Overhead Lines
R, L, G, and C are the line parameter matrices expressed in per unit length
2.2.5 Data Input and Output
Users of EMT programs obtain overhead line parameters by means of a dedicated sup-porting routine which is usually denoted “line constants” (LC) [15]. In addition, several routines are presently implemented in transients programs to create line models consider-ing different approaches [31,34,43,49]. This section describes the most basic input require-ments of LC-type routines. It is followed by an example that investigates the sensitivity of line parameters (R, L, and C) to variations in the representation of an overhead line, and shows the infl uence that some parameters can have on the transient response.
LC routine users enter the physical parameters of the line and select the desired type of line model. This routine allows users to request the following models:
Lumped-parameter equivalent or nominal pi-circuits, at the specifi ed frequency
•
Constant distributed-parameter model, at the specifi ed frequency
•
Frequency-dependent distributed-parameter model, fi tted for a given frequency
• range
In order to develop line models for transient simulations, the following input data must be available:
(
• x, y) Coordinates of each conductor and shield wire Bundle spacing, orientations
•
Sag of phase conductors and shield wires
•
FIGURE 2.9
Section of a nonhomogeneous line model.
Corrections for losses and internal flux
Zloss(ω)/2
Corrections for losses and internal flux
Zloss(ω)/2 Ideal line
(external flux)
Lext, C
Phase and circuit designation of each conductor
•
Phase rotation at transposition structures
•
Physical dimensions of each conductor
•
DC resistance of each conductor and shield wire (or resistivity)
•
Ground resistivity of the ground return path
•
Other information, such as segmented grounds, can be important.
Note that all the above information, except conductor resistances and ground resistivity, come from geometric line dimensions.
The following information can be usually provided by the routine:
The capacitance or the susceptance matrix
•
The series impedance matrix
•
Resistance, inductance, and capacitance per unit length for zero and positive
•
sequences, at a given frequency or for a specifi ed frequency range
Surge impedance, attenuation, propagation velocity, and wavelength for zero and
•
positive sequences, at a given frequency or for a specifi ed frequency range
Line matrices can be provided for the system of physical conductors, the system of equiva-lent phase conductors, or symmetrical components of the equivaequiva-lent phase conductors.
The following example is included to illustrate Proper input of physical parameters
•
Examination of LC output
•
Benchmarking impedances
• Z0, Z1/Z2
Benchmarking for frequency response
•
Application considerations
•
Example 2.1 1. Test Line
Figure 2.10 shows the geometry of the 345 kV transmission line studied in this example (dis-tances in meters). Conductor data for this line are presented in Table 2.2.
2. Sensitivity Analysis of Line Parameters
A parametric study of sequence parameters was performed. To obtain the frequency depen-dence of the resistance and the inductance of conductors, users can assume either a solid conductor or a hollow conductor and apply the skin effect correction. Skin effect entails that the highest current density is at the conductor surface. To include skin effect for hollow conductors in an LC routine, users must specify the ratio T/D, being T the thickness and D the diameter of the conductor. For the results shown in the following a solid conductor is considered.
The studies presented in the following are aimed at determining the sensitivity of line param-eters with respect to frequency, ground resistivity, skin effect, and line geometry:
Figure 2.11 shows the dependency of the series parameters (
• R, L) with respect to ground
resistivity and frequency.
Figure 2.12 shows the dependency of the series parameters (
• R, L) with respect to frequency,
using the average height of the lowest conductors above ground as a parameter. These results were deduced by assuming a ground resistivity of 100 Ω m. The average height is calculated here by h = hm + hs/3 where hm and hs correspond to the conductor height at mid-span and to the sag, respectively [15].
Figure 2.13 shows the dependence of capacitances with respect to the average height
•
of the lower conductor. Since capacitances are not frequency dependent within the range of frequencies considered in this analysis, frequency is not used as a parameter in this case.
All calculations were performed by assuming full transposition of phase conductors. One can deduce from these plots the conclusions listed in the following.
The dependence of the resistance with respect to frequency can be signifi cant, and it is
•
particularly important for the zero-sequence resistance at high frequencies, but differences between values obtained with several ground resistivities are not very signifi cant in this example below 5 kHz.
Inductance values are also frequency dependent, but their dependence is very different for
•
positive- and zero-sequence values. The positive-sequence inductance does not show large variation along the whole range of frequencies. However, for the zero-sequence inductance the frequency dependence is much larger; on the other hand, there are no signifi cant differ-ences with different ground resistivity values.
When the skin effect is included in the calculation of line parameters, differences obtained
•
by assuming either a solid or a hollow conductor are small, and negligible for frequencies below 5 kHz.
When the average height of conductors is varied (between 12 and 32 m for the lower
•
conductors), the variation of the inductance values is rather small, less than 2%, in FIGURE 2.10
Example 2.1: 345 kV single-circuit overhead line confi guration (values between brackets are midspan heights).
