaller gain margin caused by the proposed virtual
damping resistor is also visible in Fig. 19 as damped
oscillations during a transient period.
Fig. 19. LCL filter output current and its reference signal with passive
damping virtual resistance damping.
Transactions on Electrical Engineering, Vol. 5 (2016), No. 4
VI. CONCLUSIONS
The paper describes the oscillation phenomena in the
grid-connected inverter with the LCL filter. The LCL
filter resonance problem is analysed in the single-phase
inverter with the PWM control. The PR controller is used
to control the grid current. The controller is designed and
verified by simulation. It is shown that the PR controller
by its own cannot ensure the stability of the system. Thus
three different techniques are analysed and designed to
suppress the LCL filter oscillations. The passive damping
technique has the advantage of robustness and simplicity.
Unfortunately the passive damping increases the system
losses and the LCL filter capabilities of harmonics
damping are reduced. The two active damping techniques
remove these disadvantages. The notch filter at the output
of the controller can be realised in a digital form. Thus no
additional hardware costs are included. The notch filter
needs to be tuned for a particular filter and its resonant
frequency and is thus less robust. The additional current
sensor is required for the virtual resistor damping
technique. The capacitor current sensing will increase the
robustness. The simple passive damping technique can be
used to calculate the value of a virtual damping resistor.
Choosing of a suitable damping technique depends on an
application and usually an active damping together with
passive damping is used.
ACKNOWLEDGMENT
The authors wish to thank the project VEGA 1/0464/15
for its support.
The work was supported by project FEI-2015-3.
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Transactions on Electrical Engineering, Vol. 5 (2016), No. 4
97
3D Numerical Calculation of Electric Field
Intensity under Overhead Power Line Using
Catenary Shape of Conductors
Jozef Bendík 1), Matej Cenký 2) and Žaneta Eleschová 3)
1) 2) 3)
Slovak University of Technology in Bratislava, Faculty of Electrical Engineering and Information Technology,
Institute of Power and Applied Electrical Engineering, Bratislava, Slovakia,
e-mail: 1) [email protected], 2) [email protected], 3) [email protected]
Abstract — This article presents a superposition method
in combination with the Coulomb’s law and the Method of
image charges for calculation of the electric field
distribution generated by high voltage overhead power lines
above a flat surface in every dimension. Such calculations
are required to ensure the operational safety of people
exposed to the action of external electric field as well as to
reduce the cost of people protection. The method provides
options for calculation of the field around the wire of a
general shape. This substantial improvement of the method
could be applied to eliminate the usual error in the
calculation created using approximation of catenary shape
conductors by infinite straight conductors. The method has
been extensively tested on a set of shapes with known
analytical solutions. It has been shown that the numerical
solution converges uniformly to the analytical solution and
the accuracy depends only on the number of finite elements.
according to the international standards. The reference
levels for the Slovak and Czech Republic are shown in
Table 1. [4], [5]. Several national and international
organizations have formulated guidelines establishing
limits for the occupational and residential EMF exposure.
The exposure limits for EMF fields developed by ICNIRP
– formally recognized by World Health Organization
(WHO), were developed following reviews of scientific
literature, including thermal and non-thermal effects. The
standards are based on evaluations of biological effects
that have been established to have health consequences [3]
[6].
TABLE I.
LEGISLATIVE VALID REFERENCE LEVELS IN SLOVAK AND CZECH
REPUBLIC FOR EXPOSURE TO TIME-VARYING ELECTRIC AND MAGNETIC
FIELDS (UNPERTURBED RMS VALUES)
Keywords — electric field, FEM, power transmission line,
charge
I. INTRODUCTION
The overhead transmission lines are source of magnetic
as well as electric field. This electromagnetic field (EMF)
is of low frequency and it is time-varying [1]. It is
believed that the influence of the EMF field is harmful
only in certain way to the human health so the effects of
the field have to be taken in consideration in the power
transmission line project design [2]. An increasing
demand on operational safety in the vicinity of high
voltage lines calls for a more precise project preparations.
The project preparation mainly involves numerical
calculations, the result of which can facilitate and cheapen
particular project activities. An exposure to EMF field
causes flow of induced currents in living organisms, and
can have other unpleasant effects on human body [3]. In
many cases this considerations has not been proven, but
studies show a potential risk. According to this health
risks non-governmental organization International
Commission on Nonionizing Radiation Protection
(ICNIRP) established for population reference levels for
the exposure to time-varying electric and magnetic fields
shown in Table 1 [2]. These reference levels can vary
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The main conclusion from the WHO reviews is that the
electromagnetic field exposures below the limits
recommended in the ICNIRP international guidelines do
not appear to have any known consequence on health. In
the past years the European Commission gave a new
recommendation, based on the ICNIRP study, to establish
that all European Union (EU) states observe standard
reference levels of the exposure to the field. Although
reference levels vary in different countries they cannot be
lower than the EU standards. The best way to deal with
these standards is to have a truth worthy method for the
calculation of the electric field. From Maxwell`s equations
with a combination of the method of image charges
dependence of electric field from the position of the
conductor, the observer and its mirror image can be
derived. This complex approach gives in combination
with finite element method tool for calculation of the
electric field intensity.
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98
TABLE II.
REFERENCE LEVELS FOR EXPOSURE TO TIME-VARYING ELECTRIC AND MAGNETIC FIELDS (UNPERTURBED RMS VALUES) [2]
A common simplification in the evaluation is to
calculate the electric field intensity with the real conductor
approximation by the infinite straight conductor placed in
the lowest height of the sag. However this condition does
not reflect most of real situations. This simplification can
be applied only on symmetric spans.
II. TERMINOLOGY
A. Quasi-static Field
The time varying EMF is called quasi-static if we can
neglect time changes of the EMF spread by finite speed.
For a harmonic EMF field, which spread in the air and
which variables vary with angular frequency ω, the quasistatic criterion can be expressed by Eq. 1, where σ is
enviromental conductivity, f is frequency [7].
For air σ gets values from 3,10e-15 to 8,10e-15 [S/m].
The criterion is valid for both sides of the interval.
For the quasi-static field applies, that we can neglect
time derivations in I. and II. Maxwell’s equation [7].
Three orthogonal components of a vector may be
phasors with different magnitude and phase angles. These
components are called phasor-vector (Eq. 7). In this
article, a vector is indicated with an arrow and phasorvectors with a hat over the arrow [8].
In many cases the single RMS value is necessary to evaluate the field. This value is calculated from magnitudes
values of the phase-vector components as follows:
C. Catenary Shape of Conductors
Conductor attached on two sides, in this case on
transmission towers, will form curve in shape of a
catenary.[1] The catenary can be considered as symmetric,
if conductors are at their ends in the same height above a
flat surface, or asymmetric if not. Figure1 shows the
general asymmetric span, where A is the length of the
span. V1 and V2 are the heights at each end of the catenary
above the flat ground. A1 is the distance from the
beginning of the catenary to the middle of the span.
The equations simplify to the form:
B. Phasors and Vectors
The EMF field near transmission lines are described in
this article using phasors and vectors. A vector is
characterized by a magnitude and angle in space or by
three spacial components, Eq. 5 [1].
Fig. 1. Asymmetric span.
If A1 is exactly half of the A then the catenary is
symmetric. A1 is in general calculated as follows:
A phasor on the other hand is a quantity with a
sinusoidal time variation described by a magnitude and a
phase angle (Eq. 6). The angle φ describes a phase shift
[1].
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The constant c is the parameter defining the shape of
the catenary. The parameter h is the height at the lowest
point of the sag and it is calculated as follows:
99
real one. The mirror conductor has the opposite charge as
the real one, Fig. 3.
The height of the catenary y at the place x is calculated
as follows:
III. METHODOLOGY
From the III. Maxwell’s equation we can derive the
Coulomb’s law, in the integral form Eq. 12.
Let the point of an observer P be the point where we
want to calculate the electric field intensity. The vector 𝑟⃗
starts at the conductor element dl and points to the
observer P. The vector 𝑟⃗ can be written also as:
where ⃗⃗⃗⃗
𝑟𝑝 vector points from the coordinate system origin
to the observer P and ⃗⃗⃗⃗
𝑟0 points from the coordinate system
origin to the conductor element dl, Fig.2. The Coulomb’s
law equation in the analytical form is after a substitution
as follows:
Fig. 3. Example of mirror conductors above the conductive plate with
the potential φ = 0 [7].
In this new model the charge distribution in the
conductors remained unchanged and the potential on the
boundary plate is also zero as in the basic model. The
solution of the new model will have the same solution as
the initial boundary value problem. The electric field
intensity 𝐸⃗⃗ in point of the observer P, Fig. 3 equals the
superposition of electric intensities created by each
conductor, real and mirror one, Eq. 15.
B. Change from Derivations to Differences
The catenary shape of the conductor makes the
analytical calculation of Eqs. 14 and 15 too complicated.
To overcome this issue a numerical method based on the
superposition of finite elements has to be used. The core
of this method is to change the derivations to the
differences to determine the length of the conductor
element ∆l.
Fig. 2. Positions of vectors 𝑟⃗, ⃗⃗⃗⃗,
𝑟0 and ⃗⃗⃗⃗
𝑟𝑝 in relation to the observer P
and catenary.
A. Method of Mirror Images
It is not possible to find out 𝐸⃗⃗ in a dielectric
environment, where conductors hang above a conductive
plate, just by Eq. 14. The electric field in this model is
created not only by charges in conductors but also by
charges created by electrostatic induction in the
conductive plate (terrain) with potential φ = 0. The
charges distribution and density in this plate is uneven.
Solving this problem is done by the method of image
charges. The method creates a new mirror conductor
axially symmetrical according the boundary plane to each
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It is important to realize that the linear charge density
changes with every one element due to a different distance
from the ground. The vectors ⃗⃗⃗⃗⃗⃗
𝑟0𝑛 and 𝑟⃗⃗⃗⃗⃗⃗⃗
0´𝑛 will determine
the conductor element position and its image according to
the coordinate system beginning. We can determine
values of the vectors in X and Y directions by Eq. 11,
value in the Z direction equals to the overhang of the
conductor on a transmission tower. The electric field
intensity 𝐸⃗⃗ can be now determined for one conductor over
a flat surface as follows:
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100
coefficient of the conductors k and j, ℎ𝑘 is height of the
conductor k element above ground, 𝑅𝑘 is radius of the
conductor k, 𝐷𝑘𝑗 is distance between the elements of the
conductors k and j in the same distance X from of the
coordinate system beginning, 𝐷´𝑘𝑗 is the distance between
the element of the conductor k and mirror element of the
conductor j in the same distance X from the coordinate
system beginning.
A conductor of any given shape can be by this method
"chopped" to finite elements, Fig. 4.
Fig. 4. Visualization of calculation of electric field intensity by the
Finite element method and method of mirror images.
C. Calculation of Linear Charge Density
