8. CONCRECIÓN DE LAS TAREAS INTEGRADAS DEL PROYECTO TRANSVERSALES
8.1. PRIMERA FASE – LAND ART
8.1.7. DESARROLLO Y TEMPORALIZACIÓN DE LAS SESIONES
Although the lightning response of a transmission line tower is an electromagnetic phe-nomenon [65], the representation of a tower is usually made in circuit terms; that is, the tower is represented by means of several line sections and circuit elements that are assem-bled taking into account the tower structure [66–73]. There are some reasons to use this approach: such representation can be implemented in general-purpose simulation tools (e.g., EMTP-like programs), and it is easy to understand by the practical engineer.
Due to the fast-front times associated to lightning stroke currents, most tower models assume that the tower response is dominated by the transverse electromagnetic mode (TEM) wave and neglect other types of radiations. By introducing this simplifi cation, a tower can be represented as either (1) an inductance connecting shield wires to ground, (2) a constant impedance transmission line, (3) a variable impedance transmission line, or (4) a radiating structure [74].
Several tower models have been developed over the years and can be categorized in sev-eral ways, see [65,75]. Basically, two approaches have been applied [76]: models developed
1600
Example 2.2: Simulation results. (a) Energized phase response, (b) induced waveforms.
using a theoretical approach [66,67,71–73] and models based on an experimental work [70].
The simplest representation is a lossless distributed-parameter transmission line, charac-terized by a surge impedance and a travel time.
The fi rst models were deduced by assuming a vertical leader channel that hits the tower top. In fact, the response of a tower to horizontal stroke currents (i.e., the return stroke hits midway between towers) is different from the response to vertical stroke currents (i.e., the return stroke hits at the tower top). In addition, the surge impedance of the tower varies as the wave travels from top to ground. To cope with this behavior, some corrections were introduced into the fi rst models and more complicated models have been developed: They are based on nonuniform transmission lines, [77,78], or on a combination of lumped- and distributed-parameter circuit elements [70–73]. The latter approach is also motivated by the fact that in many cases it is important to obtain the lightning overvoltages across insulators located at different heights above ground; this is particularly important when two or more transmission lines with different voltage levels are sharing the same tower.
The models based on a constant-parameter circuit representation can be classifi ed into three groups: The tower is represented as a single vertical lossless line, as a multiconductor vertical line, or as a multistory model. A short summary of each approach is provided in the following [76].
a. Single vertical lossless line models
The fi rst models were developed by using electromagnetic fi eld theory, representing the tower by means of simple geometric forms, and assuming a vertical stroke to the tower top.
Wagner and Hileman used a cylindrical model and concluded that the tower impedance varies as the wave travels down to the ground [66]. Sargent and Darveniza used a conical model and suggested a modifi ed form for the cylindrical model [67]. Chisholm et al. pro-posed a modifi ed equation for the above models in front of a horizontal stroke current and recommended a new model for waisted towers [68]. These models have been implemented in the FLASH program [79]. The waist model was recommended by CIGRE [74], although the version presently implemented in the FLASH program is a modifi ed one [80].
The surge propagation velocity along tower elements can be assumed that of the light;
however, the multiple paths of the lattice structure and the crossarms introduce some time delays; consequently the time for a complete refl ection from ground is longer than that obtained from a travel time whose value is the tower height divided by the speed of light.
Therefore, the propagation velocity in some of the above models was reduced to include this effect in the tower response.
The overvoltages caused when using the above models should be the same between ter-minals of all insulator strings, since these models do not distinguish between line phases.
In fact, some differences result due to the different coupling between the shield wires and the phase conductors located at different heights above ground. The effect of crossarms was analyzed in [69] and found that they behave as short stub lines with open-circuit ends. Experimental results showed that travel times in crossarms are longer than those derived by assuming a propagation velocity equal to that of light. On the other hand, the incorporation of line sections representing crossarms reduces slightly the tower imped-ance. In general, the net effect is not signifi cant. Experimental studies of refl ections from tower bases showed that the initial refl ection differed from that predicted by assuming a lumped-circuit representation of the grounding impedance. This could be justifi ed by including transient ground-plane impedance [81]. This effect may be incorporated as an additional inductance in the grounding impedance [81–83] (Section 2.5.4.2).