8 m
8 m 8 m
16 m (10 m)
17.5 m (11.5 m)
24 m (18 m) 0.45 m
TABLE 2.2
Example 2.1: Conductor Characteristics
Diameter (cm) DC Resistance (W/km)
Phase conductors 3.0 0.0664
Shield wire conductors 1.0 1.4260
the whole range of frequencies. However, the variation is more important for resis-tance values; in fact, the positive-sequence resisresis-tance can vary more than a 50% at high frequencies.
The variation of the capacitance per unit length along a line span, see Figure 2.13, is very
• small.
From these results one can conclude that
Not much accuracy is required to specify line geometry since a rather small variation in
•
parameters is obtained for large variations in distances between conductors and heights above ground.
Since accurate frequency-dependent models are not required when simulating low- and
•
mid-frequency transients (below 10 kHz), the value of the ground resistivity is not critical.
Except for very short lines, the distributed nature of line parameters must be considered, and a rather accurate specifi cation of the ground resistivity can be required when simulating high frequency transients, as shown in the following.
3. Transient Behavior
The test line (assumed here as 80 km long) was used to illustrate the effect that losses, fre-quency dependence of parameters, and the value of the ground resistivity can have on some simple transients.
Results depicted in Figures 2.14 and 2.15 show the propagation of a step voltage on one of the outer phases of the line when this step is applied to the three-phase conductors (zero-sequence energization). Calculations presented in Figure 2.14 were performed by assuming
0
Example 2.1: Relationship between overhead line parameters and ground resistivity. (a) Zero-sequence resis-tance (Ω/km), (b) zero-sequence inducresis-tance (mH/km), (c) sequence resisresis-tance (Ω/km), (d) positive-sequence inductance (mH/km).
a constant distributed-parameter line model and calculating line parameters at power fre-quency (60 Hz) and at 5 kHz. It is obvious that the propagation takes place without too much distortion in both cases, but the attenuation is quite signifi cant when parameters are calcu-lated at 5 kHz. In addition, the propagation velocity is faster with a low ground resistivity.
When the frequency dependence of line parameters is included in the transient simulation, as shown in Figure 2.15, the propagation is made with noticeable distortion of the wave front.
The velocity of propagation decreases again as the ground resistivity is increased. This effect
0
Example 2.1: Relationship between overhead line parameters and conductor heights. (H1 = 12 m; H2 = 22 m;
H3 = 32 m). (a) Zero-sequence resistance (Ω/km), (b) zero-sequence inductance (mH/km), (c) positive-sequence resistance (Ω/km), (d) positive-sequence inductance (mH/km).
6
Example 2.1: Relationship between capacitances and conductor heights.
is due to the increase of the inductance with respect to the ground resistivity, as shown in Figures 2.11 and 2.12.
Figures 2.16 and 2.17 demonstrate again the effect of the ground resistivity on transient simulations during zero-sequence energizations. These simulations were performed by using a frequency-dependent parameter model of the line. All plots present the receiving end volt-age on one of the outer phases when the line is open. Both fi gures show that increasing the soil resistivity by a factor of 10 leads to a noticeable reduction of the dominant frequency.
It is obvious that a signifi cant attenuation can be obtained in wave propagation, even for short distances, when overhead line parameters are calculated taking into account their fre-quency dependence, and it is very evident when compared to the propagation that is obtained if this dependency is not included in calculations. However, since the highest frequency tran-sients in overhead lines usually involve the simulation of a few sections (spans) of the line, a very accurate representation of this effect is not usually needed. Accordingly, the IEEE TF on Fast Front Transients proposes to obtain line parameters at a constant frequency, between 400 and 500 kHz, for simulating lightning overvoltages [54].
–0.2 0.0
0.0 0.1 0.2
Time (ms)
0.3 0.2
0.4 0.6 0.8 1.0
20 km
Voltage (V)
40 km 60 km
–0.2 0.0
0.0 0.1 0.2
Time (ms)
0.3 0.2
0.4 0.6 0.8 1.0
20 km
Voltage (V)
40 km 60 km
(a)
(b)
100 Ω.m 1000 Ω.m
100 Ω.m 1000 Ω.m
FIGURE 2.14
Example 2.1: Zero-sequence energization of an 80 km untransposed overhead line (Constant distributed-param-eter model, Source = 1 V step). (a) Paramdistributed-param-eters calculated at 50 Hz, (b) paramdistributed-param-eters calculated at 5 kHz.
If simulations are performed without ground wires, the impact of the increased ground resistivity becomes much stronger.
Zero-sequence resistance increases with increasing ground resistivity, while zero-sequence inductance exhibits the opposite trend; this dependence being much smaller for positive-sequence quantities. The infl uence of this effect on attenuation and velocity of propagation is not negligible. Therefore some care is needed to specify the ground resistivity when high values of this parameter are possible.