It is necessary to evaluate the linear charge density τn
for each conductor element ∆l. This is possible from
known voltages on each conductor and from geometry of
each catenary element ∆l. In general for an infinite straight
conductor linear charge density can by calculated by
Eq. 18, where [𝜏𝑛 ] is one 1D matrix of the linear charges
densities at k conductors, [𝑃𝑘𝑘 ] is 2D matrix of Maxwell’s
potential coefficients, by the unit [F/m], finally [𝑈𝑘 ] is 1D
matrix of voltages at k conductors.
When calculating the field created by the conductor
catenary shape, the matrix [𝑃] will be different for every
conductor element ∆l. For the nth element from the system
of k conductors we can write:
The result is 2D matrix [𝜏𝑘𝑛 ] consisting linear charge
densities on conductor k for every nth element of the
catenary. The components of matrix [𝑃𝑘𝑗 ] for every
element can be determined by following Eqs. 20 and 21.
The matrix [𝑃𝑘𝑗 ] is symmetric and for components not in
diagonal equals 𝑃𝑘𝑗 = 𝑃𝑗𝑘 .
Distances between conductors are shown in simple
example in Fig. 5 where 𝑃𝑘𝑘 is the self-potential
coefficient of the conductor k, 𝑃𝑘𝑗 is the mutual potential
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Fig. 5. Conductor positions and their mutual distances for calculation of
coefficients 𝑃𝑘𝑗 .
D. Generalization
So far we have analyzed the calculation of 𝐸⃗⃗ only as a
stationary field formed by a constant voltage U. In the
calculation of harmonically oscillating field where the
intensity 𝐸⃗⃗̂ is the phasor-vector unit we shall use as input
̂.
phasor of effective phase voltage of each conductor ⃗⃗⃗⃗⃗
𝑈
𝑘
The linear charge density will also harmoniously change
so Eq. 22 can be rewritten as follows:
The final equation for the electric field intensity under a
power transmission line consisting of k conductors in
shape of catenary at point of the observer P with use of
FEM method is as follows:
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101
IV. ANALYTICAL VERIFICATION
The numerical method has been tested on a set of
conductor shapes which analytical solutions are known
[9]. It was shown that the numerical solution converges
uniformly to the analytical solution and the accuracy
depends only on the number of finite elements, Fig. 4. It is
important to note that the conductor segment vector ∆l is
in each case computed differently according to the
conductor shape. In both cases the τ = 1.10e-4 C/m and
the analytical results are RMS values [10].
Fig. 7. Circle loop conductor.
Fig. 6. Dependence of the numerical error from the length of the
element for given conductor shapes.
Fig. 8. Two infinite straight
conductors.
V. FIELD UNDER POWER LINE
In this farther example of results it is shown the
calculation of the electric field intensity under the power
line of the type 2×400 kV DONAU with two ground
wires, Fig. 9. The distance between towers is 350 m. The
parameters of towers for calculation are in Tab. 3. The
lowest conductor is set to be 11.5 m above the ground.
This minimum distance is no longer determined by the
standard, which is 8 m above the ground, but according to
the value of the field. It can vary approximately from 10
m to 12 m according to the mutual phase location, terrain
curvature and many more factors. The current in each
phase bundle is 2400 A and the phase voltage is set on
420 kV.
Circle loop conductor: Conductor in shape of a circle
loop with the radius R = 100 m, Eq. 24. Point of the
observer is in distance 10 m from the circle at the central
axis, Fig. 7.
Two infinite straight conductors: The distance between
conductors is a = 10 m, Eq. 25, and the point of the
observer is placed at 𝑟𝑝𝑥 = 5 m, 𝑟𝑝𝑧 = 0 m from the
coordinate system beginning, Fig. 8.
Fig. 9. Transmission tower type 2×400$ kV DONAU with two ground
wires (green). Figure shows positions of phases L1 (white), L2 (black)
and L3 (red).
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Fig. 10. 𝐸𝑅𝑀𝑆 shown in 3D graph in horizontal plane in a constant
height above ground, 1.8 m . The axis Z and colours display the field
value.
TABLE III.
HANGING POINTS OF THE CONDUCTORS ON TOWER
102
Fig. 11. 𝐸𝑅𝑀𝑆 in a vertical plane crossing the transmission line.
Calculation for the lowest distance of the conductors to the ground 𝐸𝑅𝑀𝑆
is indicated by colour from dark to red.
VI. SHIELDING EFFECT OF THE GROUND WIRES
In past years it have been discussions in the power line
community about the posibility of shielding the
electromagnetic field by ground wires placed beneath the
phase conductors. As it is shown in Figs. 12. and 13., the
transmission tower Portál 400 kV with minimum distance
of the phase conductors above the ground 12 m , such
effect can be achieved, Fig. 13.
Figures 11 and 10 show results that are calculated using
the EMFTsim ultimate software (all graphical results are
made by Dislin graphical library) which was built for this
specific type of calculations by the authors of this article.
Fig. 12. 𝐸𝑅𝑀𝑆 in a vertical plane crossing the transmission line of the
type Portál 400 kV. Calculation for the lowest distance of the
conductors to the ground. 𝐸𝑅𝑀𝑆 is indicated by colour from dark to red.
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103
including human itself. This means more study is needed
to be done for ensuring operational safety of the
transmission line but also for reducing the project costs.
For a complex computation of the electromagnetic filed a
simulation software EMFTsim Ultimate was developed.
All graphical result in this paper are made by this
software.
ACKNOWLEDGEMENT
This research was supported by Nadácia Tatra banky.
REFERENCES
Fig. 13. 𝐸𝑅𝑀𝑆 in a vertical plane crossing the transmission line of the
type Portál 400 kV . Calculation for the lowest distance of the
conductors to the ground. 𝐸𝑅𝑀𝑆 is indicated by colour from dark to red.
As it is shown, the electric field can be dramaticly
reduced, however shielding in reality will be highly
unpractical, due to additional costs for development of
new transmission towers. Also to maintain addition
seperation distances between shielding wires and phase
conductors and ground will in reality result into a
transmission tower raising.
VII. CONCLUSION
Enumeration of the electromagnetic field generated by a
power transmission line is one of crucial conditions for
completing total project of the line. This article explained
a numerical method for calculation of the electric field
intensity in a complex way. The described method also
tries to eliminate most of common approximations. As the
paper shows, this method can be fully applied for any
conductor shape. The results shown in this paper were
verified not only analytically, but also using calculations
of similar computation softwares. The value of electric
field intensity is effected also by ground and all subjects
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[1] EPRI, “AC transmission line reference book – 200 kV and above.”
[2] WHO, “Environmental Health Criteria 238 – Extremely Low
Frequency Fields.”
[3] D. M. Repacholi and E. Vandeventer, “WHO Framework for
Developing EMF Standards,” pp. 1–13, October 2003.
[4] “Vyhláška Ministerstva zdravotníctva Slovenskej republiky o
podrobnostiach o požiadavkách na zdroje elektromagnetického
žiarenia a na limity expozície obyvateľov elektromagnetickému
žiareniu v životnom prostredí,” Zbierka zákonov č. 534/2007, pp.
3812–3816, 2007.
[5] Sbírka zákonů České republiky, ročník 2008. “Nařízení vlády o
ochraně zdraví před neionizujícím zářením,” Částka 1, pp. 2–29,
2008.
[6] ICNIRP, “Guidelines for limiting exposure to time-varying electric
and magnetic fields (1 Hz to 100 kHz).” Health physics, vol. 99,
no.6, pp. 818–36, Dec. 2010.
[7] D. Mayer and J. Polák, Metody řešení elektrických a magnetických
polí, 1983.
[8] EPRI, “AC Transmission Line Reference Book – 345 kV and
above,” Chapter 7, pp. 329–417, 1982.
[9] D. Mayer, Aplikovaný elektromagnetizmus: úvod do makroskopické teorie elektromagnetického pole pro elektrotechnické
inženýry, 2012.
[10] J. Bendík, M. Cenký, and Ž. Eleshová, “Complex calculation of
intensity of electric field under power transmission line using
catenary shape of conductors and flat surface”, in Power
engineering 2016: Energy-Ecology-Economy 2016: 13th
International scientific conference. Tatranske Matliare, Slovakia.
May 31 – June 2, 2016. 1. vyd. Bratislava: Slovak University of
Technology, 2016, s. 139–144. ISBN 978-80-89402-85-4
Transactions on Electrical Engineering, Vol. 5 (2016), No. 4
104
Comparison of Reconfigurations Using
Deterministic Approach for Global Assessment
of Operational State in Power System
Boris Cintula 1), Žaneta Eleschová 2), Anton Beláň 3) and Peter Janiga 4)
1) 2) 3) 4)
Slovak University of Technology in Bratislava, Faculty of Electrical Engineering and Information Technology,
Institute of Power and Applied Electrical Engineering, Bratislava, Slovak Republic,
e-mail: 1) [email protected], 2) [email protected], 3) [email protected], 4) [email protected]
Abstract — This article deals with the analysis of impact
reconfiguration on the power system operational state. The
stated analysis assess results of the simulated calculations of
N-1 events with the aim to obtain a more complex view of
the security criterion N-1 use in comparison with the
current methods and procedures being in practice.
Methodologies based on the deterministic approach arising
from calculations of steady states with the global assessment
of the power system operational states are presented. The
article objective is to comprehensively compare the selected
operational states especially different reconfiguration
variants in a power system.
various steady states with regard to changes of load,
transit, topology and power system development.
In general, the deterministic approach is possible to
define as a principle where effect of each kind is possible
to determine entirely and definitely.
Subject of the deterministic approach for a
classification of the operational state is the assessment of
consequences after N-1 events (contingencies) on the
basis of limitations and criteria determined in advance.
The proposed methodology using the deterministic
approach is based on the following procedure:
Keywords — security criterion N-1, deterministic approach,
reconfiguration, global assessment, operational state, power
system
Step 1
Simulation calculations of the N-1 security criterion
(internal contingency), i.e. repeated calculation of the
reference steady state for outage of each power line within
the transmission power system. The aim of the
calculations is to obtain the loading values in N state and
N-1 states of all power lines (after all contingencies) in the
responsibility area.
I. INTRODUCTION
Currently a great attention is paid to the increase of
security and operation of power systems. Several large
system failures occurring worldwide in the previous
decades affirm significance and need to develop this
philosophy. There are examples of it such as the power
system failures of blackout type in USA, Italy or a
splitting into islanding operations in a synchronically
interconnected power system in Europe. The reason for
occurrence of such failures is a conjunction of several
events; nevertheless, all the cases show the only common
violated indicator which is a failure to meet the
performance of the security criterion N-1.
Nowadays, within synchronously interconnected
system the European Awareness System is used for
exchange of online information evaluating the security
criterion N-1 analysis among other operating quantities.
Based on the monitoring of the operational states,
particular categories of the operational states are
distinguished by means of the traffic light. In order to
comply with the security criterion N-1as one of the input
parameters to determine the overall operational state of the
power system: this criterion is assessed “binary”, i.e.
“condition of criterion N-1 is met” or “condition of
criterion N-1 is not met”.
II. METHODOLOGY OF GLOBAL ASSESSMENT
The methodology of global assessment of the power
system operational state is based on an analysis of
simulation results of the security criterion N-1 considering
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Step 2
Based on the contingency simulation results in the
responsibility area LODF values (Line Outage
Distribution Factor) are subsequently calculated. The
LODF value determines the percentage of power flow on
the present power line that will be shown up on other
transmission power lines after the outage of this line.
Simply, LODFs are a sensitivity measure of how a change
in a line status affects the flows on other lines in the power
system.
For the purpose of methodology proposal the LODF is
calculated according to the following formula:
 PVy  PVy