The surge impedance, measured in Ω, for the most common tower shapes can be calculated, according to the above references, by means of the following expressions:
Cylindrical tower (Figure 2.26)
•
⎛ ⎛ ⎞ ⎞
=60⋅⎜⎝ln 2 2⎜⎝ h⎟⎠−1⎟⎠
Z r (2.54)
where
h is the tower height, in m r is the tower base radius, in m
Conical tower (Figure 2.27)
•
⎛ ⎛ ⎞ ⎞
⎜ ⎟
= ⋅ ⎜⎝ ⎜ ⎟⎝ ⎠ + ⎟⎠
2
60 ln 2 h 1
Z r (2.55)
where
h is the tower height, in m r is the tower base radius, in m
Since h > r, this expression can be usually approximated by
⎛ ⎞
≈60 ln⋅ ⎜⎝ 2h⎟⎠
Z r (2.56)
The above expressions were derived by assuming a vertical path of the lightning stroke.
When the stroke hits somewhere in midspan, the lightning current approaches the tower following a horizontal path. Chisholm et al. recommended a different expression for the average surge impedance [69].
Cylindrical tower
•
⎛ ⎛ θ⎞ ⎞
=60⋅⎜⎝ln cot⎜⎝ ⎟⎠−1⎟⎠
Z 2 (2.57)
Conical tower
•
⎛ θ⎞
=60 ln cot⋅ ⎜⎝ ⎟⎠
Z 2 (2.58)
where
θ =tan−1r
h (2.59)
h
2r FIGURE 2.26
Cylindrical tower shape.
h
2r FIGURE 2.27 Conical tower shape.
Chisholm et al. proposed also an expression for the surge impedance of a general transmission-tower shape using the geometry shown in Figure 2.28, see [69]. The surge impedance would be obtained by defi ning a weighted average of the tower radius
+ +
= 1 2 2 3 1 = +
av r h r h r h ( 1 2)
r h h h
h (2.60)
and using the expression for a conical tower over a ground plane (Equation 2.58),
where
− ⎛ ⎞
θ = ⎜⎝ ⎟⎠
1 av
tan r
h (2.61)
h1 is the height from base to waist, in m h2 is the height from waist to top, in m r1 is the tower top radius, in m
r2 is the tower radius at waist, in m r3 is the tower base radius, in m
The new approach was implemented into FLASH as a new model for representing waisted tower shapes. The formula for the calculation of this surge impedance was further modifi ed. The expression presently implemented into FLASH is the following one [75]:
⎛ ⎞
π ⎛ θ⎞
= 60⋅⎜⎝ln cot⎜⎝ ⎟⎠−ln 2⎟⎠
4 2
Z (2.62)
where θ is obtained according to Equations 2.60 and 2.61.
Table 2.4 presents a summary of tower models presently implemented in the FLASH program [79,82,84]. Note that a fourth confi guration, the H-frame has been included in the table.
b. Multiconductor vertical line models
Each segment of the tower between crossarms is represented as a multiconductor vertical line, which can be reduced to a single conductor. The tower model is then a single-phase line whose section increases from top to ground, as shown in Figure 2.29. This represen-tation has been analyzed in several Refs. [71–73], using each one a different approach to obtain the parameters of each section.
The modifi ed model presented in [72] included the effect of bracings (represented by loss-less lines in parallel to the main legs) and crossarms (represented as lossloss-less line branched at junction points). As an example, the development of this model, proposed by Hara and Yamamoto, is presented in the following paragraphs.
h2
h1
2r1
2r2
2r3
h
FIGURE 2.28 Waist tower shape.
TABLE 2.4 Tower Models
Tower Waveshape Diagram Surge Impedance—Travel Time
Cylindrical
FIGURE 2.29
Multiconductor tower model.
DT1
DT2
DT3
DT4
D΄B DB
rT1
rT2
rT3
rT4
rB
r4 r3
r2
r1 ZT1
ZT2
ZT3
ZT4 h1 h2 h3 h4
1. The following empirical equation is used for the surge impedance of a cylinder [71]:
⎛ ⎛ ⎞ ⎞
= ⋅⎜⎝ ⎜⎝ ⎟⎠− ⎟⎠
T 60 ln 2 2h 2
Z r (2.63)
where r and h are the radius and height of the cylinder, respectively. Note that this expression is different from Equation 2.54.