As a further experiment, the test line is used to illustrate the effect of ground resistivity and frequency dependence of the line parameters when it is sequentially energized from a three-phase balanced sinusoidal 60 Hz source with an internal resistance of 0.001 Ω and having the receiving end open. Phases a, b, and c are closed at 3, 5, and 8 ms, respectively. Figure 2.18a shows the voltage of phase a using a frequency-dependent line model with different ground resistivity. Figure 2.18b presents again the voltage of phase a obtained by means of both a constant-parameter (at 5 kHz) and a frequency-dependent line model. The effect of ground
–0.2 0.0
0.0 0.1 0.2
Time (ms)
0.3 0.2
0.4 0.6 0.8 1.0
20 km
Voltage (V)
40 km 60 km
–0.2 0.0
0.0 0.1 0.2
Time (ms)
0.3 0.2
0.4 0.6 0.8 1.0
20 km
Voltage (V)
40 km
60 km (a)
(b)
100 Ω.m 1000 Ω.m
100 Ω.m 1000 Ω.m
FIGURE 2.15
Example 2.1: Zero-sequence energization of an 80 km untransposed overhead line (Frequency-dependent dis-tributed-parameter model). (a) Source = 1 V step, (b) source = 1 V, 20 μs pulse.
wires on a sequential closure is illustrated in Figure 2.18c for the test line represented by a frequency-dependent line model.
4. Discussion [55]
The above simulations were performed without including corona effect. This effect can have a strong infl uence on the propagation of waves when the phase conductor voltage exceeds the so-called corona inception voltage, see Section 2.3. Corona causes additional attenuation and distortion, mainly on the wave front and above the inception voltage, so a noncorona model will provide conservative results. Some programs allow users to include this effect in transient simulations. Several approaches can be considered. The simplest one includes corona effect from line geometry, although some models also consider the air density factor and even an irregularity factor. In fact, corona is a very complex phenomenon whose accurate representa-tion should be based on a distributed-hysteresis behavior. Perhaps the most important study for which corona can have a strong infl uence is the determination of incoming surges in
2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0
0.2
0.0 0.002 0.004 0.006 0.008 0.010
Time (s) (a)
(b)
Voltage (V)
100 Ω.m 1000 Ω.m
2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0
0.2
0.0 0.002 0.004 0.006 0.008 0.010
100 Ω.m 1000 Ω.m
Time (s)
Voltage (V)
FIGURE 2.16
Example 2.1: Zero-sequence energization of an 80 km untransposed overhead line (Source = 1 V step). (a) With ground wires, (b) without ground wires.
Time (ms) –1.00
(a)
(b)
1 2 3 4
–1.5 0.0 0.5 1.0 1.5
Voltage (V)
100 Ω.m 1000 Ω.m
1.0 0.8 0.6 0.4 0.2 0.0 –0.2 –0.4
0 1 2 3 4
Time (ms)
Voltage (V)
100 Ω.m 1000 Ω.m
FIGURE 2.17
Example 2.1: Zero-sequence energization of an 80 km untransposed overhead line (Source = 1 V, 50 μs pulse). (a) With ground wires, (b) without ground wires.
substations [3]. When an accurate representation of the corona effect is possible, then addi-tional input parameters are required for a full characterization of the model [9,10].
The concept of nonuniform line is used to deal with line geometries where the longitudinal variation of line parameters can be signifi cant. Examples of this type of line are lines crossing rivers or entering substations. In such cases, nonuniformities can be very important for surge propagation [7]. When corona effect (which represents a distributed nonlinearity) is included or a nonuniform line model (where the line parameters are distance dependent) is assumed, the line has to be subdivided. Thus fi nite differences based methods are preferred. On the other hand, the line can be subdivided into uniform line subsegments to represent such phe-nomena (nonlinearity and nonuniformity); however, the resultant model is prone to numerical oscillations [10].
Line confi gurations more complex than the one used in this example must be often simu-lated. In all cases, the input data to be specifi ed for these lines are similar to that required for the test line. And the main conclusions from a transient study would be similar to those derived in this example.
400 200 0
−200
−400
−600 0.0 (a)
(b)
(c)
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Time (s)
100 Ω.m 1000 Ω.m
Voltage (kV)
600 400 200 0
−200
−400
−600
−800
0.0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Frequency dependent Constant parameters
Time (s)
Voltage (kV)
600 400 200 0
−200
−400
−600
−800
0.0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Time (s)
Voltage (kV)
With ground wires Without ground wires
FIGURE 2.18
Example 2.1: Sequential closure of an 80 km untransposed overhead line (60 Hz balanced source). (a) Frequency-dependent model with different ground resistivity, (b) constant-parameters versus frequency-Frequency-dependent model, (c) effect of ground wires.
Phase-conductor resistances depend on temperature. This effect can add a nonnegligible increase to the resistance value. It can be easily included by specifying the correct value of conductor resistances.
From the above results one can conclude that, when only phase conductors and shield wires are to be included in the line model, the line parameters can be calculated from the line geometry, as well as from physical properties of phase conductors, shield wires, and ground.
A great accuracy is not usually required when specifying input values if the goal is to duplicate low frequency and slow front transients, but more care is needed, mainly with the ground resistivity value, if the goal is to simulate fast transients.
2.3 Corona Effect