LODFVxVy   n 1 Vx n .100 
 P

n


(1)
where
PVyn-1 is a loading of the assessed power line “Vy” after
outage of the power line “Vx“,
PVyn – a loading of the assessed power line “Vy“ in time
without power lines being turned off (state N),
PVxn – a loading of the power line “Vx“ in time without
power lines being turned off (state N).
Transactions on Electrical Engineering, Vol. 5 (2016), No. 4
Within the contingency list (N-1 events) all power lines
in the responsibility area are included while the ODF
calculation is performed only for loading changes of the
system power lines, i.e. loading changes of the radial
power lines (radial feeder to generation and consumption)
are not considered due to an high impact of the result
distortion. Among the partial distortion causes of the
methodology results are included:
• Small loading changes of the radial power lines after
the outage of the system power lines – consequence of
the voltage fluctuation in the end node of the power
line.
• Significant loading changes after the outage of the
radial power lines (accepted in cases for parallel
connection of the radial power lines).
Step 3
Further, in the methodology of global assessment only
positive loading changes are considered. It is necessary to
distinguish significant loading changes from the less
significant applying the data filter.
The data filter of increased loading changes is defined
by common meeting of the below stated conditions where
(2) defines a minimum loading change of the power line
after contingency (N-1 event) against state N and (3) a
minimum loading value of the given power line after
contingency (N-1 calculation) with respect to a nominal
load.
PVN 1
 0,05
(2)
IVN 1  0,3.I nV
(3)
PVN
Step 4
An overall assessment of the power system operational
states is based on perspectives defining four weight factors
(WF):
 Perspective on power lines loading
 