2. The total impedance of n parallel cylinders is given by the following expression:
( )
= ⋅ 11+ 12+ + 1 + 22+ + 2 + 1
n n n
Z Z Z Z Z Z
n (2.64)
where
n is the number of conductors
Zkk is the self-surge impedance of the kth conductor
Zkm is the mutual surge impedance between the kth and the mth conductors 3. Based on Equation 2.63, the values of these impedances are obtained as follows:
⎛ ⎛ ⎞ ⎞
= ⋅⎜⎝ ⎜⎝ ⎟⎠− ⎟⎠
60 ln 22 2
kk h
Z r (2.65a)
⎛ ⎛ ⎞ ⎞
= ⋅⎜⎝ ⎜⎝ ⎟⎠− ⎟⎠
60 ln 2 2 2
km
km
Z h
d (2.65b)
FIGURE 2.30
Different confi gurations of conductor systems. (a) Two conductors, (b) three conductors, (c) four conductors.
rT
DB DT
rB
DT
DB rB
rT
DT rT
rB DB
(a) (b) (c)
where
h is the conductor height r is the conductor radius
dkm is the distance between the kth and the mth conductors
4. The surge impedance for vertical parallel multiconductor systems consisting of two or more conductors is obtained as follows:
⎛ ⎛ ⎞ ⎞
= ⋅⎜⎜⎝ ⎜⎜⎝ eq⎟⎟⎠− ⎟⎟⎠
60 ln 22 2
n h
Z r (2.66)
where req is the equivalent radius, which is given by
⎧ ⋅ =
=⎪⎨ ⋅ =
⎪ ⋅ ⋅ =
⎩
1/2 1/2 1/3 2/3 eq
1/8 1/4 3/4
2 3
2 4
r D n
r r D n
r D n
(2.67)
where
r is the radius of the conductors
D is the distance between two neighbor conductors
5. For a geometry as that depicted in Figure 2.30, the expressions of the surge imped-ances are still valid, and they may be used after replacing r and D by the following results:
=3 T⋅ B2
r r r (2.68)
=3 T⋅ 2B
D D D (2.69)
where
rT and rB are the radii at the top and the base of the conductors, respectively DT and DB are the distances between two adjacent conductors at the top and the
base, respectively
6. The tower model shown in Figure 2.29 can be divided into several sections (four in this case), being the surge impedance of each section obtained as follows:
⎛ ⎛ ⎞ ⎞
According to the authors, these equations are applicable to towers fabricated with tubular components. When the tower is constructed with angle sections, the values of rT and rB (see Figure 2.30) must be replaced by half of the side length of the angle section.
7. The measured results show that the surge impedance of conductors is reduced about 10% by adding the bracings to the main legs. This performance is repre-sented by adding in parallel to the line that represents each tower section another line of the same length and with the following surge impedance:
= ⋅
Lk 9 Tk
Z Z (2.72)
8. Finally, the crossarms are represented by line sections branched at the junction points, being the corresponding surge impedances obtained from the expression
of a conventional horizontal conductor:
= ⋅ A =
where hAk and rAk are respectively the height and the equivalent radius of the kth crossarm.
The full model of the transmission tower would be that shown in Figure 2.31. See Ref. [72] for more details.
c. Multistory model
It is composed of four sections that represent the tower sections between crossarms [70]. Each section consists of a lossless line in series with a parallel R–L circuit, included for attenuation of the traveling waves, see Figure 2.32.
The parameters of this model were deduced from experimental results. The values of the parameters, and the model itself, have been revised in more recent years [75]. The approach was originally developed for representing towers of ultra high voltage (UHV) transmission lines.
Equivalent model of a transmission tower.
FIGURE 2.32
Multistory model of a transmission tower. (From Ishii, M. et al., IEEE Trans. Power Deliv., 6, 1327, July 1991;
Yamada, T. et al., IEEE Trans. Power Deliv., 10, 393, January 1995. With permission.) 2r1
2r2
2r3
h1
L1
L2
L3
L4 h2
h3
h4 R1
ZT1, h1, c
ZT1, h2, c
ZT2, h3, c
ZT1, h4, c R2
R3
R4
The surge impedance of each line section is obtained as in the previous model, while the propagation velocity is that of light. Without including the representation of the crossarms, the multistory model is as shown in the fi gure. The damping resistances and inductances are deduced according to the following equations [70,85]:
− ⋅ ⋅ γ
= =
+ +
T1
1 2 3
2 ln
( 1, 2, 3)
i i
R Z h i
h h h (2.74)
= − ⋅ ⋅ γ
4 2 T2 ln
R Z (2.75)
= α ⋅ ⋅2 =
( 1, 2, 3, 4)
i i h
L R i
c (2.76)
where
ZT1 is the surge impedance of the three upper tower sections ZT2 is the surge impedance of the lower tower section hi is the height of each tower section
γ is the attenuation coeffi cient α is the damping coeffi cient
The attenuation coeffi cient is between 0.7 and 0.8, while unity has been the value usually chosen for the damping coeffi cient [70,85].