N
 VF1
(4)
Vi  1;51  i  1,2,3,4,5,6,7,8,9,15,16
Vj  1;51
Where
i is a p.u. value of the power line loading,
Vi – number of the system power lines in the
responsibility area,
Vj – total number of the power lines in the responsibility
area,
N – state N.
ii. WF2 is determined by a p.u. value of the most loaded
power line after all contingencies calculation in the
responsibility area related to the nominal loading.
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 
Vi
max iVj
N 1
 VF 2
(5)
Vi  1;51  i  1,2,3,4,5,6,7,8,9,15,16
Vj  1;51
Where
i is a p.u. value of the power line loading,
Vi – number of the system power lines in the
responsibility area,
Vj – total number of the power lines in the responsibility
area,
N-1 – state after all contingencies.
 System perspective on the number of the most
affected and effecting power lines
iii. WF3 is determined by a p.u. value of the number of
the affected power lines after all contingencies
calculation related to the number of the system power
lines.
nV _ POVP
nV _ CPSV
 VF 3
(6)
Where
nV_POVP is a number of the affected power lines, where the
affected power line is considered a power line with at least
one significant positive loading change after any
contingency,
nV_CPSV – number of the system power lines in the
responsibility area.
iv. WF4 is determined by a p.u. value of the number of
the effecting power lines after all contingencies
related to the total number of the power lines.
nV _ PVPL
nV _ CPV
i. WF1 is determined by a p.u. value of the most loaded
power line in the steady state (N state) related to the
nominal loading.
Vi
max iVj
105
 VF 4
(7)
Where
nV_PVPL is a number of the effecting power lines, where the
effecting power line is considered a power line which
outage will cause at least one significant positive loading
change of any assessed power line,
nV_CPV– total number of the power lines in the
responsibility area.
For the above described p.u. values weight factors
values determining their sizes and severity levels (TABLE
I, TABLE II, TABLE III) are appointed. The definition of
the severity levels considers for the power line loading an
uncertainty of mathematical models (model accuracy,
scheduled loading, etc.) and the system perspective is
based on the analysis of a large number results of the
simulation calculations of the steady states.
A proposal of weight factor values can be adjusted in
accordance to the purpose of assessment, e.g. the
operational planning and real time operation for on-line
monitoring of the power system operation, defence plan or
development of the power system.
Transactions on Electrical Engineering, Vol. 5 (2016), No. 4
TABLE I.
SIZES AND SEVERITY LEVELS OF WEIGHT FACTOR WF1 EXPRESSING
POWER LINES LOADING IN STATE N
Intervals (p.u.) of Power Line Loading in State N
WF1
<1;∞)
1
< 0.8 ; 1 )
0.8
< 0.6 ; 0.8 )
0.6
< 0 ; 0.6 )
0.5
TABLE II.
SIZES AND SEVERITY LEVELS OF WEIGHT FACTOR WF2 EXPRESSING
POWER LINES LOADING AFTER CONTINGENCY
Intervals (p.u.) of Power Line Loading after Contingency
WF2
<1;∞)
1
< 0.9 ; 1 )
0.8
< 0.8 ; 0.9 )
0.6
< 0 ; 0.8 )
0.5
TABLE III.
SIZES AND SEVERITY LEVELS OF WEIGHT FACTORS WF3, WF4
EXPRESSING SYSTEM PERSPECTIVE ON NUMBER OF AFFECTED AND
EFFECTING POWER LINES
106
TABLE V.
SIZES AND SEVERITY LEVELS OF WEIGHT FACTOR WF2 EXPRESSING
POWER LINES LOADING AFTER CONTINGENCY CONSIDERING
SENSITIVITY ANALYSIS
Sensitivity Analysis
Intervals (p.u.) of Power Line Loading after Contingency
<1;∞)
a
WF2
1a
1b
< 0.9 ; 1 )
0.9
< 0.8 ; 0.9 )
0.7
< 0.7 ; 0.8 )
0.5
< 0 ; 0.7 )
0.4
Criterion N-1 performance is not met for 1 power line
b
Criterion N-1 performance is not met for 2 power lines at least
TABLE VI.
SIZES AND SEVERITY LEVELS OF WEIGHT FACTORS WF3, WF4
EXPRESSING SYSTEM PERSPECTIVE ON NUMBER OF AFFECTED AND
EFFECTING POWER LINES CONSIDERING SENSITIVITY ANALYSIS
Sensitivity Analysis
Intervals (p.u.) of Number of Affected and Effecting Power
Lines
WF3, WF4
< 0.75 ; 1 >
1
Intervals (p.u.) of Number of Affected and Effecting
Power Lines
WF3, WF4
< 0.7 ; 0.75 )
0.9
< 0.75 ; 1 >
1
< 0.65 ; 0.7 )
0.8
< 0.65 ; 0.75 )
0.8
0.7
< 0.5 ; 0.65 )
0.6
< 0.6 ; 0.65 )
< 0.5 ; 0.6 )
< 0 ; 0.5 )
0.5
< 0.4 ; 0.5 )
0.5
< 0 ; 0.4 )
0.4
Step 5
The limit of the weight factor values is reassessed on
the basis of the sensitivity analysis for considering of
severity extent of the particular weight factors. Based on
the sensitivity analysis of several steady states results the
weight factor intervals are reassessed. Furthermore, the
inter-levels of weight factors are expressing an
approximation of p.u. values from the margin of the
nearest worse level of the weight factors (TABLE IV,
TABLE V, TABLE VI). The result of the sensitivity
analysis consideration is more precise partial as well as
the overall assessment of the power system operation
state. In this manner it is possible to prevent determination
of a less serious state to be serious and clearly differentiate
more serious state from other states.
TABLE IV.
SIZES AND SEVERITY LEVELS OF WEIGHT FACTOR WF1 EXPRESSING
POWER LINES LOADING IN STATE N CONSIDERING SENSITIVITY
ANALYSIS
0.6
Step 6
Eventually, the overall assessment of the operational
state is determined based on the product of all weight
factors. The proposed methodology includes two
exemptions for the global assessment of the operational
state. By the exception it is meant the direct determination
of the overall assessment of the operational state, i.e. it
determines the overall assessment as “emergency” in the
case of validity of any following conditions:
• If the loading value of the most loaded power line in
state N is higher than 100 %,
• If at least two WFs equal 1, then the overall
assessment of the operation state determined by the
product of all WFs is one state worse.
TABLE VII.
CLASSIFICATION OF OVERALL GLOBAL ASSESSMENT OF POWER
SYSTEM OPERATIONAL STATE
<1;∞)
1
< 0.9 ; 1 )
0.9
Overall
Assessment of
Power System
Operational
State
Normal
< 0.0256 ; 0.05 >
Green
< 0.8 ; 0.9 )
0.8
Alarm
( 0.05 ; 0.2058 >
Yellow
< 0.7 ; 0.8 )
0.7
Alert
( 0.2058 ; 0.5832 >
Orange
< 0.6 ; 0.7 )
0.6
Emergency
( 0.5832 ; 1 >
Red
< 0.5 ; 0.6 )
0.5
< 0 ; 0.5 )
0.4
Sensitivity Analysis
Intervals (p.u.) of Power Line Loading in State N
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WF1
Intervals of Products Size
(WF1 – WF4)
Color
Determination
Transactions on Electrical Engineering, Vol. 5 (2016), No. 4
Fig. 1. Flowchart of methodology of global assessment.
III. RECONFIGURATION ASSESSMENT
Among the measures to solve situation of the N-1
criterion meeting possible implemented by transmission
system operators (TSOs) belong as follows:
• cancellation of the scheduled maintenance or break of
all works in real-time operation utilizing standby time,
• coordinated topology changes (network configuration),
• use of phase shifter transformers,
• contracted generation re-dispatch within the TSO own
control areas,
• reduction of interconnection capacities,
• manual load shedding (consumption limiting plan in
SR).
The reconfigurations are effective remedial actions to
restore the N-1 criterion meeting and therefore the
objective of the article is to focus on the reconfiguration
107
and comprehensively compare selected different
reconfiguration variants in the power system.
Verification of the proposed methodology using the
deterministic approach was gradually performed for a
large number of different operational states.
The aim of this chapter is to verify the methodology for
system topology changes and result comparison with the
steady state before the reconfiguration. Simulations are
performed by means of the mathematical model of the
power system in accordance with the following
topologies:
• Reference model of the power system (Fig. 2),
• Reconfiguration in the substation Rz18_1 (Fig. 3),
• Reconfiguration in the substation Rz14 (Fig. 3),
• Reconfiguration in the substation Rz6_1 (Fig. 3),
• Reconfiguration in the substation Rz6_1+Rz14
(Fig. 3),
• Reconfiguration in the substation Rz18_1+Rz6_1+
Rz14 (Fig. 3),
• Reconfiguration in the substation Rz6_2 (Fig. 4),
• Reconfiguration in the substation Rz18_2+Rz34
(Fig. 5),
• Reconfiguration in the substation Rz18_1+Rz6_1
(Fig. 3),
• Reconfiguration in the substation Rz18_1+Rz6_2
(Fig. 4).
Based on the result analysis of the steady state before
the reconfiguration values for loading of the system power
lines in the state N and after all contingencies (N-1
simulations), as well as values of number of the affected
and effecting power lines under the present methodology
are determined (TABLE VIII).
Partial results: the most loaded power line in the state N
(70,7 %), the most loaded power line after all
contingencies (101,6 %), number of the significantly
affected power lines (52,5 %) and number of the
significantly effecting power lines (58,8 %). The steady
state before the reconfiguration is close beyond the N-1
security criterion meeting. Above, there are stated only
summary results, but the methodology also provides
detailed identification of the affected and effecting power
lines.
Fig. 2. Reference model topology of the power system.
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Transactions on Electrical Engineering, Vol. 5 (2016), No. 4
Fig. 3. Reconfiguration in particular substations (and its combinations).
Fig. 4. Reconfiguration in particular substations (and its combinations).
Fig. 5. Reconfiguration in particular substations (and its combinations).
Fig. 6. Impact of reconfiguration on power flow change through profiles.
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108
Transactions on Electrical Engineering, Vol. 5 (2016), No. 4
109
Fig. 7. Impact of reconfiguration on transit and losses.
TABLE VIII.
GLOBAL ASSESSMENT RESULTS OF STEADY STATE BEFORE RECONFIGURATION AND DIFFERENT RECONFIGURATION VARIANTS IN THE POWER
SYSTEM
Variant
Steady State before Reconfiguration:
Pgen=4241MW, Transit=2044MW,
Pcon=4577MW
Reconfiguration in Rz18_1
Reconfiguration in Rz14
Reconfiguration in Rz6_1
Reconfiguration in Rz6_1+Rz14
Reconfiguration in Rz18_1+Rz6_1+Rz14
Reconfiguration in Rz6_2
Reconfiguration in Rz18_2+Rz34
Reconfiguration in Rz18_1+Rz6_1
Reconfiguration in Rz18_1+Rz6_2
Note:
* - product result has only information character
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p.u. Values
Weight Factor Values
Sensitivity Analysis
p.u. Values
Weight Factor Values
Sensitivity Analysis
p.u. Values
Weight Factor Values
Sensitivity Analysis
p.u. Values
Weight Factor Values
Sensitivity Analysis
p.u. Values
Weight Factor Values
Sensitivity Analysis
p.u. Values
Weight Factor Values
Sensitivity Analysis
p.u. Values
Weight Factor Values
Sensitivity Analysis
p.u. Values
Weight Factor Values
Sensitivity Analysis
p.u. Values
Weight Factor Values
Sensitivity Analysis
p.u. Values
Weight Factor Values
Sensitivity Analysis
WF1
WF2
WF3
WF4
0,707
0,6
0,7
0,677
0,6
0,6
0,695
0,6
0,6
0,740
0,6
0,7
0,744
0,6
0,7
0,726
0,6
0,7
0,637
0,6
0,6
0,668
0,6
0,6
0,701
0,6
0,7
0,572
0,5
0,5
1,016
1
1
0,886
0,6
0,7
1,032
1
1
1,030
1
1
1,034
1
1
1,059
1
1
0,931
0,8
0,9
0,948
0,8
0,9
0,950
0,8
0,9
0,852
0,6
0,7
0,525
0,6
0,6
0,625
0,6
0,7
0,550
0,6
0,6
0,575
0,6
0,6
0,625
0,6
0,7
0,625
0,6
0,7
0,625
0,6
0,7
0,575
0,6
0,6
0,625
0,6
0,7
0,650
0,8
0,8
0,588
0,6
0,6
0,569
0,6
0,6
0,667
0,8
0,8
0,667
0,8
0,8
0,647
0,6
0,7
0,647
0,6
0,7
0,667
0,8
0,8
0,686
0,8
0,8
0,569
0,6
0,6
0,647
0,6
0,7
Overall
Assessment
Alert
Alarm
Alert
Alert
Alert
Alert
Alert
Alert
Alert
Alarm
WF Product
0,222*
0,216*
0,252
0,213*
0,130*
0,176
0,263*
0,288*
0,288
0,292*
0,288*
0,336
0,311*
0,216*
0,343
0,311*
0,216*
0,343
0,247*
0,230*
0,302
0,250*
0,230*
0,259
0,237*
0,173*
0,265
0,205*
0,144*
0,196
Transactions on Electrical Engineering, Vol. 5 (2016), No. 4
110
TABLE IX.
SUMMARY OF POWER GENERATION, CONSUMPTION AND TRANSMISSION – STEADY STATE BEFORE RECONFIGURATION AND DIFFERENT
RECONFIGURATION VARIANTS
Reconfiguration in
Substation/Values
Steady State
before
Reconfiguration
Total Generation
4240,6
[MW]
4576,8
Total Load [MW]
67,36
Total Losses [MW]
Total Load
4644,16
Considering Total
Losses [MW]
-336,2
Balance [MW]
2044,16
Transit [MW]
Power Flow: Assessed
Power System -1686,55
Neighbouring Control
Area 1 [MW]
Power Flow: Assessed
Power System -841,46
Neighbouring Control
Area 2 [MW]
Power Flow: Assessed
Power System 1432,13
Neighbouring Control
Area 3 [MW]
Power Flow: Assessed
Power System 612,03
Neighbouring Control
Area 4 [MW]
Notes:
Pgen – Generation in Responsibility Area
Transit – Transit through Responsibility Area
Pcon - Consumption in Responsibility Area
Rz18_1
Rz14
Rz6_1
Rz6_1
+Rz14
Rz18_1
+Rz6_1
+Rz14
Rz6_2
Rz18_2
+Rz34
Rz18_1
+Rz6_1
Rz18_1
+Rz6_2
4240,6
4240,6
4240,6
4240,6
4240,6
4240,6
4240,6
4240,6
4240,6
4576,8
79
4576,8
69,26
4576,8
71,1
4576,8
70,97
4576,8
86,05
4576,8
80,27
4576,8
70,54
4576,8
85,34
4576,8
89,8
4655,8
4646,06
4647,9
4647,77
4662,85
4657,07
4647,34
4662,14
4666,6
-336,2
1948,69
-336,2
2042,98
-336,2
2008,68
-336,2
2006,64
-336,2
1861,64
-336,2
1959,78
-336,2
2009,81
-336,2
1888,38
-336,2
1881,34
-1686,55
-1594,92
-1547,64
-1545,75
-1614,41
-1365,03
-1639,05
-1620,72
-1455,9
-677,34
-853,52
-868,34
-868,06
-669,48
-1011,22
-777,5
-689,2
-851,44
1514,45
1441,83
1368,17
1362,81
1403,62
1408,5
1453,71
1441,55
1484,66
434,24
601,15
640,51
643,83
458,02
551,28
556,1
446,83
396,68
TABLE X.
SUMMARY OF VOLTAGE PHASE CHANGES IN PARTICULAR SUBSTATIONS AFTER RECONFIGURATION
Steady State before
Reconfiguration:
Pgen=4241MW,
Transit=2044MW,
Pcon=4577MW
Reconfiguration
in Rz18_1
Reconfiguration
in Rz14
Reconfiguration
in Rz6_1
Reconfiguration
in Rz6_1+Rz14
Reconfiguration
in Rz18_1+Rz6_1+Rz14
Reconfiguration
in Rz6_2
Reconfiguration
in Rz18_2+Rz34
Reconfiguration
in Rz18_1+Rz6_1
Reconfiguration
in Rz18_1+Rz6_2
Transit [MW]
Transit
Decrease
[MW]
Voltage
Phase in
1st
Substation
on BusBar x_a
[°]
Voltage
Phase in
1st
Substation
on BusBar x_b
[°]
Voltage
Phase in
2nd
Substation
on BusBar x_a
[°]
Voltage
Phase in
2nd
Substation
on BusBar x_b
[°]
Voltage
Phase in
3rd
Substation
on BusBar x_a
[°]
Voltage
Phase in
3rd
Substation
on BusBar x_b
[°]
Δ Voltage
Phase in 1st
Substation
[°]
Δ Voltage
Phase in 2nd
Substation
[°]
Δ Voltage
Phase in 3rd
Substation
[°]
2044,16
-
-
-
-
-
-
-
-
-
-
1948,69
-95,47
-11,898
-22,741
-
-
-
-
10,843
-
-
2042,98
-1,18
-12,281
-15,864
-
-
-
-
3,583
-
-
2008,68
-35,48
-9,282
-14,91
-
-
-
-
5,628
-
-
2006,64
-37,52
-9,147
-15,123
-15,307
-14,695
-
-
5,976
0,612
-
1861,64
-182,52
-10,427
-23,953
-6,851
-17,189
-17,444
-12,547
13,526
10,338
4,897
1959,78
-84,38
-19,978
-7,706
-
-
-
-
12,272
-
-
2009,81
-34,35
-14,445
-17,699
-19,23
-15,932
-
-
3,254
3,298
-
1888,38
-155,78
-11,249
-23,653
-8,004
-15,436
-
-
12,404
7,432
-
1881,34
-162,82
-15,112
-24,876
-19,057
-7,705
-
-
9,764
11,352
-
Results of the power line loading in the state N, after all
contingencies, number of the affected and effecting power
lines are determined by the values of weight factors: WF1WF4. Based on their product the overall assessment of the
power system operation for a particular reconfiguration
stated in TABLE VIII is determined.
The global assessment of the simulated operational
states is classified as “alarm” for the reconfiguration in the
substations Rz18_1 and simultaneous reconfigurations in
Rz18_1+Rz6_2.Other simulated operational states of the
reconfigurations are classified as “alert“. Based on the
summary of the survey results (TABLE IX) it is not
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obvious decrease of the power transit through the
responsibility area for all reconfiguration variants. A
detailed overview of the transit changes through the power
system and power flows in particular profiles after a
reconfiguration is shown in Figs. 6 and 7.
The primary objective of the reconfiguration is to
ensure regular N-1 security criterion meeting. According
to the results in TABLE VIII the reconfiguration objective
is not met for variants in the substation Rz6_1 (one
overloaded power line), Rz14, Rz6_1+Rz14 and
Rz18_1+Rz6_1+Rz14Rz6_1 (two overloaded power
lines).
Transactions on Electrical Engineering, Vol. 5 (2016), No. 4
111
One of the serious disadvantages is the significant
increase of losses in the transmission system in the control
area. Stated fact was determined by all simulated
operational states (Fig. 7.). Other potential disadvantages
include the reliability reduction of the transmission system
and distribution system in the control area, maintenance
restriction and reconstruction work restriction. There is
also a potential problem with synchronization conditions
in case of return to the base state, but this case was not
confirmed by simulated operational states (TABLE X).
Under the current rules of the power system control the
operation is considered secured after the restore of the N-1
security criterion meeting. However, this idea has not to
be correct, as confirmed by the results of the global
assessment according to the proposed methodology. Based
on the analysis of the weight factor values for all
reconfiguration variants it is possible to establish that in
all stated cases the weight factor values VF3 and VF4 are
considerably increased, i.e. there is a significant increase
of the number of the affected and effecting power lines.
That means, the power system is substantially
“vulnerable“, e.g. in case of other events the situation
could lead to the “emergency” of the power system. This
fact is confirmed also by the weight factor product, which
is for the above mentioned reconfigurations higher than in
the steady state before the reconfiguration. Although, the
security criterion N-1 is met after the reconfiguration, in
fact the operational state of the power system is worse
than before the reconfiguration.
Eventually, results obtained by applying the
methodology of the global assessment using the
deterministic approach do not lead only to classification of
the overall operational state, but also provide a
comprehensive view on the power system operation
expressed by the weight factors.
power system, preparation of defence plans as well as
with a proposal of power system development plans.
A significant advantage of the proposed methodology is
definitely an optimization possibility on the basis of
weight factor size in accordance with various levels of a
dispatching regulation. It is possible to consider a more
conservative perspective for a scheduled operation,
defence plan and development of the power system, with a
strict modification of single weight factor intervals
especially for the reason of reserve due to the mathematic
models uncertainty. For the operational regulation it is
possible to consider a sensitive margin modification of
single weight factor intervals as far as a dispatcher should
be informed about a warning state only in case of a higher
risk during the power system operation.
Results of the global assessment of the power system
topology changes refer to identification of the suitability
and successfulness of the reconfigurations regard to the
restore of the N-1 criterion meeting. The added value of
the methodology is to provide comprehensive information
about
the power system operational state after the
reconfiguration (restoration of the N-1 meeting).
IV. CONCLUSION
The submitted paper deals with the methodology of the
global assessment of the power system operational states
using the deterministic approach and the N-1 security
calculations. The present methodology is based on more
complex results using of the N-1 security calculations.
The proposed methodology does not give only the answer
whether the N-1 criterion is met but also how the criterion
is met or not met. Based on the results it is possible to
differentiate and compare the power system states where
N-1 is not met and to formulate which “bad“ state is a
worse one or, vice versa, to compare the states when N-1
is met and formulate which “good“ state is a better one.
Its strong side is universality approved by verification
of a large number of operational states and its universal
characteristic is a definition of the weight factors by
means of which the operational states are classified.
Furthermore, sizes and severity levels of the weight factor
need to be adjusted for a given regulation area where the
methodology will be used. The weight factors need to be
identified on the basis of sensitivity analysis of results of a
large number of steady states due to a uniqueness of each
power system characterized by e.g. extensity,
geographical location, structure of power resources and
many other attributes.
In accordance with the universality of the proposed
methodology it is a wide utilization in practice, especially
in the dispatching regulation, operational regulation of the
[1] P. Kundur, at al, “Definition and classification of power system
stability,” IEEE Transactions on Power System, vol. 19, no. 3,
August 2004, pp. 1387 - 1401. [Online]. Available: IEEE Xplore
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July. 2016].
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Stability and Control, 2nd ed., John Wiley & Sons, Ltd.:
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Brussels - Belgium: ENTSO-E, 2009.
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ENTSO-E, 2013.
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podmienky prístupu a pripojenia, pravidlá prevádzkovania
prenosovej sústavy. Bratislava: SEPS, a.s., 2014.
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ACKNOWLEDGMENT
These publications are the result of implementation of
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Research & Development Operational Programme funded
by the ERDF.
REFERENCES
Transactions on Electrical Engineering, Vol. 5 (2016), No. 4
112
Impact of Asymmetry of Over-Head Power Line
Parameters on Short-Circuit Currents
Žaneta Eleschová 1) and Marián Ivanič 2)
1) 2)
Slovak University of Technology in Bratislava, Faculty of Electrical Engineering and Information Technology,
Institute of Power and Applied Electrical Engineering, Bratislava, Slovakia,
e-mail: 1) [email protected], 2) [email protected]
Abstract — This paper analyses the impact of asymmetry
of over-head power line parameters on short circuit
currents when three-phase fault and phase-to-ground fault
occur. The calculation results with consideration of an
asymmetry of the power line parameters are confronted
with the calculation in accordance with the Slovak standard
STN EN 60909 which does not consider asymmetry of
equipment parameters in the power system. The calculation
of short-circuit conditions was carried out for two types of
400 kV power line towers on which is a considerably
different arrangement of phase conductors.
Keywords — over-head lines, power line parameters,
asymmetry, short-circuit current.
I. INTRODUCTION
Short-circuit currents are addressed according to the
Slovak standard STN EN 60909. The calculation
according to this standard does not consider asymmetry of
power system parameters and uses the method of
symmetrical components instead. The objective of this
paper is to compare the calculation considering
asymmetry of the power system parameters with the
calculation in accordance with the standard.
Over-head power lines represent a dominant part of the
power system and if the phases are not transposed on
towers, one of the equipment in the power system contains
a considerable asymmetry of electrical parameters.
Therefore, the calculation of short-circuit conditions was
carried out on a simplified model: an ideal source and
power line. The power line was modelled for two types of
400 kV power line towers with a considerable
arrangement of phase conductors.
The following values of the short-circuit current have
been evaluated from the simulation results:
 RMS value which represents the initial shortcircuit current I k ,
 peak value with consideration of the DC
component maximum value which represents
peak short-circuit current ip.
II. RESULTS OF SHORT-CIRCUIT SIMULATIONS WITH
CONSIDERATION OF ASYMMETRY OF POWER LINE
PARAMETERS
The calculation of the power line parameters as well as
short-circuit simulations were carried out in EMTP
(Electro-Magnetic Transient Program). A simplified
model for calculation of the short-circuit currents is
depicted in Fig. 1.
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Fig.1. Simplified model in EMTP.
A. Input Parameters
The power line was modelled with the following two
types of towers:
Fig. 2. Types of towers.
The first type of tower represents a deployment of
conductors equally distant but in a different height above
the ground. The conductors of the second type of tower
have the same height but marginal conductors are placed
in a double distance.
The following table illustrates the conductor
parameters:
TABLE I.
PARAMETERS OF CONDUCTORS
ACSR conductor
[Ω/km]
[-]
[mm]
Ground conductor
120 AlFe6
0,225
0,809
16
Soil resistivity was 100 Ω/m.
Phase conductor
450 AlFe6
0,065
0,818
29,76
Transactions on Electrical Engineering, Vol. 5 (2016), No. 4
Parameters of the asymmetrical (untransposed) power
line are provided in Table II and parameters of the
transposed power line in Table III which indicates values
in symmetrical component systems while the length of the
power line is 60 km.
TABLE II.
PARAMETERS OF UNTRANSPOSED LINES
Tower
Phase
R
[Ω]
Type 1
113
currents in individual phases while untransposed power
line was considered, calculated according to the following
relation:
I I
(1)
  UL TL 100
I TL
UL – untransposed line
TL – transposed line
Type 2
L1
L2
L3
L1
L2
L3
L1
3,65
2,51
2,35
3,70
2,43
2,35
L2
2,51
4,08
2,51
2,43
3,79
2,43
L3
2,35
2,51
3,65
2,35
2,43
3,70
TABLE IV.
RMS VALUES OF SHORT-CIRCUIT CURRENTS
Tower type
Type 1
Type 2
TL
14,2
13,4
L1
13,8
12,9
I [kA]
X
[Ω]
k
[nF]
L1
26,29
7,84
8,62
26,21
8,60
6,47
L2
14,8
14,8
L2
7,84
22,85
7,84
8,60
25,35
8,60
L3
14,1
13,0
L3
8,62
7,84
26,29
6,47
8,60
26,21
L1
-2,6
-4,0
L2
4,3
10,4
L3
-0,4
-2,9
L1
683,74 -97,22
-61,25 672,61 -84,04
L2
-97,22 695,02 -97,22
L3
-61,25
-84,04 690,60
-97,22 683,74 -19,34
-84,04
-19,34
-84,04
672,62
TABLE III.
PARAMETERS OF TRANSPOSED LINES IN SYMMETRICAL COMPONENTS
Tower
Type 1
Type 2
R0 [Ω]
8,76
8,60
R1 [Ω]
1,36
1,34
X0 [Ω]
41,40
41,77
X1 [Ω]
17,05
18,03
C0 [nF]
518,25
555,05
C1 [nF]
773,70
742,06
B. Results of Short-Circuit Simulations
As mentioned above, the model in EMTP consisted of
an ideal power source and power line. The source voltage
was established in accordance with the Slovak standard
STN EN 60909 as follows: cU n 3 , where c = 1,05 and
Un = 400 kV.
It was made a simulation of the three-phase fault and
phase-to-ground fault at the end of the power line while
the RMS value and peak value were evaluated.
C. Results for Three-Phase Fault
The three-phase fault is a symmetrical type of fault. The
short-circuit current is the same in all three phases when
asymmetry of power system parameters was not
considered. Following the simulations results, it is evident
that the short-circuit currents are different in case of this
symmetrical fault when the asymmetry of the power line
parameters was considered, the short-circuit currents are
different also in case of this symmetrical fault. The shortcircuit current RMS values are included in Table IV.
For the transposed (symmetrical) power line, the RMS
value is 14,2 kA for the first type of tower and 13,4 kA for
the second type of tower. Table IV includes also values of
relative deviations from RMS values of the short-circuit
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δ [%]
The biggest difference in RMS values of the shortcircuit current (10,4 %) is in the middle phase for tower of
the second type, as can be seen from the results.
The peak values of the short-circuit currents were also
evaluated while the development of the maximum DC
component value was considered. The DC component is
dependent on the moment of the short-circuit occurrence.
The time behaviour of the short-circuit currents with the
maximum DC component are shown in the following
figures.
Fig. 3. The time behaviour of short-circuit current upon three-phase
fault.
The amplitudes of the short-circuit currents are shown
in the table below.
TABLE V.
PEAK VALUES OF SHORT-CIRCUIT CURRENTS WITH CONSIDERATION OF
MAXIMUM DC COMPONENT
Tower type
Type 1
Type 2
TL
35,9
34,2
L1
34,9
32,8
L2
36,4
37,3
L3
35,9
33,4
L1
-2,9
-4,1
L2
1,3
9,0
L3
-0,1
-2,4
Imax [kA]
δ [%]
Transactions on Electrical Engineering, Vol. 5 (2016), No. 4
114
There are differences even in peak values of the shortcircuit currents, in percentage terms they are a bit different
than in case of the RMS values.
The difference is caused by different time constants of
the DC components in individual phases, i.e. different
decay. It means that the asymmetry of the power line
parameters has the impact not only on the AC component
value of the short-circuit current but also on time constant
of the DC component of the short-circuit current.
D. Results for Phase-to-Ground Fault
The short-circuit current for the transposed line in case
of the phase-to-ground fault is the same for every single
phase. For the untransposed line, simulations of the phaseto-ground fault were carried out for each phase separately.
TABLE VI.
RMS VALUES OF SHORT-CIRCUIT CURRENTS
Tower type
Type 1
Type 2
TL
9,5
9,3
L1
9,2
9,2
L2
10,4
9,5
L3
9,2
9,2
L1
-3,9
-1,0
L2
8,8
2,1
L3
-3,9
-1,0
I [kA]
δ [%]
The highest difference is in the middle phase for the
tower of the first type which has the biggest height above
the ground but is the closest to the ground conductor. The
results of the short-circuit currents for the phase-to-ground
fault are almost identical for the tower of the second type.
Similar results are valid also for the peak values of the
short-circuit currents and again (as for three-phase fault),
the percentage differences of the peak values are not the
same as for the RMS values.
TABLE VII.
PEAK VALUES OF SHORT-CIRCUIT CURRENTS WITH CONSIDERATION OF
MAXIMUM DC COMPONENT
Tower type
Type 1
Type 2
TL
22,1
21,7
L1
21,3
21,4
L2
24,4
21,9
L3
21,3
21,4
L1
-3,8
-1,2
L2
10,2
1,1
L3
-3,8
-1,2
Imax [kA]
δ [%]
Fig. 4. The time behaviour of short-circuit current upon
phase-to-ground fault.
III. CALCULATION ACCORDING TO THE SLOVAK
STANDARD STN EN 60909
The transposed line parameters of EMTP, i.e. from
Table III were considered for calculation of the shortcircuit impedance.
The initial short-circuit currents and peak short-circuit
currents for the three-phase fault and phase-to-ground
fault were calculated. The calculation according to the
Slovak standard STN EN 60909 is based on the method of
symmetrical component systems.
The initial short-circuit current is given by:
 Three-phase fault:
I k3 