A further experimental investigation found that an adequate calculation for both ZT1 and ZT2 could be based on Jordan’s formula:
FIGURE 2.33
Geometry for equivalent radius calculation.
r1
h1
req r2
h2
h r3
⎛ ⎛ ⎞ ⎞
=60 ln⎜⎜⎝ ⎜⎜⎝ eqh ⎟⎟⎠−1⎟⎟⎠
Z r (2.77)
where
h is the tower height
req is the equivalent radius obtained from the geometry shown in Figure 2.33 and given by [85]
+ +
= 1 2 2 3 1 = +
eq ( 1 2)
2 r h r h r h
r h h h
h (2.78)
The propagation velocity is that of the light.
A study presented in [86] concluded that the multistory model with surge impedance values originally proposed in [70] was not adequate for representing towers of lower volt-age transmission lines. According to this study, the tower model for shorter towers can be simpler than that assumed by the multistory model; that is, a single lossless line for each tower section, whose surge impedance was calculated from Equations 2.77 and 2.78 would suffi ce.
Antenna theory approach might be used for an accurate computation of lightning volt-ages, and for validation and improvement of simpler circuit and transmission-line models, as discussed in [65]. This reference also discussed possible improvements, as well as the use of nonuniform lines for transmission tower modeling.
Example 2.3
Figure 2.34 shows the tower design of the lines analyzed in this example. Main characteristics of phase conductors and shield wires are presented in Table 2.5. The aim is to simulate and compare the maximum lightning overvoltages that will occur across the insulator strings of each test line, considering different lightning stroke waveshapes.
The following tower models are used with each test line:
Towers of the test line 1 are represented using the conical and the waisted model, as it is
•
presently implemented into FLASH [79,80] and as proposed by CIGRE [74].
Towers of the test line 2 are represented using the conical model implemented into FLASH
•
[67], the waisted model recommended by CIGRE [74], and the multistory model, whose parameters were calculated according to [85].
FIGURE 2.34
Example 2.3: 400 kV line confi gurations (values within parenthesis are midspan heights). (a) Test line 1, (b) test line 2.
Example 2.3: Characteristics of Wires and Conductors
Type Diameter (mm) DC Resistance (W/km)
Phase conductors CURLEW 31.63 0.05501
Shield wires 94S 12.60 0.642
The models of both test lines are created considering the following values and modeling guidelines:
Each line is represented by means of three 400 m spans plus a 3 km section as line
termina-•
tion at each side of the point of impact. Each line section is represented as a constant dis-tributed-parameter model whose values are calculated at 400 kHz. Soil resistivity, needed to obtain the electrical parameters, is 200 Ω m.
The insulators are represented as open switches since the main goal is to compare the
over-•
voltages that will be caused in both lines with different models of the towers.
The grounding impedance of all towers of both lines is represented as a constant
•
resistance.
The lightning stroke is represented as an ideal current source (infi nite parallel impedance)
•
with a double ramp waveform, defi ned by the peak current magnitude, I100, the time-to-crest (also known as rise time or front time), tf, and the tail time, th, which is the time, mea-sured along the tail, needed to reach 50% of the peak value.
Power-frequency voltages at phase conductors are neglected.
•
Figure 2.35 shows the equivalent circuit that results for any line after following the guidelines detailed above. Tables 2.6 and 2.7 summarize the different approaches applied to represent each transmission tower, as well as the resulting parameters. The tower models used in simulations have some minor differences with respect to those depicted in tables since small resistors have been added to separate phases and shield wires when the tower is represented as a single-phase lossless line.
Some simulation results derived with the two test lines are shown in Figure 2.36. These plots depict the overvoltages caused by two different return strokes across the insulator strings of the outer phase of the fi rst test line and an upper phase of the second test line, respectively. One can observe that the differences between simulation results that correspond to one line depend on the tower model and the rise time of the return stroke current (tf): As the rise time of the return stroke waveshape increases, the differences between the results derived with different tower models decrease, and the differences are more important between results obtained with the tower models used for the second line.
On the other hand, the results derived with the models tested in this example are similar to the two lightning current waveshapes, except when the multistory approach is applied, which pro-vides much larger overvoltages. This is according to the conclusions of [86].
In fact the tower representation has less infl uence than the tower grounding as can be seen from the results derived with different grounding resistances. It is worth mentioning that a constant resistance is a very conservative approach to represent the grounding of a transmission line, as discussed in the following section.
2.5 Transmission Line Grounding