cU n
(2)
3 Z1
Phase-to-ground fault:
I k1 
c 3U n
Z1  Z 2  Z 0
c 3U n

2Z1  Z 0
Z1
– positive sequence short-circuit impedance
Z2
– negative sequence short-circuit impedance
Z0
– zero sequence short-circuit impedance
The value of the peak short-circuit current is
determined according to the following relation:
i p  2 Ik K
K  1,02  0,98e
3R k
Xk
A. Calculation for Tower Type 1
TABLE VIII.
SHORT-CIRCUIT IMPEDANCES
positive sequence
zero sequence
17,103 Ω 85,46°
42,317 Ω 78,05°
TABLE IX.
SHORT-CIRCUIT CURRENTS FOR THREE PHASE FAULT
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(3)
I k3
14,18 kA
K
ip
1,792
35,93 kA
(4)
(5)
Transactions on Electrical Engineering, Vol. 5 (2016), No. 4
TABLE X.
SHORT-CIRCUIT CURRENTS FOR PHASE TO GROUND FAULT
I k1
9,526 kA
K
ip
1,641
22,11 kA
B. Calculation for Tower Type 2
TABLE XI.
SHORT-CIRCUIT IMPEDANCES
positive sequence
zero sequence
18,08 Ω 85,74°
42,65 Ω 78,37°
TABLE XII.
SHORT-CIRCUIT CURRENTS FOR THREE PHASE FAULT
I k3
13,41 kA
K
ip
1,804
34,21 kA
TABLE XIII.
SHORT-CIRCUIT CURRENTS FOR PHASE TO GROUND FAULT
I k1
9,25 kA
K
ip
1,654
21,64 kA
The above mentioned results prove that the results of
the short-circuit currents are the same as the results of
simulations for the transposed power line which verifies
the results of the simulation.
IV. CONCLUSION
This paper deals with the impact of an asymmetry of
over-head power line parameters on the short circuit
currents in its individual phases. The objective of the
paper is to compare calculation of currents with
consideration of asymmetry of the power line parameters,
with calculation in accordance with the valid standard
STN EN 60909 on which the asymmetry of parameters is
neglected. The important result is that the short-circuit
currents in some of the phases are higher than the shortcircuit current calculated in accordance with the standard
considering untransposed power line.
Taking into consideration the three-phase fault, for the
second type of tower the difference was up to 10,4 % for
the middle phase. The positive sequence impedance
applies only with three-phase fault, therefore compared to
the transposed line the biggest difference of the shortcircuit current is in the middle phase which has the
shortest distance against other two phases. The distances
of the phase conductors on the first type of tower are
approximately the same and the currents in phases within
the three-phase fault are approximately the same as well.
The calculation of the phase-to-ground fault proved that
TELEN2016020
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115
the biggest difference (8,8 %) was for the first type of
tower, again in the middle phase. Also the zero sequence
is included into calculation for the phase-to-ground fault
which depends on the distance of the phase conductors
from the ground conductor. The phase conductors on the
second type of tower have approximately the same
distance from the ground conductor and the short-circuit
currents in individual phases for the phase-to-ground fault
are approximately the same. The arrangement of
conductors on the first type of tower is different, the
middle phase is the closest to the ground conductor and
the short-circuit current for phase-to-ground fault is the
highest in this phase.
The other important observation from the results is that
the asymmetry of parameters influences time constant of
the DC component and so influences other short-circuit
quantities such as: peak short-circuit current, thermal
short-circuit current, unsymmetrical breaking current.
Thus, these quantities are influenced by various values of
the initial short-circuit current in individual phases on one
hand and by various time constants of the DC component
for individual phases on the other hand.
This implies that correction factors taking into account
asymmetry of parameters should be defined when
calculating short-circuit currents.
ACKNOWLEDGMENT
These publications are the result of implementation of
the project: “Increase of Power Safety of the Slovak
Republic” (ITMS: 26220220077) supported by the
Research & Development Operational Programme funded
by the ERDF.
REFERENCES
[1] D. Reváková, Ž. Eleschová, and A. Beláň, Prechodné javy v
elektrizačných sústavách. Bratislava: Slovenská technická
univerzita v Bratislave, 2008.
[2] M. Ivanič and Ž. Eleschová, “The faults with consideration of line
asymetry parameters,” in Power engineering 2016: Control of
Power Systems 2016: 12th International scientific conference.
Tatranské Matliare, Slovakia. May 31 - June 2, 2016. 1. vyd.
Bratislava: Slovak University of Technology, 2016, s. 73-77. ISBN
978-80-89402-84-7.
[3] STN EN 60909-1, “Výpočet skratových prúdov v trojfázových
striedavých sústavách.”
[4] Š. Fecko, D. Reváková., L. Varga, J. Lago, and S. Ilenin, Vonkajšie
elektrické vedenia. Bratislava: STU FEI, 2010.
[5] A. J. Schwab, Elektroenergiesysteme. Erzeugung, Transport,
Übertragung und Verteilung elektrischer Energie. Berlin
Heidelberg, Springer, 2012.
Transactions on Electrical Engineering, Vol. 5 (2016), No. 4
116
Numerical Analysis and Experimental
Verification of Eigenfrequencies of Overhead
ACSR Conductor
Justín Murín 1), Juraj Hrabovský 1), Roman Gogola 1), Vladimír Goga 1) and František Janíček 2)
1)
Slovak University of Technology in Bratislava, Faculty of Electrical Engineering and Information Technology,
Department of Applied Mechanics and Mechatronics, Bratislava, Slovakia, e-mail: [email protected]
2)
Slovak University of Technology in Bratislava, Faculty of Electrical Engineering and Information Technology,
Department of Electrical Power Engineering, Bratislava, Slovakia, e-mail: [email protected]
Abstract — This contribution deals with the modal
analysis of ACSR conductor using the finite element method
(FEM) and experimental measurements of eigenfrequencies.
In numerical experiments for the modelling of the conductor
the material properties of the chosen conductor crosssection are homogenized by the Representative Volume
Element (RVE) method. The spatial modal analysis of the
power line is carried out by means of our new 3D FGM
beam finite element and by standard beam finite element of
the
commercial
software
ANSYS.
Experimental
measurements are also carried out for verification of the
numerical calculation accuracy.
Keywords — ACSR conductor, finite element method, modal
analysis, experimental measurements
I. INTRODUCTION
Vibration of overhead power lines is a very dangerous
problem because it can cause collapse of overhead power
lines or collapse of the whole transmission system. The
overhead power lines are exposed to dynamic loads (air
flow, ice-shedding, etc.) in addition to the static ones.
From the mechanical point of view the conductor is a 3D
system, so it can vibrate in longitudinal, horizontal and
vertical directions. The torsional vibrations are possible
as well. For calculation of eigenfrequencies and
eigenmodes the numerical methods are the most effective,
over all the finite element method. For the modal analysis
the beam finite element is preferable.
The material of the conductor is inhomogeneous,
therefore simplified models obtained by homogenization
of material properties are used [1, 2, 3]. The
heterogeneous cross-sections of several ACSR conductors
are shown in Fig.1.
II. HOMOGENIZATION OF MATERIAL PROPERTIES
One important goal of mechanics of heterogeneous
materials is to derive their effective properties from the
knowledge of the constitutive laws and complex microstructural behaviour of their components.
The methods based on the homogenization theory (e.g.
the mixture rules [5]) have been designed and successfully
applied to determine the effective material properties of
heterogeneous materials from the corresponding material
behaviour of the constituents (and of the interfaces
between them) and from the geometrical arrangement of
the phases. In this context, the microstructure of the
material under consideration is basically taken into
account by the Representative Volume Element (RVE).
The homogenization techniques derived at our
department (Department of Applied Mechanics and
Mechatronics) for modelling the Functionally Graded
Material (FGM) [1, 2] can also be used for
homogenization of the ACSR conductors. In case of the
conductor, the material properties vary layer-wise in the
radial direction (the longitudinal variation is not assumed).
The effective homogenized material properties (electric
conductance, thermal conductance, thermal expansion,
stiffness) are calculated from condition, that the relevant
material property of the cross-section with real
construction (Fig. 2) is equal to the material property of
the homogenized cross-section.
Fig. 2. Conductor cross-section.
Fig. 1. Construction of ACSR conductor.
Results of the modal analysis are obtained using the
commercial finite element software ANSYS and by a new
3D finite element [4]. An experimental measurement was
done to verify and to compare the effectiveness and
accuracy of each numerical calculation.
TELEN2016021
DOI 10.14311/TEE.2016.4.116
The real cross-section parameters of the ACSR
conductors are: Ri is pitch circle of the kth layer, di is wire
diameter, φi is the angle of circumferential position of the
wire, zi and yi are the distances of the wire from the centre
of the conductor cross-section. These distances of each
wire can be calculated as follows:
Transactions on Electrical Engineering, Vol. 5 (2016), No. 4
yi  Ri sin i
117
n
(1)
zi  Ri cos i
E LNH

th
Then the quadratic moment of the i wire cross-sectional
area Ai  di2 / 4 according the axis y and according the
axis z can be calculated by equations [1]:
I yi 
d i 4
64
 zi
d i 2
2
I zi 
4
d i 4
64
 yi
2
d i 2
4
(2)
(3)
Since the Young’s modulus multiplied by the crosssectional area defines the axial stiffness and multiplied by
the quadratic moment of the cross-section area defines the
bending stiffness, we have to distinguish homogenized
effective Young’s modulus for axial loading E LNH and
homogenized effective Young’s modulus for bending
M H
E L y and E LM z H .
We assume that the maximum and minimum elasticity
moduli for lateral and transversal bending can be
calculated by equations:
n
M yH
E L max

,
 I yi
zH
E LMmax
i 1
Here, Ei is the elasticity modulus of the ith wire, and n is
number of the wires.
The effective elasticity modulus for lateral and
transversal shears
M H
y
E L min


i 1
k zsm A
n
H
, GLz

 Ei I zi
i 1
n
 I zi
4
4
Fe d Fe E Fe  n Al d Al E Al
(4)
n
64
4
Fe d Fe E Fe
4
 n Al d Al
E Al
Where Gi  Ei / 21  i  is the shear modulus of the ith
n
wire, A   Ai is the cross-sectional area of the whole
i 1
real cross-section and  i is its Poisson’s ratio. Again,
k ysm,i and k ysm are the shear correction factors for the ith
wire and the whole cross-section, respectively. These
constants have to be calculated by a special method [4].
The effective elasticity modulus for torsion is:
GLM x H x  
 Gi I pi
i 1
n
 I pi
 LNH 

(5)
 i Ai
i 1
n
 Ai
(11)
i 1
n
and the effective mass density for torsional vibration is:
i 1
 i I pi
n
M H
y
y
E L max
 E L min
2
(6)
E LM z H 
zH
zH
E LMmax
 E LMmin
2
(7)
The effective elasticity modulus for axial loading is:
TELEN2016021
DOI 10.14311/TEE.2016.4.116
(10)
n
 I zi
M H
k zsm A
The effective mass density for axial beam vibration is:


M yH
(9)
k 1
i 1
 I yi
i 1
 k zsm,i Gi Ai
n
where nFe is the number of steel wires and nAl is the
number of aluminium wires. The maximum elasticity
modulus represents the case, when all wires are fixed
together (e.g. after several years of lifetime), and the
minimum elasticity modulus represents the case, when
wires can slide over each other. In practice the effective
elasticity modulus for lateral and transversal bending is
assumed as average value of the maximum and minimum
elasticity moduli [4]:
EL
H
GLz

 k zsm,i Gi Ai
n

zH
E LMmin
n
i 1
n
64


(8)
 Ai
n
 Ei I yi
i 1
n
i 1
n
i 1
The polar moment of the wire cross-sectional area to
origin of the coordinate system x, y is:
I pi  I yi  I zi
 Ei Ai
 LM x H 
i 1
n
 I pi
(12)
i 1
where,  i is the mass density of the ith wire.
III. 3D FGM BEAM FINITE ELEMENT EQUATIONS
Let us consider a 3D straight finite beam element
(Timoshenko beam theory and Saint-Venant torsion
theory) of a doubly symmetric cross-section. The nodal
degrees of freedom at the node i are: the displacements ui,
vi, wi in the local axis direction x, y, z, and the crosssectional area rotations –  x,i ,  y ,i ,  z ,i . The degrees of
freedom at the node j are denoted in a similar manner. The
internal forces at the node i are: axial force Ni, transversal
forces Ry,i and Rz,i, bending moments My,i and Mz,i, and
Transactions on Electrical Engineering, Vol. 5 (2016), No. 4
torsion moment Mx,i. Establishing of the local 3D FGM
 N i   B1,1
R  
 y ,i  
 Rz ,i  

 
 M x ,i  
 M y ,i   S

 
 M z ,i  
 N 
 j  
 Ry , j  

 
 Rz , j  
M x, j  

 
M y , j  
M  
 z, j  
0
B2 ,2
0
0
0
B3,3
118
beam finite element equations is presented in [4]:
0
0
B1,7
0
0
0
B2 ,6
0
0
B3,5
0
0
B4 ,4
0
B5 ,5
0
0
Y
B6 ,6
M
0
0
B2 ,8
0
0
0
0
B3,9
0
B3,11
0
0
0
0
0
B5 ,9
B4 ,10
0
0
B5 ,11
0
B6 ,8
0
0
0
B7 ,7
0
0
0
0
0
0
0
B9 ,9
0
B9 ,11
M
B8 ,8
E
T
0
B10,10
R
0
B11,11
Y
0   ui 


B2 ,12   vi 
0   wi 

 
0    x ,i 
0   y ,i 

 
B6 ,12    z ,i 

0   uj 

 
B8 ,12   v j 

 
0   wj 
0   x , j 

 
0   y , j 
B12,12   z , j 
(13)
In (13), the terms Bi,j contain the linear and linearized
geometric non-linear stiffness terms – containing the axial
force effect on the flexural beam stiffness matrix K and
consistent mass matrix M [4]:
B  K   2M
(14)
where  is the natural frequency. The shear correction is
accounted as well. The global stiffness matrix of the beam
structures can be established by classical methods.
Establishing of the local and global stiffness matrices as
well as the whole solution procedure were coded by the
software MATHEMATICA [12].
IV. NUMERICAL SIMULATIONS AND EXPERIMENTAL
MEASUREMENTS
For the numerical simulations and experimental
measurements the single power line with the span length
L = 19.9 m and the height difference between the points of
attachment yh = 0.8 m has been considered (Fig. 3). In this
case the maximum deflection of the power line [6, 7] is
minimal and therefore was not calculated, because the
span is small. The constant tensile force in the conductor
for each numerical calculations and experimental
measurements were: FH1 = 1.65 kN, FH2 = 4.75 kN and
FH3 = 6.68 kN.
Fig. 4. Cross-section of the used ACSR conductor.
Material properties of the material from which the
conductor is made are [9, 10]:
•
Steel:
elasticity modulus EFe = 207000 MPa,
Poisson’s ratio Fe = 0.28,
mass density Fe = 7780 kg.m-3;
•
Aluminium:
elasticity modulus EAl = 69000 MPa,
Poisson’s ratio Al = 0.33,
mass density Al = 2703 kg.m-3.
For numerical simulations a simplified model was used.
For simplifying the model of the ACSR conductor the
homogenized material properties are calculated [1, 2, 3].
The effective cross-sections of the conductor parts are:
AFe = 7.07 mm2, AAl = 42.41 mm2 and the effective crosssectional area of the whole conductor is A = 49.48 mm2.
The effective quadratic moments of the conductor crosssectional area are: Iz = Iy = 218.68 mm4. The effective
circular cross-section of the conductor is constant with
diameter def = 7.94 mm. The effective material properties
of the used conductor are:
ELNH  88714.29 MPa
Fig. 3. Model of overhead power line for modal analysis.
A symmetric conductor marked as AlFe 42/7 which is
constructed from 1 steel wire in the centre of the
conductor and 6 aluminium wires (see Fig. 4) has been
used. The diameter of the steel wire is dFe = 3 mm and the
diameter of the aluminium wires is dAl = 3 mm. The rated
tensile strength (RTS) of the chosen conductor is
FRTS = 15.27 kN [8].
TELEN2016021
DOI 10.14311/TEE.2016.4.116
M yH
EL
 ELM z H  40704.43 MPa
GLHy  GLHz  34120.91 MPa
M xH
GL
x  
27503.49 MPa
 LNH  3460.49 kgm-3
M H
 L x  2795.31 kg.m-3
 LNH  0.323
Transactions on Electrical Engineering, Vol. 5 (2016), No. 4
where E LNH is
M yH
EL
, ELM z H
the
elastic
modulus
for
tension,
are the elastic moduli for bending around
axis y and z, respectively. GLHy , GLHz are the effective shear
moduli, GLM x H x  is the effective elasticity modulus for
torsion,  LNH is the effective mass density for axial beam
vibration,  LM x H is the effective mass density for torsional
vibration and  LNH is the effective Poisson’s ratio. These
calculated effective material properties have been used in
the modal analyses of the single power lines. The first
three flexural eigenfrequencies f [Hz] in the plane xy
(vertical) and the first three flexural eigenfrequenciesf
[Hz] in the plane xz (horizontal) have been found with a
mesh 200 of BEAM188 elements of the FEM program
ANSYS [11]. The same problem has been solved using
the new 3D beam finite element (3D NFE) for the modal
analysis of composite beam structures [4] with a mesh 80
of 3D FGM elements (the calculation is performed using
the software MATHEMATICA).
119
Two bolted strain clamps were used for fixing the
conductor on two ends of the span; two IEPE piezoelectric
accelerometers with the range of  50 g (Fig. 5) were
used for experimental modal analyses to determine the
flexural eigenfrequencies. For scanning the signals from
the accelerometers 2 way oscilloscope with USB
connection to the PC was used. The range of the
oscilloscope is 20 MHz. The tension in the conductor is
measured with one load cell with sensing range
Fmax = 10 kN (Fig 6), which is close to the conductor
attachment point.
Fig. 6. Attaching of the load cell to sensing the axial force in conductor.
Fig. 5. Piezoelectric accelerometer attached on the conductor.
The data from the accelerometers placed on the
conductor is shown in Fig. 7.
To obtain the frequency spectrum (Fig. 8) the Fast
Fourier Transformation (FFT) of the measured data was
realized by software LabView [13]. The flexural mode
shapes were evaluated using software ANSYS. The results
of numerical analyses and experimental measurements are
presented in Tab. 1-3. First three flexural eigenfrequencies
in horizontal and three flexural eigenfrequencies in
vertical plane were investigated.
Fig. 7. Measured data of acceleration of the overhead ACSR conductor.
TELEN2016021
DOI 10.14311/TEE.2016.4.116
Transactions on Electrical Engineering, Vol. 5 (2016), No. 4
120
1stflexural
2ndflexural
3rdflexural
Fig. 8. Measured flexural eigenfrequencies of the used ACSR conductor for the tension FH = 6.68kN at times t = 3s and t = 7 s.
TABLE I.
FIRST THREE HORIZONTAL AND VERTICAL MEASURED AND NUMERICALLY CALCULATED EIGENFREQUENCIES OF THE USED ACSR POWER LINE
AT THE TENSION FH = 1,65KN
fmeas [Hz]
fans [Hz]
f3D [Hz]
ANS [%]
3D [%]
horizontal
2,23
2,49
2,47
11,66
10,74
vertical
2,38
2,54
2,58
6,72
8,34
horizontal
4,85
4,98
4,94
2,68
1,85
vertical
4,91
4,98
4,94
1,43
0,61
horizontal
7,23
7,47
7,41
3,32
2,51
vertical
7,24
7,47
7,42
3,18
2,44
f [Hz]
1st
2nd
3rd
TABLE II.
FIRST THREE HORIZONTAL AND VERTICAL MEASURED AND NUMERICALLY CALCULATED EIGENFREQUENCIES OF THE USED ACSR POWER LINE
AT THE TENSION FH = 4,75 KN
fmeas [Hz]
fans [Hz]
f3D [Hz]
ANS [%]
3D [%]
horizontal
3,88
4,31
4,20
11,08
8,34
vertical
3,97
4,32
4,21
8,82
6,08
horizontal
8,58
8,63
8,41
0,58
-2,01
vertical
8,73
8,63
8,41
-1,15
-3,69
horizontal
12,48
12,95
12,61
3,77
1,05
vertical
12,64
12,95
12,61
2,45
-0,21
f [Hz]
1st
2nd
3rd
TABLE III.
FIRST THREE HORIZONTAL AND VERTICAL MEASURED AND NUMERICALLY CALCULATED EIGENFREQUENCIES OF THE USED ACSR POWER LINE
AT THE TENSION FH = 6,68 KN
fmeas [Hz]
fans [Hz]
f3D [Hz]
ANS [%]
3D [%]
horizontal
4,51
5,13
4,98
13,75
10,46
vertical
4,72
5,13
4,99
8,69
5,62
horizontal
9,98
10,25
9,96
2,71
-0,16
vertical
10,23
10,25
9,96
0,20
-2,60
horizontal
14,67
15,35
14,95
4,64
1,89
vertical
14,97
15,38
14,95
2,74
-0,15
f [Hz]
1st
2nd
3rd
TELEN2016021
DOI 10.14311/TEE.2016.4.116
USUM
(AVG)
RSYS=0
DMX =.767418
1 SMX =.767418
NODAL SOLUTION
Transactions on Electrical
Engineering, Vol. 5 (2016), No. 4
STEP=1
SUB =5
FREQ=15.3547
USUM
(AVG)
RSYS=0
Y
DMX =.775861
1 SMX =.775861
MN X
Z
NODAL SOLUTION
STEP=1
SUB =6
FREQ=15.3831
USUM
(AVG)
RSYS=0
DMX =.767401
MN X
Y
SMX =.767401
MAR 22 2016
15:15:42
121
MX
MAR 22 2016
15:15:58
nd
a) 2 eigenmode in vertical plane
MX
Z
b) 3rd eigenmode in horizontal plane
MX
0
Y
MN X .085269
Z
.170537
.255806
.341075
.426343
.511612
.59688
.682149
.767418
c) 3rd eigenmode in vertical plane
0
Fig. 8. Eigenmodes
of the used ACSR
conductor for .517241
the tension FH = 6.68
kN.
.172414
.344827
.689654
.086207
.25862
.431034
.603447
.775861
V. CONCLUSION
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[1] J. Murin, V. Kutis, Improved mixture rules for the composite
In the presented contribution the numerical simulation
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and experimental measurements
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selected ACSR
0
.341067 647-650..511601
.682134
conductor is presented. The numerical
.085267 simulations
.2558were
.426334
.596868
.767401
[2]
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ACKNOWLEDGMENT
This work was supported by the Slovak Research and
Development Agency under the contract No. APVV-150326. This work was also supported by the Slovak
Research and Development Agency under the contract
No. APVV-0246-12 and APVV-14-0613, by Grant
Agency VEGA, grant No. 1/0228/14 and 1/0453/15.
Authors are also grateful to the companies SAG
Elektrovod a.s. Bratislava, Elba a.s. Kremnica and
Laná a.s. Žiar nad Hronom for sponsorship materials
needed for measurements.
TELEN2016021
DOI 10.14311/TEE.2016.4.116
[7] Š. Fecko, D. Reváková, L. Varga, J. Lago, S. Ilenin, Vonkajšie
elektrické vedenia, Bratislava: Renesans, s.r.o., 2010.
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zlanovaných kruhových drôtov, 2001.
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drôty, 2001.
[10] STN EN 60889, Tvrdo ťahané hliníkové drôty pre vodiče
nadzemných elektrických vedení, 2001.
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Houston, PA 15342/1300, USA.
[12] National Instruments Corporation, LabView, 11500 Mopac Expwy,
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Transactions on Electrical Engineering, Vol. 5 (2016), No. 4
Pástor, M., Dudrik, J.: Stability of Grid-Connected Inverter with LCL Filter
The paper analyses oscillations in a single-phase grid connected inverter with the LCL output filter. Passive and
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proportional resonant controller.
Bendík, J., Cenký, M., Eleschová, Ž.: 3D Numerical Calculation of Electric Field Intensity under Overhead
Power Line Using Catenary Shape of Conductors
This article presents a superposition method in combination with the Coulomb’s law and the Method of image
charges for calculation of the electric field distribution generated by high voltage overhead power lines above a
flat surface in every dimension. Such calculations are required to ensure the operational safety of people exposed
to the action of external electric field as well as to reduce the cost of people protection. The method provides
options for calculation of the field around the wire of a general shape. This substantial improvement of the
method could be applied to eliminate the usual error in the calculation created using approximation of catenary
shape conductors by infinite straight conductors. The method has been extensively tested on a set of shapes with
known analytical solutions. It has been shown that the numerical solution converges uniformly to the analytical
solution and the accuracy depends only on the number of finite elements.
Cintula, B., Eleschová, Ž., Beláň, A., Janiga, P.: Comparison of Reconfigurations Using Deterministic
Approach for Global Assessment of Operational State in Power System
This article deals with the analysis of impact reconfiguration on the power system operational state. The stated
analysis assess results of the simulated calculations of N-1 events with the aim to obtain a more complex view of
the security criterion N-1 use in comparison with the current methods and procedures being in practice.
Methodologies based on the deterministic approach arising from calculations of steady states with the global
assessment of the power system operational states are presented. The article objective is to comprehensively
compare the selected operational states especially different reconfiguration variants in a power system.
Eleschová, Ž., Ivanič, M.: Impact of Asymmetry of Over-Head Power Line Parameters on Short-Circuit
Currents
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Murín, J., Hrabovský, J., Gogola, R., Goga, V., Janíček, F.: Numerical Analysis and Experimental
Verification of Eigenfrequencies of Overhead ACSR Conductor
This contribution deals with the modal analysis of ACSR conductor using the finite element method (FEM) and
experimental measurements of eigenfrequencies. In numerical experiments for the modelling of the conductor
the material properties of the chosen conductor cross-section are homogenized by the Representative Volume
Element (RVE) method. The spatial modal analysis of the power line is carried out by means of our new 3D
FGM beam finite element and by standard beam finite element of the commercial software ANSYS.
Experimental measurements are also carried out for verification of the numerical calculation accuracy.
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TRANSACTIONS ON ELECTRICAL ENGINEERING VOL. 5, NO. 4 WAS PUBLISHED ON 31TH OF DECEMBER 2016